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+[    {        "title": "2103.09557v1.Spin_injection_efficiency_at_metallic_interfaces_probed_by_THz_emission_spectroscopy.pdf",        "content": "Spin injection e\u000eciency at metallic interfaces probed by THz emission spectroscopy\nJacques Hawecker1, T. H. Dang2, Enzo Rongione2, James Boust2, Sophie Collin2, Jean-Marie\nGeorge2, Henri-Jean Drouhin3, Yannis Laplace3, Romain Grasset3, Jingwei Dong3, Juliette\nMangeney1, Jerome Tignon1, Henri Ja\u000br\u0012 es2, Luca Perfetti3\u0003and Sukhdeep Dhillon1\n1Laboratoire de Physique de l'Ecole normale sup\u0013 rieure, ENS, Universit\u0013 e PSL,\nCNRS, Sorbonne Universit\u0013 e, Universit\u0013 e de Paris, F-75005 Paris, France\n2Unit\u0013 e Mixte de Physique, CNRS, Thales, Universit\u0013 e Paris-Sud,\nUniversit\u0013 e Paris-Saclay, F-91767 Palaiseau, France and\n3Laboratoire des Solides Irradi\u0013 es, CEA/DRF/lRAMIS, Ecole Polytechnique,\nCNRS, Institut Polytechnique de Paris, F-91128 Palaiseau, France\nTerahertz (THz) spin-to-charge conversion has become an increasingly important process for THz\npulse generation and as a tool to probe ultrafast spin interactions at magnetic interfaces. However,\nits relation to traditional, steady state, ferromagnetic resonance techniques is poorly understood.\nHere we investigate nanometric trilayers of Co/X/Pt (X=Ti, Au or Au0:85W0:15) as a function of\nthe 'X' layer thickness, where THz emission generated by the inverse spin Hall e\u000bect is compared\nto the Gilbert damping of the ferromagnetic resonance. Through the insertion of the 'X' layer\nwe show that the ultrafast spin current injected in the non-magnetic layer de\fnes a direct spin\nconductance, whereas the Gilbert damping leads to an e\u000bective spin mixing-conductance of the\ntrilayer. Importantly, we show that these two parameters are connected to each other and that spin-\nmemory losses can be modeled via an e\u000bective Hamiltonian with Rashba \felds. This work highlights\nthat magneto-circuits concepts can be successfully extended to ultrafast spintronic devices, as well\nas enhancing the understanding of spin-to-charge conversion processes through the complementarity\nbetween ultrafast THz spectroscopy and steady state techniques.\nPACS numbers:\nI. INTRODUCTION\nWhen a pure spin current pass through materials with\nlarge spin-orbit coupling, it can generate a transverse\ncharge current1,2by means of the Inverse Spin-Hall-E\u000bect\n(ISHE). A \rurry of activity on this topic has been mo-\ntivated by the intimate relation between ISHE and the\ndirect SHE3. The latter can be very e\u000eciently employed\nto generate a spin transfer torque capable of switching\nthe magnetization of ferromagnetic thin \flms4,5. Most\nexperiments in this \feld have been performed by spin-\npumping viaferromagnetic resonance while some works\nhave investigated the ultrafast regime6,7. More recently,\nother authors have proven that ISHE can be employed to\ngenerate an intense THz radiation. This breakthrough\nhighlighted that interfaces leading to large spin trans-\nfer torque are also excellent emitters of electromagnetic\nwaves8{12. Theoretical simulations based on superdi\u000bu-\nsive transport equations have successfully reproduced the\nobserved emission13,14. However, these frameworks do\nnot cover the impact of the electronic transmission at in-\nterfaces, neither the discussion of the particular role of\nthe interfacial spin-orbit \felds originating from charge\ntransfer and symmetry breaking15. To this end, a tighter\nconnection with steady state spintronics is highly desir-\nable. For example, the magnetocircuits analogies are\n\u0003The authors to whom the correspondence show be addressed are\nluca.perfetti@polytechnique.edu, henri.ja\u000bres@cnrs-thales.fr and\nsukhdeep.dhillon@phys.ens.frwidely employed to de\fne the e\u000eciency of the spin-to-\ncharge conversion16,17. An extension of such formalism\nto impulsive excitations has been discussed in the case\nof ultrafast spin-Seebeck e\u000bect18but not yet for the spin\ncurrent injected from ferromagnetic transition metals.\nIn the \feld of spintronics, the optimal e\u000eciency of spin\norbit torque (SOT) requires the engineering of metallic\ninterfaces favoring a higher spin-current generation. Re-\ncently, many authors have tackled this issue by inserting\ntransion metals19,20or noble metals21between cobalt (or\nCoFe, CoFeB) and platinum. Experiments with di\u000ber-\nent interlayers have shown clear correlations between the\nspin-transfer-torque, magnetoresistance19, perpendicular\nanisotropy15,19and spin memory loss22. Here, we in-\nvestigate this topic by comparing the THz emission ef-\n\fciency detected by Time Domain Spectroscopy (TDS)\nwith Ferro-Magnetic Resonance (FMR) spectroscopy of\ntrilayers Co/X/Pt. The insertion of an atomically thick\ninterlayer of X=Ti, Au or Au 0:85W0:15modi\fes the ca-\npability of the interface to generate spin currents. This\nproperty a\u000bects, on the same footing, the emission of\nelectromagnetic radiation as well as the Gilbert damping\nof the multilayer. Our result show that the THz-TDS\nemission spectroscopy is contactless and non-destructive\nmethod that can give an accurate and reliable esti-\nmate of the spin-injection e\u000eciency at spintronic in-\nterfaces. We discuss the data in the magnetocircuit\nformalism16{18, by assuming that spin-currents follow the\nevolution of magnetic \ructuations. The average spin-\nconductance ( g\"+g#)=2 characterizes the ultrafast cur-\nrents in platinum18whereas an e\u000bective spin conductancearXiv:2103.09557v1  [cond-mat.mes-hall]  17 Mar 20212\ng\"#\ne\u000bdescribes the damping torque in the ferromagnetic\nmaterial ( in our case is cobalt )16. These two quantities\nwould be proportional to each other if the spin \row was\nconserved at the interface17. In reality, strong spin-orbit\nassisted scattering processes generate a sink of angular\nmomentum and limit the spin \row that can propagate\nin platinum23{26. The comparison between ( g\"+g#)=2\nandg\"#\ne\u000bshows that the fraction of spin current lost at the\ninterface25is proportional to the spin conductance and\nmay exceed 40% in the Co/Pt bilayer. Our measure-\nments highlight that passivation of the interface by dif-\nferent compounds follows a common trend and suggests\nthe existence of a general relation between spin memory\nloss and spin-conductance. We prove this claim by choos-\ning inter-layer materials with very di\u000berent properties:\nTi is more chemically reactive and has small spin-orbit\ncoupling whereas Au and Au 0:85W0:15are less chemically\nreactive and hosts a larger spin-orbit interaction (espe-\ncially the Au:W alloy).\nFIG. 1: A) Detection of spin-to-charge conversion in a spin-\ntronic emitter. The cobalt layer has magnetization ~Mpar-\nallel to the external magnetic \feld ~Hand is in contact with\nthe platinum layer. An ultrashort laser pulse photoexcites\nthe sample generates, in the Pt side of the interface, a spin\ncurrent~Jsthat is proportional to the direct spin conductance\n(g\"+g#)=2. The inverse spin Hall e\u000bect of Pt leads to a trans-\nverse charge current ~Jc. Being shorter than one picosecond,\nthe~Jcpulse emits radiation in the THz spectral range. B)\nE\u000bect of the platinum layer on the ferromagnetic resonance of\nthe underlying cobalt. The magnetization precession driven\nby a radiofrequency \feld ~hrfinduces a spin current. The in-\ncrease of Gilbert damping due to the ~Jsinjection in the Pt\nlayer is proportional to the e\u000bective spin-mixing conductance.\nII. GENERAL FRAMEWORK OF SPINTRONIC\nTHZ EMISSION\nA framework building on few hypothesis connects\nthe spin conductance to the emitted THz radiation.In the thin \flm limit, the THz electric \feld of a\nplane wave at the surface of the sample is given\nby~ET(!) =eZR~Jc(!;z)dz. This expression links\n~ET(!) to the charge current density ~Jcvia an e\u000bective\nimpedance8,27,28:\nZ=Z0\n1 +n+Z0R\n\u001b(z)dz; (1)\nwherenis the refractive index of the substrate, zis the\ncoordinate perpendicular to the interface, Z0= 377\n is\nthe vacuum impedance andR\n\u001b(z)dzis the local conduc-\ntivity integrated over the total thickness of the multi-\nlayer. The charge current ~Jc(z) arises in platinum be-\ncause of the inverse-spin-Hall-e\u000bect acting on the spin\ncurrent \row ~Js(z) along the normal direction to the \flm\nplane. The latter decreases exponentially over a distance\nequal to the spin di\u000busion length. It follows that:\nZdPt\n0+~Jc(z)dz=Js(0+)(~ en\u0002~ es)\u0015Pt\ns\u0002stanhdPt\n2\u0015Pts;(2)\nwhere~ enis a unitary vector normal to the interface, ~ es\nis the polarization direction of the spin current, dPtis\nthe thickness of platinum layer, \u0015Pt\nsis the spin di\u000busion\nlength in platinum, \u0002 sis the spin-Hall-angle of platinum\nandJs(0+) is the magnitude of spin current density gen-\nerated in the ferromagnet, penetrating into the heavy\nmetal, and thus responsible for the charge current os-\ncillations at the platinum side of the interface. At this\nstage, it is important to recall that the magnitude of\ntheJspropagating in platinum can be smaller than the\none generated in the ferromagnet. The discontinuity of\nspin current between the two sides of the active interface\nis generally ascribed to the spin-decoherence induced by\nlocal spin-orbit \felds (also known as spin-memory-loss)23\nand has been recently proved viare\fned spin-orbit torque\nexperiments15.\nEmission over an ultrabroad spectral range8,18and\ntheoretical modeling13,14,18show thatJsevolves on a\ntimescale comparable to the energy and momentum re-\nlaxation of hot electrons. We make use of the magneto-\ncircuit formalism to write to the spin current in terms of\nspin conductance parameters17. The ultrafast generation\nof a spin accumulation on the ferromagnetic side leads to\na longitudinal component whereas the spin accumulation\non the Pt side induces a transverse component18. The\nresulting expression reads:\n~Js=~\n4\u0019\u0012g\"+g#\n2h@tM\nM^mi+g\"#h^m\u0002@t^mi\u0013\n;(3)\nwhereMis the magnetization magnitude in the very\nproximity of the interface, ^mis the local magnetization\ndirection,g\"(g#) is the spin conductance parallel (an-\ntiparallel) to the magnetization and g\"#is the spin mix-\ning conductance. The longitudinal component is propor-\ntional to (g\"+g#)=2 and to the relative demagnetiza-\ntion@tM=M . This term is driven by the quasi-ballistic3\nFIG. 2: A) THz waveforms emitted by a set of di\u000berent tri-\nlayers Co/Au 0:85W0:15(d)/Pt. B) Spin current generated in\nCo/Ti(d)/Pt, Co/Au( d)/Pt and Co/Au 0:85W0:15(d)/Pt tri-\nlayers of di\u000berent thickness. The Co and Pt layer have \fxed\nthickness of 2 nm and 5 nm, respectively. The thickness dof\nthe X=Ti,Au,Au 0:85W0:15layer is instead varied between 0\nnm and 2 nm. The parameter \u0011(d) has been extracted from\nthe THz signal via Eqn. [5] and can be considered as a nor-\nmalized spin current density in the platinum layer.\ntransport of highly excited electrons from the Cobalt to\nPlatinum and it represents the dominant contribution in\nthe case of the spin current that are generated by ultra-\nfast laser pulse14. Accordingly, the THz emission from\nCo/Pt is many orders of magnitude more intense8than\nthe one observed from an interface where the longitudinal\ncomponent is inactive18.\nOwing to the quasi-ballistic nature of the injection, the\nspin current arises the spin accumulation taking place\non a length scale 1 :4 nm18,27,28. As a consequence, the\nstrength of the emitted THz radiation scales as the en-\nergy density injected by the pump pulse8,27,28\nj@tM\nMj/ABFI\nd+dPt+dCo; (4)\nwhereABis the absorbed fraction of pump pulse in the\nmultilayer, FIis the incident \ruence of the pump pulse,\ndPt= 5 nm is the thickness of platinum layer, dCo= 2\nnm is the thickness of cobalt layer, dis the thickness\nof the X = Ti, Au or Au 0:85W0:15layer. To investigate\nthis, we have prepared Co/X( d)/Pt trilayers on glass and\nhighly resistive Si(111) substrates by sputtering deposi-\ntion at room temperature with standard experimental\nconditions. The Au 0:85W0:15material has been obtained\nvia the evaporation of a rod containing 85% of gold and\n15% of tungsten. The thickness dof the interlayer is typ-\nically varied between 0 and 2 nm. Within this range of\nd, theABcoe\u000ecient can be considered constant8,27,28.\nMorever the incident laser \ruence FIhas been kept \fxed\nand stable.The THz TDS system is placed in a re\rection geom-\netry where the generated THz pulses are collected from\nthe same surface of the spin-emitter as the excitation\n(i.e. no beam passes through the substrate). The emit-\nters are mounted with small magnetic \feld parallel ( \u0018=10\nmT) to the spin interface. We veri\fed that a switch-\ning of the ~Morientation reverses the direction of the\nemitted THz \feld, thereby con\frming that charges cur-\nrents arise from the ISHE. Fig. 2A) displays a set of THz\ntraces emitted from Co/Au 0:85W0:15(d)/Pt multilayers\nwith di\u000berent values of the Au 0:85W0:15thicknessd. The\nTHz traces recorded for di\u000berent values of dhold nearly\nidentical waveforms (see also supplementary information\n\fle29). SinceET(t;d)\u0018=ET(d)f(t) (and equivalently\nET(!;d)\u0018=ET(d)f(!)), we assume that spin \ructua-\ntions, spin mixing conductance and spin-Hall-angle have\nnegligible frequency dependence within the bandwidth of\nthe detected THz. As observed experimentally, the drop\nof THz signal as a function of dis mainly due to a de-\ncreasing spin-conductance. The latter is related to the\ndetection of the THz \feld via Eqs. [1-4]. By solving for\nthe spin conductance we obtain:\n\u0011(d) =g\"(d) +g#(d)\ng\"(0) +g#(0)=ET(d)\nET(0)Z(0)\nZ(d)d+dPt+dCo\ndPt+dCo;(5)\nwhere the impedance Z(d) has been calculated by as-\nsuming the THz conductivity in thin \flms27,28,30\u001bCo=\n3\u0002106S/m,\u001bPt= 4\u0002106S/m,\u001bAu= 4\u0002106S/m,\n\u001bAu:W= 1:2\u0002106S/m and\u001bTi= 0:5\u0002106S/m. Di\u000ber-\nences of these conductivities with respect to bulk values\nare due to strong charge scattering at the landscape of\nthe interface and to the formation of small grains30. As\na matter of facts, the factor Z(0)=Z(d) remains close to\nunity, owing to the fact that metallic interlayers with\nnanometric thickness have small parallel conductivity.\nIII. DATA ANALYSIS AND DISCUSSION\nThe parameter \u0011(d) of Eqn. [5] re\rects the relative\nreduction of the spin-injection e\u000eciency in Pt if an in-\nterlayer of thickness dis grown between Co and Pt. As\nshown by Fig. 2B), \u0011(d) follows nearly an exponential de-\ncay exp(\u0000d=lX), with characteristic length lAu= 4 nm\nfor X=Au or X=Au 0:85W0:15andlTi= 1:5 nm for X=Ti.\nAs can be observed, Ti a\u000bects the spin mixing conduc-\ntance much more e\u000bectively than Au or Au 0:85W0:15do.\nRecent experiments have shown that a submonolayer\nof Ti can indeed substantially modify the spin-transfer\ntorque of the CoFeB/Pt20and Co/Pt31interfaces. The\ninsertion of the chemically reactive Ti alters the spin\ndependent transmission/re\rection probabilities that fa-\nvor the transport of one spin \ravor with respect to the\nother. Furthermore, the surface passivation by Ti atoms\nmay modify the spin-\rip scattering potential at the in-\nterface. Although the microscopic mechanisms leading to\nthe large reduction of spin conductance is still debated,4\nFIG. 3: A) Derivative of the spin susceptibility vs inten-\nsity of the static magnetic \feld Hin the reference bi-\nlayer Co/Pt. The di\u000berent curves correspond to hrffre-\nquencies of 4-18 GHz, with step of 2 GHz. B) Varia-\ntion of resonance frequency as a function the static mag-\nnetic \feldHin the Co/Au 0:85W0:15(d)/Pt trilayers. C) Full\nwidth at half maximum of the ferromagnetic resonance in\nCo/Au 0:85W0:15(d)/Pt trilayers. The Co layer has thick-\nness of 15 nm, the Pt layer has thickness of 5 nm and the\nAu0:85W0:15layer has thickness dvarying between 0 nm and\n1.5 nm.\na systematic investigation of spin orbit torque with dif-\nferent transition metals concluded that the d-orbital \fll-\ning has a stronger in\ruence on charge-to-spin conversion\nthan the atomic number19. Our measurements corrob-\norate this \fnding: the passivation of Co/Pt interface is\nmore e\u000bective in the case of a transition metal with in-\ncomplete 3d-shell like titanium than in the case of an\nalloy with larger atomic number but closed 5 dshell like\nAu. Moreover, the larger spin-orbit interaction of W in\nthe Au 0:85W0:15does not seem to make any appreciable\ndi\u000berence with respect to pure gold. Our model in the\nlast section of this article will further clarify this, some-\nhow surprising, result.\nNext, we discuss the e\u000bective spin mixing conduc-\ntance that is measured by means of FerroMagnetic Res-\nonance (FMR)12. Samples made with 5 nm of Pt and\nthicker Co \flms (15nm) were deposited on highly resis-\ntive Si/SiO2(111) substrates before lithography pattern-\ning. The thicker ferromagnetic layer provides a clearer\nFIG. 4: A) Gilbert damping and spin conductance in\nCo/Ti(d)/Pt, Co/Au( d)/Pt and Co/Au 0:85W0:15(d)/Pt tri-\nlayers as a function of thickness d. B) Gilbert damping and\nspin conductance of the two trilayer set plot against the \u0011pa-\nrameter extracted from the emitted THz. The green dot cor-\nresponding to vanishing THz emission is the intrinsic Gilbert\ndamping measured on cobalt capped by 2 nm of alumina.\nA model that includes the spin memory loss is calculated via\nEq. [9] and superimposed (solid line) to the experimental data.\nresonance spectrum compared to a 2 nm layer. Fig. 3A)\ndisplays the di\u000berential susceptibility of the Co/Pt bi-\nlayer as a function of the external magnetic \feld H.\nCurves of di\u000berent colors stand for increasing frequency\nof radiofrequency \feld hrf. We show in Fig. 3B) the reso-\nnance frequency !ras a function of Hfor the multilayers\nCo/Au 0:85W0:15(d)/Pt. The FMR theory predicts:\n!r=\r\u00160p\nH(H+M); (6)\nwhere\ris the gyromagnetic ratio and \u00160vacuum perme-\nability and Mis the saturation magnetization. By \ftting\nthe data with Eq. [6], it is possible to extract the satura-\ntion magnetization M= 1500\u000650 emu/cm3. The damp-\ning term can be quanti\fed by measuring the half width\nat half maximum \u0001 Hof FMR linewidth. As shown by\nFig. 3C) the linear regression\n\u0001H= \u0001H0+!r\u000b\n\r\u00160; (7)5\nprovides the Gilbert damping \u000b(d) for the\nCo/Au 0:85W0:15(d)/Pt series. Likewise, this proce-\ndure is applied to extract the Gilbert damping of\nCo/Ti(d)/Pt trilayers. Moroever, the larger thickness of\ncobalt layer ( dCo= 15 nm in FMR experiments instead\nofdCo= 2 nm chosen for the THz emission experiment)\nminimize the extra contribution of two-magnons scatter-\ning to the\u000bvalue. Since two-magnon scattering scales\nas 1=d2\nCo, the associated damping term26should not\nexceed 8\u000210\u00004and it has been neglected. Therefore,\n\u000bdi\u000bers from the intrinsic \u000b0only by a term arising\nfrom the injected spin current. The e\u000bective spin mixing\nconductance g\"#\ne\u000bis obtained via23,25:\n\u0001\u000b=\u000b\u0000\u000b0=g\u0016B\n4\u0019Md Cog\"#\ne\u000b; (8)\nwheregstands for Land\u0013 e factor of the electron and \u0016B\nis the Bohr magnetron. The value \u000b0= 5\u000210\u00003is\nobtained by measuring the Gilbert damping of a 15 nm\ncobalt capped by 2 nm of alumina.\nFig. 4A) shows \u000bandg\"#\ne\u000bfor the two trilayer series as a\nfunction of interlayer thickness d. Similarly to THz mea-\nsurements, the drop of spin mixing conductance is faster\nin Co/Ti(d)/Pt than in Co/Au 0:85W0:15(d)/Pt samples.\nThis \fnding highlights the \frst important outcome of\nthis work: an intimate connection between ( g\"+g#)=2\nobtained by ultrafast currents in the THz spectral range,\nwithg\"#\ne\u000bextracted from the FMR damping linewidth.\nWe \fnd phenomenologically the universal relation:\ng\"#\ne\u000b/\u0011(d)\n1\u0000\u0018(d): (9)\nThe solid line of Fig. 4B is calculated from Eq. [9] with\nparameters g\"#\ne\u000b(0) = 75nm\u00002and\u0018(d) = 0:4\u0011(d). From\ntheir dependence on the transmission coe\u000ecient at the\ninterface17, we evince that g\",g#and (g\"+g#)=2<\ng\"#\u0018=g\"should scale as \u0011(d) upon the insertion of\nthe interlayer. Namely, we assume that g\"(d)=g\"(0) =\ng#(d)=g#(0) =g\"#(d)=g\"#(0) =\u0011(d). Moreover, we set\ng\"#= (1\u0000\u0018)g\"#\ne\u000b, where the parameter \u0018 <1 arises from\nthe spin-memory-loss22,23,25. Due the spin scattering at\nthe interface, the spin-current leading to THz emission\nin platinum is 1\u0000\u0018times smaller than the spin current\na\u000becting the ~Mprecession. The second important result\nof our work is that \u0018is proportional to the spin conduc-\ntance at the interface. The more e\u000ecient the generation\nof spin current, the higher the spin memory loss. When\nexpressed in terms of relative variation of spin conduc-\ntance, the spin memory loss \u0018appears to be insensitive to\nthe compound and thickness that has been employed to\nperform the passivation of the interface. We now turn on\nto the modeling of the spin memory loss through Rashba\n\felds at the interface.IV. MODELING OF SPIN MEMORY LOSS\nTHROUGH RASHBA SPIN-ORBIT\nINTERACTION AT THE INTERFACE.\nA. Electronic quantum transmission with\nspin-orbit interaction\nThe insertion of an interlayer X at the Co/Pt interface\nhas two mains e\u000bects: i)the formation of a thin potential\nbarrier is accompanied by smaller the spin-transmission\nvsCo/Pt. Indeed Co/Pt is known to build an excellent\nmatching for the majority spin channel near the Fermi\nlevel whereas a larger chemical mismatch may take place\nin the case of Co/X/Pt with X=Ti, Au or Au 0:85W0:15\nand; ii)since the Ti or pure Au lack the open 5 dshell of\nPt, the presence of an interlayer has to reduce spin orbit\ninteraction (SOI) at the interface32.\nIn the following, we consider a simpli\fed SOI assisted\nquantum transmission model that has been recently im-\nplemented with success for the description of SOT33{37.\nThis model will \frst highlight the role of i)andii)in the\ndescription of our data. The interface is treated as an\nideal trilayer structure Co/X/Pt with a spin current Js\npropagating along the ~ endirection, normal to the layers\n(CPP geometry). Jsis computed from the propagation\nof selected plane waves with in-plane conserved wavevec-\ntorkk, and normal wavector kzalong~ en. The quantum\ntransmission is summed hereafter over the Fermi surface,\nas it is required within an extended Landauer treatment.\nWe obtain theJs(z) pro\fle across the interface viaa\nre\fned model involving a Rashba-like term33{36. We re-\nstrict the electronic states to two electron bands with spin\npolarized states. The partitioned Hamiltonian in Co and\nPt reads:\n^H=^p2\n2m\u0003+ \u0001 e^m\u0001^\u001b+^V (10)\nwhere ^p=\u0000i~rzis the impulsion operator, m\u0003is the\ne\u000bective mass, ^mis the magnetization direction, \u0001 e'\n2 eV is the exchange coupling for Co, and ^V=^VCo= 0\nrepresents the energy position of the bottom of the spin-\naveraged 3 dCo bands. Along the same lines, we set for\nPt an exchange coupling \u0001 e= 0 and ^V=^VPt'\u00001 eV.\nThe potential di\u000berence ^VCo\u0000^VPtis representative of\nthe workfunction o\u000bset between the two metals.\nThe addition of an interlayer is simulated by an inter-\nfacial potential ^VSthat is expressed by34,36,37:\ntI^VS\u000e(z) =tIh\nVX+\u000bR\n~\u0010\n^~ p\u0002~ en\u0011\n\u0001^\u001bi\n\u000e(z); (11)\nwherezis the coordinate along the direction ~ en, the func-\ntion\u000e(z) is Dirac delta function and tIis the e\u000bective\ninterface thickness. The operator ^VSis de\fned via:VX\nis the average interface of an unpolarized potential bar-\nrier and\u000bRis the strength of Rashba interaction. We\nintroduce the two parameters having the dimension of\ninverse length. The quantity kX=VXtIm\u0003=~2tunes the\ntransmission trough the barrier and kso=\u000bRkFtIm\u0003=~26\nFIG. 5: A) Pro\fle of spin current Jsin the Co/X/Pt surface\nat the vicinity of the Co/Pt interface for 3 di\u000berent cases:\nno scattering potential (black curve), Rashba scattering only\nwithkso= 2\u0017A\u00001(red curve) and Rashba scattering plus a\npotential barrier ( kX= 3\u0017A\u00001). The spin memory loss \u0018=\n(Js(0\u0000)\u0000Js(0+))=Js(0\u0000) is the relative discontinuity of Js\nat the interface. B) Spin memory loss \u0018as a function of\npotential barrier kXfor three di\u000berent strengths of the Rashba\nscattering. C) Spin memory loss \u0018as a function of e\u000bective\nspin-conductance ge\u000bobtained by varying kXand with spin\norbit parameter equal to kso= 2\u0017A\u00001(blue circles). As a term\nof comparison we also show the relation extracted from the\nexperimental data (green dashed line).\nrule the strength of the spin-orbit scattering (see also\nsupplementary information \fle29).\nB. Results of the model\nOur model provides the pro\fle of a normalized spin-\npolarized current originating from Co (where it is normal-\nized to unity) and propagating through a Co/X/Pt tri-\nlayer. Figure 5A) depicts three speci\fc cases, correspond-\ning to: no interfacial potentials ( kX= 0 andkso= 0),\na pure Rashba interaction ( kX= 0 andkso= 2\u0017A\u00001)\nand, both a potential barrier and a Rashba interaction\n(kX= 3\u0017A\u00001andkso= 2\u0017A\u00001). The spin current is always\nmaximal in the bulk of Co, while it goes towards zero\nwhen penetrating in the non-magnetic Pt layer and mov-ing away from the interface. In the absence of the scat-\ntering potential ^VS(black curve in Fig. 5A)) the Js(0)\nvalue at the Co/Pt interface results from an equilibrium\ncondition between bulk spin-\rip rates in the two regions.\nThe spin-current is continuous everywhere (no spin-orbit\nscattering) and its value Js(0)\u00190:6 coincides with the\nprediction of a pure di\u000busive spin-model. This agreement\ncorroborates the validity of our quantum transmission\nmodel in the absence of any ^VSscattering.\nAdding a Rashba interaction kso= 2\u0017A\u00001(red curve\nin Fig. 5A)) leads to the spin-memory loss. Indeed\nthe Rashba \felds are not collinear to the incoming spin\nand induce a local spin-precession. Only a fraction of\nspin current coming from the Co reservoir is injected\ninto the Pt layer so that Js(z) displays a sizable dis-\ncontinuity at the interface22. In order to quantify this\ne\u000bect, we introduce the memory loss parameter \u0018=\n(Js(0\u0000)\u0000Js(0+))=Js(0\u0000), where 0\u0000and 0 +are the lim-\niting values reached by approaching the interface from\nthe Co and Pt side, respectively. From the chosen pa-\nrameters we extract \u0018= 0:6, which is only 50% higher\nthan our experimental value and in agreement with pre-\nvious FRM estimates23.\nThe presence of an additional unpolarized scattering\npotential with kX= 3\u0017A\u00001(Blue curve in Fig. 5A) has\ntwo main e\u000bects. On one hand, the larger back\row of Js\nin the Co layer leads to a smaller ejection of spin-current\nfrom the ferromagnet. On the other hand, an unchanged\nstrength of the Rashba \feld results in a smaller jump of\ntheJscurrent at the interface. As shown in Fig. 5B), the\nmonotonic reduction of spin memory loss as a function\nofkXtakes place for two representatives values of the\ninverse spin length kso.\nWe extract the e\u000bective spin conductance from the\nrescaled ratio between the spin current Js(0\u0000) obtained\nin the presence of an interlayer (i.e. for kX>0) and\ntheJs(0\u0000) obtained for the bare Co/Pt interface (i.e.\nforkX= 0). Figure 5B) shows the calculated \u0018vs.ge\u000b\nwhen the potential barrier kXis increased linearly to\n5\u0017A\u00001while the value ksois kept \fxed to 2 \u0017A\u00001. Note\nthat the spin memory loss display the same trend of the\ncurve that is extracted by combining FMR-spin-pumping\nand THz methods (green dashed line). This shows that\nan interposition of Ti, Au or Au 0:85W0:15introduces a\nchemical barrier at the interface. The enhanced back-\nward di\u000busion of electrons has the e\u000bect of decreasing\nboth the spin mixing conductance and the spin memory\nloss. This e\u000bect takes place even if the spin dependent\nscatteringksoremains equal to the pristine value.\nV. CONCLUSIONS AND\nACKNOWLEDGMENTS.\nIn conclusion, we report that the spin-conductance can\nbe extracted from broadband THz spectroscopy. The in-\nvestigation of Co/X( d)/Pt trilayers with X=Ti, Au and\nAu0:85W0:15show that in all cases, an interlayer reduces7\nthe spin-to-charge conversion. THz experiments have\nbeen bench-marked with the e\u000bective spin-mixing con-\nductance extracted by FerroMagnetic Resonance mea-\nsurements. A model including spin memory loss show\nthat the relative drop of spin current at the interface is\nproportional to the spin conductance and attains \u0018= 0:4\nat the Co/Pt interface. The simulations indicate that\nmodi\fed spin transmission probabilities at the interface\ncan explain this correlation. Our \fndings are very gen-\neral and show that a combination of THz emission with\nFMR spectroscopy can bring accurate characterizations\nand provide new insights into spintronic multilayers.\nWe acknowledge E. Jacquet for his contribution in\nthe thin \flm growth and M. Cosset-Cheneau for hishelp in the FMR experiments. We are very thank-\nful to Tobias Kampfrath and Marco Battiato for the\nenlightening discussions on the interpretation of THz\nemission mechanism. Synchrotron Soleil hosts a THz\nsetup where some transmission measurements have been\ndone. Financial support has been provided by the DGA\nproject ITEHR (No. 2018600074) as well as ANR Project\nTOPRISE No. ANR-16-CE24-0017. We acknowledge\nthe Horizon2020 Framework Programme of the European\nCommission under FET-Proactive Grant agreement No.\n824123 (SKYTOP). This project has received funding\nfrom the H2020 research and innovation programme s-\nNebula under grant agreement No.0863155.\n1E. Saitoh, M. Ueda, H. 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Stiles, \\Interfa-\ncial spin-orbit torques,\" Journal of Applied Physics 128,\n151101 (2020)."    },    {        "title": "2104.10918v1.Impact_of_Fe___80__B___20___insertion_on_the_properties_of_dual_MgO_perpendicular_magnetic_tunnel_junctions.pdf",        "content": "1\nImpact of Fe 80B20 insertion on the properties of dual-MgO perpendicular \nmagnetic tunnel junctions \nEnlong Liu1, Taeyoung Lee2 and Hyunsoo Yang1 \n1 Department of Electrical and computer Engineering, National University of Singapore, \n117576, Singapore \n2 GLOBALFOUNDRIES Singapore Pte. Ltd., Singapore 738 406, Singapore \nE-mail: eleyang@nus.edu.sg \n\nAbstract \nWe explore the impact of Fe 80B20 inserted at both Co 20Fe60B20/MgO interfaces of dual-MgO \nfree layers (FLs) in bottom-pinned magnetic tunne l junctions (MTJs). MTJ stacks are annealed \nfor 30 min at 350 °C and 400 °C in a vacuum after film deposition. Current-in-plane tunneling \nmeasurements are carried out to characteri ze magnetotransport properties of the MTJs. \nConventional magnetometry measurements and ferromagnetic resonance are conducted to \nestimate the saturation magnetization, the effective perpendicular anisotropy field and the Gilbert damping of dual-MgO FLs as a function of the Fe\n80B20 thickness and annealing \ntemperatures. With ultrathin Fe 80B20 (0.2  0.4 nm)  inserted, perpendicular magnetic \nanisotropy (PMA) of FLs increases with si milar tunnel magneto-resistance (TMR) and low \ndamping values. As Fe 80B20 layer thickness further increases (0.6  1.2 nm), both TMR and \nPMA degrade, and damping increases dramatically. This study demonstrates a novel approach \nto tune properties of MTJ stacks with dual-MgO FLs up to 400 °C annealing, which enables \nMTJ stacks for various applications. \n\n1. Introduction \nMagnetic tunnel junctions (MTJs) with perpendi cular magnetic anisotropy (PMA) have been \nstudied in the recent decades as the crucial el ement for next generation memory applications, \nsuch as spin-transfer-torque and spin-orb it-torque magnetic random access memory \n(STT/SOT-MRAM), due to their non-volatility, en ergy effectiveness, high endurance, and \nscalability [1–3]. Many efforts have been made in the engineering of the data-storage layer in MTJs, i.e. the free layer (FL), whose magnetic moment can be switched by the writing current. \nEspecially, dual-MgO FLs with a structure as MgO/CoFeB/spacer/CoFeB/MgO have been under intense development [4–6]. On the one hand, dual-MgO FLs can provide high PMA and \nlow damping to guarantee the high thermal stability and low switching current, after scaling-\ndown of MTJ devices [7]. On the other hand, the aformentioned properties of dual-MgO FLs \ncan be maintained after post-annealing up to 400 °C, which is required for the CMOS back-end-of-line (BEOL) process [8]. \nTo further optimize dual-MgO FLs performanc e, previous studies focused on different \ntopics. Among them, non-magnetic spacer engineering and element composition effects  have \ndrawn lots of interest. Researches on non-ma gnetic spacer sandwiched between two CoFeB \nlayers in dual-MgO FLs have been widely conducte d. Materials such as Mo [9–11], Ta [8,12], \nand W [13–15] were explored as the spacer to identify its impact on PMA and damping before 2\nand after annealing. Other works examined the effect of element (Fe or B) composition in \nCoFeB layers in MTJ stacks on several parameters, including tunnel magneto-resistance \n(TMR) [16], PMA [17–19], and annealing stability  [20]. It has been demonstrated that a \nthickness gradient in the B content can modify the properties of CoFeB/MgO bilayer system \nsuch as damping and anisotropy [21]. PMA of th e FL was also reported to be improved with \nincreasing the Fe composition in CoFeB, on which its TMR is almost independent [22].\nHowever, it is still an open question how a gradient of Fe in dual-MgO FLs impacts the overall \nproperties of MTJ stacks. In such a case, th e PMA of dual-MgO FLs would benefit from an \nincreased Fe concentration, while the TMR of the MTJs is expected to be improved due to the \nformation of Fe/MgO/Fe interface after annealing [23].  \nHere we propose an insertion of ultrathin Fe 80B20 (hereafter FeB) at the interface of \nCo20Fe60B20/MgO in dual-MgO FLs to achieve tunable magnetic properties and annealing \nstability. By optimizing the thickness of the FeB insertion layers at both CoFeB/MgO \ninterfaces, a large PMA can be obtained after 350 °C annealing and further improved at 400 \n°C, which is accompanied with a low damping constant. The result of a low saturation \nmagnetization, large anisotropy field and low damping at the same time in the FeB-inserted dual-MgO FLs makes it promising for low switching currents in MTJ devices without reducing \nthe thermal stability [24]. In addition, the tunability of FL performance by FeB insertion enlarges its potential for various spintronic appl ications where CoFeB-based MTJs are present, \nsuch as SOT-MRAM, spin logic devices and STT nano-oscillators. \n2. Experimental \nBottom-pinned perpendicular MTJs with [Co/Pt ] multilayers as a perpendicular synthetic \nantiferromagnet (p-SAF) were in-situ  deposited at room temperature by magnetron sputtering \non W/Ru/W/Ru/W bottom electrodes (BE) and ca pped by the Ta/Ru top electrode (TE) in a \nULVAC Magest S200 multi-chamber machine. All samples were first annealed with a 0.5 T \nmagnetic field perpendicular to film plane in a magnetic vacuum annealing oven at 350 °C for 30 min. The TMR and resistance-area product (RA) of the MTJ stack was measured via current-\nin-plane tunnelling method (CIPT) [25]. The hyst eresis loops of blanket stacks were measured \nby a vibrating sample magnetometer (VSM) with the magnetic field perpendicular to the \nsample plane. Field-modulated ferromagneti c resonance (FMR) measurements with the \nfrequency range of 10-25 GHz were conducted to ex tract the resonance field and linewidth of \nthe FL versus the frequency, from which the effective perpendicular anisotropy field and Gilbert damping of the FL can be estimated.  All FMR measurements were conducted with \nsamples placed film-side down on a coplanar waveguide in an electromagnet with a field range \nup to 0.5 T and perpendicular to the sample pl ane.  The same batch of samples were then \nannealed at 400 °C for 30 min to study the annealing impact. \n3. Results and discussion \n3.1 Stack characterization without FeB insertion \nThe detailed stack structure of MTJs used in this  study is provided schematically in figure 1(a). \nIt consists of (thickness in nm): \n Hard layer (HL): Pt (5)/[Co (0.25)/Pt (0.2)]/Co (0.6) \n Reference layer (RL): Co (0.6)/Pt (0.2)/ Co (0.3)/Pt (0.2)/Co (0.5)/W (0.3)/Co\n20Fe60B20 \n(0.8) 3\n Free layer (FL): Co 20Fe60B20 (1.2)/W (0.4)/ Co 20Fe60B20 (0.8). \n \nThe FL is sandwiched between the MgO tunnel barrier and the 2nd MgO layer to form a dual-\nMgO structure. The magnetic hysteresis  loop of  the full stack in figure 1(b) indicates good \nPMA in each functional layer after 350 °C annealing. From the minor hysteresis loop shown \nin figure 1(c), PMA of the FL is still maintained  after 400 °C annealing, and the coercive field \nalso increases slightly. The reduction of the saturation magnetization per area ( 𝑀௦∙𝑡) of the \nFL after 400 °C annealing can be attributed to magnetic dead layer formation, which will be \ndiscussed in the following sections.   \nAfter 400 °C annealing, a sloped plateau is observed around 500 mT in the red curve in \nfigure 1(b), indicating a decreased PMA in the RL  [26]. In addition, the TMR of the stack is \nreduced from 110% to 50%. Since the FL PMA is maintained, the TMR reduction is attributed \nto a PMA loss in the RL. Thus, the impact of Fe B insertion in dual-MgO FLs on TMR is studied \nin stacks after 350 °C annealing. \n3.2 Impact of FeB insertion on TMR and RA \nFigure 2(a) and (b) show schematically the FeB insertion position in the dual-MgO FL. Top \nFeB is inserted between the 2nd MgO layer and CoFeB above W spacer, while bottom FeB is \nbetween the MgO tunnel barrier and CoFeB below W spacer. To eliminate the difference in \nthe thickness of FL after insertion, the total thickness of FeB insertion plus remaining CoFeB \nis kept at 0.8 nm and 1.2 nm for layers above  and below W spacer, respectively. As such, the \nthickness of the top FeB insertion layer is chosen  as 0.2, 0.4, 0.6 and 0.8 nm. For the bottom \nFeB insertion layer, its thickness options are 0.2, 0.4, 0.6, 0.8, 1.0 and 1.2 nm. \nThe TMR and RA values as a function of the FeB insertion layer at different CoFeB/MgO \ninterfaces are summarized in figure 2(c) and (d), respectively. The trend for both TMR and RA \nis similar regardless of FeB insertion position. The TMR value is similar for the thickness of FeB < 0.4 nm, but the RA value reduces for thin FeB insertion < 0.6 nm and increases with \nthicker FeB. In addition, it is found that top FeB insertion leads to a more pronounced RA \nreduction. This phenomenon indicates that with top FeB insertion, the contribution to RA from \nthe 2\nnd MgO layer can be reduced. Since the MgO tunnel barrier (0.85 nm) is thicker than the \n2nd MgO layer (0.7 nm), its RA change due to bottom FeB insertion is not as significant as in \nthe top FeB case. Overall, a similar TMR value at low RA is realized when ultrathin FeB (0.2 \n 0.4 nm) is inserted, suggesting that th e formation of Fe/MgO interface benefits \nmagnetotransport properties of MTJ stacks. \nThe TMR starts to drop when FeB thicker than 0.6 nm was inserted at the bottom \nCoFeB/MgO interface, but the RA value does not show any significant change. This TMR drop \nis attributed mainly to a lower spin polarization when thicker FeB replaces CoFeB in the FL \n[27]. \n3.3 Impact of FeB insertion and annealing on FL magnetic properties \nIn figure 1(c), an example of hysteresis loop of  the FL is shown. The saturation magnetization \nper area of the FL can be estimated by using 𝑀௦∙𝑡ൌ𝑚 /𝐴, where 𝑚 is the magnetic moment, \n𝑡 is the thickness of FL, and 𝐴 is the sample area. The effective anisotropy field ( 𝜇𝐻) is \nderived from FMR measurements, as shown by exemplary data in figure 3(a) from dual-MgO \nFLs without FeB insertion, with 0.2 nm top FeB insertion and 0.2 nm bottom FeB insertion, all \nafter 350 C annealing. For each sample, the power absorption by FL versus the applied field 4\nscan is measured at various frequencies, fro m which the ferromagne tic resonance field ( 𝜇𝐻௦) \nand linewidth ( 𝜇Δ𝐻) is estimated. The relation between 𝜇𝐻௦ and frequency 𝑓 is described \nby the Kittel equation for the out-of-plane applied field [28]: \n𝑓ൌఊ\nଶగ൫𝜇𝐻௦𝜇𝐻൯                             (1) \nwhere 𝛾 is the gyromagnetic ratio. From figure 3(a), the x-intercept is the 𝜇𝐻. From 𝑀௦∙𝑡 \nand 𝜇𝐻, the effective perpendicular anisotropy energy can be calculated as \n𝐾∙𝑡ൌଵ\nଶ𝜇𝐻ሺ𝑀௦∙𝑡ሻ.                        (2)  \nFigure 4 summarized the results calculated by the above method. First, 𝑀௦∙𝑡 of FLs can be \ndescribed simply by 𝑀௦∙𝑡ൌ𝑀 ௦ி∙2.0െ𝑡ி∙ሺ𝑀௦ிെ𝑀௦ிሻ, where 𝑡ி is the \ninserted thickness of FeB. It is found in figure 4(a) and (b) that 𝑀௦∙𝑡 decreases with thicker \ninserted FeB, i.e. the slope is negative. Thus, in our stack 𝑀௦ி is smaller than 𝑀௦ி. Next, \nfor the top FeB insertion case in figure 4(a), the amount of change in 𝑀௦∙𝑡 at the same FeB \ninsertion thickness after 350 °C annealing is larger  than that of the bottom insertion case in \nfigure 4(b). This indicates that the top FeB inserted layer is more damaged than the bottom \ninserted FeB. Another major difference between top and bottom insertion cases is a larger 𝑀௦∙\n𝑡 reduction after 400 ºC annealing compared to that after 350 °C annealing, with thick bottom \nFeB inserted (> 0.4 nm). As the bottom-inserted FeB is less damaged, it still reduces the 𝑀௦∙𝑡 \nvalue on further annealing at 400 ºC. However, for the top inserted case, there is not much \nchange after 400 ºC annealing, supporting that the top FeB insertion layer is damanged, mainly \ndue to the 2nd MgO deposition.  \nThe FeB thickness dependence of  𝜇𝐻 and 𝐾∙𝑡 can be discussed together. In the top \nFeB insertion case after 350 ºC annealing, 𝜇𝐻 in figure 4(c) increases monotonically with \nFeB insertion. Due to the reduction in 𝑀௦∙𝑡, however, this increase cannot lead to a higher \nPMA. In figure 4(e), the PMA of FL after 350 ºC  annealing changes little, and even slightly \ndecreases with thicker FeB. However, after 400 ºC annealing, 𝜇𝐻 overall increases without \nany clear dependence on the FeB thickness, and the PMA of FL reaches the maximum value \nwhen 0.2 nm FeB is inserted.  \nOn the other hand, the behavior of 𝜇𝐻 and 𝐾∙𝑡 in bottom-inserted FeB cases is \ndifferent. In general, the PMA can be improved more significantly with bottom-inserted FeB \nthan top-inserted FeB cases. In 350 ºC  annealing cases, both 𝜇𝐻 and 𝐾∙𝑡 increase till \n1.0 nm FeB insertion as shown in figure 4(d) and figure 4(f), respectively. While in 400 ºC \nannealing cases, 𝐾∙𝑡 is significantly improved in thin FeB (0.2 – 0.6 nm) cases. To \nsummarize, the impact of FeB on the PMA of FL differs according to the insertion position. \nRegardless of annealing temperature, FeB insertion at the bottom CoFeB/MgO interface \ninduces a larger PMA, probably due to less damage and the formation of Fe/MgO interface. By changing the FeB insertion position and thickness, magnetic properties of dual-MgO FL \ncan be tuned in a wide range.  \nIt can be noticed that the vertical error bar of  anisotropy field is huge when the FeB thickness \nis 0.8 nm in the top insertion case, and 1.0 or 1.2 nm in bottom insertion cases. It reflects that the resonance field is difficult to be determined precisely in those cases, and thus the fits of \nEq.(1) contains large uncertainties. It is probabl y due to a large damping constant in thick FeB \ninsertion cases, which will be discussed in the following section.  \n3.4 Impact of FeB insertion layer on FL damping 5\n \nIn order to evaluate the effect of FeB insertion on the FL damping, FMR measurements are \nconducted. In figure 3(b), 𝜇Δ𝐻 versus the external excitation frequency for the three samples \nwas plotted. The linewidth of the resonance is linear in frequency: \n𝜇Δ𝐻 ൌ 𝜇 Δ𝐻ସగఈ\nఊ𝑓                          (3) \nwhere 𝜇Δ𝐻 is the inhomogeneous linewidth broadening and 𝛼 is the Gilbert damping \ncoefficient.  \nFigure 5 summarizes 𝛼 as a function of top or bottom FeB insertion thickness under different \nannealing conditions. For top FeB insertion (figure 5(a)), the influence of FeB insertion on 𝛼 \nshows a moderate dependence on its thickness after 350 C annealing. For bottom FeB insertion \n(figure 5(b)), however, 𝛼 reaches the minimum at 0.4 nm FeB insertion and increases \ndramatically beyond the measurement range w ith the FeB thickness in both annealing \nconditions. For those cases, the resonance is to o broadened to be resolved, reflecting a very \nlarge damping [21]. It also leads to a huge uncertainty in the resonance field and hence 𝜇𝐻 \ndetermination, as mentioned in the preivous section.  \nAn increase of 𝛼 is observed at 0.6 nm FeB in the 400 C annealing condition. However, a \nreduction in 𝛼 after 400 C annealing is obtained when ultrathin FeB (0.2  0.4 nm) is inserted \nat either interface. This suggests that the annealing treatment has different effects on 𝛼 of the \nsamples with various FeB insertion thicknesses. Perhaps different amount of FeB insertion leads to changes in the Fe concentration, micr ostructures and crystallization of dual-MgO FLs \nafter boron depletion upon annealing and thereby to different damping behaviors [29,30]. \nFinally, Table 1 summarizes and compares MTJ stacks after 350 C annealing with FeB \ninsertion at top, bottom, or both CoFeB/MgO interfaces. As the FeB insertion thickness and \nposition differ, the PMA of the dual-MgO FL can be tuned in a wide range, while the damping \nis almost independent. From the systematic studies on insertion thickness in the previous \nsections, top and bottom FeB are optimized to be 0.2 nm and 0.4 nm, respectively. As a result, \nthe MTJ stack with such FeB insertion at both interfaces in dual-MgO FLs can be engineered \nwith a high TMR, low RA, large 𝐾\n∙𝑡, low 𝑀௦∙𝑡, high 𝜇𝐻, and low damping constant.  \n4. Conclusion \nIn this paper, we explore the impact of Fe 80B20 layer inserted at two interfaces of \nCo40Fe60B20/MgO in dual-MgO FLs in MTJ stacks and its annealing stability. With ultrthin \nFeB (0.2  0.4 nm) inserted at the top or bottom CoFeB/MgO interface, the TMR can be \nmaintained with lower RA values, while the top-FeB insertion results in a more RA drop with \na similar TMR. In both cases, the FL saturation magnetization reduces with increasing the Table1. Comparison of magnetotransport and magnetic proper ties of dual-MgO FLs with different FeB insertion \nafter 350 C 30 min annealing. \n \nFL types TMR RA 𝑀௦∙𝑡 𝜇𝐻 𝐾∙𝑡  \n% m2 10-5 A mT mJ∙m2 10-3 \nMgO/CoFeB(1.2)/W/CoFeB(0.8)/MgO 109.3 8.5 184.0  2.7 320  1 0.294  0.004 12.5  2.6 \nMgO/CoFeB(1.2)/W/CoFeB(0.6)/ FeB(0.2) /MgO 114.9 6.9 177.7  3.1 340  5 0.302  0.007 15.0  4.9 \nMgO/ FeB(0.2) /CoFeB(1.0)/W/CoFeB(0.8)/MgO 108.6 8.3 181.2  1.7 349  2 0.316  0.006 11.3  5.0 \nMgO/ FeB(0.2) /CoFeB(1.0)/W/CoFeB(0.6)/ FeB(0.2) /MgO 108.2 7.5 175.7  2.6 381  2 0.335  0.005 12.1  0.7 \nMgO/ FeB(0.4) /CoFeB(0.8)/W/CoFeB(0.6)/ FeB(0.2) /MgO 110.5 8.6 168.3  2.9 434  2 0.365  0.006  4.4  0.7 \n 6\ninserted FeB thickness, while the FL effective anisotropy field increases. However, the PMA \nof dual-MgO FLs with FeB inserted at the bo ttom interface shows a larger improvement than \nits top FeB insertion counterpart, even after 400 C annealing. At the same time, the FeB (0.2 \n 0.4 nm) insertion at either interface reduces the damping constant in the FL. By optimizing \nthe FeB insertion layer thickness, the dual-MgO FL with a low saturation magnetization, high \neffective anisotropy field and low damping can be achieved after 400 C annealing. However, \nthe performance degrades if a thicker FeB is used to replace CoFeB in dual-MgO FLs. \nThis study demonstrates a novel approach to tune dual-MgO FL properties other than typical \nboron composition or non-magnetic spacer engineering. By using the FeB insertion layer at \nCoFeB/MgO interfaces, magnetic properties of the FL and the magnetotransportation of MTJs \ncan be engineered in a wide range, which enables MTJs to meet different performance requirements for various spintronic applications. \nAcknowledgements \nThis work was supported by NRF In vestigatorship (NRFI06-2020-0015). \nReferences \n[1] Ikeda S, Miura K, Yamamoto H, Mizunuma K, Gan H D, Endo M, Kanai S, Hayakawa \nJ, Matsukura F and Ohno H 2010 A perpendicular-anisotropy CoFeB-MgO magnetic \ntunnel junction. Nat. 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Express  6 073002 \n\n9\n \nFigure1. (a) Stack layout of blanket MTJs without FeB insertion. Free layer (FL), reference \nlayer (RL) and hard layer (HL) are indicated. Thicknesses of sublayers are shown in nm with \nparentheses. Blanket films were annealed first at 350 C and then at 400 C, both for 30 min. \n(b) Major loop and (c) minor loop of the stack in  (a) measured by VSM after different annealing \nconditions with the magnetic field perpendicular to the sample plane. \n\n \nFigure2. Schematic of FeB insertion at (a) top and (b) bottom CoFeB/MgO interface in dual-\nMgO FL in the stack shown in figure 1(a). The total thickness of inserted FeB plus remaining \nCoFeB is kept at 0.8 nm and 1.2 nm for layers above and below W spacer, respectively. The \nimpact of FeB insertion on TMR (c) a nd RA (d) of the MTJ stacks after 350 C annealing are \nshown. \n\n10\n \nFigure3. (a) The external excitation frequency as a function of ferromagnetic resonance field \nand (b) the linewidth versus frequency of FL in MTJ stacks without FeB insertion (black open \ncircles), with 0.2 nm top FeB insertion (red open triangles), and with 0.2 nm bottom FeB \ninsertion (blue open squares). Solid lines are fits. \n \n \nFigure4. Effect of FeB insertion and annealing conditions on 𝑀௦∙𝑡 ((a) and (b)), 𝜇𝐻 ((c) \nand (d)), and 𝐾∙𝑡 ((e) and (f)). (a), (c) and (e) show th e impact from FeB insertion at the \ntop CoFeB/MgO interface, while (b), (d) and (f) show the impact from bottom interface. \n11\n\n \nFigure5. Gilbert damping as a function of FeB insertion thickness at (a) top interface and (b) \nbottom interface under two annealing conditions. \n\n"    },    {        "title": "2105.03576v1.A_second_order_numerical_method_for_Landau_Lifshitz_Gilbert_equation_with_large_damping_parameters.pdf",        "content": "A SECOND-ORDER NUMERICAL METHOD FOR\nLANDAU-LIFSHITZ-GILBERT EQUATION WITH LARGE\nDAMPING PARAMETERS\nYONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE\nAbstract. A second order accurate numerical scheme is proposed and imple-\nmented for the Landau-Lifshitz-Gilbert equation, which models magnetization\ndynamics in ferromagnetic materials, with large damping parameters. The\nmain advantages of this method are associated with the following features:\n(1) It only solves linear systems of equations with constant coe\u000ecients where\nfast solvers are available, so that the numerical e\u000eciency has been greatly im-\nproved, in comparison with the existing Gauss-Seidel project method. (2) The\nsecond-order accuracy in time is achieved, and it is unconditionally stable for\nlarge damping parameters. Moreover, both the second-order accuracy and the\ngreat e\u000eciency improvement will be veri\fed by several numerical examples\nin the 1D and 3D simulations. In the presence of large damping parameters,\nit is observed that this method is unconditionally stable and \fnds physically\nreasonable structures while many existing methods have failed. For the do-\nmain wall dynamics, the linear dependence of wall velocity with respect to the\ndamping parameter and the external magnetic \feld will be obtained through\nthe reported simulations.\n1.Introduction\nFerromagnetic materials are widely used for data storage due to the bi-stable\nstates of the intrinsic magnetic order or magnetization. The dynamics of magneti-\nzation has been modeled by the Landau-Lifshitz-Gilbert (LLG) equation [9,13]. In\nparticular, two terms are involved in the dynamics of the LLG equation: the gyro-\nmagnetic term, which is energetically conservative, and the damping term, which\nis energetically dissipative.\nThe damping term is important since it strongly a\u000bects the energy required and\nthe speed at which a magnetic device operates. A recent experiment on a magnetic-\nsemiconductor heterostructure [25] has indicated that the Gilbert damping constant\ncan be adjusted. At the microscopic level, the electron scattering, the itinerant\nelectron relaxation [11], and the phonon-magnon coupling [16, 17] are responsible\nto the damping, which can be obtained from electronic structure calculations [19].\nFor the application purpose, tuning the damping parameter allows one to optimize\nthe magneto-dynamic properties in the material, such as lowering the switching\ncurrent and increasing the writing speed of magnetic memory devices [23].\nWhile most experiments have been devoted to small damping parameters [4,14,\n22], large damping e\u000bects are observed in [10,18]. The magnetization switching time\nDate : May 11, 2021.\n2010 Mathematics Subject Classi\fcation. 35K61, 65N06, 65N12.\nKey words and phrases. Micromagnetics simulations, Landau-Lifshitz-Gilbert equation,\nsecond-order method, large damping parameter.\n1arXiv:2105.03576v1  [physics.comp-ph]  8 May 20212 Y. CAI, J. CHEN, C. WANG, AND C. XIE\ntends to be shorter in the presence of the large damping constant [18]. Extremely\nlarge damping parameters ( \u00189) are presented in [10].\nThe LLG equation is a vectorial and nonlinear system with the \fxed length of\nmagnetization in a point-wise sense. Signi\fcant e\u000borts have been devoted to design\ne\u000ecient and stable numerical methods for micromagnetics simulations; see [6, 12]\nfor reviews and references therein. Among the existing numerical works, semi-\nimplicit schemes have been very popular since they avoid a complicated nonlinear\nsolver while preserving the numerical stability; see [2, 7, 24], etc. In particular,\nthe second-order accurate backward di\u000berentiation formula (BDF) scheme is con-\nstructed in [24], with a one-sided interpolation. In turn, a three-dimensional lin-\near system needs to be solved at each time step, with non-constant coe\u000ecients.\nMoreover, a theoretical analysis of the second order convergence estimate has been\nestablished in [5] for such a BDF2 method. As another approach, a linearly implicit\nmethod in [2] introduces the tangent space to deal with the length constraint of\nmagnetization, with the \frst-order temporal accuracy. As a further extension, high-\norder BDF schemes have been constructed and analyzed in a more recent work [1].\nAn unconditionally unique solvability of the semi-implicit schemes has been proved\nin [1,5], while the convergence analysis has required a condition that the temporal\nstep-size is proportional to the spatial grid-size. However, an obvious disadvantage\nhas been observed for these semi-implicit schemes: the vectorial structure of the\nLLG equation leads to a non-symmetric linear system at each time step, which\ncannot be implemented by an FFT-based fast solver. In fact, the GMRES is often\nused, while its e\u000eciency depends heavily on the temporal step-size and the spatial\ngrid-size, and extensive numerical experiments have indicated much more expensive\ncomputational costs than standard Poisson solvers [24].\nThe Gauss-Seidel projection method (GSPM) is another popular set of numerical\nalgorithms since only linear systems with constant coe\u000ecients need to be solved at\neach time step [8,15,21]. This method is based on a combination of a Gauss-Seidel\nupdate of an implicit solver for the gyromagnetic term, the heat \row of the harmonic\nmap, and a projection step to overcome the sti\u000bness and the nonlinearity associated\nto the LLG equation. In this numerical approach, the implicit discretization is only\napplied to the scalar heat equation implicitly several times; therefore, the FFT-\nbased fast solvers become available, due to the symmetric, positive de\fnite (SPD)\nstructures of the linear system. The original GSPM method [20] turns out to be\nunstable for small damping parameters, while this issue has been resolved in [8] with\nmore updates of the stray \feld. Its numerical e\u000eciency has been further improved\nby reducing the number of linear systems per time step [15]. One little de\fciency\nof GSPM is its \frst-order accuracy in time.\nMeanwhile, in spite of these improvements, the GSPM method is computation-\nally more expensive than the standard Poisson solver, because of the Gauss-Seidel\niteration involved in the algorithm. An additional de\fciency of the GSPM is its\n\frst-order accuracy in time. Moreover, most of the above-mentioned methods have\nbeen mainly focused on small damping parameters with the only exception in a\ntheoretical work [1]. In other words, there has been no numerical method designed\nspeci\fcally for real micromagnetics simulations with large damping parameters. In\nthis paper, we propose a second-order accurate numerical method to solve the LLG\nequation with large damping parameters, whose complexity is also comparable toA SECOND-ORDER METHOD FOR LLG EQUATION 3\nsolving the scalar heat equation. To achieve this goal, the LLG system is refor-\nmulated, in which the damping term is rewritten as a harmonic mapping \row. In\nturn, the constant-coe\u000ecient Laplacian part is treated by a standard BDF2 tem-\nporal discretization, and the associated dissipation will form the foundation of the\nnumerical stability. Meanwhile, all the nonlinear parts, including both the gyro-\nmagnetic term and the remaining nonlinear expansions in the damping term, are\ncomputed by a fully explicit approximation, which is accomplished by a second\norder extrapolation formula. Because of this fully explicit treatment for the nonlin-\near parts, the resulting numerical scheme only requires a standard Poisson solver at\neach time step. This fact will greatly facilitate the computational e\u000borts, since the\nFFT-based fast solver could be e\u000eciently applied, due to the SPD structure of the\nlinear system involved at each time step. In addition, the numerical stability has\nbeen demonstrated by extensive computational experiments, and these experiments\nhas veri\fed the idea that the dissipation property of the heat equation part would\nbe able to ensure the numerical stability of the nonlinear parts, with large damping\nparameters.\nThe rest of this paper is organized as follows. In section 2, the micromagnetics\nmodel is reviewed, and the numerical method is proposed, as well as its comparison\nwith the GSPM and the semi-implicit projection method (SIPM). Subsequently,\nthe numerical results are presented in section 3, including the temporal and spa-\ntial accuracy check in both the 1D and 3D computations, the numerical e\u000eciency\ninvestigation (in comparison with the GSPM and SIPM algorithms), the stability\nstudy with respect to the damping parameter, and the dependence of domain wall\nvelocity on the damping parameter and the external magnetic \feld. Finally, some\nconcluding remarks are made in section 4.\n2.The physical model and the numerical method\n2.1.Landau-Lifshitz-Gilbert equation. The LLG equation describes the dy-\nnamics of magnetization which consists of the gyromagnetic term and the damping\nterm [3,13]. In the nondimensionalized form, this equation reads as\nmt=\u0000m\u0002he\u000b\u0000\u000bm\u0002(m\u0002he\u000b) (2.1)\nwith the homogeneous Neumann boundary condition\n(2.2)@m\n@\u0017\f\f\f\n@\n= 0;\nwhere \n is a bounded domain occupied by the ferromagnetic material and \u0017is unit\noutward normal vector along @\n.\nIn more details, the magnetization m: \n\u001aRd!R3;d= 1;2;3 is a three-\ndimensional vector \feld with a pointwise constraint jmj= 1. The \frst term on the\nright-hand side in (2.1) is the gyromagnetic term and the second term stands for\nthe damping term, with \u000b>0 being the dimensionless damping coe\u000ecient.\nThe e\u000bective \feld he\u000bis obtained by taking the variation of the Gibbs free energy\nof the magnetic body with respect to m. The free energy includes the exchange\nenergy, the anisotropy energy, the magnetostatic energy, and the Zeeman energy:\n(2.3)F[m] =\u00160M2\ns\n2\u001aZ\n\n\u0000\n\u000fjrmj2+q\u0000\nm2\n2+m2\n3\u0001\n\u00002he\u0001m\u0000hs\u0001m\u0001\ndx\u001b\n:4 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nTherefore, the e\u000bective \feld includes the exchange \feld, the anisotropy \feld, the\nstray \feldhs, and the external \feld he. For a uniaxial material, it is clear that\nhe\u000b=\u000f\u0001m\u0000q(m2e2+m3e3) +hs+he; (2.4)\nwhere the dimensionless parameters become \u000f=Cex=(\u00160M2\nsL2) andq=Ku=(\u00160M2\ns)\nwithLthe diameter of the ferromagnetic body and \u00160the permeability of vacuum.\nThe unit vectors are given by e2= (0;1;0),e3= (0;0;1), and \u0001 denotes the\nstandard Laplacian operator. For the Permalloy, an alloy of Nickel (80%) and\nIron (20%), typical values of the physical parameters are given by: the exchange\nconstantCex= 1:3\u000210\u000011J/m, the anisotropy constant Ku= 100 J/m3, the sat-\nuration magnetization constant Ms= 8:0\u0002105A/m. The stray \feld takes the\nform\nhs=1\n4\u0019rZ\n\nr\u00121\njx\u0000yj\u0013\n\u0001m(y)dy: (2.5)\nIf \n is a rectangular domain, the evaluation of (2.5) can be e\u000eciently done by the\nFast Fourier Transform (FFT) [20].\nFor brevity, the following source term is de\fned\nf=\u0000Q(m2e2+m3e3) +hs+he: (2.6)\nand the original PDE system (2.1) could be rewritten as\nmt=\u0000m\u0002(\u000f\u0001m+f)\u0000\u000bm\u0002m\u0002(\u000f\u0001m+f): (2.7)\nThanks to point-wise identity jmj= 1, we obtain an equivalent form:\n(2.8)mt=\u000b(\u000f\u0001m+f) +\u000b\u0000\n\u000fjrmj2\u0000m\u0001f\u0001\nm\u0000m\u0002(\u000f\u0001m+f):\nIn particular, it is noticed that the damping term is rewritten as a harmonic map-\nping \row, which contains a constant-coe\u000ecient Laplacian di\u000busion term. This fact\nwill greatly improve the numerical stability of the proposed scheme.\nFor the numerical description, we \frst introduce some notations for discretization\nand numerical approximation. Denote the temporal step-size by k, andtn=nk,\nn\u0014\u0004T\nk\u0005\nwithTthe \fnal time. The spatial mesh-size is given by hx=hy=hz=\nh= 1=N, andmn\ni;j;`stands for the magnetization at time step tn, evaluated at the\nspatial location ( xi\u00001\n2;yj\u00001\n2;z`\u00001\n2) withxi\u00001\n2=\u0000\ni\u00001\n2\u0001\nhx,yj\u00001\n2=\u0000\nj\u00001\n2\u0001\nhyand\nz`\u00001\n2=\u0000\n`\u00001\n2\u0001\nhz(0\u0014i;j;`\u0014N+ 1). In addition, a third order extrapolation\nformula is used to approximate the homogeneous Neumann boundary condition.\nFor example, such a formula near the boundary along the zdirection is given by\nmi;j;1=mi;j;0;mi;j;N +1=mi;j;N:\nThe boundary extrapolation along other boundary sections can be similarly made.\nThe standard second-order centered di\u000berence applied to \u0001 mresults in\n\u0001hmi;j;k=mi+1;j;k\u00002mi;j;k+mi\u00001;j;k\nh2x\n+mi;j+1;k\u00002mi;j;k+mi;j\u00001;k\nh2y\n+mi;j;k+1\u00002mi;j;k+mi;j;k\u00001\nh2z;A SECOND-ORDER METHOD FOR LLG EQUATION 5\nand the discrete gradient operator rhmwithm= (u;v;w )Treads as\nrhmi;j;k=2\n64ui+1;j;k\u0000ui\u00001;j;k\nhxvi+1;j;k\u0000vi\u00001;j;k\nhxwi+1;j;k\u0000wi\u00001;j;k\nhxui;j+1;k\u0000ui;j\u00001;k\nhyvi;j+1;k\u0000vi;j\u00001;k\nhywi;j+1;k\u0000wi;j\u00001;k\nhyui;j;k +1\u0000ui;j;k\u00001\nhzvi;j;k +1\u0000vi;j;k\u00001\nhzwi;j;k +1\u0000wi;j;k\u00001\nhz3\n75:\nSubsequently, the GSPM and the SIPM numerical methods need to be reviewed,\nwhich could be used for the later comparison.\n2.2.The Gauss-Seidel projection method. The GSPM is based on a combi-\nnation of a Gauss-Seidel update of an implicit solver for the gyromagnetic term,\nthe heat \row of the harmonic map, and a projection step. It only requires a series\nof heat equation solvers with constant coe\u000ecients; as a result, the FFT-based fast\nsolvers could be easily applied. This method is \frst-order in time and second-order\nin space. Below is the detailed outline of the GSPM method in [8].\nStep 1. Implicit Gauss-Seidel:\ngn\ni= (I\u0000\u000f\u0001t\u0001h)\u00001(mn\ni+ \u0001tfn\ni); i= 2;3;\ng\u0003\ni= (I\u0000\u000f\u0001t\u0001h)\u00001(m\u0003\ni+ \u0001tf\u0003\ni); i= 1;2; (2.9)\n(2.10)0\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:\nStep 2. Heat \row without constraints:\n(2.11) f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +h\u0003\ns+he;\n(2.12)0\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:\nStep 3. Projection onto S2:\n(2.13)0\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:\nHerem\u0003denotes the intermediate values of m, and stray \felds hn\nsandh\u0003\nsare\nevaluated at mnandm\u0003, respectively.\nRemark 2.1. Two improved versions of the GSPM have been studied in [15], which\nturn out to be more e\u000ecient than the original GSPM. Meanwhile, it is found that\nboth improved versions become unstable when \u000b > 1, while the original GSPM\n(outlined above) is stable even when \u000b\u001410. Therefore, we shall use the original\nGSPM in [8]for the numerical comparison in this work.6 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n2.3.Semi-implicit projection method. The SIPM has been outlined in [5,24].\nThis method is based on the second-order BDF temporal discretization, combined\nwith an explicit extrapolation. It is found that SIPM is unconditionally stable and\nis second-order accurate in both space and time. The algorithmic details are given\nas follows.\n(2.14)8\n>>>>>><\n>>>>>>:3\n2~mn+2\nh\u00002mn+1\nh+1\n2mn\nh\nk=\u0000^mn+2\nh\u0002\u0000\n\u000f\u0001h~mn+2\nh+^fn+2\nh\u0001\n\u0000\u000b^mn+2\nh\u0002\u0010\n^mn+2\nh\u0002(\u000f\u0001h~mn+2\nh+^fn+2\nh)\u0011\n;\nmn+2\nh=~mn+2\nh\nj~mn+2\nhj;\nwhere ~mn+2\nhis an intermediate magnetization, and ^mn+2\nh,^fn+2\nhare given by the\nfollowing extrapolation formula:\n^mn+2\nh= 2mn+1\nh\u0000mn\nh;\n^fn+2\nh= 2fn+1\nh\u0000fn\nh;\nwithfn\nh=\u0000Q(mn\n2e2+mn\n3e3) +hn\ns+hn\ne. The presence of cross product in the\nSIPM yields a linear system of equations with non-symmetric structure and vari-\nable coe\u000ecients. In turn, the GMRES solver has to be applied to implement this\nnumerical system. The numerical evidence has revealed that, the convergence of\nGMRES solver becomes slower for larger temporal step-size kor smaller spatial\ngrid-sizeh, which makes the computation more challenging.\n2.4.The proposed numerical method. The SIPM in (2.14) treats both the\ngyromagentic and the damping terms in a semi-implicit way, i.e., \u0001 mis computed\nimplicitly, while the coe\u000ecient functions are updated by a second order accurate,\nexplicit extrapolation formula. The strength of the gyromagnetic term is controlled\nby \u0001m+fsince the length of mis always 1. Meanwhile, the strength of the\ndamping term is controlled by the product of \u0001 m+fand the damping parameter\n\u000b. For small \u000b, say\u000b\u00141, it is reasonable to treat both the gyromagentic and\nthe damping terms semi-implicitly. However, for large \u000b, an alternate approach\nwould be more reasonable, in which the whole gyromagentic term is computed by\nan explicit extrapolation, while the nonlinear parts in the damping term is also\nupdated by an explicit formula, and only the constant-coe\u000ecient \u0001 mpart in the\ndamping term is implicitly updated. This idea leads to the proposed numerical\nmethod. To further simplify the presentation, we start with (2.8), and the numerical\nalgorithm is proposed as follows.\n(2.15)8\n>>>>>>>>>><\n>>>>>>>>>>:3\n2~mn+2\nh\u00002mn+1\nh+1\n2mn\nh\nk=\u0000^mn+2\nh\u0002\u0010\n\u000f\u0001h^mn+2\nh+^fn+2\nh\u0011\n+\u000b\u0010\n\u000f\u0001h~mn+2\nh+^fn+2\nh\u0011\n+\u000b\u0010\n\u000fjrh^mn+2\nhj2\u0000^mn+2\nh\u0001^fn+2\nh\u0011\n^mn+2\nh;\nmn+2\nh=~mn+2\nh\nj~mn+2\nhj;A SECOND-ORDER METHOD FOR LLG EQUATION 7\nwhere\n^mn+2\nh= 2mn+1\nh\u0000mn\nh;\n^fn+2\nh= 2fn+1\nh\u0000fn\nh:\nTable 1 compares the proposed method, the GSPM and the SIPM in terms\nof number of unknowns, dimensional size, symmetry pattern, and availability of\nFFT-based fast solver of linear systems of equations, and the number of stray \feld\nupdates. At the formal level, the proposed method is clearly superior to both the\nGSPM and the SIPM algorithms. In more details, this scheme will greatly improve\nthe computational e\u000eciency, since only three Poisson solvers are needed at each\ntime step. Moreover, this numerical method preserves a second-order accuracy in\nboth space and time. The numerical results in section 3 will demonstrate that the\nproposed scheme provides a reliable and robust approach for micromagnetics simu-\nlations with high accuracy and e\u000eciency in the regime of large damping parameters.\nTable 1. Comparison of the proposed method, the Gauss-Seidel\nprojection method, and the semi-implicit projection method.\nProperty or number Proposed method GSPM SIPM\nLinear systems 3 7 1\nSize N3N33N3\nSymmetry Yes Yes No\nFast Solver Yes Yes No\nAccuracy O(k2+h2)O(k+h2)O(k2+h2)\nStray \feld updates 1 4 1\nRemark 2.2. To kick start the proposed method, one can apply a \frst-order al-\ngorithm, such as the \frst-order BDF method, in the \frst time step. An overall\nsecond-order accuracy is preserved in this approach.\n3.Numerical experiments\nIn this section, we present a few numerical experiments with a sequence of damp-\ning parameters for the proposed method, the GSPM [8] and the SIPM [24], with\nthe accuracy, e\u000eciency, and stability examined in details. Domain wall dynamics\nis studied and its velocity is recorded in terms of the damping parameter and the\nexternal magnetic \feld.\n3.1.Accuracy and e\u000eciency tests. We set\u000f= 1 andf= 0 in (2.8) for conve-\nnience. The 1D exact solution is given by\nme= (cos(X) sint;sin(X) sint;cost)T;\nand the corresponding exact solution in 3D becomes\nme= (cos(XYZ ) sint;sin(XYZ ) sint;cost)T;\nwhereX=x2(1\u0000x)2,Y=y2(1\u0000y)2,Z=z2(1\u0000z)2. In fact, the above exact\nsolutions satisfy (2.8) with the forcing term g=@tme\u0000\u000b\u0001me\u0000\u000bjrmej2+me\u0002\n\u0001me, as well as the homogeneous Neumann boundary condition.8 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nFor the temporal accuracy test in the 1D case, we \fx the spatial resolution\nash= 5D\u00004, so that the spatial approximation error becomes negligible. The\ndamping parameter is taken as \u000b= 10, and the \fnal time is set as T= 1. In the 3D\ntest for the temporal accuracy, due to the limitation of spatial resolution, we take\na sequence of spatial and temporal mesh sizes: k=h2\nx=h2\ny=h2\nz=h2= 1=N0\nfor the \frst-order method and k=hx=hy=hz=h= 1=N0for the second-\norder method, with the variation of N0indicated below. Similarly, the damping\nparameter is given by \u000b= 10, while the \fnal time Tis indicated below. In turn,\nthe numerical errors are recorded in term of the temporal step-size kin Table 2. It\nis clear that the temporal accuracy orders of the proposed numerical method, the\nGSPM, and the SIPM are given by 2, 1, and 2, respectively, in both the 1D and\n3D computations.\nThe spatial accuracy order is tested by \fxing k= 1D\u00005,\u000b= 10,T= 1 in 1D\nandk= 1D\u00003,\u000b= 10,T= 1 in 3D. The numerical error is recorded in term of\nthe spatial grid-size hin Table 3. Similarly, the presented results have indicated\nthe second order spatial accuracy of all the numerical algorithms, including the\nproposed method, the GSPM, and the SIPM, respectively, in both the 1D and 3D\ncomputations.\nTo make a comparison in terms of the numerical e\u000eciency, we plot the CPU time\n(in seconds) vs. the error norm kmh\u0000mek1. In details, the CPU time is recorded\nas a function of the approximation error in Figure 1a in 1D and in Figure 1b in\n3D, with a variation of kand a \fxed value of h. Similar plots are also displayed in\nFigure 1c in 1D and Figure 1d in 3D, with a variation of hand a \fxed value of k. In\nthe case of a \fxed spatial resolution h, the proposed method is signi\fcantly more\ne\u000ecient than the GSPM and the SIPM in both the 1D and 3D computations. The\nSIPM is slightly more e\u000ecient than the GSPM, while such an advantage depends\non the performance of GMRES, which may vary for di\u000berent values of kandh. In\nthe case of a \fxed time step size k, the proposed method is slightly more e\u000ecient\nthan the GSPM, in both the 1D and 3D computations, and the GSPM is more\ne\u000ecient than the SIPM.\n3.2.Stability test with large damping parameters. To check the numerical\nstability of these three methods in the practical simulations of micromagnetics with\nlarge damping parameters, we consider a thin \flm of size 480 \u0002480\u000220 nm3with\ngrid points 100\u0002100\u00024. The temporal step-size is taken as k= 1 ps. A uniform\nstate along the xdirection is set to be the initial magnetization and the external\nmagnetic \feld is set to be 0. Three di\u000berent damping parameters, \u000b= 0:01;10;40,\nare tested with stable magnetization pro\fles shown in Figure 2. In particular, the\nfollowing observations are made.\n\u000fThe proposed method is the only one that is stable for very large damping\nparameters;\n\u000fAll three methods are stable for moderately large \u000b;\n\u000fThe proposed method is the only one that is unstable for small \u000b.\nIn fact, a preliminary theoretical analysis reveals that, an optimal rate convergence\nestimate of the proposed method could be theoretically justi\fed for \u000b>3. Mean-\nwhile, extensive numerical experiments have implied that \u000b > 1 is su\u000ecient to\nensure the numerical stability in the practical computations.A SECOND-ORDER METHOD FOR LLG EQUATION 9\nTable 2. The numerical errors for the proposed method, the\nGSPM and the SIPM with \u000b= 10 andT= 1. Left: 1D with\nh= 5D\u00004; Right: 3D with k=h2\nx=h2\ny=h2\nz=h2= 1=N0\nfor GSPM and k=hx=hy=hz=h= 1=N0for the proposed\nmethod and SIPM, with N0speci\fed in the table.\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 4.459D-4 5.226D-4 5.588D-4 1/20 6.171D-4 4.240D-4 4.246D-4\n2.0D-2 1.147D-4 1.345D-4 1.436D-4 1/24 4.381D-4 3.010D-4 3.014D-4\n1.0D-2 2.899D-5 3.402D-5 3.631D-5 1/28 3.268D-4 2.245D-4 2.248D-4\n5.0D-3 7.192D-6 8.529D-6 9.119D-6 1/32 2.531D-4 1.739D-4 1.741D-4\n2.5D-3 1.699D-6 2.321D-6 2.518D-6 1/36 2.017D-4 1.386D-4 1.387D-4\norder 2.007 1.961 1.957 { 1.902 1.903 1.903\n(a)Proposed method\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=h2k\u0001k1k\u0001k 2k\u0001kH1\n2.5D-3 2.796D-4 2.264D-4 1.445D-3 1/36 4.194D-4 2.683D-4 2.815D-4\n1.25D-3 1.425D-4 1.174D-4 7.720D-4 1/64 2.388D-4 1.399D-4 1.500D-4\n6.25D-4 7.170D-5 5.940D-5 4.026D-4 1/144 1.069D-4 6.106D-5 6.736D-5\n3.125D-4 3.591D-5 2.971D-5 2.069D-4 1/256 6.021D-5 3.442D-5 3.860D-5\n1.5625D-4 1.799D-5 1.488D-5 1.054D-4 1/400 3.855D-5 2.208D-5 2.501D-5\norder 0.991 0.984 0.945 { 0.992 1.032 1.000\n(b)GSPM\n1D 3D\nkk\u0001k1k\u0001k 2k\u0001kH1k=hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 4.315D-4 5.111D-4 8.774D-4 1/20 6.170D-4 4.240D-4 4.249D-4\n2.0D-2 1.128D-4 1.334D-4 2.255D-4 1/24 4.380D-4 3.010D-4 3.016D-4\n1.0D-2 2.872D-5 3.399D-5 5.706D-5 1/28 3.268D-4 2.245D-4 2.251D-4\n5.0D-3 7.174D-6 8.552D-6 1.433D-5 1/32 2.531D-4 1.739D-4 1.743D-4\n2.5D-3 1.721D-6 2.333D-6 3.784D-6 1/36 2.017D-4 1.386D-4 1.389D-4\norder 1.991 1.951 1.969 { 1.902 1.903 1.902\n(c)SIPM\nUnder the same setup outlined above, we investigate the energy dissipation of\nthe proposed method, the GSPM, and the SIPM. The stable state is attainable at\nt= 2 ns, while the total energy is computed by (2.3). The energy evolution curves\nof di\u000berent numerical methods with di\u000berent damping parameters, \u000b= 2;5;8;10,\nare displayed in Figure 3. One common feature is that the energy dissipation rate\nturns out to be faster for larger \u000b, in all three schemes. Meanwhile, a theoretical\nderivation also reveals that the energy dissipation rate in the LLG equation (2.1)\ndepends on \u000b, and a larger \u000bleads to a faster energy dissipation rate. Therefore,\nthe numerical results generated by all these three numerical methods have made a\nnice agreement with the theoretical derivation.10 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nTable 3. The numerical errors of the proposed method, the\nGSPM and the SIPM with \u000b= 10 andT= 1. Left: 1D with\nk= 1D\u00005; Right: 3D with k= 1D\u00003.\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4\n1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4\n5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5\norder 2.000 2.000 2.000 { 2.035 2.047 2.037\n(a)Proposed method\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.244D-3 1/2 4.256D-3 2.470D-3 2.470D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.810D-4 5.589D-4 5.744D-4\n1.0D-2 4.619D-4 4.622D-4 5.158D-4 1/8 2.447D-4 1.388D-4 1.423D-4\n5.0D-3 1.153D-4 1.156D-4 1.302D-4 1/16 6.103D-5 3.468D-5 3.613D-5\norder 2.000 2.000 1.995 { 2.037 2.047 2.030\n(b)GSPM\n1D 3D\nhk\u0001k1k\u0001k 2k\u0001kH1hk\u0001k1k\u0001k 2k\u0001kH1\n4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3\n2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4\n1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4\n5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5\norder 2.000 2.000 2.000 { 2.035 2.047 2.037\n(c)SIPM\nMeanwhile, we choose the same sequence of values for \u000b, and display the energy\nevolution curves in terms of time up to T= 2 ns in Figure 4. It is found that the\nproposed method have almost the same energy dissipation pattern with the other\ntwo methods for moderately large damping parameters \u000b= 2;5;8. In the case of\n\u000b= 10, the SIPM has a slightly di\u000berent energy dissipation pattern from the other\ntwo numerical methods.\n3.3.Domain wall motion. A Ne\u0013 el wall is initialized in a nanostrip of size 800 \u0002\n100\u00024 nm3with grid points 128 \u000264\u00024. An external magnetic \feld of he= 5 mT\nis then applied along the positive xdirection and the domain wall dynamics is\nsimulated up to 2 ns with \u000b= 2;5;8. The corresponding magnetization pro\fles are\nvisualized in Figure 5. Qualitatively, the domain wall moves faster as the value of\n\u000bincreases. Quantitatively, the corresponding dependence is found to be linear;\nsee Figure 6. The slopes \ftted by the least-squares method in terms of \u000bandhe\nare recorded in Table 4.A SECOND-ORDER METHOD FOR LLG EQUATION 11\n10-610-510-410-3100101102\nProposed method\nGSPM\nSIPM\n(a)Varyingkin 1D up to\nT= 1\n1.8 2 2.2 2.4 2.6 2.8 3 3.2\n10-7101102103\nProposed method\nGSPM\nSIPM(b)Varyingkin 3D up to T=\n0:1\n10-510-410-310-2101102103\nProposed method\nGSPM\nSIPM\n(c)Varyinghin 1D up to\nT= 1\n10-410-310-210-1100101102103\nProposed method\nGSPM\nSIPM(d)Varyinghin 3D up to\nT= 1\nFigure 1. CPU time needed to achieve the desired numerical ac-\ncuracy, for the proposed method, the GSPM and the SIPM, in\nboth the 1D and 3D computations. The CPU time is recorded as\na function of the approximation error by varying korhindepen-\ndently. CPU time with varying k: proposed method <SIPM<\nGSPM; CPU time with varying h: proposed method /GSPM<\nSIPM.\n4.Conclusions\nIn this paper, we have proposed a second-order accurate numerical method to\nsolve the Landau-Lifshitz-Gilbert equation with large damping parameters. For the\nnumerical convenience, the LLG system is reformulated so that in which the damp-\ning term is rewritten as a harmonic mapping \row .This numerical scheme is based on\nthe second-order backward-di\u000berentiation formula approximation for the temporal\nderivative, combined with an implicit treatment of the constant-coe\u000ecient di\u000busion\nterm, and the fully explicit extrapolation approximation of the nonlinear terms, in-\ncluding the gyromagnetic term and the nonlinear part of the harmonic mapping\n\row. Thanks to the large damping parameter, the proposed method is veri\fed12 Y. CAI, J. CHEN, C. WANG, AND C. XIE\nFigure 2. Stable structures in the absence of magnetic \feld at\n2 ns when\u000b= 0:01;10;40. The color denotes the angle between the\n\frst two components of the magnetization vector. Top: Proposed\nmethod; Middle: GSPM; Bottom: SIPM. Left: \u000b= 40; Middle:\n\u000b= 10; Right: \u000b= 0:01.\n(a)Proposed\n (b)GSPM\n (c)SIPM\nFigure 3. Energy evolution curves of three numerical methods,\nwith di\u000berent damping constants, \u000b= 2;5;8;10, up tot= 2 ns in\nthe absence of external magnetic \feld. Left: Proposed numerical\nmethod; Middle: GSPM; Right: SIPM. One common feature is\nthat the energy dissipation rate is faster for larger \u000b, which is\nphysically reasonable.A SECOND-ORDER METHOD FOR LLG EQUATION 13\n(a)\u000b= 2\n (b)\u000b= 5\n(c)\u000b= 8\n (d)\u000b= 10\nFigure 4. Energy evolution curves in terms of time, for the nu-\nmerical results created by three numerical methods up to t= 2 ns\nin the absence of external magnetic \feld for (a) \u000b= 2, (b)\u000b= 5,\n(c)\u000b= 8, and (d) \u000b= 10. The energy dissipation pattern of the\nproposed method is consistent with the other two methods for (a),\n(b), and (c), and the SIPM has a slightly di\u000berent energy dissipa-\ntion pattern from the other two methods for (d).\nto be unconditionally stable. The proposed method is much more e\u000ecient than\nother semi-implicit schemes since only symmetric, positive de\fnite linear systems\nof equations with constant coe\u000ecients need to be solved. Meanwhile, the proposed\nmethod is more accurate than the standard Gauss-Seidel projection method, due\nto its second-order accuracy in time. Numerical results in 1D and 3D are pro-\nvided to demonstrate the accuracy and the e\u000eciency of the proposed numerical\nmethod. In addition, micromagnetics simulations using the proposed method have\nprovided physically reasonable structures and captured the linear dependence of\nthe domain wall velocity with respect to the damping parameter. Therefore, the\nproposed method could be e\u000eciently used for challenging practical simulations of\nmicromagnetics with large damping parameters.14 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n(a)Magnetization for initial state\n (b)Magnetization with \u000b= 2 at\n2 ns\n(c)Magnetization with \u000b= 5 at\n2 ns\n(d)Magnetization with \u000b= 8 at\n2 ns\nFigure 5. Magnetization pro\fles of Ne\u0013 el wall motion in the pres-\nence of a magnetic \feld he= 5 mT, with \u000b= 2;5;8 at 2 ns for\nthe proposed numerical method. The in-plane arrow denotes the\n\frst two components of the magnetization vector. The wall moves\nfaster for larger values of \u000band its velocity depends linearly on \u000b.\nFigure 6. Linear dependence of the wall velocity with respect to\nthe damping parameter \u000b(left) and the external magnetic \feld he\n(right).\nAcknowledgments\nThis work is supported in part by the grants NSFC 11971021 (J. Chen), NSF\nDMS-2012669 (C. Wang), NSFC 11771036 (Y. Cai).A SECOND-ORDER METHOD FOR LLG EQUATION 15\nTable 4. Linear dependence of the domain wall velocity Vin\nterms of the external magnetic \feld heand the damping parameter\n\u000b.\n\u000bV(m/s) he(mT)5 6 7 8 9 10 Slope\n3 76 91 109 123 139 154 1.024\n4 105 118 139 157 179 196 0.928\n5 129 145 169 192 217 244 0.932\n6 153 169 200 227 256 286 0.927\n7 177 196 232 263 294 333 0.927\n8 200 222 263 303 333 385 0.954\n9 230 250 294 345 385 435 0.954\n10 253 270 323 370 417 476 0.943\nSlope 0.984 0.910 0.910 0.933 0.917 0.950 {\nReferences\n[1] G. Akrivis, M. Feischl, B. Kov\u0013 acs, and C. Lubich, Higher-order linearly implicit full dis-\ncretization of the Landau-Lifshitz-Gilbert equation , Math. Comp. 90(2021), 995{1038.\n[2] F. Alouges and P. Jaisson, Convergence of a \fnite element discretization for the Landau-\nLifshitz equations in micromagnetism , Math. Models Methods Appl. Sci. 16(2006), no. 02,\n299{316.\n[3] W.F. Brown, Micromagnetics , Interscience Tracts on Physics and Astronomy. Interscience\nPublishers (John Wiley and Sons), New York-London, 1963.\n[4] S. Budhathoki, A. Sapkota, K.M. Law, B. Nepal, S. Ranjit, S. Kc, T. Mewes, and A. Hauser,\nLow Gilbert damping and linewidth in magnetostrictive FeGa thin \flms , J. Magn. Magn.\nMater. 496(2020), 165906.\n[5] J. Chen, C. Wang, and C. Xie, Convergence analysis of a second-order semi-implicit pro-\njection method for Landau-Lifshiz equation , Appl. Numer. Math. (2021). Submitted and in\nreview.\n[6] I. Cimr\u0013 ak, A survey on the numerics and computations for the Landau-Lifshitz equation of\nmicromagnetism , Arch. Comput. Methods Eng. 15(2008), no. 3, 277{309.\n[7] H. Gao, Optimal error estimates of a linearized Backward Euler FEM for the Landau-Lifshitz\nequation , SIAM J. Numer. Anal. 52(2014), no. 5, 2574{2593.\n[8] C.J. Garc\u0010a-Cervera and W. E, Improved Gauss-Seidel projection method for micromagnetics\nsimulations , J. Comput. Phys. 171(2001), no. 1, 357{372.\n[9] T.L. Gilbert, Phys. Rev. 100(1955), 1243. [Abstract only; full report, Armor Research Foun-\ndation Project No. A059, Supplementary Report, May 1, 1956 (unpublished)].\n[10] T.L. Gilbert and J.M. Kelly, Anomalous rotational damping in ferromagnetic sheets , Armour\nResearch Foundation of Illinois Institute of Technology (1955). (unpublished).\n[11] B. Heinrich, D. Fra\u0013 \u0010tov\u0013 a, and V. Kambersk\u0013 y, The in\ruence of s-d exchange on relaxation of\nmagnons in metals , Phys. Stat. Solidi B-basic Solid Stat. Phys. 23(1967), 501{507.\n[12] M. Kruz\u0013 \u0010k and A. Prohl, Recent developments in the modeling, analysis, and numerics of\nferromagnetism , SIAM Rev. 48(2006), no. 3, 439{483.\n[13] L.D. Landau and E.M. Lifshits, On the theory of the dispersion of magnetic permeability in\nferromagnetic bodies , Phys. Z. Sowjet. 63(1935), no. 9, 153{169.\n[14] D.M. Lattery, D. Zhang, J. Zhu, X. Hang, J. Wang, and X. Wang, Low Gilbert damping\nconstant in perpendicularly magnetized W/CoFeB/MgO \flms with high thermal stability ,\nSci. Rep. 8(2018), 13395.\n[15] P. Li, C. Xie, R. Du, J. Chen, and X. Wang, Two improved Gauss-Seidel projection methods\nfor Landau-Lifshitz-Gilbert equation , J. Comput. Phys. 401(2020), 109046.\n[16] T. Nan, Y. Lee, S. Zhuang, Z. Hu, J. Clarkson, X. Wang, C. Ko, H. Choe, Z. Chen, D. Budil,\nJ. Wu, S. Salahuddin, J. Hu, R. Ramesh, and N. Sun, Electric-\feld control of spin dynamics\nduring magnetic phase transitions , Sci. Adv. 6(2020), no. 40, eabd2613.16 Y. CAI, J. CHEN, C. WANG, AND C. XIE\n[17] H. Suhl, Theory of the magnetic damping constant , IEEE Trans. Magn. 34(1998), 1834{\n1838.\n[18] T. Tanaka, S. Kashiwagi, Y. Otsuka, Y. Nozaki, Y. Hong, and K. Matsuyama, Microwave-\nassisted magnetization reversal of exchange-coupled composite nanopillar with large Gilbert\ndamping constant , IEEE Tran. Magn. 50(2014), 1{3.\n[19] H. Tang and K. Xia, Gilbert damping parameter in MgO-based magnetic tunnel junctions\nfrom \frst principles , Phys. Rev. Applied 7(2017), 034004.\n[20] C. Wang and J.-G. Liu, Convergence of gauge method for incompressible \row , Math. Comp.\n69(2000), 1385{1407.\n[21] X. Wang, C.J. Garc\u0013 \u0010a-Cervera, and W. E, A Gauss-Seidel projection method for micromag-\nnetics simulations , J. Comput. Phys. 171(2001), no. 1, 357{372.\n[22] R. Weber, D. Han, I. Boventer, S. Jaiswal, R. Lebrun, G. Jakob, and M. Kl aui, Gilbert\ndamping of CoFe-alloys , J. Phys. D 52(2019), 325001.\n[23] D. Wei, Micromagnetics and recording materials , Springer Briefs in Apllied Sciences and\nTechnology, Springer Berlin Heidelberg, 2012.\n[24] C. Xie, C.J. Garc\u0013 \u0010a-Cervera, C. Wang, Z. Zhou, and J. Chen, Second-order semi-implicit\nmethods for micromagnetics simulations , J. Comput. Phys. 404(2020), 109104.\n[25] D. Zhang, M. Li, L. Jin, C. Li, Y. Rao, X. Tang, and H. Zhang, Extremely large magnetiza-\ntion and Gilbert damping modulation in NiFe/GeBi bilayers , ACS Appl. Electron. Mater. 2\n(2020), no. 1, 254{259.\nSchool of Mathematical Sciences, Beijing Normal University, Beijing, China.\nEmail address :yongyong.cai@bnu.edu.cn\nSchool of Mathematical Sciences, Soochow University, Suzhou, China.\nEmail address :jingrunchen@suda.edu.cn\nMathematics Department, University of Massachusetts, North Dartmouth, MA 02747,\nUSA.\nEmail address :cwang1@umassd.edu\nSchool of Mathematical Sciences, Soochow University, Suzhou, China.\nEmail address :20184007005@stu.suda.edu.cn"    },    {        "title": "2105.07376v1.Anatomy_of_inertial_magnons_in_ferromagnets.pdf",        "content": "arXiv:2105.07376v1  [cond-mat.mes-hall]  16 May 2021Anatomy of inertial magnons in ferromagnetic nanostructur es\nAlexey M. Lomonosov1,∗Vasily V. Temnov2,3,†and Jean-Eric Wegrowe3‡\n1Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991, Moscow, Russia\n2Institut des Mol´ ecules et Mat´ eriaux du Mans, UMR CNRS 6283 ,\nLe Mans Universit´ e, 72085 Le Mans, France and\n3LSI, Ecole Polytechnique, CEA/DRF/IRAMIS, CNRS,\nInstitut Polytechnique de Paris, F-91128, Palaiseau, Fran ce\n(Dated: May 18, 2021)\nWe analyze dispersion relations of magnons in ferromagneti c nanostructures with uniaxial\nanisotropy taking into account inertial terms, i.e. magnet ic nutation. Inertial effects are\nparametrized by damping-independent parameter β, which allows for an unambiguous discrimi-\nnation of inertial effects from Gilbert damping parameter α. The analysis of magnon dispersion\nrelation shows its two branches are modified by the inertial e ffect, albeit in different ways. The up-\nper nutation branch starts at ω= 1/β, the lower branch coincides with FMR in the long-wavelength\nlimit and deviates from the zero-inertia parabolic depende nce≃ωFMR+Dk2of the exchange\nmagnon. Taking a realistic experimental geometry of magnet ic thin films, nanowires and nanodiscs,\nmagnon eigenfrequencies, eigenvectors and Q-factors are found to depend on the shape anisotropy.\nThe possibility of phase-matched magneto-elastic excitat ion of nutation magnons is discussed and\nthe condition was found to depend on β, exchange stiffness Dand the acoustic velocity.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nAfter the first description of the dynamics of the mag-\nnetization by Landau and Lifshitz [1], Gilbert proposed\nan equation that contains a correction due to the preces-\nsional damping [2, 3]. Since then, the so-called Landau-\nLifshitz-Gilbert (LLG) equation is known to give an ex-\ncellent description of the dynamics of the magnetization,\nincluding ferromagnetic resonance (FMR) and magneto-\nstatic waves [4, 5], as well as the magnetization reversal\n[6, 7]. Ferromagnetic resonance and time-resolved mag-\nnetization measurementsallow its spatially homogeneous\nprecession ( k= 0) but also non-uniform modes of the\nmagnetizationprecession( k∝ne}ationslash= 0, where kisthe wavevec-\ntor of spin waves) to be measured [8–10]. During the last\ndecades, these techniques have been advanced in the con-\ntext of ultrafast demagnetization dynamics [11, 12] that\npaved the way for the description of new physics at the\nsub-picosecond regime. High-frequency resonant modes\nof exchange magnons have been measured with ultrafast\ntime-resolved optical techniques [8, 10, 13]. Therefore,\nthe validity of the LLG equations has been confirmed\ndown to the picosecond time scale and below.\nHowever, limitations of LLG equations has been es-\ntablished in the stochastic derivation performed by W.\nF. Brown in a famous paper published in 1963 [14]. This\nlimit is due to the hypothesis that the typical time scales\nof magnetization dynamics are much longer than those\nof other degrees of freedom forming the dissipative envi-\nronment. In analogy to the common description of the\ndiffusion process of a Brownianparticle, the inertial (mo-\nmentum) degrees of freedom are supposed to relax much\n∗lom@kapella.gpi.ru\n†vasily.temnov@univ-lemans.fr\n‡jean-eric.wegrowe@polytechnique.edufaster than its spatial coordinate. This means that the\ndegrees of freedom related to the linear momentum (in\nthe case of the usual diffusion equation), or to the an-\ngular momentum (in the case of the magnetization) are\nincluded into the heat bath. As a consequence, the iner-\ntial terms do not explicitly appear in the equations, but\nare considered to be part of the damping term [15].\nThe possibility of measuring the contribution to iner-\ntial degrees of freedom led to a generalizationof the LLG\nequation with an additional term, incorporating the sec-\nond time-derivative of magnetization:\n˙m=−γm×Heff+αm×˙m+βmרm,(1)\nwherem=M/Msis the unit magnetization vector\nthatgivesthedirectionofthemagnetizationateachpoint\n(andMsis the modulus of the magnetization, which is\nconstant), γ=γ0µ0is the gyromagnetic ratio, αstands\nfor the Gilbert damping. Inertial effects are character-\nized by the parameter β, which is introduced in a phe-\nnomenological way, i.e. independent on αandγ[16].\nThis generalized LLG equation has been derived in the\nframework of different and independent theoretical con-\ntexts [15, 17–32]; its solutions have been studied in a\nseries of publications [33–36]. The main consequence of\ninertia for the uniform magnetization (magnon with the\nwavevector k= 0) is the existenceofnutation oscillations\nthat are superimposed to the precession. This leads to\nan appearance of the second resonance peak at a higher\nfrequency in FMR spectra. The direct measurement of\nnutation has been reported recently [37, 38].\nThe goal of the present report is to study the conse-\nquences of these inertial effects on the exchange magnons\n(i.e.k∝ne}ationslash= 0 modes), in the perspective of experimental\nstudies. Magnons are defined as linear magnetic exci-\ntations propagating in ferromagnets at the micromag-\nnetic limit. This work completes the first description2\nFMR \nmagnon ys(t) s(t) \nNutation \n   magnon (a)                                                                   (b) \nxΨz\nk\nFIG. 1. (a) Inertial magnons propagating in ferromagnetic\nnanostructures with wavevector kalong the zdirection under\nan external magnetic field Hresult in complex magnetization\ndynamics. (b) They can be decomposed in FMR magnon\nand nutation magnon precessing in opposite directions on el -\nliptical trajectories at different frequencies, giving ris e to a\ncharacteristic flower-shaped trajectory.\npublished in 2015, Section IV of the remarkable work of\nToru Kikuchi and Gen Tatara [22], and independently\nreconsidered by Makhfudz et al. in 2020 [39].\nThe paper is organized as follows. Section II presents\nthe derivation of the linear magnetic excitations deduced\nfrom (1). Section III describes the dispersion relation\nin a simple case of zero dipolar field (spherical symme-\ntry). The first consequence of the inertia is that the dis-\npersion relation splits in two branches: the lower one\ns1exp(ikz−iω1t) (FMR magnons ) and the upper one\nfors2exp(ikz−iω2t) (nutation magnons ). The second\nconsequence is that the quality factor Qincreases with\nthek-vector. SectionIVgeneralizesthedescriptiontothe\ncaseofauniaxialanisotropyquantifiedbythe dimension-\nless (shape) anisotropy parameter ξ. In the anisotropic\ncase the trajectories of both FMR magnons and nuta-\ntion magnons become elliptical and rotating in opposite\ndirections at each point in space. For a given k-vector\nthe magnetization vector corresponding to a superposi-\ntion of both magnons draws a typical trochoidal trajec-\ntory (see Fig. 1). Section V discusses the conditions\nfor phase-matched excitation the nutation magnons by\nco-propagatinglongitudinalacoustic phonons, illustrated\nby the material parameters for Gd-doped Permalloy thin\nfilms [13].\nII. EXCHANGE MAGNONS IN FERROMAGNETIC\nTHIN FILMS WITH MAGNETIC INERTIA\nWe start with the LLG equation for unit magnetiza-\ntion vector mwith aneffective field Heff, which includes\nexchange interactions with stiffness D, an external field\nH= (Hx,0,Hz) and a demagnetizing field induced by\nthe shape anisotropy Hd=−MS/hatwideNm. The demagne-\ntization tensor /hatwideNdepends on the specific shape of the\nferromagnetic sample. Hereafter we assume the diagonal\nformof/hatwideNwithdiagonalelements Nx,NyandNz. Damp-ing of the magnetization dynamics is described by the\nconventional Gilbert term with parameter α. In addition\nto the conventional LLG equation we take into account\nthe inertial effect characterized by the independent pa-\nrameterβ. Then the inertial LLG equation (ILLG) takes\nthe form of Eq.(1) with Heff=H+D∆m+Hd.\nThe coordinate system was chosen such that the ex-\nternal field lies in the y= 0 plane, as is shown in Fig. 1.\nThe material is assumed to be magnetically isotropic, so\nthat the unperturbed magnetization vector also lies in\nthey= 0 plane. We seek for time- and space-dependent\nsolutions in the form m=m0+s(z,t) with spin-wave\nsolutions\ns(z,t) = (sx,sy,sz)exp(ikz−iωt) (2)\npropagating as plane waves with a real wave vector k\nalong the z-axis, see Fig. 1. Substitution m(z,t) into\nequation (1) and its linearization with respect to small\nperturbations sx,sy,szresults in a homogeneous system\nof three linear equations:\n/hatwideA\nsx\nsy\nsz\n= 0 (3)\nwhere the matrix Ais given by\n\n−iω A 12(ω,k) 0\nA21(ω,k)−iω A 23(ω,k)\n0A32(ω,k)−iω\n (4)\nwith coefficients Aij(ω,k) defined as:\nA12=mz(γDk2+γMSξyz−iαω−βω2)+γHz\nA21=−mz(γDk2+γMSξxz−iαω−βω2)+γHz\nA23=mx(γDk2+γMSξzx−iαω−βω2)+γHx(5)\nA32=−mx(γDk2+γMSξyx−iαω−βω2)−γHx\nwhere coefficients ξij=Ni−Njcharacterize the shape\nanisotropy. The condition for the nontrivial solution of\nthehomogeneoussystem(3) toexist, i.e. det A= 0, gives\nrise to the secular equation\nω2+A12(ω,k)A21(ω,k)+A23(ω,k)A32(ω,k) = 0 (6)\nwhich is used to calculate the spin wave dispersion re-\nlationω(k) for different shapes/symmetries, i.e.charac-\nterized by different types of the /hatwideNtensor.\nIII. INERTIAL EXCHANGE MAGNONS IN SAMPLES\nWITH SPHERICAL SYMMETRY\nExamples of such symmetry are infinite homogeneous\nisotropic ferromagnetic media, or any spherical body. In3\nthesecasesthedemagnetizationtensor /hatwideNisdiagonalwith\nall nonzero elements equal 1 /3, so that its contribution\nto the magnetization dynamics (1) and correspondingly\nto the wave matrix components(5) vanishes. The secular\nequation (6) takes a concise form:\n/parenleftbig\nγH+γDk2−βω2−iαω+ω/parenrightbig\n×(7)\n×/parenleftbig\nγH+γDk2−βω2−iαω−ω/parenrightbig\n= 0.\nDue to the symmetry of /hatwideN, equation (7), and hence all\nits roots, remains independent on the direction of Hand\nthe equilibrium magnetization m0with respect to the\nwave propagation direction along the z-axis. For each\npositive wavenumber k, the determinant (7) is solved\nforω. Given that the presumed solution has a form\n∼exp(ikz−iωt), positive ωdesignates the waves trav-\nelling in the positive direction. The two positive roots\ncorresponding to the first parenthesis in (7) have the fol-\nlowing forms:\nω1=1\n2β/parenleftig\n−1−iα+/radicalbig\n4γβ(Dk2+H)+(1+iα)2/parenrightig\n(8)\nω2=1\n2β/parenleftbigg\n1−iα+/radicalig\n4γβ(Dk2+H)+(1−iα)2/parenrightbigg\n(9)\nThefirstrootisthelowermagnonbranchorprecession,\nslightly modified by the inertial term and the second one\nexhibits the inertial magnon branch or nutation. It is\nconvenient to split these roots into real and imaginary\nparts:ω1,2=ω′\n1,2+iω′′\n1,2. Taylor series approximation of\nthose roots assuming the smallness of γβDk2,γβH,α ≪\n1 results in the following expressions for their real parts:\nω′\n1≈γ[Dk2+H−2βγHDk2+...] (10)\nω′\n2≈1\nβ+ω′\n1 (11)\nIn this approximation the nutation magnon branch\nis simply shifted by 1 /βwith respect to FMR magnon\nbranch. The validity of this approximation is illustrated\nin Fig. 2, where the exact roots given by (8) and (9) are\ndepicted with solid lines, whereas dashed lines represent\nthe power series approximation of (10) and (11).\nLower branch emerges from the Larmor’s frequency\nγHand grows parabolically with k. Effect of inertia re-\nduces the coefficient at the term quadratic in k. Upper\nbranch is simply displaced by +1 /βand has the simi-\nlar shape. Imaginary parts of the roots ω′′\n1,2represent\nattenuation of the corresponding magnetization dynam-\nics in time as ∝exp/parenleftbig\nω′′\n1,2t/parenrightbig\n, and therefore they must be\nnegative. In the frequency domain they characterize the\nwidth ∆f=|ω′′|/π(FWHM) of the Lorentzian spectral\nline.0.3 0.5 0.8 1.0 1.3 1.5 0.25 0.50 0.75 1.00 1.25 \nβ= 0.17ps \nβ= 0.27ps β= 0.17ps Frequency, THz\nwavenumber  k, rad/nm β= 0.27ps \n0.25 0.50 0.75 1.00 1.25 1.50 10 20 30 \nβ= 0β= 0\nα= 0.023 α= 0.023 \nα= 0.01 \nα= 0.01 Line Width, GHz\nwavenumber  k, rad/nm β= 0.27ps \nFIG. 2. (a) Dispersion of the inertial magnon (red, magenta)\nand conventional magnon (blue,green). Dashed curves shows\nthe approximations by (10) and (11). (b) corresponding line\nwidths. In order to indicate the effect of inertia on the pre-\ncession, dashed curves show the line width without inertia\nβ= 0.\nω′′\n1≈ −αγ/bracketleftbig\nDk2+H−6βγ2HDk2+.../bracketrightbig\n(12)\nω′′\n2≈ −α\nβ−ω′′\n1 (13)\nNote that in the limiting case of (12), (13) field and\nexchange stiffness have opposite effects on the attenua-\ntion of the two magnon branches: they increase the at-\ntenuation in the lower branch ω1and decrease it for the\ninertial branch ω1. In the other limiting case of large\nfield and large kattenuation of both branches tends to\nexp/parenleftig\n−α\n2βt/parenrightig\n. The damping for both branches appears\nto be naturally proportional to the Gilbert damping pa-\nrameterα. A conspicuous decrease of nutation linewidth\nω′′\n2(k= 0) with growing αreported by Cherkasski et\nal. [36] roots back to the parametrization of the nu-\ntation phenomenon in terms of a product ατ, whereτ\ndenotes the characteristic nutation lifetime. Within this\nparametrization, a variation of α, while keeping τcon-\nstant, leads to the simultaneous decrease of the nutation4\n0.2 0.4 0.6 0.810203040Q\nk, nm-11/2H=0 2 T5 T\n0.2 0.4102030in a thin film\nFIG. 3. Q-factor dependencies on kfor different values of the\nexternal field. Dashed line shows the quadratic in kapprox-\nimation for H= 0. Inset shows Q-factors for the ordinary\nmagnon (blue curve) and inertial magnon (red curve) in a\nthin film, H= 0\nfrequency 1 /β= 1/(ατ) rendering the analysis of damp-\ning extremely difficult. An alternative notation in terms\nofαandβ, introduced in this paper, resolves this prob-\nlem and allows for an independent investigation of iner-\ntial and damping effects.\nAnother parameter, which characterizes the resonant\nspectral line centered at frequency f0, is its quality factor\ndefined as Q=f0/∆f=ω′/(2ω′′). As can be seen from\nequations (8) and (9), Q-factors for both branches coin-\ncide within the accuracy of ∼(ωα)2. Dependence of the\nQ-factor on the wavenumber klooks counterintuitive in\nthat it essentially grows with k. Assuming for simplicity\nH= 0, for small kthe Q-factor can be approximated by\nexpansion of ω′andω′′in the power series in k, which\nresults in:\nQ(k)∼1\n2α1+γβDk2\n1−γβDk2+...∼1\n2α/parenleftbig\n1+2γβDk2/parenrightbig\n(14)\nExact values for the Q-factor in comparison to the\nestimate of (14) are shown in Fig. 3 for the external\nfield ranging from 0 to 5 T.\nField effect for small kcan be approximated as Q∼\n1/(2α)(1+2γβH).\nIV. INERTIAL EXCHANGE MAGNONS IN SAMPLES\nWITH CYLINDRICAL SYMMETRY\nExamplesofsuchbodies aredisks, wires, infinite plates\nand films. Axial symmetry about the z-axis retains the\ndiagonal form of /hatwideNwith the diagonal elements satisfying\nthefollowingconditions: Nx=NyandNx+Ny+Nz= 1.\nAs a result, components of the matrix /hatwideAgiven by (5)acquire terms proportional to γMS. Lack of symmetry\nmakes the magnon propagation dependent on the orien-\ntation of vectors m0andHwith respect to the z-axis.\nWe consider two limiting cases: collinear arrangement\nwithm0parallel to the axis of symmetry (Θ = Ψ = 0\nin Fig. 1); and orthogonal arrangement with m0paral-\nlel to the x-axis and Θ = Ψ = π/2. In the collinear\ncase the demagnetizing field acts simply against the ex-\nternal field, hence the secularequation remains similarto\n(7), but with field Hsubstituted with the reduced field\nH′=H−ξMS:\n/parenleftbig\nγH′+γDk2−βω2−iαω+ω/parenrightbig\n×(15)\n×/parenleftbig\nγH′+γDk2−βω2−iαω−ω/parenrightbig\n= 0\nwhereξ=Nz−Nx=Nz−Nycharacterizes the shape\neffect on demagnetizing, so that in an infinite wire ξ=\n−1/2, in the spherical symmetric (or unbounded) body\nξ= 0 and in the infinite film ξ= 1. Correspondingly the\nroots to (15) are similar to ones given in (8) and (9) with\nmodified field:\nω1=1\n2β/parenleftbigg\n−1−iα+/radicalig\n4γβ(Dk2+H′)+(1+iα)2/parenrightbigg\n(16)\nω2=1\n2β/parenleftbigg\n1−iα+/radicalig\n4γβ(Dk2+H′)+(1−iα)2/parenrightbigg\n(17)\nAt the low- klimit the lower branch roughly tends\nto the Larmor’s frequency γ(H−ξMS) and the upper\nbranch limit is 1 /β+γ(H−ξMS), which is similar to the\ncase of spherical symmetry, but with modified ��eld. In\nthe orthogonal configuration with m0andHperpendic-\nular to the axis of symmetry and to the magnon propa-\ngation direction, roots of the determinant (4) generally\ncannot be found in an analytical form. Therefore we\nfirstconsideranapproximatesolutions,andthendescribe\nbrieflythe numericalgorithmforobtainingthedispersion\ncurves. By neglecting the Gilbert attenuation ( α= 0),\nthe approximate solutions to (4) for the in-plane mag-\nnetization and field can be found in a concise analytical\nform:\nω1,2=1\nβ√\n2{2γβ(Dk2+H)+γβξM s+1\n∓/radicalbig\n4γβ(Dk2+H)+(γβξM s+1)2}1/2(18)\nHere indices 1 and 2 denote the frequencies of the\nconventional and inertial magnons respectively, sign ‘-\n‘ prior to the square root in (18) corresponds to the\nlower branch ω1; sign ‘+’ denotes the inertial branch\nω2. Numerical procedure for building the dispersion\nrelations of the magnonic modes for nonzero αor ar-\nbitrary orientation of the external field H starts with\ncalculation of the stationary equilibrium magnetization5\nm0= (mx,my,mz). This can be done by solving (1)\nin its stationary form, i.e.with all time derivatives set\nzero. In a thin film, for example, quantities Hiandmj\nare related by MSmxmz+Hxmz−Hzmx= 0. Thus\nobtained stationary magnetization components are then\nsubstituted into (5) and (4). At some fixed small kthe\ndeterminant (4) as a function of complex-valued ωpos-\nsesses two minima, which correspond to the FMR and\nnutational branches. Their exact locations can be evalu-\natedbyanumericalroutinewhichminimizestheabsolute\nvalue of the determinant (4) in the vicinity of the guess\nvalues for those branches, for example given by equa-\ntions (19) and (20). Then we give ka small increment\nand repeat the extremum search using the ωs obtained\nat the previous step as guess values, and so on. As a\nresult, calculated values for ω1andω2follow the disper-\nsion curves of both branches. Note that for nonzero α\nrootsω1,2possess imaginary parts, which determine the\nline width and Q-factor for each mode. Let us consider\nthe magnetization behavior in a thin film in more detail.\nFor this geometry /hatwideNpossesses the only nonzero compo-\nnentNz= 1, and correspondingly ξ= 1. In the small- k\nlimit, the lower branch approaches the Kittel’s frequency\nωFMR=γ/radicalbig\nH(H+MS) from below as β,k→0:\nω1≈ωFMR−1\n2γωFMR(2H+MS)β+...(19)\nEffect of the demagnetizing field on the inertial branch\nis exhibited by an upward shift by1\n2γMS; whereas effect\nof inertia is opposite:\nω2≈1\nβ+γH+1\n2γMS−/parenleftbigg\nω2\nFMR+1\n8γ2M2\nS/parenrightbigg\nβ+...(20)\nFor the orthogonal configuration, when both m0and\nHlie in the film plane, we can estimate the trajectories\nof the magnetization dynamics of both modes for small\nkand small α. For each root given by (18) we solve\nthe homogeneous equation for perturbations s(3). Nor-\nmalization of the solutions can be chosen in an arbitrary\nway,here for simplicity we define sz= 1. In orthogonal\ngeometry sxcomponent is obviously negligible or equals\nzero, so the system reduces to two equations in syand\nsz. Results shown as a power series expansion for small\ninertiaβω≪1 for the precession:\nsp=\n0\n−i/radicalig\n1+ξMS\nDk2+H/parenleftig\n1+βγξM S\n2/parenrightig\n1\nexp(−iω1t)\n(21)\nand nutation:\nsn=\n0\ni(1−βγξM S/2)\n1\nexp(−iω2t) (22)0.1 0.2 0.3 0.4 0.5 0.61.01.21.41.61.82.0\n1 T0.2 Tellipticity sy’/sx’\nk, nm-1sx’ sy’0.1 T\nFMR\nnutation\nFIG. 4. Ratio of the polarization axes for the external field\nof 0.1T, 0.2T, and 1T. Handm0are parallel to the inward\nnormal to the figure plane.\nwith the parameter of anisotropy ξ= 1 for a thin film\nnormal to the z−axis and ξ= 1/2 for a thin wire spread\nalong the z−axis. Both perturbations exhibit elliptical\npolarization within the ( x′,y′) plane. Precession trajec-\ntory is deformed by the demagnetizing effect so that the\ny-axis of the ellipse is stretched with the/radicalbig\n1+MS/H\nfactor due to demagnetizing effect, and in addition on\naccount of inertial effect. On the contrary, the nutation\nellipse is squeezed along the y-axis proportionally to the\ninertial parameter β. Ellipticity of the lower branch de-\npends on the external field (21), whereas that of the up-\nper branch in this approximation shows no dependence\non the field. Signs of sycomponents are opposite for\nnutation and precession, this indicates that they are ro-\ntating in the opposite directions. Exact polarizationscan\nbe found numerically for a reasonable set of material pa-\nrameters and fields, as is shown in Fig. 4.6\nV. EXCITATION MECHANISMS OF INERTIAL\nEXCHANGE MAGNONS\nThe only experimental evidence of inertial effects in\nferromagnets has been reported for k= 0 nutation man-\ngons in Py-thin films resonantly excited with a mag-\nnetic field of an intense quasi-monochromatic THz pulse\n[37]. In order to excite k∝ne}ationslash= 0 exchange magnon modes\none would need to have either spatially localized and\ninstantaneous stimuli [10] or any other source of effec-\ntive magnetic field characterized by spectral and spatial\noverlap with investigated magnon modes. The letter can\nbe provided through ultrashort large-amplitude acoustic\npulses [40, 41] producing effective magneto-elastic fields\nrapidly varying in time and space [42]. Acoustic pulses\npropagating through a thin ferromagnetic sample at an\nacoustic velocity vare quantified by a linearized disper-\nsion relation ωac=vk. Crossing between acoustic and\nmagnon brunches, i.e. satisfying the phonon-magnon\nphase-matchingcondition, usually facilitatesthe acoustic\nexcitation of magnetization dynamics [43, 44]. A ques-\ntion arises under which conditions the crossing between\ndispersion curves for longitudinal phonons and inertial\nmagnons can occur. Whereas for realistic magnetic fields\nthe acoustic dispersion always intersects the lower FMR-\nbranch at a frequency close to FMR frequency [42], the\ncrossing of the upper nutation brunch is less obvious.\nFIG. 5. The magneto-acoustic phase matching condition for\nnutation magnons can be tuned vie the reduction of exchange\nstiffness in Gd-doped Py samples. Gd concentration x varies\nfrom 0 to 13%. The dashed line displays the acoustic disper-\nsion relation ωac/(2π). Magneto-elastic coupling with inertial\nmangon is efficient when the dashed line lies within the pink\ntinted area. Material parameters are taken from Ref. [13] an d\nβ=0.276 ps.\nIt is possible to quantify the criterion for magneto-\nelastic crossing with nutation magnons analytically. To\ndo that we note that for larger wavenumbers ksatisfying\nDk2≫H,MStheexchangetermplaysthedominantrole\nand the asymptotic behaviourfor both branchesbecomeslinear in k:\nω1,2≈ ∓1\n2β+k/radicaligg\nγD\nβ. (23)\nIt follows from (23) that the condition for the nutation\nmagnon branch to intersect the acoustical dispersion re-\nlationωac(k), requires the asymptotic slope of ω2(k) to\nbe smaller than the acoustic velocity v:\n/radicaligg\nγD\nβ< v. (24)\nThis expression shows that for a given βthe magneto-\nelastic crossing is facilitated by small exchange stiffness\nDand small acousticvelocity. This approximateanalysis\nbreaks down for acoustic frequencies in above-THz spec-\ntral range, where the acoustic dispersion starts deviating\nfrom its linear approximation.\nFigure 5 highlights the remarkable role of exchange\nstiffness to achieve the dispersion crossing between nuta-\ntionmagnonsandlongitudinalacousticphonons. Doping\nPy thin films with Gadolinium has been shown to gradu-\nally reduce the exchange stiffness upon Gd-concentration\nfrom 300 to 100 [meV ·˚A2] [13]. For a fixed value of in-\nertial parameter β=0.276 ps, nutation magnons for pure\nPy samples do not display any crossing with acoustic\nphonons within the displayed range of k-vectors but the\nGd-doped Py with 13% Gd concentration does. The nu-\ntation magnon-phonon crossing point occurs at 0.75 THz\nfrequency and k= 0.85 nm−1(magnon wavelength of\napproximately 5 nm), i.e. magnon parameters readily\naccessible in ultrafast magneto-optical experiments [10].\nVI. CONCLUSIONS\nIn this paper we have theoretically studied exchange\ninertial magnons in ferromagnetic samples of different\nshapes under the action of an external magnetic field.\nThe parametrizationof magnetization dynamics in terms\nof two independent parameters, the Gilbert damping α\nand the inertial time β, allows for unambiguous discrim-\nination between the inertial and damping effects as well\nas their impact on both branches of magnon dispersion.\nInertial effects are found to strongly effect not only the\nfrequencies (magnon eigenvalues) of both branches but\nalso result in a monotonous increase of the Q-factor as\na function of the external magnetic field and magnon k-\nvector. The two magnon branchesare found to precessin\nopposite directions along the elliptical trajectories with\nperpendicularlyorientedlongaxisoftheellipses(magnon\neigenvectors). Their ellipticity is found to depend on the\ncomponents of the demagnetizing tensor. 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Matsuo2;3;4;5\n1Institute for Solid State Physics,\nThe University of Tokyo, Kashiwa 277-8581, Japan\n2Kavli Institute for Theoretical Sciences,\nUniversity of Chinese Academy of Sciences,\nBeijing 100190, China\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences,\nBeijing 100190, China\n4Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai 319-1195, Japan\n5RIKEN Center for Emergent Matter Science (CEMS),\nWako, Saitama 351-0198, Japan\n(Dated: August 10, 2021)\nWe theoretically consider spin pumping in a junction between a ferromagnetic insulator (FI) and\na two-dimensional electron gas (2DEG) in which the Rashba and Dresselhaus spin-orbit interactions\ncoexist. Using second-order perturbation theory, we derive an increase in linewidth in the case of\nan interfacial exchange coupling in a ferromagnetic resonance (FMR) experiment. We clarify how\nthe enhancement of Gilbert damping depends on the resonant frequency and spin orientation of the\nFI. We show that this setup of an FMR experiment can provide information on the spin texture of\n2DEG at the Fermi surface.\nI. INTRODUCTION\nSpin pumping has been studied intensively in the \feld\nof spintronics as a versatile method to generate spin cur-\nrent using magnetization dynamics1,2. Spin pumping is\nused for injecting spins from a ferromagnet into an ad-\njacent material in various heterojunction systems, such\nas ferromagnetic metal/normal metal (NM) junctions3{6\nand ferromagnetic insulator (FI)/NM junctions7. As the\nbackaction of the spin injection, the Gilbert damping of\nthe ferromagnet is modulated. Therefore, spin pumping\ncan be used as a probe of nonequilibrium spin states of\nmaterials in the sense that information on spin transport\ndue to an adjacent material is re\rected in the modulation\nof the Gilbert damping.\nAn attractive strategy is spin pumping of semiconduc-\ntor microstructures because highly developed semicon-\nductor technologies can be utilized8,9. In particular, a\ntwo-dimensional electron gas (2DEG) in a semiconduc-\ntor heterostructure is an easily controlled physical sys-\ntem that has been used in spintronics devices10{13. A\n2DEG system has two types of spin-orbit interaction,\nRashba14,15and Dresselhaus16,17. By combining these\ntwo interactions, we can control the spin polarization of\nconduction electrons in a way that is dependent on the\npropagation direction18. In fact, electron transport re-\n\recting the spin texture on the Fermi surface has been\nobserved through, e.g., the Aharonov-Casher e\u000bect19{24\nand the persistent spin helix state25{31.\nIn our work, we focus on spin pumping of a 2DEG\nin semiconductor heterostructures. It is remarkable that\nthere already exist experimental studies on injection of\nspins into two-dimensional electron systems with theRashba spin-orbit interaction in surfaces32,33, atomic\nlayer materials34,35, transition oxides36,37, and bulk semi-\nconductors38,39. However, to the best of our knowledge,\nspin pumping in a junction system composed of an FI and\na 2DEG in which the Rashba and Dresselhaus spin-orbit\ninteractions coexist has not been studied yet.\nIn this study, we consider a planar junction composed\nof an FI and a 2DEG in a semiconductor heterostructure\n(see Fig. 1). By taking both Rashba and Dresselhaus\nspin-orbit interactions in 2DEG into account, we theoret-\nically show an enhancement in Gilbert damping in the FI\nwithin the second-order perturbation with respect to in-\nterfacial exchange coupling40{45. We investigate how the\nlinewidth increase in a ferromagnetic resonance (FMR)\nexperiment depends on the resonant frequency and the\n2DEGFerromagneticinsulator (FI)Semiconductorheterostructure\nmicrowave\nFIG. 1. Schematic picture of junction composed of a fer-\nromagnetic insulator (FI) and two-dimensional electron gas\n(2DEG) in a semiconductor heterostructure. Spin current is\ninduced at the interface under microwave irradiation, whose\nfrequency is chosen to be near the frequency of the spin pre-\ncession of the FI.arXiv:2105.11193v3  [cond-mat.mes-hall]  7 Aug 20212\nspin orientation of the FI. We show that a spin pump-\ning measurement can directly detect the complex spin\ntexture of a 2DEG with spin-orbit interactions.\nThe rest of this work is organized as follows. In Sec. II,\nwe theoretically show an enhancement in Gilbert damp-\ning due to the exchange interaction with a 2DEG within\nsecond-order perturbation. In Sec. III, we show how the\nenhancement of the Gilbert damping depends on the spin\norientation of the FI. We also discuss the experimental\nrelevance of our result. Finally, we summarize our results\nin Sec. IV. In the subsequent four appendices, we give a\ndetailed derivation of the equations in Sec. II.\nII. FORMULATION\nIn this section, we analytically formulate the damping\nrate of the spin precession in a FMR experiment when\nthe FI is coupled with a 2DEG through an interfacial\ninteraction. First, we introduce the model of the 2DEG\nand the FI in Sec. II A and Sec. II B. Next, we consider\nthe interfacial coupling between the 2DEG and the FI in\nSec. II C. Finally, we calculate the second-order perturba-\ntion with respect to the interfacial coupling in Sec. II D.\nThroughout this paper, we will use the laboratory coor-\ndinates shown in Fig. 1; the xyplane is parallel to the\n2DEG, while the z-axis is perpendicular to the junction\narea.\nA. Two-dimensional electron gas\nWe consider a 2DEG with both Rashba and Dressel-\nhaus interactions. The Hamiltonian for the kinetic en-\nergy and the spin-orbit interactions is given as\nHkin=X\nk(cy\nk\"cy\nk#)^hk\u0010ck\"\nck#\u0011\n; (1)\n^hk=\u0010~2(k2\nx+k2\ny)\n2m\u0003\u0000\u0016\u0011\n^I+\u000b(ky\u001bx\u0000kx\u001by)\n+\f(kx\u001bx\u0000ky\u001by); (2)\nwhere ^hkis a 2\u00022 matrix,\u001ba(a=x;y;z ) are the Pauli\nmatrices, ^Iis an identity matrix, and ck\u001bis the annihila-\ntion operator of conduction electrons with wavenumber\nk= (kx;ky) andz-component of a spin, \u001b(=\";#). The\n\frst term of ^hkdescribes the kinetic energy of an electron\nwith chemical potential \u0016and e\u000bective mass m\u0003. The\nsecond and third terms of ^hkdescribe the Rashba and\nDresselhaus spin-orbit interactions, respectively, and \u000b\nand\fdenote the amplitudes of the respective spin-orbit\ninteractions.\nHereafter, we assume that the Fermi energy is much\nlarger than the other energy scales, such as the tem-\nperature and the amplitude of the spin-orbit interac-\ntions. The energy dispersion is then approximated as\n\u0018k\u0011~2k2=2m\u0003\u0000\u0016'vF(jkj\u0000kF), wherekFand\nvFare the Fermi wavenumber and the Fermi velocity,\nrespectively. We also approximate the spin-orbit in-\nteraction using ( kx;ky)'(kFcos';kFsin'), where'\nis the azimuth angle of the conduction electrons. Ac-\ncordingly, the Hamiltonian matrix can be rewritten as\nFIG. 2. Schematic picture of the e\u000bective magnetic \feld\nhe\u000b(') that acts on the spins of conduction electrons.\n^hk=\u0018k^I\u0000he\u000b\u0001\u001b, where\nhe\u000b'kF(\u0000\u000bsin'\u0000\fcos';\u000bcos'+\fsin';0);(3)\nis the e\u000bective magnetic \feld that acts on the spin of\na conduction electron propagating in the direction of '.\nNote that the e\u000bective magnetic \feld he\u000bcharacterizes\nthe spin texture of conduction electrons as a function of\nthe azimuth angle of the electron wavenumber. Fig. 2 (a)\nand (b) shows pro\fles of the e\u000bective magnetic \feld on\nthe Fermi surface46for\u000b=\f= 1 and 3. The spin texture\nis simpli\fed for the special case of \u000b=\f= 1; the direction\nof the e\u000bective \feld is always parallel to the straight line\nky=\u0000kx.\nBy diagonalizing ^hk, we obtain the spin-dependent elec-\ntron dispersion,\nE\u0006\nk=\u0018k\u0006he\u000b('); (4)\nhe\u000b(')\u0011jhe\u000b(')j'kFp\n\u000b2+\f2+ 2\u000b\fsin 2': (5)\nNote that 2 he\u000b(') corresponds to the spin-splitting en-\nergy for a conduction electron propagating at the azimuth\nangle'. When\u000b=\f(\u000b=\u0000\f), the spin-splitting en-\nergy vanishes at '= 3\u0019=4 ('=\u0019=4).\nGreen's function at \fnite temperature is de\fned as a\n2\u00022 matrix ^g(k;\u001c), whose elements are\ng\u001b\u001b0(k;\u001c) =\u0000~\u00001hck\u001b(\u001c)cy\nk\u001b0i; (6)\nfor 0< \u001c < ~\f, whereck\u001b(\u001c) =eHkin\u001c=~ck\u001be\u0000Hkin\u001c=~\nis the Heisenberg representation for the imaginary time\nevolution and \fis inverse temperature. Note that there\nare o\u000b-diagonal components in the Green's function that\nare due to spin-\rip processes by the spin-orbit interac-\ntions. The Fourier transform of the Green's function is\nde\fned as\n^g(k;i!n) =Z~\f\n0d\u001cei!n\u001c^g(k;\u001c); (7)\nwhere!n=\u0019(2n+ 1)=~\fare the Matsubara frequen-\ncies for fermions. For the Hamiltonian Hkin, the Green's\nfunction is calculated as\n^g(k;i!n) =(i~!n\u0000\u0018k)^I\u0000he\u000b\u0001\u001b\n(i~!n\u0000E+\nk)(i~!n\u0000E\u0000\nk): (8)3\nFIG. 3. Relation between laboratory coordinates ( x;y;z ) and\nmagnetization-\fxed coordinates ( x0;y0;z0) for FI.\nWe also consider the e\u000bect of impurity scattering in a\n2DEG by the Hamiltonian,\nVimp=uX\ni2impX\n\u001b\ty(Ri)\t(Ri) (9)\n\t(r) =1p\nAX\nkck\u001beik\u0001r; (10)\nwherei2imp indicates the impurity site, uis the\nstrength of the impurity potential, Ais the junction area,\nandRiis the position of the impurity site. The total\nHamiltonian for the 2DEG is thus H2DEG =Hkin+Vimp.\nThrough second-order perturbation with respect to the\nimpurity potential, Green's function is calculated as\n^g(k;i!n) =(i~!n\u0000\u0018k+i\u0000sgn(!n)=2)^I\u0000he\u000b\u0001\u001bQ\n\u0017=\u0006(i~!n\u0000E\u0017\nk+i\u0000sgn(!n)=2);\n(11)\nwhere \u0000 = 2 \u0019niu2D(\u000fF) is level broadening and D(\u000fF) =\nkF=(2\u0019~vF) is the density of states per spin per unit vol-\nume. For a detailed derivation, see Appendix A.\nB. Ferromagnetic insulator\nWe assume that the magnetization of the FI is in the\nxyplane parallel to the 2DEG. We de\fne the direction\nof the in-plane ordered spin measured from the x-axis as\n\u0012. Accordingly, the expectation value of the ordered spin\nin FI is expressed as\nhSii= (hSx\nii;hSy\nii;hSz\nii) = (S0cos\u0012;S0sin\u0012;0);(12)\nwhereiis the index of a localized spin site, and S0\nis the amplitude of the average spin per site. To sim-\nplify the subsequent calculation, we introduce a new co-\nordinate (x0;y0;z0), for which the direction of the or-\ndered spin is \fxed to the x0-axis, as shown in Fig. 3.\nThe average spin is expressed in this new coordinate as\n(hSx0\nii;hSy0\nii;hSz0\nii) = (S0;0;0). The spin operators for\nthe new coordinates are related to the ones for the orig-\ninal coordinates as\nSx0\ni= cos\u0012Sx\ni+ sin\u0012Sy\ni; (13)\nSy0\ni=\u0000sin\u0012Sx\ni+ cos\u0012Sy\ni; (14)\nSz0\ni=Sz\ni: (15)The Hamiltonian of the FI is written in the new coordi-\nnates as\nHFI=X\nhi;jiJij(Sx0\niSx0\nj+Sy0\niSy0\nj+Sz0\niSz0\nj)\n\u0000~\rhdcX\niSx0\ni; (16)\nwherehi;jiindicates a pair of nearest-neighbor sites, Jij\nis the ferromagnetic exchange coupling, \r(<0) is the\ngyromagnetic ratio, and hdcis the static magnetic \feld.\nWe assume hdc<0, for which the direction of the spin\n(the magnetization) becomes + x0(\u0000x0).\nWe further assume that the temperature is much lower\nthan the magnetic transition temperature. As well, we\nassumeS0\u001d1, to which the spin-wave approximation\ncan be applied. The Hamiltonian of the FI can then be\nrewritten as\nHFI=X\nk~!kby\nkbk; (17)\n~!k=Dk2+~\rhdc; (18)\nwherebkis the annihilation operator of a magnon, and\nDis spin sti\u000bness. A detailed derivation is given in Ap-\npendix B.\nWe de\fne the imaginary-time spin correlation function\nof the FI as\nG(k;\u001c) =\u00001\n~hSx0+\nk(\u001c);Sx0\u0000\nk(0)i; (19)\nG(k;i!n) =Z~\f\n0d\u001cG(k;\u001c)ei!n\u001c; (20)\nwhere!n= 2\u0019n=~\fare the Matsubara frequencies for\nbosons. This spin correlation function can be calculated\nwithin the spin-wave approximation as\nG(k;i!n) =2S0=~\ni!n\u0000!k+i\u000bGj!nj: (21)\nHere, we have introduced a phenomenological dimen-\nsionless parameter \u000bGthat describes the strength of the\nGilbert damping.\nC. Interfacial exchange interaction\nWe consider an interfacial exchange interaction be-\ntween the FI and the 2DEG with the Hamiltonian,\nHint=X\nk(TkSx0+\nksx0\u0000\nk+T\u0003\nkSx0\u0000\nksx0+\nk): (22)\nwhereTkis an exchange interaction at the interface, and\nSx0\u0006\nk=Sy0\nk\u0006iSz0\nkare creation and annihilation opera-\ntors of the localized spins in the FI. Here, the interface\nis assumed to be su\u000eciently \rat so that the momentum\nof spin excitation is conserved. The e\u000bective \feld (the\nexchange bias) which acts on the conduction electrons\nvia the interfacial coupling is also assumed to be much\nsmaller than the e\u000bective magnetic \feld he\u000b. The spin4\nladder operators for conduction electrons, sx0\u0006\nk, are de-\n\fned as follows. First, we de\fne the spin operators in\nthe magnetization-\fxed coordinates as\nsx0\nk= cos\u0012sx\nk+ sin\u0012sy\nk; (23)\nsy0\nk=\u0000sin\u0012sx\nk+ cos\u0012sy\nk; (24)\nsz0\nk=sz\nk; (25)\nusing the spin operators in the original coordinates de-\n\fned as\nsa\nk=X\n\u001b\u001b0X\nk0cy\nk0\u001b(\u001ba)\u001b\u001b0ck0+k\u001b;(a=x;y;z ):(26)\nHere,\u001baare the Pauli matrices. We de\fne the spin ladder\noperators as\nsx0+\nk\u0011sy0\nk+isz0\nk; (27)\nsx0\u0000\nk\u0011(sx0+\nk)y=sy0\n\u0000k\u0000isz0\n\u0000k: (28)\nUsing Eqs. (23)-(26), these ladder operators are rewritten\nas\nsx0\u0006\nk=1\n2X\n\u001b;\u001b0X\nk0cy\nk0\u001b(^\u001bx0\u0006)\u001b\u001b0ck0\u0006k\u001b0; (29)\n^\u001bx0\u0006=\u0000sin\u0012\u001bx+ cos\u0012\u001by\u0006i\u001bz: (30)\nD. Second-order perturbation\nLet us consider a second-order perturbation with re-\nspect to the interfacial exchange interaction Hintto the\nbulk system described by H2DEG +HFI. The spin corre-\nlation function is written in the form,\nG(k;i!n) =1\n(G0(k;i!n))\u00001\u0000\u0006(k;i!n); (31)\n\u0006(k;i!n) =jTkj2\n4\fX\nk0;i!mTr\"\n^\u001bx0\u0000^g(k0;i!m)\n\u0002^\u001bx0+^g(k0+k;i!m+i!n)#\n; (32)\nwhereG0(k;i!n) is the unperturbed part of the Green's\nfunction given in Eq. (21), and \u0006( k;i!n) is the self-\nenergy due to the interfacial exchange coupling. Since a\nuniform spin precession is induced in FMR experiments,\nwe only calculate the self-energy for k=0. To simplify\nthe notation, we will rewrite Green's function ^ g(k;i!n)\nas\n^g(k;i!n) =A(i!n)^I\u0000he\u000b\u0001\u001b\nD(i!n); (33)\nA(i!n) =i~!n\u0000\u0018k+i\u0000sgn(!n)=2; (34)\nD(i!n) =Y\n\u0017=\u0006(i~!n\u0000E\u0017\nk+i\u0000sgn(!n)=2): (35)By a straightforward calculation using the algebra of the\nPauli matrices, we obtain\n\u0006(q=0;i!n)\n=jT0j2\n\fX\nk;i!mA\u0000he\u000b\u0001^m\nDA0+he\u000b\u0001^m\nD0; (36)\nwhere ^m= (cos\u0012;sin\u0012;0) indicates the spin orientation\nof the FI,A=A(i!m),A0=A(i!m+i!n),D=D(i!m),\nandD0=D(i!m+i!n) (for the detailed derivation, see\nAppendix C). For further calculation, we evaluate the\nsum with respect to the Matsubara frequency !mby us-\ning the standard procedure based on the contour integral.\nThen, we obtain the retarded component by analytic con-\ntinuation:i!n!!+i\u000e(the details of the calculation\nare in Appendix D).\nWe will focus on the increase in the FMR linewidth due\nto the exchange coupling with the 2DEG and consider\nonly the imaginary part of the self-energy. Using the\nunperturbed spin correlation function Eq. (21), we obtain\nthe retarded component of the spin (magnon) correlation\nfunction as\nGR(q=0;!)'2S0=~\n!\u0000!q=0+i(\u000bG+\u000e\u000bG)!; (37)\n\u000e\u000bG(!)\u0011\u00002S0\n~!Im \u0006R(q=0;!): (38)\nFor\u000bG+\u000e\u000bG\u001c1 that holds for a standard setup of\nspin pumping, the linewidth of the ferromagnetic reso-\nnance is su\u000eciently small, allowing us to replace !with\nthe resonance frequency !q=0to express the enhance-\nment of the damping constant in Eq. (38). Hereafter,\nthe resonance frequency is simply written as \n( \u0011!q=0).\nThen, Im \u0006R(q=0;\n) corresponds to the increase in\nthe linewidth in the FMR experiment. Analytic calcula-\ntion of Im \u0006( q=0;\n) gives a correction for the Gilbert\ndamping as\n\u000e\u000bG(\n)'\u00002S0\n~\nIm \u0006R(q=0;\n)\n=\u000bG;0X\n\u0017;\u00170=\u00061Z2\u0019\n0d'\n2\u0019F\u0000\n~\n + (\u0017\u0000\u00170)he\u000b\u0001\n\u00021\u0000\u0017^he\u000b(')\u0001^m\n21 +\u00170^he\u000b\u0001^m\n2;(39)\nF(x) =\u0000=\u0019\u00010\n(x=\u00010)2+ (\u0000=\u00010)2; (40)\nwhere'describes the propagation direction of conduc-\ntion electrons, \u000bG;0= 2\u0019S0jT0j2AD(\u000fF)=\u00010is a dimen-\nsionless parameter, Ais the junction area, and ^he\u000b(') =\nhe\u000b(')=he\u000b(') is a unit direction vector of the e\u000bective\nmagnetic \feld. Here, we have introduced the unit of en-\nergy \u0001 0, which is the amplitude of the Dresselhaus spin-\norbit interaction, kF\f. The enhancement of the Gilbert5\ndamping can be separated into three parts:\n\u000e\u000bG=\u000e\u000bG;1+\u000e\u000bG;2+\u000e\u000bG;3 (41)\n\u000e\u000bG;1=\u000bG;0Z2\u0019\n0d'\n2\u0019F(~\n)1\u0000(^he\u000b\u0001^m)2\n2; (42)\n\u000e\u000bG;2=\u000bG;0Z2\u0019\n0d'\n2\u0019F(~\n\u00002he\u000b)(1 + ^he\u000b\u0001^m)2\n4;\n(43)\n\u000e\u000bG;3=\u000bG;0Z2\u0019\n0d'\n2\u0019F(~\n + 2he\u000b)(1\u0000^he\u000b\u0001^m)2\n4;\n(44)\nThese expressions indicate the physical mechanism of the\nenhancement of the Gilbert damping as follows. The\n\frst contribution \u000e\u000bG;1comes from elastic spin \ripping\nof conduction electrons caused by the transverse com-\nponent of the e\u000bective magnetic \feld he\u000bvia the inter-\nfacial exchange coupling. In fact, \u000e\u000bG;1vanishes when\nhe\u000bis parallel or anti-parallel to the magnetization of FI,\n^m. Since this process is elastic, the frequency-dependent\npart is just a Lorentzian form F(~\n), which has a peak\nat \n = 0. The second contribution \u000e\u000bG;2originates from\nthe magnon absorption process. This is a dynamical pro-\ncess as indicated by the peak shift of the Lorentzian form;\nthe peak of F(~\n\u00002he\u000b(')) is shifted to \n = 2 he\u000b(')=~,\nat which the magnon energy coincides with the spin-\nsplitting energy gap of conduction electrons. The second\ncontribution takes a maximum when ^he\u000bis parallel to\n^m. This is consistent with the fact that a spin of a con-\nduction electron is converted from a low-energy state ^he\u000b\nto a higher state \u0000^he\u000bby receiving a spin carried by a\nmagnon, that is in the direction of \u0000^m. We should also\nnote that the second contribution vanishes when ^he\u000bis\nanti-parallel to ^m. The third contribution \u000e\u000bG;3comes\nfrom the magnon emission process. This process is usu-\nally not important in FMR experiments, because the fre-\nquency \n is taken as positive.\nIII. RESULTS\nA. Enhancement of Gilbert damping\nFig. 4 illustrates the enhancement of the Gilbert damp-\ning,\u000e\u000bG, as a function of the resonant frequency \n for\n\u0000=\u00010= 0:5. In particular, Fig. 4 (a) and (b) shows the\nenhancement for \u000b=\f = 1 and 3, respectively. We note\nthat\u000e\u000bGdepends on the resonant frequency in contrast\nwith the Gilbert damping coe\u000ecient for the bulk FI, \u000bG.\nIn both cases, the enhancement \u000e\u000bGclearly depends on\nthe spin orientation of the FI. A peak at \n = 0 is caused\nby the static part \u000e\u000bG;1, while the structure at a \fnite\n\n comes from the magnon-absorption process described\nby\u000e\u000bG;2. Note that the broad structure indicated by\n\u000e\u000bG;2is produced by the variation in the energy split-\nting 2he\u000b(') when the azimuth angle 'changes from 0\nto 2\u0019. The range of 2 he\u000b(') is obtained from Eq. (5) as\n0\u00142he\u000b\u00144\u00010for\u000b=\f= 1 and 4\u0001 0\u00142he\u000b\u00148\u00010for\n\u000b=\f= 3. This variation in the spin splitting corresponds\nto the range of the broad structure at \fnite \n shown in\nFIG. 4. Enhancement of Gilbert damping, \u000e\u000bG, due to cou-\npling to 2DEG plotted as a function of FMR resonance fre-\nquency \n. (a) \u000b=\f= 1. (b)\u000b=\f= 3.\nFig. 4 (a) and (b).\nWhen the azimuth angle of the spin polarization in the\nFI,\u0012, is varied, the enhancement of the Gilbert damping\nis modi\fed, as shown in Fig. 4. Its general properties are\nsummarized as follows. First, \u000e\u000bGis independent of the\nspin orientation of the FI for \u000b= 0 or\f= 0. Second,\nthe result for \u000b<0 is related to that for \u000b>0 through\n\u000e\u000bG(\u000b;\u0012;\n) =\u000e\u000bG(\u0000\u000b;\u0012\u0000\u0019=2;\n): (45)\nThird,\u000e\u000bGhas symmetry relations with respect to \u0012as\n\u000e\u000bG(\u0012;\n) =\u000e\u000bG(\u0012+\u0019;\n) =\u000e\u000bG(\u0019=2\u0000\u0012;\n):(46)\nTo see the spin-orientation dependence in more detail, we\nshow a contour plot of \u000e\u000bGas a function of \u0012and \n in\nFig. 5. For \u000b= 0,\u000e\u000bGis independent of \u0012(Fig. 5 (a)).\nThe same result is obtained for \f= 0 after replacing\n\u00010withkF\u000b. The spin-orientation dependence becomes\nstrongest for \u000b=\f= 1 (Fig. 5 (b)). For this special case,\nthe direction of ^he\u000bis \fxed:\n^he\u000b=\u0006(1=p\n2;\u00001=p\n2;0): (47)\nFor the spin texture of this special case, see Fig. 2 (a).\nTherefore, the \u0012-dependent part in Eqs. (42)-(44) can be6\nFIG. 5. Contour plot of enhancement of Gilbert damping in 2DEG for (a) \u000b=\f = 0, (b)\u000b=\f = 1, and (c) \u000b=\f = 3. The\nhorizontal axis is the FMR frequency \n and the vertical axis is the azimuth angle of the spin orientation of the FI, \u0012.\ntaken out of the integrals:\n\u000e\u000bG;1/1\u0000(^he\u000b\u0001^m)2\n2; (48)\n\u000e\u000bG;2;\u000e\u000bG;3/1 + ( ^he\u000b\u0001^m)2\n4; (49)\nwhere we have used the fact that the term proportional\nto^he\u000b\u0001^mvanishes after the integration with respect to\n'. From this expression, the spin-orientation dependence\nshown in Fig. 5 (b) can be explained as follows. The peak\nat \n = 0 that is caused by \u000e\u000bG;1takes a maximum (a\nminimum) when ^he\u000b?^m(^he\u000bk^m) or equivalently\nwhen\u0012=\u0019=4;5\u0019=4 (\u0012= 3\u0019=4;7\u0019=4). This observation\nsupports the conclusion that the enhancement in Gilbert\ndamping at \n = 0 is induced by the transverse com-\nponent of the e\u000bective magnetic \feld he\u000b. In contrast,\nthe broad structure at \fnite frequencies in the range of\n0\u0014~\n\u00144\u00010, that is caused by \u000e\u000bG;2, takes a max-\nimum (a minimum) when ^he\u000bk^m= 0 ( ^he\u000b?^m).\nThis is consistent with the fact that this contribution\ncomes from the magnon absorption accompanying spin-\n\rips of the conduction electrons. Fig. 5 (c) shows the\nspin-orientation dependence for \u000b=\f = 3. Although the\n\u0012dependence cannot be expressed in a simple form for\n\u000b=\f= 3, the qualitative features are the same as in the\ncase of\u000b=\f= 1, as indicated by comparing Fig. 5 (b) and\n(c) except that the \fnite-frequency bread structure shifts\ntoward the high-frequency region 4\u0001 0\u0014~\n\u00148\u00010.\nB. Relevance to experiments\nOur results indicate that the spin-orientation depen-\ndent provides information on spin-orbit interactions in\n2DEG, in which both the Rashba and Dresselhaus spin-\norbit interactions coexist. Let us estimate a necessary\ncondition for observation of the present result. For\nGaAs/AlGaAs heterostructures47, the magnitude of the\nspin-orbit interactions is given as \u000b\u0018\f\u00184 meV\u0001\u0017A,\nleading to \u0001 0=kF\f\u00180:10 meV for the electron den-\nsity 5\u00021011cm\u00002. Because the FMR frequency for YIG\nunder a magnetic \feld of 1 T is about ~\n = 0:06 meV,\nthe ratio ~\n=\u00010is of order 1. This indicates that both\nthe elastic contribution \u000e\u000bG;1and the magnon absorp-\ntion contribution \u000e\u000bG;2can be observed experimentally\nusing a magnetic \feld of a few tesla. Note that theRashba spin-orbit interaction can be controlled by ap-\nplying an electric \feld to the sample. The amplitude of\nthe spin-orbit interactions depends on the aspects of bulk\nsemiconductors as well as on sample fabrication consid-\nerations. For example, in asymmetric InAs heterostruc-\ntures48,49, the magnitude of the Rashba spin-orbit inter-\naction is about \u000b\u0018400 meV\u0001\u0017A, leading to kF\u000b\u001814 meV\nfor the electron density 1012cm\u00002. In this case, the de-\npendence of the spin-orientation of FI is governed by the\nelastic contribution \u000e\u000bG;1. However, by using symmet-\nric InAs heterostructures50, it is possible to reduce the\nmagnitude of the Rashba spin-orbit interaction down to\nthe same order as in GaAs/AlGaAs heterostructures. In\nsuch heterostructures, we can also observe the contribu-\ntion from magnon absorption, \u000e\u000bG;2.\nIV. SUMMARY\nWe theoretically investigated spin pumping from a fer-\nromagnetic insulator (FI) into a two-dimensional gas\n(2DEG) with both Rashba and Dresselhaus spin-orbit\ninteractions. We considered the interfacial coupling\nthrough the tunnel Hamiltonian in which the momentum\nof spin excitation is conserved and derived an increase in\nthe linewidth in a ferromagnetic resonance (FMR) exper-\niment that is induced by the 2DEG within a second-order\nperturbation with respect to the interfacial coupling. We\nfound that there are three processes that enhance the\nGilbert damping: (a) an elastic process, (b) a magnon\nabsorption process, and (c) a magnon emission process.\nThe elastic process is induced by spin-\rips through the\ntransverse component of the e\u000bective magnetic \feld felt\nby conduction electrons that originate from the spin-orbit\ninteraction in the 2DEG. This elastic process is dominant\nwhen the FMR frequency is su\u000eciently low compared\nwith the energy scale of the spin-orbit interaction. In\ncontrast, the magnon absorption/emission process is a\ndynamical one that changes the number of magnons in\nthe FI and a\u000bects the Gilbert damping when the FMR\nfrequency is comparable to the spin splitting energy by\nspin-orbit coupling in the 2DEG. We discussed how these\nthree processes of enhancing the Gilbert damping depend\non the spin orientation in the FI. We also showed that\nour results can be detected in an FMR experiment using\na GaAs/AlGaAs heterostructure under a magnetic \feld7\nFIG. 6. Feynman diagram of second-order perturbation with\nrespect to the impurity potential.\nof a few tesla. Our work provides a helpful experimental\nmethod for the detection of spin texture of conduction\nelectrons at the Fermi surface.\nACKNOWLEDGMENTS\nWe would like to thank Dr. Y. Ominato for fruit-\nful discussions. T. K. acknowledges support from the\nJapan Society for the Promotion of Science (JSPS KAK-\nENHI Grant No. JP20K03831). M. M. is \fnancially sup-\nported by a Grant-in-Aid for Scienti\fc Research (Grants\nNo. JP20H01863 and No. JP21H04565) from MEXT,\nJapan.\nA. IMPURITY SCATTERING\nIn this study, we consider the e\u000bect of impurity scat-\ntering within a second-order perturbation with respect to\nan impurity potential by taking a random average. This\napproximation corresponds to the Born approximation,\nwhose diagram is shown in Fig. 6. In this approximation,\nthe temperature Green's function is written as\n(^g(k;i!n))\u00001= (^g0(k;i!n))\u00001\u0000^\u0000(i!n); (A1)\n^\u0000(i!n) =niu2Zd2k\n(2\u0019)2^g0(k;i!n); (A2)\nwhereniis the number of impurity sites. We assume that\nthe scattering rate is much smaller than the bandwidth\nof the conduction electrons. Accordingly, the self-energy\nis calculated as\n^\u0000(i!n) =\u0000iniu2kF\n2vFsgn(!n)^I\u0011\u0000i\u0000\n2sgn(!n)^I;(A3)\nwhere \u0000 denotes the impurity scattering rate. Using\nthe Dyson equation (A1), the retarded component of the\nGreen's function is obtained as Eq. (11).\nB. SPIN-WAVE APPROXIMATION\nWe derive the Hamiltonian within the spin-wave ap-\nproximation by using the Holstein-Primakov transforma-\ntion. ForS0\u001d1, it is written as\nSx0\u0000\ni'p\n2S0by\ni; (B1)\nSx0+\ni'p\n2S0bi; (B2)\nSx0\ni=S0\u0000by\nibi; (B3)wherebi(by\ni) is an annihilation (creation) operator de-\n\fned at site i. We replace the spin operators with these\nboson operators and take the Fourier transform,\nbi=1pNFX\nkeik\u0001ribk; (B4)\nwhereNFis the number of unit cells in the FI. The\nHamiltonian of the FI is modi\fed into Eqs. (17) and (18).\nWhen we consider the cubic lattice model with only the\nnearest-neighbor exchange coupling J, the dispersion is\ngiven as ~!k=~!(0)\nk+~\rhdc, where\n~!(0)\nk= 2JS0(3\u0000cos(kxa)\u0000cos(kya)\u0000cos(kza))\n'JS0a2k2: (B5)\nandais a lattice constant of the FI. The last equation is\nthe long-wavelength approximation.\nC. DERIVATION OF EQUATION (36)\nHere, we derive Eq. (36). We rewrite Green's function\nof the conduction electrons as\n^g(k;i!n) =1\nD(i!n)h\nA(i!n)^I+b\u0001\u001bi\n; (C1)\nwherea= (\u0000sin\u0012;cos\u0012;i) andb=\u0000he\u000b. Then, the\ntrace in Eq. (32) is rewritten as\nI\u0011Trh\n^\u001bx0\u0000^g(k;i!m)^\u001bx0+^g(k;i!m+i!n)i\n=1\nDD0Trh\na\u0003\u0001\u001b(A^I+b\u0001\u001b)a\u0001\u001b(A0^I+b\u0001\u001b)i\n:(C2)\nUsing the identity,\n(a\u0001\u001b)(b\u0001\u001b) = (a\u0001b)^I+i(a\u0002b)\u0001\u001b; (C3)\nTr [\u001ba] = 0; Tr [^I] = 2; (C4)\na straightforward calculation gives\nI=2\nDD0h\nAA0a\u0003\u0001a+iA0(a\u0003\u0002b)\u0001a+iAa\u0003\u0001(a\u0002b)\n\u0000(a\u0003\u0002b)\u0001(a\u0002b) + (a\u0003\u0001b)(a\u0001b)i\n: (C5)\nWe obtain Eq. (36) by substituting the explicit forms of\naandb.\nD. ANALYTIC CONTINUATION\nHere, we perform the summation in the self-energy by\nusing analytic continuation. Using the identities,\nA\nD=1\n2X\n\u0017=\u00061\ni~!m\u0000E\u0017\nk+i\u0000=2 sgn(!m); (D1)\nhe\u000b\u0001^m\nD=1\n2X\n\u0017=\u0006\u0017^he\u000b\u0001^m\ni~!m\u0000E\u0017\nk+i\u0000=2 sgn(!m);(D2)8\nFIG. 7. Contour on the complex plane.\nand the counterparts for A0andD0, the self-energy is\nrewritten as\n\u0006(0;i!n) =jT0j2\n4X\nkX\n\u0017=\u0006X\n\u00170=\u0006(1\u0000\u0017^he\u000b\u0001^m)\n\u0002(1 +\u00170^he\u000b\u0001^m)Ik\u0017\u00170;(D3)\nIk\u0017\u00170=1\n\fX\ni!m1\ni~!m\u0000E\u0017\nk+i\u0000=2 sgn(!m)\n\u00021\ni~!m+i~!n\u0000E\u00170\nk+i\u0000=2 sgn(!m+!n);(D4)\nwhere ^he\u000b=he\u000b=he\u000b. By using analytic continuation,\nIk\u0017\u00170can be expressed as a contour integral,\nIk\u0017\u00170=\u0000Z\nCdz\n2\u0019if(z)1\nz\u0000E\u0017\nk+i\u0000=2 sgn(Imz)\n\u00021\nz+i~!n\u0000E\u00170\nk+i\u0000=2 sgn(Imz+!n);(D5)\nwheref(z) = (e\fz+1)\u00001andCis a contour surrounding\nthe poles of f(z).\nWe modify the contour Cto be a sum of C1,C2,C3,\nandC4, as shown in Fig. 7, and change the integration\nvariable to z=E+i\u0011forC1,z=E\u0000i\u0011forC2,z=\nE\u0000i!n+i\u0011forC3, andz=E\u0000i!n\u0000i\u0011forC4, where\n\u0011is a positive in\fnitesimal. Then, we obtain\nIk\u0017\u00170=\u0000ZdE\n2\u0019if(E)\n\u0002\"\n\u0000i\u0000\n(E\u0000E\u0017\nk)2+ (\u0000=2)2\u00021\nE+i~!n\u0000E\u00170\nk+i\u0000=2\n+1\nE\u0000i~!n\u0000E\u0017\nk\u0000i\u0000=2\u0002\u0000i\u0000\n(E\u0000E\u00170\nk)2+ (\u0000=2)2#\n:\n(D6)By changing the variable to E0=E\u0000E\u0017\nkfor the \frst\nterm and to E0=\u0000(E\u0000E\u00170\nk) for the second term, we\nobtain\nIk\u0017\u00170=\u0000ZdE0\n2\u0019i\u0000i\u0000\nE02+ (\u0000=2)2\n\u0002\"\nf(E0+E\u0017\nk)\u0000f(\u0000E0+E\u00170\nk)\nE0+i~!n+E\u0017\nk\u0000E\u00170\nk+i\u0000=2#\n:(D7)\nThe summation with respect to the wavenumber can be\nreplaced with an integral,\n1\nAX\nkIk\u0017\u00170'D(\u000fF)Z1\n\u00001d\u0018Z2\u0019\n0d'\n2\u0019Ik\u0017\u00170; (D8)\nwhereAis the junction area and \u0018\u0011\u0018k. 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Phys. 3, 650 (2007)."    },    {        "title": "2106.03332v1.Voltage_control_of_damping_constant_in_magnetic_insulator_topological_insulator_bilayers.pdf",        "content": "arXiv:2106.03332v1  [cond-mat.mes-hall]  7 Jun 2021Voltage-control of damping constant in\nmagnetic–insulator/topological–insulator bilayers\nTakahiro Chiba,1Alejandro O. Leon,2and Takashi Komine3\n1)National Institute of Technology, Fukushima College, 30 Na gao, Kamiarakawa, Taira, Iwaki, Fukushima 970-8034,\nJapan\n2)Departamento de F´ ısica, Facultad de Ciencias Universidad Tecnol´ ogica Metropolitana, Las Palmeras 3360, ˜Nu˜ noa 780-0003,\nSantiago, Chile\n3)Graduate School of Science and Engineering, Ibaraki Univer sity, 4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511,\nJapan\n(Dated: 8 June 2021)\nThe magnetic damping constant is a critical parameter for ma gnetization dynamics and the e fficiency of memory devices\nand magnon transport. Therefore, its manipulation by elect ric fields is crucial in spintronics. Here, we theoretically\ndemonstrate the voltage-control of magnetic damping in fer ro- and ferrimagnetic–insulator (FI) /topological–insulator\n(TI) bilayers. Assuming a capacitor-like setup, we formula te an effective dissipation torque induced by spin-charge\npumping at the FI/TI interface as a function of an applied voltage. By using rea listic material parameters, we find that\nthe effective damping for a FI with 10 nm thickness can be tuned by one order of magnitude under the voltage with\n0.25 V . Also, we provide perspectives on the voltage-induce d modulation of the magnon spin transport on proximity-\ncoupled FIs.\nV oltage or electric-field control of magnetic properties\nis fundamentally and technologically crucial for energeti -\ncally efficient spintronic technologies,1,2such as magnetic\nrandom-access memories (MRAMs),3spin transistors,4,5and\nspin-wave-based logic gates.6In these technologies, voltage-\ncontrol of magnetic anisotropy (VCMA) in thin ferromag-\nnets7–9promises energy-efficient reversal of magnetization by\na pulsed voltage10–12and manipulation of propagating spin\nwaves with lower power consumption.13The control of mag-\nnetic damping is also highly desirable to increase the per-\nformance of spintronic devices. For instance, low magnetic\ndamping allows small critical current densities for magnet i-\nzation switching and spin-wave excitation by current-indu ced\nspin-transfer14and spin-orbit torques.15,16On the other hand,\na high magnetic damping can be beneficial in reducing the\ndata writing time in MRAM devices. For magnonic devices,\nmagnetic damping is a key factor because it governs the life-\ntime of spin waves or magnons as information carriers.17Even\nif the magnetic damping is a vital material parameter that go v-\nerns magnetization dynamics in several spintronic devices , its\nvoltage-control is not fully explored except for a few exper i-\nments with ferro- and ferrimagnets.18–22\nThe main origin of magnetic dissipation is the spin-orbit\ninteraction (SOI), which creates relaxation paths of the sp in-\nangular momentum into conduction electrons and the lattice .\nHence, potential candidates to achieve the voltage-contro l of\nmagnetic damping are magnetic materials and /or strong SOI\nsystems. Three-dimensional topological insulators (3D TI s),\nsuch as Bi 2Se3, are characterized by band inversion due to\na strong SOI23,24and possess an ideally insulating bulk and\nspin–momentum locked metallic surface states. Recently,\nBi2−xSbxTe3−ySey(BSTS)25and Sn-doped Bi 2−xSbxTe2S26\nhave been reported to be ideal 3D TIs with two-dimensional\n(2D) Dirac electrons on the surface and a highly insulating\nbulk. For spintronics, the interface between a ferromagnet\nand a TI can enhance the magnitude of both spin and charge\ncurrents.27,28Some experiments reported29–32the spin-chargeconversion at room temperature33,34in a bilayer of TI/ferro-\nand ferrimagnetic-insulator (FI) such as Y 3Fe5O12(YIG) with\nvery low Gilbert damping constant ( α). An essential feature\nof the FI/TI bilayer is that the TI bulk behaves as a semicon-\nductor, enabling the control of the surface carrier density by\na voltage.35Also, magnetically doped TI exhibit VCMA.12,36\nHence, TIs are a promising candidate to achieve the voltage-\ncontrol of magnetic damping.\nIn this work, we theoretically demonstrate the voltage-\ncontrol of magnetic damping in FI /TI bilayers. We formulate\nan effective dissipation torque induced by spin-charge pump-\ning at the FI/TI interface as a function of a gate voltage VG.\nOur main result is that the voltage changes the e ffective damp-\ning by one order of magnitude for a FI with a perpendicular\nmagnetization configuration and 10 nm thickness. Also, we\nprovide perspectives on the modification of magnon scattrin g\ntime in a FI-based magnonic device.\nTo study the effective damping torque, we consider 2D\nmassless Dirac electrons on the TI surface with the magnetic\nproximity effect,29–32i.e., coupled to the magnetization of an\nadjacent FI. The exchange interaction between the surface\nelectrons and the FI magnetization is modeled by a constant\nspin splitting along the magnetization direction with unit vec-\ntorm=M/Ms(in which Mis the magnetization vector with\nthe saturation magnetization Ms).37Then, the following 2D\nDirac Hamiltonian provides a simple model for the FI /TI in-\nterface state38:\nˆH=vFσ·(ˆp׈z)+∆σ·m, (1)\nwhere/planckover2pi1is the reduced Planck constant, vFis the Fermi veloc-\nity of the Dirac electrons at zero applied voltage, ˆp=−i/planckover2pi1∇\nis the momentum operator, {ˆx,ˆy,ˆz}are the unit vectors along\nthe respective Cartesian axes, σ=(σx,σy,σz) is the vector\nof Pauli matrices for the spin, and ∆is the exchange interac-\ntion constant. For simplicity, we ignore here the particle– hole\nasymmetry and the hexagonal warping e ffect in the surface\nbands. Also,∆andvFare assumed to be temperature indepen-2\nNFUBMVG\nd\n5* m\n'* \nxz\ny\nNFUBM VG\n5* '* ). ). µL\tB\n \n\tC\nµRBeffh(t)\nd\nFIG. 1. (a) Schematic geometry (side view) of a capacitor-li ke de-\nvice comprising a ferromagnetic insulator (FI) film (with th ickness d)\nsandwiched by a TI and a normal metal as a top electric gate wit hVG.\nThe yellow line corresponds to the TI surface state. The red a rrow\ndenotes the precessional magnetization directions in the F MR driven\nby static ( Beff) and oscillating ( h(t)) magnetic fields. (b) Schematic\ngeometry of a transistor-like device comprising a FI film san dwiched\nby a TI and a normal metal. The red wave arrow represents a magn on\ncurrent driven by the di fference in spin accumulation ( µL−µR) in the\nattached left and right heavy metal (HM) leads.\ndent.40Note that we operate in the weak magnetic coupling\nlimit, and therefore self-consistent treatment for the ind uced\ngap (∆)39is not necessary.\nLet us begin by calculating the dissipation torque induced\nby the spin-charge pumping28,41–43of a dynamic magnetiza-\ntion in FI/TI bilayers. A precessing magnetization, driven by\nferromagnetic resonance (FMR), as shown in Fig. 1, can be\nregarded as an effective vector potential Aeff(t)=∆/(evF)ˆz×\nm(t) with the electron charge −e(e>0), which drives a\ncharge current via an e ffective electric field Eeff=−∂tAeff\n(see the supplementary material), i.e.,\nJP=∆\nevF/parenleftBigg\nσAH∂m\n∂t−σLˆz×∂m\n∂t/parenrightBigg\n, (2)\nwhereσLandσAH are longitudinal and transverse\n(anomalous-Hall) conductivities, respectively, and depe nd on\nthez−component of the magnetization ( mz)38. From the\nHamiltonian ( 1), the velocity operator ˆv=∂ˆH/∂ˆp=vFˆz×σ\ndepends linearly on σ. Therefore, the nonequilibrium spin\npolarization µP(in units of m−2) is a linear function of the\ncharge current JPon the TI surface, i.e., µP=ˆz×JP/(evF).This nonequilibrium spin polarization µPexerts a dissipa-\ntion torque on the magnetization, TSP=−γ∆/(Msd)µP×m,\nnamely\nTSP=(−αAHmz+αLm×)/parenleftBigg∂m\n∂t−∂mz\n∂tˆz/parenrightBigg\n, (3)\nwith\nαL(AH)=γ∆2\ne2v2\nFMsdσL(AH), (4)\nwhereγis the gyromagnetic ratio and dis the thickness of the\nFI layer. Equation ( 3) is equivalent to the charge-pumping-\ninduced damping-like torque that Ndiaye et al. derived using\nthe Onsager reciprocity relation for a current-induced spi n-\norbit torque.43The first term in Eq. ( 3) originates from the\nmagnetoelectric coupling (the Chern–Simons term)37,44and\nrenormalizes the gyromagnetic ratio. By using parameters\nlisted in TABLE I,∆=40 meV , and d=10 nm,αAH≈10−4is\nestimated even by using σAHat 0 K as the upper value.38Thus,\nwe disregard the renormalization of the gyromagnetic ratio . In\ncontrast, the second term in Eq. ( 3) stems from the Rashba–\nEdelstein effect due to the spin-momentum locking on the TI\nsurface41and contributes to magnetic damping. Since we are\ninterested in voltage-control of magnetic damping, we here -\nafter focus onαLin this study.\nAccording to Eq. ( 4), the electric field e ffect on the con-\nductivityσLcan be used to control the magnetic dissipation.\nNamely, the voltage–induced change of the interfacial dens ity\nof states inσLrenders the TI a more or less e fficient spin sink.\nThe damping enhancement αLdepends on the chemical po-\ntentialµ, measured from the original band-touching (Dirac)\npoint. At room temperature, or below it, the thermal energy i s\nmuch smaller than the Fermi one, kBT≪EF, with Tthe tem-\nperature and kBthe Boltzmann constant. Then, we can use the\nfollowing Sommerfeld expansion of the chemical potential µ\nµ(T)≈EF1−π2\n6/parenleftBiggkBT\nEF/parenrightBigg2 (5)\nwith the voltage-dependent Fermi energy,12EF=µ(0), given\nby\nEF(VG)=/planckover2pi1vF/radicalBigg\n4π/parenleftBigg\nnint+∆2\n4π(/planckover2pi1vF)2+ǫ\nedVG/parenrightBigg\n, (6)\nwhereǫis the permittivity of a FI and nint=/parenleftBig\nE2\nF(0)−∆2/parenrightBig\n//parenleftBig\n4π/planckover2pi12v2\nF/parenrightBig\nis the intrinsic carrier density, i.e., at\nVG=0. Note that we can define a voltage-dependent sur-\nface electron density nV(VG)≡nint+ǫVG/(ed)that shows\nthe underlying mechanism behind the voltage-control of in-\nterfacial phenomena in insulating bilayers with surface ca r-\nriers, which goes beyond topological materials. Namely, a\nvoltage increases or decreases the e ffective electron density\nand therefore enhances or weakens all e ffects that depend on\nthis density, including isotropic45and anisotropic46exchange\ninteractions, emergence of magnetization in metals,47perpen-\ndicular magnetic anisotropy,3,9and spin-orbit torques.12The3\nЋ-\u0001\tʷ\u0012\u0011 \u000e\u0014 \n\tB\n \tC\n \nЋ-\u0001\tʷ\u0012\u0011 \u000e\u0014 \n\tD\n \nEF\n/g21/g39 \n#VML\u0001DPOEVDUJPO\n#VML\u0001WBMFODF EF2Ec\nFIG. 2. Effective damping enhancement αLof a TI/FI bilayer as functions of VGandTforEF(VG=0)=140 meV: (a) mz=1, (b) mz=0.\n(c) V oltage modulation of EFin a TI. Insets represent schematic of massless (dashed line ) and massive (solid line) surface state dispersions in\nthe bulk band gap. In these graphs, we use parameters listed i n Table I,∆= 40 meV , and d=10 nm for a FI thickness. The details of the\ncalculations are given in the text.\nvoltage-generated change in the surface density is equival ent\nto an interfacial Fermi energy shift. In this work, we predic t\nthat the spin-charge pumping e fficiency is also modulated, an\neffect that may also appear in usual FI |normal metal bilayers\nsince the spin-mixing conductance depends on the electroni c\ndensity.48\nWe investigate the e ffect of electric-gate on the e ffective\ndampingαLso that we assume hereafter that the low-energy\nDirac Hamiltonian ( 1) is an accurate description for a mo-\nmentum cut kc=/radicalbig\nE2c−∆2/(/planckover2pi1vF), in which 2 Ecis the bulk\nbandgap of TIs50(see Fig. 2(c)). Sufficiently far from the\nDirac point (/planckover2pi1τ/EF≪1,τis the transport relaxation time),\nthe electron scattering can be treated by the first Born appro x-\nimation.51With this, the longitudinal conductivity reads52\nσL=e2\n2h/integraldisplayEc\n−EcdEkEkτ(Ek,T)\n/planckover2pi1E2\nk−∆2m2\nz\nE2\nk+3∆2m2z/parenleftBigg\n−∂fFD\n∂Ek/parenrightBigg\n,(7)\nwhere fFD=/bracketleftbigexp{(Ek−µ)/(kBT)}+1/bracketrightbig−1is the Fermi-Dirac\ndistribution, the energy Ekis the eigenvalue of Eq. ( 1), and\nτ(Ek,T) is the transport relaxation time of massless Dirac\nelectrons within the Born approximation for impurity and\nphonon scatterings. By applying the Matthiessen rule,\n1\nτ(Ek,T)=Ek(a+bkBT), (8)\nwhere a=nV2\n0//parenleftBig\n4/planckover2pi13v2\nF/parenrightBig\n(in units of eV−1s−1) parameterize\ncontribution of the impurity scattering,53,54nis the impurity\nconcentration, and V0is the scattering potential. Also, contri-\nbution to the transport relaxation time from the phonon scat -\ntering53,54can be approximated by b=D2\n0//parenleftBig\n4/planckover2pi13v2\nFρtsv2\nL/parenrightBig\n(in\nunits of eV−2s−1), whereρis the mass density of the quintu-\nple layer (QL) in the TI crystal structure, tsis the thickness of\none atomic layer in 1 QL of TIs, vLis the longitudinal phonon\nvelocity, and D0is the deformation potential constant.\nFigures 2(a) and (b) show the VGandTdependence of the\neffective damping enhancement αLfor out-of-plane ( mz=1)\nand in-plane ( mz=0) magnetization configurations, respec-\ntively. Also, Fig. 2(c) illustrates the voltage modulation ofTABLE I. Material parameters for the TI /FI bilayer.\nSymbol Value Unit\naBSTS Fermi velocity vF 4.0×105ms−1\naBSTS bulk band gap 2 Ec 300 meV\nbYIG gyromagnetic ratio γ 1.76×1011T−1s−1\nbYIG Gilbert damping constant α 6.7×10−5\nbYIG saturation magnetization Ms 1.56×105Am−1\ncYIG relative permittivity ǫ/ǫ 0 15\naReference 25,bReference 33,cReference 49.\nEFin TI. The bulk damping constant can be influenced by\nmaterial and device parameters, such as SOI and magnetic\nanisotropies34. However, we predict the voltage-modulation\nof the damping enhancement by spin-charge pumping. There-\nfore, our results are independent of the intrinsic dissipat ion\nmechanisms. At the FI /TI interface, orbital hybridization be-\ntween TI and the 3 dtransition metal in FI, such as YIG, de-\nforms the TI surface states, which might shift the Dirac poin t\nto the lower energy and lift up EF,55so that we consider rel-\natively high value EF(VG=0)=140 meV with the corre-\nsponding carrier density of the order of 1012cm−2. Also,∆is\nused within the values reported experimentally in FI-attac hed\nTIs.56,57For impurity parameters, we use n=1011cm−2and\nV0=0.15 keVÅ2based on an analysis of the transport prop-\nerties of a TI surface.52We could not find estimates of the\nphonon scattering for BSTS in the literature so that we adopt\nthose of non-substituted Bi 2Te3being vL=2.9×105ms−1,\nD0=35 eV , ts=0.16 nm, andρ=7.86×103kgm−3\nin Ref. 58. These scattering parameters describe a relatively\nclean interface with the sheet resistance ∼1 kΩ, which is one\norder less than that of experiments. In Figs. 2(a) and (b),αL\nmonotonically decreases with increasing Tat a fixed VGwhile\nit has peaks for changing VGat a fixed T>0 (see also the\ninset of Fig. 3). This feature reflects thermal excitation of sur-\nface carriers into the bulk states ( Ek>Ec), reducing the spin-\ncharge-pumping contribution. With the out-of-plane config u-4\nration,αLcan be tuned by one order of magnitude under the\nvoltage, whileαLchanges by less than a factor two with the in-\nplane state, which suggests that the out-of-plane configura tion\nis superior in controllability. The calculated Tdependence of\ndamping enhancement at VG=0 for the in-plane configura-\ntion agrees with a few experiments with the FI /TI bilayer.31,59\nNote that at much lower than EF(VG=−250 mV)≈40 meV ,\nour calculation with the in-plane configuration breaks down\nbecause of the finite level broadening due to the higher-orde r\nimpurity scattering.60The VG–dependent FMR is character-\nized by the Landau-Lifshitz-Gilbert theory in the suppleme n-\ntary material.\nThe electric manipulation of magnon spin transport is a rel-\nevant topic in spintronics. For example, in YIG with an in-\njector and a detector Pt contact, changes of the magnon spin\nconductivity can be obtained by using a third electrode that\nchanges the magnon density,61–63potentially providing a func-\ntionality similar to the one a field-effect transistors . Damping\ncompensation by current-driven torques64,65in magnetic het-\nerostructures also influences magnon transport. Here, we pr o-\nvide a perspective on the electric-field-induced modulatio n of\nmagnon scattering time, τm. Magnons can be injected and\ndetected by their interconversion with charge currents in a dja-\ncent heavy metals (HMs) through the direct and inverse spin-\nHall effects.33Similar to charge transport induced by an elec-\ntrochemical potential gradient, a magnon spin current can b e\ndriven by the gradient of a magnon chemical potential inject ed\nby an external source.66,67Magnon transport through a FI can\nbe controlled by the gate voltage that modulates the e ffective\ndamping in Eq. ( 4).\nSo far, the magnon spin transport in the FI /TI bilayer lacks\nmicroscopic theory with few exceptions.68,69However, from\nthe bulk of magnon spin transport,66the control ofτmresults\nin the modification of all transport properties, including t he\nmagnon spin conductivity. In the presence of a TI contact, in -\nterfacial magnons are scattered by conducting Dirac electr ons\non the TI surface.70Considering a very thin ferromagnet that\ncan be modeled by a 2D magnet. The inset of Fig. 3shows\nthat the damping enhancement is at least one order of magni-\ntud larger than the bulk one of YIG.33,34Accordingly, let us\nassume that interfacial magnons are absorbed by transferri ng\ntheir energy and angular momentum to Dirac electrons at a\nrate 1/τm∝αL.66While there is no know microscopic expres-\nsion for the magnon spin conductivity in the present system,\nbulk magnon transport obeys the relationship σm∝τm,61,62\nwhereσmis the magnon spin conductivity. In our case, the\nscattering timeτmis dominated by the magnon-relaxation pro-\ncess into the FI/TI interface. To estimate an e ffect of electric-\ngate on the magnon spin transport, we define the modulation\nefficiency\nηm=τm(VG)−τm(Vmax)\nτm(Vmax)=αL(Vmax)\nαL(VG)−1, (9)\nwhere Vmax(≈−68 mV for Fig. 3) gives the maximum value\nofαL(and therefore the minimum value of τm). In principle,\nτmdepends on VGthrough not onlyαLbut also via magnon\ndispersion relation, /planckover2pi1ωq,66including a VG–dependent mag-\nnetic anisotropy. However, this VG–dependence is quite smallFIG. 3. Modulation e fficiency (ηm) as a function of the gate volt-\nage. Inset shows the corresponding behavior of the e ffective damp-\ning enhancementαL. In these graphs, we use parameters listed in\nTable I,∆= 40 meV , and d=2 nm for a FI thickness. We also set\nEF(VG=0)=140 meV and T=300 K.\neven for a FI with 2 nm thickness (see the supplementary ma-\nterial), so that we disregard the influence of the magnon gap\nin the following calculation. Figure 3shows VG-dependence\nof the modulation e fficiency at room temperature in which the\nstrongly nonlinear behavior is interpreted as follows. Dow n\ntoVG≈−130 mV,αLis affected by the thermal excitation of\nsurface carriers, which makes a peak around VG≈−70 mV.\nFrom−130 mV to−250 mV, the thermal excitation is sup-\npressed, so thatαLmonotonically decreases with |VG|due to\nthe reduction of the Fermi surface. Hence, in this regime,\none can effectively modulate the magnon spin transport by the\nvoltage.\nIn summary, we have theoretically demonstrated the\nvoltage-control of magnetic damping in ferro- ferrimagnet ic\ninsulator (FI)/topological insulator (TI) bilayers. Assuming\na capacitor-like setup, we formulate an e ffective damping\ntorque induced by spin-charge pumping at the FI /TI interface\nas a gate voltage function. The presence of a perpendicular\nelectric field results in a shift of the Fermi level or, equiv-\nalently, a modified interfacial electron density, increasi ng or\ndecreasing the efficiency of the pumping process. We stud-\nied the consequences of this damping enhancement using re-\nalistic material parameters for FI and TI. We found that the\neffective damping with the out-of-plane magnetization con-\nfiguration can be modulated by one order of magnitude under\nthe voltage with 0.25 V . The present results motivate an ap-\nplication: the magnon scattering time can be tuned by a gate\nvoltage, potentially allowing for a magnon transistor type of\napplication. A complete quantitative description of the la t-\nter requires a microscopic theory of magnon spin transport\nin FI/TI bilayers, which might remain an unexplored issue.\nThe voltage-control of magnetic damping paves the way for\nlow-power spintronic and magnonic technologies beyond the\ncurrent-based control.\nSee the supplementary material for the calculation of the5\nspin-charge pumping in FI /TI bilayers, the characterization of\nthe FMR under several values of the applied voltage, the in-\nfluence of VG–dependence of the anisotropy in the magnon\ndispersion.\nWe thank Camilo Ulloa and Nicolas Vidal-Silva for fruitful\ndiscussions. This work was supported by Grants-in-Aid for\nScientific Research (Grant No. 20K15163 and No. 20H02196)\nfrom the JSPS and Postdoctorado FONDECYT 2019 Folio\n3190030.\nDATA AVAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\n1H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. 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In the limit of a strong exchange interaction, the oscillatory dynamics of the\norder parameter in these AFMs, which have opposite chiralities, could be mapped to that of two\ndamped-driven pendulums with signi\fcant di\u000berences in the magnitude of the threshold currents\nand the range of frequency of operation. The theoretical framework allows us to identify the in-\nput current requirements as a function of the material and geometry parameters for exciting an\noscillatory response. We also obtain a closed-form approximate solution of the oscillation frequency\nfor large input currents in case of both Mn 3Ir and Mn 3Sn. Our analytical predictions of threshold\ncurrent and oscillation frequency agree well with the numerical results and thus can be used as com-\npact models to design and optimize the auto oscillator. Employing a circuit model, based on the\nprinciple of tunnel anisotropy magnetoresistance, we present detailed models of the output power\nand e\u000eciency versus oscillation frequency of the auto oscillator. Finally, we explore the spiking\ndynamics of two unidirectional as well as bidirectional coupled AFM oscillators using non-linear\ndamped-driven pendulum equations. Our results could be a starting point for building experimen-\ntal setups to demonstrate auto oscillations in metallic AFMs, which have potential applications in\nterahertz sensing, imaging, and neuromorphic computing based on oscillatory or spiking neurons.\nI. INTRODUCTION\nTerahertz (THz) radiation, spanning from 100 Giga-\nhertz (GHz) to 10 THz, are non-ionizing, have short\nwavelength, o\u000ber large bandwidth, scatter less, and are\nabsorbed or re\rected di\u000berently by di\u000berent materials.\nAs a result, THz electronics can be employed for safe\nbiomedical applications, sensing, imaging, security, qual-\nity monitoring, spectroscopy, as well as for high-speed\nand energy-e\u000ecient non-von Neumann computing (e.g.,\nneuromorphic computing). THz electronics also has po-\ntential applications in beyond-5G communication sys-\ntems and Internet of Things. Particularly, the size of\nthe antennae for transmitting the electromagnetic signal\ncould be signi\fcantly miniaturized in THz communica-\ntion networks [1{8]. These aforementioned advantages\nand applications have led to an intense research and de-\nvelopment in the \feld of THz technology with an aim\nto generate, manipulate, transmit, and detect THz sig-\nnals [3, 9]. Therefore, the development of e\u000ecient and\nlow power signal sources and sensitive detectors that op-\nerate in the THz regime is an important goal [3].\nMost coherent THz signal sources can be categorized\ninto three types | particle accelerator based sources,\nsolid state electronics based sources, and photonics based\nsources [3, 9]. Particle accelerator based signal genera-\ntors include free electron lasers [10], synchrotrons [11],\n\u0003ankits4@illinois.edu\nyrakheja@illinois.eduand gyrotrons [12]. While particle accelerator sources\nhave the highest power output, they require a large\nand complex set-up [13]. Solid state generators include\ndiodes [14{16], transistors [17, 18], frequency multipli-\ners [19], and Josephson junctions [20], whereas photonics\nbased signal sources include quantum cascade lasers [21],\ngas lasers [22], and semiconductor lasers [23]. Solid-state\ngenerators are e\u000ecient at microwave frequencies whereas\ntheir output power and e\u000eciency drop signi\fcantly above\n100 GHz [13]. THz lasers, on the other hand, provide\nhigher output power for frequencies above 30 THz [24],\nhowever, their performance for lower THz frequencies\nis plagued by noise and poor e\u000eciency [13]. Here, we\npresent the physics, operation, and performance bench-\nmarks of a new type of nanoscale THz generator based\non the ultra-fast dynamics of the order parameter of an-\ntiferromagnets (AFMs) when driven by spin torque.\nSpin-transfer torque (STT) [25, 26] and spin-orbit\ntorque (SOT) [27] enable electrical manipulation of ferro-\nmagnetic order in emerging low-power spintronic radio-\nfrequency nano-oscillators [28]. When a spin current\ngreater than a certain threshold (typically around 108\u0000\n109A=cm2[28, 29]) is injected into a ferromagnet (FM)\nat equilibrium, the resulting torque due to this current\npumps in energy which competes against the intrinsic\nGilbert damping of the material. When the spin torque\nbalances the Gilbert damping, the FM magnetization un-\ndergoes a constant-energy steady-state oscillation around\nthe spin polarization of the injected spin current. Such\noscillators are nonlinear, current tunable with frequencies\nin the range of hundreds of MHz to a few GHz with out-\nput power in the range of nano-Watt (nW). They are alsoarXiv:2106.08528v2  [cond-mat.mes-hall]  15 Jan 20222\ncompatible with the CMOS technology [28]; however, the\ngeneration of the THz signal using FMs would require\nprohibitively large amount of current, which would lead\nto Joule heating and degrade the reliability of the elec-\ntronics. It would also lead to electromigration and hence\nirreversible damage to the device set-up [30].\nAFM materials, which are typically used to exchange\nbias [31] an adjacent FM layer in spin-valves or magnetic\ntunnel junctions for FM memories and oscillators, have\nresonant frequencies in the THz regime [32{35] due to\ntheir strong exchange interactions. It was suggested that\nSTT could, in principle, be used to manipulate the mag-\nnetic order in conducting AFMs [36], leading to either\nstable precessions for their use as high-frequency oscilla-\ntors [37, 38] or switching of the AFM order [39] for their\nuse as magnetic memories in spin-valve structures. The\nSOT-based spin Hall e\u000bect (SHE), on the other hand,\ncould enable the use of both conducting [40{44] and in-\nsulating [29, 43, 45, 46] AFMs in a bilayer comprising an\nAFM and a non-magnetic (NM) layer, for its use as a\nhigh frequency auto-oscillator [47].\nTable I lists the salient results from some of the re-\ncently proposed AFM oscillators. These results, how-\never, are reported mainly for collinear AFMs, while de-\ntailed analyses of the dynamics of the order parameter in\nthe case of non-collinear AFMs is lacking. In this paper,\n\frstly, we \fll this existing knowledge gap in the model-\ning of auto oscillations in thin-\flm non-collinear copla-\nnar AFMs like Mn 3Ir;Mn3Sn, or Mn 3GaN under the ac-\ntion of a dc spin current. Secondly, we compare their\nperformance (generation and detection) against that of\ncollinear AFMs such as NiO for use as a THz signal\nsource. In the case of NiO, inverse spin Hall e\u000bect (iSHE)\nis employed for signal detection, whereas in this work we\nutilize the large magnetoresistance of metallic AFMs. Fi-\nnally, we investigate these auto oscillators as possible can-\ndidates for neuron emulators. Considering that the spin\npolarization is perpendicular to the plane of the AFM\nthin-\flm, three possible device geometries are identi\fed\nand presented in Fig. 1 for the generation and detection\nof auto oscillations in metallic AFMs.\nFigure 1(a) is based on the phenomena of spin injec-\ntion and accumulation in a local lateral spin valve struc-\nture. Charge current, I write, injected into the structure\nis spin-polarized along the magnetization of FM 1(FM 2)\nand gets accumulated in the NM. It then tunnels into the\nAFM with the required perpendicular spin-polarization\n(adapted from Ref. [50]). The bottom MgO layer used\nhere would reduce the leakage of charge current into\nthe metallic AFMs considered in this work. This would\nreduce chances of Joule heating in the AFM thin-\flm\nlayer. On the other hand, in Fig. 1(b), spin \flter-\ning [51] technique is adopted wherein, a conducting AFM\nis sandwiched between two conducting FMs. Di\u000ber-\nent scattering rates of the up-spins and down-spins of\nthe injected electron ensemble at the two FM interfaces\nresults in a perpendicularly polarized spin current as\nshown. The structure in Fig. 1(c) generates spin polar-\nFM1\nIwrite\nFM2NMJs\nnpAFMMgOPtLoadIdeal Bias Tee\nFM1FM2\nIwriteAFM\nMgOIdeal Bias Tee\nLoad\nJsnpIread\nFM1\nNMLoadIdeal Bias Tee(a)\n(b) (c)xyzIread\nIwriteIread\nPt\nMgO\nJsnpAFMMgOLbtCbt\nCbt Lbt LbtCbtFIG. 1. Device geometries to inject perpendicularly polar-\nized spin current in thin-\flm metallic AFMs. In all the cases,\nIwrite is the charge current injected to generate spin current,\nwhereas, I readis the charge current injected to extract the\noscillations as a transduced voltage signal using the princi-\nples of tunnel anisotropy magnetoresistance (TAMR). (a) Lat-\neral spin valve structure leads to spin accumulation in NM\nfollowed by injection into the AFM. (b) Perpendicular spin\nvalve structure spin \flters the injected charge current. (c)\nFM/NM/AFM trilayer structure generates spin current due\nto interfacial spin-orbit torque.\nization perpendicular to the interface due to the interfa-\ncial SOTs generated at the FM/NM interface (adapted\nfrom Ref. [52, 53]). In this case, the spin current injected\ninto the AFM has polarization along both yandzdirec-\ntion; however, the interface properties could be tailored\nto suppress the spin polarization along y[52]. In order\nto extract the THz oscillations of the order parameter as\na measurable voltage signal, the tunnel anisotropy mag-\nnetoresistance (TAMR) measurements are utilized [54].\nIn this work, we establish the micromagnetic model\nfor non-collinear coplanar AFMs with three sublattices\nalong with the boundary conditions in terms of both the\nsublattice magnetizations (Section II A), as well as the\nN\u0013 eel order parameter (Section II B). We show that in the\nmacrospin limit the oscillation dynamics correspond to\nthat of a damped-driven pendulum (Section III A and\nSection III B). The oscillation dynamics of AFM mate-\nrials with two di\u000berent chiralities in then compared in\nSection III C. We use the TAMR detection scheme to\nextract the oscillations as a voltage signal and present\nmodels of the output power and e\u000eciency as a function\nof the oscillator's frequency (Section IV). This is followed\nby a brief investigation of the e\u000bect of inhomogeneity due\nto the exchange interaction on the dynamics of the AFM\norder (Section V). Finally, we discuss the implication of\nour work towards building coherent THz sources in Sec-3\nTABLE I. Recent numerical studies on electrically controlled AFM THz oscillators. The investigated AFM materials, the\ndirection of their uniaxial anisotropy axis ueand that of the spin polarization of the injected spin current npare listed. Salient\nresults along with the schemes to extract the oscillation as a voltage signal are also brie\ry stated. Ref. [44] does not provide\nthe name of a speci\fc AFM, however, an AFM with uniaxial anisotropy is considered.\nRef. AFM material ue np Salient Features Detection Schemes\n[45] NiO x x a) THz oscillations for current above a threshold iSHE\nb) Feedback in AFM/Pt bilayer sustains oscillation\n[29] NiO x -z a) Hysteretic THz oscillation in a biaxial AFM iSHE\nb) Threshold current dependence on uniaxial anisotropy\n[40]\u000b\u0000Fe2O3 a)z y a) Monodomain analysis of current driven oscillations in -\nAFM insulators with DMI\nb)y y Similar to [45] -\n[41]\u000b\u0000Fe2O3 x y a) Canted net magnetization due to DMI Dipolar radiation\nb) Small uniaxial anisotropy leads to low power THz frequency\n[42] CuMnAs ;Mn2Auy z a) Low dc current THz signal generation due to N\u0013 eel SOT -\nb) Phase locked detector for external THz signal\n[43] NiO x Varied a) Comparison of analytical solutions to micromagnetic results AMR/SMR\nb) E\u000bect of DMI on hysteretic nature of dynamics\n[48] Mn 2RuxGa z y a) Generation of spin current in single AFM layer AMR\nb) Oscillation dependence on reactive and dissipative torques\n[44] Uniaxial Ani. z Varied a) Non-monotonic threshold current variation with np -\nb) E\u000bects of anisotropy and exchange imperfections\n[49] Mn 3Sn x-y plane z a) E\u000bective pendulum model based on multipole theory AHE\n[46] NiO;Cr2O3 x Varied a) General e\u000bective equation of a damped-driven pendulum iSHE\nb) Analytic expression of threshold current and frequency\nOur Mn 3Sn;Mn3Ir x-y plane z a) Di\u000berent numerical and analytic models TAMR\nWork b) Inclusion of generation current for TAMR e\u000eciency\nc) Non-linear dynamics of bidirectional coupled oscillators\ntion VI, and towards hardware neuron emulators for neu-\nromorphic computing architecture in Section VII. Some\nof the salient results from this work are listed in Table I.\nII. THEORY\nA. Magnetization Dynamics\nWe consider a micromagnetic formalism in the con-\ntinuum domain [43, 55] under which a planar non-\ncollinear AFM is considered to be composed of three\nequivalent interpenetrating sublattices, each with a con-\nstant saturation magnetization Ms[56]. Each sublattice,\ni(= 1, 2 or 3), is represented as a vector \feld mi(r;t)\nsuch that for an arbitrary r=r0,kmi(r0;t)k= 1. The\ndynamics of the AFM under the in\ruence of magnetic\n\felds, damping, and spin torque is assumed to be gov-\nerned by three Landau-Lifshitz-Gilbert (LLG) equations\ncoupled by exchange interactions. For sublattice i, the\nLLG is given as [57]\n@mi\n@t=\u0000\r\u00160\u0000\nmi\u0002He\u000b\ni\u0001\n+\u000bi\u0012\nmi\u0002@mi\n@t\u0013\n\u0000!smi\u0002(mi\u0002np)\u0000\f!s(mi\u0002np);(1)\nwheretis time in seconds, He\u000b\niis the position dependent\ne\u000bective magnetic \feld on i,\u000biis the Gilbert dampingparameter for i, and\n!s=~\n2e\rJs\nMsda(2)\nis the frequency associated with the input spin current\ndensity,Js, with spin polarization along np. Here,dais\nthe thickness of the AFM layer, ~is the reduced Planck's\nconstant,\u00160is the permeability of free space, eis the el-\nementary charge, and \r= 17:6\u00021010T\u00001s\u00001is the gyro-\nmagnetic ratio. For all sublattices, the spin polarization,\nnp, is assumed to be along the zaxis. Finally, \fis a mea-\nsure of the strength of the \feld-like torque as compared\nto the antidamping-like torque. The e\u000bect of \feld-like\ntorque on the sublattice vectors here is the same as that\nof an externally applied magnetic \feld|canting towards\nthe spin polarization direction. Results presented in the\nmain part of this work do not include the e\u000bect of the\n\feld-like torque; however, a small discussion on the same\nis presented in the supplementary material [58].\nThe e\u000bective magnetic \feld, He\u000b\niat each sublattice,\nincludes contributions from internal \felds as well as ex-\nternally applied magnetic \felds and is obtained as\nHe\u000b\ni(r;t) =\u00001\n\u00160Ms\u000eF\n\u000emi(r;t); (3)\nwhere\u000e\n\u000emi=@\n@mi\u0000r\u0001@\n@(rmi), andFis the energy\ndensity of the AFM, considered in our work. It is given\nas4\nF=X\nhi;ji\u0000\nJmi\u0001mj+Aijrmi\u0001rmj\u0001\n+Aii3X\ni=1(rmi)2+3X\ni=1Khm2\ni;z\u0000Ke(mi\u0001ue;i)2\n+DX\nhi;jiz\u0001(mi\u0002mj) +Dii3X\ni=1(mi;zr\u0001mi\n\u0000(mi\u0001r)mi;z) +DijX\nhi;ji((mi;zr\u0001mj\n\u0000(mi\u0001r)mj;z)\u0000(mj;zr\u0001mi\u0000(mj\u0001r)mi;z))\n\u00003X\ni=1\u00160MsHa\u0001mi;(4)\nwherehi;jirepresents the sublattice ordered pairs (1 ;2),\n(2;3) and (3;1).\nThe \frst three terms in Eq. (4) represent exchange en-\nergies. HereJ(>0) is the homogeneous inter-sublattice\nexchange energy density whereas Aii(>0) andAij(<\n0) are the isotropic inhomogeneous intra- and inter-\nsublattice exchange spring constants, respectively. The\nnext two terms in Eq. (4) represent magnetocrystalline\nanisotropy energy for biaxial symmetery upto the low-\nest order withKe(>0) andKh(>0) being the easy and\nhard axes anisotropy constants, respectively. We assume\nthat the easy axes of sublattices 1, 2 and 3 are along\nue;1=\u0000(1=2)x+ (p\n3=2)y,ue;2=\u0000(1=2)x\u0000(p\n3=2)y\nandue;3=x, respectively, and an equivalent out of plane\nhard axis exists along the zaxis. The next three terms\nrepresent the structural symmetry breaking interfacial\nDzyaloshinskii-Moriya Interaction (iDMI) energy density\nin the continuum domain. Its origin lies in the interaction\nof the antiferromagnetic spins with an adjacent heavy\nmetal with a large spin-orbit coupling [59, 60]. Here, we\nassume the AFM crystal to have Cnvsymmetry [61] such\nthat the thin-\flm AFM is isotropic in its plane, and D,\nDii, andDijrepresent the e\u000bective strength of homoge-\nneous and inhomogeneous iDMI, respectively, along the\nzdirection. Finally, the last term in Eq. (4) represents\nthe Zeeman energy due to an externally applied mag-\nnetic \feld Ha. Now, using Eq. (4) in Eq. (3) we get the\ne\u000bective \feld for sublattice ias\nHe\u000b\ni=X\nj\nj6=i\u0012\n\u0000J\n\u00160Msmj+Aij\n\u00160Msr2mj\u0013\n+2Aii\n\u00160Msr2mi\n\u00002Kh\n\u00160Msmi;zz+2Ke\n\u00160Ms(mi\u0001ue;i)ue;i\n+Dz\u0002(mj\u0000mk)\n\u00160Ms\u00002Dii\n\u00160Ms((r\u0001mi)z\u0000rmi;z)\n\u0000Dij\n\u00160Ms((r\u0001(mj\u0000mk))z\u0000r(mj;z\u0000mk;z))\n+Ha;\n(5)where (i;j;k ) = (1;2;3);(2;3;1);or (3;1;2), respectively.\nIn order to explore the dynamics of the AFM, we adopt\na \fnite di\u000berence discretization scheme and discretize the\nthin-\flm of dimension L\u0002W\u0002dainto smaller cells of\nsizesL\u0002sW\u0002sd. Each of these cells is centered around\nposition rsuch that mi(r;t) denotes the average mag-\nnetization of the spins within that particular cell [43].\nFinally, we substitute Eq. (5) in Eq. (1) and use fourth-\norder Runge-Kutta rule along with the following bound-\nary conditions for sublattice iof the thin-\flm considered\n(see supplementary material [58]):\n2Aii@mi\n@\u0011\u0011\u0011+AijX\nj\nj6=imi\u0002\u0012@mj\n@\u0011\u0011\u0011\u0002mi\u0013\n+Diimi\u0002(\u0011\u0011\u0011\u0002z)\n+Dijmi\u0002(mi\u0002((\u0011\u0011\u0011\u0002z)\u0002(mk\u0000mj))) = 0;(6)\nwhere\u0011\u0011\u0011is the normal vector perpendicular to a surface\nparallel to xory. The above equation ensures that the\nnet torque due to the internal \felds on the boundary\nmagnetizations of each sublattice is zero in equilibrium\nas well as under current injection [62]. For the energy\ndensity presented in Eq. (4), the \felds at the boundary\nare non-zero only for inhomogeneous inter- and intra-\nsublattice exchange, and Dzyaloshinskii-Moriya interac-\ntions. Finally, in the absence of DMI, we have the Neu-\nmann boundary condition@mi\n@\u0011\u0011\u0011= 0, which implies that\nthe boundary magnetization does not change along the\nsurface normal \u0011\u0011\u0011.\nFor all the numerical results presented in this work\nwe solve the system of Eqs. (1), (5), and (6) with the\nequilibrium state as the starting point. The equilibrium\nsolution in each case was arrived at by solving these three\nequations for zero external \feld and zero current with a\nlarge Gilbert damping of 0 :5.\nB. N\u0013 eel Order Dynamics\nThe aforementioned micromagnetic modeling ap-\nproach assuming three sublattices is extremely useful in\nexploring the physics of the considered AFM systems. It\nis, however, highly desirable to study an e\u000bective dynam-\nics of the AFMs under the e\u000bect of internal and external\nstimuli in order to gain fundamental insight. Therefore,\nwe consider an average magnetization vector mand two\nstaggered order parameters n1andn2to represent an\nequivalent picture of the considered AFMs. These vec-\ntors are de\fned as [37, 56, 63]\nm\u00111\n3(m1+m2+m3); (7a)\nn1\u00111\n3p\n2(m1+m2\u00002m3); (7b)\nn2\u00111p\n6(\u0000m1+m2); (7c)5\nsuch thatkmk2+kn1k2+kn2k2= 1. The energy land-\nscape (Eq. (4)) can then be represented as\nF\n3=3J\n2m2+Am(rm)2+An\n2\u0010\n(rn1)2+ (rn2)2\u0011\n+Kh\u0000\nm2\nz+n2\n1;z+n2\n2;z\u0001\n\u0000Ke\n2\u00123\n2(n1;x\u0000n2;y)2\n+1\n2(n1;y+n2;x)2+mx\u0010\nmx\u0000p\n2(n1;x\u0000n2;y)\u0011\n+my\u0010\nmy+p\n2(n1;y+n2;x)\u0011\n+ 4n1;xn2;y\u0011\n+p\n3Dz\u0001(n1\u0002n2) +Dii(mzr\u0001m\u0000(m\u0001r)mz\n+n1;zr\u0001n1\u0000(n1\u0001r)n1;z+n2;zr\u0001n2\n\u0000(n2\u0001r)n2;z) +p\n3Dij(n1;zr\u0001n2\u0000(n1\u0001r)n2;z\n\u0000n2;zr\u0001n1+ (n2\u0001r)n1;z)\u0000\u00160MsHa\u0001m;\n(8)\nwhereAm=\u0000\nAii+Aij\u0001\nandAn=\u0000\n2Aii\u0000Aij\u0001\n.\nAn equation of motion involving the staggered order\nparameters can be obtained by substituting Eq. (1) in the\n\frst-order time derivatives of Eq. (7) and evaluating each\nterm carefully (see supplementary material [58]). How-\never, an analytical study of such an equation of motion\nthat consists of contributions from all the energy terms\nof Eqs. (4) or (8) would be as intractable as the dynam-\nics of individual sublattices itself. Therefore, we consider\nthe case of AFMs with strong inter-sublattice exchange\ninteraction such that J \u001djDj\u001dK e. This corresponds\nto systems with ground state con\fned to the easy-plane\n(x\u0000yplane) and those that host kmk\u001c1 (weak ferro-\nmagnetism), n1?n2, andkn1k\u0019kn2k\u00191=p\n2 [56, 63].\nHowever, when an input current is injected in the system,\nthe sublattice vectors cant towards the spin polarization\ndirection leading to an increase in the magnitude of m\nwhile decreasing that of n1andn2. Spin polarization\nalong the zdirection and an equal spin torque on each\nsublattice vector ensures that n1andn2have negligible\nzcomponents at all times (Eqs. (7b), (7c)). Therefore,\nwe consider\nn1(r;t) =\u00150\n@p\n1\u0000n2\n1zcos'(r;t)p\n1\u0000n2\n1zsin'(r;t)\nn1z(r;t)1\nA; (9a)\nn2(r;t) =\u00150\n@p\n1\u0000n2\n2zcos('(r;t)\u0006\u0019=2)p\n1\u0000n2\n2zsin('(r;t)\u0006\u0019=2)\nn2z(r;t)1\nA;(9b)\nwhere'is the azimuthal angle from the xaxis and\njn1zj;jn2zj\u001c1.\nThe two choices for n2correspond to two di\u000berent\nclasses of materials|one with a positive (+ \u0019=2) chiral-\nity and the other with a negative ( \u0000\u0019=2) chirality [56].\nMaterials that have a negative (positive) value of Dcor-\nrespond to + \u0019=2(\u0000\u0019=2) chirality because the respective\ncon\fguration reduces the overall energy of the system.\nL12phase of AFMs like Mn 3Ir;Mn3Rh, or Mn 3Pt is ex-\npected to host + \u0019=2 chirality whereas the hexagonalphase of AFMs like Mn 3Sn;Mn3Ge, or Mn 3Ga is ex-\npected to host\u0000\u0019=2 chirality [56, 64].\nIII. SINGLE DOMAIN ANALYSIS\nA. Positive Chirality\nDe\fning n3= (n1\u0002n2)=\u0015, and considering the case\nof +\u0019=2 chirality, it can be shown that mis just a de-\npendent variable of the N\u0013 eel order dynamics. To a \frst\norder, mcould be expressed as [56, 63]\nm=\u00001\n!E(n1\u0002_ n1+n2\u0002_ n2+n3\u0002_ n3\u0000\r\u00160Ha\n\u0000\f!snp);\n(10)\nwhere!E= 3\rJ=Ms. One can then arrive at the equa-\ntion of motion for the N\u0013 eel vectors as\nn1\u0002\u0002\n n1\u0000c2r2n1\u0000!E!Kn1+!E!Kh(n1\u0001z)z\n+!E!D(n2\u0002z) +!E!ii\nDn1+!E!ij\nDn2+\u000b!E_n1\n\u0000!E!s(n1\u0002np)] +n2\u0002\u0002\n n2\u0000c2r2n2\u0000!E!Kn2\n+!E!Kh(n2\u0001z)z\u0000!E!D(n1\u0002z) +!E!ii\nDn2\n\u0000!E!ij\nDn1+\u000b!E_n2\u0000!E!s(n2\u0002np)i\n+n3\u0002n3\n+\r\u00160(n1\u0002_n1+n2\u0002_n2+n3\u0002_n3)\u0002Ha\n\u0000\r\u00160_Ha= 0;\n(11)\nwherec=p\n!E\rAn=Ms,!Kh= 2\rKh=Ms,!K;n1=\n!K\n4\u0000\u0000\nn1;y+n2;x+p\n2my\u0001\n^ x+ (n1;x+ 3n2;y\n+p\n2mx\u0001\n^ y\u0001\n,!D=p\n3\rD\nMs,!K;n2=!K\n4((n1;y+n2;x\n+p\n2my\u0001\n^ x+\u0000\nn1;x+ 3n2;y+p\n2mx\u0001\n^ y\u0001\n,\n!ii\nD;ni=2\rDii\nMs((r\u0001ni)z\u0000rni;z),\nand!ij\nD;ni=p\n3\rDij\nMs((r\u0001ni)z\u0000rni;z).\nThe equations of motion (Eqs. (10) and (11)) derived\nhere are useful in the numerical study of textures like\ndomain walls, skyrmions, and spin-waves in AFMs with\nbiaxial anisotropy under the e\u000bect of external magnetic\n\feld and spin current. However, here we are interested\nin analytically studying oscillatory dynamics of the order\nparameter in thin-\flm AFMs, therefore, we neglect in-\nhomogeneous interactions compared to the homogeneous\n\felds. Using Eq. (9) in Eq. (11) and neglecting the time\nderivative of n1zandn2z, we have\n'+\u000b!E_'+!E!K\n2sin 2'+!E!s= 0; (12)\nwhere!K= 2\rKe=Ms. This indicates that in the limit\nof strong exchange interaction, the dynamics of the stag-\ngered order parameters is identical to that of a damped-\ndriven non-linear pendulum [65]. This equation is iden-\ntical to the case of collinear AFMs such as NiO when the6\n(c) (d)\n(a) (b)Negative Chirality Positive Chirality\nFIG. 2. Stationary solution for AFMs with positive (a, b) and negative chirality (c, d). (a) Sublattice magnetization for currents\nbelow the threshold current Jth1\ns. WhenJs= 0 (equilibrium state), the sublattice vectors micoincide with the easy axes ue;i,\nwhereas for a non-zero current smaller than Jth1\ns, the macrospins have stationary solutions other than the equilibrium solution,\nas depicted by dashed and dotted line. The zcomponent of these vectors is zero. (b) An equivalent representation of (a) through\nthe staggered order parameters n1andn2. They are perpendicular to each other and have zero out-of-plane component. The\nthinner dash-dotted gold arrows through the thicker arrows of n1represent the analytic expression of the stationary solution\n'=\u00001\n2sin\u00001\u0010\n2!s\n!K\u0011\n. The average magnetization mis vanishingly small, as can also be noticed from Eq. (10). (c) Sublattice\nmagnetization at equilibrium ( Js= 0). Thezcomponent of these vectors is zero. Here, only the sublattice vector m3coincides\nwith its corresponding easy axis. On the other hand, m1andm2are oriented such that the energy due to DMI is dominant\nover anisotropy. (d) An equivalent representation of (c) through the staggered order parameters n1andn2, and average\nmagnetization m.n1andn2are almost perpendicular to each other with a negative chirality, as assumed in Eq.(9). A small\nin-plane net magnetization (shown by the magni\fed green arrow) also exists in this case [56].\n(a) (b) (c) (d)\n(e) (g) (f) (h)Positive Chirality Negative Chirality\nα = 0.01α = 0.01 α = 0.01\nα = 0.01da = 4 nm da = 4 nm\nda = 4 nm da = 4 nm\nFIG. 3. Upper panel (a-d) shows the time averaged frequency as a function of input spin current, whereas the lower panel (e-h)\nshows the FFT of the oscillations corresponding to the cases marked by the dashed red boxes above. In the time averaged\nfrequency response in the upper panel, the dashed black lines denote the analytic expression of frequency (Eq. (17)). (a), (c)\nFrequency response for di\u000berent \flm thicknesses for \u000b= 0:01. (e), (g) FFT of the signal corresponding to Js=Jth2\ns. (b),\n(d) Frequency response for di\u000berent damping constants for da= 4 nm. (f), (h) FFT of the signal corresponding to \u000b= 0:1\nandJth1\ns, respectively. Positive chirality: The numerical values of the average frequency match very well against the analytic\nexpression for lower damping and large current. On the other hand, non-linearity and, hence, higher harmonics are observed\nfor small current and large damping. Negative chirality: The numerical values of the average frequency exactly match against\nthe analytic expression for all values of damping and input current considered here.\ndirection of spin polarization is perpendicular to the easy-\nplane [29, 44, 66]. However, the dynamics of the non-\ncollinear coplanar AFMs discussed here is signi\fcantlydi\u000berent in the direction of the spin torques, magnitude\nof threshold currents as well as the range of possible fre-\nquencies. Here, the sin 2 'dependence signi\fes a two-fold7\nanisotropy symmetric system.\nB. Negative Chirality\nFor the case of\u0000\u0019=2 chirality, it can be shown that m\nis a dependent variable of the N\u0013 eel order; however, in this\ncase there are additional in-plane terms that arise due to\na competition between the DMI, exchange coupling and\nmagnetocrystalline anisotropy. To a \frst order, mis ex-\npressed as [56, 67] (also see supplementary material [58])\nm=\u00001\n!E(n1\u0002_ n1+n2\u0002_ n2+n3\u0002_ n3\u0000\r\u00160Ha\n\u0000\f!snp)\u0000!K\n2!E(cos'x\u0000sin'y);\n(13)\nwhich is used to arrive at the equation of motion for the\nN\u0013 eel vectors as\nn1\u0002\u0002\n n1\u0000c2r2n1\u0000!E!Kn1+!E!Kh(^ n1\u0001z)z\n+!E!D(n2\u0002z) +!E!ii\nDn1+!E!ij\nDn2+\u000b!E_n1\n\u0000!E!s(n1\u0002np)] +n2\u0002\u0002\n n2\u0000c2r2n2\u0000!E!Kn2\n+!E!Kh(^ n2\u0001z)z\u0000!E!D(n1\u0002z) +!E!ii\nDn2\n\u0000!E!ij\nDn1+\u000b!E_n2\u0000!E!s(n2\u0002np)i\n+n3\u0002n3\n+\r\u00160(n1\u0002_n1+n2\u0002_n2+n3\u0002_n3)\u0002Ha\u0000\r\u00160_Ha\n\u0000!K\n2(sin'x+ cos'y) _'\u0000\r\u00160!K\n2(Ha;zsin'x\n+Ha;zcos'y\u0000(Ha;xsin'+Ha;ycos')z) = 0:\n(14)\nSimilar to the previous case, we are interested in a the-\noretical analysis of the oscillation dynamics in thin \flm\nAFMs with negative chirality. Therefore, we use Eq. (9)\nin Eq. (14) and neglect all the inhomegeneous interac-\ntions to arrive at a damped-driven linear pendulum equa-\ntion given as\n'+\u000b!E_'+!E!s= 0: (15)\nHere the dependence of the dynamics on anisotropy is\nnot zero but very small, and it scales proportional to\n!3\nK\n!2\nEcos 6'[67]. However, for a \frst-order approxima-\ntion in mand dynamics in the THz regime, it can be\nsafely ignored. The cos 6 'dependence implies that these\nmaterials host a six-fold anisotropic symmetry. Though\nthis equation is similar to that obtained for the case of\na collinear AFM with spin polarization along the easy\naxis [44], the dynamics is signi\fcantly di\u000berent from that\nof the collinear AFM.\nC. Comparison of Dynamics for Positive and\nNegative Chiralities\nHere, we contrast the dynamics of AFM order param-\neter for positive and negative chiralities. The numericalresults presented in this section are obtained in the single-\ndomain limit assuming thickness da= 4 nm,\u000b= 0:01,\nMs= 1:63 T,Ke= 3 MJ=m3,J= 2:4\u0002108J=m3,\nD=\u000020 MJ=m3for positive chirality or 20 MJ =m3for\nnegative chirality [56], unless speci\fed otherwise.\nFigure 2 shows the stationary solutions of the thin-\n\flm AFM system with di\u000berent chiralities. For the case\nof positive chirality, it can be observed from Fig. 2(a)\nthat in equilibrium the sublattice vectors micoincide\nwith the easy axes ue;i. When a non-zero spin current is\napplied, the equilibrium state is disturbed; however, be-\nlow a certain threshold, Jth1\ns, the system dynamics con-\nverge to a stationary solution in the easy-plane of the\nAFM, indicated by dashed blue, and dotted red set of\narrows. An equivalent representation of the stationary\nsolutions in terms of the staggered order parameters is\npresented in Fig. 2(b). n1andn2are perpendicular to\neach other with zero out-of-plane component for all val-\nues of the input currents. The gold dash-dotted arrows\npassing through n1correspond to the stationary solu-\ntions given as '=\u00001\n2sin\u00001\u0010\n2!s\n!K\u0011\n, obtained analytically\nby setting both _ 'and 'as zero in Eq. (12). In positive\nchirality material, the average magnetization mis van-\nishingly small in the stationary state. This can also be\nperceived from Eq. (10) as we do not consider any exter-\nnal \feld. Since these materials have a two-fold symmetry,\nthey also host '=\u0019\u00001\n2sin\u00001\u0010\n2!s\n!K\u0011\nstationary states.\nFor the case of negative chirality, it can be observed\nfrom Fig. 2(c) that in equilibrium only the sublattice\nvector m3coincides with the its corresponding easy axis,\nwhereas both m1andm2are oriented such that the en-\nergy due to DMI is dominant over anisotropy, which in\nturn lowers the overall energy of the system. It can be\nobserved from Fig. 2(d) that n1andn2are almost per-\npendicular to each other. A small in-plane net magneti-\nzation exists in this case and is shown here as a zoomed\nin value (zoom factor = 100x) for the sake of compari-\nson to staggered order parameters. Due to the six-fold\nanisotropy dependence other equilibrium states, wherein\neither of m1orm2coincide with their easy axis while the\nother two sublattice vectors do not, also exist. Finally,\ndue to the small anisotropy dependence, non-equilibrium\nstationary states exist for much lower currents [68] than\nthose considered here, and therefore are not shown.\nFor materials with positive chirality, the system be-\ncomes unstable when the input spin current exceeds the\nthreshold,Jth1\ns. The resultant spin torque pushes the\nN\u0013 eel vectors out of the easy-plane, and they oscillate\naround the spin polarization axis, np=z;with THz fre-\nquency due to strong exchange. This threshold current\nis given as [29, 44, 46]\nJth1\ns=da2e\n~Ms\n\r!K\n2=da2e\n~Ke; (16)\nwhile the frequency of oscillation in the limit of large\ninput current (neglecting the sin 2 'term) from Eq. (12)8\n(a) (b)\n(c) (d)Positive Chirality Negative Chirality\nFIG. 4. The out-of-plane (z) component of the average magnetization, m, and n3for di\u000berent values of input currents for both\npositive and negative chirality. Positive chirality: (a) When current is increased from zero but to a value below the threshold\n(0:95Jth1\ns),mzis zero. However, it increases to a large value when Js=Jth1\ns.mzdecreases again to a smaller value when\nthe current is decreased to Js= 0:86Jth1\ns. Finally, when the current is further reduced below the lower threshold to 0 :9Jth2\ns,\nmzbecomes zero again. (b) n3is initially equal to 1 =p\n2, but decreases in magnitude during the AFM dynamics since the\nmagnitude of mincreases when the sublattice vectors move out of the plane. As soon as the current is lowered below Jth2\ns,\nthe system goes to a stationary state and n3= 1=p\n2. Negative chirality: (c) Since the threshold current in this case is small,\nnon-zeromzis observed for all values of current considered here. (d) n3decreases in magnitude when current increases but\napproaches\u00001=p\n2 for lower values of current. Here \u000b= 0:01, andda= 4 nm.\nis given as[44, 46]\nf=1\n2\u0019!s\n\u000b=1\n2\u0019~\n2e\rJs\nMsda1\n\u000b: (17)\nAdditionally, for small damping, there exists a lower\nthreshold,Jth2\ns< Jth1\ns, which is equal to the current\nthat pumps in the same amount of energy that is lost in\none time period due to damping [29, 44, 46]. The lower\nthreshold current is given as\nJth2\ns=da2e\n~Ms\n\r2\u000b\n\u0019p!E!K=da2e\n~2\u000b\n\u0019p\n6JKe:(18)\nThe presence of two threshold currents enables energy-\ne\u000ecient operation of the THz oscillator in the hysteretic\nregion [29].\nThe average frequency response as a function of the\ninput spin current and the Fourier transform of the os-\ncillation dynamics is plotted in the left panel of Fig. 3\nfor materials with positive chirality. It can be observed\nfrom Fig. 3(a) that the fundamental frequency for dif-\nferent \flm thicknesses scales as predicted by Eq. (17)\nexcept for low currents near Jth2\nswhere non-linearity in\nthe form of higher harmonics appears as seen from the\nFFT response in Fig. 3(e). Next, Figs. 3(b), (f) show\nthat the non-linearity in the frequency response for low\ninput current increases as the value of the damping co-\ne\u000ecient increases. This is expected as the contribution\nfrom the uniaxial anisotropy (sin 2 'term) becomes sig-\nni\fcant owing to large damping and low current making\nthe motion non-uniform (  '6= 0) [29].\nIn the case of materials with negative chirality, small\nequivalent anisotropy suggests that the threshold current\nfor the onset of oscillations is very small, while the fre-\nquency of oscillations increases linearly with the spin cur-\nrent considered here and is given by Eq. (17). Indeed the\nsame can be observed from Figs. 3(c), (d) where the re-\nsults of numerical simulations exactly match the analyticexpression. The FFT signal in Figs. 3(g), (h) contains\nonly one frequency corresponding to uniform rotation of\nthe order ( '= 0). This coherent rotation of the order\nparameter with such small threshold current [68] in AFM\nmaterials with negative chirality opens up the possibil-\nity of operating such AFM oscillators at very low en-\nergy for frequencies ranging from MHz-THz. It can also\nbe observed from Fig. 3(b), (d) that for lower values of\ndamping, such as \u000b= 0:005, the frequency of oscillations\nsaturates for input current slightly above Jth1\ns. This is\nbecause the energy pumped into the system is larger than\nthat dissipated by damping. As a result the sublattice\nvectors move out of the easy-plane and get oriented along\nthe spin polarization direction (slip-\rop). For larger val-\nues of damping, the same would be observed for larger\nvalues of current. Finally, we would like to point out that\nthe values of both Jth1\nsandJth2\nsobserved from numerical\nsimulations were slightly di\u000berent from their analytical\nvalues for di\u000berent damping constants, similar to that\nreported in Ref. [43] for collinear AFMs.\nFigure 4 shows the out-of-plane (z) components of m\nandn3for non-zero currents for AFMs with di\u000berent chi-\nralities. For negative chirality materials, the steady-state\nzcomponent of both mandn3does not oscillate with\ntime ( '\u00190), whereas, for the case of positive chirality,\nthe steady-state zcomponents of both mandn3show\nsmall oscillations with time (  '6= 0) similar to the case\nof NiO with spin polarization along the hard axis [29].\nIt can be observed from Fig. 4(a) that for positive chi-\nrality, as current increases from below the upper thresh-\nold current (0.95 Jth1\ns) toJth1\ns, the out-of-plane compo-\nnent of magnetization vectors and hence the average mag-\nnetization mincreases from zero to a larger value. Due\nto the hysteretic nature of the AFM oscillator, the mag-\nnitude of mreduces when current is lowered but is non-\nzero as long as the input current is above Jth2\ns. Similarly,\nit can be observed from Fig. 4(b) that the out-of-plane9\ncomponent of n3, which was initially 1 =p\n2, decreases\nas the current increases above Jth1\ns. When the current\nis lowered to a value below Jth1\ns(0:86Jth1\nshere) , the\nmagnitude of the out-of-plane component of n3increases\nagain and eventually saturates to 1 =p\n2 when the current\nis lowered further below Jth2\ns(0:9Jth2\ns, here).\nIt can also be observed from Fig. 4(c) that for negative\nchirality AFMs, the out-of-plane component of the av-\nerage magnetization although small is non-zero even for\nsmall currents due to the lower value of threshold current.\nOn the other hand, nz\n3in Fig. 4(d) decreases in magnitude\nfrom an initial value of \u0015= 1=p\n2 to\u0015<1=p\n2 as current\nincreases sincekmkincreases. The values of current are\nassumed to be the same for both positive and negative\nchirality AFMs for the sake of comparison. Next, using\n(a)\n(b)\n(c)\n(d)\nFIG. 5.mzfor four di\u000berent values of input current Js.mz\nincreases with current and so does the frequency of oscillation.\nHere (a)Js= 1:6Jth2\ns, (b)Js= 0:9Jth1\ns. They are both inside\nthe hysteretic region bounded by Jth2\nsandJth1\ns. (c)Js=Jth1\ns,\nand (d)Js= 1:5Jth1\nslies outside the hysteretic region. These\nresults correspond to \u000b= 0:01, andda= 4 nm.\nEq. (9) in Eq. (10), it can be shown that _ 'is directly pro-\nportional to mz[56]. Therefore, to present the features\nof angular velocity with input current, we show mzfor\nfour di\u000berent values of input current Jsfor positive chi-\nrality material in Fig. 5 . Here, Figs. 5(a)-(c) correspond\nto the hysteretic region, whereas Fig. 5(d) is for current\noutside the hysteretic region. As mentioned previously,\nan increase in current increases the spin torque on the\nsublattice vectors which leads to an increase in mzand\nhence _'.\nIV. SIGNAL EXTRACTION\nAn important requirement for the realization of an\nAFM-based auto-oscillator is the extraction of the gen-\nerated THz oscillations as measurable electrical quanti-ties viz. voltage and current. It is expected for the ex-\ntracted voltage signal to oscillate at the same frequency\nas that of the N\u0013 eel vector and contain substantial out-\nput power ( >1\u0016W) [69]. In this regard, the landmark\ntheoretical work on NiO based oscillator [29] suggested\nthe measurement of spin pumped [70] time varying in-\nverse spin Hall voltage [71] across the heavy metal (Pt)\nof a NiO=Pt heterostructure. However, the time varying\nvoltage at THz frequency requires an AFM with signif-\nicant in-plane biaxial anisotropy [29, 69], thus limiting\nthe applicability of this scheme to only select AFM ma-\nterials. In addition, the output power of the generated\nsignal is sizeable (above 1 \u0016W) only for frequencies below\n0:5 THz [69]. A potential route to overcoming the afore-\nmentioned limitations is coupling the AFM signal genera-\ntor to a high-Q dielectric resonator, which would enhance\nthe output power even for frequencies above 0 :5 THz [41].\nThis method, however, requires devices with sizes in the\n10's micrometers range for frequencies above 2 THz and\nfor the AFMs to possess a tilted net magnetization in\ntheir ground state [69]. A more recent theoretical work\non collinear AFM THz oscillators [43] suggested employ-\ning Anisotropy Magnetoresistance (AMR) or Spin Mag-\nnetoresistance (SMR) measurements in a four terminal\nAFM/HM spin Hall heterostructure. This would enable\nthe extraction of the THz oscillations as longitudinal or\ntransverse voltage signals. However, the reported values\nof both AMR and SMR at room temperatures in most\nAFMs is low and would, in general, require modulating\nthe band structure for higher values [72].\nIdeal Bias \nRLC\nLR(t)\nA TJ\nCbtLbt\nPac(a)\nRLPac (b)\nZth\nUthIread\nFIG. 6. (a) An equivalent circuit representation of Fig. 1\n(adapted from [69]). The generation (write) current is not\nshown in the circuit, although its e\u000bect is included as a varia-\ntion in the resistance R(t) through its frequency dependence.\n(b) Thevenin equivalent of (a).\nA recent theoretical work [69] proposed employing a\nfour terminal AFM tunnel junction (ATJ) in a spin Hall\nbilayer structure with a conducting AFM to e\u000bectively\ngenerate and detect THz frequency oscillations as vari-\nations in the tunnel anisotropy magnetoresistance [54].\nA DC current passed perpendicularly to the plane of\nthe ATJ generates an AC voltage, which is measured\nacross an externally connected load. It was shown that\nboth the output power and its e\u000eciency decrease as fre-\nquency increases, nevertheless, it was suggested that this\nscheme could be used for signal extraction in the fre-\nquency range of 0 :1\u000010 THz, although the lateral size10\nof the tunnel barrier required for an optimal performance\ndepends on the frequency of oscillations (size decreases as\nthe frequency increases) [69]. The analysis presented in\nRef. [69], however, neglects the generation current com-\npared to the read current while evaluating the e\u000eciency\nof power extraction. But it can be observed from the\nresults in Section III C that the threshold current, and,\ntherefore, the generation current, depend on AFM ma-\nterial properties, such as damping, anisotropy, and ex-\nchange constants, and could be quite large. Therefore, in\nour work we include the e\u000bect of the generation current\nto accurately model the power e\u000eciency of the TAMR\nscheme.\n(a) (b)\nFIG. 7. (a) Output power and (b) e\u000eciency dependence on\nthe area of cross-section of the tunnel barrier for di\u000berent fre-\nquencies. The thickness of the barrier is \fxed to db= 1 nm.\nThe e\u000bect of write current and the input power associated\nwith it is not considered here, therefore, these results are in-\ndependent of the choice of the AFM material.\nIn order to evaluate the performance of the TAMR\nscheme, an equivalent circuit representation (adapted\nfrom Ref. [69]) of the device setup of Fig. 1 is shown\nin Fig. 6(a), while its Thevenin equivalent representation\nis shown in Fig. 6(b). The circuits in Fig. 6 only repre-\nsent the read component, while the THz generation com-\nponent is omitted for the sake of clarity. In Fig. 6(a),\nthe dashed red box encloses a circuit representation of\nthe ATJ, comprising a series combination of an oscillat-\ning resistance R(t) =R0+ \u0001Rcos!tand inductance\nL=\u00160db, connected in parallel to a junction capacitor\nC=Ac\u000f\u000f0=db(assumed parallel plate). The constant\ncomponent, R0, in the oscillating resistance, R(t), is the\nequilibrium resistance of the MgO barrier and is given\nasR0=RA(0) exp(\u0014db)\nAc. Here,RA(0) is the resistance-\narea product of a zero-thickness tunnel barrier, \u0014is the\ntunneling parameter, dbis the barrier thickness, and Ac\nis the cross-sectional area. The pre-factor, \u0001 R, of the\ntime varying component of R(t) is the resistance variation\ndue to the oscillation of the magnetization vectors with\nrespect to the polarization axis. \u0001 R= (\u0011=(2 +\u0011))R0,\nwhere\u0011is the TAMR ratio of the barrier and depends\non the temperature and material properties.\nDue to the \row of the DC current, Iread, an alter-\nnating voltage develops across the ATJ, which is mea-\nsured across an externally connected load RL, separated\nfrom the ATJ via an ideal bias tee (enclosed in the green\ndashed box). The bias-tee, characterized by an induc-\ntanceLbtand a capacitance Cbt, and assumed to have noTABLE II. List of common antiferromagnetic materials Mn 3X\nand their associated parameters. Here Msis in Tesla,Keis\nin kJ=m3, andJis in MJ=m3. Sign ofDwhich decides the\nchirality is also mentioned.\nX\u000b M sKeJ D Ref.\nIr 0:01 1:63 3000 240 - [56]\nPt 0:013 1:37 10 280 - [73]\nRh 0:013 2:00 10 230 - [73{75]\nGa 0:008 0:54 100 110 + [75{78]\nSn 0:003 0:50 110 59 + [67, 75, 77, 79]\nGe 0:0009 0:28 1320 77 + [75, 77, 78, 80]\nGaN 0:1 0:69 10 280 - [81, 82]\nNiN 0:1 1:54 10 177 - [81, 82]\nvoltage drop across it, blocks any DC current from \row-\ning into the external load. Therefore, the AC voltage of\nthe ATJ is divided only into its impedance (a combina-\ntion ofR0,L, andC) and that of the load RL[69].\nNext, we simplify the ATJ circuit into a Thevenin\nimpedance Zthand voltage Uthas shown in Fig. 6(b).\nThey are evaluated as\nZth=R0+j!L\n(1\u0000\u0018) +j\f; (19)\nand\nUth=Uac\n(1\u0000\u0018) +j\f; (20)\nwherej=p\u00001,!= 2\u0019f,\u0018=!2LC,\f=!R0C, and\nUac=Iread\u0001R. The output voltage and average power\nacross the load can then be obtained as\nUL=UthRL\nZth+RL=Uacr\n1 +jp+r(1\u0000\u0018+j\f);(21)\nand\nPL=1\n2jULj2\nRL=U2\nac\n2RLr2\n1 +qr2+ 2r+p2; (22)\nwherer=RL=R0,q= (1\u0000\u0018)2+\f2, andp=!L\nR0. Finally,\nthe e\u000eciency of the power extraction can be obtained as\n\u0010=PL\nPin\n=0:5r\n1 +qr2+ 2r+p21\nI2\nwriteRGenR0=U2ac+ 1;(23)\nwhereRGenis the resistance faced by the generation cur-\nrent. It can be observed from Eqs. (21)-(23) that the\noutput voltage, output power, and the e\u000eciency of power\nextraction decrease with an increase in frequency since \u0018,\n\f,q, andpincrease with ![69, 83].\nConsidering that the load impedance is \fxed to 50 \nby the external circuit, one can only optimize the source\nimpedance to achieve PL>1\u0016W andUL>1 mV. In\nthis regard, the resistance of the source tunnel barrier can11\nTABLE III. Material Parameters of the NM, and at the\nNM/AFM interface.\nParameters Values Ref.\ngM 3:8\u00021010S/m2[50]\ngm 3:8\u0002109S/m2[50]\n\u001aCu 6\u000210\u00009\n m2[50]\ntCu 5 nm [50]\nbe altered by either varying the thickness of the tunnel\nbarrier,db, or its cross-sectional area, Ac. However, the\noptimum values of dbandAcfor the desired output sig-\nnals is frequency dependent, and, therefore, tunnel barri-\ners of di\u000berent sizes would be required for di\u000berent oper-\nating frequencies [69, 83]. For all estimates, we consider\ndb= 1 nm,\u0011= 1:3,\u0014= 5:6 nm\u00001,RA(0) = 0:14 \n\u0016m2,\nand\u000f= 9:8 [69]. For reliable operation of the tunnel bar-\nrier, we consider the electric \feld across the barrier to be\nE= 0:3 V=nm [69], which is below the barrier break-\ndown \feld. Ignoring the e\u000bect of the generation current\nin Eq. (23), as suggested in Ref. [69], we deduce from\nFig. 7 that the optimal cross-sectional area Ac\u00190:36\n\u0016m2forf= 0:1 THz,Ac\u00190:25\u0016m2forf= 1 THz,\nAc\u00190:16\u0016m2forf= 10 THz.\nIrSnPtRhGeGaGaNNiN\nX10-710-510-3ζ(%)\nf=2.0THz\nFIG. 8. Power e\u000eciency for di\u000berent materials Mn 3X listed in\nTable II. The dashed horizontal line shows the expected e\u000e-\nciency of\u0011= 0:011% for the optimized geometry ( db= 1 nm,\nandAc= 0:24\u0016m2) if write current is neglected. The e\u000e-\nciency, however, decreases signi\fcantly due to the inclusion\nof write current. Here the AFM thin-\flm thickness dais as-\nsumed to be 4 nm.\nTable II lists the material properties of various con-\nducting AFMs. Depending on the sign of their DMI\nconstant, these AFMs could host moments with either\na positive or a negative chirality. The closed-form model\npresented in Eq. (17) can be used to evaluate the re-\nquired spin current for frequency f= 2 THz, regardless\nof the chirality since frequency scales linearly with theinput current in this region (see Fig. 3). For a given spin\ncurrent density ( Js), the charge current density ( Jwrite)\nfor the lateral spin-valve structure of Fig. 1(a) is given as\nJwrite =gM+gm\ngM\u0000gmJs: (24)\nwheregMandgmare the conductance of the majority-\nand minority-spin electrons at the NM (Cu)/FM inter-\nface. The input power required to start the oscilla-\ntions is given as ( JwriteAc)2RCu, whereRGen=RCu=\n\u001aCuLCu\nACu=\u001aCupAc\ntCupAcis the resistance of the copper\n(NM) underneath the bottom MgO. In order to evaluate\nthe resistance of the copper layer, we have assumed its\nlength and width to be the same as MgO and the AFM\nthin-\flm.\nThe e\u000eciency of power extraction for the listed AFM\nmaterials is presented in Fig. 8. The dashed horizon-\ntal line denotes the expected e\u000eciency of \u0011= 0:011% if\nthe e\u000bect of generation current is neglected and the area\nof cross-section of MgO is optimized for f= 2:0 THz.\nHowever, it can be observed that the e\u000eciency decreases\nsigni\fcantly i.e. by a few orders when the input power\ndue to the generation current in included in the analy-\nsis. For materials with large damping and large uniaxial\nanisotropy constants, the required generation current is\nhigher leading to lower e\u000eciency. This result shows that\nfurther optimization of the device geometry for di\u000berent\nmaterials is required to increase the e\u000eciency.\nThis method of power extraction could be more suit-\nable for materials with negative chirality. We can observe\nfrom Fig. 9 that the output power as well as the e\u000eciency\nfor both Mn 3Sn and Mn 3Ge for frequencies between 0.1\nTHz and 2.0 THz are signi\fcant. The required genera-\ntion current for Mn 3Ge is smaller than that for Mn 3Sn,\ntherefore, the e\u000eciency is higher for the former. Also,\nthe e\u000eciency of power extraction increases with decrease\nin area of cross-section in both the cases but this is ac-\ncompanied by a decrease in output power.\nIt might be possible to increase the output power and\noverall e\u000eciency of the system if the material properties\nof the tunnel barrier such as \u0011;\u0014, andRA(0) could be\naltered. Large room temperature tunneling magnetore-\nsistance in an ATJ is feasible either by using a tunnel\nbarrier other than MgO [84] or inserting a di\u000busion bar-\nrier to enhance magneto-transport [85]. Here we adopted\nthe TAMR extraction scheme because we have consid-\nered metallic AFMs so a DC current through the ATJ\nstructure can be easily applied. In addition, the three-\nor four-terminal compact ATJ structure along with its\nsmall lateral size enables dense packing of several such\nTHz oscillators on a chip accompanied with a net in-\ncrease in the output power and e\u000eciency of the oscillator\narray [72]. For example, with an array of 10 \u000210 such\nAFM oscillators excited in parallel, the output power and\ne\u000eciency could be scaled up by 100 \u0002compared to the\nresults presented in Figs. 7 and 8.12\n(a) (b)\n(c) (d)\nFIG. 9. Upper panel: (a) Output power and (b) e\u000eciency for\nMn3Sn. Lower panel: (c) Output power and (d) e\u000eciency for\nMn3Ge. Two di\u000berent cross-section size of the MgO barrier is\nconsidered. The output power depends only on the frequency\nof oscillation and therefore is same for both the materials.\nThe e\u000eciency of power extraction depends on the generation\ncurrent, which is lower for Mn 3Ge, leading to a higher value\nof e\u000eciency in that case.\nV. EFFECTS OF INHOMOGENEITY DUE TO\nEXCHANGE INTERACTION\nThe results presented in Section III C correspond to\nthe case of a single-domain AFM particle and are, there-\nfore, independent of the lateral dimensions of the thin-\n\flm. This can also be deduced from the equations of the\nthreshold current and the average oscillation frequency.\nHowever, when the lateral dimensions of the AFM thin-\n\flm exceed several 10's of nm, micromagnetic analysis\nmust be carried out. In this section, we analyze the dy-\nnamics in thin-\flm AFMs of varying dimensions within a\nmicromagnetic simulation framework. We consider AFM\nthin-\flms of dimensions 50 nm \u000250 nm and investigate\nthe e\u000bect of the inhomogeneity due to exchange interac-\ntions. In each case, the thin-\flm was divided into smaller\ncubes, each of size 1 nm \u00021 nm\u0002danm, since the do-\nmain wall width \u0001 0=p\n(2Aii\u0000Aij)=(2Ke)>1 nm for\nKecorresponding to Mn 3Ir as listed in Table II. It can\nbe observed from Fig. 10 that for materials with positive\nchirality the e\u000bects of inhomogeneity becomes important\nfor low currents. On the other hand, for materials with\nnegative chirality, inhomogeneities do not appear to have\nany e\u000bect. For positive chirality materials, the numerical\nvalues of frequency for di\u000berent spring constants deviates\nsigni\fcantly from that obtained from the single domain\nsolution, as well as analytic results. In this case, the hys-\nteretic region reduces in size since the lower threshold\ncurrent increases in magnitude as compared to the the-\noretical prediction as can be observed from Fig. 10(a).\nPositive Chirality Negative Chirality\n(a) (b)\n(c)                                               (d)FIG. 10. Frequency vs. input current for di\u000berent values\nof inhomogeneous exchange constants (intra-sublattice (a, c),\ninter-sublattice (b, d)) for both positive and negative chirality.\nIn all cases \u000b= 0:01, andda= 4 nm. Other parameters\ncorrespond to those of Mn 3Ir as listed in Table II for both\npositive and negative chirality materials with the exception\nof the sign ofDfor the latter.\nWhile we have not included the e\u000bect of inhomogeneous\nDMI in our work, we expect such interactions to lead to\nthe formation of domain walls in the thin-\flm similar to\nthe case of collinear AFMs [43]. A more detailed analy-\nsis of the dynamics of the positive chirality materials due\nto variation in exchange interaction as well as inhomoge-\nneous DMI would be carried out in a future publication.\nVI. DISCUSSION\nWe focused on the dynamics of the order parameters\nin exchange dominant non-collinear coplanar AFMs with\nboth positive (+ \u0019=2) and negative (\u0000\u0019=2) chiralities as-\nsociated to the orientation of equilibrium magnetization\nvectors. In both these classes of AFMs, the exchange en-\nergy is minimized for a 2 \u0019=3 relative orientation between\nthe sublattice vectors. Next, the negative (positive) sign\nof the iDMI coe\u000ecient minimizes the system energy for\ncounterclockwise (clockwise) ordering of m1;m2, andm3\nin the x\u0000yplane leading to positive (negative) chiral-\nity. Finally, all the sublattice vectors coincide with their\nrespective easy axis only in the case of the positive chiral-\nity materials due to the relative anticlockwise orientation\nof the easy axes. On the other hand, the negative chi-\nrality materials have a six-fold symmetry wherein only\none of the sublattice vectors can coincide with its respec-\ntive easy axis. As a result, these AFM materials with\ndi\u000berent chiralities have signi\fcantly distinct dynamics\nin the presence of an input spin current. For AFM ma-\nterials with + \u0019=2 chirality, oscillatory dynamics are ex-13\ncited only when the injected spin current overcomes the\nanisotropy, thus indicating the presence of a larger cur-\nrent threshold. Moreover, the dynamics in such AFMs is\nhysteretic in nature. Therefore, it is possible to sustain\noscillations by lowering the current below that required\nto initiate the dynamics as long the energy pumped in by\nthe current overcomes that dissipated by damping. On\nthe other hand, in the case of \u0000\u0019=2 chirality AFMs, os-\ncillations can be excited when signi\fcantly smaller spin\ncurrent with appropriate spin polarization is injected into\nthe AFM. Hence, \u0000\u0019=2 chirality AFMs may be more\namenable to tuning the frequency response over a broad\nfrequency range, from the MHz to the THz range [68].\nThe oscillation of the AFM N\u0013 eel vectors can be mea-\nsured as a coherent AC voltage with THz frequencies\nacross an externally connected resistive load through the\ntunnel anisotropic magnetoresistance measurements for\nboth +\u0019=2 and\u0000\u0019=2 chirality materials. In general, as\nthe frequency increases, the magnitude of both the out-\nput power and the e\u000eciency of power extraction decrease,\nhowever, it is possible to enhance both these quantities\nby optimizing the cross-sectional area of the tunnel junc-\ntion. This, however, is limited due to larger threshold\ncurrent requirement for materials with large damping.\nTherefore, a hybrid scheme of electrically synchronized\nAFM oscillators on a chip could be used to further en-\nhance the power and e\u000eciency [86, 87].\n(a) (b)\nFIG. 11.mzfor larger damping, \u000b= 0:1, andda= 4 nm. (a)\nNon-coherent (spike-like) signals near the threshold current\nJs= 1:1Jth1\ns. (b) Coherent signal for larger current Js=\n1:5Jth1\ns. The angular frequency is directly proportional to\nmz, and therefore it would show the exact same features (in\nthe absence of any external \feld) for the chosen values of\ncurrent.\nMetallic AFMs such as Mn 3Ir and Mn 3Sn could be\nconsidered as examples of + \u0019=2 and\u0000\u0019=2 chiralities,\nrespectively. Recently, thin-\flms with di\u000berent thickness\nranging from 1 nm to 5 nm of both these materials have\nbeen grown using UHV magnetron sputtering [88{91].\nIn addition, di\u000berent values of damping constants have\nbeen reported for Mn 3Sn [56, 77]. Therefore, we expect\nthe results presented in Sections III C, IV and V to be\nuseful for benchmarking THz dynamics in experimental\nset-ups with such thin \flms metallic antiferromagnets.\n(a) (b)\n(c) (d)FIG. 12. Time dynamics (single and train of spikes) of a single\n\\neuron\" for di\u000berent input currents and frequencies. The net\ninput current should be greater than the threshold current\n(\u0015 > 0:2) for a non-zero dynamics. For an input current\nabove the threshold, as the external frequency increases the\ndynamics changes from (a) bursts of spikes to (c) single spikes\nto (d) no spikes ( \u0015= 0:3). As the input current increases to\n\u0015= 0:4 the range of external frequency where the spiking\nbehaviour is observed increases.\nVII. POTENTIAL APPLICATIONS\nNeurons in the human brain could be thought of as a\nnetwork of coupled non-linear oscillators, while the stim-\nuli to excite neuronal dynamics is derived from the neigh-\nboring neurons in the network [8, 92{94]. For materials\nwith +\u0019=2 chirality, a non-linear behaviour was observed\nfor large damping, and input currents near the threshold\ncurrent,Jth1\ns, in Fig. 3(b), (f). This non-linearity cor-\nresponds to Dirac-comb-like magnetization dynamics, as\nshown in Fig. 11(a), and is similar to the dynamics of\nbiological neurons in their spiking behaviour as well as\na dependence on the input threshold. However, unlike a\nbiological neuron which shows various dynamical modes\nsuch as spiking, bursting, and chattering [95], the dy-\nnamics here shows only spikes and does not show any\nrefractory (\\resting\") period. Recent works [96, 97] have\nshown that it is possible to generate single spiking as\nwell as bursting behaviours using NiO-based AFM oscil-\nlators by considering an input DC current below Jth1\ns,\nand superimposing it with an AC current. As the AC\ncurrent changes with time, the total current could either\ngo above the threshold, thereby triggering a non-linear\nresponse, or below the threshold current resulting in a\n\\resting\" period. Here we explore the possibility of spik-\ning behaviours in + \u0019=2 chirality materials such as Mn 3Ir\nunder the e\u000bect of an input spin current. We use the non-\nlinear pendulum model of Eq. (12) and study the possible\ndynamics in case of a single oscillator, two unidirectional\ncoupled oscillators, and two bidirectional coupled oscil-14\nlators.\nA. Ultra-fast Hardware Emulator of Neurons\nWe consider a large damping of \u000b= 0:1 while the other\nmaterial parameters correspond to that of Mn 3Ir as listed\nin Table II. Next we choose an input current Js(t) =\nJdc\ns+Jac\ns(t), whereJdc\ns= 0:8Jth1\nsis the dc component of\nthe input current, superimposed with a smaller ac signal\nJac\ns(t) =\u0015Jth1\nscos(2\u0019fact). The time dynamics of this\nnon-linear oscillator is governed by\n'+\u000b!E_'+!E!K\n2sin 2'+!E!s(t) = 0; (25)\nwhere!s(t)(/Js(t)) is the time varying input current.\n(a) (b)\nFIG. 13. The dynamics of two neuron system with unidirec-\ntional coupling at fac= 60 GHz, and \u0015= 0:3. The dotted\nblue curve corresponds to the \frst neuron. (a) Second neuron\nshows no spike for \u0014= 0:028 but a single spike for \u0014= 0:032.\n(b) The single spiking behaviour changes to bursts with three\nspikes as\u0014increases and coupling strengthens.\nFigure 12 presents the dynamics of Eq. (25) for dif-\nferent input current and frequencies. Firstly, it can be\nobserved that the input current must be greater than\nthe threshold current to excite any dynamics viz. \u0015\nmust be greater than 0.2 (dotted line corresponding to\n\u0015= 0:2 shows no spikes for any value of external fre-\nquency). Secondly, for input currents above the thresh-\nold viz.\u0015=f0:3;0:4g, a train of spikes is observed for\nlower frequency of 20 GHz in Fig. 12(a). However, as\nfrequency of the input excitation increases the number of\nobserved spikes decreases for both values of current con-\nsidered here (Fig. 12(b, c)). Finally, it can be observed\nfrom Fig. 12(d) that for very large frequency the spiking\nbehaviour vanishes for lower current ( \u0015= 0:3) but per-\nsists for higher current ( \u0015= 0:4). For higher values of\ncurrent, the cut-o\u000b frequency is higher. This observed\nspiking behaviour is indeed similar to that of biological\nneurons [95]. Here, however, the observed dynamics is\nvery fast in the THz regime and thus the AFM oscilla-\ntors could be used as the building blocks of an ultra-high\nthroughput brain-inspired computing architecture.\n(a)\n(b)\n(c)\n(d)FIG. 14. The dynamics of two neuron system with unidirec-\ntional coupling at fac= 180 GHz. The dashed blue curve\ncorresponds to the \frst neuron. \u0015= 0:3: (a) Single spike\nfor\u0014= 0:04 but not for \u0014= 0:036. (b) The single spik-\ning behaviour become prominent as the coupling strengthens.\n\u0015= 0:4: (c) Single spike for \u0014= 0:032 in response to a double\nspiking behaviour of the \frst neuron. (d) For larger \u0014second\nneuron shows bursting dynamics with two spikes.\nB. Two unidirectional coupled arti\fcial neurons\nA network composed of interacting oscillators forms\nthe basis of the oscillatory neurocomputing model pro-\nposed by Hoppensteadt and Izhikevich [98]. In such a\nnetwork, the dynamics of an oscillating neuron (or a\n\\node\") is controlled by the incoming input signal as\nwell as its coupling to neighboring neurons. To inves-\ntigate this coupling behaviour we consider a system of\ntwo unidirectional coupled neurons. The \frst neuron is\ndriven by an external signal and its dynamics is governed\nby Eq. (25). The dynamics of the second neuron, on the\nother hand, depends on the output signal of the \frst neu-\nron as well as the coupling between the two neurons. It\nis governed by\n'j+\u000b!E_'j+!E!K\n2sin 2'j+!E!s\n\u0000\u0014ij!E_'isgn(!s) = 0;\n(26)\nwhere\u0014ij=\u0014is the unidirectional coupling coe\u000ecient\nfrom neuron i= 1 toj= 2. There is no feedback from\nthe second neuron to the \frst and therefore \u0014ji= 0. In\naddition to the input from the \frst neuron, the second\nneuron is also driven by a constant DC current Jdc\ns2(/!s\nin Eq. (26)). We choose this DC current to be the same\nas that for the \frst neuron viz. Jdc\ns2= 0:8Jth1\ns. The\ndynamics of the second neuron for two di\u000berent external\ninput currents ( \u0015=f0:3;0:4g) and frequencies ( fac=\nf60;180gGHz) is presented in Figs. 13 and 14.15\nFirstly, it can be observed that in all the cases the sec-\nond neuron shows a spiking behaviour only for \u0014above\na certain value. Secondly, for \u0015= 0:3 andfac= 60 GHz,\nwherein the \frst neuron shows bursting behaviour con-\nsisting of three spikes, the second neuron shows a single\nspike (Fig. 13(a)) for lower value of \u0014, and three spikes\nfor stronger coupling (Fig. 13(b)). This behaviour is due\nto the threshold dependence of the second neuron as well\nas due to its inertial dynamics. Similar behaviour is also\nobserved for \u0015= 0:4, andfac= 180 GHz in Figs. 14(c),\n(d). Thirdly, for \u0015= 0:3 andfac= 180 GHz, wherein the\n\frst neuron shows a single spike, Fig. 14(a) shows that\ncompared to the case of fac= 60 GHz a slightly higher\nvalue of\u0014is now required to excite the second neuron.\nThe single spiking behaviour of the second neuron be-\ncomes more prominent as the coupling strength increases\nbecause of a stronger input as shown in Fig. 14(b). Re-\ncently, it was suggested that this coupled behaviour of\nTHz arti\fcial neurons could be used to build ultra-fast\nmulti-input AND, OR, and majority logic gates [96].\n(a)\n(b)\n(c)\n(d)\nFIG. 15. The dynamics of two neuron system with bidirec-\ntional coupling at fac= 180 GHz, and \u0015= 0:3. First neu-\nron shows bursting behaviour in this system while the second\nneuron follows the \frst neuron for all values of \u0014. As the cou-\npling between the two neurons increase the number of spikes\nfor both the neurons increases.\nC. Two bidirectional coupled arti\fcial neurons\nIn some circuits it is possible that the coupling be-\ntween any two neurons is bidirectional. In such cases, in\naddition to a forward coupling from the \frst neuron to\nthe second, a feedback exists from the second neuron to\nthe \frst. The dynamics of each neuron of this coupled\nsystem is governed by Eq. (26), however, !s=!s(t) for\nthe \frst neuron, as discussed previously. We consider\n\u001412=\u001421=\u0014. Figures 15 and 16 show the dynamics of\nthe two neurons of this coupled system with the coupling\u0014atfac= 180 GHz for \u0015= 0:3 and 0.4, respectively.\n(a) (b)\n(c) (d)\nFIG. 16. The dynamics of two neuron system with bidirec-\ntional coupling at fac= 180 GHz, and \u0015= 0:4. Second\nneuron \fres when the coupling is above a certain threshold\nwhich in turn leads to another spike for the \frst neuron. As\nthe coupling between the two neurons increase the number of\nspikes for both the neurons increases.\nFirstly, Fig. 15(a) shows that for \u0014= 0:04 the dynam-\nics of both _'1and _'2are almost similar to that presented\nin Fig. 14(a), viz. the e\u000bects of coupling is very small.\nHowever, as the coupling between the two neurons in-\ncreases (Fig. 15(b)-(d)), a positive feedback is established\nbetween the two neuron leading to dynamics with two or\nmore spikes, in general. This is observed after the sec-\nond neuron has \fred, at least once, because the positive\nfeedback leads to a net input greater than the threshold\ncurrent to the \frst neuron, even though the external in-\nput has reduced below the threshold. Similar behavior is\nalso observed in the case of \u0015= 0:4, although at lower\nvalues of coupling, as presented in Fig. 16. The results\nallude to the threshold behaviour of the neurons, inertial\nnature of the dynamics, and a dependence of the dynam-\nics on the phase di\u000berence between the two neurons. The\ndynamics of two bidirectional coupled arti\fcial neurons\npresented here could be the \frst step towards building\nAFM-based recurrent neural networks or reservoir com-\nputing [99], instead of the slower FM-based coupled os-\ncillator systems [100, 101].\nVIII. CONCLUSION\nIn this work, we numerically and theoretically explore\nthe THz dynamics of thin-\flm metallic non-collinear\ncoplanar AFMs such as Mn 3Ir and Mn 3Sn, under the\naction of an injected spin current with spin polarization\nperpendicular to the plane of the \flm. Physically, these\ntwo AFM materials di\u000ber in their spin con\fguration viz.\npositive chirality for Mn 3Ir, and negative chirality for16\nMn3Sn. In order to explore the dynamics numerically,\nwe solve three LLG equations coupled to each other via\ninter-sublattice exchange interactions. We also analyze\nthe dynamics theoretically in the limit of strong exchange\nand show that it can be mapped to that of a damped-\ndriven pendulum if the e\u000bects of inhomogeneity in the\nmaterial are ignored. We \fnd that the dynamics of Mn 3Ir\nis best described by a non-linear pendulum equation and\nhas a hysteretic behaviour, while that of Mn 3Sn in the\nTHz regime is best described by a linear pendulum equa-\ntion and has a signi\fcantly small threshold for oscillation.\nThe hysteretic dynamics in the case of Mn 3Ir allows for\npossibility of energy e\u000ecient THz coherent sources. On\nthe other hand, a small threshold current requirement\nin the case of Mn 3Sn indicates the possibility of e\u000e-\ncient coherent signal sources from MHz to THz regime.\nWe employ the TAMR detection scheme to extract the\nTHz oscillations as time-varying voltage signals across\nan external resistive load. Including inhomogeneous ef-\nfects leads to a variation in the dynamics | the lowerthreshold current for sustaining the dynamics increases,\nthe hysteretic region reduces, and the frequency of oscil-\nlation decreases for lower current levels. Finally, we also\nshow that the non-linear behaviour of positive chirality\nmaterials with large damping could be used to emulate\narti\fcial neurons. An interacting network of such oscil-\nlators could enable the development of neurocomputing\ncircuits for various cognitive tasks. 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Rusconi5, 6\n1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.\n2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.\n3Max Planck Institute for the Science of Light, Staudtstra\u0019e 2, 91058 Erlangen, Germany\n4Department of Physics, University of Erlangen-N urnberg, Staudtstra\u0019e 7, 91058 Erlangen, Germany\n5Max-Planck-Institut f ur Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany.\n6Munich Center for Quantum Science and Technology,\nSchellingstrasse 4, D-80799 M unchen, Germany.\n(Dated: June 1, 2022)\nIn the absence of dissipation a non-rotating magnetic nanoparticle can be stably levitated in a\nstatic magnetic \feld as a consequence of the spin origin of its magnetization. Here we study the\ne\u000bects of dissipation on the stability of the system, considering the interaction with the background\ngas and the intrinsic Gilbert damping of magnetization dynamics. At large applied magnetic \felds\nwe identify magnetization switching induced by Gilbert damping as the key limiting factor for\nstable levitation. At low applied magnetic \felds and for small particle dimensions magnetization\nswitching is prevented due to the strong coupling of rotation and magnetization dynamics, and\nthe stability is mainly limited by the gas-induced dissipation. In the latter case, high vacuum\nshould be su\u000ecient to extend stable levitation over experimentally relevant timescales. Our results\ndemonstrate the possibility to experimentally observe the phenomenon of quantum spin stabilized\nmagnetic levitation.\nI. INTRODUCTION\nThe Einstein{de Haas [1, 2] and Barnett e\u000bects [3] are\nmacroscopic manifestations of the internal angular mo-\nmentum origin of magnetization: a change in the mag-\nnetization causes a change in the mechanical rotation\nand conversely. Because of the reduced moment of in-\nertia of levitated nano- to microscale particles, these ef-\nfects play a dominant role in the dynamics of such sys-\ntems [4{10]. This o\u000bers the possibility to harness these\ne\u000bects for a variety of applications such as precise magne-\ntometry [11{16], inertial sensing [17, 18], coherent spin-\nmechanical control [19, 20], and spin-mechanical cool-\ning [21, 22] among others. Notable in this context is\nthe possibility to stably levitate a ferromagnetic parti-\ncle in a static magnetic \feld in vacuum [23, 24]. Stable\nlevitation is enabled by the internal angular momentum\norigin of the magnetization which, even in the absence of\nmechanical rotation, provides the required angular mo-\nmentum to gyroscopically stabilize the system. Such a\nphenomenon, which we refer to as quantum spin stabi-\nlized levitation to distinguish it from the rotational stabi-\nlization of magnetic tops [25{27], relies on the conserva-\ntive interchange between internal and mechanical angular\nmomentum. Omnipresent dissipation, however, exerts\nadditional non-conservative torques on the system which\nmight alter the delicate gyroscopic stability [26, 28]. It\nthus remains to be determined if stable levitation can\nbe observed under realistic conditions, where dissipative\ne\u000bects cannot be neglected.\nIn this article, we address this question. Speci\f-\ncally, we consider the dynamics of a levitated magnetic\nnanoparticle (nanomagnet hereafter) in a static magnetic\n\feld in the presence of dissipation originating both fromthe collisions with the background gas and from the\nintrinsic damping of magnetization dynamics (Gilbert\ndamping) [29, 30], which are generally considered to be\nthe dominant sources of dissipation for levitated nano-\nmagnets [8, 13, 31{33]. Con\fned dynamics can be ob-\nserved only when the time over which the nanomagnet is\nlevitated is longer than the period of center-of-mass os-\ncillations in the magnetic trap. When this is the case, we\nde\fne the system to be metastable . We demonstrate that\nthe system can be metastable in experimentally feasible\nconditions, with the levitation time and the mechanism\nbehind the instability depending on the parameter regime\nof the system. In particular, we show that at weak ap-\nplied magnetic \felds and for small particle dimensions\n(to be precisely de\fned below) levitation time can be\nsigni\fcantly extended in high vacuum (i.e. pressures be-\nlow 10\u00003mbar). Our results evidence the potential of\nunambiguous experimental observation of quantum spin\nstabilized magnetic levitation.\nWe emphasize that our analysis is particularly timely.\nPresently there is a growing interest in levitating and con-\ntrolling magnetic systems in vacuum [9, 34, 35]. Current\nexperimental e\u000borts focus on levitation of charged para-\nmagnetic ensembles in a Paul trap [19, 36, 37], diamag-\nnetic particles in magneto-gravitational traps [38{40], or\nferromagnets above a superconductor [14, 20, 41]. Lev-\nitating ferromagnetic particles in a static magnetic trap\no\u000bers a viable alternative, with the possibility of reaching\nlarger mechanical trapping frequencies.\nThe article is organized as follows. In Sec. II we in-\ntroduce the model of the nanomagnet, and we de\fne\ntwo relevant regimes for metastability, namely the atom\nphase and the Einstein{de Haas phase. In Sec. III and\nIV we analyze the dynamics in the atom phase and thearXiv:2106.14858v3  [cond-mat.mes-hall]  31 May 20222\nFigure 1. (a) Illustration of a spheroidal nanomagnet levi-\ntated in an external \feld B(r) and surrounded by a gas at the\ntemperature Tand the pressure P. (b) Linear stability dia-\ngram of a non-rotating nanomagnet in the absence of dissipa-\ntion, assuming a= 2b. Blue and red regions denote the stable\natom and Einstein{de Haas phase, respectively; hatched area\nis the unstable region. Dashed lines show the critical values\nof the bias \feld which de\fne the two phases. In particular,\nBEdH,1\u00115\u0016=[4\r2\n0(a2+b2)M],BEdH,2\u00113 [\u0016B02=(4\r0M)]1=3,\nandBatom = 2kaV=\u0016. Numerical values of physical parame-\nters used to generate panel (b) are given in Table I.\nEinstein{de Haas phase, respectively. We discuss our re-\nsults in Sec. V. Conclusions and outlook are provided in\nSec. VI. Our work is complemented by three appendices\nwhere we de\fne the transformation between the body-\n\fxed and laboratory reference frames (App. A), analyze\nthe e\u000bect of thermal \ructuations (App. B), and provide\nadditional \fgures (App. C).\nII. DESCRIPTION OF THE SYSTEM\nWe consider a single domain nanomagnet levitated in\na static1magnetic \feld B(r) as shown schematically in\nFig. 1(a). We model the nanomagnet as a spheroidal\nrigid body of mass density \u001aMand semi-axes lengths a;b\n(a > b ), having uniaxial magnetocrystalline anisotropy,\nwith the anisotropy axis assumed to be along the major\nsemi-axisa[42]. Additionally, we assume that the mag-\nnetic response of the nanomagnet is approximated by a\npoint dipole with magnetic moment \u0016of constant mag-\nnitude\u0016\u0011j\u0016j, as it is often justi\fed for single domain\nparticles [42, 43]. Let us remark that such a simpli\fed\nmodel has been considered before to study the classical\ndynamics of nanomagnets in a viscous medium [31, 44{\n49], as well as to study the quantum dynamics of mag-\nnetic nanoparticles in vacuum [5, 13, 50, 51]. Since the\nmodel has been successful in describing the dynamics of\nsingle-domain nanomagnets, we adopt it here to inves-\ntigate the stability in a magnetic trap. In particular,\nour study has three main di\u000berences as compared with\n1We denote a \feld static if it does not have explicit time depen-\ndence, namely if @B(r)=@t= 0.Table I. Physical parameters of the model and the values used\nthroughout the article. We calculate the magnitude of the\nmagnetic moment as \u0016=\u001a\u0016V, where\u001a\u0016=\u001aM\u0016B=(50amu),\nwith\u0016Bthe Bohr magneton and amu the atomic mass unit.\nParameter Description Value [units]\n\u001aM mass density 104[kg m\u00003]\na;b semi-axes see main text [m]\n\u001a\u0016 magnetization 2 :2\u0002106[J T\u00001m\u00003]\nka anisotropy constant 105[J m\u00003]\n\r0 gyromagnetic ratio 1 :76\u00021011[rad s\u00001T\u00001]\nB0 \feld bias see main text [T]\nB0\feld gradient 104[T m\u00001]\nB00\feld curvature 106[T m\u00002]\n\u0011 Gilbert damping 10\u00002[n. u.]\nT temperature 10\u00001[K]\nP pressure 10\u00002[mbar]\nM molar mass 29 [g mol\u00001]\n\u000bc re\rection coe\u000ecient 1 [n. u.]\nprevious work. (i) We consider a particle levitated in\nhigh vacuum, where the mean free path of the gas parti-\ncles is larger than the nanomagnet dimensions (Knudsen\nregime [52]). This leads to gas damping which is gen-\nerally di\u000berent from the case of dense viscous medium\nmostly considered in the literature. (ii) We consider\ncenter-of-mass motion and its coupling to the rotational\nand magnetic degrees of freedom, while previous work\nmostly focuses on coupling between rotation and mag-\nnetization only (with the notable exception of [48]). (iii)\nWe are primarily interested in the center-of-mass con\fne-\nment of the particle, and not in its magnetic response.\nWithin this model the relevant degrees of freedom of\nthe system are the center-of-mass position r, the linear\nmomentum p, the mechanical angular momentum L, the\norientation of the nanomagnet in space \n, and the mag-\nnetic moment \u0016. The orientation of the nanomagnet\nis speci\fed by the body-\fxed reference frame Oe1e2e3,\nwhich is obtained from the laboratory frame Oexeyez\naccording to ( e1;e2;e3)T=R(\n)(ex;ey;ez)T, where\n\n= (\u000b;\f;\r )Tare the Euler angles and R(\n) is the\nrotational matrix. We provide the expression for R(\n)\nin App. A. The body-\fxed reference frame is chosen such\nthate3coincides with the anisotropy axis. The magnetic\nmoment\u0016is related to the internal angular momentum F\naccording to the gyromagnetic relation \u0016=\r0F, where\n\r0is the gyromagnetic ratio of the material2.\n2The total internal angular momentum Fis a sum of the individ-\nual atomic angular momenta (spin and orbital), which contribute\nto the atomic magnetic moment. For a single domain magnetic\nparticle, it is customary to assume that Fcan be described as\na vector of constant magnitude, jFj=\u0016=\r0(macrospin approxi-\nmation) [43].3\nA. Equations of Motion\nWe describe the dynamics of the nanomagnet in the\nmagnetic trap with a set of stochastic di\u000berential equa-\ntions which model both the deterministic dissipative evo-\nlution of the system and the random \ructuations due to\nthe environment. In the following it is convenient to de-\n\fne dimensionless variables: the center-of-mass variables\n~r\u0011r=a,~p\u0011\r0ap=\u0016, the mechanical angular momen-\ntum`\u0011\r0L=\u0016, the magnetic moment m\u0011\u0016=\u0016, and\nthe magnetic \feld b(~r)\u0011B(a~r)=B0, whereB0denotes\nthe minimum of the \feld intensity in a magnetic trap,\nwhich we hereafter refer to as the bias \feld. Note that\nwe choose to normalize the position r, the magnetic mo-\nment\u0016and the magnetic \feld B(r) with respect to the\nparticle size a, the magnetic moment magnitude \u0016, and\nthe bias \feld B0, respectively. The scaling factor for an-\ngular momentum, \u0016=\r0, and linear momentum, \u0016=(a\r0),\nfollow as a consequence of the gyromagnetic relation.\nThe dynamics of the nanomagnet in the laboratory\nframe are given by the equations of motion\n_~r=!I~p; (1)\n_e3=!\u0002e3; (2)\n_~p=!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p+\u001ep(t); (3)\n_`=!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`+\u0018l(t); (4)\n_m=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)\n+\u0010b(t)]: (5)\nHere we de\fne the relevant system frequencies: !I\u0011\n\u0016=(\r0Ma2) is the Einstein{de Haas frequency, with M\nthe mass of the nanomagnet, !L\u0011\r0B0is the Larmor\nfrequency,!A\u0011kaV\r0=\u0016is the anisotropy frequency,\nwithVthe volume of the nanomagnet and kathe ma-\nterial dependent anisotropy constant [43], !\u0011I\u00001L\nis the angular velocity, with Ithe tensor of inertia,\nand!e\u000b\u00112!A(m\u0001e3)e3+!Lb(~r). Dissipation is\nparametrized by the dimensionless Gilbert damping pa-\nrameter\u0011[29, 53], and the center-of-mass and rotational\nfriction tensors \u0000 cmand \u0000 rot, respectively [32]. The e\u000bect\nof stochastic thermal \ructuations is represented by the\nrandom variables \u001ep(t) and\u0018l(t) which describe, respec-\ntively, the \ructuating force and torque exerted by the\nsurrounding gas, and by \u0010b(t) which describes the ran-\ndom magnetic \feld accounting for thermal \ructuations\nin magnetization dynamics [54]. We assume Gaussian\nwhite noise, namely, for X(t)\u0011(\u001ep(t);\u0018l(t);\u0010b(t))Twe\nhavehXi(t)i= 0 andhXi(t)Xj(t0)i\u0018\u000eij\u000e(t\u0000t0).\nEquations (1-4) describe the center-of-mass and rota-\ntional dynamics of a rigid body in the presence of dis-\nsipation and noise induced by the background gas [32].\nThe expressions for \u0000 cmand \u0000 rotdepend on the parti-\ncle shape { here we take the expressions derived in [32]for a cylindrical particle3{, and on the ratio of the sur-\nface and the bulk temperature of the particle, which\nwe assume to be equal to the gas temperature T. Fur-\nthermore, they account for two di\u000berent scattering pro-\ncesses, namely the specular and the di\u000busive re\rection\nof the gas from the particle, which is described by a\nphenomenological interpolation coe\u000ecient \u000bc. The or-\nder of magnitude of the di\u000berent components of \u0000 cmand\n\u0000rotis generally well approximated by the dissipation\nrate \u0000\u0011(2Pab=M )[2\u0019M=(NAkBT)]1=2, wherePand\nMare, respectively, the gas pressure and molar mass,\nkBis the Boltzmann constant and NAis the Avogadro\nnumber. The magnetization dynamics given by Eq. (5)\nis the Landau-Lifshitz-Gilbert equation in the laboratory\nframe [8, 57], with the e\u000bective magnetic \feld !e\u000b=\r0.\nWe remark that Eqs. (1-5) describe the classical dynam-\nics of a levitated nanomagnet where the e\u000bect of the\nquantum spin origin of magnetization, namely the gy-\nromagnetic relation, is taken into account phenomeno-\nlogically by Eq. (5). This is equivalent to the equations\nof motion obtained from a quantum Hamiltonian in the\nmean-\feld approximation [24].\nLet us discuss the e\u000bect of thermal \ructuations on\nthe dynamics of the nanomagnet at subkelvin temper-\natures and in high vacuum. These conditions are com-\nmon in recent experiments with levitated particles [58{\n60]. The thermal \ructuations of magnetization dy-\nnamics, captured by the last term in Eq. (5), lead\nto thermally activated transition of the magnetic mo-\nment between the two stable orientations along the\nanisotropy axis [54, 61]. Such process can be quan-\nti\fed by the N\u0013 eel relaxation time, which is given by\n\u001cN\u0019(\u0019=! A)p\nkBT=(kaV)ekaV=(kBT). Thermal acti-\nvation can be neglected when \u001cNis larger than other\ntimescales of magnetization dynamics, namely the pre-\ncession timescale given by \u001cL\u00111=j!e\u000bj, and the Gilbert\ndamping timescale given by \u001cG\u00111=(\u0011j!e\u000bj). Con-\nsidering for simplicity j!e\u000bj\u00182!A, for a particle size\na= 2b= 1 nm and temperature T= 1 K, and the\nvalues of the remaining parameters as in Table I, the ra-\ntio of the timescales is of the order \u001cN=\u001cL\u0018103, and\nit is signi\fcantly increased for larger particle sizes and\nat smaller temperatures. We remark that, for the val-\nues considered in this article, \u001cNis much larger than the\nlongest dynamical timescale in Eqs. (1-5) which is associ-\nated with the motion along ex. Thermal activation of the\nmagnetic moment can therefore be safely neglected. The\nstochastic e\u000bects ascribed to the background gas, cap-\ntured by the last terms in Eqs. (3-4), are expected to be\nimportant at high temperatures (namely, a regime where\nMkBT\r2\n0a2=\u00162&1 [32]). At subkelvin temperatures and\nin high vacuum these \ructuations are weak and, con-\nsequently, they do not destroy the deterministic e\u000bects\n3The expressions for \u0000 cmand \u0000 rotfor a cylindrical particle\ncapture the order of magnitude of the dissipation rates for a\nspheroidal particle [55, 56].4\ncaptured by the remaining terms in Eqs. (1-5) [33]. In-\ndeed, for the values of parameters given in Table I and\nfora= 2b,MkBT\r2\n0a2=\u00162\u00190:8T=(a[nm]). For sub-\nkelvin temperatures and particle sizes a>1 nm, thermal\n\ructuations due to the background gas can therefore be\nsafely neglected.\nIn the following we thus neglect stochastic e\u000bects by\nsetting\u001ep=\u0018l=\u0010b= 0, and we consider only the de-\nterministic part of Eqs. (1-5) as an appropriate model\nfor the dynamics [8, 33, 54]. In App. B we carry out\nthe analysis of the dynamics including the e\u000bects of gas\n\ructuations in equations (1-5), and we show that the\nresults presented in the main text remain qualitatively\nvalid even in the presence of thermal noise. For the mag-\nnetic \feld B(r) we hereafter consider a Io\u000be-Pritchard\nmagnetic trap, given by\nB(r) =ex\u0014\nB0+B00\n2\u0012\nx2\u0000y2+z2\n2\u0013\u0015\n\u0000ey\u0012\nB0y+B00\n2xy\u0013\n+ez\u0012\nB0z\u0000B00\n2xz\u0013\n;(6)\nwhereB0;B0andB00are, respectively, the \feld bias, gra-\ndient and curvature [62]. We remark that this is not a\nfundamental choice, and di\u000berent magnetic traps, pro-\nvided they have a non-zero bias \feld, should result in\nsimilar qualitative behavior.\nB. Initial conditions\nThe initial conditions for the dynamics in Eqs. (1-5),\nnamely at time t= 0, depend on the initial state of the\nsystem, which is determined by the preparation of the\nnanomagnet in the magnetic trap. In our analysis, we\nconsider the nanomagnet to be prepared in the thermal\nstate of an auxiliary loading potential at the temperature\nT. Subsequently, we assume to switch o\u000b the loading\npotential at t= 0, while at the same time switching\non the Io\u000be-Pritchard magnetic trap. The choice of the\nauxiliary potential is determined by two features: (i) it\nallows us to simply parametrize the initial conditions by a\nsingle parameter, namely the temperature T, and (ii) it is\nan adequate approximation of general trapping schemes\nused to trap magnetic particles.\nRegarding point (i), we assume that the particle is lev-\nitated in a harmonic trap, in the presence of an external\nmagnetic \feld applied along ex. This loading scheme\nprovides, on the one hand, trapping of the center-of-mass\ndegrees of freedom, with trapping frequencies denoted by\n!i(i=x;y;z ). On the other hand, the magnetic moment\nin this case is polarized along ex. The Hamiltonian of the\nsystem in such a con\fguration reads Haux=p2=(2M) +P\ni=x;y;zM!2\nir2\ni=2+LI\u00001L=2\u0000kaVe2\n3;x\u0000\u0016xBaux, where\nBauxdenotes the magnitude of the external magnetic\n\feld, which we for simplicity set to Baux=B0in all our\nsimulations. At t= 0 the particle is released in the mag-\nnetic trap given by Eq. (6). For the degrees of freedomx\u0011(~r;~p;`;mx)T, we take as the initial displacement\nfrom the equilibrium the corresponding standard devia-\ntion in a thermal state of Haux. More precisely, xi(0) =\nxi;e+ (hx2\nii\u0000hxii2)1=2, wherexi;edenotes the equilib-\nrium value, andhxk\nii\u0011Z\u00001R\ndxxk\niexp[\u0000Haux=(kBT)],\nwithk= 1;2 and the partition function Z. For the Eu-\nler angles \nwe use \n 1(0)\u0011cos\u00001[\u0000p\nhcos2\n1i] and\n\ni(0)\u0011cos\u00001[p\nhcos2\nii] (i= 2;3). The initial condi-\ntions for e3follow from \nusing the transformation given\nin App. A.\nRegarding point (ii), the initial conditions obtained in\nthis way describe a trapped particle prepared in a ther-\nmal equilibrium in the presence of an external loading\npotential where the center of mass is decoupled from the\nmagnetization and the rotational dynamics. It is outside\nthe scope of this article to study in detail a particular\nloading scheme. However, we point out that an auxil-\niary potential given by Hauxcan be obtained, for exam-\nple, by trapping the nanomagnet using a Paul trap as\ndemonstrated in recent experiments [19, 21, 37, 63{70].\nIn particular, trapping of a ferromagnetic particle has\nbeen demonstrated in a Paul trap at P= 10\u00002mbar,\nwith center-of-mass trapping frequency of up to 1 MHz,\nand alignment of the particle along the direction of an\napplied \feld [19]. We note that particles are shown to\nremain trapped even when the magnetic \feld is varied\nover many orders of magnitudes or switched o\u000b. We re-\nmark further that alignment of elongated particles can\nbe achieved using a quadrupole Paul trap even in the\nabsence of magnetic \feld [55, 71].\nC. Linear stability\nIn the absence of thermal \ructuations, an equilibrium\nsolution of Eqs. (1-5) is given by ~re=~pe=`e= 0 and\ne3;e=me=\u0000ex. This corresponds to the con\fguration\nin which the nanomagnet is \fxed at the trap center, with\nthe magnetic moment along the anisotropy axis and anti-\naligned to the bias \feld B0. Linear stability analysis of\nEqs. (1-5) shows that the system is unstable, as expected\nfor a gyroscopic system in the presence of dissipation [28].\nHowever, when the nanomagnet is metastable, it is still\npossible for it to levitate for an extended time before\nbeing eventually lost from the trap, as in the case of a\nclassical magnetic top [25{27]. As we show in the fol-\nlowing sections, the dynamics of the system, and thus its\nmetastability, strongly depend on the applied bias \feld\nB0. We identify two relevant regimes: (i) strong-\feld\nregime, de\fned by bias \feld values B0> B atom, and\n(ii) weak-\feld regime, de\fned by B0< B atom, where\nBatom\u00112kaV=\u0016. This di\u000berence is reminiscent of the\ntwo di\u000berent stable regions which arise as a function of\nB0in the linear stability diagram in the absence of dis-\nsipation [see Fig. 1(b)] [23, 24]. In Sec. III and Sec. IV\nwe investigate the possibility of metastable levitation by\nsolving numerically Eqs. (1-5) in the strong-\feld and\nweak-\feld regime, respectively.5\nIII. DYNAMICS IN THE STRONG-FIELD\nREGIME: ATOM PHASE\nThe strong-\feld regime, according to the de\fnition\ngiven in Sec. II C, corresponds to the blue region in the\nlinear stability diagram in the absence of dissipation,\nshown in Fig. 1(b). This region is named atom phase\nin [23, 24], and we hereafter refer to the strong-\feld\nregime as the atom phase. This parameter regime corre-\nsponds to the condition !L\u001d!A;!I. In this regime, the\ncoupling of the magnetic moment \u0016and the anisotropy\naxise3is negligible, and, to \frst approximation, the\nnanomagnet undergoes a free Larmor precession about\nthe local magnetic \feld. In the absence of dissipation,\nthis stabilizes the system in full analogy to magnetic trap-\nping of neutral atoms [72, 73].\nIn Fig. 2(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 20\nnm and the bias \feld B0= 200 mT. As evidenced by\nFig. 2(a), the magnetization mxof the particle changes\ndirection. During this change, the mechanical angu-\nlar momentum lxchanges accordingly in the manifesta-\ntion of the Einstein{de Haas e\u000bect, such that the to-\ntal angular momentum m+`is conserved4. The dy-\nnamics observed in Fig. 2(a) is indicative of Gilbert-\ndamping-induced magnetization switching, a well-known\nphenomenon in which the projection of the magnetic mo-\nment along the e\u000bective magnetic \feld !e\u000b=\r0changes\nsign [30]. This is expected to happen when the applied\nbias \feldB0is larger than the e\u000bective magnetic \feld\nassociated with the anisotropy, given by \u0018!A=\r0. Mag-\nnetization switching displaces the system from its equi-\nlibrium position on a timescale which is much shorter\nthan the period of center-of-mass oscillations, estimated\nfrom [24] to be \u001ccm\u00181\u0016s. The nanomagnet thus shows\nno signature of con\fnement [see Fig. 2(b)].\nThe timescale of levitation in the atom phase is given\nby the timescale of magnetization switching, which we\nestimate as follows. As evidenced by Fig. 2(a-b), the\ndynamics of the center of mass and the anisotropy axis\nare approximately constant during switching, such that\n!e\u000b\u0019!e\u000b(t= 0). Under this approximation and as-\nsuming\u0011\u001c1, the magnetic moment projection mk\u0011\n!e\u000b\u0001m=j!e\u000bjevolves as\n_mk\u0019\u0011[!L+ 2!Amk](1\u0000m2\nk): (7)\nAccording to Eq. (7) the component mkexhibits switch-\ning ifmk(t= 0)&\u00001 and!L=2!A>1 [30], both of\nwhich are ful\flled in the atom phase. Integrating Eq. (7)\nwe obtain the switching time \u001c[de\fned as mk(\u001c)\u00110],\nwhich can be well approximated by\n\u001c\u0019ln\u0000\n1 +jmk(t= 0)j\u0001\n2\u0011(!L+ 2!A)\u0000ln\u0000\n1\u0000jmk(t= 0)j\u0001\n2\u0011(!L\u00002!A):(8)\n4We always \fnd the transfer of angular momentum to the center\nof mass angular momentum r\u0002pto be negligible.\nFigure 2. Dynamics in the atom phase. (a) Dynamics of\nthe magnetic moment component mx, the mechanical angular\nmomentum component lx, and the anisotropy axis component\ne3;xfor nanomagnet dimensions a= 2b= 20 nm and the bias\n\feldB0= 200 mT. For the initial conditions we consider\ntrapping frequencies !x= 2\u0019\u00022 kHz and!y=!z= 2\u0019\u000250\nkHz. Unless otherwise stated, for the remaining parameters\nthe numerical values are given in Table I. (b) Center-of-mass\ndynamics for the same case considered in (a). (c) Dynamics of\nthe magnetic moment component mk. Line denoted by circle\ncorresponds to the case considered in (a). Each remaining\nline di\u000bers by a single parameter, as denoted by the legend.\nDotted vertical lines show Eq. (8). (d) Switching time given\nby Eq. (8) as a function of the bias \feld B0and the major\nsemi-axisa. In the region left of the thick dashed line the\ndeviation from the exact value is more than 5%. Hatched\narea is the unstable region in the linear stability diagram in\nFig. 1.(b).\nThe estimation Eq. (8) is in excellent agreement with\nthe numerical results for di\u000berent parameter values [see\nFig. 2(c)].\nMagnetization switching characterizes the dynamics of\nthe system in the entire atom phase. In particular, in\nFig. 2(d) we analyze the validity of Eq. (8) for di\u000berent\nvalues of the bias \feld B0and the major semi-axis a, as-\nsumingb=a=2. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time, as\nestimated from the full dynamics of the system, by 5%;\nleft of this line the deviation becomes increasingly more\nsigni\fcant, with Eq. (8) predicting up to 20% larger val-\nues close to the stability border (namely, for bias \feld\nclose toBatom = 90 mT). We believe that the signi\f-\ncant deviation close to the border of the atom phase is\ndue to the non-negligible coupling to the anisotropy axis,6\nFigure 3. Dynamics in the Einstein{de Haas phase. (a) Motion of the system in the ey-ezplane until time t= 5\u0016s for\nnanomagnet dimensions a= 2b= 2 nm and the bias \feld B0= 0:5 mT. For the initial conditions we consider trapping\nfrequencies !x= 2\u0019\u00022 kHz and !y=!z= 2\u0019\u00021 MHz. For the remaining parameters the numerical values are given\nin Table I. (b) Dynamics of the projection mkand (c) dynamics of the anisotropy axis component e3;x, for the same case\nconsidered in (a). (d) Dynamics of the center-of-mass component ryand (e) dynamics of the magnetic moment component\nmxon a longer timescale, for the same values of parameters as in (a). (f) Escape time t?as a function of gas pressure P, for\ndi\u000berent con\fgurations in the Einstein{de Haas phase. Circles correspond to the case considered in (a). Each remaining case\ndi\u000bers by parameters indicated by the legend. (g) Escape time t?as a function of the major semi-axis a, with the values of\nthe remaining parameters as in (a). Dashed vertical line denotes the upper limit of the Einstein{de Haas phase, given by the\ncritical \feld BEdH,1 [see Fig. 1(b)].\nwhich results in additional mechanisms not captured by\nthe simple model Eq. (7). In fact, it is known that cou-\npling between magnetization and mechanical degrees of\nfreedom might have an impact on the switching dynam-\nics [74]. As demonstrated by Fig. 2(d), the switching\ntime is always shorter than the center-of-mass oscillation\nperiod\u001ccm, and thus no metastability can be observed in\nthe atom phase.\nLet us note that the conclusions we draw in Fig. 2\nremain valid if one varies the anisotropy constant ka,\nGilbert damping parameter \u0011, and the temperature T,\nas we show in App. C. Finally, we note that the dis-\nsipation due to the background gas has negligible ef-\nfects. In particular, for the values assumed in Fig. 2(a-b)\nthe timescale of the gas-induced dissipation is given by\n1=\u0000 = 440\u0016s.\nIV. DYNAMICS IN THE WEAK-FIELD\nREGIME: EINSTEIN{DE HAAS PHASE\nWe now focus on the regime of weak bias \feld, cor-\nresponding to the condition !L\u001c!A. In this regime\nmagnetization switching does not occur, and the dynam-\nics critically depend on the particle size. In the follow-\ning we focus on the regime of small particle dimensions,i.e.!L\u001c!I, which, as we will show, is bene\fcial for\nmetastability. In the absence of dissipation, this regime\ncorresponds to the Einstein{de Haas phase [red region\nin Fig. 1(b)] [23, 24]. The hierarchy of energy scales in\nthe Einstein{de Haas phase (namely, !L\u001c!A;!I) man-\nifests in two ways: (i) the anisotropy is strong enough to\ne\u000bectively \\lock\" the direction of the magnetic moment \u0016\nalong the anisotropy axis e3(!A\u001d!L), and (ii) accord-\ning to the Einstein{de Haas e\u000bect, the frequency at which\nthe nanomagnet would rotate if \u0016switched direction is\nsigni\fcantly increased at small dimensions ( !I\u001d!L),\nsuch that switching can be prevented due to energy con-\nservation [4]. In the absence of dissipation, the combina-\ntion of these two e\u000bects stabilizes the system.\nIn Fig. 3(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 2 nm\nand the bias \feld B0= 0:5 mT. The nanomagnet is\nmetastable, as evidenced by the con\fned center-of-mass\nmotion shown in Fig. 3(a). In Fig. 3(b-c) we show the\ndynamics of the magnetic moment component mkand\nthe anisotropy axis component e3;x, respectively, which\nindicates that no magnetization switching occurs in this\nregime. We remark that the absence of switching can-\nnot be simply explained on the basis of Eqs. (7-8). In\nfact, the simple model of magnetization switching, given7\nby Eq. (7), assumes that the dynamics of the rotation\nand the center-of-mass motion happen on a much longer\ntimescale than the timescale of magnetization dynam-\nics. However, in this case rotation and magnetization\ndynamics occur on a comparable timescale, as evidenced\nby Fig. 3(b-c). The weak-\feld condition alone ( !L\u001c!A)\nis thus not su\u000ecient to correctly explain the absence of\nswitching, and the role of particle size ( !L\u001c!I) needs\nto be considered.\nLet us analyze the role of Gilbert damping in this case.\nSince in the Einstein{de Haas phase mk\u00181, we de\fne\nm\u0011e3+\u000em, where\u000emrepresents the deviation of m\nfrom the anisotropy axis e3, and we assumej\u000emj\u001cje3j\n[see Fig. 3(b)]. This allows us to simplify Eq. (5) as\n\u000e_m\u0019!e\u000b\u0002\u000em\u0000\u0011[2!A+!3e3\u0001(m+`)]\u000em;(9)\nwhere!3\u0011\u0016=(\r0I3), withI3the principal moment of\ninertia along e3. As evidenced by Eq. (9), the only e\u000bect\nof Gilbert damping is to align mande3on a timescale\ngiven by\u001c0\u00111=[\u0011(2!A+!3)], irrespective of the dy-\nnamics of e3. For the values of parameters considered in\nFig. 3(a-c), \u001c0= 5 ns, and it is much shorter than the\ntimescale of center-of-mass dynamics, given by \u001ccm\u00181\n\u0016s. For all practical purposes, the magnetization in the\nEinstein{de Haas phase can be considered frozen along\nthe anisotropy axis. The nanomagnet in the presence of\nGilbert damping is therefore equivalent to a hard magnet\n(i. e.ka!1 ) [24].\nThe main mechanism behind the instability in the\nEinstein{de Haas phase is thus gas-induced dissipation.\nIn Fig. 3(d-e) we plot the dynamics of the center-of-\nmass component ryand the magnetic moment compo-\nnentmxon a longer timescale, for two di\u000berent values of\nthe pressure P. The e\u000bect of gas-induced dissipation is\nto dampen the center-of-mass motion to the equilibrium\nposition, while the magnetic moment moves away from\nthe equilibrium. Both processes happen on a timescale\ngiven by the dissipation rate \u0000. When ex=mx\u00190, the\nsystem becomes unstable and ultimately leaves the trap\n[see arrow in Fig. 3(d)]. We de\fne the escape time t?as\nthe time at which the particle position is y(t?)\u00115y(0),\nand we show it in Fig. 3(f) as a function of pressure Pfor\ndi\u000berent con\fgurations in the Einstein{de Haas phase,\nand forb=a=2. Fig. 3(f) con\frms that the dissipation\na\u000bects the system on a timescale which scales as \u00181=P.\nThe metastability of the nanomagnet in the Einstein{de\nHaas phase is therefore limited solely by the gas-induced\ndissipation, which can be signi\fcantly reduced in high\nvacuum. Finally, in Fig. 3(g) we analyze the e\u000bect of\nparticle size on metastability. Speci\fcally, we show the\nescape time t?as a function of the major semi-axis aat\nthe bias \feld B0= 0:5 mT, forb=a=2. The escape time\nis signi\fcantly reduced at increased particle sizes. This\ncon\frms the advantage of the Einstein{de Haas phase to\nobserve metastability, even in the presence of dissipation.V. DISCUSSION\nIn deriving the results discussed in the preceding sec-\ntions, we assumed (i) a single-magnetic-domain nanopar-\nticle with uniaxial anisotropy and constant magnetiza-\ntion, with the values of the physical parameters summa-\nrized in Table I, (ii) deterministic dynamics, i. e. the\nabsence of thermal \ructuations, (iii) that gravity can be\nneglected, and (iv) a non-rotating nanomagnet. Let us\njustify the validity of these assumptions.\nWe \frst discuss the values of the parameters given in\nTable I, which are used in our analysis. The material pa-\nrameters, such as \u001aM,\u001a\u0016,kaand\u0011, are consistent with,\nfor example, cobalt [75{78]. We remark that the uniax-\nial anisotropy considered in our model represents a good\ndescription even for materials which do not have an in-\ntrinsic magnetocrystalline uniaxial anisotropy, provided\nthat they have a dominant contribution from the uniaxial\nshape anisotropy. This is the case, for example, for fer-\nromagnetic particles with a prolate shape [75]. We point\nout that the values used here do not correspond to a spe-\nci\fc material, but instead they describe a general order\nof magnitude corresponding to common magnetic materi-\nals. Indeed, our results are general and can be particular-\nized to speci\fc materials by replacing the above generic\nvalues with exact numbers. As we show in App. C, the re-\nsults and conclusions presented here remain unchanged\neven when di\u000berent values of the parameters are con-\nsidered. The values used for the \feld gradient B0and\nthe curvature B00have been obtained in magnetic mi-\ncrotraps [62, 79{82]. The values of the gas pressure P\nand the temperature Tare experimentally feasible, with\nnumerous recent experiments reaching pressure values as\nlow asP= 10\u00006mbar [58, 68, 70, 83{85]. All the values\nassumed in our analysis are therefore consistent with cur-\nrently available technologies in levitated optomechanics.\nThermal \ructuations can be neglected at cryogenic\nconditions (as we argue in Sec. II A), as their e\u000bect is\nweak enough not to destroy the deterministic e\u000bects cap-\ntured by Eqs. (1-5). In particular, thermal activation of\nthe magnetization, as quanti\fed by the N\u0013 eel relaxation\ntime, can be safely neglected due to the large value of\nthe uniaxial anisotropy even for the smallest particles\nconsidered. As for the mechanical thermal \ructuations,\nwe con\frm that they do not modify the deterministic\ndynamics in App. B, where we simulate the associated\nstochastic dynamics.\nGravity, assumed to be along ex, can be safely ne-\nglected, since the gravity-induced displacement of the\ntrap center from the origin is much smaller than the\nlength scale over which the Io\u000be-Pritchard \feld signi\f-\ncantly changes [24]. Speci\fcally, the gravitational poten-\ntialMgx shifts the trap center from the origin r= 0\nalong exby an amount rg\u0011Mg= (\u0016B00), wheregis\nthe gravitational acceleration. On the other hand, the\ncharacteristic length scales of the Io\u000be-Pritchard \feld\nare given by \u0001 r0\u0011p\nB0=B00for the variation along\nex, and \u0001r0\u0011B0=B00for the variation o\u000b-axis. When-8\neverrg\u001c\u0001r0;\u0001r0, gravity has a negligible role in the\nmetastable dynamics of the system. In the parameter\nregime considered in this article, this is always the case.\nWe note that the condition to neglect gravity is the same\nas for a magnetically trapped atom, since both Mand\u0016\nscale with the volume.\nFinally, we remark that the analysis presented here\nis carried out for the case of a non-rotating nanomag-\nnet5. The same qualitative behavior is obtained even in\nthe presence of mechanical rotation (namely, considering\na more general equilibrium con\fguration with `e6= 0).\nThe analysis of dynamics in the presence of rotation is\nprovided in App. C. In particular, the dynamics in the\nEinstein{de Haas phase remains largely una\u000bected, pro-\nvided that the total angular momentum of the system is\nnot zero. In the atom phase, mechanical rotation leads to\ndi\u000berences in the switching time \u001c, as generally expected\nin the presence of magneto-mechanical coupling [74, 88].\nVI. CONCLUSION\nIn conclusion, we analyzed how the stability of a nano-\nmagnet levitated in a static magnetic \feld is a\u000bected by\nthe most relevant sources of dissipation. We \fnd that in\nthe strong-\feld regime (atom phase) the system is un-\nstable due to the Gilbert-damping-induced magnetiza-\ntion switching, which occurs on a much faster timescale\nthan the center-of-mass oscillations, thereby preventing\nthe observation of levitation. On the other hand, the sys-\ntem is metastable in the weak-\feld regime and for small\nparticle dimensions (Einstein{de Haas phase). In this\nregime, the con\fnement of the nanomagnet in a mag-\nnetic trap is limited only by the gas-induced dissipation.\nOur results suggest that the timescale of stable levitation\ncan reach and even exceed several hundreds of periods of\ncenter-of-mass oscillations in high vacuum. These \fnd-\nings indicate the possibility of observing the phenomenon\nof quantum spin stabilized magnetic levitation, which we\nhope will encourage further experimental research.\nThe analysis presented in this article is relevant for\nthe community of levitated magnetic systems. Speci\f-\ncally, we give precise conditions for the observation of\nthe phenomenon of quantum spin stabilized levitation\nunder experimentally feasible conditions. Levitating a\nmagnet in a time-independent gradient trap represents a\nnew direction in the currently growing \feld of magnetic\nlevitation of micro- and nanoparticles, which is interest-\ning for two reasons. First, the experimental observation\nof stable magnetic levitation of a non-rotating nanomag-\nnet would represent a direct observation of the quantum\nnature of magnetization. Second, the observation of such\n5Rotational cooling might be needed to unambiguously identify\nthe internal spin as the source of stabilization. Subkelvin cooling\nof a nanorotor has been recently achieved [86, 87], and cooling\nto\u0016K temperatures should be possible [56].phenomenon would be a step towards controlling and us-\ning the rich physics of magnetically levitated nanomag-\nnets, with applications in magnetometry and in tests of\nfundamental forces [9, 11, 34, 35].\nACKNOWLEDGMENTS\nWe thank G. E. W. Bauer, J. J. Garc\u0013 \u0010a-Ripoll, O.\nRomero-Isart, and B. A. Stickler for helpful discussions.\nWe are grateful to O. Romero-Isart, B. A. Stickler and\nS. Viola Kusminskiy for comments on an early ver-\nsion of the manuscript. C.C.R. acknowledges funding\nfrom ERC Advanced Grant QENOCOBA under the EU\nHorizon 2020 program (Grant Agreement No. 742102).\nV.W. acknowledges funding from the Max Planck So-\nciety and from the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) through Project-\nID 429529648-TRR 306 QuCoLiMa (\"Quantum Cooper-\nativity of Light and Matter\"). A.E.R.L. thanks the AMS\nfor the \fnancial support.\nAppendix A: Rotation to the body frame\nIn this appendix we de\fne the transformation ma-\ntrix between the body-\fxed and the laboratory reference\nframes according to the ZYZ Euler angle convention,\nwith the Euler angles denoted as \n= (\u000b;\f;\r )T. We\nde\fne the transformation between the laboratory frame\nOexeyezand the body frame Oe1e2e3as follows,\n0\n@e1\ne2\ne31\nA=R(\n)0\n@ex\ney\nez1\nA; (A1)\nwhere\nR(\n)\u0011Rz(\u000b)Ry(\f)Rz(\r) =0\n@cos\rsin\r0\n\u0000sin\rcos\r0\n0 0 11\nA\n0\n@cos\f0\u0000sin\f\n0 1 0\n\u0000sin\f0 cos\f1\nA0\n@cos\u000bsin\u000b0\n\u0000sin\u000bcos\u000b0\n0 0 11\nA:(A2)\nAccordingly, the components vj(j= 1;2;3) of a vector\nvin the body frame Oe1e2e3and the components v\u0017\n(\u0017=x;y;z ) of the same vector in the laboratory frame\nOexeyezare related as\n0\n@v1\nv2\nv31\nA=RT(\n)0\n@vx\nvy\nvz1\nA: (A3)\nThe angular velocity of a rotating particle !can be writ-\nten in terms of the Euler angles as != _\u000bez+_\fe0\ny+ _\re3,\nwhere ( e0\nx;e0\ny;e0\nz)T=Rz(\u000b)(ex;ey;ez)Tdenotes the\nframeOe0\nxe0\nye0\nzobtained after the \frst rotation of the9\nlaboratory frame Oexeyezin the ZYZ convention. By\nusing (A1) and (A2), we can rewrite angular velocity in\nterms of the body frame coordinates,\n!= _\u000b2\n4R(\n)\u000010\n@e1\ne2\ne31\nA3\n5\n3+_\f2\n4R(\r)\u000010\n@e1\ne2\ne31\nA3\n5\n2+ _\re3;\n(A4)\nwhich is compactly written as ( !1;!2;!3)T=A(\n)_\n,\nwith\nA(\n) =0\n@\u0000cos\rsin\fsin\r0\nsin\fsin\rcos\r0\ncos\f 0 11\nA: (A5)\nAppendix B: Dynamics in the presence of thermal\n\ructuations\nIn this appendix we consider the dynamics of a lev-\nitated nanomagnet in the presence of stochastic forces\nand torques induced by the surrounding gas. The dy-\nnamics of the system are described by the following set\nof stochastic di\u000berential equations (SDE),\nd~r=!I~pdt; (B1)\nde3=!\u0002e3dt; (B2)\nd~p= [!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p] dt+p\nDcmdWp;(B3)\nd`= [!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`] dt+p\nDrotdWl;\n(B4)\ndm=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)]dt;\n(B5)\nwhere we model the thermal \ructuations as uncorrelated\nGaussian noise represented by a six-dimensional vector\nof independent Wiener increments (d Wp;dWl)T. The\ncorresponding di\u000busion rate is described by the tensors\nDcmandDrotwhich, in agreement with the \ructuation-\ndissipation theorem, are related to the corresponding dis-\nsipation tensors \u0000 cmand \u0000 rotasDcm\u00112\u0000cm\u001f;D rot\u0011\n2\u0000rot\u001f, where\u001f\u0011MkBT\r2\n0a2=\u00162.\nIn the following we numerically integrate Eqs. (B1-B5)\nusing the stochastic Euler method implemented in the\nstochastic di\u000berential equations package in MATLAB. As\nthe e\u000bect of thermal noise is more prominent for small\nparticles at weak \felds, we focus on the Einstein-de Haas\nregime considered in Sec. IV. We show that even in this\ncase the e\u000bect of thermal \ructuations leads to dynamics\nwhich are qualitatively very close to the results obtained\nin Sec. IV. In Fig. 4 we present the results of the stochas-\ntic integrator by averaging the solution of 100 di\u000berent\ntrajectories calculated using the same parameters consid-\nered in Fig. 3(a-c). The resulting average dynamics agree\nqualitatively with the results obtained by integrating the\ncorresponding set of deterministic equations Eqs. (1-5)\nFigure 4. Stochastic dynamics of a nanomagnet for the same\nparameter regime as considered in Fig. 3. (a) Average motion\nof the system in the y-zplane until time t= 5\u0016s. (b) Dy-\nnamics of center of mass along the ey(top) and ez(bottom)\ndirections. (c) Dynamics of the anisotropy axis component\ne3;x. (d) Numerical error as function of time. The simulations\nshow the results of the average of 100 di\u000berent realizations of\nthe system dynamics. In panels (b-d) the solid dark lines are\nthe average trajectories, while the shaded area represents the\nstandard deviation.\n[cfr. Fig. 3(a-c)]. The main e\u000bect of thermal excitations\nis to shift the center of oscillations of the particle's de-\ngrees of freedom around the value given by the thermal\n\ructuations. This is more evident for the dynamics of\ne3[cfr. Fig. 4(c) and Fig. 3(c)]. We thus conclude that\nthe deterministic equations Eqs. (1-5) considered in the\nmain text correctly capture the metastable behavior of\nthe system. We emphasize that the results presented in\nthis section include only the noise due to the surround-\ning gas. Should one be interested in simulating the ef-\nfect of the \ructuations of the magnetic moment, the Eu-\nler method used here is not appropriate, and the Heun\nmethod should be used instead [89].\nLet us conclude with a technical note on the numerical\nsimulations. In the presence of dissipation and thermal\n\ructuations the only conserved quantity of the system is\nthe magnitude of the magnetic moment ( jmj= 1). We\nthus use the deviation 1 \u0000jmj2as a measure of the numer-\nical error in both the stochastic and deterministic sim-\nulations presented in this article. For the deterministic\nsimulations the error stays much smaller than any other\nphysical degree of freedom of the system during the whole\nsimulation time. The simulation of the stochastic dynam-\nics shows a larger numerical error [see Fig. 4(d)], which\ncan be partially reduced by taking a smaller time-step\nsize. We note that, for the value of magnetic anisotropy\ngiven in Table I, the system of SDE is sti\u000b. This, together\nwith the requirement imposed on the time-step size by10\nthe numerical error, ultimately limits the maximum time\nwe can simulate to a few microseconds. However, this is\nsu\u000ecient to validate the agreement between the SDE and\nthe deterministic simulations presented in the article.\nAppendix C: Additional \fgures\nIn this appendix we provide additional \fgures.\n1. Dynamics in the atom phase\nIn Fig. 5 we analyze magnetization dynamics in the\natom phase as a function of di\u000berent system parame-\nters. In Fig. 5(a) we show how magnetization switching\nchanges as the anisotropy constant kais varied. We con-\nsider the bias \feld B0= 1100 mT, which is larger than\nthe value considered in the main text. This is done to en-\nsure thatB0>B atom for all anisotropy values. Fig. 5(a)\ndemonstrates that the switching time \u001c, given by Eq. (8),\nis an excellent approximation for the dynamics across a\nwide range of values for the anisotropy constant ka. The\nlarger discrepancy between Eq. (8) and the line showing\nthe case with ka= 106J/m3is explained by the prox-\nimity of this point to the unstable region (in this case\ngiven by the critical \feld Batom = 900 mT), and better\nagreement is recovered at larger bias \feld values.\nIn Fig. 5(b) we analyze the validity of Eq. (8) for dif-\nferent values of the Gilbert damping parameter \u0011and the\ntemperature T. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time by\n5%; below this line the deviation becomes increasingly\nmore signi\fcant. As evidenced by Fig. 5(b), \u001cshows lit-\ntle dependence on T; its order of magnitude remains con-\nstant over a wide range of cryogenic temperatures. On\nthe other hand, the dependence on \u0011is more pronounced.\nIn fact, reducing the Gilbert parameter signi\fcantly de-\nlays the switching time, leading to levitation times as\nlong as\u00181\u0016s.\nAdditionally, we point out that \u001cdepends on the \feldgradientB0and curvature B00through the initial con-\nditionmk(t= 0). In particular, magnetization switch-\ning can be delayed by decreasing B0, as this reduces\nthe initial misalignment of the magnetization and the\nanisotropy axis (i. e. jmk(t= 0)j!1).\n2. Dynamics in the presence of rotation\nIn Fig. 6 we consider a more general equilibrium con-\n\fguration, namely a nanomagnet initially rotating such\nthat in the equilibrium point Le=\u0000I3!Sex, with!S>0\ndenoting the rotation in the clockwise direction. This\nequilibrium point is linearly stable in the absence of dis-\nsipation [23, 24], with additional stability of the system\nprovided by the mechanical rotation, analogously to the\nclassical magnetic top [25{27].\nIn Fig. 6(a) we analyze how magnetization switching\nin the atom phase changes in the presence of rotation for\ndi\u000berent values of parameters. The rotation has a slight\ne\u000bect on the switching time \u001c, shifting it forwards (back-\nwards) in case of a clockwise (counterclockwise) rotation.\nThis is generally expected in the presence of magneto-\nmechanical coupling [74, 88].\nIn Fig. 6(b) we show the motion in the y-zplane in\nthe Einstein{de Haas phase for both directions of rota-\ntion. This can be compared with Fig. 3(a). 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Mech. , P09008\n(2014)."    },    {        "title": "2107.00982v3.Anomalous_Gilbert_Damping_and_Duffing_Features_of_the_SFS___boldmath___varphi_0___Josephson_Junction.pdf",        "content": "arXiv:2107.00982v3  [cond-mat.supr-con]  25 Aug 2021Anomalous Gilbert Damping and Duffing Features of the SFS ϕ0Josephson Junction\nYu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4\n1BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2Dubna State University, Dubna, 141980, Russia\n3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan\n4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran\n(Dated: August 26, 2021)\nWe demonstrate unusual features of phase dynamics, IV-char acteristics and magnetization dy-\nnamics of the ϕ0Josephson junction at small values of spin-orbit interacti on, ratio of Josephson to\nmagnetic energy and Gilbert damping. In particular, an anom alous shift of the ferromagnetic reso-\nnance frequency with an increase of Gilbert damping is found . The ferromagnetic resonance curves\nshow the Duffing oscillator behaviour, reflecting the nonline ar nature of Landau-Lifshitz-Gilbert\n(LLG) equation. Based on the numerical analysis of each term in LLG equation we obtained an\napproximated equation demonstrated both damping effect and Duffing oscillator features. The re-\nsulting Duffing equation incorporates the Gilbert damping in a special way across the dissipative\nterm and the restoring force. A resonance method for the dete rmination of spin-orbit interaction in\nnoncentrosymmetric materials which play the role of barrie r inϕ0junctions is proposed.\nIntroduction. The Josephson junctions (JJ) with the\ncurrent-phaserelation I=Icsin(ϕ−ϕ0), wherethephase\nshiftϕ0is proportional to the magnetic moment of ferro-\nmagneticlayerdetermined bythe parameterofspin-orbit\ninteraction, demonstratea number ofunique featuresim-\nportant for superconducting spintronics, and modern in-\nformation technology [1–6]. The phase shift allows one\nto manipulate the internal magnetic moment using the\nJosephson current, and the reverse phenomenon which\nleads to the appearance of the DC component in the su-\nperconducting current [7–9].\nInteractive fields can bring nonlinear phenomena of\nboth classical, and quantum nature. A basic example\nis the magnons strongly interacting with microwave pho-\ntons [10]. As a result we could name Bose-Einstein con-\ndensation of such quasiparticles, i.e. magnons [11, 12],\nand synchronization of spin torque nano-oscillators as\nthey coherently emit microwave signals in response to\nd.c. current [13]. It is interesting that (semi)classical an-\nharmonic effects in the magnetodynamics described by\nthe Landau-Lifshitz-Gilbert (LLG) model in thin films or\nheterostructures [14, 15], and the quantum anharmonic-\nity in the cavity mangnonics [16] can well be modeled\nby so simple a nonlinear oscillator as Duffing. The cor-\nresponding Duffing equation contains a cubic term and\ndescribesthe oscillationsofthe variousnonlinearsystems\n[17].\nDespite the fact that nonlinear features of LLG are\nstudied often during a long time and in different systems,\nmanifestation of the Duffing oscillator behavior in the\nframeworkofthisequationisstill notcompletelystudied.\nCloser to our present investigation, in the study of the\ndynamics of antiferromagnetic bimeron under an alter-\nnatingcurrent,Duffingequationformsagoodmodel, and\nthis has applications in weak signal detection [14, 18, 19].\nAs another application with Duffing oscillator at work,\nwe can mention the ultra thin Co 20Fe60B20layer, andits largeangle magnetizationprecessionunder microwave\nvoltage. There are also ‘foldover’ features, characteris-\ntic of the Duffing spring, in the magnetization dynamics\nof the Co/Ni multilayer excited by a microwave current\n[15, 20, 21]. But nonlinear features of ϕ0Josephson junc-\ntions have not been carefully studied yet. In this Letter,\nwe show that the Duffing oscillator helps in the under-\nstanding of the nonlinear features of ϕ0Josephson junc-\ntions at small values of system parameters.\nCoupling of superconducting current and magnetiza-\ntion and its manifestation in the IV-characteristics and\nmagnetizationdynamicsopensthedoorfortheresonance\nmethod determination of spin-orbit intensity in noncen-\ntrosymmetric materials playing the role of barrier in ϕ0\njunctions. As it is well known, the spin-orbit interaction\nplays an important role in modern physics, so any novel\nmethod for its determination in real materials would be\nvery important. There are a series of recent experiments\ndemonstrating the modification of Gilbert damping by\nthe superconducting correlations (see Ref.[22] and cita-\ntionstherein). Inparticular, the pronouncedpeaksin the\ntemperature dependence of Gilbert damping have been\nobserved for the ferromagnetic insulator/superconductor\nmultilayers [23] which might be explained by the pres-\nence of spin relaxation mechanisms like the spin-orbit\nscattering [22]. Here, we use the noncentrosymmetric\nferromagnetic material as a weak link in ϕ0junctions.\nThe suitable candidates may be MnSi or FeGe, where\nthe lack of inversion center comes from the crystalline\nstructure [8].\nThe Gilbert damping determines the magnetization\ndynamics in ferromagnetic materials but its origin is not\nwell understood yet. Effect of nonlinearity on damp-\ning in the system is very important for application of\nthese materials in fast switching spintronics devices. Our\nstudy clarifies such effects. In Ref.[24] the authors dis-\ncuss the experimental study of temperature-dependent2\nGilbert damping in permalloy (Py) thin films of varying\nthicknesses by ferromagnetic resonance, and provide an\nimportant insight into the physical origin of the Gilbert\ndamping in ultrathin magnetic films.\nIn this Letter we demonstrate an anomalous depen-\ndence of the ferromagnetic resonance frequency with an\nincrease of the Gilbert damping. We find that the reso-\nnance curves demonstrate features of Duffing oscillator,\nreflecting the nonlinear nature of LLG equation. The\ndamped precession of the magnetic moment is dynami-\ncally driven by the Josephson supercurrent, and the res-\nonance behavior is given by the dynamics of the Duffing\nspring. The resonance methods for the determination of\nspin-orbit interaction in the ϕ0junction are proposed.\nModel and Methods. In the considered SFS ϕ0junc-\ntion (see Fig.1) the superconducting phase difference ϕ\nand magnetization Mof the F layer are two coupled dy-\nnamical variables. Based on the LLG equation for the\nFigure 1: Schematic view of SFS ϕ0Josephson junction. The\nexternal current applied along xdirection, ferromagnetic easy\naxis is along zdirection.\nmagnetic moment Mwith effective magnetic field Heff,\nresistively capacitively shunted junction (RCSJ) model,\nand Josephson relation for the phase difference ϕ, we de-\nscribe dynamics of the SFS ϕ0junction by the system of\nequations in normalized variables\ndm\ndt=ωFheff×m+α/parenleftbigg\nm×dm\ndt/parenrightbigg\n,\nheff=Grsin(ϕ−rmy)/hatwidey+mz/hatwidez, (1)\ndV\ndt=1\nβc[I−V+rdmy\ndt−sin(ϕ−rmy)],\ndϕ\ndt=V,\nwheremis vector of magnetization with components\nmx,y,z, normalized to the M0=/bardblM/bardbland and satisfy-\ning the constraint/summationtext\ni=x,y,zm2\ni(t) = 1,ωF= ΩF/ωc,\nΩF=γK/νis ferromagnetic resonance frequency, γis\nthe gyromagnetic ratio, Kis an anisotropic constant, ν\nis the volume of the ferromagnetic F layer, αis the phe-\nnomenologicaldamping constant(Gilbert damping), heff\nis the vector of effective magnetic field, normalized to\ntheK/M0(heff=HeffM0/K),G=EJ/(Kν) relation\nof Josephson energy to magnetic one, ris a parameter\nof spin-orbit coupling, ϕis phase difference of JJ, Vis\nvoltage normalized to the Vc=IcR,Iccritical current\nof JJ,Rresistance of JJ, βc= 2eIcCR2//planckover2pi1is McCumberparameter, Cis capacitance of JJ, Iis bias current nor-\nmalized to the Ic. In this system of equation time tis\nnormalized to the ω−1\nc, whereωc= 2eIcR//planckover2pi1is character-\nistic frequency. In the chosen normalization, the average\nvoltage corresponds to the Josephson frequency ωJ.\nFerromagnetic resonance in ϕ0junction. The ferro-\nmagnetic resonance features are demonstrated by aver-\nage voltage dependence of the maximal amplitude of the\nmycomponent ( mmax\ny), taken at each value of bias cur-\nrent. To stress novelty and importance of our finding,\nwe first present the analytical results for average volt-\nage dependence of mmax\nyalong IV-characteristics in the\nferromagnetic resonance region. As it was discussed in\nRefs.[8, 25, 26], in case Gr≪1,mz≈1, and neglecting\nquadratic terms mxandmy, we get\n/braceleftBigg\n˙mx=ξ[−my+GrsinωJt−αmx]\n˙my=ξ[mx−αmy],(2)\nwhereξ=ωF/(1 +α2). This system of equations can\nbe written as the second order differential equation with\nrespect to the my\n¨my=−2αξ˙my−ξ2(1+α2)my+ξ2GrsinωJt.(3)\nCorresponding solution for myhas the form\nmy(t) =ω+−ω−\nrsinωJt−α++α−\nrcosωJt,(4)\nwhere\nω±=Gr2ωF\n2ωJ±ωF\n((ωJ±ωF)2+(αωJ)2),(5)\nand\nα±=Gr2ωF\n2αωJ\n((ωJ±ωF)2+(αωJ)2).(6)\nSo,mydemonstrates resonance with dissipation when\nJosephson frequency is approaching the ferromagnetic\none (ωJ→ωF). The maximal amplitude mmax\nyas a\nfunction of voltage (i.e., Josephson frequency ωJ) at dif-\nferentα, calculated using (4), is presented in Fig.2 (a).\nWe see the usual characteristicvariation of the resonance\ncurve with an increase in dissipation parameter when the\nmaximal amplitude and position of resonance pick cor-\nresponds to the damped resonance. We note that the\nanalytical result (4) were obtained in the case Gr≪1.\nPresented in Fig.2(b) results of numerical simulations\nmmax\ny(V) dependence at different values of dissipation\nparameter αdemonstrate the essential differences with\nthe results followedfrom the analytical consideration(4).\nWe note also that the strong coupling of the supercon-\nducting phase difference ϕand magnetization Mof the\nF layermanifests itself by appearanceof subharmonics of\nthe resonance at ω= 1/2,1/3,1/4 demonstrated in the\ninset to Fig.2(b).3\nFigure 2: (a) Analytical results for maximal amplitude mmax\ny\nin the ferromagnetic resonance region for different α; (b)\nNumerical results for maximal amplitude of magnetization\nmy−component at each values of bias current and voltage\nalong IV-characteristics of the ϕ0junction in the ferromag-\nnetic resonance region for various α. Inset shows the man-\nifestation of the resonance subharmonics. Parameters are:\nβc= 25, G=0.05, r=0.05, ωF= 0.5.\nWe stress two important features followed from the\npresented results. First, the ferromagnetic resonance\ncurves show the foldover effect, i.e., the features of Duff-\ning oscillator. Different from a linear oscillator, the non-\nlinear Duffing demonstrates a bistability under external\nperiodic force [27]. Second, the ferromagnetic resonance\ncurves demonstrate an unusual dependence of the reso-\nnance frequency as a function of Gilbert damping α. As\nshown in Fig. 3(a), an increase in damping leads to a\nnonuniform change in the resonant frequency, i.e., with\nan increase in damping the resonance maximum shifts\ntoωFat small α, but then moves to the opposite side,\ndemonstrating the usual damped resonance. So, with\nan increase in α, unusual dependence of the resonance\nvoltage transforms to the usual one. For the parameters\nchosen, the critical value of this transformation is around\nα= 0.02−0.03. We call this unusual behaviour of the\nresonance maximum of mmax\nyas an “α-effect”. Both the\nα−effect and Duffing features in our system appear due\nto the nonlinear features of the system dynamics at small\nFigure 3: (a) α-dependence of the resonance curve mmax\ny(V)\npeak presented in Fig.2 in the damping parameter interval\n[0.006 – 0.2]. Dashed line indicates ferromagnetic resonan ce\nposition; (b) Comparison of the resonance curves mmax\ny(V)\ncalculated by full LLG equation (1) and the approximate\nequation (8).\nG,r,α≪1. To prove it, we have carried out the nu-\nmerical analysis of each term of LLG full equation (first\ntwo equations in (1)) for the set of model parameters\nG= 0.05,r= 0.05α= 0.005. After neglecting the\nterms of order 10−6, we have\n˙mx\nξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2\nz,\n˙my\nξ=mxmz−αmym2\nz, (7)\n˙mz\nξ=−Grmxsin(ϕ−rmy)+αmz(m2\nx+m2\ny),\nInthisapproximationweobserveboththe“ α–effect”and\nDuffing oscillator features. Neglecting here the last term\nαmz(m2\nx+m2\ny) in third equationfor ˙ mz, which is orderof\n10−4, leadstothe losingoftheDuffing oscillatorfeatures,\nbut still keeps alpha-effect. We note that equation (7)\nkeeps the time invariance of the magnetic moment, so\nthat term plays an important role for manifestation of\nDuffing oscillator features by LLG equation.\nThe generalized Duffing equation for ϕ0junction.\nThe LLG is a nonlinear equation and in case of simple\neffective field it can be transformed to the Duffing equa-\ntion [14, 17]. Such transformation was used in Ref.[17]\nto demonstrate the nonlinear dynamics of the magnetic\nvortex state in a circular nanodisk under a perpendicular\nalternating magnetic field that excites the radial modes\nof the magnetic resonance. They showed Duffing-type\nnonlinear resonance and built a theoretical model corre-\nsponding tothe Duffing oscillatorfromthe LLG equation\nto explore the physics of the magnetic vortex core polar-\nity switching for magnetic storage devices.\nThe approximated LLG system of equations (7)\ndemonstrates both α-effect and features of Duffing os-\ncillator. As demonstrated in the Supplemental Materials\n[28], the generalizedDuffing equation forthe ϕ0junction,\n¨my+2ξα˙my+ξ2(1+α2)my\n−ξ2(1+α2)m3\ny=ξ2GrsinωJt.(8)4\ncan be obtained directly from the LLG system of equa-\ntions.\nAs we see, for small enough Gandr, it is only the\ndimensionless damping parameter αin LLG that plays a\nrole in the dynamics of the system. We can think of a\nharmonic spring with a constant that is hardened or soft-\nened by the nonlinear term. For a usual Duffing spring,\nwith independent coefficients of the various terms, the\nresonancepeak relative to the harmonic (linear) resonant\nfrequency folds over to the smaller (softening) or larger\n(hardening) frequencies. In the frequency response, the\ninterplay of the specific dependence of each coefficient on\nαplays an important role and as Fig.3(a) shows, there is\na particular αthat brings the resonant frequency closest\nto ferromagnetic resonance.\nSimulations of the mydynamics in the framework of\nDuffing equation can explain observed foldover effect in\nthe frequency dependence of mmax\ny. Comparison the re-\nsults followed from analytical approximate equation (8)\nand results of full equation (1) for maximal amplitude of\nmmax\nyin the ferromagnetic resonance region is presented\nin Fig.3(b). So, the magnetization dynamics in the SFS\nϕ0-junction due to the voltage oscillations can effectively\nbe described by a scalar Duffing oscillator, synchronizing\nthe precession of the magnetic moment with the Joseph-\nson oscillations.\nEffect of spin-orbit interactions. As we mentioned\nabove, the spin-orbit interaction plays an important role\nin different fields of modern physics. Here we have sug-\ngested a novel method for its determination in real non-\ncentrosymmetric ferromagnetic materials like MnSi or\nFeGe, where the lack of inversion center comes from\nthe crystalline structure Ref.[8] and which play role a\nweak link in ϕ0junctions. Based on the obtained re-\nsults, presented in Fig.4, we propose different versions of\nthe resonance method for the determination of spin-orbit\ninteraction in these materials. Particularly, in Fig.4(a)\nwe present the simulation results of maximal amplitude\nmmax\nybased on (1) at G= 0.05,α= 0.01 at different\nvalues of spin-orbit parameter rin the ferromagnetic res-\nonance region. This case corresponds to the nonlinear\napproximation leading to the Duffing equation (8). The\nsame characteristics calculated by equation (1) for larger\nvalueα= 0.1, i.e. corresponding to the linear approxi-\nmation (3) are presented in Fig.4(b). As it was expected,\nin caseα= 0.01 the foldover effect is more distinct.\nIn Fig.4(c) the r-dependence of the resonancepeak po-\nsition, obtained from the simulation results of full equa-\ntion atα= 0.01 andα= 0.1 for the same set of model\nand simulation parameters is demonstrated. We stress\nhere that nonlinear features of LLG equation leading to\nthe Duffing’s shift of the mmax\nypeak of main harmonic\nwith r presented in Fig.4(c) show the manifestation of\nnonlinearity.\nDespite the noted differences between results for α=\n0.01 andα= 0.1 , we see in both cases a monotonic\nFigure 4: (a) Voltage dependence of mmax\nyin the ferromag-\nnetic resonance region at different values of spin-orbit int er-\naction based on (1) at G= 0.05,α= 0.01. Inset enlarges\nthe main harmonic; (b) The same as in (a) for α= 0.1; (c)\nShift ofmmax\nypeak as a function of spin-orbit interaction at\ntwo values of Gilbert damping; (d) r-dependence of the main\nharmonic and subharmonics peaks in case (a); (e) The same\nas in (d) for the case (b).\nlinear increase of mmax\nypeak of main harmonic and sub-\nharmonics with rdemonstrated in Fig.4(d) and Fig.4(e).\nSuch lineardependence canbe noted fromEq. (6) ofRef.\n[14], but the authors did not discuss it. This dependence\nmight serve as a calibrated curve for spin-orbit interac-\ntion intensity, thus creating the resonance methods for r\ndetermination.\nConclusions. Based on the reported features of the\nϕ0Josephson junction at small values of spin-orbit in-\nteraction, ratio of Josephson to magnetic energy and\nGilbert damping, we have demonstrated that the cou-\npled superconducting current and the magnetic moments\nin theϕ0-junction result in the current phase relation in-5\ntertwining with the ferromagnetic LLG dynamics. The\nferromagnetic resonance clearly shows this interplay. In\nparticular, an anomalous shift of the ferromagnetic res-\nonance frequency with an increase of Gilbert damping\nis found. The ferromagnetic resonance curves demon-\nstrate features of Duffing oscillator, reflecting the nonlin-\near nature of LLG equation. The obtained approximated\nequation demonstrates both damping effect and Duffing\noscillator features. We have shown that due to the non-\nlinearity, asmodeledbythe generalizedDuffing equation,\nthe parameters of the system can compensate each other\nresulting in unusual response. The position of the maxi-\nmum can shift towards and then away from the expected\nresonant frequency, as the damping is decreased. There\nare also foldover effects that was explained by the pro-\nposed model. A resonance method for the determination\nof spin-orbit interaction in noncentrosymmetric materi-\nals which play the role of barrier in ϕ0junctions was\nproposed.\nThe experimental testing of our results would in-\nvolve SFS structures with ferromagnetic material having\nenough small value of Gilbert damping. Potential candi-\ndate for experimental realization could be ferromagnetic\nmetals or insulators which have small values of damping\nparameter ( α∼10−3−10−4). In Ref.[29] the authors\nreport on a binary alloy of cobalt and iron that exhibits\na damping parameterapproaching10−4, which is compa-\nrable to values reported only for ferrimagnetic insulators\n[30, 31]. Using superconductor-ferromagnetic insulator-\nsuperconductor on a 3D topological insulator might be\na way to have strong spin-orbit coupling needed for ϕ0\nJJ and small Gilbert dissipation for α-effect [5]. We note\nin this connection that the yttrium iron garnet YIG is\nespecially interesting because of its small Gilbert damp-\ning (α∼10−5). The interaction between the Joseph-\nson current and magnetization is determined by the ra-\ntio of the Josephson to the magnetic anisotropy energy\nG=EJ/(Kν) and spin-orbit interaction r. The value of\nthe Rashba-type parameter rin a permalloy doped with\nPt[32] and in the ferromagnets without inversion sym-\nmetry, like MnSi or FeGe, is usually estimated to be in\nthe range 0 .1−1. The value of the product Grin the ma-\nterialwith weakmagneticanisotropy K∼4×10−5KA−3\n[33], and a junction with a relatively high critical current\ndensity of (3 ×105−5×106)A/cm2[34] is in the range\n1−100. It givesthe set offerromagneticlayerparameters\nand junction geometry that make it possible to reach the\nvalues used in our numerical calculations for the possible\nexperimental observation of the predicted effect.\nNumerical simulations were funded by the project 18-\n71-10095oftheRussianScientificFund. A.J.andM.R.K.\nare grateful to IASBS for financial support.[1] Jacob Linder and W. A. Jason Robinson, Nature Physics\n11, 307 (2015).\n[2] Yu. M. Shukrinov, Accepted for UFN.\nDOI:https://doi.org/10.3367/UFNe.2020.11.038894\n[3] A.A. Mazanik, I.R. Rahmonov, A.E. Botha, and Yu.M.\nShukrinov, Phys. Rev. Applied 14, 014003 (2020).\n[4] M. Nashaat and Yu. M. Shukrinov, Physics of Particles\nand Nuclei Letters, 17, 79. (2020).\n[5] I. V. Bobkova , A. M. Bobkov, I. R. Rahmonov, A. 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Cros, and A. Fert, Appl. Phys.\nLett.103, 082408 (2013).[31] M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M.\nKl¨ aui, A. V. Chumak, B. Hillebrands, and C. A. Ross,\nAPL Mater. 2, 106102 (2014).\n[32] A. Hrabec, F. J. T. Goncalves, C. S. Spencer, E. Aren-\nholz, A. T. N’Diaye, R. L. Stamps, and C. H. Marrows,\nPhys. Rev. B 93, 014432 (2016).\n[33] A. Yu. Rusanov, M. Hesselberth, J. Aarts, and A. I.\nBuzdin, Phys. Rev. Lett. 93, 057002 (2004).\n[34] J. W. A. Robinson, F. Chiodi, M. Egilmez, G. B. Hal´ asz\nand M. G. Blamire, Scientific Report 2, 699 (2012).arXiv:2107.00982v3  [cond-mat.supr-con]  25 Aug 2021Supplemental Material to “Anomalous Gilbert Damping and Du ffing Features of the\nSFSϕ0Josephson Junction”\nYu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4\n1BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2Dubna State University, Dubna, 141980, Russia\n3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan\n4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran\n(Dated: August 26, 2021)\nHere, we demonstrate by numerical methods that a generalize d Duffing equation can be obtained\ndirectly from LLG system of equations, for small system para meters of S/F/S junction.\nBoth the α−effect and Duffing features obtained by\nLLG system of equations appear due to the nonlinear\nfeatures of its dynamics at small G,r,α≪1. To proveit,\nwe have carried out the numerical analysis of each term\nof LLG full equation (first two equations in the equation\n(1) of the main text) for the set of model parameters\nG= 0.05,r= 0.05α= 0.005. After neglecting the\nterms of order 10−6, we have\n˙mx\nξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2\nz,\n˙my\nξ=mxmz−αmym2\nz, (1)\n˙mz\nξ=−Grmxsin(ϕ−rmy)+αmz(m2\nx+m2\ny),\nThe procedure is as follows. Expanding mn\nzin a series\nwith the degree of ( mz−1) we can find\nmn\nz=nmz−(n−1). (2)\nFrom expression m2\nx+m2\ny+m2\nz= 1 and (2), we obtain\nmz=2−m2\ny\n2. (3)\nUsing approximation sin( ϕ−rmy) = sin(ωJt) in (1),\ndifferentiatingsecondequationofthe system(1) andsub-\nstituting ˙ mx,mxand ˙mzfrom first second and third\nequations of the system (1), respectively and using the\nexpression (2), (3) and assuming mz= 1 only in denom-\ninators, we come to a second order differential equation\nwith respect to my\n¨my=a1˙m3\ny+a2my˙m2\ny+a3m4\ny˙my+a4m2\ny˙my+a5˙my\n+a6m5\ny+a7m3\ny+a8my−c1˙m2\nysinωJt (4)\n+c2m4\nysinωJt+c3m2\nysinωJt+AsinωJt.The numerical calculation for the used set of model\nparameters allows us to estimate each of the terms in the\nequation, as presented in Table I.\nNow, if we neglect those terms smaller than 10−4, the\nequation (4) takes on the form of Duffing equation with\nTable I: Numerical analysis of equation (4) terms.\na1α\nξa1˙m3\ny∼1.76×10−5\na2 α2a2my˙m2\ny∼3.4×10−8\na3ξα3a3m4\ny˙my∼7.7×10−12\na4ξ(3α−α3)a4m2\ny˙my∼2×10−5\na52ξα a5˙my∼6×10−4\na6ξ2(α2+2α4)a6m5\ny∼5.56×10−9\na7ξ2(1+α2−α4)a7m3\ny∼3.7×10−3\na8ξ2(1+α2)a8my∼6.1×10−2\nc1 Gr c1˙m2\nysinϕ∼3.6×10−5\nc22ξ2α2Grc2m4\nysinϕ∼5.3×10−11\nc3ξ2Gr(α2−2)c3m2\nysinϕ∼4.5×10−5\nAξ2Gr AsinωJt∼6.25×10−4\ndamping dependent coefficients, i.e., we have a general-\nization of the Duffing equation\n¨my+2ξα˙my+ξ2(1+α2)my\n−ξ2(1+α2)m3\ny=ξ2GrsinωJt.(5)"    },    {        "title": "2107.11699v1.Electron_Phonon_Scattering_governs_both_Ultrafast_and_Precessional_Magnetization_Dynamics_in_Co_Fe_Alloys.pdf",        "content": "1 \n Electron -Phonon Scattering governs both Ultrafast and Precessional Magnetization \nDynamics in Co -Fe Alloys  \nRamya Mohan1, Victor  H. Ortiz2, Luat Vuong2, Sinisa Coh2, Richard B. Wilson1,2* \n1Materials Science & Engineering Program, University of California, Riverside  \n2Department of Mechanical Engineering, University of California, Riverside  \n* Corresponding Author : rwilson@ucr.edu  \n \n \nAbstract  \nRecent investigations have advanced the understanding of how structure -property relationships \nin ferromagnetic metal alloys affect the magnetization dynamics on nanosecond time -scales . A \nsimilar understanding for magnetization dynamics on femto - to pico -second time -scales does not \nyet exist. To address this, we perform time -resolved magneto optic Kerr effect (TRMOKE) \nmeasurements of magnetization dynamics in Co -Fe alloys on femto - to nano-second  regimes . \nWe show that Co -Fe compositions that exhibit low Gilbert damping parameters also feature \nprolonged ultrafast demagnetization upon photoexcitation. We analyze our experimental TR -\nMOKE data with the three-temperature -model (3TM) and the Landau -Lifshitz -Gilbert equation . \nThese analyses  reveal a strong compositional dependence of the dynamics across all time -scales \non the strength of electron -phonon interactions. Our findings are beneficial to the spintronics  and \nmagnonics community , and wil l aid in the quest  for energy -efficient magnetic storage  \napplications.  \nIntroduction  \nLaser excitation of  a magnetic metal causes energy to cascade from photoexcited electrons into \nspin and vibrational degrees of freedom1–3. In ferromagnetic 3d transiti on metals such as Fe, Co, \nand Ni, the rapid increase in thermal energy stored by spin degrees of freedom causes  \nfemtosecond   quenching of the magnetization2,3, followed by a partial recover over the next few \npicoseconds . Subsequently, on nanosecond time-scales , a temperature induced change in \nequilibrium properties causes  oscillatory precessions of the magnetic moment .  \nBoth ultrafast  and precessional magnetization dynamics involve energy exchange between  \nmagnetic and vibrational degrees of freedom . The energy exchange is mediated by quasi -particle \ninteractions . The strength of quasi -particle  interactions in a ferromagnet depends on e lectronic \nband structure4,5. In 3d ferromagnetic alloys, the electronic energy bands  near the Fermi -level \nvary strong ly with composition6. Several  recent investigations  of nanosecond precessional \ndynamics  in ferromagnetic alloy s have explored  the relationship between electronic band \nstructure, quasi -particle interactions , and magnetic damping6–8. Schoen et al.  report a n intrinsic 2 \n damping parameter less than 10-3 for Co 0.25Fe0.756, which is unusually low for  a metal . They \nconclude  that the low damping  in Co0.25Fe0.75 is a result of  a minimization in the density of states \nat the Fermi -level, which decreases the rate of electron -phonon scattering . \nResearchers have not yet reached a  unified understanding of how quasi -particle interactions \ngovern  the magnetization dynamics in the  femtosecond regime2,9–15. Some studies have \nhypothesized that spin-flips caused by electron -phonon interactions are key drivers of \nfemtose cond  magnetization dynamics9,11. Other experimental and theoretical studies have \nexplored the importance of electron -magnon interactions12–15. Encouraged by the recent \nadvances  in the materials science of nanosecond precessional dynamics6–8, we study the \ncompositional dependence of ultrafast magnetization dynamics in Co -Fe alloys. Our  study’s  goal \nis to understand the relationship between electronic band structure , quasi -particle interactions, \nand femto -magnetism properties of ferromagnetic metal alloys.   \nWe perform time -resolved magneto optic Kerr effect  (TR-MOKE)  measurements to characterize  \nthe magnetization dynamics of thin CoxFe1-x alloy  films (capped and seeded with Ta/Cu layers on \na sapphire substrate)  on femto - to nanosecond time-scale s. See Methods for details on sample \ngeometry.  We observe that the ultrafast magnetization dynamics are a strong function of Co -\nconcentration , see Figure . 1a. The ultrafast dynamics of Co xFe1-x differ most significantly from \nthose of Co and Fe at a composition of  x = 0.25.  We also analyze the time -resolved macroscopic \nprecessional dynamics and report the effective damping parameter of our samples , see Figure  2a. \nAfter linewidth analyses, f or CoxFe1-x, we observe  that the  Gilbert  damping parameter varies \nfrom  3.6 ×10−3 to 5.6 ×10−3 for compositions between x = 0 and 1, with a minimum value of \n1.5 ×10−3 at x = 0.25 , in good agreement with previously reported results , see Figure 3b .  \nTo determine the strength and composition dependence of electron -magnon and electron -phonon  \nquasi -particle interactions , we analyze our ultrafast magnetization dynamics  data with a three-\ntemperature -model (3TM)2,16. Our results reveal a strong composition al dependence of  the \nelectron -phonon energy transfer coefficient, 𝑔𝑒𝑝,  suggesting that  the variation in the ultrafast \ndynamics  in Co xFe1-x alloys  occurs primarily due to electron -phonon scattering. We draw this \nconclusion because t he value of 𝑔����𝑝 depends on the rate of phonon emission by hot electrons 17. \nElectron -phonon scattering is also predicted  to govern the dampin g of nanosecond precessional \ndynamics  6,18,19. Therefore, o ur results  demonstrate  that the same  microscopic electron -phonon \ninteractions  responsible for Gilbert damping also play a dominant role in femto -magnetism \nproperties of ferromagnetic alloys.  \nResults  \nUltrafast  Magnetization  Dynamics  \nWe plot the normalized u ltrafast magnetization dynamics  response , ∆M(t), for Co, Fe, and \nCo0.25Fe0.75 as a function of time delay in Figure . 1a. Data for the rest of the Co -Fe compositions \nare plotted  in Supplementary Figure  1. All our measurements were performed with an incident 3 \n laser fluence less than ~15 J/m2. This is a sufficiently  small fluence  for the dynamics in our \nexperiments to follow a  linear regime.  In other words, decreasing the incident f luence by a factor \nof two decreases the optical signal by a factor of two, but does not change the time-dependence  \nof the signal .    \nWe use a polar TR -MOKE configuration t o measure  the ultrafast magnetization dynamics  at \nfemtosecond time delays.  A schemati c of our experimental setup is shown in Supplementary  \nFigure  2a. We apply a n external 2.2 Tesla (T) field perpendicular to the plane of the sample  \nusing  an electromagnet (GMW 3480). This external field is strong enough to effectively \novercome the in-plane shape anisotropy of the Co -Fe alloys and saturate the moment in the out -\nof-plane direction. Since the equilibrium orientation of the moment is in the out -of-plane \ndirection , both, before and after laser irradiation, this geometry allows us to  quan tify the  \nfemtosecond  demagnetization  response  of the Co -Fe alloys , without the presence of macroscopic  \nprecessional dynamics , see schematic in Fig ure 1b.  \nUpon excitation with the pump pulse, the magnetic moment decreases on a sub -picosecond time-\nscale due to the flow of energy from electrons to magnons2,3,16,20,21. Then, on picosecond time-\nscale s, the magnetization partially recovers as energy is transferr ed to the lattice  and temperature \ngradients across the film thickness relax.  After a few picoseconds, the  magnetic film reaches  a \nnew equilibrium at an elevated temperature.  Ultrafast dynamics with sub -picosecond  \ndemagnetization followed by picosecond  re-magnetization  are commonly categorized as “type I” \ndynamics , and are characteristic of 3d ferromagnetic metals such as Fe, Co, and Ni9.  \nTo elucidate  how the de - and re -magnetization dynamics change with composition, we define \ntwo data descriptors : τD and R. We define  the demagnetization time , τD, as the delay time where \nd∆M(t)/dt reaches its maximum value.  We define R as the ratio of the maximum of 𝛥𝑀(𝑡) to \n𝛥𝑀(𝑡≈10ps). We plot τD  and R as a function of composition in Figure  3a. τD varies weakly \nwith composition and has a minimum  value  of 40 fs at x = 0.25. In contrast , we observe that R \nvaries strongly with composition and is a maximum  of 4 at x = 0.25.  \nNanosecond Precessional Dynamics   \nWe show measurements of the  macroscopic  precessional dynamics of Fe, Co, and Co 0.25Fe0.75 in \nFigure  2a. Data for the other Co -Fe compositions are plotted in Supplementary  Figure  3. We use \na polar TR -MOKE experimental setup, with an obliquely angled external magnetic field, to \nmeasure the macroscopic precessional dynamics of our samples. A schematic of our \nexperimental setup is shown in Supplementary  Figure 2b. Tilting the electromagnet to an angle \nof 11° , with respect to the plane of the sample, allows us to apply a canted  external  magnetic \nfield so that the magnetic moment has an out -of-plane component. The equilibrium orientation of \nthe moment depends on the balancing between the applied  external  field and the thin -film shape \nanisotropy field.  The shape anisotropy field in the z -direction is proportional to the  out-of-plane \ncomponent of the  magnetic moment. Upon heating, the  total magnetic moment decreases . This \ndecrease results in an ultrafast change to the  out-of-plane  anisotropy field  and equilibrium 4 \n orientation . As a result, t he magnetic moment will precess to a new equilibrium orientation , see \nschematic in Figure  2b. Our polar TR -MOKE setup detects  changes in the out -of-plane moment , \nso we can sensitively measure the frequency and amplitude of the precessional dynamics.  \nWe collect between  6 and 12 TR-MOKE scans  of precessional dynamics for each sample .  Each \nof these scans is co llected  with a different applied external magnetic field , ranging from 0. 2 T to \n2.2 T. The TR -MOKE signals include precessional dynamics  in addition with a background \nrelated to temperature -induced demagnetization. To analyze the precessional dynamics,  we \nsubtract  the background with  a biexponential  decay  function . We fit the resulting dataset with a \ndamped harmonic function,  V(t)=Asin(ωt+∅)exp (−t/τ). Our fits yield unique  values of A \n(amplitude),  ∅ (the initial phase of the oscillation), T (period), and τ (the exponential decay time \nof the precession). Using these values,  we determine  the effective dimensionless damping \nparameter , αeff = ω.τ-1. \nThe resonance frequency  is a function of applied external magnetic field  and magnetic moment,  \n𝜔=γ √Heff(Heff+μ0Ms). Here, ɣ is the gyromagnetic ratio,  μ0 is the vacuum permeability, \nHeff is the out -of-plane component of the external magnetic field  as measured by a Hall probe , \nand Ms is the saturation magnetization of the sample . We derive the magnetic moment of the \nsample by treating Ms as a fit parameter . We also perform VSM measurements of the moment  of \nsome of the samples and find that the magnetic moment obtained is in good agreement with the \nvalue that we derive by fitting our precessional dynamics data . See Supplementary  Figure 4 for \nmore details . \nThe effective damping parameter α eff that we deduce from our precessional dynamics \nmeasurements includes effects from damping and  inhomogeneous broadening. The effect of \ninhomogeneous  broadening is independent of the applied field at high frequencies22. To obtain \nthe Gilbert  damping parameter  intrinsic to the sample  geometry  (not intrinsic to the material) , we \nplot the effective linewidth, αeff∙f,  as a function of frequency, and linearly fit  to the equation , \nαeff∙f=α∙f+∆H, where ∆H  is the inhomogeneous broadening component and α is the Gilbert \ndamping parameter . Further details can be found in Supplementary Figure 5. \nIn contra st to prior investigations that performed FMR measurements in the frequency range \nfrom 16 -18 GHz8 and 40 GHz6, our TR -MOKE experimental setup allows us to study dynamics \nat frequencies as large as 90 GHz. At such high frequency, we can be confident that our \nmeasured Gilbert damping parameter is dominate d by the intrinsic linewidth over \ninhomogeneous  broadening effects.  \nThe Gilbert damping parameter we observe of α = 1.5 ×10−3 for Co 0.25Fe0.75 is amongst the \nlowest ever reported for a ferromagnetic metal. Schoen et al. report α=2.1 ×10−3 for \nCo0.25Fe0.75.  After accounting for radiative and spin -pumping contributions, they estimate an \nintrinsic damping parameter for Co0.25Fe0.75 to be αint=5 ×10−4 .  Lee et al. 8 performed FMR \nmeasurements of Co0.25Fe0.75 epitaxial films and report α=1.4 ×10−3. Wei et al. report α=5 \n 1.5 ×10−3 for Fe 0.75Al0.25 films 7. We note that our measured damping parameter likely \nincludes  significant contributions from spin -pumping into the adjoining Ta /Cu layers, but we  did \nnot experimentally examine the effect s of spin -pumping in our samples.  \nAnalysis  and Discussion  \nThe c omparison of 𝑅 and 𝛼 in Figure  3a and Figure  3b reveals  that the two quantities depend on \ncomposition in a similar manner.  R is at a maximum and 𝛼 is at a minimum at x = 0.25 .  Fe and \nCoxFe1-x alloys with x ≥ 0.5 have small R and high 𝛼.  Alternatively, C oxFe1-x  alloys with  0.1< x \n< 0.5 have both high 𝑅 and low 𝛼. To confirm this correlation , we performed a hierarchical \ncluster analysis of the raw data  at both femtosecond and nanosecond time-scale s. The clustering \nalgorithm divides  the Co -Fe alloys into groups based on similarit ies in the dynamics data . The \nclustering results as a function of composition are nearly  identical  when based on the femto -\n/pico -second time -scale data vs. the nanosecond time -scale data.  We include further details on \nthe clustering analysis in  Supplementary  Note 1 and Supplementary Figure 6 .  \nWe now explain the correlation between ultrafast and precessional dynamics by considering how \nelectronic scattering processes depend on composition. Similar to prior studies of damping in \nCo-Fe alloys6,7,23, our results for  𝛼 vs. x are in good agreement with the “breathing Fermi \nsurface ” model for damping24. In this model , spin -orbit coupling causes the Fermi -level to shift \nwith the precessi ons of the magnetic moment25. A shift in the equilibrium Fermi -level leads to a \nnonequilibrium electron population . As the Fermi -level repopulates, i ntra-band electron -phonon \nscattering  transfers energy to the lattice .  The “breathing Fermi surface” model predicts  that the  \ndamping parameter  is directly  proportional to 𝐷(𝜀𝑓), because  more electronic states near 𝜀𝑓 leads \nto higher  rates of electron -phonon scattering . We observe  that the  𝛼 value  for Co0.25Fe0.75 is \n~2.5x lower th an 𝛼 for Fe. Density functional theory predicts a ~2x difference in 𝐷(𝜀𝑓) for \nCo0.25Fe0.75 vs. Fe, see Supplementary  Note 2  or Ref.6. Therefore, like prior studies of Co -Fe \nalloys6,7,23, we conclude that intra -band electron -phono n scattering governs precessional \ndamping.  \nTo better understand how composition affects  electron -magnon and electron -phonon energy \ntransfer  mechanisms , we analyze our 𝛥𝑀(𝑡) data with a phenomenological three temperature \nmodel  (3TM) , see Figure  4. The 3TM describes how heat flows  between electrons, phonons, and \nmagnons after laser excitation of the Co-Fe sample . (See Methods for  additional  details. ) The \n3TM predicts  that τD depends on two groupings of model parameters: 𝜏𝑒𝑚≈𝐶𝑚/𝑔𝑒𝑚 and 𝜏𝑒𝑝≈\n𝐶𝑒/𝑔𝑒𝑝.  Here 𝐶𝑚 and 𝐶𝑒 are the magnon and electron heat-capacity per unit volume, and 𝑔𝑒𝑚 \nand 𝑔𝑒𝑝 are the energy transfer coefficients from electrons to magnons an d phonons, \nrespectively.  We estimate v alues for 𝐶𝑒 vs. composition using the Sommerfeld model together \nwith the electronic density of states vs. composition reported in Ref.6. The 3TM also predicts that \nthe parameter R is determined by the following grouping of parameters: 𝑅= 𝐶𝑝𝑔𝑒𝑚/𝐶𝑚𝑔𝑒𝑝 16, \nwhere 𝐶𝑝 is the phonon heat -capacity per unit volume .  We assume  that the  value of  𝐶𝑝 is 3.75 6 \n MJ m-3 K-1 for Co, Fe and Co -Fe alloys.  With these estimates for 𝐶𝑒 and 𝐶𝑝, and other relevant \nmodel parameters, summarized in Supplementary  Table 1, we can deduce unique values for \n𝐶𝑚/𝑔𝑒𝑚 and 𝐶𝑝/𝑔𝑒𝑝  as a function of composition from our TR-MOKE  data, see Figure  4b. \nBased on our 3TM analysis, we conclude that the strong composition dependence of R is due to \nthe composition dependence of 𝑔𝑒𝑝. Boltzmann rate -equation modelling of the nonequilibrium \nelectron dynamics after photoexcitation predicts that the electron -phonon energy -trans fer \ncoefficient  is 𝑔𝑒𝑝=[𝜋ℏ𝑘𝐵𝐷(𝜀𝐹)]𝜆⟨𝜔2⟩ 5. Here,  𝜆⟨𝜔2⟩ is the second frequency moment of the \nEliashberg function  and is a measure of the strength of electron -phonon interactions . Most of the \ncomposition al dependence we observe in 𝑔𝑒𝑝 is explained by the composition al dependence of \n𝐷(𝜀𝑓). To show this, we include a prediction for 𝑔𝑒𝑝 in Figure  4b.  Our prediction uses the \n𝐷(𝜀𝑓) vs. x reported  in6  and treats 𝜆⟨𝜔2⟩ as a composition independent fit parameter . We find \n𝜆⟨𝜔2⟩=260 meV2 provides an excellent fit to our data .  The best-fit value for 𝜆⟨𝜔2⟩ is in good \nagreement with  𝜆⟨𝜔2⟩≈𝜆𝑅Θ𝐷22⁄=280 meV2.  Here, 𝜆𝑅 is derived from electrical resistivity \ndata for Fe  26, and Θ𝐷=470𝐾 is the Debye temperature of Fe.   \nBefore beginning our experimental study, we hypothesized that the energy transfer coefficient \nbetween electrons and magnons, \nemg , would be correlated with the phase -space for electron -\nmagnon scattering . We expected the phase -space for electron -magnon scattering to be a  strong \nfunction of band -structure near the Fermi -level 12–15.  We also expected the phase -space to be \nminimized at a composition of x = 0.25, because of the minimum in the density of states at the \nfermi -level.   To explore how the phase -space for electron -magnon scattering depends on \ncomposition, we performed density functional theory calculations for the electronic band \nstructure with x = 0 and x = 0.25, see Supplementa ry Note 2.  Our DFT calculations suggest  that \nthe phase -space for electron -magnon scattering is an order of magnitude higher for x = 0 vs. \n0.25.  However, we do not see evidence  that this large theoretical difference in electron -magnon \nscattering phase -space  affects ultrafast dynamics . The time -scale for magnons to heat up after \nphotoexcitation, \n/em m emCg , decreases monotonically with increasing x, and does display  \nstructure near x ~ 0.25.  \nSeveral  theoretical models predict a strong correlation between τ D and αint. For example,  \nKoopmans et al.  predicts τ D will be inversely proportional to α by assuming  that the  dissipative \nprocesses responsible for damping also drive ultrafast demagnetization  27.  Alternatively, Fähnle \net al.  predict s that  τD should be proportional to αint 28. In our experiments on Co -Fe thin films, w e \nobserve only a weak correlation between τD and αint. While α int varies  with composition  by a \nfactor of three , τD for 8 of the 9 compositions we study fall within 20% of 75 fs.  The τD value we \nobtained for Fe (=  76 fs) agrees well with experimental results reported in 9,12,29. \n \n 7 \n Conclusions  \nWe have measured the magnetization dynamics of Co xFe1-x thin-films , and we observe that  both \nultrafast and precessional dynamics of Co 0.25Fe0.75 differ significantly from Co and Fe . When the \nmoment of Co0.25Fe0.75 is driven away from its equilibrium  orientation , the time -scale for  the \nmoment  to return to equilibrium is 3 -4x as long as for Fe or Co.  Similarly, when spins of \nCo0.25Fe0.75 are driven into a nonequilibrium state by ultrafast laser heating, the time -scale for \nthermalization with the lattice is 2 -3x as long as for Fe or Co. Through  3TM analyses, we \ndemonstrate that this occurs  primarily  due to the effect of  the electronic band -structure on \nelectron -phonon interactions , consistent with the “breathing Fermi surface” theory . Our findings \nare of fundamental importance to the field of ul trafast magnetism, which seeks to control \nmagnetic order on femto - to picosecond time-scale s. Such control requires  a thorough  \nunderstanding of how and why energy is exchanged between electronic, spin, and vibrational \ndegrees of freedom. Prior studies have  shown  that 𝑔𝑒𝑝 is correlated with a wide range of physical \nproperties, e.g the superconducting transition temperature30, electrical resistivity 26, \nphotoelectron emission31, and the laser fluence required for ablation32. To our knowledge, o ur \nstudy provides the first demonstration that 𝑔𝑒𝑝 in ferromagnetic metals is also correlated to the \nGilbert damping parameter 𝛼.  \nOur findings  also have implications for the ongoing search for magnetic materials with ultrafast \nmagnetic switching functionality. Atomistic spin dynamics simulations predict that the energy \nrequired for ultrafast electrical or optical switching of rare -earth ferromagnetic  alloys, e.g. \nGdFeCo, is governed by the electron -phonon energy transfer coefficient33. To date, most studies \naimed at exploring the materials science of ultrafast switching have used alloy composition as a \nway to control magnetic  properties  34–37. Our work  suggests  an alternative strategy for reducing \nthe energy requirements for ultrafast magnetic switching. The alloy composition should be \nchosen  to minimize the electronic density of states at the Fermi -level. Such metals will have \nlower electron -phonon energy trans fer coefficients, and therefore more  energy efficient ultrafast \nswitching  33.  \nFinally , our findings  offer a new route for discovering ferro magnetic materials with ultra -low \ndamping  as a result of low 𝑔𝑒𝑝. Current methods for identifying low damping materials involve \nlabor -intensive ferromagnetic resonance measurements of one alloy composition at a time.  \nAlternatively, high-throughput  localized measurements of  ultrafast demagnetization  dynamics  of \nsamples produced using combinatorial techniques38 would allow promising alloy compounds \nwith weak electron -phonon interactions to be rapidly identified  39–41.  \n \n \n \n 8 \n Materials and Methods  \nSample Preparation  \nWe sputter deposit the Co -Fe samples onto sapphire substrates with a direct current (DC) \nmagnetron sputtering system (Orion, AJA International). The base pressure prior to deposition is \nless than 3.5  × 10-7 torr. We sputter with an Ar gon pressure of ~3.5 × 10-3 torr. The geometry of \nthe samples is sapphire/Ta(2nm)/Cu(3nm)/Co xFe1-x(15nm)/Cu(3nm)/Ta(1nm). The Co xFe1-x layer \nis deposited  by co -sputtering two 4N purity Co and Fe targets at different powers. We chose this \nfilm geometry to mimic the samples in Ref.6 which demonstrated  low damping at x = 0.25.  \nTo ensure an accurate thickness of each layer in our samples, we calibrate the deposition rates of \neach metal by sputtering individual Co, Fe, Ta, and Cu films onto SiO 2/Si substrates and/or BK -7 \nglass substrates. We use picosecond acoustics42 and time-domain thermo -reflectance  (TDTR) \nmeasurements43,44 to determine the thicknesses of these individual films. We validate the \ncomposition of the Co -Fe alloy layer by perf orming Energy Dispersive X -Ray Spectroscopy \n(EDS) analyses with a scanning electron microscope ( FEI Nova Nano SEM 450) at an operating \nvoltage of 15 kV and working distance of 14 mm. We analyze the EDS data using Aztec Synergy  \nsoftware ( Oxford Instruments ). \nTime -Resolved MOKE Experimental Setup  \nWe use a pump/probe laser system to perform TR -MOKE measurements of the magne tization \ndynamics. The pulsed laser is a Ti:sapphire oscillator with an 80 MHz repetition rate. The laser \nbeam is split into a pump and probe beam, that are modulated to frequencies of 10.7 MHz and \n200 Hz , respectively. A time -delayed  pump beam irradiates the sample surface and heats the \nmetal film. The ultrafast heating causes a change in the magnetic moment. We measure the time -\nevolution of the magnetic moment by monitoring the polarization of the probe beam reflected of f \nthe sample surface. The reflected probe beam’s polarization state is affected by the out -of-plane \nmagnetic moment of the sample due to the polar Kerr effect. Additional details about the MOKE \nexperiment set -up are in Ref.45. \nThe t ime-resolution of our experiment is controlled by the convolution of the intensity vs.  time \nof the pump and probe pulses. The wavelength of our pump and probe beams is tunable. \nEmploying a red  (900 nm ) pump and  blue (450 nm ) probe yields  higher time-resolution  \ncapabilities , allowing us to accurately measure the ultrafast magnetization at fe mtosecond time \ndelays . We measure the full-width -at-half-maximum ( FWHM ) of the convolution of the pump \nand probe pulses by performing an inverse Faraday effect  (IFE)  measurement on Pt . We obtain a \nFWHM  value  of 390 fs for  the convoluted pulses , and a pulse duration of 2 10 fs for the 900 nm \npump/450 nm probe beam setup . For further details on our IFE measurements and pulse duration \ncalculations, please refer to  Supplementary Figure 8.  9 \n To investigate  the precessional dynami cs on longer time -scales, we use a pump and probe \nwavelength of 783 nm. The pulse duration  for this setup  is 610 fs due to pulse broadening from a \ntwo-tint setup we use to prevent pump light from reaching the balanced detector45,46.  \nThree Temperature Modeling  \nTo determine  the electron, phonon, and magnon energy transfer coefficients,  we use t he \nphenomenological  three -temperature model  (3TM),  given by the following set of equations :  \n𝐶𝑒𝑑𝑇𝑒\n𝑑𝑡=𝑔𝑒𝑝(𝑇𝑝−𝑇𝑒)+ 𝑔𝑒𝑚(𝑇𝑚−𝑇𝑒)+Ʌ𝑒𝑑2𝑇𝑒\n𝑑𝑧2+𝑆(𝑧,𝑡)    (1) \n𝐶𝑝𝑑𝑇𝑝\n𝑑𝑡=𝑔𝑒𝑝(𝑇𝑝−𝑇𝑒)+Ʌ𝑝𝑑2𝑇𝑝\n𝑑𝑧2     (2) \n𝐶𝑚𝑑𝑇𝑚\n𝑑𝑡=𝑔𝑒𝑚(𝑇𝑚−𝑇𝑒)+Ʌ𝑚𝑑2𝑇𝑚\n𝑑𝑧2   (3) \n𝑆(𝑧,𝑡)= 𝑆0𝑃(𝑡)𝐴(𝑧)   (4) \nEquations  1 – 3  describe the temperature evolution of electrons (e), phonons (p) and magnons \n(m), as a function of time delay  (t). C, T, and Ʌ are the heat capacity per unit volume, \ntemperature, and thermal conductivity, respectively.  We use the density of states (DOS) at the \nFermi level  as a function of Co -concentration6 to calculate the electronic heat capacity (C e) using \nthe Sommerfeld model . We assume that the phonon -magnon energy transfer is negligible \ncompared to electron -magnon coupling, and  thus,  neglect  𝑔𝑝𝑚.  \nWe calculate the laser energy absorption by electrons  (S), as a function of depth  (z) and time \ndelay  (t), as described in Equation 4. The terms P(t) and A(z) denote the time -dependent laser \npulse intensity and the optical absorption profile as a function of stack thickness. We calculat e \nA(z) us ing the refractive indices of each metal constituent of the stack47–49. The material \nparameters that are used to numerically solve equations 1 – 4 are listed in Supplementary Table \n1. \n \n \n \n \n \n \n \n 10 \n Figures:  \n                   \nFigure 1. Ultrafast magnetization dynamics of Co, Fe, and Co 0.25Fe0.75 thin films (a) Polar TR -\nMOKE data showing ultrafast demagnetization behavior at short delay times. (b) Schematic \nillustration of the three phases of an ultrafast magnetization dynamics experiment. Stage I: A large \nexternal magnetic field  oriented normal to the plane of t he sample leads to an equilibrium moment , \n𝑀⃗⃗  in the out -of-plane direction. Stage II: Upon heating with a pump beam, ultrafast \ndemagnetization ( 𝑀′⃗⃗⃗⃗ ) occurs within ~100s of fs. Energy from hot electrons is transferred to the \nmagnons, increasing the amplitude of precession. Stage III: Over the next few picoseconds, energy \nis transferred from magnons and electrons to the lattice. Additionally, spatial t emperature gradients \nrelax. As a result, magnons cool, i.e. the average precessional amplitude of individual spins \ndecreases. As a result, the magnetization partially recovers to 𝑀′′⃗⃗⃗⃗⃗⃗ . The time -scale for the partial \nrecovery in stage III depends strongly o n the composition.    \n11 \n   \nFigure 2 . Precessional dynamics in Co, Fe, and Co 0.25Fe0.75 thin films (a)  Polar TR -MOKE data \non sub -nanosecond time-scale s. (b) Illustration of the three stages for precessional dynamics after \nlaser excitation . Stage I: Prior to laser excitation, the presence of a canted external magnetic field, \n𝐻𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗ , oriented at an angle θ. This results in the orientation of the out -of-plane moments, 𝑀⃗⃗ 𝑧. Stage \nII: Laser -induced photoexcitation leads to the disorder of the magnetic moment, causing a decay \nin the net magnetization , denoted by 𝑀′⃗⃗⃗⃗ . The net torque imba lance causes macroscopic precessions \nof the magnons, towards equilibrium, 𝐻′𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ , over several ~100s of picoseconds . Stage III : \nEventually, after ~1 ns, the magnetic moment re -equilibrates to 𝐻′𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ . The lifetime of the magnetic \nprecessions depends o n the effective damping parameter, α eff.  The time -scale for the precessional \ndynamics to cease ( in stage III) depends strongly on composition, and is a maximum for x = 0.25.  \n12 \n  \nFigure 3.  Compositional dependence of descriptors for the ultrafast dynamics  data . (a) R \ndescribes the maximum change in the magnetic moment, i.e. how far from equilibrium spin -\ndegrees of freedom are driven after ultrafast excitation. τD describes the lag between zero delay \ntime and demagnetization, as a function of Co -concentration.  (b) α denotes the Gilbert damping \nparameter, as a function of Co concentration. Data obtained from our TR -MOKE experiments \ndescribed in this study (plotted in orange), agree reasonably with data from Ref. [6] (plotted in \ngreen). Co 0.25Fe0.75 features the largest deviation in R and α, when compared to its constituent \nelements Co and Fe.  \n \n \n \n \n \n \n \n \n \n \n \n13 \n  \nFigure 4. Analyses of Ultrafast Demagnetization Results using the Three Temperature Model \n(3TM) in Co -Fe alloys . (a) Polar TR -MOKE dataset of the Co 0.25Fe0.75 composition (black circles) \nwith best -fit results of the 3TM. The 3TM describes the temperature excursions of the electrons \n(blue curve), magnons (red curve) and phonons (green curve) after laser excitation. (b) We treat \n𝑔𝑒𝑝 and 𝑔𝑒𝑚 as fit parameters when solving the 3TM. Using literature values of C p and C m (further \ndetails available in Supplementary Table 1), we calculate and plot the electron -phonon (τ ep) and \nelectron -magnon (τ em) relaxation times, as a function of Co -concentration. The red -line is a best -\nfit value for the electron -phonon relaxation time as a function of composition, with the assumption \nof a composition -independent value for the electron -phonon coupling parameter λ . \n \n \n \n \n \n \n \n \n \n \n \n14 \n References:  \n1. Kirilyuk, A., Kimel, A. V & Rasing, T. 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Optical constants of transition metals. Phys. Rev. B  9, \n5056 –5070 (1974).  \n48. P. B. Johnson and R. W. Chri sty. Optical Constant of the Nobel Metals. Phys. Rev. B  6, 17 \n 4370 –4379 (1972).  \n49. Ordal, M. A., Bell, R. J., Alexander, R. W., Newquist, L. A. & Querry, M. R. Optical \nproperties of Al, Fe, Ti, Ta, W, and Mo at submillimeter wavelengths. Appl. Opt.  27, 1203 \n(1988).  \n \n \n \nAcknowledgements  \nThe work by R. M., V. H. O, and R. B. W. was primarily supported by the U.S. Army Research \nLaboratory and the U.S. Army Research Office under contract/grant number W911NF -18-1-\n0364 and W911NF -20-1-0274. R. M. and R. B. W. also acknowledge support by NSF (C BET – \n1847632). The work by L. V. and S. C. was supported by the U.S. Army Research Laboratory \nand U.S. Army Research Office under contract/grant number W911NF -20-1-0274. Energy \nDispersive X -Ray Spectroscopy (EDS) analyses  were performed at the Central Fac ility for \nAdvanced Microscopy and Microanalysis (CFAMM) at UC Riverside.  \nAuthor Contributions  \nR. M. and R. B. W. designed the experiments. R. M. prepared all the samples and characterized \nthem , and performed TR-MOKE experiments . V. H. O performed VSM measurements. L. V. \nperformed hierarchical clustering analyses. S. C. performed DFT calculations. R. M. and R . B. \nW. analyzed the data and wrote the manuscript, with discussions and contributions from L. V. \nand S. C . \nAdditional Information:  Supplementary information is provided with this manuscript.  \nCompeting Interests:  The authors declare no competing interest.  \nData Availability:  The data that supports the findings of this paper are available from the \ncorresponding author upon reasonable request.  \nCorrespondence:  Correspondence and request for additional information must be addressed to \nrwilson@ucr.edu   "    },    {        "title": "2108.02380v1.Spin_transfer_torque_driven_localized_spin_excitations_in_the_presence_of_field_like_torque.pdf",        "content": "arXiv:2108.02380v1  [cond-mat.stat-mech]  5 Aug 2021Spin-transfer torque driven localized spin excitations in\nthe presence of field-like torque\nM. Lakshmanana, R. Aruna, Avadh Saxenab,\naDepartment of Nonlinear Dynamics, School of Physics, Bhara thidasan University,\nTiruchirappalli 620024, Tamil Nadu, India\nbTheoretical Division and Center for Nonlinear Studies, Los Alamos National\nLaboratory, Los Alamos, NM 87545, USA\nAbstract\nWestudy theexistence of localizedone-spinexcitation intheHeisenb erg one-\ndimensional ferromagnetic spin chain in the presence of perpendicu lar and\nparallel external magnetic fields and current with spin-transfer t orque and\nfield-like torque. The Landau-Lifshitz-Gilbert-Slonczewski (LLGS ) equation\nisexactlysolvedfortheonespinexcitationintheabsenceofonsitea nisotropy\nfor the excitations of spin with fields perpendicular and parallel to th e chain.\nWe show the removal of damping in the spin excitations by appropriat ely\nintroducing current and also the enhancement of angular frequen cy of the\noscillations due to field-like torque in the case of both perpendicular a nd par-\nallel field. The exactness of the analytical results is verified by matc hing with\nnumerical counterparts. Further, we numerically confirm the exis tence of in-\nphase and anti-phase stable synchronized oscillations for two spin- excitations\nin the presence of current with perpendicular field and field-like torq ue. We\nalso show that the one-spin excitation is stable against thermal nois e and\ngets only slightly modified against thermal fluctuations.\nKeywords: Spin torque, Spin transfer nano-oscillator, LLGS equation,\nPT-symmetry, Spin excitations.\n1. Introduction\nBeing an important aspect in applied magnetism [1, 2], both from theo-\nretical and application points of view [3, 4], the study of dynamics of c lassical\nEmail address: avadh@lanl.gov (Avadh Saxena)\nPreprint submitted to Physica A August 22, 2021Heisenberg ferromagnetic spin chain with anisotropic interaction is o f funda-\nmental interest in the context of spin waves in arbitrarily shaped ma gnetic\nstructures [5] and spin-transfer torque in ferromagnetic layers [28]. Although\nthere are continuum cases which are completely integrable with solito n so-\nlutions, no discrete integrable case has been studied except for th e Ishimori\nlattice with a modified version [7]. Two of the present authors have ide ntified\na number of special solutions in the discrete spin chain case under va rious\nsituations such as with external magnetic field and onsite anisotrop y [8]. Un-\nderstanding such classes of solutions in these classical spin chain sy stems is\none of the important areas of investigations in spin dynamics.\nDue to its practical relevance [9, 10] the occurrence of localized b reathers\noroscillationsinferromagneticspinchainswithsuitableonsiteanisotr opyhas\nbeen studied for several years and recently two ofus andSubash have studied\ntheexcitationsofone, twoandthreespinsalongwiththeirlinearsta bilityand\nhave obtainedexplicit analytical solutions fortheHeisenberg anisot ropicspin\nchain in the presence of external magnetic field with onsite anisotro py [11].\nAlso, the two of us have studied the dynamics of one/many spin excit ations\nand the impact of spin current torque in an anisotropic Heisenberg f erromag-\nnetic spin chain in a constant/variable external magnetic field analyt ically\nand numerically. We have also proved analytically that the spin curren t can\nbalance the damping effect and have extended the study of such mo del to\nshow that in (parity-time reversal or) PT-symmetric magnetic nan ostruc-\ntures the gain/loss terms are canceled by the ferromagnetic coup ling which\nleads to spin oscillations [12].\nOn the other hand the spin transfer torque has been a promising ca n-\ndidate, starting from its discovery by Berger [13] and Slonczewski [14] the-\noretically as well as experimentally, for magnetization switching and m ag-\nnetization oscillations with their corresponding applications in the writeop-\neration for nanomagnetic memory storage [15] and microwave gene ration\n[16]. The dynamics of magnetization driven by spin transfer torque c an be\ninvestigated numerically and analytically by solving the governing Land au-\nLifshitz-Gilbert-Slonczewski (LLGS) equation [2, 17]. The role of an addi-\ntional torque, known as field-like torque [18, 19, 20], in magnetizat ion has\nbeen examined recently for its fruitful outcomes such as zero field oscillations\n[21, 22] and elimination of steady state motion in coupled spin torque o scilla-\ntors [23], respectively. While the spin transfer torque transfers s pin angular\nmomentum from a pinned layer to a free layer of the spin valve system , the\nfield-like torque arises due to the precession of spin polarized electr ons, from\n2the pinned layer, around the free layer’s magnetic moment. The field -like\ntorque finds technological applications in domain wall reflection [24] a s well\nas in magnetization uniformity in heavy metal/ferromagnetic metal/h eavy\nmetal layer structures [25].\nThe dynamics of the Heisenberg one dimensional discrete ferromag netic\nspin chain for the localized excitations of the one or more spins driven by\nspin-transfer torque in the presence of field-like torque has not b een studied\nyet to the best of our knowledge.\nInthispaperweanalyticallysolve theLLGSequationalongwithfield-like\ntorque for the components of spin for the one-spin excitation and numerically\nsolve for the two-spin excitations in the presence of field both para llel and\nperpendicular to the direction of spin chain. Also, we show the enhan cement\nof the angular frequency of the oscillations due to the field-like torq ue and\nanisotropy. Further, we identify the conditions among field, curre nt and\nthe magnitude of field-like torque to obtain the undamped oscillations for\ndifferent cases. We also confirm the stable nature of the one-spin e xcitations\nagainst thermal fluctuations.\nThe organization of the paper is as follows. In Sec. 2 we introduce th e\nHamiltonianmodelofthespinchainsystem. Wesolvetheone-spinexc itation\nin the presence of field in Sec. 3 and current and field-like torque in Se c. 4.\nIn Sec. 5 we solve the general case with perpendicular field, curren t and\nfield-like torque. In Sec. 6 we deduce the one-spin excitation for th e case\nof parallel field, current and field-like torque. We briefly study the t wo-spin\nexcitationsinthepresence ofperpendicular field, current andfield -like torque\nin Sec. 7. In Sec. 8, we discuss the influence of thermal noise on one -spin\nexcitation. Finally, in Sec. 9 we present our main conclusions.\n2. Model for spin chain\nThe Hamiltonian corresponding to the evolution of Nnumber of spins of\na one-dimensional anisotropic Heisenberg ferromagnetic spin chain is given\nby\nH=−N/summationdisplay\n{n}(ASx\nnSx\nn+1+BSy\nnSy\nn+1+CSz\nnSz\nn+1)−D/summationdisplay\nn(Sz\nn)2−H./summationdisplay\nnSn,(1)\n3whereSx\nn,Sy\nnandSz\nnare the spin components of the classical unit spin vector\n/vectorSn, satisfying the condition\n(Sx\nn)2+(Sy\nn)2+(Sz\nn)2= 1, n= 1,2,...,N. (2)\nHereA,BandCare the exchange interaction parameters, Dis the onsite\nanisotropy parameter and His the external magnetic field. The Landau-\nLifshitz equation of motion for the nthspin of the chain, specified by the\nHamiltonian (1), is deduced by introducing appropriate spin Poisson b racket\nrelations as[26]\ndSn\ndt=Sn×Heff+αSn×(Sn×Heff), n= 1,2,...,N, (3)\nwhereHeff=−δH/δSnis the effective field and αis the Gilbert damping\nparameter.\n3. One-spin excitation in the presence of perpendicular fiel d\nConsider a one-dimensional spin chain with the excitation of one spin S0\nas follows:\n.....(1,0,0),(1,0,0),(Sx\n0,Sy\n0,Sz\n0),(1,0,0),(1,0,0).... (4)\nThe Hamiltonian for this system, with external field H= (0,0,H) alongz\ndirection, is written from Eq.(1) as\nH=−[(N−3)A+2ASx\n0]−D(Sz\n0)2−HSz\n0. (5)\nThe effective field is derived from the Hamiltonian given in Eq.(5) as\nHeff= 2Aˆi+[2DSz\n0+H]ˆk, (6)\nwhereˆiandˆkare unit vectors along positive xandzdirections, respectively.\nBy substituting Eq.(6) in Eq.(3), the equations of motion for the exc ited spin\nare obtained as\ndSx\n0\ndt= 2DSy\n0Sz\n0+HSy\n0+α/bracketleftbig\n−2A(1−(Sx\n0)2)+2DSx\n0(Sz\n0)2+HSx\n0Sz\n0/bracketrightbig\n,(7)\ndSy\n0\ndt= 2ASz\n0−2DSx\n0Sz\n0−HSx\n0+α/bracketleftbig\n2ASx\n0Sy\n0+2DSy\n0(Sz\n0)2+HSy\n0Sz\n0/bracketrightbig\n,(8)\ndSz\n0\ndt=−2ASy\n0+α/bracketleftbig\n2ASx\n0Sz\n0−2DSz\n0(1−(Sz\n0)2)−H(1−(Sz\n0)2)/bracketrightbig\n.(9)\n4From Eqs.(7),(8) and (9), one can verify that\nSx\n0dSx\n0\ndt+Sy\n0dSy\n0\ndt+Sz\n0dSz\n0\ndt= 0, (10)\nto confirm that S2= (Sx\n0)2+(Sy\n0)2+(Sz\n0)2= constant = 1 is conserved. By\nconsidering the case where the onsite anisotropy is zero, we can wr ite the\ndynamical equations from Eqs.(7),(8) and (9), for the one-spin ex citation as\ndSx\n0\ndt= HSy\n0+α/bracketleftbig\n−2A(1−(Sx\n0)2)+HSx\n0Sz\n0/bracketrightbig\n, (11a)\ndSy\n0\ndt= 2 ASz\n0−HSx\n0+α[2ASx\n0Sy\n0+HSy\n0Sz\n0], (11b)\ndSz\n0\ndt= −2ASy\n0+α/bracketleftbig\n2ASx\n0Sz\n0−H(1−(Sz\n0)2)/bracketrightbig\n. (11c)\nFigure 1: (Color online) Undamped oscillations of (a) Sx\n0, (b)Sy\n0and (c)Sz\n0forA=\n0.1,D= 0,H= 0.1 andα= 0. Here the red lines and black dots are plotted from\nanalytical (Eq.(19)) and numerical (Eq.(11)) results, respective ly. The initial conditions\nare (0.10,0.00,0.99). (d) The three-dimensional trajectory of S0.\n5Figure 2: (Color online) Damped oscillations of (a) Sx\n0, (b)Sy\n0and (c)Sz\n0forA= 0.1,D=\n0,H= 0.1 andα= 0.005 plotted from Eq.(19). In the insets for the intermediate range\nof time, the red lines and black dots are plotted from analytical (Eq.( 19)) and numer-\nical (Eq.(11)) results, respectively. The initial conditions are (0.10 ,0.00,0.99). (d) The\nthree-dimensional trajectory of S0. The insets show the oscillations of the corresponding\ncomponents between t= 4000 and t= 4250.\nSince the coupled system of Eqs.(11) is difficult to solve as such, they are\ntransformedintermsofthestereographic complex variable ωanditscomplex\nconjugate ω∗as\nω=Sx\n0+iSy\n0\n1+Sz\n0, ω∗=Sx\n0−iSy\n0\n1+Sz\n0, (12)\nusing the following transformations,\nSx\n0=ω+ω∗\n1+ωω∗, Sy\n0=−iω−ω∗\n1+ωω∗, Sz\n0=1−ωω∗\n1+ωω∗, i=√\n−1,(13)\nas follows:\ndω\ndt=A(α−i)/bracketleftbigg\nω2+H\nAω−1/bracketrightbigg\n=A(α−i)(ω−ω+)(ω−ω−),(14)\n6whereω±= (1/2A)(−H±Ω), Ω =√\nH2+4A2. Eq.(14) can be exactly\nsolved as\nω(t) =ω+−C ω−eΩ(α−i)t\n1−C eΩ(α−i)t, ω∗(t) =ω+−C∗ω−eΩ(α+i)t\n1−C∗eΩ(α+i)t,(15)\nwhereCandC∗are arbitrary complex constants which can be obtained by\nusing Eq.(15) as\nC=Cr+iCi=ω+−ω(0)\nω−−ω(0), C∗=Cr−iCi=ω+−ω∗(0)\nω−−ω∗(0).(16)\nThe constants CrandCican be reexpressed using Eq.(12) as follows:\nCr=(H2−Ω2)(1+Sz\n0(0))2+4AH(1+Sz\n0(0))Sx\n0(0)+1−(Sz\n0(0))2\n4A2(ω−(1+Sz\n0(0)−Sx\n0(0))2+(Sy\n0(0))2),(17)\nCi=Ω(1+Sz\n0(0))Sy\n0(0)\nA(ω−(1+Sz\n0(0)−Sx\n0(0))2+(Sy\n0(0))2). (18)\nThe components of S0can be obtained by substituting Eqs.(15) and (16) in\nEqs.(13) as\nSx\n0=2/braceleftbig\nω+e−2αΩt+(H/A)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+ω−(C2\nr+C2\ni)/bracerightbig\n(1+ω2\n+)e−2αΩt−2(1+(H2−Ω2)/4A2)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+(1+ ω2\n−)(C2r+C2\ni),\n(19a)\nSy\n0=2(Ω/A)e−αΩt[Crsin(Ωt)−Cicos(Ωt)]\n(1+ω2\n+)e−2αΩt−2(1+(H2−Ω2)/4A2)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+(1+ ω2\n−)(C2r+C2\ni),\n(19b)\nSz\n0=(1−ω2\n+)e−2αΩt−2(1−(H2−Ω2)/4A2)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+(1−ω2\n−)(C2\nr+C2\ni)\n(1+ω2\n+)e−2αΩt−2(1+(H2−Ω2)/4A2)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+(1+ ω2\n−)(C2\nr+C2\ni).\n(19c)\nFrom Eqs.(19), we can observe that continuous oscillations are pos sible only\nin the absence of damping ( α= 0). Also, in the presence of damping, when\nt→ ∞\nSx\n0(∞) =2ω−\n1+ω2\n−, Sy\n0(∞) = 0, Sz\n0(∞) =1−ω2\n−\n1+ω2\n−.(20)\nEqs.(20) clearly show that in the presence of damping the external field\nenablesS0to reach the steady state in the xz-plane. The time period of the\n7oscillations can be determined as T= 2π/Ω. The spin excitations of Sx\n0,Sy\n0\nandSz\n0in the absence and presence of damping are plotted in Figs.1 and 2\nrespectively by using the expressions given in Eqs.(19). The dots co rrespond\nto the numerical results plotted from Eqs.(11). Fig.1(d) confirms t he closed\nperiodic oscillations of the components in the absence of damping.\n4. One-spin excitation in the presence of spin-transfer tor que and\nfield-like torque\nBy considering the one-dimensional spin chain in the free layer of a sp in-\nvalve (tri-layer) structure, the dynamics of the nthspin in the presence of\ncurrent is governed by the following LLGS equation[21, 27],\ndSn\ndt=Sn×Heff+αSn×(Sn×Heff)+jSn×(Sn×Sp)+j βSn×Sp,(21)\nwherejis the magnitude of the spin-transfer torque [28] which can also be\nequivalently called the damping-like torque [29] and βis the magnitude of\nthe field-like torque. Sp= (1,0,0) is the polarization vector of the pinned\nlayer. The equations of motion of S0in the absence of onsite anisotropy and\nperpendicular field are obtained by substituting Eq.(6) in Eq.(21) as\ndSx\n0\ndt=−2αA2(1−(Sx\n0)2), (22a)\ndSy\n0\ndt= 2A1Sz\n0+α[2A2Sx\n0Sy\n0], (22b)\ndSz\n0\ndt=−2A1Sy\n0+α2A2Sx\n0Sz\n0, (22c)\nwhereA1=A+(jβ/2) andA2=A+(j/2α). Eqs.(22) can be transformed\ninto stereographic form using Eqs.(13) as follows:\ndω\ndt=−i(A1+iαA2)(ω2−1). (23)\nEq.(23) can be solved as\nω=1+C e2(αA2−iA1)t\n1−Ce2(αA2−iA1)t, ω∗=1+C∗e2(αA2+iA1)t\n1−C∗e2(αA2+iA1)t, (24)\n8whereC=Cr+iCi=ω(0)−1\nω(0)+1andC∗=Cr−iCi=ω∗(0)−1\nω∗(0)+1are arbitrary\nconstants. CrandCiare obtained as\nCr=(Sx\n0(0)−Sz\n0(0)−1)(Sx\n0(0)+Sz\n0(0)+1)+( Sy\n0(0))2\n(Sx\n0(0)+Sz\n0(0)+1)2+(Sy\n0(0))2,(25)\nCi=2Sy\n0(0)(1+Sz\n0(0))\n(Sx\n0(0)+Sz\n0(0)+1)2+(Sy\n0(0))2. (26)\nBy substituting Eqs.(24) in Eqs.(13) we can get the components of S0in the\npresence of spin-transfer and field-like torques as\nSx\n0=e−2(2αA+j)t−(C2\nr+C2\ni)\ne−2(2αA+j)t+(C2r+C2\ni), (27a)\nSy\n0=2e−(2αA+j)t(Cicos[(2A+jβ)t]−Crsin[(2A+jβ)t])\ne−2(2αA+j)t+(C2\nr+C2\ni),(27b)\nSz\n0=−2e−(2αA+j)t(Crcos[(2A+jβ)t]+Cisin[(2A+jβ)t])\ne−2(2αA+j)t+(C2r+C2\ni).(27c)\nFromEq.(27a)onecanverifythatirrespective ofthefield-like torq ue(asymp-\ntotically for large t)S0approaches (-1,0,0) or (1,0,0) when 2 αA+j >0 or\n2αA+j <0, respectively. Also, thecurrent damps outthesystem even inth e\nabsence of damping. Further, the periodic oscillations appear when the con-\ndition 2αA+j= 0 is satisfied. From Eqs.(27b) and (27c) it can be observed\nthat the angular frequency of the oscillations is 2 A+jβ, which interestingly\nimplies that the current can enhance the frequency of the oscillatio ns only in\nthe presence of field-like torque.\n5. Dynamics of one-spin excitation in the combined presence of\nperpendicular field, spin-transfer torque and field-like to rque\nEquations of motion in the presence of perpendicular magnetic field H=\n(0,0,H), spin-transfer torque and field-like torque without onsite anisot ropy\ncan be written from Eq.(21) as\ndSx\n0\ndt=HSy\n0+α/bracketleftbig\n−2A2(1−(Sx\n0)2)+HSx\n0Sz\n0/bracketrightbig\n, (28a)\ndSy\n0\ndt= 2A1Sz\n0−HSx\n0+α[2A2Sx\n0Sy\n0+HSy\n0Sz\n0], (28b)\ndSz\n0\ndt=−2A1Sy\n0+α/bracketleftbig\n2A2Sx\n0Sz\n0−H(1−(Sz\n0)2)/bracketrightbig\n, (28c)\n9Figure 3: (Color online) Undamped oscillations for (a) Sx\n0, (b)Sy\n0and (c) Sz\n0when\nA= 0.1,D= 0, H= 0.1,α= 0.005, j=−0.00125016 and β= 0.1. Here the red\nlines and black dots are plotted from analytical (Eq.(34)) and numer ical (Eq.(28)) re-\nsults, respectively. The initial conditions are (0.10,0.40,0.91). (d) Th e three-dimensional\ntrajectory of S0.\nwhereA1=A+(jβ/2) andA2=A+(j/2α). Eqs.(28) are transformed into\na stereographic equation using Eqs.(13) as follows:\ndω\ndt= (A2α−iA1)/bracketleftbigg\nω2+H(1+iα)\n(A1+iαA2)ω−1/bracketrightbigg\n= (A2α−iA1)(ω−ω+)(ω−ω−), (29)\nwhere now\nω±=/braceleftBigg\nPr±/radicalBigg\n(Q2r+Q2\ni)1/2\n1+(Qi/2Qr)2/bracerightBigg\n+i/braceleftBigg\nPi±(Qi/2Qr)/radicalBigg\n(Q2r+Q2\ni)1/2\n1+(Qi/2Qr)2/bracerightBigg\n,\n(30)\n10and\nPr=−H(A2α2+A1)\n2(A2\n2α2+A2\n1), Pi=−αH(A1−A2)\n2(A2\n2α2+A2\n1),\nQr=H2(A2α2+A1)2−H2α2(A1−A2)2+4(A2\n2α2+A2\n1)2\n4(A2\n2α2+A2\n1),\nQi=2αH2(A2α2+A1)(A1−A2)\n4(A2\n2α2+A2\n1)2.\nBy solving Eq.(29), we get\nω(t) =ω+−C ω−e(K+iΩ)t\n1−C e(K+iΩ)t, ω∗(t) =ω∗\n+−C∗ω∗\n−e(K−iΩ)t\n1−C∗e(K−iΩ)t,(31)\nwhereω∗\n+andω∗\n−are complex conjugates of ω+andω−, respectively. Cand\nC∗can be derived from Eq.(31) as C=Cr+i Ci=ω+−ω(0)\nω−−ω(0),C∗=Cr−i Ci=\nω∗\n+−ω(0)∗\nω∗\n−−ω(0)∗, whereω(0) =Sx\n0(0)+i Sy\n0\n1+Sz\n0(0)andω(0)∗=Sx\n0(0)−i Sy\n0\n1+Sz\n0(0). HereKand Ω are\ngiven by\nK= 2/radicalBigg\n(Q2\nr+Q2\ni)1/2\n1+(Qi/2Qr)2[A2α+A1(Qi/2Qr)], (32)\nΩ = 2/radicalBigg\n(Q2\nr+Q2\ni)1/2\n1+(Qi/2Qr)2[A2α(Qi/2Qr)−A1]. (33)\nFrom Eqs.(31), the components of S0can be determined using Eq.(13) as\nSx\n0= 2/braceleftbiggT1+T2e−2Kt+2[T3sin(Ωt)−T4cos(Ωt)]e−Kt\nT5+T6e−2Kt+2[T7sin(Ωt)+T8cos(Ωt)]e−Kt/bracerightbigg\n,(34a)\nSy\n0= 2/braceleftbiggT9+T10e−2Kt+2[T11sin(Ωt)+T12cos(Ωt)]e−Kt\nT5+T6e−2Kt+2[T7sin(Ωt)+T8cos(Ωt)]e−Kt/bracerightbigg\n,(34b)\nSz\n0=/braceleftbiggT13+T14e−2Kt+2[T15sin(Ωt)+T16cos(Ωt)]e−Kt\nT5+T6e−2Kt+2[T7sin(Ωt)+T8cos(Ωt)]e−Kt/bracerightbigg\n,(34c)\nwhere the explicit forms of T1,T2,...,T16are given in the Appendix.\nFrom Eqs.(34) we can identify that S0damps out and reaches steady\nstate when K/negationslash= 0. The steady state values of Sx\n0,Sy\n0andSz\n0are given by\nSx\n0=T1\nT5, Sy\n0=T9\nT5, Sz\n0=T13\nT5,asttends to ∞whenK >0,\nSx\n0=T2\nT6, Sy\n0=T10\nT6, Sz\n0=T14\nT6,asttends to ∞whenK <0.\n11 0 0.6 1.2\n-1 -0.5  0  0.5  1Ω\nβ\nFigure 4: (Color online) Enhancement of angular frequency of unda mped oscillations by\nthe field-like torque when A= 0.1,D= 0,α= 0.005 and j= 0.1.\nWhenK= 0, the undamped oscillations appear. The values of current,\nfield and field-like torque for which undamped oscillations are possible c an\nbe obtained from Eq.(32) as follows:\nA2α+A1(Qi/2Qr) = 0,\nH2+4A2/parenleftbig\nA12+α2A22/parenrightbig2\nA13+3α2A1A22+α4A23−α2A23= 0. (35)\nThe angular frequency of the undamped oscillations is derived by usin g\nEq.(35) in Eq.(33) as\nΩ =−2/bracketleftBigg\n(A1−A2)(A1+α2A2)/parenleftbig\nA12+α2A22/parenrightbig/radicalbig\nA12+4α2A22\nA13+3α2A1A22+α2(α2−1)A23/bracketrightBigg1/2\n.(36)\nThe existence of undamped oscillations is confirmed by plotting Sx\n0,Sy\n0and\nSz\n0in Figs.3(a), (b) and (c) respectively when H= 0.1, β= 0.1 andj=\n−0.00125. Also, Fig.3(d) shows the three-dimensional trajectory of S0of the\nundamped oscillations. The enhancement of angular frequency of u ndamped\noscillations by field-like torque, plotted from Eq.(36), is shown in Fig.4 w hen\nA= 0.1,D= 0,α= 0.005 and j= 0.1. Also, the enhancement of angular\nfrequency by the introduction of onsite anisotropy Din Eqs.(28) is shown\nappropriately in Fig.5 when A= 0.1,H= 0.1,α= 0.005,β= 0.1 and\nj=−0.01, from appropriate numerical analysis.\n12 0.4 0.8 1.2 1.6\n 0  0.4  0.8  1.2  1.6Ω\nD\nFigure 5: (Color online) Enhancement of frequency of undamped os cillations by the\nanisotropy DwhenA= 0.1,α= 0.005,β= 0.1,H= 0.1 andj=−0.01.\n6. One-spin excitation in the combined presence of parallel field,\nspin-transfer torque and field-like torque\nWhentheexternalmagneticfieldisappliedparalleltothechain, i.e. alo ng\nx-axis, the dynamical equations can be obtained from Eq.(21) by co nsidering\nH= (H,0,0) as\ndSx\n0\ndt= 2DSy\n0Sz\n0−[j+α(2A+H)](1−(Sx\n0)2)+2αDSx\n0(Sz\n0)2,(37a)\ndSy\n0\ndt= (2A+H+jβ)Sz\n0+[j+α(2A+H)]Sx\n0Sy\n0+2αDSy\n0(Sz\n0)2,(37b)\ndSz\n0\ndt=−(2A+H+jβ)Sy\n0+[j+α(2A+H)]Sx\n0Sz\n0−2αDSz\n0(1−(Sz\n0)2).\n(37c)\nThe above Eqs.(37) for D= 0 are transformed into a stereographic equation\nusing Eq.(13) as\ndω\ndt=−1\n2[α(2A+H)+j−i(2A+H+jβ)](1−ω2).(38)\nEq.(38) is solved as\nω=1+Ce[j+α(2A+H)−i(2A+H+jβ)]t\n1−Ce[j+α(2A+H)−i(2A+H+jβ)]t, ω∗=1+C∗e[j+α(2A+H)+i(2A+H+jβ)]t\n1−C∗e[j+α(2A+H)+i(2A+H+jβ)]t,\n(39)\n13whereC=Cr+iCi=ω(0)−1\nω(0)+1andC∗=Cr−iCi=ω∗(0)−1\nω∗(0)+1are arbitrary\nconstants. Here CrandCican be obtained by using Eqs.(12) as in Eqs.(25)\nand (26). By substituting Eqs.(39) into Eqs.(13) we can derive\nSx\n0=1−(C2\nr+C2\ni)e2[j+α(2A+H)]t\n1+(C2r+C2\ni)e2[j+α(2A+H)]t, (40a)\nSy\n0= 2e[j+α(2A+H)]t/braceleftbiggCicos([2A+H+jβ]t)−Crsin([2A+H+jβ]t)\n1+(C2r+C2\ni)e2[j+α(2A+H)]t/bracerightbigg\n,\n(40b)\nSz\n0=−2e[j+α(2A+H)]t/braceleftbiggCrcos([2A+H+jβ]t)+Cisin([2A+H+jβ]t)\n1+(C2r+C2\ni)e2[j+α(2A+H)]t/bracerightbigg\n.\n(40c)\nFrom Eqs.(40), one can understand that the spin Sx\n0switches asymptotically\n(t→ ∞) to +1 or −1 whenj+α(2A+H)<0 orj+α(2A+H)>0,\nrespectively. Theangularfrequencyoftheoscillationsisgivenby2 A+H+jβ,\nwhich implies that the angular frequency is independent of the curre nt in the\nabsence of field-like torque. Further, it can be noticed that the un damped\noscillations in the presence of parallel field are possible when the cond ition\nj+α(2A+H) = 0 is satisfied.\nManipulation of single electron spin states in solids is receiving much\nattention for quantum computing [30, 31], mainly for localized electro n spins\nin solids which show long relaxation and coherence times and their stat es can\nbe easily manipulated via microwave or radio frequency pulses [32]. Also ,\nsingle spin dynamics in a Heisenberg XXZ spin chain has been studied for a\nquantum transistor [33] and coherent manipulation of a single spin st ate by\nmicrowave pulses has been investigated [34].\nThe formation of localized spin excitations in a magnetic layer is exper-\nimentally possible. It has been proved that by means of antiferroma gnetic\ncoupling a reference layer with fixed magnetization direction can be f ormed\nfrom an oppositely magnetized pinned layer. Thus, it is possible to for m a\nferromagnetic layer with fixed direction of magnetization [35]. It ha s been\nexperimentally proved that by placing a nano-contact in this fixed lay er, the\nlocalized region of magnetization beneath it can be excited by passing a cur-\nrent [36, 37]. These works demonstrate the possibility of exciting loc alized\nspins without altering the spins outside of the localized region and red ucing\nthe number of spins by reducing the cross-sectional area of the n ano-contact.\n14Thus, thespin transfer torquecannot affect thespins other tha nthe localized\nspins.\n7. Two-spin excitation in the presence of perpendicular fiel d, spin-\ntransfer torque and field-like torque\nThe studies onone-spinexcitation canbeextended intomulti-spin ex cita-\ntionsingeneral. Inthissection we numerically study thetwo-spin exc itations\nin the presence of perpendicular field, current and field-like torque . The case\nof parallel field can also be similarly analyzed. Considering the one dimen -\nsional spin chain with the excitation of two spins S0andS1as follows,\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).....,(41)\nthe Hamiltonian for this system, with perpendicular external field H=\n(0,0,H) along positive zdirection, is written from Eq.(1) as\nH=−[(N−4)A+ASx\n0+ASx\n0Sx\n1+ASx\n1+BSy\n0Sy\n1+CSz\n0Sz\n1]\n−D(Sz\n0)2−D(Sz\n1)2−HSz\n0−HSz\n1. (42)\nThe corresponding effective fields for the two spins S0andS1can be derived\nas\nHeff,S0=A(1+Sx\n1)ˆi+BSy\n1ˆj+[CSz\n1+H+2DSz\n0]ˆk, (43)\nHeff,S1=A(1+Sx\n0)ˆi+BSy\n0ˆj+[CSz\n0+H+2DSz\n1]ˆk. (44)\nThe LLGS equations corresponding to the spins Sn, n= 0,1,in the presence\nof field and current are given by\ndSn\ndt=Sn×Heff,Sn+αSn×(Sn×Heff,Sn)+jSn×(Sn×Sp)+j βSn×Sp.(45)\nThe corresponding dynamical equations for the components of S0andS1\n15Figure 6: (Color online) Undamped oscillations of S0(black line) and S1(red dots). (a)\nSx\n0,Sx\n1(b)Sy\n0,Sy\n1, (c)Sz\n0,Sz\n1and (d) magnetization trajectory when A= 0.1,B=\n0.1,C= 0.1,D= 0, H= 0.1414,α= 0.005, j=−0.001, β= 0.1. The initial conditions\nare (0.6,0.8,0.0).\nwithSp= (1,0,0) can be derived as\ndSx\n0\ndt=CSy\n0Sz\n1+2DSy\n0Sz\n0+HSy\n0−BSz\n0Sy\n1\n+α[−A(1−(Sx\n0)2)(1+Sx\n1)+BSx\n0Sy\n0Sy\n1+CSx\n0Sz\n0Sz\n1+2DSx\n0(Sy\n0)2+HSx\n0Sz\n0]\n−j(1−(Sx\n0)2), (46a)\ndSy\n0\ndt=ASz\n0(1+Sx\n1)−CSx\n0Sz\n1−2DSx\n0Sz\n0−HSx\n0\n+α[ASx\n0Sy\n0(1+Sx\n1)−BSy\n1(1−(Sy\n0)2)+CSy\n0Sz\n0Sz\n1+2DSy\n0(Sz\n0)2+HSy\n0Sz\n0]\n+jSx\n0Sy\n0+jβSz\n0, (46b)\ndSz\n0\ndt=BSx\n0Sy\n1−ASy\n0(1+Sx\n1)\n+α[ASx\n0Sz\n0(1+Sx\n1)+BSy\n0Sz\n0Sy\n1−(CSz\n1+2DSz\n0+H)(1−(Sz\n0)2)]\n+jSx\n0Sz\n0−jβSy\n0, (46c)\n16-1 0 1\n 19500  19750  20000(a)\nS0x, S1x\nt-1 0 1\n 19500  19750  20000(b)\nS0y, S1y\nt-1 0 1\n 19500  19750  20000(c)\nS0z, S1z\nt\nFigure 7: (Color online) Anti-phase synchronized oscillations of S0(black line) and S1(red\nline). (a) Sx\n0,Sx\n1(b)Sy\n0,Sy\n1and (c) Sz\n0,Sz\n1whenA= 0.1,B= 0.1,C= 0.1,D=\n0, H= 0.1414,α= 0.005, j=−0.001, β= 0. The initial conditions are (0.6,0.8,0.0) and\n(0.61,0.79,0.0).\ndSx\n1\ndt=CSy\n1Sz\n0+2DSy\n1Sz\n1+HSy\n1−BSz\n1Sy\n0\n+α[−A(1−(Sx\n1)2)(1+Sx\n0)+BSx\n1Sy\n1Sy\n0+CSx\n1Sz\n1Sz\n0+2DSx\n1(Sy\n1)2+HSx\n1Sz\n1]\n−j(1−(Sx\n1)2), (47a)\ndSy\n1\ndt=ASz\n1(1+Sx\n0)−CSx\n1Sz\n0−2DSx\n1Sz\n1−HSx\n1\n+α[ASx\n1Sy\n1(1+Sx\n0)−BSy\n0(1−(Sy\n1)2)+CSy\n1Sz\n1Sz\n0+2DSy\n1(Sz\n1)2+HSy\n1Sz\n1]\n+jSx\n1Sy\n1+jβSz\n1, (47b)\ndSz\n1\ndt=BSx\n1Sy\n0−ASy\n1(1+Sx\n0)\n+α[ASx\n1Sz\n1(1+Sx\n0)+BSy\n1Sz\n1Sy\n0−(CSz\n0+2DSz\n1+H)(1−(Sz\n1)2)]\n+jSx\n1Sz\n1−jβSy\n1. (47c)\nEqs.(46) and (47) for the case D= 0 can be transformed into the stereo-\n17Figure 8: (Color online) Damped oscillations of (a) Sx\n0(b)Sy\n0and (c)Sz\n0forA= 0.1,B=\n0.1,C= 0.1,D= 0, H= 0.1414,α= 0.005, j=−0.001, β= 0. The initial conditions\nfor the two spins are (0.6,0.8,0.0). The insets for the intermediate ra nge of time show\nsynchronization of the respective components of the two spins.\ngraphic form as\ndω0\ndt=−A\n2(α−i)(1−ω2\n0)/parenleftbigg\n1+ω1+ω∗\n1\n1+ω1ω∗\n1/parenrightbigg\n−B\n2(α−i)(1+ω2\n0)/parenleftbiggω1−ω∗\n1\n1+ω1ω∗\n1/parenrightbigg\n+C(α−i)ω0/parenleftbigg1−ω1ω∗\n1\n1+ω1ω∗\n1/parenrightbigg\n+H(α−i)ω0, (48a)\ndω1\ndt=−A\n2(α−i)(1−ω2\n1)/parenleftbigg\n1+ω0+ω∗\n0\n1+ω0ω∗\n0/parenrightbigg\n−B\n2(α−i)(1+ω2\n1)/parenleftbiggω0−ω∗\n0\n1+ω0ω∗\n0/parenrightbigg\n+C(α−i)ω1/parenleftbigg1−ω0ω∗\n0\n1+ω0ω∗\n0/parenrightbigg\n+H(α−i)ω1. (48b)\nEqs.(46) and (47) are numerically solved and the undamped in-phase\nsynchronized oscillations of spins S0andS1are plotted in Figs.6 when A=\n0.1,B= 0.1,C= 0.1,D= 0, H= 0.1414,α= 0.005, j=−0.001, β=\n180.1 for the same initial conditions (0.6,0.8,0.0). Interestingly, the two- spin\nsystem shows anti-phase synchronized oscillations when the initial c onditions\nare slightly different. Figs.7 show the undamped anti-phase synchro nized\noscillations of spins S0andS1whenA= 0.1,B= 0.1,C= 0.1,D= 0, H=\n0.1414,α= 0.005, j=−0.001, β= 0.1 for the different initial conditions\n(0.6,0.8,0.0) and (0.61,0.79,0.0). Damped oscillations in the absence of fie ld-\nlike torque are shown in Figs.8. Same results are obtained by solving th e\nsystem (48) as well.\n8. Effect of thermal noise on one-spin excitation in the prese nce of\nperpendicular field\n-1 0 1\n 0  50  100(a)\nS0x\nt(ns) 0 1\n 99  99.5  100(b)S0x\nt(ns)\n-1 0 1\n 99  99.5  100(c)\nS0y\nt(ns)\nFigure 9: (Color online) Numerically plotted temporal evolutions of (a ) & (b)Sx\n0and (c)\nSy\n0whenA= 1000 Oe, D=0,H= 1000 Oe (perpendicular field), α=0.005,j=-12.5016\n(-0.1043 mA) and β= 0.1. Here the red and black lines are plotted in the presence ( T=\n300 K) and absence ( T= 0 K) of thermal noise, respectively. The initial conditions are\n(0.10,0.40,0.91). (d) The three-dimensional trajectory of S0with (red line) and without\n(black line) the thermal noise.\n19Wealso investigate nowtheeffectofthermalfluctuationsonthedy namics\nof one-spin excitation. It is carried out by including the thermal field due to\nthermal noise in the effective field as follows [23]:\nHeff= 2Aˆi+[2DSz\n0+H]ˆk+Hth, (49)\nwhere the thermal field is given by\nHth=√\nFG, F=2αkBT\n(1+α2)Msµ0V△t. (50)\nIn the above equation, Gis the Gaussian random number generator vector\nof the oscillator with components ( Gx,Gy,Gz), which satisfies the statistical\nproperties /angbracketleftGm(t)/angbracketright= 0 and/angbracketleftGm(t)Gn(t′)/angbracketright=δmnδ(t−t′) for allm,n=x,y,z.\nHerekBis the Boltzmann constant, Tis the temperature, Ms= 1448.4\nemu/cc is the saturation magnetization, µ0is the magnetic permeability in\nfree space, V = 2.5 ×64×64 nm3[17] is a typical volume of the free layer and\n△tis the step size of the time scale used in the simulation.\nThe temporal evolution of Sx\n0is plotted in Fig.9(a) for the time range t\n= 0 to 100 ns, where the black and red solid lines are plotted in the abse nce\n(T= 0 K) and presence ( T= 300 K) of thermal noise, respectively, for the\nparameters [21, 23] A= 1000 Oe, D=0,H= 1000 Oe (perpendicular field),\nα=0.005,j=-12.5016 (-0.1043 mA) and β= 0.1 (See Appendix B). In Fig.9\nwe observe that there is a slight variationin the oscillation boundary w ithout\nany change in the amplitude. The smooth oscillations even in the prese nce\nof thermal noise are confirmed by plotting the temporal evolutions ofSx\n0,Sy\n0\nandSz\n0in Figs.9(b), (c) and (d), respectively. From these figures we obse rve\nthat the thermal noise only very slightly affects the time evolution of the\nspin.\n9. Conclusions\nBy solving the LLGS equation along with field-like torque we have an-\nalytically deduced the expressions for one-spin excitation in the pre sence of\nperpendicular/parallel fields. It has been observed that the field- like torque\nis essential to enhance the frequency of the oscillations and it incre ases the\nfrequency of oscillations for both the cases of perpendicular and p arallel\nmagnetic fields. Relevant conditions have been obtained among the c urrent,\nmagnetic field and field-like torque to obtain the undamped oscillations . The\n20numerical study has been extended to the case of two-spin excita tions and\nthe possibility of undamped in-phase and anti-phase synchronized o scilla-\ntions has been shown between the two spins. The investigations on o ne-spin\nexcitation against thermal fluctuations show that the system is on ly slightly\naffected by thermal noise. Our results are potentially important fo r under-\nstanding the spin dynamics in relevant magnetic materials and struct ures\n[1, 5].\n21Appendix A\nIn this appendix, we provide the full expressions of various parame ters\nTi, i= 1, 2, ...,16, given in Eq.(34):\nT1= (C2\nr+C2\ni)(Pr/radicalbig\n1+(Qi/2Qr)2−(Q2\nr+Q2\ni)1/4),\nT2= (Q2\nr+Q2\ni)1/4+Pr/radicalbig\n1+(Qi/2Qr)2,\nT3=CiPr/radicalbig\n1+(Qi/2Qr)2−Cr(Q2\nr+Q2\ni)1/4(Qi/2Qr),\nT4=CrPr/radicalbig\n1+(Qi/2Qr)2+Ci(Q2\nr+Q2\ni)1/4(Qi/2Qr),\nT5= (C2\nr+C2\ni)/bracketleftbig\n−2(Q2\nr+Q2\ni)1/4(Pr+Pi(Qi/2Qr))\n+/radicalbig\n1+(Qi/2Qr)2(1+P2\nr+P2\ni+/radicalBig\nQ2r+Q2\ni)/bracketrightbigg\n,\nT6= 2(Q2\nr+Q2\ni)1/4(Pr+Pi(Qi/2Qr))\n+/radicalbig\n1+(Qi/2Qr)2(1+P2\nr+P2\ni+/radicalBig\nQ2\nr+Q2\ni),\nT7= 2Cr(Q2\nr+Q2\ni)1/4(Pi−Pr(Qi/2Qr))\n+2Ci/radicalbig\n1+(Qi/2Qr)2(1+P2\nr+P2\ni+/radicalBig\nQ2\nr+Q2\ni),\nT8= 2Ci(Q2\nr+Q2\ni)1/4(Pi−Pr(Qi/2Qr))\n−2Cr/radicalbig\n1+(Qi/2Qr)2(1+P2\nr+P2\ni+/radicalBig\nQ2r+Q2\ni),\nT9= (C2\nr+C2\ni)(Pi/radicalbig\n1+(Qi/2Qr)2−(Q2\nr+Q2\ni)1/4(Qi/2Qr)),\nT10= (Q2\nr+Q2\ni)1/4(Qi/2Qr)+Pi/radicalbig\n1+(Qi/2Qr)2,\nT11=Cr(Q2\nr+Q2\ni)1/4+CiPi/radicalbig\n1+(Qi/2Qr)2,\nT12=Ci(Q2\nr+Q2\ni)1/4−CrPi/radicalbig\n1+(Qi/2Qr)2,\nT13= (C2\nr+C2\ni)/bracketleftBig\n2(Q2\nr+Q2\ni)1/4(Pr+Pi/radicalbig\n1+(Qi/2Qr)2)\n−/radicalbig\n1+(Qi/2Qr)2(1−P2\ni−P2\nr−(Q2\nr+Q2\ni)1/4)/bracketrightBig\n,\nT14=−2(Q2\nr+Q2\ni)1/4(Pr+Pi/radicalbig\n1+(Qi/2Qr)2)\n+/radicalbig\n1+(Qi/2Qr)2(1−P2\ni−P2\nr−(Q2\nr+Q2\ni)1/4),\nT15= 2Cr(Q2\nr+Q2\ni)1/4(Pr(Qi/2Qr)−Pi)−Ci/radicalbig\n1+(Qi/2Qr)2(P2\ni+P2\nr−1−(Q2\nr+Q2\ni)),\nT16= 2Cr(Q2\nr+Q2\ni)1/4(Pr(Qi/2Qr)−Pi)+Ci/radicalbig\n1+(Qi/2Qr)2(P2\ni+P2\nr−1−(Q2\nr+Q2\ni)).\n22Appendix B: Comparison of numerical parameters with realis tic\nmaterial parameters\nHere we will briefly explain the procedure to deduce the expressions for\ncurrentjandthe coefficient of field-like torque βby comparing Eq. (21) with\nthe standard form of Landau-Lifshitz-Gilbert-Slonczewski (LLG S) equation\nutilized for the spin torque nano oscillator (STNO) that consists of a ferro-\nmagnetic free layer and pinned layer with a nonmagnetic conducting s pacer\nlayer which separates the ferromagnetic free and pinned layers.\nAs discussed in Sec. 4 the LLGS equation for a spin in the presence of\nperpendicular field, current and field-like torque is given by\ndS\ndt=S×Heff+αS×(S×Heff)+jS×(S×Sp)+j βS×Sp,(A.1)\nwhere\nHeff= 2Aˆi+Hˆk. (A.2)\nUsing the orthogonality relation S.dS\ndt= 0, one can deduce from Eq.(A.1) the\nfollowing equation:\nS×dS\ndt=S×(S×Heff)−αS×Heff−jS×Sp+jβS×(S×Sp).\n(A.3)\nFrom Eq.(A.3) we can derive,\nS×(S×Heff) =S×dS\ndt+αS×Heff+jS×Sp−jβS×(S×Sp).\n(A.4)\nBy substituting Eq.(A.4) in Eq.(A.1) we obtain,\ndS\ndt= (1+α2)S×Heff+αS×dS\ndt+j(1−αβ)S×(S×Sp)+j(α+β)S×Sp.\n(A.5)\nWith a rescaling of time t→ −γ\n1+α2t, we get\ndS\ndt=−γS×Heff+αS×dS\ndt−γj1−αβ\n1+α2S×(S×Sp)−γjα+β\n1+α2S×Sp,\n(A.6)\n23The standard form of LLGS equation used for studying the unit mag netiza-\ntion vector mof the free layer of the STNO is given by [21, 23]\ndm\ndt=−γm×H′\neff+αm×dm\ndt+γHsm×(m×mp)−γHsβ′S×mp,\n(A.7)\nwhere\nH′\neff= (Hx+Kxmx)ˆi+(Hy+Kymy)ˆj+[Hz+(Kz−Nz)mz]ˆk,(A.8)\nand\nHs=¯hηI\n2eMsV. (A.9)\nHereH′effis the effective field that includes the external fields Hx,Hyand\nHzalongx,yandzdirections, respectively, anisotropy fields Kx,Kyand\nKzalongx,yandzdirections, respectively, and demagnetization field Nz\nin the free layer, γis the gyromagnetic ratio, αis the damping constant,\nthe unit vector mp= (1,0,0) is along the polarization of the pinned layer\nandβ′is field-like torque, ¯ h(=h/2π) is the reduced Planck’s constant, η\nis the dimensionless parameter which determines the magnitude of th e spin\ntransfer torque, Iis the current flowing through the free layer, eis charge\nof the electron, Msis the saturation magnetization and Vis the volume of\nthe free layer. Here, the demagnetization field has been included on ly for\nz-direction since the normal of the free layer plane is along the z-direction.\nBy comparing Eqs.(A.6) and (A.7) we obtain the relations\nI=−2eMsVj\n¯hη/parenleftbigg1−αβ\n1+α2/parenrightbigg\n, β′=α+β\n1−αβ, (A.10)\nand similarly by comparing Eqs.(A.2) and (A.8) we get\nHx= 2A, Hy= 0, Hz=H, Kx= 0, Ky= 0, Kz−Nz= 0.(A.11)\nThe material parameters are adopted from Refs. [21, 23], and are given\nbyα= 0.005, |β′| ≤0.5 (which gives the condition -0.506 ≤β≤0.493),η\n= 0.54,Ms= 1448.3 emu/c.c., V= 2.5×64×64 nm3. To verify the impact\nof thermal fluctuations we have numerically plotted the temporal e volutions\nofSand spin trajectory using Eq.(A.7) in Figs.9 with ( T= 300 K) and\nwithout ( T= 0 K) the thermal noise for the choice of the parameters [21, 23]\nA= 1000Oe, D=0,H= 1000Oe(perpendicular field), α=0.005,j=-12.5016\n(I = -0.1043 mA) and β= 0.1.\n24Acknowledgements\nThe research work of ML and RA was supported by a DST-SERB Distin -\nguished Fellowship (No.: SERB/F/6717/2017-18). ML also wishes to t hank\nthe Center for Nonlinear Studies, Los Alamos National Laboratory , USA for\nits warm hospitality during his visit in the summer of 2019. This work was\nsupported in part by the U.S. Department of Energy.\nReferences\nReferences\n[1] B. Hillerbrands, K. Ounadjela, Spin Dynamics in Confined Magnetic\nStructures, vols. I & II, Springer, Berlin, 2002.\n[2] M. Lakshmanan, Philos. Trans. R. Soc. A 369 (2011) 1280.\n[3] B. Georges, V. Cros, and A. Fert, Phys. Rev. B 73 (2006) 0604 R.\n[4] Z. 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Islama,<\naPhysics Discipline, Khulna University, Khulna 9208, Bangladesh\nARTICLE INFO\nKeywords :\nMagnetization reversal\nShape anisotropy\nEnergy barrier\nStimulatedenergyabsorption _emissionABSTRACT\nWe investigate the influence of shape anisotropy on the magnetization reversal of a single-domain\nmagnetic nanoparticle driven by a circularly polarized linear down-chirp microwave field pulse\n(DCMP). Based on the Landau-Lifshitz-Gilbert equation, numerical results show that the three con-\ntrollingparametersofDCMP,namely,microwaveamplitude,initialfrequencyandchirprate,decrease\nwith the increase of shape anisotropy. For certain shape anisotropy, the reversal time significantly\nreduces. Thesefindingsarerelatedtothecompetitionofshapeanisotropyanduniaxialmagnetocrys-\ntalline anisotropy and thus to the height of energy barrier which separates the two stable states. The\nresultofdampingdependenceofmagnetizationreversalindicatesthatforacertainsampleshape,there\nexistsanoptimaldampingsituationatwhichmagnetizationisfastest. Moreover,itisalsoshownthat\ntherequiredmicrowavefieldamplitudecanbeloweredbyapplyingthespin-polarizedcurrentsimulta-\nneously. Theusageofanoptimumcombinationofbothmicrowavefieldpulseandcurrentissuggested\ntoachievecostefficiencyandfasterswitching. Sothesefindingsmayprovidetheknowledgetofabri-\ncate the shape of a single domain nanoparticle for the fast and power-efficient magnetic data storage\ndevice.\n1. Introduction\nObtaining fast and energy-efficient magnetization rever-\nsalofsingle-domainoftheperpendicularlymagnetizednanopar-\nticle is an interesting issue owing to its potential applica-\ntion in high-density data storage devices [1, 2, 3] and rapid\ndata access [4]. For high density, high thermal stability and\nlow error rate in device application, high anisotropy mate-\nrials [5] are needed. But it is a challenge to find out a way\nwith low energy to achieve the fastest magnetization rever-\nsalforhigh-anisotropymagneticnanoparticle. Overthepast\ntwo decades, several controlling parameters _driving forces\narediscoveredtoachievefastestmagnetizationreversalwith\nlow cost. Namely, magnetization reversal using a constant\nmagnetic field [6, 7], reversal by microwave field of con-\nstant frequency or time dependent frequency, either with or\nwithout a polarized electric current [8, 9, 10, 11, 12, 13,\n14, 15, 16, 17, 18, 19] and by spin transfer torque (STT)\nor spin orbit torque (SOT) [20, 21, 22, 23, 24, 25, 26, 27,\n28, 29, 30, 31, 32, 33, 34]. All methods mentioned above\naresufferingfromspecificdrawbacks[6,35,36,37,38,39,\n40, 41]. For example, magnetic field or microwave field\ndriven magnetization is not energy efficient and fast as ex-\npected. Incaseofcurrent(bySTT)orSOTdriven,thehigher\nthreshold current requirement is a bottleneck. Later on, re-\nsearchers are digressed to employ the microwave chirped\npulses (microwave with time-dependent frequency) which\ninduce fast and energy-efficient magnetization reversal [13,\n14, 15, 16, 17, 19] but still the required field amplitude, ini-\ntial frequency and chirp rate are not small as desired prac-\n<Corresponding authors:\ntorikul@phy.ku.ac.bd (M.T. Islam)\nORCID(s): 0000-0002-4168-8695 (Z.K. Juthy); 0000-0002-7500-1572\n(M.A.J. Pikul); 0000-0002-6742-2158 (M.A.S. Akanda);\n0000-0002-3846-4009 (M.T. Islam)tically. Recent studies [42, 43] have demonstrated that the\nfastmagnetizationreversalofacubicnanoparticle(i.e.,with\nzerodemagnetization _shapeanisotropy)obtainedbythelin-\neardownchirpmicrowavepulse(DCMP).Thephysicalmech-\nanismofthismodelisthattheDCMP(whosefrequencylin-\nearlydecreasesfrominitial +f0tofinal*f0)stimulatesmi-\ncrowave energy absorption (emission) by the magnetic mo-\nment before (after) crossing the energy barrier. The effi-\nciencyofthemicrowaveenergyabsorption _emissiondepends\nonhowcloselythemicrowavefrequencyandmagnetization\nintrinsic frequency match during reversal. This frequency-\nmatching is related to the energy barrier [44, 45]. However,\ninsmall-scalenanoparticle,demagnetizationfield _shapeanisotropy\n(whichwasexcludedpreviously)hasagreatimportance[46,\n47,44]because,itinteracts _opposesthedominateduniaxial\nmagnetocrystalline anisotropy and thus reduces the energy\nbarrier between two stable states. So, the magnetization re-\nversal of a nanoparticle with finite shape anisotropy might\nbefasterandenergyefficient. Inaddition,fabricatingperfect\ncubicnanoparticle(i.e.,withzerodemagnetizationfield)isa\nchallengingissuefrompracticalpointofview. Forthemost\ndevice applications, the desired microwave pulse (DCMP)\nshould be withsmaller amplitude, initial frequency and low\nchirp rate.\nTherefore,thisstudyfocusesonhowtheshapeanisotropy\nHshapeaffects the energy consumption i.e., the amplitude\nHmw, initial frequency foand pulse chirp rate \u0011of DCMP\nand the magnetization switching time ts. Interestingly, we\nfoundthattheparametersofDCMPi.e., Hmw,fo,and\u0011de-\ncrease with the increase of shape anisotropy Hshape. In ad-\ndition,thereisacritical Hshapeorcross-sectionalarea A,for\nwhichorlarger, tssignificantlyreducesto0.346nswhichis\nclose to the theoretical limit [40]. For instance, in case of\na sample of cross-sectional area 3232nm2, orHshape=\n0:598T, we estimated ts= 0:346ns,Hmw= 0:0187T,\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 1 of 7arXiv:2108.10965v2  [cond-mat.mes-hall]  2 Oct 2021Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\nfo=5:2GHz,andoptimal \u0011=24:96ns*2whicharesignif-\nicantly smaller than ts=0:56ns,Hmw=0:045T,fo=21\nGHz,andoptimal \u0011=67:2ns*2forthecubicshapedorzero\nHshape. Thephysicalreasonofthesefindingsisrelatedtothe\ncompetitionbetweenshapeanisotropyandtheuniaxialmag-\nnetocrystalline anisotropy. The findings of damping depen-\ndence of magnetization reversal indicate that, for a specific\nAorHshape,thereexistsaoptimaldampingsituation. More-\nover,thisstudyshowsthattherequiredmicrowavefieldam-\nplitude of DCMP can be lowered by applying direct current\nsimultaneously along with DCMP. Therefore, an optimum\ncombination of microwave field pulse and current could be\napplied to achieve faster and power efficient magnetization\nreversal. So these findings might be significant to design\ntheshapeofsingledomainnanoparticletoproducethelow-\nenergy cost magnetic storage device with high speed data\nprocessing.\n2. Analytical Model and Method\nWe consider a spin valve system that consists of a free\nand fixed ferromagnetic layers by sandwiching a nonmag-\nneticspacer,asshownschematicallyinFigure1(a). Thefree\nlayerisasquarenanoparticleofarea Aandthickness d. The\nmagnetization direction of the fixed and free layer is along\nzdirection, i.e., both are perpendicularly magnetized. The\nmagnetizationofthefreelayercanbetreatedasamacrospin\nrepresented by the unit moment mwith total magnetization\nAdMs, whereMsrepresents the saturation magnetization.\nThe demagnetization field, which generated by the magne-\ntization of nanoparticle, can be approximated by an easy-\nplane shape anisotropy. The demagnetization field _shape\nanisotropy field is expressed as Hshape=Hshapemzzsuch\nthatHshape=*\u00160.Nz*Nx/Ms,istheshapeanisotropyco-\nefficient,where NzandNxaredemagnetizationfactors[45]\nand\u00160=4\u001910*7N_A2is the vacuum magnetic perme-\nability. Because of the strong uniaxial magnetocrystalline\nanisotropy Hani=Hanimzz, the magnetization of the free\nlayer prefers to stay in two ground states, i.e., mparallel to\nzand*z.\nUnderthecircularlypolarizedDCMPandspinpolarized\ncurrent, the magnetization dynamics mis governed by the\nLandau-Lifshitz-Gilbert (LLG) equation [7, 9, 37, 48]\ndm\ndt=*\rmHeff+\u000bmdm\ndt*\rhsm.pm/(1)\nwhere\u000bisthedimensionlessGilbertdampingconstantand,\n\risgyromagneticratio. Heffisthetotaleffectivefieldwhich\ncontains the microwave magnetic field Hmw, the exchange\nfield2A\nMs(2mwhereAistheexchangestiffnessconstant,and\ntheeffectiveanisotropyfield Hkalongzdirection,i.e., Hk=\nHkmzz.hsis the strength of spin transfer torque(STT),\nhs=`PJ\n2e\u0016oMsd(2)\nwhereJ,e,`,P,\u00160andddenote the current density, elec-\ntron charge, the Planck’s constant, spin polarization of cur-\nmf0\n0-f(b) (a)\nτ\n-\n+\npFigure 1: (a) Schematic diagram of the system in where mand\npdenote the unit vectors of the magnetization of free layer\nand fixed layer, respectively. A linear down-chirp microwave\nfield and an electric current are applied on a single domain\nnanoparticle. (b) The frequency profile illustrates (sweeping\nfrom+f0to*f0) the linear down-chirp microwave field.\nrent,thevacuumpermeability,andthicknessofthefreelayer,\nrespectively.\nTheHkhas two components: an uniaxial magnetocrys-\ntalline anisotropy Haniand a shape anisotropy Hshapefield\nwhich can be expressed as Hk=Hani+Hshape= [Hani*\n\u00160.Nz*Nx/Ms]mzz. Thus, the resonant frequency of the\nnanoparticlecanbefoundfromtheKittelformula f0=\r\n2\u0019[Hani*\n\u00160.Nz*Nx/Ms]:\nWefirstapplyonlythemicrowavefield-drivenmagneti-\nzationreversal,i.e.,withoutelectriccurrent(SST)term,the\nrate of energy change is expressed as\n \u000f=*\u000b\róómHeffóó2*m\fHmw (3)\nSince damping is positive, so, the first term is always nega-\ntive. For time-dependent external microwave field, depend-\ning on the angle between the instantaneous magnetization\nmandHmw, the second term can be either positive or neg-\native, in other words, the microwave field pulse can trigger\nstimulated energy absorption or emission.\nIt has been shown that the fast magnetization reversal is\nachieved by DCMP [42] with the physical picture: the fre-\nquency of DCMP matches the frequency change of magne-\ntizationprecession. Hence,ittriggersstimulatedmicrowave\nabsorption (emission) by (from) the magnetization before\n(after) it crosses over the energy barrier.\nThemicrowavefieldthatwehaveappliedtakestheform,\nHmw=Hmw\u0004cos\u001e.t/x+sin\u001e.t/y\u0005(4)\nwhereHmwistheamplitudeofthemicrowavefieldand \u001e.t/\nis the phase. In this paper, we use a DCMP whose instanta-\nneous frequency is defined as f.t/ =1\n2\u0019d\u001e\ndt, which linearly\ndecreases with time at constant rate \u0011(in units of ns*2) as\nshown in Figure 1(b). The phase \u001e.t/and the instantaneous\nfrequencyf.t/are given by,\n\u001e.t/=2\u0019.f0t*\u0011\n2t2/;f.t/=f0*\u0011t (5)\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 2 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\n/s40/s98/s41/s40/s99/s41\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s49/s50/s116\n/s115/s32/s40/s110/s115/s41\n/s104 /s32/s40/s110/s115/s45/s50\n/s41/s32/s32 /s65\n/s48/s32\n/s32/s32 /s65\n/s49/s32/s32/s32\n/s32/s32 /s65\n/s50/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s49/s53/s51/s48/s52/s53/s54/s48/s104 /s32/s40/s110/s115/s45/s50\n/s41\n/s72\n/s115/s104/s97/s112/s101/s40/s84/s41/s48/s46/s48 /s48/s46/s51 /s48/s46/s54 /s48/s46/s57/s45/s49/s48/s49/s109\n/s122\n/s116/s32 /s40/s110/s115/s41/s32/s32 /s65\n/s48\n/s32/s32 /s65\n/s49/s32/s32\n/s32/s32 /s65\n/s50/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52\n/s32/s32 /s65\n/s53\n/s32/s32 /s65\n/s54\n/s32/s32 /s65\n/s55\n/s32/s32 /s65\n/s56/s40/s97/s41\nFigure 2: Model parameters of the free layer, Ms=106A/m,Aex=1310*12J_m,Hk=0:75T,\r=1:761011rad/(T \fs),\nand\u000b= 0:01. (a) Temporal evolution of mzinduced by DCMP with Hmw=0.045 T for the samples of different A. (b) The\ndependence of switching time tson the chirp rate \u0011for different cross-sectional areas while keeping the microwave field amplitudes\nHmwconstant. The vertical dashed lines indicate the lower and upper limits of \u0011for magnetization switching. (c) The dependence\nof chirp rate \u0011on shape anisotropy Hshapefor switching time within 1 ns.\n/s40/s99/s41/s40/s98/s41\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/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/s45/s57/s48 /s48 /s57/s48/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51\n/s115/s115/s115/s115/s115\n/s105/s105/s105/s105/s105/s105/s101 /s47/s77\n/s115/s72\n/s107\n/s113 /s32/s40/s100/s101/s103/s41/s32/s32 /s65\n/s48\n/s32/s32 /s65\n/s50/s32/s32\n/s32/s32 /s65\n/s52/s32\n/s32/s32 /s65\n/s53/s32\n/s32/s32 /s65\n/s54\n/s32/s32 /s65\n/s55\n/s32/s32 /s65\n/s56/s105/s115/s40/s97/s41\nFigure 3: (a) The potential \"along the line \u001e= 0 for the\nsamples of different A. Angles\u0012and\u001eare defined as cos*1mz\nandtan*1.mx_my/. The symbols i and s represent the initial\nstateandthesaddlepoint,respectively. Magnetizationreversal\ntrajectories of (b) A0=88nm2and (c)A8=3232 nm2.\nwherefoistheinitialfrequencyat t=0. Thedurationofthe\nmicrowavepulseis \u001c=2fo\n\u0011whichmeansthefinalfrequency\nis*fo. AccordingtotheappliedDCMP,thesecondtermof\nEq. (3) can be expressed as\n \"=*Hmwsin\u0012.t/sin\b.t/d\u001e\ndt(6)\nwhere\b.t/istheanglebetween mt(thein-planecomponent\nofmandHmw). Therefore, the microwave field pulse can\ntrigger the stimulated energy absorption (for *\b.t/) which\nis occurred before crossing the energy barrier and emission\n(for\b.t/) after crossing the energy barrier.\nFor this study, the parameters of Permalloy are chosen\nfrom typical experiments on microwave-driven magnetiza-\ntion reversal as Ms=106A_m,Hani=0:75T,\r=1:76\n1011rad_.T\fs/,Aex= 1310*12J_m and\u000b= 0:01. Al-\nthough, the demonstrated strategy and findings would still\nbe valid for other materials. The cell size used throughout\nthis study is .222/ nm3. We solved the LLG equa-tionnumericallyusingtheMUMAX3package[49]fortime-\ndependent circularly polarized microwave field. We con-\nsider the switching time window of 1ns and least magne-\ntization reversal mz=*0:9.\n3. Numerical Results\nIn a small-scale nanoparticle, demagnetization field or\nshapeanisotropyfield Hshapeisanimportantpropertytotake\ninto account in the investigation of the magnetization rever-\nsal. In order to approximate demagnetization field _shape\nanisotropyHshape,wechoosecuboidshapedsample _nanoparticle\nofvolumeV=Ad,whereAisthecross-sectionalareaand d\nistheheight. Toincreasethedemagnetizationfieldstrength\norHshape,Aisgraduallyincreasedwhilethethickness d=8\nnm is kept fixed. Specifically, we intend to investigate the\nmagnetization reversal of the samples, A1=1010 ,A2=\n1212,A3=1414 ,A4=1616 ,A5=2020 ,A6=\n2424,A7=2828 ,A8=3232 nm2. Foreachsample,\nby finding the demagnetization factors [45] NzandNx, the\nshape anisotropy coefficient Hshape=\u00160.Nz*Nx/Msare\ncalculated analytically. Hshapeactually acts along *zi.e.,\nit opposes the intrinsic easy axis anisotropy field Haniand\nhence reduces the effective anisotropy. Therefore, for each\nA,theresonance _theoreticalfrequencyisestimatedbywell-\nknownKittel-formula f0=\r\n2\u0019\u0004Hani*\u00160.Nz*Nx/Ms\u0005as\nshown in the Table 1.\nToinvestigatetheDCMP-drivenmagnetizationreversals\nof nanoparticles of different A, by keeping Hmw=0:045T\nandf0(close to resonant frequency corresponding to Aor\nHshape) fixed, we estimate the optimal \u0011for each sample or\nHshape. Figure2(b)showsthe \u0011dependenceof tsforseveral\nsamples orHshape. It is noted that there exists a window of\n\u0011denoted by vertical dashed line. For a small range of \u0011in\nthe right edge of window, the tsis minimal. It means that\nthereisaflexiblespacetochoose \u0011. Inaddition,theoptimal\n\u0011.Hshape/decreaseswith HshapeasshownintheFigure2(c).\nThese findings are useful in device application since it was\nreported that generating DCMP with higher chirp rate is a\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 3 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\nTable 1\nSimulated values of the controlling parameters for different samples of cross-sectional area A\nCross*sectional area Shape anisotropy field Natural frequency Simulated minimal frequency\nA(nm2) hshape(T) f0(GHz) f0(GHz)\nA1 0.09606 18.3 18.3\nA2 0.17718 16 17\nA3 0.2465 14.1 14.\nA4 0.3064 12.4 12.4\nA5 0.4049 9.6 10.2\nA6 0.4827 7.49 9\nA7 0.5457 5.72 7.5\nA8 0.5980 4.26 5.2\nchallenge [42]. Afterward, DCMP (with Hmw= 0:045,f0\ncorrespondingto Hshapeandoptimal\u0011(Hshape))drivenmag-\nnetization reversals are investigate and thus the time evolu-\ntions ofmzfor different HshapeorAare shown in the Fig-\nure 2(a). It is found that with the increase of Hshape, the\nmagnetization switching time tsremains almost same until\nA7= 2828 nm2but forA8g3232nm2, the mag-\nnetizationreversesverysmoothlyi.e., tsdropsto=0.34 ns\nwhichisclosetothetheoreticallimit(0.39 ns)[40]. So,itis\nmentionedthatthereisacritical Aandforlargervaluesthan\nit,theswitchingtimebecomessignificantlysmallerwhichis\nan ultimate demand in device application.\nTo understand the reason why fast reversal is obtained,\none can recall that the energy barrier is proportional to the\neffectiveanisotropyfield[39,40]. Sincethe Hshapeopposes\nthe anisotropy field Haniand thus reduces the height of the\nenergy barrier (energy difference between the initial state\nand saddle point). This reduction of energy barrier for dif-\nferentAispresentedintheFigure3(a). Therefore,themag-\nnetizationreversalbecomeseasierandfasterbecausethemi-\ncrowave frequency and magnetization precession frequency\nclosely matches as expected. To be more explicit, one may\nlookatthetrajectoriesofmagnetizationreversalfortwocases\nofAorHshapein Figure 3(b) and 3(c). For A0= 88\nnm2orHshape= 0, the magnetization takes a longer time\nbecause of some additional processions while crossing the\nhigher energy barrier. However, for A8= 3232 nm2\norHshape= 0:598T, the magnetization reversal is fairly\nsmoothi.e.,withoutanyadditionalprocessionssincetheen-\nergy barrier is reduced by Hshape. For more justification,\nFig4(a)showsthechangetheangle \b(blue)andmz(black\nline)withtimefor A8=3232 nm2. Notethattheangle \b\nremains almost 90oduring magnetization reversal which is\ndesiredfortriggeringanefficientmicrowaveenergyabsorp-\ntion (emission) by magnetization. Thus the corresponding\nenergychangingrate  \u000f(red),inFig4(c)isobtainedwithab-\nsorption and emission peaks (no distortion) which lead fast\nreversal. However, Fig 4(b), for A0= 88nm2i.e., for\nzeroHshape, shows the  \u000f(red line) absorption and emission\npeaks (distorted) indicating inefficient i.e., slower magneti-\nzation reversal (black line).\nThen,fortheDCMPwithfixed Hmw=0:045Tandop-\ntimal chirp rate \u0011.Hshape/, we obtain the fast magnetization\n/s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s45/s49/s48/s49\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s50/s55/s48/s45/s49/s56/s48/s45/s57/s48/s48/s57/s48/s49/s56/s48/s50/s55/s48/s70 /s32/s40/s100/s101/s103/s41\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s45/s49/s48/s49\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s49/s48/s49/s101 /s32/s40/s49/s48/s49/s53\n/s74/s47/s115/s32/s109/s51\n/s41/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s49/s48/s49\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51/s101 /s32/s40/s49/s48/s49/s53\n/s74/s47/s115/s32/s109/s51\n/s41/s40/s97/s41Figure 4: (a) ForA8= 3232 nm2, the plot of the relative\nangle\bagainst time (blue line) and the time dependence of\nmz(black line). Plots of the energy changing rate  \u000fof mag-\nnetization against time (red line) and the time dependence of\nmz(black line) for the cross-sectional area (b) A0=88nm2\nand (c)A8=3232 nm2\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s53/s49/s48/s49/s53/s50/s48/s32/s102\n/s111/s32/s40/s84/s104/s101/s111/s114/s101/s116/s105/s99/s97/s108/s41\n/s32/s102\n/s111/s32/s40/s83/s105/s109/s117/s108/s97/s116/s101/s100/s41\n/s32/s72\n/s109/s119\n/s72\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/s52/s48/s46/s48/s50/s56/s48/s46/s48/s51/s50/s72\n/s109/s119/s32/s40/s84/s41\nFigure 5: The dependence of initial frequency foand min-\nimum required field amplitude Hmwas a function of shape\nanisotropyHshapefor switching time within 1 ns.\nreversal of different Afor a certain range of f0around each\nresonance_naturalfrequency. Thereexistaminimal f0,i.e.,\nminimalf0correspondingtoeachsample,forwhichswitch-\ning time is lowest. Figure 5 shows the simulated minimal\nfrequencyf0(bluecircles)asafunctionof Hshapeandfound\nthatthesimulated f0(bluecircles)ofDCMPdecreaseswith\nincreasingHshape. These can be explained as the Hshapere-\nduces the effective anisotropy Hk. For further justification,\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 4 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\n/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s116\n/s115/s40/s110/s115/s41\n/s97/s32/s32 /s65\n/s48/s32\n/s32/s32 /s65\n/s49/s32/s32\n/s32/s32 /s65\n/s50/s32/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52/s32\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/s40/s99/s41/s40/s98/s41\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/s40/s97/s41\nFigure 6: (a) Minimal switching time for the samples of dif-\nferentAas a function of Gilbert damping \u000b. Magnetization\nreversal trajectories of A3=1414 nm2for damping (b) \u000b=\n0.010 and (c) \u000b=0:017.\nthetheoreticallycalculatedfrequencies(blackline)areplot-\nted in same Figure 5 and noted that the decreasing trends\nare consistent as expected. Later on, by keeping the min-\nimalf0.Hshape/and optimal \u0011.Hshape/(i.e., corresponding\ntodifferentAorHshape)fixed,weinvestigatetheminimally\nrequiredamplitude Hmw(redsquare)asafunctionof Hshape.\nFigure5showsthat Hmwalsodecreaseswiththeincreaseof\nHshape. This is explained by the same reason of reducing\nthe height of the energy barrier with Hshapeas shown in the\nFigure 2(b). So even the smaller amplitude Hmwcan drive\nfaster magnetization reversal.\nTheGilbertdampingcoefficient \u000bisanimportantmate-\nrial parameter which has a significant effect on the magne-\ntization reversal process as well as switching time. In gen-\neral, damping has two influences. First, it hinders energy\nabsorption (cost more energy) during climbing the energy\nbarrier for magnetization reversal [50]. Second, after cross-\ning the energy barrier, it dissipates the energy of magne-\ntization promptly and thus it accelerates the magnetization\nto reach the ground _stable state [51, 52]. However, in the\nmagnetizationswitchingprocess,themagnetizationrequires\nto climb an energy barrier (originated from the anisotropy\nfield) and then goes to a stable state. Ideally, for the fast\nandenergy-efficientmagnetizationreversal,smaller(larger)\ndamping (\u000b) before (after) cross over the energy barrier is\nbetter. Therefore, this study emphasizes finding the optimal\n\u000bofthesampleofdifferent Aatwhichthefastestreversalis\nvalid. Accordingly, DCMP-driven magnetization as a func-\ntion of\u000bfor different Ais studied. Figure 6(a) shows the\nGilbert damping dependence of switching time for different\nA. It is found that for A0= 88nm2(uniaxial sample),\nthe range of \u000bis small but for finite shape anisotropy (biax-\nial sample), the range of \u000bis large which is useful in device\napplication. Moreover, for the sample of 1414nm2, the\nswitchingtimeislowest( ts=0:482ns)at\u000b=0:017. Tobe\nmore explicit, Figure 6(b) and 6(c) show the trajectories of\nmagnetization reversal for \u000b= 0:01and\u000b= 0:017respec-\ntively and observed that for \u000b= 0:017, the reversal path is\nshorter. This is happened because, after cross over the en-\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s72\n/s109/s119/s32/s40/s84/s41\n/s74 /s32/s40/s49/s48/s49/s50\n/s32/s65/s47/s109/s50\n/s41/s32/s32 /s65\n/s48/s32\n/s32/s32 /s65\n/s49/s32/s32\n/s32/s32 /s65\n/s50/s32/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52/s32Figure 7: Phase diagram, for magnetization switching within\n1 ns, in terms of DCMP field amplitudes Hmwand current\ndensityJfor the samples of different A.\nergy barrier, the larger \u000breduces the magnetization preces-\nsion by dissipating the magnetization energy promptly and\nthus,itleadstofastreversal. Itisindicatedthatthereisaspe-\ncific sample that has an optimal damping situation at which\nthe magnetization reversal might be fastest.\nFormostdeviceapplication,thedesiredmicrowavepulse\n(DCMP) should be of smaller amplitude Hmw, smaller ini-\ntialfrequency foandlowchirprate \u0011,butwehaveobtained,\nforasampleof A=1616 nm2,minimalHmw=0:0278T,\noptimumfo=12:4GHz, and\u0011=40:672ns*2. In order to\nfurther reduce Hmw, we simultaneously apply a direct cur-\nrent such that it assists the magnetization reversal induced\nbyDCMP.ForDCMPcontribution,wetune Hmwwhilethe\nminimalfoandoptimal \u0011correspondingtodifferent Akept\nfixed. Figure 7 shows the Hmw*Jphase diagram for the\nsample of different Aunder the condition of magnetization\nreversal time 1 ns. As an example, for A= 1616 nm2,\nwithout DCMP, the required current to obtain fast switch-\ning gets very high i.e., J= 2:31012A_m2while without\ncurrenti.e.,J=0,theHmw=0:0278Tislargefrompracti-\ncal point of view. Therefore, with simultaneous applying of\nDCMPandcurrent,therequiredvaluesof HmwandJcould\nbe smaller than the above mentioned values (for individual\ncase)whichgivesalargescopeofpracticalrealizationofthis\nDCMP-driven magnetization reversal strategy.\n4. Discussion and Conclusions\nThisstudyinvestigatestheinfluenceofshapeanisotropy\nontheDCMP-drivenmagnetizationreversaltime tsandthe\nthreecontrollingparameters( Hmw,foand\u0011)ofDCMP,and\ndamping dependence of magnetization reversal. The shape\nanisotropy interestingly assists magnetization reversal since\nthe parameters of DCMP Hmw,foand\u0011decrease with the\nthe increase of shape anisotropy Hshape. For the sample of\nA8g3232nm2,tssignificantly reduces to 0.346 ns.\nThe minimal parameters of DCMP are estimated as Hmw=\n0:0187T,fo=5:2GHz,andoptimal \u0011=24:96ns*2which\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 5 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\nare useful in device application. These findings can be at-\ntributed to the shape anisotropy Hshapewhich reduces the\neffective anisotropy and thus reduces the height of the en-\nergybarrier. TheDCMPstimulatesefficientenergyabsorp-\ntion and emission during magnetization reversal. A recent\nstudy [43] reported that the temperature effect also assists\nthe magnetization reversal, i.e., it reduces the parameters of\nmicrowave field pulse. Therefore, it is desired that, at finite\ntemperature,theestimatedvaluesof Hmw,foand\u0011wouldbe\ndecreased further (which is beyond the scope of this study).\nThe result of damping dependence of magnetization rever-\nsal indicates that for a specific AorHshape, there exists an\noptimal damping situation. Moreover, it is also shown that\nthe required microwave field amplitude can be lowered by\napplying the direct current simultaneously. The usage of an\noptimumcombinationofbothmicrowavefieldpulseandcur-\nrent is suggested to achieve cost efficient and faster switch-\ning. So these findings may provide the knowledge to fabri-\ncate the shape of a single domain nanoparticle to generate\nthe magnetic data storage device.\n5. Acknowledgements\nThisworkwassupportedbytheMinistryofScienceand\nTechnology (Grant No. 440 EAS) and the Ministry of Edu-\ncation (BANBEIS, Grant No. SD2019972).\nReferences\n[1] S. Sun, C. B. Murray, D. Weller, L. Folks, A. Moser, Monodisperse\nFePt nanoparticles and ferromagnetic FePt nanocrystal superlattices,\nScience 287 (5460) (2000) 1989–1992.\n[2] S. Woods, J. Kirtley, S. Sun, R. Koch, Direct investigation of su-\nperparamagnetism in Co nanoparticle films, Physical Review Letters\n87 (13) (2001) 137205.\n[3] D. Zitoun, M. Respaud, M.-C. Fromen, M. J. 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Fukushima, Switching time of a single spin in linearly varying\nfield, Journal of Applied Physics 111 (12) (2012) 123907.\n[52] C.Y.Mao,J.-G.Zhu,R.M.White,T.Min,Effectofdampingconstant\nontheswitchinglimitofathin-filmrecordinghead,Journalofapplied\nphysics 85 (8) (1999) 5870–5872.\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 7 of 7"    },    {        "title": "2109.03076v2.Inertial_spin_dynamics_in_epitaxial_cobalt_films.pdf",        "content": "Inertial spin dynamics in epitaxial cobalt \flms\nVivek Unikandanunni,1Rajasekhar Medapalli,2, 3Marco Asa,4Edoardo Albisetti,4\nDaniela Petti,4Riccardo Bertacco,4Eric E. Fullerton,2and Stefano Bonetti1, 5,\u0003\n1Department of Physics, Stockholm University, 10691 Stockholm, Sweden\n2Center for Memory and Recording Research, University of California San Diego, San Diego, CA 92093, USA\n3Department of Physics, Lancaster University, Bailrigg, Lancaster LA1 4YW, United Kingdom\n4Department of Physics, Politecnico di Milano Technical University, Milano, Italy\n5Department of Molecular Sciences and Nanosystems,\nCa' Foscari University of Venice, 30172 Venice, Italy\nWe investigate the spin dynamics driven by terahertz magnetic \felds in epitaxial thin \flms of\ncobalt in its three crystalline phases. The terahertz magnetic \feld generates a torque on the mag-\nnetization which causes it to precess for about 1 ps, with a sub-picosecond temporal lag from the\ndriving force. Then, the magnetization undergoes natural damped THz oscillations at a frequency\ncharacteristic of the crystalline phase. We describe the experimental observations solving the inertial\nLandau-Lifshitz-Gilbert equation. Using the results from the relativistic theory of magnetic inertia,\nwe \fnd that the angular momentum relaxation time \u0011is the only material parameter needed to\ndescribe all the experimental evidence. Our experiments suggest a proportionality between \u0011and\nthe strength of the magneto-crystalline anisotropy.\nThe fundamental understanding of magnetism has\nmuch improved since the \frst experiments on magnetic\nmaterials at femto- and picosecond time scales, the so-\ncalled ultrafast regime. The pioneering work of Beaure-\npaire et al. [1] demonstrated that sub-picosecond magne-\ntization dynamics is possible, against the prediction of\nthe textbook Landau-Lifshitz-Gilbert (LLG) equation.\nThose results have since then been con\frmed in sev-\neral experiments and using both optical [2{12] and X-\nray techniques [13{16], making the \feld of ultrafast mag-\nnetism, and of magnetization dynamics beyond the LLG\nequation, an active \feld of research with possible impli-\ncations for novel data storage technologies.\nAnother recent analysis of the LLG equation showed\nthat from a classical mechanics point of view, the LLG\nequation assumes an unphysical inertial tensor [17], also\nsuggested by Gilbert himself in a footnote almost two\ndecades ago [18]. However, Ciornei et al. [17] re-derived\nthe equation with a realistic inertial tensor which resulted\nin a slightly revised equation known as the inertial LLG\n(iLLG) equation, which predicts the appearance of a nu-\ntation resonance at a frequency much higher than the\nferromagnetic resonance (FMR) one. After that pioneer-\ning work, much theoretical e\u000bort has been performed try-\ning to identify the frequency regime where such nutation\nresonance was to be expected, with predictions varying\nover a few orders of magnitude [19{23]. The experimen-\ntal con\frmation has been achieved only recently and de-\ntected nutation dynamics in the \u00181 THz range [24].\nDespite this novel experimental evidence, and the signif-\nicant theoretical progress to understand the microscopic\norigin of inertia [25{44], a complete picture of how dif-\nferent material parameters a\u000bect the nutation dynamics\nis still missing.\n\u0003stefano.bonetti@fysik.su.seIn this Letter, we provide the \frst experimental\ndata on the dependence of inertial spin dynamics on\na key magnetic property, i.e. the magneto-crystalline\nanisotropy. It has been suggested that spin-orbit cou-\npling is the fundamental interaction needed to derive a\ncorrect inertial tensor from \frst principles. Hence, exper-\niments where the magneto-crystalline anisotropy is well\nde\fned and controlled, are expected to return impor-\ntant insights on this open question. Similar to Ref. [24],\nwe perform THz pump / optical probe time-resolved\nmagneto-optical measurements to trigger and detect in-\nertial spin dynamics. In contrast to the approach of\nRef. [24], where a narrowband (\u0001 f=f\u00190:1) THz source\nwas implemented, we use intense single-cycle terahertz\nradiation [45] whose broad band (\u0001 f=f > 1) allows to\ncover the frequency range where the nutation resonance\nis expected to appear. In addition, due to the impul-\nsive character of the driving pulse, our measurements\nare expected to detect not only the forced response of\nthe system, but also its natural one. We model the mag-\nnetization dynamics solving the inertial LLG equation\nnumerically and we contextualize our results with the\nexisting microscopic theory of magnetic inertia.\nWe choose to investigate three epitaxial cobalt thin\n\flms grown on MgO substrate with face-centered cu-\nbic (fcc), body-centered cubic (bcc), and hexagonal close\npacked (hcp) crystal structures. The fabrication details\nfor the three samples are given in the Supplemental Mate-\nrial [46]. The hcp sample has a strong in-plane magneto-\ncrystalline anisotropy characterized by an easy magne-\ntization axis along the c-direction of the hcp structure\n(which lies in the \flm plane in our \flms), and a hard axis\northogonal to it. For the two cubic crystal structures, the\nanisotropy is still in-plane, but its strength is much re-\nduced, and a hard magnetization direction is not clearly\nidenti\fed [46]. The hcp and fcc samples were grown un-\nder similar deposition conditions and are respectively 10\nnm and 15 nm in thickness, while the bcc sample wasarXiv:2109.03076v2  [cond-mat.mes-hall]  11 Sep 20212\n(c)\nMHTHz\ny\nzx\n𝜃\"#$𝜃%\n(a)(b)\n𝜃\nFIG. 1. (a) Geometry of THz pump-MOKE probe setup. (b)\nFrequency spectrum of terahertz pump pulse (c) Magnetiza-\ntion loops for fcc, bcc, and hcp cobalt measured using the\nlongitudinal MOKE.\ngrown in a di\u000berent laboratory and it has a thickness of\n8 nm. Fig. 1(a) shows the geometry of the single cycle\nTHz pump - optical probe experiment. The magnetiza-\ntionMof the sample is aligned along the x-direction by\nmeans of an external bias \feld jHextj= 100 mT, kept\nconstant during the experiment. The single-cycle THz\npump pulses are generated in the organic crystal OH1\nby the optical recti\fcation of 1300 nm radiation from an\noptical parametric ampli\fer [47]. The pump pulse has a\npeak magnetic \feld of 0.3 T parallel to the y-direction,\nwhich maximizes the torque on the magnetization, and\nimpinges on the sample at an angle of incidence \u0012inc= 45\ndegrees. Fig. 1(b) is the Fourier transform of the electro-\noptical sampling measurement in a a 50 \u0016m-thick GaP\ncrystal [48], used to characterize the THz pulse. It shows\nthat the pump \feld is peaked at around 2 THz, and has\na bandwidth exceeding 1 THz. The magnetization dy-\nnamics is probed using the time-resolved magneto-optical\nKerr e\u000bect (MOKE), in the speci\fc measuring the Kerr\nrotation angle \u0012Kof a nominally 40 fs, 800 nm probe\nbeam, using a balanced detection scheme. All radiation\nis derived from the same ampli\fed laser system ensuring\nintrinsic synchronization, with the relative delay between\nthe beams controlled by a mechanical translation stage,\nand with the pump modulated at a frequency equal to\nhalf the laser repetition rate. Fig. 1(c) shows the easy\naxis magnetization loops for the three samples investi-\ngated in this work. The coercive \feld for the hcp sample\nis about 50 mT, whereas is approximately 30 mT for both\nfcc and bcc samples [46].\nFig. 2 shows the time-resolved MOKE measurements\nof the terahertz-\feld induced dynamics in all three sam-\nples. The plotted traces represent the di\u000berence of the\ndata recorded with magnetic \felds of equal magnitude\nbut opposite polarity, ensuring the magnetic character of\nFIG. 2. Solid symbols: time-resolved Kerr rotation measure-\nments on fcc, bcc and hcp cobalt thin \flms. Dashed line: inte-\ngral of the pump THz magnetic \feld HTHz. Inset: zoomed-in\nmain panel data for t>1:7 ps. The data is shifted vertically\nfor clarity. The continuous lines the best \fts obtained using\nEq. (1).\nthe signal [49]. For all samples, the MOKE response is\ndominated by the coherent precession of the magnetiza-\ntion around the applied THz magnetic \feld HTHz, which\nin fcc and bcc \flms is larger in amplitude than in the\nhcp one. A very small demagnetization, showing up as a\nlingering non-zero average MOKE signal, is also present.\nThe presence of both coherent (precession) and incoher-\nent (demagnetization) e\u000bects in the observed THz-driven\ndynamics is consistent with Ref. [50], where it was also\nobserved that in epitaxial \flms the demagnetization sig-\nnal was negligible. Similar to Refs. [50] and [24], no\ncoherent precession is observed when HTHzkM, since\nin that case the torque acting on the magnetization is\nzero. We present this measurement in the Supplemental\nMaterial [46].\nThe dashed grey line in the same plot is the inte-\ngral ofHTHzover time, obtained numerically from the\nelectro-optic sampling measurement used to character-\nize the THz \feld. Ref. [50] demonstrated that the co-\nherent response of the magnetization to an o\u000b-resonant\nTHz \feld can be obtained simply by integrating HTHz,\nwhich is the solution of the LLG equation for small and\no\u000b-resonant excitations. However, while in Ref. [50] the\ntemporal overlap between the MOKE data and the in-\ntegral ofHTHzwas exact within the experimental error,\nhere we notice a substantial lag between them, approxi-\nmately 200 fs for the fcc and bcc samples, and 400 fs for\nthe hcp one, highlighted by the vertical lines. In other\nwords, the shape of the MOKE response is still consistent\nwith the integral of the THz \feld if properly scaled, how-3\n(b)\n(a)\nFIG. 3. (a) Symbols: time-resolved Kerr signal at t >\n1:7 ps for THz magnetic \feld values of di\u000berent maximum\namplitude. The data is vertically shifted for clarity. Solid\nlines: best \ft obtained using Eq. (1). (b) Symbols: extracted\noscillation amplitude Bas a function of THz magnetic \feld\nand corresponding standard deviation. Dashed line: linear \ft\nto the data with imposed zero o\u000bset.\never its phase is not. This phase shift is particularly dra-\nmatic for the hcp sample, where it looks as if the magne-\ntization precesses in the opposite direction as compared\nto the fcc and bcc samples. We repeated the experiment\non all samples, and the evidence is robust. The sign\nof the magneto-optical coe\u000ecient does not change either\nbetween the di\u000berent samples, as demonstrated by the\nmagneto-optical hysteresis loops in Fig. 1(c). Hence, the\nobserved phase shift is real and it appears to be strongly\ndependent on the crystalline structure of the sample.\nAnother intriguing observation from the data in Fig. 2\nis found in the inset, where we zoom in on the main panel\ndata at temporal delays t>1:7 ps. When the pump \feld\nhas left the sample, a comparatively tiny, yet detectable,\ndamped ringing of the magnetization can be observed.\nWe can \ft such behavior with the phenomenological for-\nmula\n\u0001\u0012K(t) =Ae\u0000t=\u001c1+Be\u0000t=\u001c2sin(2\u0019ft) (1)\nwhere\u001c1is the recovery time of the incoherent demagne-\ntization dynamics, \u001c2is the decay time of the sinusoidal\noscillation, fis the frequency of the oscillations, and A,\nBare the constants describing the the amplitude of the\ndemagnetization and, respectively, of the sinusoidal os-\ncillations. The \ft returns ffcc\u00191:3 THz,fbcc\u00191:4\nTHz andfhcp\u00192:1 THz.\u001c2is found to be approxi-\nmately 0.82, 0.70 and 0.72 ps for the fcc, bcc and, re-\nspectively, hcp samples, corresponding to damping coef-\n\fcients\u000b= 1=!\u001c2which are\u000bfcc\u00190:15,\u000bbcc\u00190:16\nand\u000bhcp\u00190:10.\nBefore discussing these results, we present in Fig. 3\nthe THz \feld dependent measurement on the fcc sample,\nwhich showed the largest signal in Fig. 2. The terahertz\n\feld strength is controlled through the relative orienta-\ntion of a pair of wire-grid polarizers in the THz pump\npath. The second polarizer was kept \fxed in order to\npreserve the polarization of the THz \feld impinging on\nthe sample. The time-resolved MOKE signal is shown inFig. 3(a) for the maximum \feld strength, 75% and 50%\nof it, below which we were at the noise level of our setup.\nWe used again Eq. (1) to \ft the oscillations and to ex-\ntract the amplitudes and recovery times as a function of\nTHz \feld strength. Fig. 3(b) shows the extracted oscil-\nlations amplitude Bas a function of THz \feld strength,\nwhich can be \ftted with a linear function with no o\u000bset.\nThe evidence presented so far is consistent with the\npresence of a sizeable magnetic inertia in crystalline\ncobalt \flms, manifesting itself with a lagging response\nto an external \feld and to the appearance of nutation\noscillations. In order to investigate this hypothesis thor-\noughly, we performed numerical simulations using the in-\nertial LLG equation, written in a slightly di\u000berent form\nthan the one given in Ref. [24]\ndM\ndt=\u0000j\rjM\u0002He\u000b+M\u0002\u0012\n\u000bdM\ndt\u0000\u0011d2M\ndt2\u0013\n;(2)\nwherej\rj/2\u0019= 28 GHz/T is the gyromagnetic ratio,\nHe\u000b= (Hbias+HK)x+HTHz(t)y+Hdzis the e\u000bec-\ntive magnetic \feld which comprises of the external bias\n\feldHbias, the anisotropy \feld HK, the applied THz \feld\nHTHz(t), and the demagnetizing \feld Hd;Msis the sat-\nuration magnetization of the sample, \u000bis the Gilbert\ndamping parameter, and \u0011is the angular momentum re-\nlaxation time de\fned as in Ref. [35], i.e. \u0011=\u000b\u001c. Since\n\u000b\u001c1, the absolute values \u0011are much smaller than \u001cre-\nported in Ref. [24]. The last term on the right hand side\nof Eq. (2) is the nutation term that is present only when\n\u00116= 0. In the following, we solve this equation in the\nmacrospin approximation (i.e. the sample is considered\nas a homogeneous ferromagnet) and using a conventional\nfourth-order Runge-Kutta method.\nIn Fig. 4(a)-(c), we compare the results from these nu-\nmerical simulations to the experimental results in fre-\nquency domain. For both simulations and experiments,\nwe Fourier transform the temporal traces obtained at\ntime delays t >1:7 ps, when the THz pump \feld has\nleft the sample. Including the full temporal trace would\nhide the small features below the broad single-cycle re-\nsponse. In the numerical simulations, we calculate He\u000b\nsolely from experimentally measured quantities found in\nthe Supplemental Material [46] or in previous references\n[51]. This allow us to estimate Hd\u00191:6 T for all sam-\nples, andHK\u00190:8 T for the hcp sample and one order of\nmagnitude smaller for the other two samples. We used\nthe nominal values for HbiasandHTHz. The only two\nfree parameters are then \u0011and\u000b, which can be indepen-\ndently tuned to match the peak frequency and, respec-\ntively, linewidth. Using \u0011fcc= 120 fs,\u0011bcc= 110 fs and\n\u0011hcp= 75 fs we can reproduce the main experimental\npeak frequency, and assuming \u000bfcc= 0:15,\u000bbcc= 0:16\nand\u000bhcp= 0:10 from the \fts using Eq. (1), we can also\nmatch the linewidth of the main peak. No observable\ndi\u000berence was found within 5-10 fs for \u0011and within 0.01\nfor\u000b, giving an approximate 10% relative accuracy. We\nhave performed additional simulations (not shown) and\nwe also observe that the e\u000bective \feld does not a\u000bect4\nFIG. 4. Experimental (solid) and simulated (semi-\ntransparent) Fourier transform of the magnetization dynam-\nics in (a) fcc (blue) (b) bcc (orange) and (c) hcp (green) cobalt\nthin \flms. The full experimental trace is used for the exper-\nimental data in Fig. 2, and the Mzcomponent for the simu-\nlations. (d) Solid lines: simulated response of Mzto a single\ncycle terahertz \feld HTHzin the time-domain using the same\nparameters. Dashed line: integral over time of HTHz.\nthe nutation frequency and linewidth in a noticeable way\nunless it reaches values of the order of a few Tesla.\nWe discuss below the plausibility of these values; as-\nsuming for the time being that they are reasonable, and\nlooking at Fig. 4(d), we obtain the remarkable result that\nthe inertial LLG equation is able to reproduce all the ex-\nperimental evidence of Fig. 2: the presence of a damped\nnutation oscillation and the temporal shift of the coher-\nent magnetization precession. In this small amplitude\nlimit, the inertial LLG also predicts a linear scaling of\nthe coherent precession and of the nutation amplitude\nwith terahertz \feld strength, as shown experimentally in\nFig. 3(b). None of these experimental evidences can be\nreproduced solving the standard LLG equation, proving\nthat the additional inertial term is necessary.\nThe only experimental evidence which is not repro-\nduced by the inertial LLG equation, in the currently\nknown form and in the macrospin approximation, is the\npresence of higher order harmonic peaks in the frequency\nresponse seen in Fig. 4(a)-(c). We can clearly identify\nthe second and third harmonics for the fcc and bcc sam-\nples, and the second harmonic for the hcp one. How-\never, Kikuchi et al. [22] predicted such possibility if the\nthird and other higher order time derivatives of the mag-\nnetization, not included in the standard framework of\nthe inertial LLG model, are considered. We leave thisquestion open to future theoretical and experimental in-\nvestigations, here we simply note that the presence of\nharmonics at integer multiples nof the fundamental fre-\nquency could be consistent with nutation dynamics. It is\nnot consistent with the presence of standing waves across\nthe \flm thickness, which show instead a n2dependence\ndue to con\fnement [52]. We also do not observe any\napparent inverse thickness dependence, which is instead\nexpected in the case of standing waves.\nAs a \fnal control to test the general validity of our\nexperimental results and of the inertial LLG equation,\nwe performed additional measurements using a di\u000berent\nTHz single-cycle pump \feld with a bandwidth extending\nfrom 2 to 4 THz instead, i.e. with negligible overlap with\nthe nutation resonances. This is achieved by replacing\nthe nonlinear crystal generating the THz radiation and\nby adjusting the corresponding pump wavelength, leav-\ning the rest of the setup unchanged. The results are re-\nported in the Supplemental Material [46], and they show\nthat neither THz oscillations nor phase shift of the coher-\nent precession is observed when the pump \feld does not\nmatch the nutation resonance. This is also in agreement\nwith previous measurements done in fcc cobalt driven\nby a THz \feld with similar bandwidth [53]. The inertial\nLLG equation with the same parameters reproduces even\nthese experimental data to an excellent degree, with no\nnutation oscillations nor phase shift observed in this case.\nWe now turn the discussion to the two free parame-\nters in the inertial LLG equation, namely the damping\n\u000band the angular relaxation time \u0011. In order to match\nthe experimental linewidth, we used for all three \flms\na damping parameter which is an order of magnitude\nlarger than the typical FMR Gilbert damping of these\nmaterials [51]. While we do not have a microscopic ex-\nplanation for these large values, we notice that the same\nissue was found in the \frst experimental report of nuta-\ntion in ferromagnets and left as an open question [24].\nOur experiments, which are able to observe the natu-\nral nutation oscillations, allow us to extract the damp-\ning factor directly from the data. In order to estimate\nthe magnitude of \u000bfor the inertial dynamics, a micro-\nscopic theoretical investigation is needed, which is be-\nyond the scope of our work. Our experiments suggest\nthough that a complete inertial LLG equation may con-\ntain either distinct Gilbert and inertial damping coe\u000e-\ncients, or a time-dependent one, in order to fully describe\nthe magnetization dynamics. A time-dependent \u000bcan be\nqualitatively linked to a damping mechanism dominated\nby comparatively stronger electron-phonon scattering at\nsub-picosecond time scales, and weaker spin-lattice relax-\nation at longer time scales [18, 54, 55]. A time-dependent\n\u000bhas also been recently suggested to include all damping\nmechanisms, including time-retardation e\u000bects [56].\nThe most important experimental observation of this\nwork, which is expected to contribute to a microscopic\nunderstanding of inertial dynamics, is the strong depen-\ndence of the nutation frequency on the di\u000berent cobalt\nsamples with di\u000berent magneto-crystalline anisotropy,5\nwhich in turn is dependent on the strength of the spin-\norbit coupling. The relativistic theory of magnetic in-\nertia at ultrafast time scales [29] demonstrated that the\npresence of a \fnite angular momentum relaxation time is\ndue to a spin-orbit coupling e\u000bect of order 1 =c4, whereas\nthe Gilbert damping is of order 1 =c2. These two quan-\ntities are therefore dependent on each other, and it was\nsuggested in Ref. [29] that the ratio \u0011=\u000bshould be a con-\nstant. From our data, we calculate \u0011=\u000b = 746\u000646 fs\nfrom the three cobalt \flms, which is constant within the\naccuracy of our estimates of the two parameters ( \u001810%).\nThis reduces the number of free parameters in the iner-\ntial LLG equation to only one, at least within the same\n3delement, and it further strengthens the interpretation\nof our results in terms of relativistic spin dynamics. We\nnote that the magneto-crystalline anisotropy energy is\nabout one order of magnitude larger in hcp cobalt than\nin the two cubic phases, while the nutation frequency\ndi\u000bers by less than a factor of two among them. This\npreliminary observation suggests that a relation of pro-\nportionality may exist between the frequency of nutation\nand the strength of the magneto-crystalline anisotropy,\nand that it is sub-linear. We also note that fcc and bcc\nphases are energetically close; a small change in lattice\nparameter can induce a so-called Bain transformation be-\ntween them [57]. Hence, it is not too surprising that\nalso their magneto-crystalline anisotropy and nutation\nfrequencies are similar. We anticipate that future works\nwill shed light on how to derive the nutation frequency\nfrom \frst principles or from other magnetic properties of\nthe material.In summary, we measured the temporal evolution of\nterahertz-\feld driven spin dynamics in three epitaxial\ncobalt samples with fcc, bcc and hcp crystal structures.\nWe observed the appearance of THz oscillations with\ndistinct frequencies for the three samples and of a de-\nlayed coherent magnetization response, which could be\nnaturally described in the framework of the inertial LLG\nequation assuming a magnetic damping one order of mag-\nnitude larger than the conventional Gilbert damping at\nFMR frequencies. While surprising, this evidence may\nbe consistent with recent theoretical works suggesting a\ntime-dependent damping coe\u000ecient. We could also esti-\nmate a constant ratio between the angular momentum\nrelaxation time and the measured damping, in agree-\nment with the prediction of the full relativistic theory of\nmagnetic inertia. Finally, we could observe higher har-\nmonics of the nutation oscillations, not described by the\ncurrently accepted inertial LLG equation with temporal\nderivatives up to the second order, but possibly consis-\ntent with a higher order extension of the same equation.\nOur work provides the strongest evidence for inertial spin\ndynamics so far, where all the experimental results can\nbe reproduced with a single free parameter. We envisage\nthat our results will trigger future experimental and the-\noretical investigations towards a deeper microscopic un-\nderstanding of magnetic inertia at ultrafast time scales.\nV.U. and S.B. acknowledge support from the European\nResearch Council, Starting Grant 715452 \\MAGNETIC-\nSPEED-LIMIT\". 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Hudait,3Jing Zhao,4F. Marc Michel,4Claudia Mewes,2Tim Mewes,2and Satoru Emori1\n1Department of Physics, Virginia Polytechnic Institute\nand State University, Blacksburg, VA 24061, USA\n2Department of Physics and Astronomy,\nThe University of Alabama, Tuscaloosa, AL 35487 USA\n3Department of Electrical and Computer Engineering,\nVirginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA\n4Department of Geosciences, Virginia Polytechnic Institute\nand State University, Blacksburg, VA 24061, USA\nAbstract\nWe examine room-temperature magnetic relaxation in polycrystalline Fe \flms. Out-of-plane fer-\nromagnetic resonance (FMR) measurements reveal Gilbert damping parameters of \u00190.0024 for Fe\n\flms with thicknesses of 4-25 nm, regardless of their microstructural properties. The remarkable\ninvariance with \flm microstructure strongly suggests that intrinsic Gilbert damping in polycrys-\ntalline metals at room temperature is a local property of nanoscale crystal grains, with limited\nimpact from grain boundaries and \flm roughness. By contrast, the in-plane FMR linewidths of\nthe Fe \flms exhibit distinct nonlinear frequency dependences, indicating the presence of strong\nextrinsic damping. To \ft our in-plane FMR data, we have used a grain-to-grain two-magnon scat-\ntering model with two types of correlation functions aimed at describing the spatial distribution of\ninhomogeneities in the \flm. However, neither of the two correlation functions is able to reproduce\nthe experimental data quantitatively with physically reasonable parameters. Our \fndings advance\nthe fundamental understanding of intrinsic Gilbert damping in structurally disordered \flms, while\ndemonstrating the need for a deeper examination of how microstructural disorder governs extrinsic\ndamping.\n1arXiv:2109.03684v2  [cond-mat.mtrl-sci]  24 Feb 2022I. INTRODUCTION\nIn all magnetic materials, magnetization has the tendency to relax toward an e\u000bective\nmagnetic \feld. How fast the magnetization relaxes governs the performance of a variety\nof magnetic devices. For example, magnetization relaxation hinders e\u000ecient precessional\ndynamics and should be minimized in devices such as precessional magnetic random access\nmemories, spin-torque oscillators, and magnonic circuits1{4. From the technological perspec-\ntive, it is important to understand the mechanisms behind magnetic relaxation in thin-\flm\nmaterials that comprise various nanomagnetic device applications. Among these materials,\nbcc Fe is a prototypical elemental ferromagnet with attractive properties, including high sat-\nuration magnetization, soft magnetism5, and large tunnel magnetoresistance6,7. Our present\nstudy is therefore motivated by the need to uncover magnetic relaxation mechanisms in Fe\nthin \flms { particularly polycrystalline \flms that can be easily grown on arbitrary substrates\nfor diverse applications.\nTo gain insights into the contributions to magnetic relaxation, a common approach is to\nexamine the frequency dependence of the ferromagnetic resonance (FMR) linewidth. The\nmost often studied contribution is viscous Gilbert damping8{13, which yields a linear increase\nin FMR linewidth with increasing precessional frequency. In ferromagnetic metals, Gilbert\ndamping arises predominately from \\intrinsic\" mechanisms14{16governed by the electronic\nband structure17. Indeed, a recent experimental study by Khodadadi et al.18has shown\nthat intrinsic, band-structure-based Gilbert damping dominates magnetic relaxation in high-\nquality crystalline thin \flms of Fe, epitaxially grown on lattice-matched substrates. However,\nit is yet unclear how intrinsic damping is impacted by the microstructure of polycrystalline\nFe \flms.\nMicrostructural disorder in polycrystalline Fe \flms can also introduce extrinsic magnetic\nrelaxation. A well-known extrinsic relaxation mechanism is two-magnon scattering, where\nthe uniform precession mode with zero wave vector scatters into a degenerate magnon mode\nwith a \fnite wave vector19{22. Two-magnon scattering generally leads to a nonlinear fre-\nquency dependence of the FMR linewidth, governed by the nature of magnon scattering\ncenters at the surfaces23,24or in the bulk of the \flm25{28. While some prior experiments\npoint to the prominent roles of extrinsic magnetic relaxation in polycrystalline ferromag-\nnetic \flms29{31, systematic studies of extrinsic relaxation (e.g., two-magnon scattering) on\n2polycrystalline Fe thin \flms are still lacking.\nHere, we investigate both the intrinsic and extrinsic contributions to magnetic relax-\nation at room temperature in polycrystalline Fe \flms. We have measured the frequency\ndependence of the FMR linewidth with (1) the \flm magnetized out-of-plane (OOP), where\ntwo-magnon scattering is suppressed25such that intrinsic Gilbert damping is quanti\fed re-\nliably, and (2) the \flm magnetized in-plane (IP), where two-magnon scattering is generally\nexpected to coexist with intrinsic Gilbert damping.\nFrom OOP FMR results, we \fnd that the intrinsic Gilbert damping of polycrystalline Fe\n\flms at room temperature is independent of their structural properties and almost identical\nto that of epitaxial \flms. Such insensitivity to microstructure is in contrast to disorder-\nsensitive Gilbert damping recently shown in epitaxial Fe at cryogenic temperature18. Our\npresent work implies that Gilbert damping at a su\u000eciently high temperature becomes a\nlocal property of the metal, primarily governed by the structure within nanoscale crystal\ngrains rather than grain boundaries or interfacial disorder. This implication refutes the\nintuitive expectation that intrinsic Gilbert damping should depend on structural disorder in\npolycrystalline \flms.\nIn IP FMR results, the frequency dependence of the FMR linewidth exhibits strong\nnonlinear trends that vary signi\fcantly with \flm microstructure. To analyze the nonlin-\near trends, we have employed the grain-to-grain two-magnon scattering model developed\nby McMichael and Krivosik25with two types of correlation functions for capturing inho-\nmogeneities in the \flm. However, neither of the correlation functions yields quantitative\nagreement with the experimental results or physically consistent, reasonable parameters.\nThis \fnding implies that a physical, quantitative understanding of extrinsic magnetic re-\nlaxation requires further corrections of the existing two-magnon scattering model, along\nwith much more detailed characterization of the nanoscale inhomogeneities of the magnetic\n\flm. Our study stimulates opportunities for a deeper examination of fundamental magnetic\nrelaxation mechanisms in structurally disordered ferromagnetic metal \flms.\nII. FILM DEPOSITION AND STRUCTURAL PROPERTIES\nPolycrystalline Fe thin \flms were deposited using DC magnetron sputtering at room\ntemperature on Si substrates with a native oxide layer of SiO 2. The base pressure of the\n3chamber was below 1 \u000210\u00007Torr and all \flms were deposited with 3 mTorr Ar pressure. Two\nsample series with di\u000berent seed layers were prepared in our study: subs./Ti(3 nm)/Cu(3\nnm)/Fe(2-25 nm)/Ti(3 nm) and subs./Ti(3 nm)/Ag(3 nm)/Fe(2-25 nm)/Ti(3 nm). In this\npaper we refer to these two sample series as Cu/Fe and Ag/Fe, respectively. The layer\nthicknesses are based on deposition rates derived from x-ray re\rectivity (XRR) of thick\ncalibration \flms. The Ti layer grown directly on the substrate ensures good adhesion of\nthe \flm, whereas the Cu and Ag layers yield distinct microstructural properties for Fe\nas described below. We note that Cu is often used as a seed layer for growing textured\npolycrystalline ferromagnetic metal \flms32,33. Our initial motivation for selecting Ag as an\nalternative seed layer was that it might promote qualitatively di\u000berent Fe \flm growth34,\nowing to a better match in bulk lattice parameter 𝑎between Fe ( 𝑎\u0019286\u0017A) and Ag\n(𝑎p\n2\u0019288\u0017A) compared to Fe and Cu ( 𝑎p\n2\u0019255\u0017A).\nWe performed x-ray di\u000braction (XRD) measurements to compare the structural properties\nof the Cu/Fe and Ag/Fe \flms. Figure 1(a,b) shows symmetric 𝜃-2𝜃XRD scan curves\nfor several \flms from both the Cu/Fe and Ag/Fe sample series. For all Cu/Fe \flms, the\n(110) body-center-cubic (bcc) peak can be observed around 2 𝜃=44°\u000045°(Fig. 1(a)). This\nobservation con\frms that the Fe \flms grown on Cu are polycrystalline and textured, where\nthe crystal grains predominantly possess (110)-oriented planes that are parallel to the sample\nsurface. For Ag/Fe (Fig. 1(b)), the (110) bcc peak is absent or extremely weak, from\nwhich one might surmise that the Fe \flms grown on Ag are amorphous or only possess\nweak crystallographic texture. However, we \fnd that the Ag/Fe \flms are, in fact, also\npolycrystalline with evidence of (110) texturing. In the following, we elaborate on our XRD\nresults, \frst for Cu/Fe and then Ag/Fe.\nWe observe evidence for a peculiar, non-monotonic trend in the microstructural properties\nof the Cu/Fe \flms. Speci\fcally, the height of the 𝜃-2𝜃di\u000braction peak (Fig. 1(a)) increases\nwith Fe \flm thickness up to \u001910 nm but then decreases at higher Fe \flm thicknesses. While\nwe do not have a complete explanation for this peculiar nonmonotonic trend with \flm\nthickness, a closer inspection of the XRD results (Fig. 1) provides useful insights. First, the\nFe \flm di\u000braction peak shifts toward a higher 2 𝜃value with increasing \flm thickness. This\nsigni\fes that thinner Fe \flms on Cu are strained (with the Fe crystal lattice tetragonally\ndistorted), whereas thicker Fe \flms undergo structural relaxation such that the out-of-plane\nlattice parameter converges toward the bulk value of \u00192.86 \u0017A, as summarized in Fig. 1(e).\n4354 04 55 05 5Ag/Fe2 nm6 nmIntensity [arb. unit]  \nCu/Febulk bcc Fe (110)1\n0 nm15 nm25 nm8\n nm(\na)\n4045502\nθ [deg]10 nm2\n5 nm \n2\nθ [deg]10 nm15 nm6\n nm2\n nm8 nm(\nb)\n16182022242628Ag/Fe2 nm6 nmIntensity [arb. unit]  \nCu/Febulk bcc Fe (110)1\n0 nm15 nm25 nm8\n nm(\nc)2\n5 nm \nθ\n [deg]10 nm15 nm6\n nm2\n nm8 nm(\nd)\n2.842.862.882.902.922.940\n5 10152025051015Bulk value 2.86 Cu/Fe \nAg/FeOut-of-planel\nattice parameter [Å](\ne)Crystallite size [nm]T\nhickness [nm](f)FIG. 1. (Color online) 𝜃-2𝜃X-ray di\u000braction scan curves for (a) Cu/Fe (blue lines) and (b) Ag/Fe\n(red lines) sample series. The inset in (b) is the grazing-incidence XRD scan curve for 10 nm thick\nAg/Fe \flm. Rocking curves for (c) Cu/Fe (blue lines) and (d) Ag/Fe (red lines) sample series.\n(e) Out-of-plane lattice parameter estimated via Bragg's law using the 2 𝜃value at the maximum\nof the tallest \flm di\u000braction peak. (f) Crystallite size estimated via the Scherrer equation using\nthe full-width-at-half-maximum of the tallest \flm di\u000braction peak. In (e) and (f), the data for the\nAg/Fe \flm series at a few thickness values are missing because of the absence of the bcc (110) peak\nin𝜃-2𝜃XRD scans.\nSecond, as the Fe \flm thickness approaches \u001910 nm, additional di\u000braction peaks appear to\nthe left of the tall primary peak. We speculate that these additional peaks may originate\nfrom Fe crystals that remain relatively strained (i.e., with an out-of-plane lattice parameter\nlarger than the bulk value), while the primary peak arises from more relaxed Fe crystals\n(i.e., with a lattice parameter closer to the bulk value). The coexistence of such di\u000berent\nFe crystals appears to be consistent with the rocking curve measurements (Fig. 1(c)), which\nexhibit a large broad background peak in addition to a small sharp peak for Cu/Fe \flms\nwith thicknesses near \u001910 nm. As we describe in Sec. IV, these \u001910 nm thick Cu/Fe samples\nalso show distinct behaviors in extrinsic damping (highly nonlinear frequency dependence of\n5the FMR linewidth) and static magnetization reversal (enhanced coercivity), which appear\nto be correlated with the peculiar microstructural properties evidenced by our XRD results.\nOn the other hand, it is worth noting that the estimated crystal grain size (Fig. 1(f)) {\nderived from the width of the 𝜃-2𝜃di\u000braction peak { does not exhibit any anomaly near the\n\flm thickness of\u001910 nm, but rather increases monotonically with \flm thickness.\nUnlike the Cu/Fe \flms discussed above, the Ag/Fe \flms do not show a strong (110) bcc\npeak in the 𝜃-2𝜃XRD results. However, the lack of pronounced peaks in the symmetric 𝜃-2𝜃\nscans does not necessarily signify that Ag/Fe is amorphous. This is because symmetric 𝜃-2𝜃\nXRD is sensitive to crystal planes that are nearly parallel to the sample surface, such that the\ndi\u000braction peaks capture only the crystal planes with out-of-plane orientation with a rather\nsmall range of misalignment (within \u00181°, dictated by incident X-ray beam divergence). In\nfact, from asymmetric grazing-incidence XRD scans that are sensitive to other planes, we\nare able to observe a clear bcc Fe (110) di\u000braction peak even for Ag/Fe samples that lack\nan obvious di\u000braction peak in 𝜃-2𝜃scans (see e.g. inset of Fig. 1(b)). Furthermore, rocking\ncurve scans (conducted with 2 𝜃\fxed to the expected position of the (110) Fe \flm di\u000braction\npeak) provide orientation information over an angular range much wider than \u00181°. As shown\nin Fig. 1(d), a clear rocking curve peak is observed for each Ag/Fe sample, suggesting that\nFe \flms grown on Ag are polycrystalline and (110)-textured { albeit with the (110) crystal\nplanes more misaligned from the sample surface compared to the Cu/Fe samples. The out-\nof-plane lattice parameters of Ag/Fe \flms (with discernible 𝜃-2𝜃di\u000braction \flm peaks) show\nthe trend of relaxation towards the bulk value with increasing Fe thickness, similar to the\nCu/Fe series. Yet, the lattice parameters for Ag/Fe at small thicknesses are systematically\ncloser to the bulk value, possibly because Fe is less strained (i.e., better lattice matched)\non Ag than on Cu. We also \fnd that the estimation of the crystal grain size for Ag/Fe {\nalthough made di\u000ecult by the smallness of the di\u000braction peak { yields a trend comparable\nto Cu/Fe, as shown in Fig. 1(f).\nWe also observe a notable di\u000berence between Cu/Fe and Ag/Fe in the properties of \flm\ninterfaces, as revealed by XRR scans in Fig. 2. The oscillation period depends inversely\non the \flm thickness. The faster decay of the oscillatory re\rectivity signal at high angles\nfor the Ag/Fe \flms suggests that the Ag/Fe \flms may have rougher interfaces compared to\nthe Cu/Fe \flms. Another interpretation of the XRR results is that the Ag/Fe interface is\nmore di\u000buse than the Cu/Fe interface { i.e., due to interfacial intermixing of Ag and Fe. By\n60.000.050.100.150.200.250.3010 nm2\n5 nmReflectivity [a.u.] \n(\na) Cu/Fe \nAg/Fe \nq\nz [Å-1](b)FIG. 2. (Color online) X-ray re\rectivity scans of 10 nm and 25 nm thick \flms from (a) Cu/Fe\n(blue circles) and (b) Ag/Fe (red squares) sample series. Black solid curves are \fts to the data.\n\ftting the XRR results35, we estimate an average roughness (or the thickness of the di\u000buse\ninterfacial layer) of .1 nm for the Fe layer in Cu/Fe, while it is much greater at \u00192-3 nm\nfor Ag/Fe36.\nOur structural characterization described above thus reveals key attributes of the Cu/Fe\nand Ag/Fe sample series. Both \flm series are polycrystalline, exhibit (110) texture, and\nhave grain sizes of order \flm thickness. Nevertheless, there are also crucial di\u000berences\nbetween Cu/Fe and Ag/Fe. The Cu/Fe series overall exhibits stronger 𝜃-2𝜃di\u000braction\npeaks than the Ag/Fe series, suggesting that the (110) bcc crystal planes of Fe grown on\nCu are aligned within a tighter angular range than those grown on Ag. Moreover, Fe grown\non Cu has relatively smooth or sharp interfaces compared to Fe grown on Ag. Although\nidentifying the origin of such structural di\u000berences is beyond the scope of this work, Cu/Fe\n7and Ag/Fe constitute two qualitatively distinct series of polycrystalline Fe \flms for exploring\nthe in\ruence of microstructure on magnetic relaxation.\nIII. INTRINSIC GILBERT DAMPING PROBED BY OUT-OF-PLANE FMR\nHaving established the di\u000berence in structural properties between Cu/Fe and Ag/Fe, we\ncharacterize room-temperature intrinsic damping for these samples with OOP FMR mea-\nsurements. The OOP geometry suppresses two-magnon scattering25such that the Gilbert\ndamping parameter can be quanti\fed in a straightforward manner. We use a W-band\nshorted waveguide in a superconducting magnet, which permits FMR measurements at high\n\felds ( &4 T) that completely magnetize the Fe \flms out of plane. The details of the mea-\nsurement method are found in Refs.18,37. Figure 3(a) shows the frequency dependence of\nhalf-width-at-half-maximum (HWHM) linewidth Δ𝐻OOP for selected thicknesses from both\nsample series. The linewidth data of 25 nm thick epitaxial Fe \flm from a previous study18\nis plotted in Fig. 3 (a) as well. The intrinsic damping parameter can be extracted from the\nlinewidth plot using\nΔ𝐻OOP=Δ𝐻0¸2𝜋\n𝛾𝛼OOP𝑓 (1)\nwhereΔ𝐻0is the inhomogeneous broadening38,𝛾=𝑔𝜇𝐵\nℏis the gyromagnetic ratio ( 𝛾2𝜋\u0019\n2.9 MHz/Oe [Ref.39], obtained from the frequency dependence of resonance \feld37), and\n𝛼OOP is the measured viscous damping parameter. In general, 𝛼OOP can include not only\nintrinsic Gilbert damping, parameterized by 𝛼int, but also eddy-current, radiative damping,\nand spin pumping contributions40, which all yield a linear frequency dependence of the\nlinewidth. Damping due to eddy current is estimated to make up less than 10% of the total\nmeasured damping parameter37and is ignored here. Since we used a shorted waveguide in\nour setup, the radiative damping does not apply here. Spin pumping is also negligible for\nmost of the samples here because the materials in the seed and capping layers (i.e., Ti, Cu,\nand Ag) possess weak spin-orbit coupling and are hence poor spin sinks31,41,42. We therefore\nproceed by assuming that the measured OOP damping parameter 𝛼OOP is equivalent to the\nintrinsic Gilbert damping parameter.\nThe extracted damping parameter is plotted as a function of Fe \flm thickness in Fig.\n3(b). The room-temperature damping parameters of all Fe \flms with thicknesses of 4-25\n80204060801001200306090120150180 \n25nm epitaxial Fe \n10nm Cu/Fe \n25nm Cu/Fe \n10nm Ag/Fe \n25nm Ag/FeΔHOOP [Oe]f\n [GHz](a)\n05101520250.0000.0010.0020.0030.004 epitaxial Fe \nCu/Fe \nAg/FeαOOPT\nhickness [nm](b)FIG. 3. (Color online) (a) OOP FMR half-width-at-half-maximum linewidth Δ𝐻OOPas a function\nof resonance frequency 𝑓. Lines correspond to \fts to the data. (b) Gilbert damping parameter\n𝛼𝑚𝑎𝑡ℎ𝑟𝑚𝑂𝑂𝑃 extracted from OOP FMR as a function of \flm thickness. The red shaded area\nhighlights the damping value range that contains data points of all \flms thicker than 4 nm. The\ndata for the epitaxial Fe sample (25 nm thick Fe grown on MgAl 2O4) are adapted from Ref.18.\nnm fall in the range of 0.0024 \u00060.0004, which is shaded in red in Fig. 3(b). This damping\nparameter range is quantitatively in line with the value reported for epitaxial Fe (black\nsymbol in Fig. 3(b))18. For 2 nm thick samples, the damping parameter is larger likely\ndue to an additional interfacial contribution43{45{ e.g., spin relaxation through interfacial\nRashba spin-orbit coupling46that becomes evident only for ultrathin Fe. The results in\nFig. 3(b) therefore indicate that the structural properties of the &4 nm thick polycrystalline\nbcc Fe \flms have little in\ruence on their intrinsic damping.\nIt is remarkable that these polycrystalline Cu/Fe and Ag/Fe \flms { with di\u000berent thick-\n9nesses and microstructural properties (as revealed in Sec. II) { exhibit essentially the same\nroom-temperature intrinsic Gilbert damping parameter as single-crystalline bcc Fe. This\n\fnding is qualitatively distinct from a prior report18on intrinsic Gilbert damping in single-\ncrystalline Fe \flms at cryogenic temperature, which is sensitive to microstructural disorder.\nIn the following, we discuss the possible di\u000berences in the mechanisms of intrinsic damping\nbetween these temperature regimes.\nIntrinsic Gilbert damping in ferromagnetic metals is predominantly governed by transi-\ntions of spin-polarized electrons between electronic states, within a given electronic band\n(intraband scattering) or in di\u000berent electronic bands (interband scattering) near the Fermi\nlevel15. For Fe, previous studies15,18,47indicate that intraband scattering tends to dominate\nat low temperature where the electronic scattering rate is low (e.g., \u00181013s\u00001); by contrast,\ninterband scattering likely dominates at room temperature where the electronic scattering\nrate is higher (e.g., \u00181014s\u00001). According to our results (Fig. 3(b)), intrinsic damping at\nroom temperature is evidently una\u000bected by the variation in the structural properties of the\nFe \flms. Hence, the observed intrinsic damping is mostly governed by the electronic band\nstructure within the Fe grains , such that disorder in grain boundaries or \flm interfaces has\nminimal impact.\nThe question remains as to why interband scattering at room temperature leads to Gilbert\ndamping that is insensitive to microstructural disorder, in contrast to intraband scattering\nat low temperature yielding damping that is quite sensitive to microstructure18. This dis-\ntinction may be governed by what predominantly drives electronic scattering { speci\fcally,\ndefects (e.g., grain boundaries, rough or di\u000buse interfaces) at low temperature, as opposed\nto phonons at high temperature. That is, the dominance of phonon-driven scattering at\nroom temperature may e\u000bectively diminish the roles of microstructural defects in Gilbert\ndamping. Future experimental studies of temperature-dependent damping in polycrystalline\nFe \flms may provide deeper insights. Regardless of the underlying mechanisms, the robust\nconsistency of 𝛼OOP (Fig. 3(b)) could be an indication that the intrinsic Gilbert damp-\ning parameter at a su\u000eciently high temperature is a local property of the ferromagnetic\nmetal, possibly averaged over the ferromagnetic exchange length of just a few nm48that is\ncomparable or smaller than the grain size. In this scenario, the impact on damping from\ngrain boundaries would be limited in comparison to the contributions to damping within\nthe grains.\n10Moreover, the misalignment of Fe grains evidently does not have much in\ruence on the\nintrinsic damping. This is reasonable considering that intrinsic Gilbert damping is predicted\nto be nearly isotropic in Fe at su\u000eciently high electronic scattering rates49{ e.g.,\u00181014s\u00001\nat room temperature where interband scattering is expected to be dominant15,18,47. It is\nalso worth emphasizing that 𝛼OOP remains unchanged for Fe \flms of various thicknesses\nwith di\u000berent magnitudes of strain (tetragonal distortion, as evidenced by the variation in\nthe out-of-plane lattice parameter in Fig. 1(e)). Strain in Fe grains is not expected to impact\nthe intrinsic damping, as Ref.18suggests that strain in bcc Fe does not signi\fcantly alter\nthe band structure near the Fermi level. Thus, polycrystalline Fe \flms exhibit essentially\nthe same magnitude of room-temperature intrinsic Gilbert damping as epitaxial Fe, as long\nas the grains retain the bcc crystal structure.\nThe observed invariance of intrinsic damping here is quite di\u000berent from the recent study\nof polycrystalline Co 25Fe75alloy \flms31, reporting a decrease in intrinsic damping with in-\ncreasing structural disorder. This inverse correlation between intrinsic damping and disorder\nin Ref.31is attributed to the dominance of intraband scattering, which is inversely propor-\ntional to the electronic scattering rate. It remains an open challenge to understand why the\nroom-temperature intrinsic Gilbert damping of some ferromagnetic metals might be more\nsensitive to structural disorder than others.\nIV. EXTRINSIC MAGNETIC RELAXATION PROBED BY IN-PLANE FMR\nAlthough we have shown via OOP FMR in Sec. III that intrinsic Gilbert damping is\nessentially independent of the structural properties of the Fe \flms, it might be expected\nthat microstructure has a pronounced impact on extrinsic magnetic relaxation driven by\ntwo-magnon scattering, which is generally present in IP FMR. IP magnetized \flms are more\ncommon in device applications than OOP magnetized \flms, since the shape anisotropy of\nthin \flms tends to keep the magnetization in the \flm plane. What governs the performance\nof such magnetic devices (e.g., quality factor50,51) may not be the intrinsic Gilbert damping\nparameter but the total FMR linewidth. Thus, for many magnetic device applications, it is\nessential to understand the contributions to the IP FMR linewidth.\nIP FMR measurements have been performed using a coplanar-waveguide-based spectrom-\neter, as detailed in Refs.18,37. Examples of the frequency dependence of IP FMR linewidth\n110501001502002500\n10203040506070050100150200250Cu/FeA\ng/Fe 2 nm \n6 nm \n8 nm \n10 nm \n15 nm \n25 nmΔHIP [Oe] \n12(\na) \nf\n [GHz](b)FIG. 4. (Color online) IP FMR half-width-at-half-maximum linewidth Δ𝐻IPas a function of\nresonance frequency 𝑓for (a) Cu/Fe and (b) Ag/Fe. The vertical dashed line at 12 GHz highlights\nthe hump in linewidth vs frequency seen for many of the samples.\nare shown in Fig. 4. In contrast to the linear frequency dependence that arises from in-\ntrinsic Gilbert damping in Fig. 3(a), a nonlinear hump is observed for most of the \flms\nin the vicinity of \u001912 GHz. In some \flms, e.g., 10 nm thick Cu/Fe \flm, the hump is so\nlarge that its peak even exceeds the linewidth at the highest measured frequency. Similar\nnonlinear IP FMR linewidth behavior has been observed in Fe alloy \flms52and epitaxial\nHeusler \flms53in previous studies, where two-magnon scattering has been identi\fed as a\nsigni\fcant contributor to the FMR linewidth. Therefore, in the following, we attribute the\nnonlinear behavior to two-magnon scattering.\nTo gain insight into the origin of two-magnon scattering, we plot the linewidth at 12\n122550751001251500\n5101520250255075100125150 Cu/Fe \nAg/Fe Cu/Fe \nAg/FeΔHIP @ 12 GHz [Oe](a)HC [Oe]T\nhickness [nm](b)FIG. 5. (Color online) (a) IP FMR half-width-at-half-maximum linewidth at 12 GHz { approxi-\nmately where the maximum (\\hump\") in linewidth vs frequency is seen (see Fig. 4) { as a function\nof \flm thickness for both Cu/Fe and Ag/Fe. (b) Coercivity 𝐻𝑐as a function of \flm thickness for\nboth Cu/Fe and Ag/Fe. The red shaded area highlights thickness region where the Cu/Fe sample\nseries show a peak behavior in both plots.\nGHz { approximately where the hump is seen in Fig. 4 { against the Fe \flm thickness in\nFig. 5(a). We do not observe a monotonic decay in the linewidth with increasing thickness\nthat would result from two-magnon scattering of interfacial origin54. Rather, we observe\na non-monotonic thickness dependence in Fig. 5(a), which indicates that the observed\ntwo-magnon scattering originates within the bulk of the \flms. We note that Ag/Fe with\ngreater interfacial disorder (see Sec. II) exhibits weaker two-magnon scattering than Cu/Fe,\nparticularly in the lower thickness regime ( .10 nm). This observation further corroborates\n13that the two-magnon scattering here is not governed by the interfacial roughness of Fe\n\flms. The contrast between Cu/Fe and Ag/Fe also might appear counterintuitive, since\ntwo-magnon scattering is induced by defects and hence might be expected to be stronger\nfor more \\defective\" \flms (i.e., Ag/Fe in this case). The counterintuitive nature of the\ntwo-magnon scattering here points to more subtle mechanisms at work.\nTo search for a possible correlation between static magnetic properties and two-magnon\nscattering, we have performed vibrating sample magnetometry (VSM) measurements with a\nMicrosense EZ9 VSM. Coercivity extracted from VSM measurements is plotted as a function\nof \flm thickness in Fig. 5(b), which shows a remarkably close correspondence with linewidth\nvs thickness (Fig. 5(a)). In particular, a pronounced peak in coercivity is observed for Cu/Fe\naround 10 nm, corresponding to the same thickness regime where the 12 GHz FMR linewidth\nfor Cu/Fe is maximized. Moreover, the 10 nm Cu/Fe sample (see Sec. II) exhibits a tall,\nnarrow bcc (110) di\u000braction peak, which suggests that its peculiar microstructure plays a\npossible role in the large two-magnon scattering and coercivity (e.g., via stronger domain\nwall pinning).\nWhile the trends shown in Fig. 5 provide some qualitative insights, we now attempt to\nquantitatively analyze the frequency dependence of FMR linewidth for the Cu/Fe and Ag/Fe\n\flms. We assume that the Gilbert damping parameter for IP FMR is equal to that for OOP\nFMR, i.e.,𝛼IP=𝛼OOP. This assumption is physically reasonable, considering that Gilbert\ndamping is theoretically expected to be isotropic in Fe \flms near room temperature49. While\na recent study has reported anisotropic Gilbert damping that scales quadratically with\nmagnetostriction55, this e\u000bect is likely negligible in elemental Fe whose magnetostriction is\nseveral times smaller56,57than that of the Fe 07Ga03alloy in Ref.55.\nThus, from the measured IP linewidth Δ𝐻IP, the extrinsic two-magnon scattering\nlinewidthΔ𝐻TMS can be obtained by\nΔ𝐻TMS=Δ𝐻IP\u00002𝜋\n𝛾𝛼IP (2)\nwhere2𝜋\n𝛾𝛼IPis the Gilbert damping contribution. Figure 6 shows the obtained Δ𝐻TMSand \ft\nattempts using the \\grain-to-grain\" two-magnon scattering model developed by McMicheal\nand Krivosik25. This model captures the inhomogeneity of the e\u000bective internal magnetic\n\feld in a \flm consisting of many magnetic grains. The magnetic inhomogeneity can arise\nfrom the distribution of magnetocrystalline anisotropy \feld directions associated with the\n14randomly oriented crystal grains52. In this model the two-magnon scattering linewidth\nΔ𝐻TMS is a function of the Gilbert damping parameter 𝛼IP, the e\u000bective anisotropy \feld\n𝐻𝑎of the randomly oriented grain, and the correlation length 𝜉within which the e\u000bective\ninternal magnetic \feld is correlated. Further details for computing Δ𝐻TMS are provided in\nthe Appendix and Refs.25,52,53. As we have speci\fed above, 𝛼IPis set to the value derived\nfrom OOP FMR results (i.e., 𝛼OOP in Fig. 3(b)). This leaves 𝜉and𝐻𝑎as the only free\nparameters in the \ftting process.\nThe modeling results are dependent on the choice of the correlation function 𝐶¹Rº, which\ncaptures how the e\u000bective internal magnetic \feld is correlated as a function of lateral distance\nRin the \flm plane. We \frst show results obtained with a simple exponentially decaying\ncorrelation function, as done in prior studies of two-magnon scattering25,52,53, i.e.,\n𝐶¹Rº=exp\u0012\n\u0000jRj\n𝜉\u0013\n (3)\nEquation 3 has the same form as the simplest correlation function used to model rough\ntopographical surfaces (when they are assumed to be \\self-a\u000ene\")58. Fit results with Eq. (3)\nare shown in dashed blue curves in Fig. 6. For most samples, the \ftted curve does not\nreproduce the experimental data quantitatively. Moreover, the \ftted values of 𝜉and𝐻𝑎\noften reach physically unrealistic values, e.g., with 𝐻𝑎¡104Oe and𝜉 1 nm (see Table I).\nThese results suggest that the model does not properly capture the underlying physics of\ntwo-magnon scattering in our samples.\nA possible cause for the failure to \ft the data is that the simple correlation function\n(Eq. 3) is inadequate. We therefore consider an alternative correlation function by again\ninvoking an analogy between the spatially varying height of a rough surface58and the spa-\ntially varying e\u000bective internal magnetic \feld in a \flm. Speci\fcally, we apply a correlation\nfunction (i.e., a special case of Eq. (4.3) in Ref.58where short-range roughness 𝛼=1) for\nthe so-called \\mounded surface,\" which incorporates the average distance 𝜆between peaks\nin topographical height (or, analogously, e\u000bective internal magnetic \feld):\n𝐶¹Rº=p\n2jRj\n𝜉𝐾1 p\n2jRj\n𝜉!\n𝐽0\u00122𝜋jRj\n𝜆\u0013\n (4)\nwhere𝐽0and𝐾1are the Bessel function of the \frst kind of order zero and the modi\fed Bessel\nfunction of the second kind of order one, respectively. This oscillatory decaying function is\nchosen because its Fourier transform (see Appendix) does not contain any transcendental\n15020406080100120 \nExperimental \nSelf-affine \nMoundedΔHTMS [Oe]Cu/FeA g/Fe6\n nm8\n nm1\n0 nm1\n5 nm2\n5 nm(a)( f)0\n50100150ΔHTMS [Oe](\nb)( g)0\n50100150ΔHTMS [Oe](\nc)( h)0\n255075100125ΔHTMS [Oe](\nd)( i)0\n2 04 06 0050100150200ΔHTMS [Oe]f\n [GHz](e)0\n2 04 06 0f\n [GHz](j)FIG. 6. (Color online) Extrinsic two-magnon scattering linewidth Δ𝐻TMSvs frequency 𝑓and \ftted\ncurves for 6, 8, 10, 15, and 25 nm Cu/Fe and Ag/Fe \flms. Black squares represent experimental\nFMR linewidth data. Dashed blue and solid red curves represent the \ftted curves using correlation\nfunctions proposed for modeling self-a\u000ene and mounded surfaces, respectively. In (d), (e), (h), (i),\ndashed blue curves overlap with solid red curves.\n16functions, which simpli\fes the numerical calculation. We also stress that while Eq. (4) in\nthe original context (Ref.58) was used to model topographical roughness, we are applying\nEq. (4) in an attempt to model the spatial \ructuations (\\roughness\") of the e\u000bective internal\nmagnetic \feld { rather than the roughness of the \flm topography.\nThe \ftted curves using the model with Eq. (4) are shown in solid red curves in Fig. 6. Fit\nresults for some samples show visible improvement, although this is perhaps not surprising\nwith the introduction of 𝜆as an additional free parameter. Nevertheless, the \ftted values\nof𝐻𝑎or𝜆still diverge to unrealistic values of ¡104Oe or¡104nm in some cases (see\nTable I), which means that the new correlation function (Eq. (4)) does not fully re\rect\nthe meaningful underlying physics of our samples either. More detailed characterization of\nthe microstructure and inhomogeneities, e.g., via synchrotron x-ray and neutron scattering,\ncould help determine the appropriate correlation function. It is also worth pointing out that\nfor some samples (e.g. 15 nm Cu/Fe and Ag/Fe \flms), essentially identical \ft curves are\nobtained regardless of the correlation function. This is because when 𝜆\u001d𝜉, the Fourier\ntransform of Eq. (4) has a very similar form as the Fourier transform of Eq. (3), as shown in\nthe Appendix. In such cases, the choice of the correlation function has almost no in\ruence\non the behavior of the two-magnon scattering model in the \ftting process.\nV. SUMMARY\nWe have examined room-temperature intrinsic and extrinsic damping in two series of\npolycrystalline Fe thin \flms with distinct structural properties. Out-of-plane FMR mea-\nsurements con\frm constant intrinsic Gilbert damping of \u00190.0024, essentially independent\nof \flm thickness and structural properties. This \fnding implies that intrinsic damping in\nFe at room temperature is predominantly governed by the crystalline and electronic band\nstructures within the grains, rather than scattering at grain boundaries or \flm surfaces. The\nresults from in-plane FMR, where extrinsic damping (i.e., two-magnon scattering) plays a\nsigni\fcant role, are far more nuanced. The conventional grain-to-grain two-magnon scatter-\ning model fails to reproduce the in-plane FMR linewidth data with physically reasonable\nparameters { pointing to the need to modify the model, along with more detailed character-\nization of the \flm microstructure. Our experimental \fndings advance the understanding of\nintrinsic Gilbert damping in polycrystalline Fe, while motivating further studies to uncover\n17TABLE I. Summary of IP FMR linewidth \ft results. Note the divergence to physically unreasonable\nvalues in many of the results. Standard error is calculated using equation√︁\nSSRDOF\u0002diag¹COVº,\nwhere SSR stands for the sum of squared residuals, DOF stands for degrees of freedom, and COV\nstands for the covariance matrix.\nSelf-a\u000ene Mounded\nSample\nSeriesThickness\n(nm)𝜉\n(nm)𝐻𝑎\n(Oe)𝜉\n(nm)𝐻𝑎\n(Oe)𝜆\n(nm)\nCu/Fe6 70\u000610 170\u000610 80\u000690 24\u00063 >1\u0002104\n8 200\u0006100 150\u000620 700\u00061000 25\u00062 900\u0006100\n10 140\u000640 200\u000620 160\u000650 33\u00061 800\u0006200\n15 9\u00062 800\u0006100 10\u000620 100\u000680 >1\u0002104\n25 0\u00065 >1\u000210460\u000630 >1\u000210410.41\u00060.01\nAg/Fe6 0\u000640 >1\u0002104150\u000640 >1\u000210411.7\u00060.7\n8 0\u000630 >1\u0002104170\u000650 >1\u000210412\u00064\n10 6\u00061 1500\u0006300 8\u000640 200\u0006500 >1\u0002104\n15 2\u00062 4000\u00063000 3\u00069 500\u0006900 >6\u0002103\n25 0\u00066 >1\u0002104140\u000650 >1\u000210415\u00066\nthe mechanisms of extrinsic damping in structurally disordered thin \flms.\nACKNOWLEDGMENTS\nS.W. acknowledges support by the ICTAS Junior Faculty Program. D.A.S. and S.E.\nacknowledge support by the National Science Foundation, Grant No. DMR-2003914. P.\nN. would like to acknowledge support through NASA Grant NASA CAN80NSSC18M0023.\nA. R. would like to acknowledge support through the Defense Advanced Research Project\nAgency (DARPA) program on Topological Excitations in Electronics (TEE) under Grant\nNo. D18AP00011. This work was supported by NanoEarth, a member of National Nan-\notechnology Coordinated Infrastructure (NNCI), supported by NSF (ECCS 1542100).\n18Appendix A: Details of the Two-Magnon Scattering Model\nIn the model developed by McMichael and Krivosik, the two-magnon scattering contri-\nbutionΔ𝐻TMS to the FMR linewidth is given by25,52,53\nΔ𝐻TMS=𝛾2𝐻2\n𝑎\n2𝜋𝑃𝐴¹𝜔º∫\nΛ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔\u0000𝜔𝑘ºd2𝑘 (A1)\nwhere𝜉is correlation length, 𝐻𝑎is the e\u000bective anisotropy \feld of the randomly oriented\ngrain.𝑃𝐴¹𝜔º=𝜕𝜔\n𝜕𝐻\f\f\n𝐻=𝐻FMR=√︃\n1¸¹4𝜋𝑀𝑠\n2𝜔𝛾º2accounts for the conversion between the fre-\nquency and \feld swept linewidth. Λ0𝑘represents the averaging of the anisotropy axis \ruc-\ntuations over the sample. It also takes into account the ellipticity of the precession for both\nthe uniform FMR mode and the spin wave mode52. The detailed expression of Λ0𝑘can\nbe found in the Appendix of Ref.52. The coe\u000ecients in the expression of Λ0𝑘depend on\nthe type of anisotropy of the system. Here, we used \frst-order cubic anisotropy for bcc Fe.\n𝛿𝛼¹𝜔\u0000𝜔𝑘ºselects all the degenerate modes, where 𝜔represents the FMR mode frequency\nand𝜔𝑘represents the spin wave mode frequency. The detailed expression of 𝜔𝑘can be found\nin Ref.25. In the ideal case where Gilbert damping is 0, 𝛿𝛼is the Dirac delta function. For a\n\fnite damping, 𝛿𝛼¹𝜔0\u0000𝜔𝑘ºis replaced by a Lorentzian function1\n𝜋¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻\n¹𝜔𝑘\u0000𝜔º2¸»¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻¼2,\nwhich is centered at 𝜔and has the width of ¹2𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻.\nFinally,𝐶𝑘¹𝜉º(or𝐶𝑘¹𝜉𝜆º) is the Fourier transform of the grain-to-grain internal \feld\ncorrelation function, Eq. (3) (or Eq. (4)). For the description of magnetic inhomogeneity\nanalogous to the simple self-a\u000ene topographical surface58, the Fourier transform of the\ncorrelation function, Eq. (3), is\n𝐶𝑘¹𝜉º=2𝜋𝜉2\n»1¸¹𝑘𝜉º2¼3\n2 (A2)\nas also used in Refs.25,52,53. For the description analogous to the mounded surface, the\nFourier transform of the correlation function, Eq. (4), is58\n𝐶𝑘¹𝜉𝜆º=8𝜋3𝜉2\u0010\n1¸2𝜋2𝜉2\n𝜆2¸𝜉2\n2𝑘2\u0011\n\u0014\u0010\n1¸2𝜋2𝜉2\n𝜆2¸𝜉2\n2𝑘2\u00112\n\u0000\u0010\n2𝜋𝜉2\n𝜆𝑘\u00112\u001532 (A3)\nWhen𝜆\u001d𝜉, Eq. (A3) becomes\n𝐶𝑘¹𝜉º\u00198𝜋3𝜉2\n\u0010\n1¸𝜉2\n2𝑘2\u00112 (A4)\n19100102104106108101010-2410-2210-2010-1810-1610-1410-1210-10 \nSelf-affine  \nMounded λ = 10 nm \nMounded λ = 100 nm \nMounded λ = 1000 nmCk [m2]k\n [m-1]ξ = 100 nmFIG. 7. Fourier transform of correlation function for mounded surfaces as a function of wavenumber\n𝑘for three di\u000berent 𝜆values. Fourier transform of correlation function for self-a\u000ene surfaces as a\nfunction of 𝑘is also included for comparison purpose. 𝜉is set as 100 nm for all curves.\nwhich has a similar form as Eq. (A2). This similarity can also be demonstrated graphically.\nFigure 7 plots a self-a\u000ene 𝐶𝑘curve (Eq. (A2)) at 𝜉=100 nm and three mounded 𝐶𝑘curves\n(Eq. (A3)) at 𝜆=10, 100, 1000 nm. 𝜉in mounded 𝐶𝑘curves is set as 100 nm as well. It\nis clearly shown in Fig. 7 that when 𝜆=1000 nm, the peak appearing in 𝜆=10 and 100\nnm mounded 𝐶𝑘curves disappears and the curve shape of mounded 𝐶𝑘resembles that of\nself-a\u000ene𝐶𝑘.\nThe hump feature in Fig. 4 is governed by both 𝛿𝛼and𝐶𝑘(see Eq. A1). 𝛿𝛼has the shape\nof1in reciprocal space ( 𝑘space), as shown in our videos in the Supplemental Material as\nwell as Fig. 5(b) of Ref.53and Fig 2 (b) of Ref.25. The size of the contour of the degenerated\nspin wave modes in 𝑘space increases as the microwave frequency 𝑓increases, which means\nthe number of available degenerate spin wave modes increases as 𝑓increases. As shown\nin Fig. 7, self-a\u000ene 𝐶𝑘is nearly constant with the wavenumber 𝑘until𝑘reaches\u00181𝜉.\nThis suggests that the system becomes e\u000bectively more uniform (i.e. weaker inhomogeneous\nperturbation) when the length scale falls below the characteristic correlation length 𝜉(i.e.,\n𝑘 ¡1𝜉). Because inhomogeneities serve as the scattering centers of two-magnon scattering\n20process, degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be scattered into.\nNow we consider the 𝑓dependence of the two-magnon scattering rate. When 𝑓is small,\nthe two-magnon scattering rate increases as 𝑓increases because more degenerate spin wave\nmodes become available as 𝑓increases. When 𝑓further increases, the wavenumber 𝑘of\nsome degenerate spin wave modes exceeds 1 𝜉. This will decrease the overall two-magnon\nscattering rate because the degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be\nscattered into, as discussed above. Furthermore, the portion of degenerate spin wave modes\nwith𝑘 ¡ 1𝜉increases as 𝑓continues to increase. When the impact of decreasing two-\nmagnon scattering rate for degenerate spin wave modes with high 𝑘surpasses the impact\nof increasing available degenerate spin wave modes, the overall two-magnon scattering rate\nwill start to decrease as 𝑓increases. Consequently, the nonlinear trend { i.e., a \\hump\" {\nin FMR linewidth Δ𝐻TMS vs𝑓appears in Fig. 4.\nHowever, the scenario discussed above can only happen when 𝜉is large enough, because\nthe wavenumber 𝑘of degenerate spin wave modes saturates (i.e., reaches a limit) as 𝑓\napproaches in\fnity. If the limit value of 𝑘is smaller than 1𝜉, the two-magnon scattering\nrate will increase monotonically as 𝑓increases. In that case the hump feature will not\nappear. See our videos in the Supplemental Material that display the 𝑓dependence of Λ0𝑘,\n𝛿𝛼¹𝜔\u0000𝜔𝑘º,𝐶𝑘¹𝜉º\n2𝜋𝜉2,Λ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔\u0000𝜔𝑘º\n2𝜋𝜉2 , andΔ𝐻TMS for various𝜉values.\nPrevious discussions of the hump feature are all based on the self-a\u000ene correlation func-\ntion (Eq. 3). The main di\u000berence between the mounded correlation function (Eq. 4) and the\nself-a\u000ene correlation function (Eq. 3) is that the mounded correlation function has a peak\nwhen𝜆is not much larger than 𝜉as shown in Fig. 7. This means when the wavenumber\n𝑘of degenerate spin wave modes enters (leaves) the peak region, two-magnon scattering\nrate will increase (decrease) much faster compared to the self-a\u000ene correlation function. In\nother words, the mounded correlation function can generate a narrower hump compared to\nthe self-a\u000ene correlation function in the two-magnon linewidth Δ𝐻TMS vs𝑓plot, which is\nshown in Fig. 6 (b, c).\n1Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L.-C. 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Kuanr2,* \n1School of Physical Sciences, Jawaharlal Nehru University, New Delhi, INDIA 110067  \n2Special Centre for Nanosciences, Jawaharlal Nehru University, New Delhi, INDIA 110067  \n3Morgan State University, Department of Physics, Baltimore, MD, USA 21251  \n \n \nAbstract  \nYttrium Iron Garnet (YIG) and b ismuth (Bi)  substituted YIG (Bi 0.1Y2.9Fe5O12, BYG) films are \ngrown  in-situ on single crystalline Gadolinium Gallium Garnet (GGG) substrates  [with (100) and \n(111) orientation s] using  pulsed laser deposition  (PLD ) technique .  As the orientation of the Bi-\nYIG film changes from (100) to (111) , the lattice constant  is enhanced  from 12.384  Å to 12.401 Å \ndue to  orientation dependent distribution of Bi3+ ions at dodecahedral  sites in the lattice cell. \nAtomic force microscopy (AFM) images show smooth  film surfaces  with roughness 0.308 nm in \nBi-YIG (111) . The change in substrate orientation leads to the modification of Gilbert damping \nwhich , in turn, gives rise to the enhancement of  ferromagnetic resonance (FMR) line width . The \nbest value s of Gilbert damping are found to be  (0.54±0.06 )×10-4, for YIG (100)  and \n(6.27±0.33) ×10-4, for Bi-YIG (111)  oriented  films . Angle variation ( ) measurements of the H r are \nalso performed, that shows  a four -fold symmetry for the resonance field  in the  (100) g rown film. \nIn addition, the value of  effective magnetization  (4πM eff) and extrinsic linewidth  (ΔH 0) are \nobserved to be  dependent on  substrate orientation . Hence PLD growth can assist single -crystalline \nYIG and BY G films with  a perfect interface that can be used for  spintronics  and related device \napplications.  \n \n \n \nKeyword s: Pulse Laser Deposition, Epitaxial YIG thin films, lattice strain, ferromagnetic \nresonance, Gilbert damping, inhomogeneous broa dening  \n \nCorresponding authors:  bijoykuanr@mail.jnu.ac.in , spatnaik@mail.jnu.ac.in  2 \n 1. Introduction  \nOne of the most widely studied material s for the realization of spintronic devices appears to be the \niron garnets , particularly the  yttrium iron garnet (YIG , Y3Fe5O12) [1,2] .  In thin film form  of YIG  \nseveral potential applications have been envisaged that include spin-caloritronics  [3,4] , magneto -\noptical (MO) devices, and microwave resonators, circulators, and  filters [5–8]. The attraction of \nYIG over other ferroic materials is primarily due to their strong magnet o-crystalline  anisotropy \nand low magnetization damping  [2]. Furthermore, towards  high frequency  applications, YIG’s \nmain advantage s are its electrically insulating behavior  along with low ferromagnetic resonance \nline-width (H) and low Gilbert damping  parameter [9–11]. These are important parameters for \npotential  use in high fr equency filters and actuators [12–14]. In this paper, we report optimal \ngrowth parameters for pure and Bi -doped YIG on oriented subs trates and identify the conditions \nsuitable for their  prospective  applications.  \n \nIn literature, YIG is known to be a room temperature  ferrimagnetic insulator with a  Tc near 560 K \n[15]. It has a cubic structure (space group Ia3̅d). The y ttrium  (Y) ions occupy the  dodecahedral  \n24c sites ( in the Wyckoff notation), two Fe ions  at octahedral 16a and three at tetrahedral 24d sites, \nand oxygen the 96h sites  [16,17] . The d site is resp onsible for the ferri magnetic  nature of YIG. It \nis already reported that substitution of Bi in place of Y in YIG leads to substantial improvement in \nthe magneto -optical response [7,18 –25]. It was also observed that MO performance increa ses \nlinearly with Bi/Ce doping concentration [22]. Furthermore, substitution of  Bi in YIG  (BYG)  is \ndocumented to provide growth -induced anisotropy  that is useful in applications such as magnetic \nmemory and logic devices [26–30]. The study of basic properties of  Bi -substituted YIG materials \nis of great current interest  due to their applications  in magneto -optical devices , magnon -3 \n spintronics , and related fields such as caloritronics  due to its high uniaxial anisotropy and faraday \nrotation  [21,31 –35]. The structural and magnetic pr operties can be changed via change in Bi3+ \nconcentration in YIG or via choosing a proper  substrate orientation. Therefore, t he choice of \nperfect substrate orientation is crucial for the identification of the growth of Bi  substituted YIG \nthin films.  \nIn this  work, we have studied the structural and magnetic properties  of Bi-substituted YIG \n[Bi0.1Y2.9Fe5O12 (BYG)] and YIG thin films with two different single crystalline Gadolinium \nGallium Garnet (GGG)  substrate orientation s: (100) and (111) . The YIG and BYG films of  \nthickness  ~150 nm  were grown by pulsed laser deposit ion (PLD) method  [23,36,37]  on top of  \nsingle -crystalline GGG  substrates . The structural and magnetic properties of all grown films were \ncarried out using x -ray diffraction (XRD), surface morphology by atomic force microscopy \n(AFM) , and magnetic properties via vibrating sample magnetometer (VSM) and ferromagnetic \nresonance (FMR) techniques.  The FMR is the most useful  technique to study the magnetization \ndynamics by measuring the  properties of magnetic materials through evaluation of their damping \nparameter and linewidth .  Furthermore, it provides insightful information on the static magnetic \nproperties such as the saturation magnetization and the anisotropy field. FMR is also extremely \nhelpful to study fundamentals of spin wave dynami cs and towards characte rizing the relaxation \ntime and L ande g factor of  magnetic material s [11]. \n \n2. Experiment  \n \nYIG a nd BY G target s were synthesized  via the solid -state reaction method . Briefly, y ttrium oxide \n(Y2O3) and iron oxide (Fe 2O3) powder s were ground for ~14 hours  before calcination at 1100 oC. 4 \n The calcined  powders were pressed into pellets and sintered at 1300 oC.  Using thes e YIG and \nBYG targets,  thin films of thickness ~150 nm  were grown  in-situ on (100) - and (111) -oriented \nGGG  substrate s by the PLD technique . The prepared samples have been labeled as YIG (100) , \nYIG (111) , BYG (100), and BYG (111).  GGG substrates were cleaned using acetone and \nisopropanol.  Before deposition, the deposition chamber  was thoroughly cleaned and evacuated to \na base vacuum of 2 ×10-6 mbar. We have used KrF excimer laser (248  nm), with pulse frequency \n10 Hz to ablate the target s at 300mJ  energy . During deposition , target to substrate distance, \nsubstrate temperature , and oxygen pressure w ere kept at  ~4.8 cm, 825 oC, and 0.15 mbar , \nrespectively.  Best films were grown at a rate  of 6 nm/min . The as -grown thin film s were annealed \nin-situ for 2 hours at 825  oC and cooled down to 300 oC in the presence of oxygen  (0.15 mbar)  \nthroughout the process . The structural properties of the thin film were determined  by XRD  using \nCu-Kα radiation  (1.5406  Å) and surface morphology as well as  the thickness of the film were \ncalculated with atomic force microscopy  by WITec  Gmb H, Germany . Magnetic properties were \nstudied using a 14 tesla PPMS (Cryogenic) . FMR  measurements were carried out  by the  Vector \nNetwork Analyzer ( VNA ) (Keysight , USA)  using a coplanar waveguide ( CPW ) in a flip -chip \ngeometry with dc magnetic field applied parallel  to the film plan e. \n \n3. Results and Discussion  \n3.1 Structural properties  \nThe room temperature XRD  data for the polycrystalline targets of YIG and BYG are plotted in \nfigure  1 (a) and 1  (b) respectively.  Rietveld refinement patterns after fitting XRD data are also \nincluded in the panel s. XRD peaks are indexed according to the JCPDS card no. ( # 43-0507) . Inset 5 \n of figure  1 (a) shows crystallographic sub -lattices  of YIG that elucidates  Fe13+ tetrahedral  site, \nFe23+ octahedral site , and Y3+ dodecahedral site. Inset (i) of figure  1 (b)  shows evidence for \nsuccessful incorporation of Bi into YIG ; the lattice constant  increases when Bi is substituted into \nYIG due to larger ionic radii of Bi (1.170 Å) as compared with Y (1.019 Å) [19]. From Rietveld \nrefinement  we estimate the lattice constant of YIG and BYG to be  12.377 Å [38] and 12.401 Å  \nrespectively .  \n \nFigure  2 (a) and 2  (b) show the XRD pattern of  bare (100) and (111) oriented GGG substrates . \nThis is followed by figure  2 (c)  & 2(d) for YIG and figure  2 (e)  & 2 (f) for BYG as grown thin \nfilms. XRD patterns confirm  the single -crystalline grow th of YIG and BYG thin film s over GGG \nsubstrates . The l attice  constant, lattice mismatch  (with respect to substrate) , and lattice volume \nobtain ed from XRD  data are listed in Table  1. Lattice cons tant a for the cubic structure is evaluated \nusing the [39]. \n𝒂=𝜆√ℎ2+𝑘2+𝑙2\n2sin𝜃                                                        (1) \n Where  𝜆 is the wavelength of Cu -Kα radiation , 𝜃 is the diffraction angle , and [h , k, l ] are the \nmiller indices of the corresponding XRD peak. Lattice  misfit  (𝛥𝑎\n𝑎) is evaluated using equation 2  \n[24,38] . \n𝛥𝑎\n𝑎=(𝑎𝑓𝑖𝑙𝑚− 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 )\n𝑎𝑓𝑖𝑙𝑚 100               (2) \nWhere  𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒  are the lattice constant of film and substrate respectively.  Lattice \nconstant of pure YIG bulk  is 12.377  Å, whereas we have observed  a larger value of lattice constants \nof YIG and BY G films  than th at of bulk YIG as shown in T able 1 . Sim ilarly, to these  obtained \nresults, a larger value of lattice constants than that of bulk YIG has been reported as well [40–44]. 6 \n The obtained values of the lattice constant are in agreement with the previous reports \n[18,21,25,34,45] . In the case of  BYG (111), the value of lattice constant slightly increases \ncompared to BYG (100) because the distribution of Bi3+ in the dodecahedral site depends on the \norientation of the substrate [28,46] . Inset (ii) of figure  1 (b) shows  plane (111) has more \ncontribution of Bi3+ ions [(ionic radius of bismuth (1.170 Å) is larger as compared with YIG (1.019 \nÅ) [19]]. This slight increase in the lattice constant  (in 111 direction)  implies a more lattice \nmismatch (or strain ) in BYG films . Positive  value of lattice mismatch indicates the slightly larger \nlattice constant of films (YIG and BYG) were observed as compared to substrates (GGG). We \nwould like to emphasize  that lattice plane dependence growth  is important to signify  the changes \nin the struc tural and magnetic properties.  \n3.2 Surface morphology study  \nRoom temperature AFM images  with roughness are  shown in figure 3 (a)-(d). Roughness plays \nan important role from the application prospective as it is related to  Gilbert  damping factor α. \nLower roughness (root mean square height) is observed for the (111) oriented films of YIG and \nBYG compared to those grown on (100) oriented substrates. Available literature [61] indicate that \nroughness would depend more on variation on growth parameters ra ther than on substrate \norientation.  In this sense further study is needed to clarify substrate dependence of roughness. No \nsignificant change in the roughness is observed between YIG and BYG films  [38,47] . Table 1  \ndepicts a comparison between the roughne ss measured in YIG and the BY G thin films.  7 \n 3.3 Static magnetization p roperties  \nVSM  magnetization  measurements were performed  at 300 K with magnetic field appl ied parallel \nto the film plane  (in-plane) .  Figure 3 (e) and 3  (f) show the magnetization plot s for YIG and BYG \nrespectively after careful subtraction of paramagnetic contr ibution  that is assigned to the substrate.  \nThe m easured saturation magnetization ( 4πM S) data are given in Table 1 which are in general \nagreement with the reported values [11,40,48] . Not much change in the measured  4πM S value of \nYIG and Bi -YIG films  are observed . The ferrimagnetism nature of YIG arises from super -\nexchange interaction between the non -equivalent Fe3+ ions at octahedral  and tetrahedral  sites [49]. \nBismuth located at dodecahedral site does not affect the tetrahedral and octahedral Fe3+ ions. So, \nBismuth does not show a significant change in saturation magnetization at  room temperature. I t is \nreported in literature th at Bi addition leads to increase  in Curie temperature, so in t hat sense there \nis an decreasing  trend in saturation magnetization in BYG films in contrast to YIG films [50,51] . \nError bars in saturation magnetization relate to uncertainty in sample volume.  \n3.4 Ferromagnetic r esonance properties  \nThe FMR absorption spectroscopy  is shown in figure 4. These measurements were  performed  at \nroom temperature .  The external dc magnetic field  was appli ed parallel to the plane of the film . \nLorentzian  fit of the calibrated experimental data are used to calculate t he FMR linewidth (∆H) \nand resonance  magnetic  field (H r). From the e nsemble of all the FMR data at different resonance \nfrequencies  (f = 1 GHz -12 GHz ), we have calculated the gyromagnetic ratio (γ) , effective \nmagnetization field ( 4𝜋𝑀𝑒𝑓𝑓) from the  fitting  of Kittel’s  in-plane equation  [52].  \n 8 \n In general, t he uniform precession of magnetization can be described by the Landau -Lifshitz -\nGilbert (LLG) equation of motion;  \n𝜕𝑀⃗⃗ \n𝜕𝑡=−𝛾(𝑀⃗⃗ ×𝐻⃗⃗ 𝑒𝑓𝑓)+𝐺\n𝛾𝑀𝑠2[𝑀⃗⃗ ×𝜕𝑀⃗⃗ \n𝜕𝑡]   (3) \nHere, t he first term corresponds to the precessional torque in the effective magnetic field and the \nsecond term is the Gilbert damping torque.  The gyromagnetic ratio  is given by 𝛾=𝑔𝜇𝐵/ℏ , where  \n𝑔 is the Lande’s factor, 𝜇𝐵 is Bohr magnetron and ℏ is the Planck’s constant. Similarly, 𝐺=𝛾𝛼𝑀𝑠 \nis related to the intrinsic relaxation rate in the nanocomposites and 𝛼 represents the Gilbert \ndamping constant. Ms (or 4πMs) is the saturation magnetization. It can be shown that t he solution \nfor in -plane resonance frequency can be written as;  \n𝑓𝑟=𝛾′√(𝐻𝑟)(𝐻𝑟+4𝜋𝑀𝑒𝑓𝑓)     (4), \nWhere  𝛾′=𝛾/2𝜋, 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 is the effective field and 𝐻𝑎𝑛𝑖=2𝐾1\n𝑀𝑠 is the anisotropy \nfield.  Following through, we have obtained Gilbert damping parameter (α) and inhomogeneous \nbroadening (∆H 0) linewidth from the fitting of Landau –Lifshitz –Gilbert equation (LLG) [53] \n𝛥𝐻(𝑓)=𝛥𝐻0+4𝜋𝛼\n√3𝛾𝑓                                      (5) \n \n \n \n \nDerived parameters from the FMR study are listed in T able 2 . The obtai ned Gilbert damping (α) \nis in agreement with the  reported thin films used for the study  of spin-wave  propagation  \n[2,27,54,55] .  In the case of YIG no t much change  in the value of α is seen . However , a substantial \nincrease is observed in case of BYG with (111) orientation. Qualitatively this could be assigned to  9 \n the presence of  Bi3+ ions which induce s spin-orbit coupling (SOC)  [56–58] and also  due to  electron \nscattering inside the lattice as lattice  mismatch (or  strain ) increases  [59]. We have seen  more \ndistribution of Bi3+ ions along (111) planes ( see inset (ii) of figure  1 (b) ) and also slightly larger \nlattice mismatch  in BYG (111) from our XRD  results. These results also explain higher  value of \nGilbert damping  and ΔH 0 in case of BYG (111).  The change in 4𝜋𝑀𝑒𝑓𝑓 could be attributed to \nuniaxial in -plane magnetic anisotropy . This is because no change in 4πM S is observed from \nmagnetization measurements and 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑎𝑛𝑖 [38,40,60] . The uniaxial inplane magnetic \nanisotropy  is induced due to lattice mismatch between films and GGG substrates  [38,40] . The \ncalculated gyromagnetic ratio (γ)  and ΔH 0 are also included in Table 2 . The magnitude of  ΔH 0 is \nclose to reported values for same  substrate orientation  [38]. In summary we find that YIG with \n(100) orientation yields lowest damping fact or and extrinsic contribution to  linewidth.  These are \nthe r equired optimal parameters for spintronic s application with high spin diffusion length.  \nHowever,  MOKE signal is usual ly very low in bare YIG thin films because of its lower magnetic \nanisotropy and strain [61]. But previous reports suggest that magnetic anisotropy and magnetic \ndomains formation can be achieved in YIG system by doping rare earth materials like Bi and Ce \n[18,61]. We have shown that anisotropic characteristic  with Bi doping in YIG is more pron ounced \nalong <111> direction which can lead to the enhanced MOKE signal in Bi -YIG films on <111> \nsubstrate.  \nWe have also recorded polar angle  () data of resonance field ( Hr) versus magnetic field \n(H) at frequency 12  GHz  for the BYG (100) and BYG (111)  films (figure  5 (c) & 5(d) respectively  \nwhere inset shows the azimuthal angle ( ) variation of Hr measured at frequency of 3 GHz ). The \ndata are fitted with modified Kittel equation . From figure 5 (c) & (d),  we can see that Hr increases \nup to 2.5 kOe in BYG (100) and 3.0 kOe in BYG (111)  by varying the  direction  of H from 0 to 90 10 \n degree with respect to sample surface  (inset of Fig 5 (a)) . Obtained parameters from angular \nvariation of FMR magnetic field H r (θH) are listed in the inset of figure 5 (c) & (d) . From  variation \ndata (by varying the  direction  of H from 0 to 18 0 degree with respect to sample  edge (Fig 5 (a)  \nInset ), we see clear four -fold and two -fold in-plane anisotropy in BYG (100) and BYG (111)  films  \n[61,62] . This further consolidates single -crystalline characteristics of our films. The change \nobserved in Hr with respect to  variation  is 79.52 Oe in BYG (100)  (H=0 to 45)  and 19.25 Oe in \nBYG (111) ( H=0 to 45). Thus,  during  in-plane rotation, higher change in FMR field is observed \nalong  (100) orientation . \n \n4. Conclusion  \nIn conclusion , we have  grown high quality YIG and B i-YIG thin film s on GGG substrates with \n(100)  and (111)  orientation . The films were gr own by pulsed laser deposition. The optimal \nparameters  i.e. target to substrate distance, substrate temperature, and oxygen pressure are \ndetermined to be  ~ 4.8 cm, 825 oC, and 0.15 mbar, respectively. The as  grown thin films have \nsmooth surfaces and  are found to be  phase pure  from AFM and XRD characterizations.  From FMR \nmeasurements , we have found  lower value of damping parameter in (100) YIG that indicates \nhigher spin diffusion length for potential spintronics application.  On the other -hand bismuth \nincorporation to YIG leads to dominance of anisotropic characteristics  that augers well for \napplication in magnetic bubble memory and  magneto -optic  devices . The enhanced value of α in \nBi-YIG films is ascribed to the spin orbit coupled Bi3+ ions. We also ta bulate the values of \nmagnetic parameters such as linewidth ( ∆H0), gyromagnetic ratio ( γ), and effective magnetization  \n4𝜋𝑀𝑒𝑓𝑓 with respect to substrate orientation.  Unambiguous four-fold in -plane anisotropy is \nobserved in (100) oriented films. 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Rev. Appl. 4 (2015) 14008.  \n[62] A. Krysztofik, L.E. Coy, P. Kuświk, K. Zał \\keski, H. Głowiński, J. Dubowik, Ultra -low \ndamping in lift -off structured yttrium iron garnet thin films, Appl. Phys. Lett. 111 (2017) \n192404.  \n \n \n  20 \n List of Tables  with caption  \n \nTable 1: Lattice and magnetic p arameters obtained from XRD , AFM and VSM.  \nS. No.  Sample  Lattice \nconstant \n(Å) Lattice \nMismatch \n(%) Lattice \nvolume  \n(Å3) Roughness  \n \n(nm)  4πM S \n \n(Gauss)  \n1. YIG (100)  12.403  0.42 1907.81  0.801  1670.15±83.51  \n2. YIG (111)  12.405  0.40 1909.02  0.341  1654.06±82.70  \n3. BYG (100)  12.384  0.36 1899.11  0.787  1788.50±89.43  \n4. BYG (111)  12.401  0.65 1906.93  0.308  1816.31±90.82  \n \n \nTable 2: Damping and linewidth p arameters obtained from FMR  \nS. No.  Sample  α\n(10-4) ΔH 0 \n(Oe)  4πM eff \n(Oe)  γ' \n(GHz/kOe)  \n1. YIG (100)  (0.54±0.06)  26.24±0.10  1938 .60±37.57  2.89±0.01  \n2. YIG (111)  (1.05±0.13)  26.51±0.21  2331 .38±65.78  2.86±0.02  \n3. BYG (100)  (1.66±0.10)  26.52 ±0.17  1701.67±31.87  2.89±0.11  \n4. BYG (111)  (6.27±0.33)  29.28 ±0.62  2366 .85±62.60  2.86±0.02  \n \n  21 \n Figure Captions  \nFigure 1: XRD with Rietveld refinement pattern of (a) YIG target ( inset shows  crystallographic \nsub-lattices, Fe 13+ tetrahedral  site, Fe 23+ octahedral site and Y3+ dodecahedral site ) (b) BYG target  \n(inset (i) shows effect of Bi doping into YIG , inset  (ii) shows contribution of the Bi3+ along the \n(100) and (111) planes ). \n \nFigure 2: XRD pattern of  (a) GGG (100) , (b) GGG (111), (c)  YIG (100) , (d) YIG (111) , (e) BYG \n(100) , and (f) BYG (111) . \n \nFigure 3: AFM images of  (a) YIG  (100), (b ) YIG (111) , (c) BYG (100), (d) BYG (111) and  static \nmagnetization graph of (e ) YIG (100), YIG (111);  and (f) BY G (100), BYG (111) . \n \nFigure 4: FMR absorption spectra of (a) YIG (100), (b) YIG (111), (c) BYG (100), and (d) BYG \n(111).  \n \nFigure 5: (a) FMR magnetic field Hr is plotted as a function of frequency f.  Experiment data fitted \nwith Kittel equation for YIG and BYG oriented films. Inset shows how the applied field  angle is \nmeasured from sample surface  (b) Frequency -dependent FMR linewidth data fitted with LLG \nequation for YIG and BYG oriented films.  Inset shows the magnified version to illustrate  the effect \nof Bi doping in YIG . (c) and (d) show angular variation   of FMR magnetic field (Hr (θH)) fitted \nwith modified Kittel  equation at 12  GHz frequency for BYG (100) and  BYG (111)  films .  Insets \nshow the FMR magnetic  field (H r) as a function of azimuthal angle ( ). \n 22 \n  \nFigure 1  \n \n23 \n  \nFigure 2 \n \n \n \n \n \n24 \n  \nFigure 3 \n \n \n \n \n25 \n  \n \n \n \n \nFigure 4 \n \n \n \n \n \n \n26 \n  \n \n \n \nFigure 5 \n \n \n"    },    {        "title": "2109.12071v1.Damping_in_yttrium_iron_garnet_film_with_an_interface.pdf",        "content": "arXiv:2109.12071v1  [cond-mat.mtrl-sci]  24 Sep 2021Damping in yttrium iron garnet film with an interface\nRavinder Kumar,1,2,∗B. Samantaray,1Shubhankar Das,3Kishori Lal,1D. Samal,2,4and Z. Hossain1,5,†\n1Department of Physics, Indian Institute of Technology, Kan pur 208016, India.\n2Institute of Physics, Bhubaneswar 751005, India.\n3Institute of Physics, Johannes Gutenberg-University Main z, 55099 Mainz, Germany.\n4Homi Bhabha National Institute, Anushakti Nagar, Mumbai 40 0085, India.\n5Institute of Low Temperature and Structure Research, 50-42 2 Wroclaw, Poland.\n(Dated: September 27, 2021)\nWe report strong damping enhancement in a 200 nm thick yttriu m iron garnet (YIG) film due\nto spin inhomogeneity at the interface. The growth-induced thin interfacial gadolinium iron garnet\n(GdIG) layer antiferromagnetically (AFM) exchange couple s with the rest of the YIG layer. The\nout-of-plane angular variation of ferromagnetic resonanc e (FMR) linewidth ∆ Hreflects a large in-\nhomogeneous distribution of effective magnetization ∆4 πMeffdue to the presence of an exchange\nspringlike moments arrangement in YIG. We probe the spin inh omogeneity at the YIG-GdIG inter-\nface by performing an in-plane angular variation of resonan ce fieldHr, leading to a unidirectional\nfeature. The large extrinsic ∆4 πMeffcontribution, apart from the inherent intrinsic Gilbert co n-\ntribution, manifests enhanced precessional damping in YIG film.\nI. INTRODUCTION\nThe viability of spintronics demands novel magnetic\nmaterials and YIG is a potential candidate as it ex-\nhibits ultra-low precessional damping, α∼3×10−5[1].\nThe magnetic properties of YIG thin films epitaxially\ngrown on top of Gd 3Ga5O12(GGG) vary significantly\ndue to growth tuning[ 2,3], film thickness[ 4], heavy met-\nals substitution[ 5–7] and coupling with thin metallic\nlayers[8–10]. The growth processes may also induce the\nformation of a thin interfacial-GdIG layer at the YIG-\nGGG interface[ 11–13]. The YIG-GdIG heterostructure\nderived out of monolithic YIG film growth on GGG ex-\nhibits interestingphenomenasuchasall-insulatingequiv-\nalent of a synthetic antiferromagnet[ 12] and hysteresis\nloop inversion governed by positive exchange-bias [ 13].\nThe radio frequency magnetization dynamics on YIG-\nGdIG heterostructure still remains unexplored and need\na detailed FMR study.\nThe relaxation of magnetic excitation towards equi-\nlibrium is governed by intrinsic and extrinsic mecha-\nnisms, leading to a finite ∆ H[14,15]. The former mech-\nanism dictates Gilbert type relaxation, a consequence of\ndirect energy transfer to the lattice governed by both\nspin-orbit coupling and exchange interaction in all mag-\nnetic materials[ 14,15]. Whereas, the latter mechanism\nis a non-Gilbert-type relaxation, divided mainly into two\ncategories[ 14,15]- (i) the magnetic inhomogeneity in-\nduced broadening: inhomogeneity in the internal static\nmagnetic field, and the crystallographic axis orienta-\ntion; (ii) two-magnon scattering: the energy dissipates in\nthe spin subsystem by virtue of magnon scattering with\nnonzero wave vector, k∝negationslash= 0, where, the uniform reso-\nnance mode couples with the degeneratespin waves. The\n∗ravindk@iitk.ac.in\n†zakir@iitk.ac.inangular variation of Hrprovides information about the\npresence of different magnetic anisotropies[ 4,6]. Most\nattention has been paid towards the angular dependence\nofHr[4,6], whereas, the angular variation of the ∆ H\nis sparsely investigated. The studies involving angular\ndependence of ∆ Hmay help to probe different contribu-\ntions to the precessional damping.\nInthispaper, theeffectsofintrinsicandextrinsicrelax-\nation mechanisms on precessionaldamping of YIG film is\nstudied extensively using FMR technique. An enhanced\nvalue of α∼1.2×10−3is realized, which is almost two\norders of magnitude higher than what is usually seen in\nYIGthinfilms, ∼6×10−5[1,2]. Theout-of-planeangular\nvariation of ∆ Hshows an unusual behaviour where spin\ninhomogeneity at the interface plays significant role in\ndefining the ∆ Hbroadening and enhanced α. In-plane\nangular variation showing a unidirectional feature, de-\nmandstheincorporationofanexchangeanisotropytothe\nfree energy density, evidence of the presence of an AFM\nexchangecoupling at the YIG-GdIG interface. The AFM\nexchange coupling leads to a Bloch domain-wall-like spi-\nralmoments arrangementin YIG and givesrise to a large\n∆4πMeff. This extrinsic ∆4 πMeffcontribution due to\nspin inhomogeneity at the interface adds up to the inher-\nent Gilbert contribution, which may lead to a significant\nenhancement in precessional damping.\nII. SAMPLE AND MEASUREMENT SETUPS\nWe deposit a ∼200 nm thick epitaxial YIG film on\nGGG(111)-substrate by employing a KrF Excimer laser\n(Lambda Physik COMPex Pro, λ= 248 nm) of 20 ns\npulse width. A solid state synthesized Y3Fe5O12target\nis ablated using an areal energy of 2.12 J.cm−2with a\nrepetition frequency of 10 Hz. The GGG(111) substrate\nis placed 50 mm away from the target. The film is grown\nat 800oC temperature and in-situpost annealed at the\nsame temperature for 60 minutes in pure oxygen envi-ronment. The θ−2θX-ray diffraction pattern shows epi-\ntaxial growth with trails of Laue oscillations (Fig. 3(a)\nof ref[3]). FMR measurements are performed using a\nBruker EMX EPR spectrometer and a broadband copla-\nnar waveguide (CPW) setup. The former technique uses\na cavity mode frequency f≈9.60 GHz, and enables us\nto perform FMR spectra for various θHandφHangu-\nlar variations. The latter technique enables us to mea-\nsure frequency dependent FMR spectra. We define the\nconfigurations Hparallel ( θH= 90o) and perpendicular\n(θH= 0o) to the film plane for rf frequency and angu-\nlar dependent measurements. The resultant spectra are\nobtained as the derivative of microwave absorption w.r.t.\nthe applied field H.\nIII. RESULTS AND DISCUSSION\nA. Broadband FMR\nFig.1(a) shows typical broadband FMR spectra in\na frequency frange of 1.5 to 13 GHz for 200 nm thick\nYIG film at temperature T= 300 K and θH= 90o.\nThe mode appearing at a lower field value is the main\nmode, whereas the one at higher field value represents\nsurface mode. We discuss all these features in detail in\nthe succeeding subsection IIIB. We determine the res-\nonance field Hrand linewidth (peak-to-peak linewidth)\n∆Hfrom the first derivative of the absorption spectra.\nFig.1(b) shows the rf frequency dependence of Hrat\nθH= 90oand 0o. We use the Kittel equation for fitting\nthe frequency vs. Hrdata from the resonance condi-\ntion expressed as[ 10],f=γ[Hr(Hr+4πMeff)]1/2/(2π)\nforθH= 90oandf=γ(Hr−4πMeff)/(2π) for\nθH= 0o. Where, γ=gµB/ℏis the gyromagnetic ratio,\n4πMeff= 4πMS−Haniis the effective magnetization\nconsisting of 4 πMSsaturation magnetization (calculated\nusing M(H)) and Hanianisotropy field parametrizing cu-\nbic and out-of-plane uniaxial anisotropies. The fitting\ngives 4πMeff≈2000 Oe, which is used to calculate the\nHani≈ −370 Oe.\nFig.1(c) shows the frequency dependence of ∆ Hat\nθH= 90o. The intrinsic and extrinsic damping contri-\nbutions are responsible for a finite width of the FMR\nsignal. The intrinsic damping ∆ Hintarises due to the\nGilbert damping of the precessing moments. Whereas,\nthe extrinsic damping ∆ Hextexists due to different non-\nGilbert-type relaxations such as inhomogeneity due to\nthe distribution of magnetic anisotropy ∆ Hinhom, or\ntwo-magnon scattering (TMS) ∆ HTMS. The intrinsic\nGilbert damping coefficient ( α) can be determined using\nthe Landau-Liftshitz-Gilbert equation expressed as[ 10],\n∆H= ∆Hin+ ∆Hinhom= (4πα/√\n3γ)f+ ∆Hinhom.\nConsidering the above equation where ∆ Hobeys lin-\nearfdependence, the slope determines the value of α,\nand ∆Hinhomcorresponds to the intercept on the ver-\ntical axis. We observe a very weak non-linearity in the\nfdependence of ∆ H, which is believed to be due to thecontribution of TMS to the linewidth ∆ HTMS. The non-\nlinearfdependence of ∆ Hin Fig.1(c) can be described\nin terms of TMS, assuming ∆ H= ∆Hin+ ∆Hinhom+\n∆HTMS. We put a factor of 1 /√\n3 to ∆Hdue to the\npeak-to-peak linewidth value extraction[ 14]. The TMS\ninduces non-linear slope at low frequencies, whereas a\nsaturation is expected at high frequencies. TMS is in-\nduced by scattering centers and surface defects in the\nsample. The defects with size comparable to the wave-\nlength of spin waves are supposed to act as scattering\ncentres. The TMS term at θH= 90ocan be expressed\nas[16]-\n∆HTMS(ω) = Γsin−1/radicalBigg/radicalbig\nω2+(ω0/2)2−ω0/2/radicalbig\nω2+(ω0/2)2+ω0/2,(1)\nwithω= 2πfandω0=γ4πMeff. The prefactor Γ\ndefines the strength of TMS. The extracted values are\nas follows: α= 1.2×10−3, ∆H0= 13 Oe and Γ = 2 .5\nOe. The Gilbert damping for even very thin YIG film\nis extremely low, ∼6×10−5. Whereas, the value we\nachieved is higher than the reported in the literature for\nYIG thin films[ 2]. Also, the value of Γ is insignificant,\nimplying negligible contribution to the damping.\nB. Cavity FMR\nFig.2(a) shows typical T= 300 K cavity-FMR\n(f≈9.6 GHz) spectra for YIG film performed at dif-\nferentθH. The FMR spectra exhibit some universal\nfeatures: (i) Spin-Wave resonance (SWR) spectrum for\nθH= 0o; (ii) rotating the Haway from the θH= 0o,\nthe SWR modes successively start diminishing, and at\ncertain critical angle θc(falls in a range of 30 −35o;\nshaded region in Fig. 2(b)), all the modes vanish except\na single mode (uniform FMR mode). Further rotation\nofHforθH> θc, the SWR modes start re-emerging.\nWe observe that the SWR mode appearing at the higher\nfield side for θH> θc, represents an exchange-dominated\nnon-propagating surface mode[ 17–19]. The above dis-\ncussed complexity in HrvsθHbehaviour has already\nbeen realized in some material systems[ 19], including a\nµ-thick YIG film[ 18]. The localized mode or surface\nspin-wave mode appears for H∝bardblbut not⊥to the film-\nplane[17–19]. WeassigntheSWR modesforthesequence\nn= 1,2,3,...., as it provides the best correspondence\ntoHex∝n2, where, Hex=Hr(n)−Hr(0) defines ex-\nchange field[ 20]. The exchange stiffness can be obtained\nby considering the modified Schreiber and Frait classical\napproach using the mode number n2dependence of res-\nonance field (inset Fig. 3(c))[ 20]. For a fixed frequency,\nthe exchange field Hexof thickness modes is determined\nby subtracting the highest field resonance mode ( n= 1)\nfrom the higher modes ( n∝negationslash= 1). In modified Schreiber\nand Frait equation, the Hexshows direct dependency on\nthe exchange stiffness D:µ0Hex=Dπ2\nd2n2(wheredis\n2/s40/s99/s41/s40/s98/s41/s40/s97/s41\nFIG. 1. Room temperature frequency dependent FMR measureme nts. (a) Representative FMR derivative spectra for differen t\nfrequencies at θH= 90o. (b) Resonance field vs. frequency data for θH= 90oandθH= 0oare represented using red and\nblue data points, respectively. The fitting to both the data a re shown using black lines. (c) Linewidth vs. frequency data at\nθH= 90o. The solid red circles represent experimental data, wherea s the solid black line represents ∆ Hfitting. Inhomogeneous\n(∆Hinhom), Gilbert (∆ Hα) and two-magnon scattering (∆ HTMS) contributions to ∆ Hare shown using dashed green, solid\nyellow and blue lines, respectively.\nthe film thickness). The linear fit of data shown in the\ninset of Fig. 2(b) gives D= 3.15×10−17T.m2. The ex-\nchange stiffness constant Acan be determined using the\nrelationA=D MS/2. The calculated value is A= 2.05\npJ.m−1, which is comparable to the value calculated for\nYIG,A= 3.7 pJ.m−1[20].\nYIG thin films with in-plane easy magnetization ex-\nhibit extrinsic uniaxial magnetic and intrinsic magne-\ntocrystallinecubic anisotropies[ 21]. The total free energy\ndensity for YIG(111) is given by[ 21,22]:\nF=−HMS/bracketleftbigg\nsinθHsinθMcos(φH−φM)\n+cosθHcosθM/bracketrightbigg\n+2πM2\nScos2θM−Kucos2θM\n+K1\n12/parenleftbigg7sin4θM−8sin2θM+4−\n4√\n2sin3θMcosθMcos3φM/parenrightbigg\n+K2\n108\n−24sin6θM+45sin4θM−24sin2θM+4\n−2√\n2sin3θMcosθM/parenleftbig\n5sin2θM−2/parenrightbig\ncos3φM\n+sin6θMcos6φM\n\n(2)\nThe Eq. 2consists of the following different energy\nterms; the first term is the Zeeman energy, the second\nterm is the demagnetization energy, the third term is\nthe out-of-plane uniaxial magnetocrystalline anisotropy\nenergyKu, and the last two terms are the first and sec-\nond order cubic magnetocrystalline anisotropy energies\n(K1andK2), respectively. The total free energy density\nequation is minimized by taking partial derivatives w.r.t.\ntoθMandφMtoobtaintheequilibriumorientationofthe\nmagnetization vector M(H), i.e.,∂F/∂θ M=∂F/∂φ M=\n0. Theresonancefrequencyofuniformprecessionatequi-\nlibrium condition is expressed as[ 21,23,24]:\nωres=γ\nMSsinθM/bracketleftBigg\n∂2F\n∂θ2\nM∂2F\n∂φ2\nM−/parenleftbigg∂2F\n∂θM∂φM/parenrightbigg2/bracketrightBigg1/2\n(3)\nMathematica is used to numerically solve the reso-nance condition described by Eq. 3for the energy den-\nsity given by Eq. 2. The solution for a fixed frequency\nis used to fit the angle dependent resonance data ( Hr\nvs.θH) shown in Fig. 2(b). The main mode data\nsimulation is shown using a black line. The parame-\nters obtained from the simulation are Ku=−1.45×104\nerg.cm−3,K1= 1.50×103erg.cm−3, andK2= 0.13×103\nerg.cm−3. The calculated uniaxial anisotropy field value\nisHu∼ −223 Oe.\nThe ∆Hmanifests the spin dynamics and related re-\nlaxation mechanisms in a magnetic system. The intrinsic\ncontribution to ∆ Harises due to Gilbert term ∆ Hint≈\n∆Hα, whereas, the extrinsic contribution ∆ Hextconsists\nof line broadening due to ∆ Hinhomand ∆HTMS. The\nterms representing the precessional damping due to in-\ntrinsic and extrinsic contributions can be expressed in\ndifferent phenomenologicalforms. Figure 2(c) shows∆ H\nas a function of θH. TheθHvariation of ∆ Hshows\ndistinct signatures due to different origins of magnetic\ndamping. We consider both ∆ Hintand ∆Hextmag-\nnetic damping contributions to the broadening of ∆ H,\n∆H= ∆Hα+∆Hinhom+∆HTMS. The first term can\nbe expressed as[ 14]-\n∆Hα=α\nMS/bracketleftbigg∂2F\n∂θ2\nM+1\nsin2θM∂2F\n∂φ2\nM/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(2πf\nγ)\n∂Hr/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n.(4)\nThe second term ∆ Hinhomhas a form[ 14]-\n∆Hinhom=/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr\nd4πMeff/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆4πMeff+/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr\ndθH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆θH.(5)\nWhere, the dispersion of magnitude and direction of\nthe 4πMeffare represented by ∆4 πMeffand ∆θH, re-\nspectively. The ∆ Hinhomcontribution arises due to a\nsmall spread of the sample parameters such as thickness,\ninternal fields, or orientation of crystallites within the\nthin film. The third term ∆ HTMScan be written as[ 25]-\n3/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s49/s50/s51/s32/s72\n/s101/s120/s40/s107/s79/s101/s41\n/s110/s50/s67/s114/s105/s116/s105/s99/s97/s108/s32/s97/s110/s103/s108/s101/s54/s53/s52/s51/s72\n/s114/s32/s40/s32/s107/s79/s101/s32/s41/s50/s110/s32/s61/s32/s49\n/s72/s32/s40/s32/s68/s101/s103/s114/s101/s101/s32/s41/s50 /s51 /s52 /s53 /s54/s68/s101/s114/s105/s118/s97/s116/s105/s118/s101/s32/s97/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s72 /s32/s40/s107/s79/s101/s41/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s32/s32\n/s32/s69/s120/s112/s116/s46/s32/s68/s97/s116/s97\n/s32 /s72\n/s32 /s72\n/s32 /s77\n/s101/s102/s102\n/s32\n/s32 /s72\n/s84/s77/s83/s72 /s32/s40/s79/s101/s41/s32\n/s40/s100/s101/s103/s114/s101/s101 /s41/s40/s99/s41/s40/s98/s41/s40/s97/s41\nFIG. 2. Room temperature out-of-plane angular θHdependence of FMR. (a) Derivative FMR spectra shown for diffe rentθH\nperformed at ≈9.6 GHz. (b) θHvariation of uniform mode and SWR modes of resonance field Hr. Inset: Exchange field\n(Hex) vs mode number square ( n2). (c)θHvariation of the linewidth (∆ H), where, the experimental and simulated data are\nrepresented by solid yellow circles and black line, respect ively. The different contributions ∆ Hα, ∆4πMeff, ∆θHand ∆HTMS\nare represented by gray, purple, green and red lines, respec tively.\n∆HTMS=/summationtext\ni=1Γout\nifi(φH)\nµ0γΦsin−1/radicalbigg√\nω2+(ω0/2)2−ω0/2√\nω2+(ω0/2)2+ω0/2,\nΓout\ni= Γ0\niΦA(θ−π/4)dHr(θH)\ndω(θH)/slashbigg\ndHr(θH=0)\ndω(θH=0)\n(6)\nThe prefactor Γout\nidefines the TMS strength and has\naθHdependency in this case. The type and size of the\ndefects responsible for TMS is difficult to characterize\nwhich makes it non-trivial to express the exact form of\nΓout\ni. Although, it mayhaveasimplified expressiongiven\nin Eq.6, where, Γ0\niis a constant; A(θ−π/4), a step\nfunction which makes sure that the TMS is deactivated\nforθH< π/4; anddHr(θH)/dω(θH), a normalization\nfactor responsible for the θHdependence of the Γout\ni.\nIn fig.2(c)the solid dark yellow circles and black solid\nline represent the experimental and simulated ∆ HvsθH\ndata, respectively. We also plot contributions of different\nterms such as ∆ Hα(blue color line), ∆4 πMeff(purple\ncolorline), ∆ θH(greencolorline) and ∆ HTMS(red color\nline). The fitting provides following extracted parame-\nters,α= 1.3×10−3, ∆4πMeff= 58 Oe, ∆ θH= 0.29o\nand Γ0\ni= 1.3 Oe. The precessional damping calculated\nfromthe∆ Hvs.θHcorroboratewiththevalueextracted\nfrom the frequency dependence of ∆ Hdata (shown in\nFig.1(c));α= 1.2×10−3. The ∆ Hbroadening and\nthe overwhelmingly enhanced precessional damping are\nthedirectconsequenceofcontributionsfromintrinsicand\nextrinsic damping. Usually, the Gilbert term and the\ninhomogeneity due to sample quality contribute to the\nbroadening of ∆ Hand enhanced αin YIG thin films. If\nwe interpret the ∆ HvsθHdata, it is clear that damping\nenhancement in YIG is arising from the extrinsic mag-\nnetic inhomogeneity.The role of an interface in YIG coupled with metals\nor insulators leading to the increments in ∆ Handαhas\nbeen vastly explored. Wang et. al. [ 9] studied a variety\nof insulating spacers between YIG and Pt to probe the\neffect on spin pumping efficiency. Their results suggest\nthe generation of magnetic excitations in the adjacent\ninsulating layers due to the precessing magnetization in\nYIG at resonance. This happens either due to fluctu-\nating correlated moments or antiferromagnetic ordering,\nvia interfacial exchange coupling, leading to ∆ Hbroad-\nening and enhanced precessional damping of the YIG[ 9].\nTheimpurityrelaxationmechanismisalsoresponsiblefor\n∆HbroadeningandenhancedmagneticdampinginYIG,\nbut is prominent only at low temperatures[ 16]. Strong\nenhancement in magnetic damping of YIG capped with\nPt has been observed by Sun et. al. [ 8]. They suggest\nferromagneticorderingin an atomically thin Pt layerdue\nto proximity with YIG at the YIG-Pt interface, dynam-\nically exchange couples to the spins in YIG[ 8]. In recent\nyears, some research groups have reported the presence\nof a thin interfacial layer at the YIG-GGG interface[ 11–\n13]. The 200 nm film we used in this study is of high\nquality with a trails of sharp Laue oscillations [see Fig\n3(a) in ref.[ 3]]. Thus it is quite clear that the observed\n∆Hbroadening and enhanced αis not a consequence of\nsample inhomogeneity. The formation of an interfacial\nGdIG layer at the YIG-GGG interface, which exchange\ncouples with the YIG film may lead to ∆ Hbroadening\nand increased α. Considering the above experimental ev-\nidences leading to ∆ Hbroadening and enhanced Gilbert\ndampingdueto couplingwithmetals andinsulators[ 8,9],\nit is safe to assume that the interfacial GdIG layer at the\ninterface AFM exchange couples with the YIG[ 11–13],\nand responsible for enhanced ∆ Handα.\nFig.3shows in-plane φHangular variation of Hr. We\n4/s32/s68/s97/s116/s97\n/s32/s84/s111/s116/s97/s108\n/s32/s69/s120/s99/s104/s97/s110/s103/s101/s72\n/s114/s32/s40/s79/s101/s41/s50/s48/s48/s32/s110/s109/s40/s97/s41\n/s40/s98/s41\n/s49/s48/s48/s32/s110/s109\n/s72/s32/s40/s68/s101/s103/s114/s101/s101/s41\nFIG. 3. (a) In-plane angular φHvariation of Hr. The exper-\nimental data are represented by solid grey circles. Whereas ,\nthe simulated data for total and exchange (unidirectional)\nanisotropy are represented by black and red solid lines, re-\nspectively. (a) 200 nm thick YIG sample. (b) 100 nm thick\nYIG sample.\nsimulate the in-plane HrvsφHangular variation using\nthe free energy densities provided in ref. [ 26] and an\nadditional term, −KEA.sinθM.cosφM, representing the\nexchange anisotropy ( KEA). Even though φHvaria-\ntion ofHrshown in Fig. 3(a) is not so appreciable\nas the film is 200 nm thick, a very weak unidirectional\nanisotropy trend is visible, suggesting an AFM exchange\ncoupling between the interface and YIG. It has been\nshown that the large inhomogeneous 4 πMeffis a direct\nconsequence of the AFM exchange coupling at the inter-\nface of LSMO and a growth induced interfacial layer[ 27].\nThe YIG thin film system due to the presence of a hard\nferrimagnetic GdIG interfacial layer possesses AFM ex-\nchange coupling[ 11–13]. A Bloch domain-wall-like spiral\nmoments arrangement takes place due to the AFM ex-\nchange coupling acrossthe interfacial GdIG and top bulk\nYIG layer[ 11–13]. An exchange springlike characteris-\ntic is found in YIG film due to the spiral arrangement\nof the magnetic moments [ 11–13]. The FMR measure-\nment and the extracted value of ∆4 πMeffreflect inho-\nmogeneous distribution of 4 πMeffin YIG-GdIG bilayer\nsystem. The argument of Bloch domain-wall-like spiral\narrangement of moments is conceivable, as this arrange-\nment between the adjacent layers lowers the exchange\ninteraction energy[ 27]. To further substantiate the pres-ence of an interfacial AFM exchange coupling leading\nto spin inhomogeneity at YIG-GdIG interface, we per-\nformed in-plane φHvariation of Hron a relatively thin\nYIG film ( ∼100 nm with growth conditions leading to\nthe formation of a GdIG interfacial layer[ 13]). Fig.3(b)\nshows prominent feature of unidirectional anisotropydue\nto AFM exchange coupling in 100 nm thick film. It is\nevident that the interfacial layer exchange couples with\nthe rest of the YIG film and leads to a unidirectional\nanisotropy. We observethatthe interfacialexchangecou-\npling may cause ∆ Hbroadening and enhanced αdue to\nspin inhomogeneity at the YIG-GdIG interface, even in\na 200 nm thick YIG film.\nIV. CONCLUSIONS\nThe effects of spin inhomogeneity at the YIG and\ngrowth-induced GdIG interface on the magnetization dy-\nnamics of a 200 nm thick YIG film is studied extensively\nusing ferromagnetic resonance technique. The Gilbert\ndamping is almost two orders of magnitude larger\n(∼1.2×10−3) than usually reported in YIG thin films.\nThe out-of-plane angular dependence of ∆ Hshows\nan unusual behaviour which can only be justified after\nconsidering extrinsic mechanism in combination with the\nGilbert contribution. The extracted parameters from\nthe ∆HvsθHsimulation are, (i) α= 1.3×10−3from\nGilbert term; (ii) ∆4 πMeff= 58 Oe and ∆ θH= 0.29o\nfrom the inhomogeneity in effective magnetization and\nanisotropy axes, respectively; (iii) Γ0\ni= 1.3 Oe from\nTMS. The TMS strength Γ is not so appreciable,\nindicating high quality thin film with insignificant defect\nsites. The AFM exchange coupling between YIG and\nthe interfacial GdIG layer causes exchange springlike\nbehaviour of the magnetic moments in YIG, leading to a\nlarge ∆4 πMeff. The presence of large ∆4 πMeffimpels\nthe quick dragging of the precessional motion towards\nequilibrium. A unidirectional behaviour is observed in\nthe in-plane angular variation of resonance field due to\nthe presence of an exchange anisotropy. 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B 101, 024408 (2020) .\n6"    },    {        "title": "2109.12605v1.Transition_state_dynamics_of_a_driven_magnetic_free_layer.pdf",        "content": "Transition state dynamics of a driven magnetic free layer\nJohannes M ¨ogerlea, Robin Schuldta, Johannes Rei \u000ba, J¨org Maina, Rigoberto Hernandezb,c,\u0003\naInstitut f¨ ur Theoretische Physik I, Universit¨ at Stuttgart, 70550 Stuttgart, Germany\nbDepartment of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA\ncDepartments of Chemical &Biomolecular Engineering, and Materials Science and Engineering, Johns Hopkins University, Baltimore, Maryland\n21218, USA\nAbstract\nMagnetization switching in ferromagnetic structures is an important process for technical applications such as data\nstorage in spintronics, and therefore the determination of the corresponding switching rates becomes essential. We\ninvestigate a free-layer system in an oscillating external magnetic field resulting in an additional torque on the spin.\nThe magnetization dynamics including inertial damping can be described by the phenomenological Gilbert equation.\nThe magnetization switching between the two stable orientations on the sphere then requires the crossing of a potential\nregion characterized by a moving rank-1 saddle. We adopt and apply recent extensions of transition state theory for\ndriven systems to compute both the time-dependent and average switching rates of the activated spin system in the\nsaddle region.\nKeywords: magnetization switching, ferromagnetic free-layer system, Landau–Lifshitz–Gilbert equation, transition\nstate theory, normally hyperbolic invariant manifold, stability analysis\n1. Introduction\nIn recent years, the promise of spintronics to emerging technological applications has attracted growing interest\nleading to extensive research e \u000borts in experimental [ 1–8] and theoretical physics [ 1,2,9–13]. The relative simplicity\nand accuracy of the single-domain models for ferromagnetic structures has proven to be a popular choice for charac-\nterizing such spintronics applications. Specifically, these models describe the macro spin-dynamics underlying the\nGilbert equation [ 14–16]. The landscape of the corresponding potential includes two minima at the stable spin up\nandspin down positions which are separated by a rank-1 saddle in certain configurations [ 17,18]. The typical goal\nin spintronics applications is to achieve and control the magnetization switching within a target timescale—viz., a\nspecified rate. This can be achieved, for example, through application of a spin torque [ 19]. An alternative approach\nis microwave-assisted magnetic recording—more specifically, microwave-assisted switching [ 20–23]—where a mi-\ncrowave field perpendicular to the easy axis is used in conjunction with a static external field along the easy axis in\norder to facilitate the magnetization switching. Multiple variations of this scheme have been proposed [24–27], some\nof which rely solely on rotating AC fields perpendicular to the easy axis [ 28,29]. In this paper, we focus on a single\nAC field along the easy axis without any static external fields.\nIn chemical reactions, the transition from reactants to products is typically marked by a barrier region with a\nrank-1 saddle that has exactly one unstable direction called the reaction coordinate, while the remaining orthogonal\nmodes are locally stable and are associated with other bound internal motions. The dynamical crossing of a rank-1\nsaddle in such chemical systems can be described by transition state theory ( TST) [30–36], which then allows for\nthe calculation of rate constants and the flux. However, TST is not restricted to chemical reactions as it has been\napplied in many other fields, including, e. g., atomic physics [ 37], solid state physics [ 38], cluster formation [ 39,40],\ndi\u000busion dynamics [ 41,42], and cosmology [ 43–45]. Notably, the theory has also been extended to time-dependent\n\u0003Corresponding author\nEmail address: r.hernandez@jhu.edu (Rigoberto Hernandez)\nPreprint submitted to Communications in Nonlinear Science and Numerical Simulation September 28, 2021arXiv:2109.12605v1  [cond-mat.mtrl-sci]  26 Sep 2021driven systems [ 46]. Although originally framed using perturbation theory [ 47–50], the requisite locally recrossing-free\ndividing surface ( DS) and instantaneous decay rates in TST can now be obtained with more generally-applicable\nmethods [51–55] as employed here.\nThus, the central result of this paper is the demonstration of the applicability of time-dependent TST to characterize\nthe dynamical crossing of a macrospin across a time-dependent rank-1 saddle using the recent advances cited above.\nIn the language of TST, the spin up andspin down regions can be interpreted as reactants andproducts , and the\nmagnetization switching corresponds to the “chemical” reaction. An important di \u000berence between the previous systems\nto which TST has so far been applied, and the ferromagnetic systems described by the Gilbert equation lies in the\ngeometry of the phase-space structure. Typically, a Hamiltonian system with ddegrees of freedom is described by\na (2d)-dimensional phase space with dcoordinates and dassociated velocities or momenta. The Gilbert equation,\nhowever, is a first-order di \u000berential equation for the dynamics of the magnetic moment on a sphere, i. e., there are\nno independent velocities or momenta. Therefore, the dynamics is e \u000bectively that of a one degree of freedom ( DoF)\nsystem [ 14,56–58]. Nevertheless, within this domain a DScan be associated with the neighborhood of the rank-1\nsaddle. In analogy to chemical reactions, we conjecture that the reactive flux across this DSis associated with the\ndecay rate of the spin flip. In this context, the reactive flux is that of all the trajectories that are reactants (viz., spin up )\nin the infinite past and products (viz., spin down ) in the infinite future. In transition state theory, the reactive flux is\napproximated by the sum of the positive velocities (headed in the direction of the product) over the surface, and it is\nexact if no trajectory recrosses the DS.\nWe show that recent extensions of TST for systems with time-dependent moving saddles [ 51–54] can indeed be\napplied to a ferromagnetic single-domain system with a two-dimensional phase space describing the orientation of the\nmagnetic moment on the sphere and the dynamics following the Gilbert equation. The system can even be driven by a\ntime-dependent external magnetic field. The free-layer system and the applied methods are introduced in Sec. 2. The\napplicability of TST relies on the fact that for any time t, the two-dimensional phase space exhibits a stable and unstable\nmanifold, which intersect in a point on the normally hyperbolic invariant manifold ( NHIM ). A locally recrossing-free\nDSseparating the spin down andspin up regions in phase space can be attached to this point. The time-dependent\nmoving points of the NHIM form the transition state ( TS) trajectory, which is a periodic orbit when the free-layer\nsystem is driven by an oscillating magnetic field.\nThe TStrajectories are the starting point for the calculation of rate constants, and the characterization of the\nmagnetization switching. Through application of the ensemble method and the local manifold analysis ( LMA )\ndeveloped in Ref. [ 54], we obtain the time-dependent instantaneous rates along TStrajectories at various amplitudes\nand frequencies of the driving external magnetic field. We also find in Sec. 3 that the time-averaged rates along the TS\ntrajectories depend significantly on the external driving.\n2. Theory and methods\nHere, we briefly discuss our model (cf. Fig. 1) motivated by a free-layer system [ 17] and present the equations,\nwhich describe the spin dynamics of this model including external driving. Then we introduce the basic ideas of TST\nand the methods, which will be applied for the computation of the instantaneous and average rates of the magnetization\nswitching.\n2.1. Spin dynamics in a driven free-layer model\nThe model addressed here is based on a magnetic single-domain layer with variable magnetization M, known as a\nfree layer . This layer is modeled in analogy to Stoner and Wohlfarth [ 59,60] including a demagnetization field for a\nthin film (shape anisotropy) [ 60,61]. A periodic external magnetic field is added to drive the magnetization. This field\nis intended as a generic placeholder for some externally applied torque—e. g., certain types of spin torque [ 62] such as\nthe one stated below—and must not necessarily be realized by a magnetic coil or antenna.\nFor a classical description of the spin system we start from the Gilbert equation [14, 57]\n˙M=\u0000\rM\u0002H+\u000b\nMSM\u0002˙M (1)\nto describe the motion of a magnetic moment M=\u0000\rS, with Sthe spin,\rthe gyromagnetic ratio, and MS=jMj\nthe saturation magnetization [cf. Fig. 1(b)]. The magnetic moment Mis damped by a strength proportional to the\n2HextM(t)\nfree layer(a)\nˆex\nˆeyˆezM(t)\nHdHan\nHextHeff(b)Figure 1: (a) Schematic of a magnetic single-domain layer with variable magnetization M(free layer) in an external magnetic field Hext. (b) Magnetic\nfield components governing the evolution of the free layer’s magnetization M. The e \u000bective field Hacting upon Mconsists of the demagnetization\nfield Hd=\u0000MSmxˆex, the magnetocrystalline anisotropy field Han=HKmzˆez, and the external driving Hext=Hext\nzsin(!t)ˆez. The material’s easy\naxis is aligned with the z-axis.\ncoe\u000ecient\u000band can be driven by the e \u000bective magnetic field H. Because the velocity ˙Mis orthogonal to M, the length\nof the magnetic moment is conserved and therefore we can write M=MSmwithjmj=1 and mbeing dimensionless.\nThe implicit di \u000berential equation (1)can be brought to an explicit form. Substituting ˙Mon the right-hand side\nof Eq. (1)with the equation itself and using the relation M\u0002(M\u0002˙M)=(M\u0001˙M)M\u0000M2˙M=\u0000M2\nS˙Mas well as\nM=MSmwe obtain the Landau–Lifshitz–Gilbert (LLG) equation [14, 16]\n˙m=\u0000\r\n1+\u000b2m\u0002[H+\u000b(m\u0002H)]. (2)\nHere, we investigate the motion of a magnetic moment in a free-layer model described by the potential [61]\nU=M2\nS\n2m2\nx\u0000MSHK\n2m2\nz\u0000MSHext\nzsin(!t)mz, (3)\nwhere HKis the anisotropy constant of the free layer. A magnetization switching induced by an additional torque\nmodifying the dynamics of Eq. (2), can in principle be achieved by various ways [ 20–29]. For the description of spin\ntorque in a pinned-layer system, Slonczewski introduced an additional term to the standard Gilbert equation, depending\non the polarization of the pinned layer [ 19]. In this model, the spin torque is proportional to the applied electron\ncurrent Iflowing trough the pinned layer and, thus, can in principle become oscillating if an AC-source is used [ 63,64].\nWhile this specific type of spin torque cannot be represented purely by an additional magnetic field term, others—e. g.,\nManchon and Zhang [ 62]—have suggested spin torques that can. Due to the fact that the influence of some spin torques\ncan be reformulated as an additional e \u000bective field acting on the spin dynamics [ 62,65], we directly add our applied\nfield expression into the e \u000bective field, leading to significant simplifications [ 16]. The last term in Eq. (3)describes\nsuch an oscillating external magnetic field in zdirection with amplitude Hext\nzand frequency !. The e \u000bective magnetic\nfield then reads\nH=\u00001\nMSrmU=0BBBBBBBB@\u0000MSmx\n0\nHKmz+Hext\nzsin(!t)1CCCCCCCCA. (4)\nFor the free-layer system with parameters based on Refs. [ 22,27,29] the saturation magnetization and the\ngyromagnetic ratio read\nMS=1\u0002106A m\u00001and\r=2:217\u0002105m A\u00001s\u00001, (5)\nrespectively. Using these values as units, we can set MS=1and\r=1for computations with dimensionless parameters.\nIn the following, we choose\nMS=1 ,\r=1 ,\u000b=0:01 , HK=0:5 , (6)\n3m\nx−1\n0\n1my−101mz\n−101(a)\n−π 0 +π\nϕ0π/2πθ(b)\n−0.20.00.20.4\nUFigure 2: The free-layer potential (3)(a) on the sphere and (b) in the ( ';\u0012) plane. The saddle points at \u0012=\u0019=2 and'=\u0006\u0019=2 mark the regions of the\nTS, which must be crossed for the magnetization switching. A typical trajectory with higher friction \u000b=0:1, propagated without external driving\nfrom a spin-down state to a spin-up state, is shown in cyan (or light gray in print) in both panels. Vertical markers highlight the part of the trajectory\nshown in (b).\nand an external magnetic field with amplitude and frequency\nHext\nz=0:15 and!=\u0019=8 (7)\nas reference parameters, if not stated otherwise. This corresponds to HK=5\u0002105A m\u00001,Hext\nz=1:5\u0002105A m\u00001,\nand!=2\u0019=13:86 GHz in the problem defined in Ref. [ 66] with the standard material parameters of permalloy. This\napplied field and frequency are well in the range of typical experimental conditions.\nTo take advantage of the symmetry of the system one can transform the LLG equation (2)in spherical angular\ncoordinates \u0012and', i. e.,\n˙\u0012=\r\n1+\u000b2\u0010\nH'+\u000bH\u0012\u0011\n, ˙'=\r\n1+\u000b21\nsin\u0012\u0010\n\u0000H\u0012+\u000bH'\u0011\n, (8)\nfor\u0012<f0;\u0019gand with the projections of the e \u000bective field to the basis vectors, H\u0012=H\u0001ˆe\u0012andH'=H\u0001ˆe'. The\npotential (3)on the sphere for the spin system without external driving is illustrated in Fig. 2(a). The reduced equations\nof motion (8)for the two spherical angular variables describe a two-dimensional problem with the potential U(\u0012;')\npresented in Fig. 2(b). Two energy minima are located at the poles with \u0012=0 and\u0012=\u0019, which make the potential (3)\na perfect candidate to observe magnetization switching. At the equator \u0012=\u0019=2 there are two local energy maxima for\n'=0 and'=\u0019and two rank-1 saddles for '=\u0000\u0019=2 and'= +\u0019=2.\nA typical trajectory without external driving illustrating the magnetization switching is drawn in both panels (a)\nand (b) of Fig. 2. It starts on the spin down side of the sphere ( \u0012 > \u0019= 2), crosses the saddle region of the potential\nnear\u0012=\u0019=2,'=\u0000\u0019=2 and approaches the spin up position (\u0012\u00190) on a spiral caused by the damping term in the\nGilbert equation (1). We are interested in spin-flip processes crossing the regions close to one of the rank-1 saddles,\nand investigate in the following, without loss of generality, spin flips crossing the rank-1 saddle near '= +\u0019=2.\n2.2. Transition state theory\nThe free-layer system, described by the potential (3), features a rank-1 saddle point at \u0012='=\u0019=2, as shown in\nFig. 2(b). This saddle can act as a bottleneck of the spin dynamics, which makes it a candidate for the application of\nTST models [ 30,31,33,36]. In typical scenarios for a chemical reaction, a one-dimensional reaction path—e. g., the\nminimum energy path [ 67]—characterizes the progress of the reaction. A rank-1 saddle point separates reactants from\nproducts along this unstable mode, and can be used to naively characterize the flux and associated reaction rate. In this\ncontext, it acts as a TS. In higher dimensions, the other degrees of freedom are stable and are referred to as orthogonal\nmodes. More generally, the TSmarks the transition between reactants and products through the location of a DS. Here,\nwe apply TST to a magnetization switching in the free-layer system—e. g., from the “reactant” state spin up to the\n“product” state spin down —caused by a time-dependent driving of the system via an external magnetic field. To achieve\nthis aim, we resort to recent extensions of TST to time-dependent driven systems [52–55].\n4π/4π/2 3π/4\nϕπ/4π/23π/4θ(a)\nNHIMsaddleI\nIIIII\nIVWuWs\nWs\nWuDS\n−1 0 1\n[ϕ−ϕ‡(t)]/10−4−202[θ−θ‡(t)]/10−4\n(b)\nDS\nWuWs∆ϕu ∆ϕs\n∆θu∆θs\nt0t0+ ∆tFigure 3: (a) Phase-space structure of the driven free-layer system introduced in Sec. 2.1. The stable and unstable manifolds, WsandWuseparate\nfour di \u000berent regions marked ( I)–(IV) (see text). The intersection of the manifold’s closures forms the NHIM . The DSattached to this point\nseparates the spin up andspin down regions in phase space. The external driving causes the NHIM to detach from the saddle point. (b) Schematic of\nthe geometric structure that underlies the rate constant expressions summarized in Sec. 2.2.2. Initially, an equidistant spin ensemble connecting Ws\non the reactant side with the DSparallel toWuis generated. Upon time propagation, parts of the ensemble undergo spin flips as they move through\nthe DS. The resulting ensemble is still equidistant, parallel to Wu, and connected to Ws.\n2.2.1. Phase-space structure and TS trajectory\nIn the free-layer system introduced above, the magnetization switching is related to a change in the \u0012coordinate—\ne. g., from\u0012&0 to\u0012.\u0019in an uptodown spin state. In applying TST to resolve the activated dynamics of a spin, it\nthus appears natural to take the angle \u0012as the reaction coordinate and 'as an orthogonal mode. However, an important\ndi\u000berence between the spin system described by the equations of motion in (8), and systems typically addressed by\nTST requires some considerations, discussed below, to make the analogy complete.\nIn a chemical or mechanical system with ddegrees of freedom the dynamics is typically described by dsecond-order\ndi\u000berential equations for the coordinates or, in the Hamilton formalism, by 2 dfirst-order di \u000berential equations for the\ncoordinates and canonical momenta in the 2 d-dimensional phase space. In the spin system, the LLG equation results\nin the first-order di\u000berential equations (8)for the two coordinates \u0012and', i. e., there are no canonical momenta p\u0012\nandp', which belong to these coordinates. Nevertheless, TST can be applied to this system. The crucial point is that\nthe two-dimensional phase space of the spin system consisting of the two coordinates \u0012and'is treated in formal\nmathematical analogy to the two-dimensional phase space of a one DoF Hamiltonian system with a reaction coordinate\nand the corresponding canonical momentum.\nThe phase-space structure of the driven spin system in the vicinity of the rank-1 saddle at a given time tis illustrated\nin Fig. 3(a). Note that the reaction coordinate \u0012is the ordinate and 'the abscissa, which di \u000bers from corresponding\npresentations in Refs. [ 52–55,68], where the reaction coordinate is chosen as the abscissa and the corresponding\nvelocity along the ordinate. The stable and unstable manifolds WsandWuseparate four di \u000berent regions, where\n(I) the spin stays down , (II) the spin stays up, (III) the spin switches from uptodown , and ( IV) the spin switches\nfrom down toup, when the system is propagated backwards and forwards in time. One subtlety regarding the time\npropagation of the spins should be noted: Due to the damping of the magnetic field by the term proportional to \u000bin\nEq.(1)the spin without external driving always moves towards a potential minimum, i. e., the spin up orspin down\nposition when propagated forwards in time. However, it moves towards one of the potential maxima located at \u0012=\u0019=2,\n'=0 or\u0012=\u0019=2,'=\u0019(see Fig. 2) when propagated backwards. Therefore, appropriate cuto \u000bs for the propagation of\ntrajectories must be introduced to obtain the correct classification to one of the regions ( I)–(IV) in Fig. 3(a). Failing to\ndo so can lead to visible artifacts, or it can cause the classification algorithm to not terminate. Similar problems in\ndissipative chemical systems have been discussed in Ref. [ 69]. In our case, we have found 0:1\u0019<'< 0:9\u0019to yield\nreliable results.\nThe intersection of the stable and unstable manifold is a point ( 'z;\u0012z) on the NHIM . Such points do not leave\nthe saddle region when propagated forwards or backwards in time. Therefore, these points describe spins that reside\npermanently in an unstable intermediate state roughly in xdirection that is neither spin up norspin down . Note that\nfor driven systems the points of the NHIM in general do not coincide with the time-dependent position of the saddle\n5marked by the black point in Fig. 3(a). The line with constant angle \u0012=\u0012zrepresents a recrossing-free DS, which\nseparates the “reactants” and “products” in TST, i. e., a spin with \u0012<\u0012zisspin up and a spin with \u0012>\u0012zisspin down .\nIn case of periodic driving of the spin system by a time-dependent external magnetic field, the points on the NHIM\nfollow a periodic orbit with the same period as the external driving. This orbit is called the TStrajectory, and is of\nfundamental importance for the computation of rate constants.\nFor the numerical construction of the NHIM , we resort to the binary contraction method ( BCM ) introduced in\nRef. [ 68]. For a given time t, the algorithm in the BCM is initialized by defining a quadrangle with each of its corners\nlying exclusively within one of the four regions in the ( ';\u0012) plane shown in Fig. 3(a). In each iterative step, we first\ndetermine an edge’s midpoint. Then, the adjacent corner corresponding to the same region as that midpoint is moved\nto the midpoint’s position. By repeating this interleaved bisection procedure in turn for all edges, the quadrangle\nsuccessively contracts and converges towards the intersection of the stable and unstable manifolds, i. e., a point on the\nNHIM. This method is numerically very e \u000bective and e \u000ecient for systems such as the one addressed here.\n2.2.2. Decay rates\nThree di \u000berent methods for calculating decay rates in driven systems have recently been introduced and applied in\nthe literature [ 54,55]. Here, we adopt these methods with appropriate modifications for the free-layer system. The\nresulting decay rates are a measure of the instability of specific trajectories near the saddle. They di \u000ber significantly\nfrom the Kramers rate [ 70] used in the theory of chemical reactions but nevertheless provide insight about the rate\nprocess.\nEnsemble method. The conceptually simplest method for calculating decay rates keis by means of propagation of\nan ensemble. In analogy to Ref. [ 54] we identify a line segment parallel to the unstable manifold that satisfies the\nproperty: it lies on the reactant side between the stable manifold and the DSat a distance that is small enough to allow\nfor linear response and large enough to suppress numerical instability. At t=t0, a spin ensemble is placed on this line\nas illustrated by blue dots in Fig. 3(b) and propagated in time to yield a time-dependent spin up population N\"(t). The\nensemble at time t=t0+ \u0001tis marked in Fig. 3(b) by red and orange dots. Spins, which have crossed the DS(red dots)\narespin down and thus cause a decrease of the population N\"(t) (see the orange dots) with increasing time. In principle,\none can now obtain a reaction rate constant keby fitting an exponential decay N\"(t)/exp[\u0000ke(t\u0000t0)]to the spin up\npopulation. This, however, is not possible in all systems because the decay in N\"(t) can be nonexponential. Instead, we\nuse the more general approach described in Ref. [54], which involves examining the instantaneous decays\nke(t)=\u0000˙N\"(t)\nN\"(t). (9)\nLocal manifold analysis. The ensemble method is computationally expensive because it requires the propagation\nof a large number of spins for su \u000eciently long time. An alternative method, called the LMA , can be used to obtain\ninstantaneous spin-flip rates purely from the geometry of the stable and unstable manifolds in phase space. The LMA\nis based on the observation that the equations of motion (8)can be linearized in the local vicinity of a trajectory mz(t)\non the NHIM with the Jacobian\nJ(t)=@(˙\u0012;˙')\n@(\u0012;')\f\f\f\f\f\fmz(t). (10)\nWith the (not necessarily normalized) directions of the stable and unstable manifolds WsandWuat time tgiven\nby (\u0001's;\u0001\u0012s) and ( \u0001'u;\u0001\u0012u) with \u0001\u0012s= \u0001\u0012u, as marked in Fig. 3(b), and using the linearization of the equations of\nmotion (8)with the Jacobian (10) for the propagation of the spin ensemble, we finally obtain an analytical expression\nfor the instantaneous rates of the magnetization switching\nkm(t)=\u0000˙N\"(t)\nN\"(t)=\u0000lim\n\u0001t!0N\"(t+ \u0001t)\u0000N\"(t)\n\u0001tN\"(t)=@˙\u0012\n@'\f\f\f\f\f\fmz(t) \u0001'u\n\u0001\u0012u\u0000\u0001's\n\u0001\u0012s!\n. (11)\nThese rates can be calculated independently at di \u000berent times t, which allows for computations in parallel. Note that\n\u0001's=\u0001\u0012sand\u0001'u=\u0001\u0012uare the inverse slopes of the stable and unstable manifolds WsandWuin Fig. 3(b), and thus\nthe instantaneous rate km(t) in Eq. (11) is mainly determined by the di \u000berence of these two inverse slopes. This di \u000bers\n61.4 1.6\nϕ1.251.501.752.00θ(a)\n0.00 0.25 0.50 0.75 1.00\nt/T1.341.361.381.401.42k(b) Hint\nz= 0.00 (static)\nHint\nz= 0.15,ω=π/8\nHint\nz= 0.30,ω=π/8\nHint\nz= 0.05,ω=π/8\nHint\nz= 0.15,ω=π/4\nHint\nz= 0.15,ω=π/16Figure 4: (a) A selection of TStrajectories of the free-layer system with the potential (3)described by the LLG equation (2). The static TStrajectory\nwithout external magnetic field ( Hext\nz=0) is marked by a black dot at \u0012='=\u0019=2. The TStrajectory with the reference parameters given in Eqs. (6)\nand(7)is shown as solid black line. TStrajectories with driving parameters deviating from Eq. (7)are drawn with colored dash or dash-dotted lines.\nThe elliptical shape and orientation of the TStrajectories depends strongly on the driving by the oscillating external magnetic field. (b) Instantaneous\nrates (dark lines) and mean rates (pale lines) for some of the periodic TS trajectories shown in (a).\nfrom, e. g., Ref. [ 54], where the instantaneous rate is related to the slopes of the stable and unstable manifolds; the\ninverse slopes in Eq. (11) occur because the reaction coordinate \u0012is not the abscissa but the ordinate in Figs. 3(a) and\n3(b). As discussed above, the angles \u0012and'are not canonical variables as is typical in applications of TST to systems\nwith Hamiltonian dynamics [ 45,54,55]. This manifests in a nontrivial and time-dependent prefactor (@˙\u0012.\n@')\f\f\fmz(t)\nin Eq. (11), which is an element of the Jacobian (10). In the limiting case of a Cartesian reaction coordinate xwith\ncanonical momentum p=m˙x(where mis the particle mass), the corresponding element of the Jacobian reduces to a\nconstant@˙x/@p=1=m[54].\nFloquet method. The average decay rates kFacross time-dependent barriers can also be obtained directly using a\nFloquet stability analysis [ 51,54]. While this method is computationally much cheaper than the ensemble method and\nthe LMA, it cannot yield instantaneous rates.\nTo obtain the time-independent rate constant kFfor a given TStrajectory on the NHIM , we linearize the equations\nof motion using the Jacobian (10). By integrating the di \u000berential equation\n˙\u001b(t)=J(t)\u001b(t) with \u001b(0)=1, (12)\nwe then obtain the system’s fundamental matrix \u001b(t). When considering trajectories with period T,M=\u001b(T) is called\nthe monodromy matrix. Its eigenvalues muandms, termed Floquet multipliers, can be used to determine the Floquet\nrate constant\nkF=1\nT(lnjmuj\u0000lnjmsj). (13)\nAs shown below, the Floquet rate constant kFagrees perfectly with the instantaneous rates ke(t) and km(t) when the\nlatter two are averaged over one period Tof the TS trajectory.\n3. Results and discussion\nWe now present and discuss the TStrajectories and the related instantaneous and averaged decay rates obtained for\nthe free-layer system with and without driving by an oscillating external magnetic field. TStrajectories in the ( ';\u0012)\nphase space at various amplitudes Hext\nzand frequencies !of the driving field relative to the reference system of Eq. (7)\nare shown in Fig. 4(a). The static TStrajectory without external driving is marked by a black dot at \u0012='=\u0019=2,\nwhich coincides with the position of the static saddle in Fig. 2. When driven by an oscillating external field, the TS\ntrajectories become periodic orbits with the same period as the driving. The elliptical shape and orientation of the orbits\nstrongly depend on the amplitude and frequency of the driving. The black solid line marks the TStrajectory with the\n7π/4 π/2\nω0.00.20.4Hint\nz\n1.201.251.301.351.40\n¯kFigure 5: Mean rates ¯kas function of the frequency !and amplitude Hext\nzof the external magnetic field. The diamonds mark the parameters used in\nFig. 4(b).\nreference parameters given in Eqs. (6)and(7). The dashed lines mark TStrajectories, where either the amplitude Hext\nz\n(blue lines) or the frequency !(red lines) deviates from these reference parameters. For the chosen sets of parameters\ninvestigated here, the driving frequency mostly a \u000bects the shape of the orbits, whereas the driving amplitude has a large\ninfluence on the orbit size while preserving the shape approximately.\nRate constants, which are related to the TStrajectories in Fig. 4(a), have been computed with the methods introduced\nin Sec. 2.2.2 and are shown in Fig. 4(b). The (dark lines mark the instantaneous rates obtained by the LMA as functions\noft=Twhere T=2\u0019=! is the period of the corresponding TStrajectory. As can be seen, the oscillation amplitude of\nthe instantaneous rates at high amplitude Hext\nz=0:3 of the driving field (dark blue line) is slightly higher than that of the\nsystem with Hext\nz=0:15 ( black line). This trend continues for Hext\nz=0:05, where the oscillation is almost unnoticeable.\nThe pale lines present the averaged rate constants. Here, the increase of the Hext\nzfrom 0:15(light gray line) to 0:3(pale\nblue line) causes a significant decrease in the averaged rate constant. The dark and pale red lines in Fig. 4(b) mark the\ninstantaneous and averaged decay rate of the system at lower frequency !=\u0019=16 and higher frequency !=\u0019=4 of the\noscillating magnetic field. The instantaneous rate fluctuates much stronger around the mean value. The alternation in\nthe strength of these fluctuations is strong evidence for a sign change in the modulation amplitude around the reference\nfrequency. As mentioned above, the rate constants obtained as time averages of the instantaneous rates over one period\nof the TS trajectory agree perfectly with the rate constants computed using the Floquet method.\nFinally, the dependence of the averaged decay rate ¯kon the amplitude Hext\nzand frequency !of the magnetic field is\nreported in Fig. 5. The diamonds mark the parameters of the TStrajectories shown in Fig. 4(b). A minimum in the\nrates lies near the corner with low frequencies !and high amplitudes Hext\nz.\n4. Conclusion and outlook\nWe have investigated magnetization switching in a ferromagnetic free-layer system. The dynamics of the magnetic\nmoment is described by the Gilbert equation (1). We have shown that TST can be applied to its two-dimensional phase\nspace even though the Gilbert equation does not have the expected structure of a Hamiltonian system with coordinates\nand canonical momenta. We obtained the periodic TStrajectories of the free-layer system driven by an additional\noscillating external magnetic field. In turn, these form the basis for the calculation of the instantaneous and averaged\ndecay rates. The rates significantly depend on the time-dependent driving, i. e., the amplitude and frequency of the\nexternal magnetic field. The magnetization switching can thus be controlled by the external driving.\nIn this paper, we have assumed that the time derivative ˙mof the magnetic moment follows the magnetic field\nwithout relaxation, as described by the Gilbert equation (1). In future work, the model for the free-layer system could\nbe extended by taking into account relaxation of the spins [ 71], which requires one to enlarge the phase space from two\nto four dimensions. TST will then allow us to study the influence of the relaxation on the decay rates.\nPerhaps surprisingly, an increase in the field Hext\nzmostly leads to a decrease in the mean rate in Fig. 5. This\nis perhaps a consequence of the intermediate friction regime wherein the population of activated spins—i. e., those\n8that would go over the barrier—are dampened by the dissipation. Moreover, as the driving frequency increases, the\nmoving trajectory explores a wider oscillation potentially averaging—and suppressing—the di \u000berence in the curvatures\nassociated with the stable and unstable directions that contributes to the rate. Resolution of this phenomenon remains a\nchallenge for future work.\nIn summary, this work suggests that the application of recent advances in locally nonrecrossing TST to magnetization\nswitching could be helpful in future work addressing dynamics in spintronics.\nDeclaration of competing interest\nThe authors declare that they have no known competing financial interests or personal relationships that could have\nappeared to influence the work reported in this paper.\nCRediT authorship contribution statement\nJohannes M ¨ogerle: Methodology, Software, Formal analysis, Investigation, Writing – Original Draft. Robin\nSchuldt: Methodology, Formal analysis, Investigation, Writing – Original Draft. Johannes Rei \u000b:Methodology,\nSoftware, Validation, Resources, Data Curation, Writing – Review & Editing, Visualization. J¨org Main: Conceptual-\nization, Methodology, Formal analysis, Resources, Writing – Original Draft, Writing – Review & Editing, Supervision,\nProject administration, Funding acquisition. Rigoberto Hernandez: Conceptualization, Writing – Review & Editing,\nProject administration, Funding acquisition.\nAcknowledgments\nFruitful discussions with Robin Bardakcioglu, Matthias Feldmaier, and Andrej Junginger are gratefully acknowl-\nedged. The German portion of this collaborative work was supported by Deutsche Forschungsgemeinschaft (DFG)\nthrough Grant No. MA1639 /14-1. RH’s contribution to this work was supported by the National Science Foundation\n(NSF) through Grant No. CHE-1700749. This collaboration has also benefited from support by the European Union’s\nHorizon 2020 Research and Innovation Program under the Marie Skłodowska-Curie Grant Agreement No. 734557.\nReferences\n[1]C. M. Schneider, B. Zhao, R. Kozhuharova, S. Groudeva-Zotova, T. M ¨uhl, M. Ritschel, I. M ¨onch, H. 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Ostatnický1 \n1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic \n2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic \n3Technical University Dresden, 01062 Dresden, Germany \n4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany \n5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany \n6 Department of Physics, University of Konstanz, 78457 Konstanz, Germany \n \nLaser-induced magnetization dynamics is one of the key methods of modern opto-spintronics which aims \nat increasing the spintronic device speed1,2. Various mechanisms of interaction of ultrashort laser pulses \nwith magnetization have been studied, including ultrafast spin-transfer3, ultrafast demagnetization4, \noptical spin transfer and spin orbit torques 5,6,7 , or laser-induced phase transitions8,9. All these effects can \nset the magnetic system out of equilibrium, which can result in precession of magnetization. Laser-induced \nmagnetization precession is an important research field of its own as it enables investigating various \nexcitation mechanisms and their ultimate timescales2. Importantly, it also represents an all-optical analogy \nof a ferromagnetic resonance (FMR) experiment, providing valuable information about the fundamental \nparameters of magnetic materials such as their spin stiffness, magnetic anisotropy or Gilbert damping10. \nThe “all-optical FMR” (AO-FMR) is a local and non-invasive method, with spatial resolution given by the \nlaser spot size, which can be focused to the size of few micrometers.  This makes it particularly favourable \nfor investigating model spintronic devices.  \nMagnetization precession has been induced in various classes of materials including ferromagnetic \nmetals11, semiconductors10, 12, or even in materials with a more complex spin structure, such as non-\ncollinear antiferromagnets13. Ferrimagnetic insulators, with Yttrium Iron Garnet (YIG, Y 3Fe5O12) as the \nprime representative14, are of particular importance for spintronic applications owing to their high spin \npumping efficiency15 and the lowest known Gilbert damping16. However, inducing magnetization dynamics \nin ferrimagnetic garnets using optical methods is quite challenging, as it requires large photon energies 2 \n (bandgap of YIG is Eg ≈ 2.8 eV)17. This spectral region is rather difficult to access with most common ultrafast \nlaser systems, which are usually suited for near-infrared wavelengths. Therefore, methods based mostly \non non-thermal effects, such as inverse Faraday18,19 and Cotton-Mouton effect20 or photoinduced magnetic \nanisotropy21, 22   have been used to trigger the magnetization precession in YIG so far. For these phenomena \nto occur, large laser fluences of tens of mJ/cm2 are required23. In contrast, laser fluences for a thermal \nexcitation of magnetization precession usually do not exceed tens of J/cm2 (Refs. 12, 21, 13). Using the \nlow fluence excitation regime allows for the determination of quasi-equilibrium material parameters, not \ninfluenced by strong laser pulses.  In magnetic garnets, an artificial engineering of the magnetic anisotropy \nvia the inclusion of bismuth was necessary to achieve thermally-induced magnetization precession21.  \nIn this paper, we show that magnetization precession can be induced thermally by femtosecond laser \npulses in a thin film of pure YIG only by adding a metallic capping layer. The laser pulses locally heat the \nsystem, which sets the magnetization out of equilibrium due to the temperature dependence of its \nmagnetocrystalline anisotropy. This way we generate a Kittel (n = 0, homogeneous precession) FMR mode, \nwith a precession frequency corresponding to the quasi-equilibrium magnetic anisotropy of the thin YIG \nfilm10.  We thus prove that the AO-FMR method is applicable for determining micromagnetic parameters \nof thin YIG films. Using the AO-FMR technique we revealed that at low temperature the Kittel mode \ndamping is significantly faster than at room-temperature, in accord with previous FMR experiments24,25.  \nOur experiments were performed on a 50 nm thick layer of pure YIG grown by pulsed-laser deposition on \na gadolinium-gallium-garnet (GGG) (111)-oriented substrate. One part of the film was covered by 8 nm of \nAu capping layer, the other part by Pt capping, both being prepared by ion-beam sputtering. Part of the \nsample was left uncapped as a reference.  X-ray diffraction confirmed the excellent crystal quality of the \nYIG film with a very low level of growth-induced strain, as described in detail in Ref. 26. The magnetic \nproperties were further characterized using SQUID magnetometry and ferromagnetic resonance \nexperiments, showing the in-plane orientation of magnetization (see Supplementary Material, Part 1 and \nFigs. S1 and S2). The deduced low-temperature (20 K) saturation magnetization µ0Ms   180 mT is in \nagreement with results published on qualitatively similar samples27 again confirming a good quality of the \nstudied YIG film. Magnetic anisotropy of the system at 20 K was established from an independent \nmagneto-optical experiment (Ref. 28), the corresponding anisotropy constants for cubic anisotropy of the \nfirst and second order are Kc1 = 4680 J/m3  and Kc2 = 223 J/m3, while the overall uniaxial out-of-plane \nanisotropy is vanishingly small.  3 \n Laser-induced dynamics was studied in a time-resolved magneto-optical experiment in transmission \ngeometry, as schematically shown in Fig. 1(a). An output of a Ti:Sapphire oscillator generating 200 fs laser \npulses was divided into a strong pump beam, with fluences tuned between 70 and 280 µJ/cm2, and a 20-\ntimes weaker probe beam. The beams were focused on a 30 m spot on the sample, which was placed in \na cryostat and kept at cryogenic temperatures (typically 20 K). An external magnetic field (up to 550 mT) \ngenerated by an electromagnet was applied in y direction (see Fig. 1). The wavelength of pump pulses (800 \nnm) was set well below the absorption edge of the YIG layer, as indicated in the transmission spectrum of \nthe sample in Fig. 1(b). The wavelength of probe pulses (400 nm) was tuned to match the maximum of the \nmagneto-optical response of bulk YIG [see inset in Fig. 1(b) and Ref. 29].  \nThe detected time-resolved magnetooptical (TRMO) signal corresponding to the rotation of polarization \nplane of the probe beam Δβ, was measured as a function of the time delay Δt between pump and probe \npulses. In Fig. 1(c), we show an example of TRMO signals observed in uncapped YIG and two YIG/metal \nheterostructures. Clearly, in the presence of the metallic capping layer an oscillatory TRMO signal is \nobserved, whose amplitude depends on the capping metal used. Frequency and damping of the \noscillations, on the other hand, remain virtually unaffected by the type of the capping layer, while no \noscillations are observed in the uncapped YIG sample. \nThe TRMO signals can be phenomenologically described by a damped harmonic function after removing a \nslowly varying background (see Supplementary Material, Part 2 and Fig. S3),12 \n∆𝛽(Δ𝑡)=𝐴cos(2𝜋𝑓𝛥𝑡+𝜑)exp(−𝛥𝑡 𝜏⁄ ),         (1) \nwhere A is the amplitude of precession, f its frequency, φ the phase and τ the damping time. The fits are \nshown in Fig. 1(c) as solid lines.  \nIn order to demonstrate that the TRMO signals result from (laser-induced) magnetization dynamics, we \nvaried the external magnetic field Hext and extracted the particular precession parameters by fitting the \ndetected signals by Eq. (1). As depicted in Fig. 2(a), the experimentally observed dependence of the \nprecession frequency on the applied field is in excellent agreement with the solution of Landau-Lifshitz-\nGilbert (LLG) equation, using the free energy of a [111] oriented cubic crystal [see Supplementary, section \n5, Eq. (S5) and Ref. 28]. This correspondence with the LLG model proves that our oscillatory signals reflect \nindeed the precession of magnetization in uniform (Kittel) mode in YIG. We stress that the precession \nfrequency is inherent to the YIG layer and does not depend on the type of the capping layer. 4 \n The detection of the uniform Kittel mode can be further confirmed by comparing the frequency of the \noscillatory TRMO signal with the frequency of resonance modes observed in a conventional, microwave-\ndriven ferromagnetic resonance (MW-FMR) experiment. The MW-FMR experiment was performed in the \nin-plane ( H = 0°) and out-of-plane ( H = 90°) geometry of the external field. We measured the TRMO signals \nin YIG/Au sample in a range of magnetic field angles H and modelled the angular dependency of f by LLG \nequation with the same parameters that were used in Fig. 2(a). The output of the model is presented in \nFig. 2(b), together with precession frequencies obtained from TRMO and FMR experiments. The MW-FMR \ndata fit well to the overall trend, confirming the presence of uniform magnetization precession [Ref. 30] \nTo find the exact physical mechanism that triggers laser-induced magnetization precession in our \nYIG/metal bilayers, we measured the TRMO signals at different sample temperatures T. For comparison \nwe calculated also the dependence of f on the first order cubic anisotropy constant Kc1   from the LLG \nequation, which is shown in the inset of Fig. 2(c). This graph reveals that f should be directly proportional \nto Kc1 in the studied range of temperatures . In Fig 2(c) we plot f as a function of T, together with the \ntemperature dependence of Kc1 (T) obtained from Ref. 28 and Ref. 32.  Clearly, both Kc1 and f show a similar \ntrend in temperature.  Considering also the temperature dependence of the precession amplitude [see \nFig. S5 (a) and Section 4 of Supplementary Material], we identify the pump pulse-induced heating and \nconsequent modification of the magnetocrystalline anisotropy constant Kc1 as the dominant mechanism \ndriving laser-induced magnetization precession. \nIn order to estimate the pump-induced increase in quasi-equilibrium temperature of the sample, we first \nfit the temperature dependence of the parameter Kc1 reported in literature by a second order polynomial \n[Fig. 2(c)]. Owing to the linear relation between f and Kc1 and the known temperature dependence of f, \nthe measured dependence of f on pump fluence I can be converted to the intensity dependence of the \ntemperature increase T(I), which is shown Fig. 2(d). As expected, higher fluence leads to more \npronounced heating, which results in a decrease of the precession frequency. Note that for the highest \nintensity of 300 J/cm2, the sample temperature can increase by almost 80 K. \nNature of the observed laser-induced magnetization precession was further investigated by comparing \nsamples with different capping layers. In Fig. 3(a) we show the amplitude A of the oscillatory signal in the \nYIG/Pt and YIG/Au layers as a function of I. The difference between the samples is apparent both in the \nabsolute amplitude of the precession and in its increase with I, the YIG/Pt showing stronger precession. \nFurthermore, precession damping is stronger in YIG/Au than in YIG/Pt, as apparent from Fig 3 (b) where \neffective Gilbert damping parameter eff is presented as a function of Hext. These values of eff were 5 \n obtained by fitting the TRMO data by the LLG equation, as described in the Supplementary Material \n(Section 5). Despite the relatively large fitting error, we can still see that YIG/Pt shows slightly lower    \n0.020, while the YIG/Au has   0.025. To understand these differences, we modeled the propagation of \nlaser-induced heat in GGG/YIG/Pt and GGG/YIG/Au multilayers by using the heat equation (see \nSupplementary Material, Section 7). In Fig. 3(c),  T is presented as a function of time delay t after pump \nexcitation for selected depths from the sample surface. In Fig. 3(d), the same calculations are presented \nfor variable depths and fixed t. The model clearly demonstrates that a significantly higher T can be \nexpected in the Pt-capped layer simply due to its smaller reflection coefficient as compared to Au-capping \n(see Supplementary Material, Section 7). This in turn leads to a higher amplitude of the laser-induced \nmagnetization precession in YIG/Pt compared to the YIG/Au, as apparent in Fig. 3(a).  \nAccording to our model, an extreme increase in temperature is induced in the first few picoseconds after \nexcitation, which acts as a trigger of magnetization precession. After approximately 10 ps, precession takes \nplace in quasi-equilibrium conditions. The system returns to equilibrium on a timescale of nanoseconds, \nwhich shows also in the TRMO signals as the slowly varying background (Fig. S3). The precession frequency \nwe detect reflects the quasi-equilibrium state of the system. Therefore, the temperature increase T \ndeduced from the TRMO signal can be compared with our model for large time delays after the excitation \n(t  10 ps). In YIG/Au sample, the experimental values of T = (25  10) K for excitation intensity of 150 \nJ/cm2 [see Fig. 2(d)], while the model gives us  T  5K [Fig. 3 (c)]. Clearly, the values match in the order \nof magnitude but there is a factor of  5 difference. This difference results from the boundary conditions \nof the model that assumes ideal heat transfer between the sample and the holder, which is experimentally \nrealized using a silver glue with less than perfect performance at cryogenic conditions. \nFrom Fig. 3(d) it also follows that large thermal gradients are generated across the 50 nm layer. This could \nlead to significant inhomogeneity in magnetic properties of the layer, that would increase the damping \nparameter  by an extrinsic term. In our TRMO measurements,  is indeed very large for a typical YIG \nsample (TRMO  2-2.5 x10-2) and exceeds the value obtained from room-temperature MW-FMR by almost \nan order of magnitude ( FMR  1x10-3, see Supplementary Material, Section 1b). As the modeled thermal \ngradient alone cannot account for such a large change in Gilbert damping (see Supplementary Material, \nSection 6), we attribute this increase in Gilbert damping to the difference in the ambient temperatures. \nLarge change of Gilbert damping (by a factor of 30) between room and cryogenic (20 K) temperature has \nrecently been reported on a seemingly high quality YIG thin film24. It was explained in terms of the presence \nof rare earth or Fe2+ impurities that are activated at cryogenic temperatures.  It is likely that the same 6 \n process occurs in our sample.  Even though other mechanisms related to the optical excitation can also \ncontribute to the increase in TRMO (see Supplementary Material, Section 6), the all-optical and standard \nFMR generated Kittel modes correspond very well [see Fig. 2(b)]. Furthermore, also the observed sample-\ndependent Gilbert damping is consistent with this explanation.  The YIG/Pt sample is heated to higher \ntemperature by the pump laser pulse [Fig. 3(c), (d)] than the YIG/Au sample, which according to Ref. 24 \ncorresponds to a lower Gilbert damping. It is worth noting that damping parameter can be increased also \nby spin-pumping from YIG to the metallic layer. However, this effect is expected to be significantly higher \nwhen Pt is used as a capping, which does not agree with our observations.  \nIn conclusion, we demonstrated the feasibility of the all-optical ferromagnetic resonance method in 50-\nnm thin films of plain YIG. Magnetization precession can be triggered by laser-induced heating of a metallic \ncapping layer deposited on top of the YIG film. The consequent change of sample temperature modifies \nits magnetocrystalline anisotropy, which sets the system out of equilibrium and initiates the magnetization \nprecession. Based on the field dependence of precession frequency, we identify the induced magnetization \ndynamics as the fundamental (Kittel) FMR mode, which is virtually independent of the type of capping and \nreflects the quasi-equilibrium magnetic anisotropy. The Gilbert damping parameter is influenced by line-\nbroadening mechanism due to low-temperature activation of impurities, which is an important aspect to \nbe taken into account for low-temperature spintronic device applications.  \nRegarding the efficiency of the optical magnetization precession trigger, it was found that the type of \ncapping layer strongly influences the precession amplitude. The precession in YIG/Pt attained almost twice \nthe amplitude of that in YIG/Au under the same conditions. This indicates that a suitable choice of capping \nlayer should be considered in an optimization of this local non-invasive magnetometric method. \n \nAcknowledgments: \nThis work was supported in part by the INTER-COST grant no. LTC20026 and by the EU FET Open RIA \ngrant no. 766566. We also acknowledge CzechNanoLab project LM2018110 funded by MEYS CR for the \nfinancial support of the measurements at LNSM Research Infrastructure and the German Research \nFoundation (DFG SFB TRR173 Spin+X projects A01 and B02 #268565370). \n \n 7 \n LITERATURE \n[1]  A. Hirohata et al., JMMM 509, 16671 (2020)  \n[2]  A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010) \n[3] F. Siegries et al., Nature 571, 240–244 (2019) \n[4] E. Beaurepaire  et al., Phys. Rev. Lett. 76, 4250 (1996). \n[5] P. Nemec et al., Nature Physics 8, 411-415 (2012)  \n[6] G.M. Choi et al., Nat. 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B 97, 014422 (2018) \n[22]  A. Stupakiewicz et al., Nature 542, 71 (2017) \n[23] F. Atoneche et al., Phys. Rev. B 81, 214440 (2010) \n[24] C. L. Jermain et al., PRB 95, 174411 (2017) 8 \n [25] H. Maier-Flaig et al.,   Phys. Rev. B 95, 214423 (2017)  \n[26] B. Bhoi et al.: J. Appl.Phys. 123, 203902 (2018) \n[27] J. Mendil et al.: Phys. Rev. Mat. 3, 034403 (2019) \n[28] E. Schmoranzerova et al., ArXiv XXX (2021) \n[29] E. Lišková Jakubisova et al., Appl. Phys. Lett. 108, 082403 (2016) \n[30] We note that the FMR data were obtained at room temperature while the TRMO experiment was \nperformed at 20K. However, as apparent from Fig. 2(c), the precession frequency  varies by less than \n10% between 20 K and 300 K, which is well below the experimental error of   (H). This justifies \ncomparison of the precession frequencies obtained from the TRMO experiment with the FMR data. \n[31] M. Haider et al., J. Appl. Phys. 117, 17D119 (2015) \n[32] N. Beaulieu et al., IEEE Magnetics Letters 9, 3706005 (2018) \n \n \n \nFIGURES \n \n \n9 \n Fig. 1: (a) Schematic illustration of the pump&probe experimental setup, where Eprobe is the probe beam linear \npolarization orientation which is rotated by an angle  after transmission through the sample with respect to the \norientation E’probe. An external magnetic field H ext is applied at an angle H. (b) Absorption spectrum of the studied \nYIG sample, where OD stands for the optical density defined as minus the decadic logarithm of sample \ntransmittance. The red arrow indicates the wavelength of the pump beam PUMP = 800 nm. Inset: Spectrum of Kerr \nrotation K of bulk YIG crystal29. The blue arrow shows the wavelength of the probe beam PROBE = 400 nm. (c) \nTypical time-resolved magneto-optical signals of a plain 50 nm YIG film (black dots), YIG /Pt  (green dots) and \nYIG/Au bilayer (blue dots) at 20 K and 0Hext = 100 mT, applied at an angle H = 40°.  Lines indicate fits by Eq. (1). \nThe data were offset for clarity. \n \n \n \nFig. 2: (a) Frequency f of magnetization precession as a function of magnetic field applied at an angle H = 40°, for \nYIG/Pt (blue dots) and YIG/Au (green triangles) at  T = 20 K and I = 150 J/cm2.  The line is calculated from LLG equation \n(Eq. S3) with the free energy given by (Eq.S5) (b) Field-angle dependence of f in YIG/Au sample for 0Hext = 300 mT \n(blue dots), compared to a model by LLG  model (line) and to frequencies measured by MW-FMR (red stars)32. (c) \nTemperature dependence of f in YIG/Au sample (black points), where 0Hext = 300 mT was applied at H = 40°. The \ntemperature dependence of cubic anisotropy constant Kc1 was obtained from Ref. 28 (red dots) and Ref. 32 (red star, \nT = 20 K). The data were fitted by an inverse polynomial dependence 𝐾ଵ(𝑇)= ଵ\n(ା்ା்మ), with parameters: a = 0.18 \nm2/kJ; b= 9 x 10-4 m2/kJ.K; c = 9 x 10-6m2/kJ.K2. Inset: Dependence f(Kc1) obtained from the LLG equation. (d)  f as a \nfunction of pump pulse fluence I, from which the increase of sample temperature T for the used pump fluences was \nevaluated using the f(T) dependence.  \n \n10 \n  \nFig. 3: Comparison of magnetization precession in YIG/Pt and YIG/Au samples. (a) Precession amplitude A as a \nfunction of pump fluence I (dots) with the corresponding linear fits 𝐴 = 𝑠∙𝐼.  The parameter s Pt = (1.05 0.09)x10-2  \nrad.cm2/J in the YIG/Pt, and s Au = (0.50.1)x10-2  rad.cm2/J in YIG/Au. These dependencies were measured for \n0Hext = 300 mT and T 0 = 20 K. In YIG/Pt sample the as-measured data obtained for H = 40° are shown. In the YIG/Au \nsample, the A(I) dependence was originally measured for H = 21° and recalculated to  H = 40° according to the \nmeasured angular dependence, as described in detail in Supplementary Material, Section 3. (b) Gilbert damping eff \nfor Hext applied at an angle H = 40°. The values of eff result from fitting the TRMO signals to LLG equation; I = 140 \nJ/cm2. (c) and (d) Increase in lattice temperature as a function of time delay between pump and probe pulses for \nselected depths from the sample surface (c) and as a function of depth for fixed time delays (d). I = 140 J/cm2, T0 = \n20 K. The heat capacities and conductivities of individual layers are provided in the Supplementary Material, Section \n7.  \n \n \n  \n11 \n Thermally induced all-optical ferromagnetic resonance in thin YIG films: \nSupplementary Material \nE. Schmoranzerová1*, J. Kimák1, R. Schlitz3  , S.T. B. Goennenwein3,   D. Kriegner2,3, H. Reichlová2,3, ,  Z. \nŠobáň2, G. Jakob5, E.-J. Guo5, M. Kläui5, M. Münzenberg4, P. Němec1 , T. Ostatnický1 \n1Faculty of Mathematics and Physics, Charles University, Prague, 12116, Czech Republic \n2Institute of Physics ASCR v.v.i , Prague, 162 53, Czech Republic \n3Technical University Dresden, 01062 Dresden, Germany \n4Institute of Physics, Ernst-Moritz-Arndt University, 17489, Greifswald, Germany \n5Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany \n \n6 Department of Physics, University of Konstanz, 78464 Konstanz, Germany \n \n \n1. Magnetic characterization  \n \nA. SQUID magnetometry \nA superconducting quantum device magnetometer (SQUID) was used to characterize the magnetic \nproperties of the thin YIG film at several sample temperatures. The magnetic hysteresis loops, detected \nwith magnetic field applied in [2-1-1] crystallographic direction of the YIG layer, are shown in Fig. S1. As \nexpected [t26], the saturation magnetization increases at low temperatures, which is accompanied by a \nslight increase in coercive field. At room temperature, the effective saturation magnetization is estimated \nto be Ms = 95 kA/m. This value is in good agreement with the effective magnetization Meff obtained from \nthe ferromagnetic resonance (FMR) measurement (see Section 1b), which indicates only a weak out-of-\nplane magnetic anisotropy [s1]. However, as discussed in detail in Ref. 26, the Ms from our SQUID \nmeasurement is burdened by a relatively large error. Therefore, mere comparison of SQUID and FMR \nexperiment is not sufficient to evaluate the size of the out-of-plane magnetic anisotropy. An additional \nexperiment such as static magneto-optical measurement [28] is needed in order to get more precise \nestimation of the out-of plane magnetic anisotropy. \n \nB.  FMR measurement  \nThe SQUID magnetometry was complemented by so-called broad band ferromagnetic resonance \nmeasurements using a co-planar waveguide to apply electromagnetic radiation of a variable frequency f \n=/2 to the sample. The measurement was performed at room temperature and further details on the \nmethod can be found in Ref. s2. An exemplary set of spectra showing the normalized microwave \ntransmission | S21|norm obtained at different external fields magnitudes applied in the sample plane, is \nshown in Fig. S2(a). The set of Lorentzian-shape resonances can be fitted by the equation: 12 \n |𝑆ଶଵ|୬୭୰୫=ቀഘ\nమቁమ\nቀഘ\nమഏିഘబ\nమഏቁାቀഘ\nమቁమ+𝑦          (S1) \nWhere f0 =0/2  is the FMR resonance frequency, /2 is the half width half maximum line width, B the \namplitude of the FMR line and y0 a frequency independent offset. From an automated fitting of the set of \nlines obtained at different Hext, we extract the magnetic field dependence of the resonance frequency \n0/2 (Hext) [Fig. S2(b)] and linewidth (Hext) [Fig. S2(c)].  Clearly, the resonance frequencies correspond \nto the fundamental (Kittel) mode, and can correspondingly be fitted by the Kittel formula [s3]:  \nఠబ\nଶగ=ఊ\nଶగඥ𝜇𝐻ୣ୶୲(𝜇𝐻ୣ୶୲+𝜇𝑀ୣ)  (S2) \nWhere Meff is the effective saturation magnetization that includes the out-of-plane anisotropy term, and  \nis gyromagnetic ratio. From this fit, it is possible to evaluate Meff , Kittel  = 94.9 kA/m \nFrom the linewidth dependence (Hext)=2 + 0 we can extract both the inhomogeneous line \nbroadening and the Gilbert damping parameter, as shown in Fig. S2(c) [s2]. In our experiment, the \ninhomogeneous linewidth broadening is 0 = 55.8 MHz, and the Gilbert damping parameter  = 0.001. \nBoth values are on a higher side compared e.g. with YIG prepared by liquid phase epitaxy [s8] but in good \nagreement with typical YIG thin films similar to our layers, which were prepared by pulsed laser deposition \n[27]. This again confirms the good quality of the studied thin YIG films. \n \n2. Processing of time-resolved magneto-optical data \nIn order to extract the parameters describing the precession of magnetization correctly from the time-\nresolved magneto-optical (TRMO) signals, it is first necessary to remove the slowly varying background on \nwhich the oscillatory signals are superimposed. For this purpose, we fitted the measured data by the \nsecond-order polynomial. The fitted curve was then subtracted from the measured signals, as \ndemonstrated in Fig. (S3).  \nFrom the physical point of view, the background can be attributed to a slow return of magnetization to its \nequilibrium state after the pump beam induced heating, which can take place on the timescale of tens of \nnanoseconds [10]. Since both saturation magnetization Ms and magnetocrystalline anisotropy Kc are \ntemperature-dependent, their temporal variation can in principle contribute to the background signal. \nHowever, as explained later in Section 4, the variation of Ms is very weak at cryogenic temperatures. The \nheat-induced modification of Kc, and the resulting change of the magnetization quasi-equilibrium \norientation, is, therefore, more probable origin of the slowly varying background, which is detected in the \nMO experiment by the Cotton-Mouton effect [28]. \n3. Angular dependence of precession amplitude  \nIn order to mutually compare the values of precession amplitudes measured in YIG/Pt and YIG/Au samples \nat different angles of the external magnetic field H, it is necessary to correct their values for the value of \nH. The following procedure was used to correct the data presented in Fig. 3 of the main text . \n 13 \n First, we measured in detail angular dependence of the precession amplitude in the YIG/Au layer, which is \npresented in Fig. S4. Amplitude of the oscillatory signal detected in our experiment does not depend solely \non the amplitude of the magnetization precession but also on the size of the magneto-optical (MO) effect. \nIn our experimental setup, the change of H was achieved by tilting the sample relative to the position of \nelectromagnet poles [see Fig. 1(a)]. The MO response, however, varies also with the angle of incidence \nwhich is modified simultaneously with a change of H [see Fig. 1(a)] . Therefore, it is not straightforward to \ndescribe the A(H) analytically. Instead, we fitted the measured dependence A(H) by a rational function in \na form of y = 1/(A+Bx2), which is the lowest order polynomial function that can describe the signal properly. \nFrom the fit we derived a correction factor of 1.7 by which the amplitudes A measured at H =21° has to \nbe multiplied to correspond to that measured at H =40°. This factor was then used to recalculate the \nintensity dependence of the precession amplitude A(I)  in YIG/Au measured at H =21° to the A(I) at H \n=40°, which could be directly compared to the A(I) dependence detected at YIG/Pt for H =40° - see Fig. \nS4(b). \n \n4. Temperature dependence of precession amplitude \n \nIn order to further investigate the origin of the laser-induced magnetization precession, the amplitude of \nthe oscillatory MO signal was measured as a function of the sample temperature in YIG/Pt sample, see Fig. \nS5(a). In Fig. S5 (b), we show temperature dependence of saturation magnetization Ms, as obtained from \nRef. 32 \nThe only parameter changed within this experiment was the sample temperature. It is reasonable to \nexpect that the size of the magneto-optical effect is not strongly temperature dependent in the studied \ntemperature range between 20 and 50 K (see Ref. 28) Therefore, the dependence A(T) presented in Fig S5 \ncorresponds directly to the temperature dependence of magnetization precession amplitude. By \ncomparing the Ms(T) and A(T) data, it is immediately apparent that the laser-induced heating would not \nmodify Ms enough to account for the large change of the magnetization precession amplitude with the \nsample temperature. Even assuming the most extreme laser-induced temperature increase T  80 K \nshown in Fig. 3(c), the laser-induced Ms variation would be less than 5%, while the precession amplitude \nchanges by more than 50% between 20 and 50 K. In contrast, the magnetocrystalline anisotropy Kc1 \nchanges drastically even in this relatively narrow temperature range [see Fig. 2(c)]. Consequently, the \nchange of Kc1, which leads to a significant change of the position of quasi-equilibrium magnetization \norientation in the studied sample (see Section 5) provides a more plausible explanation for the origin of \nthe laser-induced magnetization precession in the YIG/metal layer. \n \n5.  LLG equation model  \nThe data were modelled by numerical solution of the Landau-Lifshitz-Gilbert (LLG) equation, as defined in \n[s9]: \nௗ𝑴(௧)\nௗ௧= −𝜇𝛾ൣ𝑴(𝑡)×𝑯𝒆𝒇𝒇(𝑡)൧+ఈ\nெೞቂ𝑴(𝑡)×ௗ𝑴(௧)\nௗ௧ቃ, (S3) 14 \n where  is the gyromagnetic ratio,  is the Gilbert damping constant, and MS is saturated \nmagnetization.The effective magnetic field Heff is given by: \n𝑯𝒆𝒇𝒇(𝑡)=డி\nడ𝑴       (S4) \nwhere F is energy density functional that contains contributions from the external magnetic field Hext, \ndemagnetizing field and the magnetic anisotropy of the sample. We consider the form of F including first- \nand second-order cubic terms as defined in Ref. [t24]. The polar angle  is measured with respect to the \ncrystallographic axis [111] and the azimuthal angle  = 0 corresponds to the direction [21ത1ത], with an \nappropriate index referring to the magnetization position (index M) or the direction of the external \nmagnetic field (index H). The resulting functional takes the form (in the SI units): \n \n𝐹= −𝜇𝐻𝑀[sin𝜃ெsin𝜃ு+cos𝜃ெcos𝜃ுcos(𝜑ு−𝜑ெ)]+ቀଵ\nଶ𝜇𝑀ଶ−𝐾uቁsinଶ𝜃ெ \n  +𝐾c1\n12ൣ7cosସ𝜃ெ−8cosଶ𝜃ெ+4−4√2cosଷ𝜃ெsin𝜃ெcos3𝜑ெ൧ \n  +c2\nଵ଼ൣ−24cos𝜃ெ+45cosସ𝜃ெ−24cosଶ𝜃ெ+4−2√2cosଷ𝜃ெsin𝜃ெ(5cosଶ𝜃ெ−\n2)cos3𝜑ெ+cos𝜃ெcos6𝜑ெ൧ ,                       (S5) \n \nwhere 0 is the vacuum permeability and we consider the following values of constants: magnetization M \n= 174 kA/m, first-order cubic anisotropy constant Kc1=4.68 kJ/m3, second-order cubic anisotropy constant \nKc2 = 222 J/m3 [t24].  \nFor modelling the dependence of precession frequency on the external magnetic field Hext  [Fig. 2 (a)] and \non the angle H [Fig. 2 (a)], we assumed that in a steady state magnetization direction is parallel to Hext, i.e. \nM =  H, and M = H. This is surely fulfilled for large enough magnitude of Hext. Since the coercive field is \nvery small, we can assume the procedure to be correct. Further correspondence to experimental data \nEvaluation of the Gilbert damping factor from the as-measured magneto-optical oscillatory data was done \nby fitting signals by a theoretical curve calculated by solving numerically LLG equation [Eq. (S3)]. We \nconsidered the magnetization free energy density in a form of Eq. (S5) using magnitude and direction of \nthe external magnetic field from the experiment. The electron g-factor was set to 2.0 and then the Gilbert \nfactor and five parameters of the fourth-order polynomial to remove the background MO signal were the \nfitting parameters. The resulting dependence of fitted effective Gilbert factors αeff on external magnetic \nfield is displayed in Fig. 3(b) in the main text, from which the field-independent Gilbert factor α can be \nevaluated. \n \n6. Comparison of Gilbert damping parameter from MW-FMR and TRMO experiments \nThe Gilbert damping from the room-temperature FMR measurement on the YIG film   1∙10-3 and the \nresults from fits of the low-temperature pump&probe data   2∙10-2, differ by an order of magnitude. As \ndetailed in the main text, we attribute this difference to the different sample temperatures in the AO-FMR \nand MW-FMR measurements. However, one might also argue that the increased damping in the optical \nexperiments is caused either by a spatial inhomogeneity of the magnetization oscillations or it is the result \nof the perturbation of the YIG surface. 15 \n In the former case, we expect that the spatial inhomogeneity of the temperature distribution [see Fig. \n3(d)] causes the magnetization to oscillate in a form of a superposition of harmonic waves with well-\ndefined in-plane wavevectors. Considering the dispersion of the allowed oscillatory modes [s4] and \nincluding the relevant value of the exchange stiffness [s5], we revealed that neither the inhomogeneity \ndue to the finite cross section of the excitation laser beam nor the temperature gradient perpendicular to \nthe sample surface can cause such a strong decrease of the Gilbert damping factor that is observed \nexperimentally. Here, we provide an estimate on which time scales the mode dispersion influences the \ndecay of the signal if the exchange stiffness is taken into account. Following [s5], the mode dispersion is \ndescribed by the additive exchange field in the form: \n𝜇𝐻ex=𝐷ቈ𝜋ଶ\n𝑑ଶ𝑛ଶ+𝑘∥ଶ , \nwhere D ≈ 5∙10-17 T.m2 is the exchange stiffness, n is the order of the confined magnon mode, d is the YIG \nlayer thickness and k‖ is the in-plane magnon wave vector. We consider here only the n = 0 case since this \nis the only visible harmonic mode observed in the experimental MO data, as proven by the numerical \nfitting. Note that the frequency shift ∆𝜔/2𝜋=|𝛾|𝜇𝐻ex/2𝜋, where the symbol γ stands for the electron \ngyromagnetic ratio, of the n = 1 mode would be 5.5 GHz, which would be then clearly distinguishable from \nthe basic n = 0 mode in the lowest external magnetic fields. The in-plane wave vector k‖ can be calculated \nfrom the FWHM (full width at half maximum) width of the laser spot on the sample L, which is about 30 \nµm in our case, that leads to the order of magnitude k‖ ≈ (2π/L) ≈ 105 m-1. The frequency increase due to \nthe finite laser spot size can be estimated as ∆𝜔/2𝜋=|𝛾|𝐷𝑘∥ଶ/2𝜋≈14 kHz. Inverse of this value (  0.1 \nms) determines the typical time scale at which the magnon dynamics is influenced by their dispersion due \nto the finite laser spot size, which is clearly out of the range of the experimental time scale. \nThe presence of a metallic layer on the top of the YIG sample surface can result into two significant \ndamping processes. First, the magnetization oscillations (and thus oscillations of the macroscopic magnetic \nfield) are coupled to electromagnetic modes which penetrate the surrounding material and can be \neventually radiative for small magnon wave vectors. Penetration into conductive material in turn causes \nenergy dissipation through finite conductivity of such material. We checked the magnitudes of the \nadditional damping caused by the radiative field and energy dissipation in a thin metallic layer and we \nfound that these processes exist but the additional energy loss cannot explain the observed magnitude of \nthe Gilbert damping parameter. The second possible explanation of the increased precession damping due \nto the presence of the metal/YIG surface may be that there is an additional perturbation to otherwise \nhomogeneous sample due to some inhomogeneity through surface roughness or spatially inhomogeneous \nlocal spin pinning. Since both the surface roughness and spin pinning can depend on the composition of \nthe capping layer, it can also cause a minor difference in the resulting damping factor, as observed in Fig. \n3(b). \nOverall, we attribute the experimentally observed difference in Gilbert damping measured by FMR and \npump&probe techniques to the difference in ambient temperatures that were used in these experiments, \nwhich is in accord with the results of Ref. t22. \n \n7.  Heat propagation in YIG/Pt and YIG/Au 16 \n Heat propagation in our sample structures was modelled in terms of the heat equation: \nడ்\nడ௧=ఒ\n∆𝑇 ,        (S4) \nwhere T is the local temperature, λ is the local thermal conductivity, c is the heat capacity and the symbol \nΔ denotes the Laplace operator. The spatio-temporal temperature distribution in the studied sample has \nbeen calculated by a direct integration of Eq. (S4) in a time domain, assuming excitation of the metallic \nlayer by an ultrashort optical pulse [with a temporal duration of 100 fs (FWHM)]. We have taken the whole \nstructure profile of vacuum/metal/YIG/GGG into consideration, assuming that the GGG substrate had a \nperfect heat contact with the cold finger of the cryostat, which has been held on a constant temperature. \nThe respective heat conductivities ( λ) and heat capacities ( c) were set to the following values. Au: λ = 5 \nW/m K [s6], c = 1.3∙104 J/cm3, Pt: λ = 10 W/m K [s7], c = 1.2∙104 J/cm3, YIG: λ = 60 W/m K, c = 6.7∙103 J/cm3, \nGGG: λ = 300 W/m K, c = 2.1∙104 J/m3. \nTo evaluate the initial heat transfer from the optical pulses to the capping metallic layer, we considered \nthe proper geometry of our experiment, i.e. a 8 nm thick metallic layer deposited on the YIG sample, the \nincidence angle of the laser beam of 45 degrees and its p-polarization. We then used optical constants of \ngold and platinum in order to calculate transmission and reflection coefficients of a nanometer-thick \nmetallic layers by means of the transfer matrix method. From those, we estimated the efficiency of power \nconversion from the optical field to heat to be 3% for gold and 6.5% for platinum. The total amount of heat \ndensity was then calculated by multiplication of the pump pulse energy density and the above-mentioned \nefficiency. \nThe data shown in Fig. 3(c)-(d) were then extracted from the full spatio-temporal temperature distribution. \nClearly, the temperature increase in the YIG/Pt sample is approximately twice larger than that of the \nYIG/Au sample as a consequence of twice larger efficiency of the light-heat energy conversion in favour of \nplatinum. Correspondingly, also the amplitudes of the MO oscillations in Fig. 3(a) reveal the ratio  2:1. \n \n \nLITERATURE \n[s1] B. Bhoi et al.: J. Appl.Phys. 123, 203902 (2018) \n[s2] H. Maier-Flaig et al,.PRB 95, 214423 (2017) \n[s3] Ch. Kittel: “Introduction to solid state physics (8th ed.)”. New Jersey: Wiley. (2013). \n \n[s4] D. D. Stancil, A. Prabhakar: Spin waves – theory and applications (Springer, New York, 2009). \n[s5] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko, B. Hillebrands, \nand A. Conca, J. Phys. D: Appl. Phys. 48, 015001 (2014). \n[s6] G. K. White, Proc. Phys. Soc. A 66, 559 (1953). \n[s7] X. Zhang, H. Xie, M. Fujii, H Ago, K. Takahashi, T. Ikuta, T. Shimizu, Appl. Phys. Lett. 86, 171912 (2005) 17 \n  \n[s8] C. Dubs et al., Phys. Rev. Materials 4, 024416 (2020) \n[s9] J. Miltat, G. Albuquerque, and A. Thiaville, An introduction to micromagnetics in the dynamic regime, \nin Spin dynamics in confined magnetic structures I, edited by B. Hillebrands and K. Ounadjela, Springer, \nBerlin, 2002, vol. 83 of Topics in applied physics.  \n \nFIGURES \n \n \n \nFig. S1: Magnetic hysteresis loops measured by SQUID magnetometry with magnetic field Hext applied in \ndirection [2-1-1] at several sample temperatures. The saturation magnetization obtained from SQUID \nmagnetometry measurement at room temperature is roughly Ms = 95 kA/m , assuming a YIG layer thickness \nof 50 nm. \n \n18 \n  \n \nFig. S2: (a) Ferromagnetic resonance spectra measured at room temperature for several different external \nmagnetic field magnitudes µ0Hext from 0 to 540 mT applied in the sample plane. Resonance peaks were \nfitted by Eq. (S1) and the obtained resonance frequencies and  linewidths are plotted as points  in panels \n(b) and (c), respectively. The lines correspond to fit by Kittel formula [Eq. (S2)], which enables to evaluate \neffective magnetization Meff = 94.9 kA/m and Gilbert damping parameter of   = 0.001. \n \n19 \n Fig. S3: Removal of slowly varying background from time-resolved magneto-optical signals. The red dots \ncorrespond to as-measured signals, line indicates the polynomial background that is subtracted from the \nraw signals. Black dots show the signal after background subtraction, black line representing the fit by Eq. \n(1) of the main text. The data were taken at external field of 0Hext = 300 mT, temperature 20 K and pump \nfluence I = 140 J/cm2.. \n \n \nFig. S4: (a) Dependence of the amplitude A of oscillatory magneto-optical signal on the sample tilt \n(different field angles of magnetic field H ) measured in YIG/Au sample . 0Hext = 300 mT, temperature T \n= 20 K and pump fluence I = 150 J/cm2.. (b) Pump intensity dependence of A measured  for YIG/Au \nsample at H =21° (red points), the same dependence recalculated to correspond to H =40 (blue \npoints) where A(I) was measured for YIG/Pt sample (green points); T = 20 K. \n \n \nFig. S5: (a) Temperature dependence of amplitude of the time-resolved magneto-optical signals measured \nfor external field 0Hext = 300 mT applied at an angle H = 30°. (b) Temperature dependence of saturation \nmagnetization Ms obtained from Ref. 32. \n \n"    },    {        "title": "2111.03233v1.Giant_oscillatory_Gilbert_damping_in_superconductor_ferromagnet_superconductor_junctions.pdf",        "content": "Giant oscillatory Gilbert damping in \nsuperconductor/ferromagnet/superconductor junctions  \n \nAuthors  \nYunyan Yao1,2†, Ranran Cai1,2†, Tao Yu3, Yang Ma1,2, Wenyu Xing1,2, Yuan Ji1,2, Xin -Cheng \nXie1,2,4,5, See -Hun Yang6*, and Wei Han1,2* \nAffiliations   \n1International Center for Quantum Materials, School of Physics, Peking University, Beijing \n100871, China.  \n2Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.  \n3Max Planck Institute for the Structure and Dynamics of Matte r, 22761 Hamburg, Germany  \n4CAS Center for Excellence in Topological Quantum Computation, University of Chinese \nAcademy of Sciences, Beijing 100190, P. R. China  \n5Beijing Academy of Quantum Information Sciences, Beijing 100193, P. R. China  \n6IBM Research - Almaden, San Jose, California 95120, USA   \n†These authors contributed equally to the work  \n*Correspondence to: weihan@pku.edu.cn (W.H.); seeyang@us.ibm.com (S.H.Y.) . \n \n \nAbstract   \nInterfaces between materials with differently ordered phases present unique opportunities for \nexotic physical properties, especially the interplay between ferromagnetism and superconductivity \nin the ferromagnet/superconductor heterostructures. The investig ation of zero- and π-junctions has \nbeen of particular interest for both fundamental physical science and emerging technologies.  Here, \nwe report the experimental observation of giant oscillatory Gilbert damping in the superconducting \nNb/NiFe/Nb junctions wi th respect to the NiFe thickness. This observation suggests an \nunconventional spin pumping and relaxation via zero-energy Andreev bound states that exist only \nin the Nb/NiFe/Nb π-junctions, but not in the Nb/NiFe/Nb zero-junctions. Our findings could be \nimportant for further exploring the exotic physical properties of ferromagnet/superconductor \nheterostructures, and potential applications of ferromagnet π-junctions in quantum computing, \nsuch as half -quantum flux qubits.  2 \n  \nOne sentence summary: Giant oscillat ory Gilbert damping is observed in \nsuperconductor/ferromagnet/superconductor junctions with varying the ferromagnet thickness.  \n \n \nIntroduction  \nThe interplay between ferromagnetism and superconductivity  has induced many exotic and \nexciting physical properties in ferromagnet (FM)/superconductor (SC) heterostructures (1-3). Of \nparticular interest is the unconventional π-phase ground state SC/FM/SC junction  that might be \nrealized for cert ain FM thicknesses arising from the quantum intermixing of the wave functions \nbetween spin -singlet Cooper pairs in SC and spin -polarized electrons in FM (1, 3, 4). At the FM/SC \ninterface, a Cooper pair moving into the FM will ha ve a finite center -of-mass momentum, resulting \nin the oscillation of the  real part of  superconducting order parameter  (Re {Ψ})  with respect to the \nFM thickness  (Fig. 1A ) (1, 5, 6). Depending on the FM thicknesses, the Cooper pair wavefunctions \nin the two superconductors on either side of the FM can have a phase difference from zero or π, \nforming so -called zero-junctions with positive Josephson coupling (Fig. 1B ) or π-junctions with  \nthe negative Josephson coupling  (Fig. 1C) . The FM π-junctions can be used for quantum \ncomputing applications (7, 8), as half quantum flux qubits (9). Due to the scientific and technical \nimportance, the research on the FM π-junctions has been active for the last tw o decades (6, 10-13). \nPrevious experimental studies have demonstrated the switching between zero- and π-junctions in \nSC/FM/SC structures by varying the temperature and the FM thickness (11, 14-17). These reports \nmainly focus on the electrical properties of the FM zero- and π-junctions. Recently, dynamic spin \ninjection into SCs has attracted considerable interest  both the exper imentally  (18-21) and \ntheoretically (22-26). However, the spin -dependent properties in FM zero- and π-junctions have \nnot been explored yet. The investigation of the spin -dependent properties requires the spin current \nprobes, such as the dynamical spin pumping (27). Furthemore, for the application of the FM π-\njunctio ns in quantum computing technologies (9), the magnetization/spin dynamic properties are \nextremely important to be studi ed. \nHere, we report the experimental observation of giant oscillatory Gilbert damping in the \nsuperconducting Nb/NiFe/Nb junctions with respect to the NiFe thickness, which can be 3 \n qualitatively explained by the different spin pumping efficiency via the Andr eev bound states \n(ABS) of Nb/NiFe/Nb zero- and π-junctions. Using a minimal model based on the ABS, we show \nthat an unconventional spin pumping into the zero -energy ABS  penetrat ed into SCs  could occur s \nonly for the π-junctions, which can lead to the oscillatory Gilbert damping as a function of the \nNiFe thick ness. \nResults  \nFigures 1D and 1E  show the schematic s of the spin pumping, magnetization dynamics, and \nenhanced Gilbert damping in the  SC/FM/SC  zero- and π-junction s. Spin pumping refers to the \nspin-polarized current injection to non -magnetic materials from a FM with precessing \nmagnetization around its ferromagnetic resonance (FMR) conditions (28, 29). In FM and its \nheterostructures, the Gilbert damping (\n) characterizes the magnetization dynamics , as described \nby the Landau -Lifshitz -Gilbert formula with an additiona l Slonczewski -torque term (30-32): \n 𝑑𝒎\n𝑑𝑡=−𝛾𝒎×𝑯𝒆𝒇𝒇+𝛼𝒎×𝑑𝒎\n𝑑𝑡+𝛾\n𝑀𝑠𝑉(ℏ\n4𝜋𝑔↑↓𝒎×𝑑𝒎\n𝑑𝑡) (1) \nwhere 𝒎=𝑴/|𝑴| is the magnetization unit vector,  𝛾 is the gyromagnetic ratio, 𝑯𝒆𝒇𝒇 is the total \neffective magnetic field, 𝑀𝑠=|𝑴| is the saturation magnetization, and 𝑔↑↓ is the interface spin  \nmixing conductance. The pumped spin current from FM into SCs can be expressed by Js =\nℏ\n4𝜋𝑔↑↓𝒎×𝑑𝒎\n𝑑𝑡 (29). The spin pumping into the SCs  give rise  to an enhanced Gilbert damping \nconstant  that is proportional to the spin pumping current  (αsp~ J s) (29). Fig. 1E illustrates the \npumped spin current mediated by the zero -energy ABS inside the superconducting gap  in π-\njunctions , which will be discussed later in details.  While for a zero-junction, the pumped spin \ncurrent is mediated by the ABS near the superconducting gap (Fig. 1D). The ABS can be formed \nwithin the FM layer and then extended into the interface of SCs with the superconducting coherent \nlength scale (33, 34).   \nThe SC/FM/SC junctions consist of a NiFe (Ni 80Fe20) layer (thickness: ~ 5 - 20 nm) \nsandwiched by two Nb layers (thickness: 100 nm) grown by magnetron sputt ering (see Methods \nand f ig. S1). To maximize the integrity of samples for a systematic study, more than tens of \nsamples are grown in each run via rotation mask technique in a sputtering system, which is the \nsame as in the previous study of the oscillatory exchange coupling in mag netic multilayer 4 \n structures (35). The Gilbert damping  and spin pumping  are measured by the ferromagnetic \nresonance  (FMR)  technique (see Methods for details) .  \nAbove the TC of Nb, spin pumping in the Nb/NiFe/Nb junctions leads to the spin accumulation \nin Nb near the interface, which can be described by the spin -dependent chemical potentials, as \nillustrated in Fig. 2 A. The Gilbert damping of NiFe in the Nb/NiFe/Nb junctions is determined \nfrom the microwave frequen cy-dependent FMR spectra ( fig. S2). A typical FMR curve with the \nLorentzian fitting is shown in Fig. 2 B, from which the  half linewidth (ΔH) can be obtained. The \nGilbert damping can be extracted from the best linear -fitting curve of ΔH vs. f (Fig. 2 C). Figure \n2D shows the NiFe thickness dependence of the Gilbert damping in the Nb/NiFe/Nb junctions \nmeasured at T = 10, 15, and 20 K, respectively. Interestingly,  an oscillating feature of the Gilbert \ndamping is observed as a function of 𝑑NiFe in the region of 𝑑NiFe < ~15 nm. This oscillating \nbehavior can be attributed to the quantum -interference effect of angular momentum transfer \nbetween the local precessing magnetic moment and conduction electrons in thin NiFe that was \ntheoretically predicted by Mills (36), but has not been experimentally reported yet. Above TC, the \ncontinuous energy bands of Nb, similar to the normal met al in the Mills theory, overlap with both \nspin-up and spin -down bands of NiFe at the interface, thus allowing the conducting electrons in \nNiFe to flip between the spin -down and spin -up states. As illustrated in the inset of Fig. 2 D, one \nspin-down electron scatters with the local magnetic moment and then flips to the spin -up \npolarization, giving rise to the angular momentum transfer between the spin -polarized electrons \nand the magnetic moment. Besides the change of angular momentum, the momentum of the \nelect ron also changes ( ∆𝑘), due to different Fermi vectors for spin -up (𝑘𝐹↑) and spin -down ( 𝑘𝐹↓) \nelectrons with exchange splitting (Fig. 2 A). When the NiFe layer is thin enough to become \ncomparable with 1\n∆𝑘, quantum -interference effect of the spin -polariz ed electrons shows up, which \ngives rise to the oscillating spin -transfer torque to the NiFe. When the NiFe thickness is \n2𝑛𝜋/[𝑘F↑−𝑘F↓] (n is an integer), the matching of the quantum levels between the spin -up and \nspin-down electrons in NiFe induces smaller Gilbert damping. On the other hand, when the NiFe \nthickness is (2n+1)𝜋/[𝑘F↑−𝑘F↓], a larger Gilbert damping is induced. Consequently, th e \nGilbert damping in the Nb/NiFe/Nb structures oscillates with a period of 2𝜋/[𝑘F↑−𝑘F↓] \n(Supplementary Materials S1) . Experimentally, an oscillating period ( λ) of ~ 1.8 nm is identified 5 \n (see the red dashed arrow in Fig. 2 D). At T = 50 K, the oscillating f eature disappears since the \nquantum -interference effect is smeared by thermal excitations ( fig. S3) . \nNext, we investigate the spin pumping and spin transfer torque of the Nb/NiFe/Nb junctions \nin the superconducting states below TC with a superconducting gap (Fig. 3 A). TC in the \nNb/NiFe/Nb junctions is obtained from typical four -probe resistance measurement as a function \nof the temperature. A typical temperature -dependent resistance curve measured on the Nb/NiFe \n(12 nm)/Nb junction is shown in Fig. 3b, indicating the TC of ~ 8.6 K. As dNiFe changes, TC of the \nNb/NiFe/Nb junctions exhibits little variat ion between ~ 8.4 and ~ 8.9 K ( fig. S4). Similar to the \nnormal states of Nb, the Gilbert damping below TC is also obtained from the be st linear -fitting \nresult of the  half linewidth vs. frequency ( fig. S5). During the FMR measurement, TC varies a little \n(< 1 K) ( Fig. S 6). As the temperature decreases, 𝛼 decreases abruptly from ~ 0.012 to ~ 0.0036 \nacross the TC (fig. 3C), which indicates the decrease of spin current injected into Nb due to the \nformation of superconducting gap below TC. This observation is consistent with previous reports \non spin pumping into SCs where the spin current is mediated by Bogoliubov quasiparticles (18, \n19, 37). As the tem perature decreases far below the TC, the quasiparticle population dramatically \ndecreases, leading to reduced spin pumping and Gilbert damping.  \nRemarkably, the oscillating amplitude of the Gilbert damping of the Nb/NiFe/Nb junctions as \na function of the NiF e thickness is dramatically enhanced as the temperature decreases into the \nsuperconducting states of Nb (Fig. 3 D). At T = 4 K, the oscillating magnitude of the Gilbert \ndamping constant is ~ 0.005 for the first three oscillations, which is comparable to the  background \nvalue of ~ 0.006. The obtained Gilbert damping values are not affected by thermal cycles, and the \nlarge oscillating feature has been confirmed on a different set of samples. Such a giant oscillation \nof the Gilbert damping cannot be explained by  spin pumping of Bogoliubov  quasiparticle -\nmediated spin current in SCs. Since as the temperature decreases, the population of the Bogoliubov  \nquasiparticles monotonically and rapidly decreases with an increase of the SC gap, which would \nlead to lower Gilber t damping and also smaller oscillation compared to the normal states. Note \nthat the oscillating period of the Gilbert damping at T = 4 K is the same as that at T = 10 K that is \nsupposed to be 2𝜋/[𝑘F↑−𝑘F↓] due to the quantum interference effect . Such oscillating period of \n2𝜋/[𝑘F↑−𝑘F↓] is the also same as that of the zero- and π-phase ground states transitions in FM \nJosephson devices, which is equal to the coherence length in NiFe film of 2𝜋/[𝑘F↑−𝑘F↓] in the 6 \n ballistic regime (1, 11, 17), and √ℏ𝐷𝑑𝑖𝑓𝑓 ∕𝐸𝑒𝑥  in the diffusive regime ( 𝐷𝑑𝑖𝑓𝑓 is the diffusion \ncoefficient, and 𝐸𝑒𝑥 is the exchange energy ). The observed oscillating period of ~ 1.8 nm in our \nstudy is similar to the zero - 𝜋 oscillating period measured in the NiFe Josephson junctions in the \ndiffusive regime reported previously (11, 17).  \nThe Gilbert damping difference  (∆𝛼) between the zero- and 𝜋-junctions is extracted as a \nfunction of NiFe thickness, as shown in Fig. 4 A. We assume the larger Gi lbert damping for the 𝜋-\njunctions and smaller value s for the zero-junctions, which will be discuss ed later in details . The \nthickness -dependent  Gilbert damping of  the zero- and 𝜋-junctions  are expected to both behave as \nα ~ 1 ⁄ dNiFe (29). Hence, we can treat them separately, as illustrated by the guide lines in the inse t \nof Fig. 4A, and ∆𝛼 is obtained by subtracting the fitted 1/d curve for the expected  zero-junctions  \n(black  dashed line) . Clearly, there is a pronounced oscillating feature of ∆𝛼 for the Nb/NiFe/Nb \njunctions with NiFe thickness from ~ 5 nm to ~ 11 nm. When the NiFe thickness is above ~ 11 \nnm, the oscillating feature of the Gilbert damping is largel y suppressed compared to thinner NiFe \njunctions. This feature might be associated wi th the strong Josephson coupling for thin NiFe \njunctions and the exponential decaying of the Josephson coupling as the NiFe thickness increases \n(11, 17). To confirm this, the Jose phson junctions are fabricated using the shadow mask technique, \nand a Josephson coupling is observed from the Nb/NiFe (5 nm and 10 nm)/Nb junctions \n(Supplementary Materials and fig. S7). \nDiscussion  \nLet us discuss the physical mechanism that induces the giant oscillating Gilbert damping in the \nfollowing. Apart from the spin pumping via ABS  discussed above (Fig. 1 D and 1E ), the spin \ncurrent in SCs can also be mediated by Bogoliubov quasiparticles  (fig. S8A) (18, 19, 22, 23, 38), \nspin-triplet pairs  (fig. S 8B) (3). Regarding Bogoliubov  quasiparticles, they populate around the \nedge of superconducting gap at elevated temperatures close to TC (39). As shown both theoretical \nand experimental studies, the enhanced Gilbert damping in the SC/FM/SC heterostructures \nhappens around TC (18, 19, 22, 23, 38). As the temperature decreases down to 0.5 TC, the \nBogoliubov quasiparticles are mostly frozen out, for which the spin pumping is forbidden that will \nno longer contribute to the enhanced Gilbert damping. Hence, the Bogoliubov quasi particles are \nvery unlikely to account for our experimental results. Regarding  the spin-triplet pairs, it has been \nshown in previous studies that the spin -triplet current under FMR conditions and spin triplet 7 \n correlations would be different for zero- and 𝜋-junctions (4, 38, 40, 41), which might result in \ndifferent Gilbert damping theoretically. However, in our study, there are not spin sinks adjacent to \nthe Nb layers , thus not allowing the spin -triplet Cooper pairs to be relaxed in the Nb. This is \ndiffe rent from previous report on the Pt/SC/FM/SC/Pt heterostructures (20), where the Pt is used \nas the spin sink. Experimentally, as the temperature below TC, the Gilbert damping exhibits a \nmonotonic decrease for the Nb/NiFe/Nb heterostructures (Fig. 3 C), which is different from the \nenhanced Gilbert damping due to spin -triplet pairs  (20). Furthermore, no Josephson current in the \nNb/NiFe/Nb heterostructures is observed i n Nb/NiFe (30 nm)/Nb junction ( fig. S7), which  \nindicates the absence of long -range spin -triplet Josephson coupling. Both these experimental \nresults indicate that the contribution from the spin -triplet pairs is not significant to the enhanced \nGilbert damping in the superconducting Nb/NiFe/Nb junctions . \nTo our best understanding, the most reasonable mechanism is the spin pumping via the ABS, \nwhich can qualitatively  describe our experimental observation. Previous studies have \ndemonstrated that the energy of ABS inside the superconducting gap depends on the  \nsuperconducting -phase (42, 43). For the FMR measurement under  open -circuit ed conditions , the \ninversion symmetry of the current -phase  (𝜑) relationships is preserved (43-45). For 𝜋-junctions, \nthere is a 𝜋-phase shift in the current -phase relationship curves compared to zero-junctions , i.e., \nthe properties of 𝜑 = 0 of a 𝜋-junction is the same as those of 𝜑 = 𝜋 of a zero-junction. Since this \n𝜋-phase shift is already taken into account by the FM exchange field , the ABS energy of the 𝜋-\njunctions  can be obtained at 𝜑 = 0 in the ground states , which is similar to that of 𝜑 = 𝜋 of zero-\njunctions .  \nFor 𝜋-junctions, ABS is located around the  zero-energy inside of the superconducting gap (Fig. \n1D). The ABS could penetrate into the superconducting Nb films with scale of superconducting \ncoherent length (~ 30 nm), which is evanescent to dissipate the spin angular moment um (25, 26, \n44). As shown in Fig. 4 B, the transfer efficiency of spin angular momentum via the zero -energy \nABS can lead to an enhanced Gilbert damping . Whileas, for  zero-junctions, the distribution of the \nABS is near the edge of the superconducting gap (Fig. 1C) , thus, the spin pumping effic iency is \nsuppressed due to the reduced population of the ABS at low temperatures  (Fig. 4 C). Furthermore , \nthe oscil latory energy levels of the ABS  between the zero- and 𝜋-junctions is also consistent with \nthe density of states (DOS)  oscillating in supercon ductors  between the zero- and 𝜋-junctions (1, 6, 8 \n 38, 44, 46). In consequence, as the NiFe thickness increas es, the oscillatory spin pumping \nefficiency via ABS  at the FM/SC interface  (or DOS in SC s) gives rise to the oscillatory Gilbert \ndamping.  We have proposed a simplified model for the case of ideal transparency of electrons \n(Supplementary Materials S2 and f ig. S9 ). For the less transparency cases, i.e., in diffusive regime , \n(42, 43), the energy level s of the ABS in  𝜋-junctions  locates away from zero -energy, but they are \nstill much smaller than those  of the ABS in zero-junctions. Actually, the similar oscillating \nbehaviors of ABS (or DOS) can be preserved in the diffusive regime  (6, 46). Hence, an oscillating \nspin pumping efficiency would also be expected in the diffusive regime, which could lead to the \noscillating Gilbert damping  observed in our experiment . To fully understand the experimental \nobservation of the oscillatory Gilbert damping and the detailed spin relaxation process in the \ndiffusive regime, further theoretical studies are needed.  \nFurthe rmore, the control samples of bilayer Nb/NiFe heterostructur es do not exhibit the large \noscillatory feature for the Gilbert damping as the NiFe thickness varies  at T = 4 K  (fig. S10), which \nfurther presents the important role of phase difference across NiFe  in the large the oscillatory \nGilbert damping  observed in t he trilayer  Nb/NiFe/Nb heterostructures .  \nIn conclusion, giant oscillatory Gilbert damping is observed in the superconducting \nNb/NiFe/Nb junctions with respect to the NiFe thickness. To our best knowledge, neither the \nBogoliubov quasiparticles, nor the spin -triplet pairs are relevant to this observation.  The most \npossible explanation for such giant oscillatory Gilbert damping could be related to the different \nABS energy levels and the DOS at the NiFe/SC interface in zero- and π- junctions. To full y \nunderstand these results, further theoretical studies are needed. Looking forward, our experimental \nresults  might pave the way for controlling the magnetization dynamics  by the superconducting \nphase  in a FM Josephson junction  in the SQUID setup, and could be important potential \napplications of ferromagnet π-junctions in quantum computing, such as half -quantum flux qubits.  \n \nMaterials and Methods  \nMaterials growth  \nThe SC/FM/SC heterostructures consisting of Nb (100 nm) and Ni 80Fe20 (NiFe ; ~ 5 - 20 nm) were \ngrown on thermally oxidized Si substrates in a d.c. magnetron sputtering system with a base \npressure of ∼1× 10−8 torr. To systematically vary the NiFe thickness that is crucial for the quantum -9 \n size effect, we adopted the rotating multi -platter technique that allows us to grow dozens of \nNb/NiFe/Nb samples in each run  (35). The thickness of the Nb layer is fixed to be ~100 nm that is \nmuch larger than the spin diffusion length of Nb  (20, 47). After the growth, a thin Al 2O3 layer (~ \n10 nm) was deposited in situ  as a capping layer to avoid sample degradation against air/water \nexposure. The crystalline properties of Nb/NiFe/Nb heterostructures were chara cterized by X -ray \ndiffraction (fig. S1A ) and high -resolution cross -section al tra nsmission electron microscopy (f ig. \nS1B) using a 200 -kV JEOL 2010F field -emission microscope. The NiFe thickness is determined \nby the growth rate that is calibrated by TEM measurement, where the uncertainty of the NiFe \nthickness is obtained to be  smaller than ~ 0.8  nm (f ig. S1B).  The resistivity of the NiFe layers \n(thickness: 5 - 20 nm) is ranging from 60 to 35 μΩ ·cm, which corresponds to the  mean free path \nbetween 2.3 and  3.9 nm.  \n \nFerromagnetic resonance measurement.   \nThe spin pumping of Nb/NiFe/Nb heterostructures was characterized via FMR  using the coplanar \nwave guide technique connected with a vector network analyzer (VNA; Agilent E5071C) in the \nvariable temperature insert of a Physical Properties Measurement System (PPMS; Quantum \nDesign)  (19). The FMR spectra were characterized by measuring the amplitudes of forward \ncomplex transmission coefficients (S 21) as the in -plane magnetic field decreases from 4000 to 0 \nOe under the microwave power of 1 mW. The typical FMR results measured on the Nb/NiFe (12 \nnm)/Nb het erostructures are shown in the fig. S2A  (T = 10 K) and fig. S 5A (T = 4 K). Weaker \nFMR signals are observed in the superconducting states compared to the normal states.  \nThe half linewidth ( ∆𝐻) can be obtained by the Lorentz fitting of the magnetic field -dependent \nFMR sign al following the relationship ( figs. S2B  and S4 B): \n 𝑆21∝𝑆0(∆𝐻)2\n(∆𝐻)2+(𝑯−𝑯𝒓𝒆𝒔)2 (3) \nwhere 𝑆0 is the coefficient for the transmitted microwave power,  𝑯 is the external in -plane \nmagnetic field, and 𝑯𝒓𝒆𝒔 is the resonance magnetic field.  The Gilbert damping constant (α) can be \nobtained from the slope of the best linear -fitting results of the ∆𝐻 vs. the microwave frequency ( f) \n(48-51): 10 \n  ∆𝐻=∆𝐻0+(2𝜋𝛼\n𝛾)𝑓 (4) \nwhere ∆𝐻0 is the zero -frequency line broadening that is related to the inhomogeneous properties, \nand  𝛾 is the gyromagnetic ratio. From the best linearly fits of the ∆𝐻 vs. f results measured on the \ntypical Nb/Py ( 12 nm)/Nb sample (red lines in f igs. S2 C and S 5C), 𝛼 is determined to be 0.012 \nand 0.0054 at T = 10 and 4 K, respectively. A larger zero -frequency line broadening ∆𝐻0 is \nobserved for the superconducting state compared to the normal state of Nb/Py/Nb heterostructures, \nwhich could be attributed to Meissn er screening effect and the formation of trapped magnetic \nfluxes in Nb  (51). The thickness dependent ∆𝐻0 is shown in f ig. S1 1C, and no obviously \noscillatory behaviors  are observed . \nThe effective magnetization and the gyromagn etic ratio can be fitted via the in -plane Kittel \nformula (51): \n 𝑓𝑟𝑒𝑠=𝛾\n2𝜋√(𝐻𝑟𝑒𝑠+ℎ)(𝐻𝑟𝑒𝑠+ℎ+4𝜋𝑀𝑒𝑓𝑓),                                      (5)  \nwhere 𝑓𝑟𝑒𝑠 and 𝐻𝑟𝑒𝑠 are the resonant microwave frequency and magnetic field respectively , \n4𝜋𝑀𝑒𝑓𝑓 is the effective saturated magnetization , and ℎ is the shifted magnetic field induced by \nsuperconducting proximity effect. The thickness -dependent gyromagnetic ratio and eff ective \nmagnetization can be found in fig. S1 1A and fig. S1 1B. Both parameters do not exhibit  any \noscillatory features  as the Gilbert damping does (Fig . 4A), which demonstrates that the oscillatory \nGilbert damping is not caused by any unintentional experimental error . \n \nSuperconducting transition temperature measurement.  \nThe superconducting transition temperature ( TC) of the Nb/NiFe/Nb heterostructures was \ndetermined via the zero -resistance temperature measured by four -probe method in a P PMS using \nstandard a.c. lock -in technique at a low frequency of 7 Hz. The TC of Nb (100 nm)/NiFe/Nb (100 \nnm) heterostructures exhibits little variation as a f unction of the NiFe thickness ( fig. S 4). It is \nnoticed that the FMR measurement can affect the TC a little (< 1 K), as shown in fig. S6.  \n 11 \n Supplementary Materials  \nSupplementary Materials and Methods  \nfig. S1 . The crystalline properties of the Nb/NiFe/Nb heterostructures.  \nfig. S2. Gilbert dampin g measurement of Nb/NiFe/Nb heterostructures at T = 10 K.  \nfig. S3. NiFe thickness dependence of Gilbert damping at T = 50 K.  \nfig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures.  \nfig. S5. Measurement of the Gilbert damping of Nb/ NiFe/Nb heterostructures at T = 4 K.  \nfig. S6. The effect of FMR measurement on the TC of Nb/NiFe/Nb heterostructures.  \nfig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions.  \nfig. S 8. Illustration of magnetization dynamics and spin pumping i n the SC/FM/SC \nheterostructures due to Bogoliubov quasiparticles and equal spin -triplet Cooper pairs.  \nfig. S9. Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at T = 4 K.  \nfig. S10. Gilbert damping of control sample of bilayer Nb /NiFe junctions.  \nfig. S1 1. Thickness dependen ce of  gyromagnetic ratio, effective magnetization and \ninhomogeneous half -linewidth.  \n \n \nReferences and Notes:  \n1. A. I. Buzdin, Proximity effects in superconductor -ferromagnet heterostructures. Rev. 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T.Y. is \nfinancially supported by DFG Emmy Noether program (SE 2558/2 -1).  \n \nAuthor contributions:  W.H. conceived and supervised the project. Y.Y. and R.C. \nperformed the ferromagnetic resonance measurements. Y.Y. and Y.M. performed X -ray \ndiffraction measurements. T.Y. performed the theor etical calculations. S.H.Y. synthesized \nthe Nb/NiFe/Nb heterostructures. Y.Y. and W.H. wrote the manuscript with the \ncontribution from all authors. All the authors discussed the results.  \n \nCompeting interests:  The authors declare no competing interests.  \n \nData Availability: All data needed to evaluate the conclusions in the paper are present in \nthe paper and/or the Supplementary Materials.  \n  16 \n  \nFig. 1. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC \nheterostructures.  (A) The oscillatory real part of the superconducting order parameter (Re {Ψ}, \ngreen curve ) penetrated into FM leads to the zero-state and  𝜋-state. (B) The symmetric order \nparameter in the zero-junction s. (C) The anti -symmetric order parameter in the 𝜋-junction s. (D-E) \nSpin pumping via t he ABS  in SCs  in the zero- and 𝜋-junction s. M and 𝛼FM are the magnetization \nand Gilbert damping of the FM layer itself, and 𝛼sp is the enhanced Gilbert damping, which arises \nfrom the spin dissipation in SC layers during the spin pumping process.  \n \n17 \n  \nFig. 2. Oscillatory Gilbert damping of the Nb/NiFe/Nb heterostructures above TC. (A) The \nillustration of spin pumping into the normal states of Nb layers and the electronic band structure \nof NiFe with different spin-up and spin -down Fermi vectors (𝑘F↑ and 𝑘F↓) due to the exchange \nsplitting (2 𝐸𝑒𝑥). The spin pumping gives rise to the spin accumulation in the Nb layers, indicated \nby the spin -split chemical potential ( 𝜇↑ and 𝜇↓). (B) A typical FMR  curve measured with f = 12 \nGHz (black circles) and the Lorentzian fitting curve (red line) measured on Nb/Py (12 nm)/Nb . \nΔH is the half line width at the half maximum of FMR signal. (C) The determination of the Gilbert \n18 \n damping from ΔH vs. f. The red line represents the best linear -fitting curve. (D) The oscillatory \nGilbert damping as a function of NiFe thickness ( dNiFe) measured at T = 10, 15, and 20 K, \nrespectively. The experimental oscillating period ( 𝜆) is marked by the red dashed arrow. The inset: \nIllustration of the quantum -interference effect o f the angular momentum transfer between the local \nmagnetic moment and the spin -polarized electrons . When the NiFe thickness decreases to scale of \n1\n∆𝑘, the quantum -interference effect starts to be significant in the angular momentum transfer  and \nspin pumping into the Nb layers .  \n  \n \n \n \n \n \n \n \n \n \n \n 19 \n            \nFig. 3. Giant oscillatory Gilbert damping in Nb/NiFe/Nb heterostructures below TC. (A) The \nillustration of electronic band structures of Nb in the normal and superconducting states. (B) The \ndetermination of TC via zero -resistance temperature measured on the typical Nb/NiFe (12 nm)/Nb \nheterostructures. (C) Temperature dependence of Gilbert d amping of the typical Nb/NiFe (12 \nnm)/Nb heterostructures. (D) The oscillatory Gilbert damping as a function of the NiFe thickness \nin the Nb/NiFe/Nb heterostructures measured at T = 10, 7, 5, and 4 K, respectively. The oscillating \nfeature below TC (T = 4 and 5 K) is dramatically enhanced compared to that above TC (T = 10 K).  \n  \n20 \n  \nFig. 4. Physical mechanism of the giant oscillatory Gilbert dam ping in Nb/NiFe/Nb junctions. \n(A) The NiFe thickness dependence of the Gilbert damping difference ( ∆α) between the  \nNb/NiFe/Nb π- and zero-junctions  at T = 4 K . Inset: The Gilbert damping of zero- and π-junctions. \nThe solid balls represent the experimental data, the blue and black dash lines are the guide lines \nfor π- and zero-junctions, respectively. For bo th guide lines, the damping is expected to behave as \nα ~ 1 ⁄ dNiFe. (B-C) Illustration of the spin pumping via the ABS and the enhanced Gilbert damping \nfor Nb/NiFe/Nb π- and zero-junctions, respectively.  The red thick -arrows indicate the pumping \nand relaxation  of the spin current in SCs . \n21 \n Supplementary Materials for  \n \n \nGiant oscillatory Gilbert damping in \nsuperconductor/ferromagnet/superconductor junctions  \n \nAuthors  \nYunyan Yao1,2†, Ranran Cai1,2†, Tao Yu3, Yang Ma1,2, Wenyu Xing1,2, Yuan Ji1,2, Xin -Cheng \nXie1,2,4,5, See -Hun Yang6*, and Wei Han1,2* \n \n \nThis SM file includes : \n⚫ Supplementary Materials and Methods  \n⚫ fig. S1. The crystalline properties of the Nb/NiFe/Nb heterostructures.  \n⚫ fig. S2. Gilbert damping measurement of Nb/NiFe/Nb heterostructures at T = 10 K.  \n⚫ fig. S3. NiFe thickness dependence of  Gilbert damping at T = 50 K . \n⚫ fig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures.  \n⚫ fig. S5. Measurement of the Gilbert damping of Nb/NiFe/Nb heterostructures at T = 4 K.  \n⚫ fig. S6. The effect of FMR measurement on the TC of Nb/NiFe/N b heterostructures.  \n⚫ fig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions.  \n⚫ fig. S8. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC \nheterostructures due to Bogoliubov quasiparticles and equal spin -triplet Cooper pai rs.  \n⚫ fig. S9. Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at \nT = 4 K.  \n⚫ fig. S10. Gilbert damping of control sample of bilayer Nb/NiFe junctions.  \n⚫ fig. S11. Thickness dependence of gyromagnetic ratio, effective magnetization an d \ninhomogeneous half -linewidth . \n \n  22 \n Supplementary Materials and Methods  \nSection 1: Model of oscillating Gilbert damping above TC.  \nThe oscillatory Gilbert damping in normal metal (NM)/ferromagnet (FM)/NM \nheterostructures arising from quantum interference effect is analyzed based on previous theory by \nMills (36). Within the linear response theory, the enhanced Gilbert damping is related to the \ndynamical spin susceptibility ( 𝜒−+(𝛺)) of conduction electrons in a FM,  \n 𝛼sp=𝐽2𝑀𝑠𝑉\n2𝑁2ℏ3𝛾Λ2 (S1) \nwhere Λ2=Im(𝑑𝜒−+(𝛺)\n𝑑𝛺|\nΩ=0). Using one dimensional model,  we obtain   \n Λ2=1\n𝜋2∫𝑑𝑥𝑑 𝑥′Im[𝐺↑(𝑥,𝑥′,𝜖F)]Im[𝐺↓(𝑥,𝑥′,𝜖F)]\nFM (S2) \nwhere 𝐺𝜎(𝑥,𝑥′,𝜖F) is the Green’s function for conduction electrons with 𝜎-spin at the Fermi \nenergy ( 𝜖F). In a FM, 𝐺𝜎(𝑥,𝑥′,𝜖F) is related to the exchange energy.  \n [−ℏ2\n2𝑚𝑑\n𝑑𝑥2−𝜖±𝐸ex]𝐺𝜎(𝑥,𝑥′,𝜖F)=𝛿(𝑥−𝑥′) (S3) \nFor the FM film with a thickness dFM in − dFM/2 < x < dFM/2, the Green’s function satisfies the \nrelation  \n 𝐺𝜎(𝑥,𝑥′,𝜖F)=𝐺𝜎(𝑥′,𝑥,𝜖F)=𝐺𝜎(−𝑥,−𝑥′,𝜖F) (S4) \nHence, the imaginary part of the Green’s function could be expressed by  \n Im[𝐺𝜎(𝑥,𝑥′,𝜖F)]=−π{𝑁F𝜎cos [𝑘F𝜎(𝑥−𝑥′)]+𝑁F𝜎′cos [𝑘F𝜎(𝑥+𝑥′)]}     (S5) \nwhere 𝑘F𝜎=√2𝑚\nℏ2(𝜖F∓𝐸ex) is the Fermi wave -vector in the FM, 𝑁F𝜎 and 𝑁F𝜎′ are equivalent to \nthe density of states and the modulation amplitude of the local density of states, respectively. For \nthe same position of x, the local density of states is equal to  \n 𝑁𝜎(𝑥,𝜖F)=𝑁F𝜎+𝑁F𝜎′cos [2𝑘F𝜎𝑥] (S6) \nSince 𝐸ex is much smaller compared to 𝜖F, the spatial modulation of the local density of states is \nnegligible. The combination of  equations (S2) and (S5) leads to  \nΛ2=∫ 𝑑𝑥𝑑 𝑥′𝑑FM 2⁄\n−𝑑FM 2⁄{𝑁F↑cos[𝑘F↑(𝑥−𝑥′)]}∗{𝑁F↓cos[𝑘F↓(𝑥−𝑥′)]}        (S7) 23 \n                =2𝑁F↑𝑁F↓{1\n(𝑘F↑−𝑘F↓)2sin2[𝑘F↑−𝑘F↓\n2𝑑FM]+1\n(𝑘F↑+𝑘F↓)2sin2[𝑘F↑+𝑘F↓\n2𝑑FM]} \nClearly,  the enhanced  Gilbert  damping  is expected  to oscillate  as a function  of the FM \nthickness  with two periods  of  2𝜋/[𝑘F↑−𝑘F↓] and 2𝜋/[𝑘F↑+𝑘F↓]. For real FM materials,  such as \nNiFe  with (𝑘F↑+𝑘F↓)≫(𝑘F↑−𝑘F↓), the second  term in the equation  (S7) could  be negligible,  \nleaving  only one oscillating  period  of 2𝜋/[𝑘F↑−𝑘F↓]. When  the FM thickness  is equal  to \n2𝑛𝜋/[𝑘F↑−𝑘F↓], a lower  Gilbert  damping  is obtained.  On the other  hands  with FM thickness  of \n(2𝑛+1)𝜋/[𝑘F↑−𝑘F↓], a larger  Gilbert  damping  is obtained.   \nSection  2: Calculation  of the enhanced  Gilbert  damping  in Nb/NiFe/Nb  by spin pumping  via \nAndreev  bound  states  (ABS).   \nAs the reciprocal process of the spin transfer torque, conventional spin pumping is achieved \nby the magnetization torques provided by the driven quasiparticle carriers (29, 52-54), which are \nthe electrons in the normal metals. In SC/FM heterostructures, however, the quasiparticle carriers \ncan be either Bogoliubov quasiparticles or ABS (42), which lie above and within the \nsuperconducting gaps, respectively. Therefore, it is desirable to formulate and estim ate the \ncontribution to the spin pumping via the ABS (55), when the temperature is much smaller than the \nsuperconducting critical temperature.   \nWithout loss of generality, we start the analysis from a left -propagating electron of energy 𝜀 \nand spin 𝜎 = {↑, ↓} = {+, −,}. When the Zeeman splitting J is much smaller than the Fermi energy \nEF, it has momentum  \n 𝑘𝜎=𝑘𝐹+(𝜀+𝜎𝐽)/(ℏ𝜈𝐹),                                                             (S8) \nwhere 𝜈𝐹 is the Fermi velocity of the electron. When one electron goes from the FM to the SCs, it \nis reflected as a hole by the Andreev reflection at the right FM/SC interface; this hole has a phase \nshift χ=−arccos (𝜀/Δ) with respect to the electron (56), where Δ is the superconducting gap . \nSimilarly, when a hole goes from the metal to the superconductor at the left FM/SC interface, an \nelectron can be reflected. With a proper energy, the Andreev reflections can form a closed path, as \na result of which the ABS forms. This requires that the phase accumulated in the reflections \nsatisfies the Sommerfeld quantization condition, i.e. in the ballistic regime,  24 \n     𝜀𝐿\nℏ𝜈𝐹+𝜎𝐽𝑑NiFe\nℏ𝜈𝐹−arccos (𝜀\n∆)=𝑛𝜋+𝜑\n2,                                 (S9) \nwhere 𝜑 is the phase difference between the two superconductors , dNiFe is the thickness of the FM \nlayer and n is an integer. Since ℏ𝜈𝐹/∆  ≥ 100 nm with 𝜈𝐹 = 2.2 ×  105 m/s and ∆ = 1 meV at T = 4 \nK in our experiment (17, 57, 58), dNiFe < 19 nm << ℏ𝜈𝐹/∆ such that the first term in Eq. (S9) can \nbe safely disregarded. For the FMR measurements with open -circuited configuration, the junctions \nalways stay in the ground states (43-45). For 𝜋-junctions, there is a 𝜋-phase shift in the current -\nphase relationship curves compared to zero-junctions , i.e., the properties of 𝜑 = 0 of a 𝜋-junction \nis the same as those of 𝜑 = 𝜋 of a zero-junction. Since this 𝜋-phase shift is already taken into \naccount by the FM exchange field , the ABS energy of the 𝜋-junctions  can be obtained at 𝜑 = 0 in \nthe ground states , which is similar to that of 𝜑 = 𝜋 of zero-junctions . Hence, the energy of the ABS \ncan be described by   ε0=±∆cos (𝐽𝑑NiFe\nℏ𝜈𝐹) for ideal case with perfect transparency of \nelectrons/holes.  \nIn reality, the interfacial scattering and transport conditions (ballistic or diffusive regimes) of \nFM could affect th e energy of the ABS. Following previous studies (42, 59), a transmission \ncoefficient ( D) could be introduced to describe this issue , which is close to unity in the ballistic \nregime but can also be large in the  diffusive regime  with an ideal transparency at the interface (43, \n60, 61). In this work, we focus on the ideal cases with perfect transparency of electrons/holes . The \nenergy of the ABS oscillates from the ed ge of the superconducting gap to the zero -energy with \nrespect to the FM thickness (fig. S9A).   \nThe pumped spin current reads (29, 52-54), \n𝐉𝑠(𝑡)=ℏ\n4𝜋𝑔eff↑↓𝒎×𝑑𝒎\n𝑑𝑡,                                                          (S10)  \nwhere m is the magnetization unit vector, and we define the effective mixing spin conductivity \n𝑔eff↑↓ at the finite temperature via the zero -temperature one 𝑔↑↓ by (53, 55)  \n  𝑔eff↑↓=𝑛0∫𝑑𝜀𝑑𝑓(𝜀)\n𝑑𝜀Re[𝑔↑↓(𝜀)].                                              (S11)  \nHere, n0 is the number of the conduction channel that roughly corresponds the conduction electron \ndensity at the interface and 𝑓(𝜀)=1/{exp [𝜀/(𝑘𝐵𝑇)]+1} is the Fermi -Dirac distribution of \nelectron at the temperature T. Importantly, in the ballistic limit Re[𝑔↑↓(𝜀)]=1 when  𝜀=𝜀0; it 25 \n has width ∆𝜀 depending on the FM thickness dNiFe in the ballistic regime or the mean free path 𝑙𝑚 \nin the diffusive regime . By the uncertainty principle, ∆𝜀∆𝑡=2𝜋ℏ, where ∆𝑡=𝑙𝑚/𝜈𝐹 is the \npropagation time of the electron in the junction, leading to  ∆𝜀 ~ 2𝜋ℏ𝜈𝐹/𝑙𝑚. By further \nconsidering the degeneracy due to spin (× 2) and the existence of two interfaces (× 2), we thus can \nestimate  \n    𝑔eff↑↓ ~ 8𝜋𝑛0ℏ𝜈𝐹\n𝑙𝑚𝑑𝑓(𝜀0)\n𝑑𝜀.                                                         (S12)  \nThe pumped spin current carries the angular momentum away from the precessing magnetization \nand hence cause an enhanced Gilbert damping, which is described by  \n𝛿𝛼=2𝛾ℏ2𝜈𝐹\n𝑀𝑠𝑙𝑚𝑑NiFe𝑑𝑓(𝜀0)\n𝑑𝜀 ,                                                      (S13)  \nwhere 𝛾  is electron gyromagnetic ratio and 𝑀𝑠 is the saturated magnetization of the ferromagnet.  \n      We are now ready to estimate the contribution of ABS to  the Gilbert damping at T = 4 K with \nvarying transmission coefficient. We take 𝑛0 =0.5 ×  1016 m−2 following Ref. 44 , 𝑙𝑚~ 3 nm,  𝜈𝐹= \n2.2 ×  105 m/s, J = 400 meV and 𝜇0𝑀𝑠≈1 𝑇 from previous experimental results (17). With \nsuperconducting gaps ∆ ≈ 1 meV at T = 4 K for Nb (57, 58), Fig.S8A plots the normalized energy \nof ABS by the superconducting gap at T = 4 K as a function of dNiFe for the ideal transparency case . \nThe oscillation of the Gilbert damping can be resolved by using the FM exchange field -induced \nphase shift of 𝐽𝑑NiFe\nℏ𝜈𝐹 (fig. S9B) . For simplicity, we have disregarded the possible thickness \ndependence of the superconducting gaps and magnetizations. To  be noted, our theoretical \nestimation is based on a simplified model that assumes D = 1. For the diffusive regime or the case \nof non -perfect transparency of electrons at the interface  (42, 43), similar oscillating behaviors of \nABS (or DOS) in the SCs can also be preserved. For example, the oscillating ABS (or DOS) in the \nSCs have been shown to exist in the diffusive regime theoretically (6, 46), and indeed, the zero to \n𝜋 transitions have been experimentally observed in both the ballistic and diffusiv e regimes from \nthe supercurrent measurements (11, 17). To fully understand the experimental observation of the \noscillatory Gilbert damping in the diffusive regime, further theoreti cal studies are needed.  \n \n \n \n 26 \n Section 3: Measurement of the Josephson coupling in Nb/NiFe/Nb.  \nThe Nb/NiFe/Nb  Josephson devices are fabricated using the shadow mask techniques during \nthe films growth. As shown in figs. S7A and S7B, the Josephson devices have a junction area ( A) \nof ~ 80 μm  × 80 μm, and the other areas are electrically isolated by a 100 nm AlO x layer. The \nJosephson current is measured by standard a.c. lock -in technique. The normalized differential \nresistances (dV/dI) measured on the Nb/NiFe (5 nm)/Nb junction at various temperatures are \nshown in fig. S7C. The critical current ( Ic) is defined as poi nt where the differential resistance \nincreases above the value for the zero -bias current. The normal resistance (R n) is determined to be \nthe saturated value of the normal states of the Josephson coupling measurement. The measured \narea-resistance product (R nA) of ~  5×10−10 Ω𝑚2 is higher than that reported in metallic \nJosephson junction (17, 61), and comparable to that of FM Josephson junction with a thin tunnel \nbarrier (62). This behavior indicates that there is more likely a thin NiFeO x layer (indicated by Fig. \nS8B) in the junction formed during the AlO x growth step in the presence of oxygen gas. As the \ntemperature increases, I c and the characteristic voltage (I cRn) decrease (figs. S7C and S7D). Clear \nJosephson currents are observed on the Nb/NiFe (5 nm)/Nb junction and Nb/NiFe (10 nm)/Nb \njunction (figs. S7E and S78F). And the estimated SC gap energy is ~ 0.9 meV at T = 2 K (1, 43), \nwhich is comparable to the value of ~1.36 meV at T = 0 K estimated from TC of ~ 8.5 K (fig. S4). \nOn the other hand, no Josephson current could be observed in the Nb/NiFe (30 nm)/Nb junction \n(figs. S7E and S7F). The absence of Jo sephson current in Nb/NiFe (30 nm)/Nb junction indicates \nthat there is no long -range spin -triplet Josephson coupling in the Nb/NiFe/Nb heterostructures in \nour experiment.  \n  27 \n  \n \n \nfig. S1. The crystalline properties of the Nb/NiFe/Nb heterostructures. (A) The θ -2θ X -ray \ndiffraction results measured on the typical Nb/NiFe (12 nm)/Nb sample, where Nb (110) and NiFe \n(111) peaks are observed. ( B) High -resolution transmission electron micrographs mea sured on the \ntypical Nb/NiFe (12 nm)/Nb sample. The dashed lines show the interfaces between Nb and NiFe \nlayers. The red bars indicate the deviation of NiFe at the interface.  \n \n \n \n \n \n \n  \n28 \n  \n \n \nfig. S2. Gilbert damping measurement of Nb/NiFe /Nb heterostructures at T = 10 K. (A) The \ntypical FMR  spectra as a function of magnetic field with microwave frequency ( f) of 10, 12, 14, \n16, and 18 GHz, respectively. (B) The typical FMR  spectrum measured with f = 12 GHz (black \ncircles) and the Lorentz fi tting curve (red line). ΔH is the half linewidth of the FMR signal. (C) \nThe determination of the Gilbert damping from ΔH vs. f. The red line indicates the best linear -\nfitting curve. These results are obtained on the typical  Nb/Py (12 nm)/Nb sample.  \n  \n29 \n  \n \n \n \nfig. S3. NiFe thickness dependence of  Gilbert damping at T = 50 K .  \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 \n30 \n  \nfig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures. The TC is \ndetermined from the zero -resistance temperature via four -probe resistance measurement.  \n \n31 \n  \nfig. S5. Measurement of the Gilbert damping of Nb/NiFe/Nb heterostructures at T = 4 K. (A) \nThe typical FMR  spectra as a function of magnetic field with microwave  frequency ( f) of 10, 12, \n14, 16, and 18 GHz, respectively. (B) The typical FMR  spectrum measured with f = 12 GHz (black \ncircles) and the Lorentz fitting curve (red line). ΔH is the half linewidth of the FMR signal. (C) \nThe determination of the Gilbert damping from ΔH vs. f. The red line indicates the best linear -\nfitting curve. These results are obtained on the typical  Nb/Py (12 nm)/Nb sample.  \n  \n32 \n  \n \nfig. S6. The effect of FMR measurement on the TC of Nb/NiFe/Nb heterostructures. The four -\nprobe resistances vs. temperature are probed from the typical Nb/NiFe  (12 nm)/Nb sample \nwith/without the presence of the in -plane magnetic field and microwave power.   \n33 \n  \n \nfig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions.  (A) The optical \nimage of a typical Nb/NiFe/Nb Josephson device and schematic of the electrical measurement \ngeometry.  (B) The cross -section of the Josephson devices with a junction area of ~ 80 μm  × 80 \nμm. At the junction, a thin oxide layer of NiFeO x is mostly liked formed on the top surface of NiFe \nduring the growth of A lOx in the presence of oxygen. ( C) The normalized differential resistance \n(dV/dI) as a function of the bias  current  measured on the Nb/NiFe (5 nm)/Nb junction from T = 2 \nto 6 K.  (D) The temperature dependence of the characteristic voltage (I cRn) of the Nb/NiFe (5 \nnm)/Nb Josephson  junction . (E) The normalized differential resistance as a function of the bias  \ncurrent  of the Nb/NiFe/Nb junctions ( dNiFe = 5, 10 and 30 nm) at T = 2 K. ( F) The NiFe thickness \ndependence of the characteristic voltages a t T = 2 K.  \n \n34 \n  \n \nfig. S8. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC \nheterostructures due to Bogoliubov quasiparticles (A) and equal spin -triplet Cooper pairs \n(B). The dark and light balls represent the electron -like and hole-like quasiparticles respectively. \nThe red and blue arrows indicate the spin up and spin down respectively.  \n  \n35 \n   \n \nfig. S9.  Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at T \n= 4 K. (A) The normalized energy of ABS by the superconducting gap at T = 4 K as a function of \ndNiFe for the ideal transparency case. ( B) The enhanced Gilbert damping via ABS as a function of \ndNiFe.  \n  \n36 \n  \n \nfig. S10. Gilbert damping of control sample of bilayer Nb/Ni Fe junctions. (A)  Gilbert damping \nof bilayer Nb/NiFe junctions at T = 4, 5, and 10  K. (B) Comparison of the Gilbert damping of \nbilayer Nb/NiFe and trilayer Nb/NiFe/Nb junctions at T = 4 K.  \n \n \n \n \n \n \n \n \n37 \n  \nfig. S11.  Thickness dependence of gyromagnetic ratio (and g factor) (A), effective \nmagnetization (B) and inhomogeneous half -linewidth (C) . The blue, black, green and red \ndotted -lines represent to temperature of T = 10, 7, 5 and 4 K, respectively.  \n \n \n"    },    {        "title": "2111.08768v1.Ultrathin_ferrimagnetic_GdFeCo_films_with_very_low_damping.pdf",        "content": "Ultrathin ferrimagnetic GdFeCo \flms with very low damping\nLakhan Bainsla*,1,a)Akash Kumar,1Ahmad A. Awad,1Chunlei Wang,2Mohammad Zahedinejad,1Nilamani\nBehera,1Himanshu Fulara,1Roman Khymyn,1Afshin Houshang,1Jonas Weissenrieder,2and J. \u0017Akerman1,b)\n1)Physics Department, University of Gothenburg, 412 96 Gothenburg, Sweden.\n2)Department of Applied Physics, KTH Royal Institute of Technology, 106 91 Stockholm,\nSweden\nFerromagnetic materials dominate as the magnetically active element in spintronic devices, but come with\ndrawbacks such as large stray \felds, and low operational frequencies. Compensated ferrimagnets provide an\nalternative as they combine the ultrafast magnetization dynamics of antiferromagnets with a ferromagnet-like\nspin-orbit-torque (SOT) behavior. However to use ferrimagnets in spintronic devices their advantageous prop-\nerties must be retained also in ultrathin \flms ( t<10 nm). In this study, ferrimagnetic Gd x(Fe87:5Co12:5)1\u0000x\nthin \flms in the thickness range t= 2{20 nm were grown on high resistance Si(100) substrates and studied\nusing broadband ferromagnetic resonance measurements at room temperature. By tuning their stoichiometry,\na nearly compensated behavior is observed in 2 nm Gd x(Fe87:5Co12:5)1\u0000xultrathin \flms for the \frst time,\nwith an e\u000bective magnetization of Me\u000b= 0.02 T and a low e\u000bective Gilbert damping constant of \u000b= 0.0078,\ncomparable to the lowest values reported so far in 30 nm \flms. These results show great promise for the\ndevelopment of ultrafast and energy e\u000ecient ferrimagnetic spintronic devices.\nI. INTRODUCTION\nSpintronic devices utilize the spin degree of freedom for\ndata storage, information processing, and sensing1,2with\ncommercial applications such as hard drives, magnetic\nrandom access memories, and sensors. Besides conven-\ntional memory applications based on quasi-static opera-\ntion of magnetic tunnel junctions, high frequency spin-\ntronic oscillators3,4have recently been demonstrated for\nanalog computing applications such as bio-inspired neu-\nromorphic computing5,6, logic operations, energy har-\nvesting and Ising Machines.7For the \frst time, such oscil-\nlators are now used in commercial magnetic hard drives\nto facilitate writing to the disc.8The key challenges in\ndeveloping such devices is to \fnd material combinations\nwhich allow for fast operation, low-power consumption,\nnon-volatility, and high endurance. Due to their nat-\nural spin polarization and easy manipulation, ferromag-\nnetic materials (FM) dominate as active elements in these\ndevices.4However, FMs come with drawbacks such as:\n(i) large magnetic stray \felds a\u000becting the operation of\nneighbouring devices; (ii) limited scalability of magnetic\nbits in memory devices; (iii) the operating frequency of\nspin-based oscillators limited by ferromagnetic resonance\nfrequency, and (iv) slow synchronization of such oscilla-\ntors. These shortcomings drive researchers to \fnd more\nsuitable materials for future spintronic devices.\nVery recently, the interest in antiferromagnetic (AFM)\nspintronics9{11increased rapidly, as AFM materials have\nno stray \felds and can o\u000ber ultrafast spin dynamics, in-\ncluding AFM resonance frequencies in the THz region.\nIt was theoretically shown that such high-frequency ex-\ncitations are possible to achieve without any applied\nmagnetic \feld by injecting spin currents into AFM\na)Electronic mail: lakhan.bainsla@physics.gu.se\nb)Electronic mail: johan.akerman@physics.gu.sematerials.12{15Experiments have since demonstrated\npossible THz writing/reading capabilities.16However,\nthe absence of a net magnetic moment in AFMs leads\nto di\u000eculties in the read-out of the spin dynamics, in-\ncluding any microwave output signal from the AFM\noscillators.13{15\nA possible solution is presented by ferrimagnets\n(FiMs), which combine the properties of FMs and AFMs.\nFiMs posses magnetic sub-lattices in the same way as\nAFMs do, but their sub-lattices are inequivalent. The\nmagnetic sub-lattices in FiMs often consist of di\u000berent\nmagnetic ions, such as rare earth (e.g. Gd) and transi-\ntion metal (e.g. Fe, Co) alloys (RE-TM) such as CoGd,\nand as a result, a large residual magnetization remains\ndespite the two opposing sub-magnetizations. The tem-\nperature dependence of RE and TM sub-magnetizations\nin FiM can be quite di\u000berent which result in magneti-\nzations that can increase, and even change sign, with\ntemperature17,18, in stark contrast to the non-monotonic\ndecreasing temperature dependence for FMs and AFMs.\nSimilar e\u000bects could also be seen by varying the com-\nposition of ferrimagnetic alloys instead of changing the\ntemperature.19In addition, the di\u000berent properties of the\ntwo magnetic sub-lattices also results in two compen-\nsation points, namely the magnetization compensation\npointTmand the angular compensation point Ta. AtTm,\nthe two magnetic sub-lattices cancel each other, which re-\nsults in a zero net magnetic moment, while at T a, their\nnet angular momentum vanishes, as in AFMs. Therefore,\natTa, FiMs can have a near-THz resonance as in AFMs,\nwhile still having a net magnetic moment which can lead\nto strong read-out signals, including e\u000ecient microwave\nsignal output from FiM-based oscillators20, as well as ef-\n\fcient control and excitation. FiMs also show high spin\npolarization which also make them suitable candidate for\ne\u000ecient magnetic tunnel junctions.21\nDue to these unique properties, research in FiMs for\nspintronic applications is intensifying22, focusing mainly\non RE-TM based systems such as CoTb23, CoGd24, andarXiv:2111.08768v1  [cond-mat.mtrl-sci]  16 Nov 20212\nFigure 1. (a) Schematic illustration of the coplanar waveguide (CPW), the thin \flm sample and its orientation, the directions\nof the applied magnetic \feld H, the microwave \feld hrf, and the e\u000bective magnetic \feld He\u000bduring FMR measurements.\nInset shows the \flm stack. (b) FMR response (derivative of the FMR absorption) for a 10 nm Gd 12:5Fe76:1Co11:4\flm (S2)\nrecorded at di\u000berent frequencies and \ftted (solid lines) to Eq. 1. While FMR curves were recorded at 1 GHz frequency intervals\nthroughout this study, \fgure (b) only shows curves with \u0001 f= 2 GHz for clarity.\nGdFeCo25and Mn 3\u0000xPtxGa26,27based Heusler alloy.\nAmong these, GdFeCo has been studied the most with\ndemonstrations of fast domain wall motion28and ultra-\nfast spin dynamics17nearTa, large spin-orbit torques\nand their sign reversal,25,29low magnetic damping in\nthick 30 nm \flms,30and sub-picosecond magnetization\nreversal,31to name a few. What is missing, however, is a\ndemonstration that these unique material properties per-\nsist down to much thinner \flms, which will ultimately be\nneeded if FiMs are to be used in spin-Hall nano oscillators\n(SHNOs).4\nIn the present study, we systematically study the\ngrowth and functional properties of ultrathin ferrimag-\nnetic Gd x(Fe87:5Co12:5)1\u0000xthin \flms [referred to as\nGdx(FeCo) 1\u0000xhereafter]. GdFeCo thin \flms in the\nthickness range of 2{20 nm were grown on high resis-\ntance silicon (HR-Si) substrate. The atomic composi-\ntion of Gd x(FeCo) 1\u0000xwas controlled using co-sputtering\nand determined using inductively coupled plasma optical\nemission spectroscopy (ICP-OES). The magnetic prop-\nerties and Gilbert damping were studied using broad-\nband ferromagnetic resonance (FMR) measurements. We\nalso demonstrate ultra low Gilbert damping for 2 nm\nGdFeCo, near the compensation point of Gd x(FeCo) 1\u0000x.\nThese results paves the way for integration of FiMs into\nvarious spintronic devices and applications.\nII. RESULTS AND DISCUSSION\nThe growth conditions for GdFeCo were \frst optimized\nby growing four 10 nm thick Gd 12:5Fe76:1Co11:4\flms on\nHR-Si (100) substrates using di\u000berent MgO seed layer\nthicknesses: 0 nm (S1), 6 nm (S2), 10 nm (S3 & S4);in S4, the seed was annealed at 600C for 1 hour prior\nto GdFeCo deposition to check the e\u000bect of MgO crys-\ntallinity. MgO was chosen as seed since it is insulating\nand therefore will not contribute any spin sinking to the\nmagnetic damping.32\nA. Seed layer dependence on 10nm thick\nGd12:5Fe76:1Co11:4\flms\nFurther details of the growth conditions are given in\nthe experimental section. FMR measurements, on 6 \u00023\nmm2rectangular pieces cut from these \flms, were then\nperformed using a NanOsc PhaseFMR-40 FMR Spec-\ntrometer. The sample orientation on the coplanar waveg-\nuide (CPW), together with the directions of the applied\n\feld, the microwave excitation \feld hrf, and the e\u000bec-\ntive magnetic \feld He\u000b, are shown in Fig. 1(a). Typical\n(derivative) FMR absorption spectra obtained for S2 are\nshown in \fgure 1(b) together with \fts to a sum of sym-\nmetric and anti-symmetric Lorentzian derivatives:33\ndP\ndH(H) =\u00008C1\u0001H(H\u0000HR)\n[\u0001H2+ 4(H\u0000HR)2]2+2C2(\u0001H2\u00004(H\u0000HR)2)\n[\u0001H2+ 4(H\u0000HR)2]2\n(1)\nwhereHR, \u0001H,C1, andC2represent the resonance \feld,\nthe full width at half maximum (FWHM) of the FMR ab-\nsorption, and the symmetric and anti-symmetric \ftting\nparameters of the Lorentzian derivatives, respectively.\nThe extracted values of HRvs.fare shown in \fgure\n2 (b) together with \fts to Kittel's equation:34\nf=\r\u00160\n2\u0019q\n(HR\u0000Hk)(HR\u0000Hk+Meff) (2)3\nFigure 2. (a) Seed layer dependence of frequency vs resonance \feld of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid\nsymbols and solid lines are the experimental data points and \ftting with equation (2), respectively. (b) Resonance linewidth\n(\u0001H)vs.frequency of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid symbols and solid lines are the experimental data\npoints and \ftting with equation (3), respectively. The e\u000bective Gilbert damping constant values of all the samples are given in\n\fgure 2 (b). The black and violet dotted lines in \fgure 2(b) shows the \ftting of equation (3) in low and high frequency regions,\nrespectively.\nwhere,\r,HkandMe\u000bare the gyromagnetic ratio, the\nin-plane magnetic anisotropy \feld, and the e\u000bective mag-\nnetization of the sample, respectively, all allowed to be\nfree \ftting parameters. Values for \randHkonly showed\nminor variation between the four samples, with \r=2\u0019=\n29.4-30.0 GHz/T and Hk= 66-104 Oe. Me\u000bvaried more\nstrongly, with values of 0.79, 1.19, 0.71 and 0.76 T ob-\ntained for S1, S2, S3 and S4, respectively.\nThe e\u000bective Gilbert damping constant \u000bcan then be\nobtained from \fts of \u0001 Hvs.fto:35\n\u0001H= \u0001H0+4\u0019\u000bf\n\r\u00160(3)\nwhere the o\u000bset \u0001 H0represents the inhomogeneous\nbroadening. Equation (3) is well \ftted to the experimen-\ntal values, using \u0001 H0and\u000bas adjustable \ftting param-\neters for all the four samples, as shown in the \fgure 2(b).\n\u0001H0= 2{4 mT is essentially sample independent within\nthe measurement accuracy. In contrast, the obtained val-\nues of\u000bvary quite strongly and are given inside \fgure\n2(b). The GdFeCo grown with 6 nm MgO seed layer (S2)\nclearly shows the lowest value of \u000b= 0:0055, although\nthis might be a\u000bected by the slight non-linear behavior\naround 10 to 15 GHz. However, when only the high-\feld\ndata is \ftted, the extracted damping of \u000b= 0.0076 is\nstill the lowest and at all frequencies the linewidth of S2\nlies well below all the other samples. As damping is one\nof the most important parameters for spintronic devices,\nwe hence chose the growth conditions of S2 for all subse-\nquent \flms in this study.B. Thickness dependence on Gd 12:5Fe76:1Co11:4\flms\nAfter optimizing the growth conditions for\nGd12:5Fe76:1Co11:4, the thickness dependence of the\n\flms was studied with the same composition using the\ngrowth conditions of sample S2. The FMR linewidth\n\u0001Hvs. f is shown in \fgure 3(a) and exhibits a relatively\nstrong dependence on thickness. It is noteworthy\nthat the 4 nm \flm shows the narrowest linewidth at\nall frequencies, clearly demonstrating that very low\ndamping can be achieved also in ultra-thin GdFeCo.\nThe extracted Me\u000band\u000bare shown vs.thickness in\n\fgure 3(b), both showing a strong thickness dependence.\nDamping as low as \u000b= 0:0055 is obtained for the 10\nnm thick \flms. If only the high-\feld portion of the data\nis \ftted, the extracted damping increases to 0.0076,\nwhich is still about an order of magnitude lower than\nany literature value on 10 or 30 nm \flms.19,36Both\nthe 10 and 20 nm \flms showed a minor nonlinearity in\n\u0001Hvs.fdata and were therefore analysed by \ftting\nthe data in both the low and the high \feld regions\nseparately, as shown by the dotted lines in \fgure 3(a).\nThe\u000bvalue for the 20 nm \flm increased slightly from\n0.0098 to 0.0109 if only high \feld data is used for\nanalysis. The relatively higher damping for the 20 nm\n\flm might be due to the radiative damping mechanism\nwhich increases proportionally with magnetic layer\nthickness.37We conclude that 2 nm ultrathin \flms can\nindeed be grown with reasonably low damping. Since\nthe damping is strongly thickness dependent in this\nregime, the optimum thickness for devices may likely be\nfound in the 2{4 nm range.4\nFigure 3. (a) FMR linewidth \u0001 Hvs.ffor four Gd 12:5Fe76:1Co11:4\flms with di\u000berent thicknesses, together with linear \fts to\nequation (3). The dotted lines show \fts for the 20 nm \flm in its low and high frequency regions, respectively. (b) E\u000bective\nmagnetization and e\u000bective Gilbert damping constant vs.thickness; lines are guides to the eye.\nC. Composition dependence on 2nm thick \flms\nTo \fnally investigate whether we can achieve a com-\npensated ferrimagnetic behavior also in ultra-thin \flms,\nwe grew 2 nm Gd x(FeCo) 1\u0000x\flms in the composition\nrange 12{27 at.% Gd. The \flms were characterized using\nFMR spectrometry as described above and the extracted\nresults are shown in \fgure 4.\nThe extracted Me\u000band\u000bfollow a similar trend as re-\nported earlier for one order of magnitude thicker GdFeCo\n\flms characterized using an all-optical pump-probe tech-\nnique.17We \frst note that we can indeed reach an es-\nsentially fully compensated antiferromagnetic behavior in\ntwo \flms around a composition of 25 at.% Gd. We have\nmarked this compensation point with xmand a dashed\nline in \fgure 4 (c). Both \flms show very low damping of\n0.0078 and 0.009 respectively. However, just below this\ncomposition, the damping shows a peak, which is con-\nsistent with an angular compensation point, which we\ndenote byxa. It is noteworthy that the extracted damp-\ning value of \u000b= 0.0142 is still more than an order of\nmagnitude lower than \u000b= 0.45 of 30 nm \flms measured\nusing FMR spectrometry19and\u000b= 0.20 of 20 nm \flms\nmeasured using an optical pump-probe technique.17\nIII. CONCLUSION\nIn view of the potential application of compensated\nferrimagnets to spintronic devices, we prepared ferri-\nmagnetic thin \flms of Gd x(FeCo) 1\u0000xon high resistance\nSi(100) substrates and studied them using the FMR mea-\nsurements. Their growth conditions were optimized us-\ning 10 nm thick Gd 12:5Fe76:1Co11:4\flms, after which\nthickness dependent studies were done on the same com-\nposition in the thickness range of 2{20 nm. Composi-\ntion dependence studies were \fnally done on 2 nm thick\nGdx(FeCo) 1\u0000x\flms and an essentially compensated fer-rimagnetic behavior was observed for the \frst time in\nultrathin 2 nm \flms. The angular momentum compensa-\ntion and magnetic compensation points observed in this\nwork are very close to those reported earlier on much\nthicker \flms in the literature. A record low \u000bvalue of\nabout 0.0078 is obtained near the magnetic compensa-\ntion point, which is an order of magnitude lower than\nthe values reported in the literature using similar analysis\nmethods. The observation of compensated ferrimagnetic\nbehavior in ultrathin \flms together with very low value\nof\u000bare promising results for the future development of\nultrafast and energy e\u000ecient ferrimagnetic spintronic de-\nvices.\nEXPERIMENTAL SECTION\nA. Thin \flms growth and composition analysis\nAll the samples were prepared on high resistivity\nSi(100) substrates using a magnetron sputtering sys-\ntem with a base pressure of less than 2 \u000210\u00008torr.\nThin \flms of Gd x(FeCo) 1\u0000xwere deposited using the\nco-sputtering of high purity (more than 99.95%) Gd\nand Fe 87:5Co12:5targets, and composition analysis\nwas done using the inductively coupled plasma mass\nspectroscopy (ICP-MS). Thin \flms stacking structure\nof Si(100)/MgO(t)/Gd 12:5Fe76:1Co11:4(10)/SiO 2(4)\nwere used for seed layer dependence studies, here,\nthe number in the bracket is the thickness of the\nlayer in nm, where t=0, 6 and 10 nm. Four sam-\nples, namely S1 to S4 were prepared to obtain the\nbest conditions to grow Gd 12:5Fe76:1Co11:4(10) \flms.\nFor S1, Gd 12:5Fe76:1Co11:4(10) was grown directly\nover HR-Si (100) substrates, while in both S2 and\nS3 Gd 12:5Fe76:1Co11:4were grown with MgO seed\nlayer of 6 and 10 nm, respectively. All the lay-\ners in S1-S3 were grown at room temperature and5\nFigure 4. (a) Frequency vs.resonance \feld and (b) resonance linewidth vs.frequency, of 2 nm thick Gd x(FeCo) 1\u0000x\flms as\na function of Gd content in atomic %. (c) E\u000bective magnetization and e\u000bective Gilbert damping constant vs.Gd content.\nSolid symbols represent the values obtained by \ftting the experimental FMR data in (a) and (b) using the equation (2) and\n(3), respectively; solid lines in (c) are guides to the eye. xaand xmshow the angular and magnetic compensation points,\nrespectively, obtained from the literature17,19.\nno further heat treatment was given to them. In\nS4, 10 nm MgO seed layer were grown over HR\nSi(100) substrates at RT and followed by a in-situ\npost-annealing at 600C for 1 hour, and after that\nGd12:5Fe76:1Co11:4were deposited. The stacking struc-\nture of Si(100)/MgO(6)/Gd 12:5Fe76:1Co11:4(m)/SiO 2(4)\nwere used for thickness dependence studies, where\nm is the thickness of Gd 12:5Fe76:1Co11:4layer,\nand varied from 2 to 20 nm. For composi-\ntion dependence studies, stacking structure of\nSi(100)/MgO(6)/Gd x(FeCo) 1\u0000x(2)/SiO 2(4) were used,\nwhere xvaried from 12.5 to 26.7. The composition of\nGdx(FeCo) 1\u0000x\flms was varied by changing the sput-\ntering rate of Fe 87:5Co12:5target, while keeping the Gd\nsputtering rate \fxed for most \flms. All the samples for\nthickness dependence and composition dependence were\ngrown at room temperature and no post-annealing was\nused. Layer thicknesses were determined by estimating\nthe growth rate using the Dektak pro\fler on more than\n100 nm thick \flms.B. Inductively coupled plasma mass spectroscopy\n(ICP-MS) measurements\nThe elemental composition (Co, Fe, and Gd) of the\nthin \flm samples was determined by inductively coupled\nplasma optical emission spectroscopy (ICP-OES) using a\nThermo Fisher Scienti\fc iCAP 6000 Series spectrometer.\nEach thin \flm sample was exhaustively extracted in 5 mL\nHNO3 (65%, Supelco, Merck KgaA, Sigma-Aldrich) for\na duration of 30 min. 5 mL ultrapure MilliQ-water (18\nM\ncm) was added to the solution and the extract was al-\nlowed to rest for 30 minutes. The extract was transferred\nto a 100 mL volumetric \rask. The extracted sample was\nthen rinsed for several cycles in ultrapure water. The\nwater used for rinsing was transferred to the same volu-\nmetric \rask. The extract was diluted to 100 mL for ICP\nanalysis. ICP check standards were prepared from stan-\ndard solutions (Co and Fe: Merck, Germany; Ga: Accu-\nstandard, USA). The relative standard deviation (from\nthree individual injections) were within 1%.6\nTable I. The obtained values of e\u000bective Gilbert damping constant \u000bat room temperature (RT) in this work and comparison\nwith the lowest values reported so far in the literature at RT and also at their respective angular momentum compensation\n(Ta) and magnetic compensation (T m) points.\nFilm composition Film thickness \u000b Measurement technique Analysis method Reference\nGd23:5Fe68:9Co7:6 30 \u00180.45 (at RT) FMR Kittel's FMR19\n\u00180.35 (at RT) Pump-probe\nGd22Fe74:6Co3:4 20 \u00180.21 (at T a) Pump-probe -do-17\n\u00180.13 (at T m)\nGd25Fe65:6Co9:4 10 \u00180.07 (at RT) Spin torque FMR -do-36\n\u00190.01 (at RT) Spin torque FMR Ferrimagnetc resonance\nGd23:5Fe66:9Co9:6 30 0.0072 (at RT) Domain wall (DW) Field driven DW30\nmotion mobility\nGd12:5Fe76:1Co11:4 10 0.0055 Broadband FMR Kittel's FMR This work\n0.0076 (HF data) -do- -do- This work\nGd12:5Fe76:1Co11:4 4 0.0064 -do- -do- This work\nGd12:5Fe76:1Co11:4 2 0.0101 -do- -do- This work\nGd23:4Fe67:0Co9:6 2 0.0141 -do- -do- This work\nGd24:4Fe66:1Co9:5 2 0.0078 -do- -do- This work\nC. Ferromagnetic resonance (FMR) measurements\nRectangular pieces of about 6 \u00023 mm2were cut from\nthe blanket \flms and broadband FMR spectroscopy was\nperformed using a NanOsc Phase FMR (40 GHz) system\nwith a co-planar waveguide for microwave \feld excita-\ntion. Microwave excitation \felds hrfwith frequencies up\nto 30 GHz were applied in the \flm plane, and perpendic-\nular to the applied in-plane dc magnetic \feld H. All the\nFMR measurements were performed at the room tem-\nperature. The schematic of FMR measurement setup is\nshown in 1(a), and further details about the measure-\nments are given in Section 2 (results and discussions).\nSUPPORTING INFORMATION\nSupporting Information is available from the Wiley\nOnline Library or from the corresponding author.ACKNOWLEDGEMENTS\nLakhan Bainsla thanks MSCA - European Commission\nfor Marie Curie Individual Fellowship (MSCA-IF Grant\nNo. 896307). This work was also partially supported\nby the Swedish Research Council (VR Grant No. 2016-\n05980) and the Horizon 2020 research and innovation\nprogramme (ERC Advanced Grant No. 835068 \"TOP-\nSPIN\").\nCONFLICT OF INTEREST\nThe authors declare no con\rict of interest.\nAUTHOR CONTRIBUTIONS\nL.B. and J. \u0017A. planned the study. L.B. grew the \flms,\nperformed the FMR measurements and analysed the ob-\ntained FMR data. J.W. helped with ICP-MS measure-\nments and analysis. L.B. wrote the original draft of the\npaper. J. \u0017A. coordinated and supervised the work. All\nauthors contributed to the data analysis and co-wrote\nthe manuscript.7\nDATA AVAILABILITY STATEMENT\nThe data that support the \fndings of this study are\navailable from the corresponding author on reasonable\nrequest.\nREFERENCES\n1S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, v. S. von\nMoln\u0013 ar, M. Roukes, A. Y. Chtchelkanova, and D. 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This equation arises in the ap proximation of\nHelmholtz equation by the WKBJ or EWT methods. We obtain the c omplex-\nvalued solutions (called eikonals) as parametrizations in a complex variable.\nWe consider both the cases ofconstant and non-constant inde x ofrefraction. In\nboth cases, the relevant parametrizations depend on some ho lomorphic func-\ntion. In the case of non-constant index of refraction, the pa rametrization also\ndepends on some extra exponential complex-valued function and on a quasi-\nconformal homeomorphism. This is due to the use of the theory of pseudo-\nanalytic functions and the related similarity principle. T he parametrizations\ngive information about the formation of caustics and the lig ht and shadow\nregions for the relevant eikonals.\n1.Introduction\nLetnbe some continuous strictly positive real-valued function of x∈RNand\nconsider the following nonlinear first-order differential equation\n(1.1)N/summationdisplay\nj=1/parenleftBig∂φ\n∂xj/parenrightBig2\n=n2;\n(1.1) is the so-called eikonal equation that arises in many a field of science, such as\noptics, acoustics, nuclear physics, control theory, to name a fe w.\nAn important motivation to study (1.1) is that its solutions help to und erstand\nthe asymptotic behaviour of solutions of the Helmholtz equation (that ensues from\nMaxwell’s system),\n(1.2) ∆ W+k2n2W= 0,\nfor large values of the wave number k.Here,nhas the meaning of an index of\nrefraction , whose reciprocal is proportional to the velocity of propagation o f waves\nthrough the medium.\nAn expansion, originating from the so-called WKBJ method (after We ntzel,\nKramers, Brillouin, and Jeffreys) represents solutions of (1.2), as ymptotically as\nk→+∞, by\n(1.3) W≃eikφ∞/summationdisplay\nm=0Am(ik)−m.\n2010Mathematics Subject Classification. 78A05, 35F20, 35C05, 35A22, 35A30, 30G20.\nKey words and phrases. Complex eikonal equation, geometrical optics, explicit so lutions,\npseudo-analytic functions.\n12 R. MAGNANINI\nHere,φis a real-valued phase function , which is a solution of (1.1), and the Am’s\nare solutions of the so-called transport equations :\n2∇φ·∇A0+∆φA0= 0,\n2∇φ·∇Am+∆φAm= ∆Am−1,(m= 1,2,...);\nsee e.g. [17].\nThe expansion (1.3) is successful in accounting for those phenome na pertaining\ngeometrical optics , such as the insurgence of caustics via the mechanism of rays.\nHowever, the expansion cannot describe those phenomena, such as the development\nofevanescent waves , that take place past a caustic and create a somewhat blurred\nshadow region .\nMore information can be derived by the theory of Evanescent Wave Tracking\n(EWT)proposed by Felsen and coworkers (see [3]-[12]). In EWT, (1.3) is rep laced\nby the following expansion\n(1.4) W=eλ+kv/braceleftBig\ncos(µ+ku)+O/parenleftBig1\nk/parenrightBig/bracerightBig\nask→ ∞.\nThe real-valued functions u,v,λ,µ are all independent of kand turn out to obey\nboth\n|∇u|2−|∇v|2−n2= 0,∇u·∇v= 0,\nand\n∇u·∇λ+∇v·∇µ+1\n2∆u= 0,∇v·∇λ−∇u·∇µ+1\n2∆v= 0.\nNoticethattheformersystemfor uandvtellsusthat φ=u+ivisacomplex-valued\nsolution of (1.1).\nObserve that in the region in which v≡0,φis real-valued and the character\nof the expansion in (1.4) is purely oscillatory. This corresponds to th e regime of\ngeometrical optics (the light region) in which light propagates along r ays. When\nv<0 instead, an inspection of (1.4) tells us that Wdecays exponentially for large\nvalues ofk. This creates the evanescent waves, which vanish exponentially in t he\nshadow region near the caustic.\nOne more sophisticated asymptotic expansion, which powerfully help s to repre-\nsent solutions of (1.2) on both sides of a caustic, and once more sho ws the utility of\ncomplex-valued solutions of (1.1), is developed in a theory of Kravts ov and Ludwig\n(see [13], [14],[16],[18],[19],[20],[3]). It reads as\nW≃Ai(k2\n3v)eikU∞/summationdisplay\nm=0Am(ik)−m+ik−1\n3Ai′(k2\n3v)eikU∞/summationdisplay\nm=0Bm(ik)−m,\nask→ ∞. Thereal-valuedfunctions UandVandthe complex-valuedones Amand\nBmare independent on k;Ai(·) denotes the Airy function defined by the problem\nAi′′(s)−sAi(s) = 0,Ai(0) = 3−2\n3Γ(2/3)−1,Ai′(0) =−3−1\n3Γ(1/3)−1.\nHere, Γ is Euler’s gamma function. By Kravtsovand Ludwig’s expansio n and (1.2),\none gets that\n|∇U|2−V|∇V|2−n2= 0,∇U·∇V= 0.\nAgain observe that the complex-valued function U+i√\nVsatisfies (1.1).\nComplex-valued solutions of (1.1) can be obtained by means of the me thod of\ncomplex rays (see [7], [21]). As an illustration, we carry out here the simplest case\nin whichN= 2 andn≡1. Given a pair ( a,b), consider the parametrization\n(1.5) x=a+φcosγ, y=b+φsinγ.\nWhena,b∈R, (1.5) defines a real-valued solution φof (1.1) as the distance of the\npoint (x,y) to the point ( a,b). If either aorbis not real, then a complex-valuedSOLUTIONS OF THE COMPLEX EIKONAL EQUATION 3\nsolutionφof (1.1) can be constructed via (1.5) by making complex the paramet er\nγ, but by keeping xandyreal-valued. For instance, for a= 0 andb=i,we obtain:\n(1.6) φ=/radicalbig\nx2+(y−i)2=/radicalbig\n(z+1)(z−1).\nUnfortunately, this approachappearsto be difficult to use when th e refractiveindex\nnis non-constant.\nIn this paper, we shall present another way to construct explicit c omplex-valued\nsolutions of (1.1) in the plane ( x,y). The method can be extended to the case of\nnon-constant index of refraction. The solutions are obtained as p arametrizations\nof the following type\nC⊃A∋ζ/ma√sto→(z(ζ),φ(ζ))∈C2.\nIn fact, ifwe introduce the complex variable z=x+iyand its conjugate z=x−iy,\n(1.1) can be rewritten as:\n(1.7) 4 φzφz=n(z)2.\nWe then observe that, for fixed z, the point ( φz,φz) belongs to a hyperbola (in C2)\nthat, for instance, we can parameterize by a function ζ=ζ(z,z) as\nφz=n\n2ζ, φz=n\n2ζ.\nThis assumption gives a differential constraint on the parameter fu nctionζ(z,z),\nwhich can be converted in an equation for its inverse z=z(ζ,ζ). This can be solved\nand, by rather simple manipulations, we obtain solutions of (1.7) para metrized as\nz=z(ζ,ζ), φ=φ(ζ,ζ).\nWe begin with Section 2, in which we explain in detail the method, when nis\nconstant. As a result, we obtain the (local) explicit parametrization ofφ:\n(1.8)z=f(ζ)+ζ2f(ζ)\n1−|ζ|4,\nφ=1\n1−|ζ|4/braceleftBig\nζf(ζ)+f(ζ)\nζ/bracerightBig\n−f(ζ)\n2ζ+1\n2/integraldisplayf(ζ)\nζ2dζ.\nHere — and this is remarkable — f(ζ) is some analytic function of the variable ζ.\nIn the case in which nis non-constant, the situation is more complicated, but\nperfectly consistent. We present this case in Section 3. The releva nt parametriza-\ntion this time is\n(1.9) z=Z(χ(ζ)), φ= Φ(ζ,χ(ζ)),\nwhere\nZ(χ) =es(χ)f(χ)+κ(χ)es(χ)f(χ)\n1−|κ(χ)|2forχ∈C.\nFor the details on how to construct Φ, we refer the reader to Coro llary 3.6.\nThe functions χ(ζ),s(χ), andκ(χ) are related to the derivatives of the function\nℓ= logn, as explained in Section 3. In particular, χ(ζ) is a homeomorphism\nassociated to a metric induced by the derivatives of ℓ, i.e. it is a univalent solution\nof theBeltrami equation:\nχζ=σχζ.\nHere,σis a complex-valued function that depends on the derivatives of ℓ.\nIn order to obtain the parametrization 1.9, it was crucial to use the elegant\nand now classical theory of pseudoanalytic functions developed by L. Bers and\nI. N. Vekua (see [2], [27], and [1] for another application). If ℓis constant, it turns\nout thatσ≡0, so thatχ(ζ) can be taken to be the identity χ(ζ) =ζ. Accordingly,\none can compute that s≡0 andκ(χ) =χ2, so that (1.9) reduces to (1.8).4 R. MAGNANINI\nIn Section 4, we analyse (1.8) and (1.9) and use them to give geometr ical infor-\nmation about the set of critical points of v= Im(φ), and accordinglyof caustics and\nthe related light and shadow regions. This analysis goes along with and extends\nthat developed (in the case of constant index of refraction) by th e author and his\nco-authors in the series of papers [21]-[25], and [4]. In fact, in that c ase, it confirms\nthat the set of critical points of v= Im(φ) is always a continuum , made either\nof a finite number of segments or even of regions with non-empty int erior (which\ncorrespond to the light regions). When the index of refraction is no n-constant,\nthe segments and light regions are distorted by the mapping z/ma√sto→χ(ζ(z,z)), which\ntakes into account the effect of the index of refraction n.\nIn our presentation, regularity of solutions is not an issue. The fun ctions we\nconsider should be intended as sufficiently regular as to make our com putations\nwork. Thus, in our statements we shall not specify regularity assu mptions. Also,\nthe parametrizations we obtain should also be intended to have a loca l character.\n2.Explicit solutions by Legendre transformation:\nconstant index of refraction\nIn this section, we will consider equation (1.1) in the plane and with a ho moge-\nneous index of refraction nthat we normalize to 1.\nIntroducing the complex variable z=x+iyand its conjugate z=x−iygives\nthat (1.1) is equivalent to the following equation\n(2.1) 4 φzφz= 1,\nsince∂x=∂z+∂zand∂y=i(∂z−∂z).\nEquation (2.1) tells us that the point ( φz,phiz) belongs to a hyperbola, that we\nchoose to parameterize by\n(2.2) φz=1\n2ζ, φz=1\n2ζ,\nfor some non-constant parameter ζ=ζ(z,z).\nA sufficiently smooth function φsatisfies Schwarz’s lemma on second derivatives\n(φzz=φzz), hence (2.2) gives that\n(2.3) ζ2ζz+ζz= 0.\nWe now change variables — in fact operating a Legendre transformation — and\ninvert the roles of z,zandζ,ζ. Since it holds that\nζzzζ+ζzzζ= 1,\nζzzζ+ζzzζ= 0,\nwe have that\n(2.4)ζz=zζ\n|zζ|2−|zζ|2, ζz=−zζ\n|zζ|2−|zζ|2,\nzζ=ζz\n|ζz|2−|ζz|2, zζ=−ζz\n|ζz|2−|ζz|2.\nNotice that (2.3) gives that\n|ζz|2−|ζz|2=|ζz|2(1−|ζ|4) =|ζz|2(1+|ζ|2)|(1−|ζ|2).\nSinceζz/ne}ationslash= 0 (otherwise ζwould be constant, thanks to (2.3)), we have that the\njacobian determinant |ζz|2−|ζz|2is zero if and only if |ζ|2= 1. This holds if and\nonly if∇v= 0. In fact, we compute that\nζ−1\nζ= 2φz−2φz= 2i(vx+ivy),SOLUTIONS OF THE COMPLEX EIKONAL EQUATION 5\nso that\n(2.5) ζ(|ζz|2−|ζz|2) =−2i|ζz|2(|ζ|2+1)(vx+ivy).\nThus, away from the critical points of v,ζ=ζ(z,z) is locally invertible by a\nfunctionz=z(ζ,ζ), and (2.3) and (2.4) give:\n(2.6) ζ2zζ−zζ= 0.\n(For global invertibility, see [15], for instance.) Since (2.6) can be re written as\n∂ζ(z−ζ2z) = 0,\nwe infer that z−ζ2zmust be an analytic function f(ζ): i.e. we have that\n(2.7) z−ζ2z=f(ζ),\nlocally.\nExample 2.1. Letf(ζ) =f0+f1ζ+f2ζ2— a second degree polynomial in ζ.\nThen, we obtain from (2.7) that\n(f2+z)ζ2+f1ζ+f0−z= 0,\nand hence\nζ=−f1+/radicalbig\nf2\n1+4(z−f0)(z+f2)\n2(z+f2),\nwhere the square root is the multi-valued complex-valued function. Plugging this\nexpression of ζinto (2.2) and integrating gives that\nφ=1\n2/radicalBig\nf2\n1+4(z−f0)(z+f2)−1\n2f1log/radicalbig\nf2\n1+4(z−f0)(z+f2)+f1\nz−f0+c,\nwherecis an arbitrary constant. Notice that, if we choose f(ζ) =−1−ζ2, we\nrecover (1.6).\nIn general, we cannot expect to always obtain solutions in the explicit form as\nin this example. However, we will next show that solutions in parametr ic form can\nalways be constructed by this method.\nTheorem 2.2. LetΩ⊂Cbe a simply connected open domain. Suppose that\n∇v/ne}ationslash= 0inΩ. Then,φ=u+ivis a solution in Ωof(1.1)if and only if there\nexists an analytic function f(ζ)such that\n(2.8)z=f(ζ)+ζ2f(ζ)\n1−|ζ|4,\nφ=1\n1−|ζ|4/braceleftBig\nζf(ζ)+f(ζ)\nζ/bracerightBig\n−f(ζ)\n2ζ+1\n2/integraldisplayf(ζ)\nζ2dζ.\nProof.In fact, let us go back to (2.7). By conjugating, we obtain the linear system\ninzandz,\nz−ζ2z=f(ζ),\nζ2z−z=−f(ζ),\nthat is solved by the zgiven in (2.8) and its conjugate.\nNow, since φζ=φzzζ+φzzζandφζ=φzzζ+φzzζ, from (2.2) and (2.8) we\nobtain that\nφζ=1+|ζ|4\n(1−|ζ|4)2/braceleftBigf′(ζ)\n2ζ(1−|ζ|4)+ζ2f(ζ)+f(ζ)/bracerightBig\n,\nφζ=2|ζ|2\n(1−|ζ|4)2/braceleftBigf′(ζ)\n2ζ(1−|ζ|4)+ζ2f(ζ)+f(ζ)/bracerightBig\n.6 R. MAGNANINI\nThese equations can be locally integrated to get the second equatio n in (2.8).\nViceversa, thanks to (2.4) we have that\nφz=φζζz+φζζz=φζzζ−φζzζ\n|zζ|2−|zζ|2, φz=φζζz+φζζz=−φζzζ+φζzζ\n|zζ|2−|zζ|2.\nThus, we compute φζ,φζ,zζ, andzζfrom (2.8) and discover that 4 φzφz= 1, after\ntedious calculations. /square\n3.Explicit solutions by Legendre transformation:\nnon-constant index of refraction\nIf the index of refraction in (1.1) is non-constant, then solutions o f (1.1) still\nadmit a parametrization in terms of some analytic function. In this se ction, we\nshall show how to construct the necessary modifications.\nRemark 3.1. There is a special case in which one can easily contruct solutions,\nwhen the index of refraction in (1.1) is non-constant. In fact, let w: Ω→w(Ω) be\na bi-holomorphic homeomorphism and suppose that nhas the special form:\nn(z) =|w′(z)|forz∈Ω.\nLetφbe a solution of (1.1) in Ω. Notice that the function ψ, defined in w(Ω) by\nψ(w) =φ(z) forz∈Ω, is such that\nφz=ψww′(z), φz=ψww′(z).\nThus, we have that\n|w′(z)|2= 4φzφz= 4ψwψw|w′(z)|2.\nSincew′(z)/ne}ationslash= 0, we infer that 4 ψwψw= 1 inw(Ω). Therefore, in this case, we can\nsimply reduce our problem to the case of constant index of refract ion.\nIn general, however, n(z) may not be the modulus of an analytic function. In\nwhat follows, we shall describe how to proceed. In Section 4, we sha ll see that, even\nif the procedure can hardly be used to construct explicit solutions, it still provides\nuseful geometric information about caustics, shadow regions, an d light regions.\nWe start with the following simple proposition.\nProposition 3.2. LetN= 2and suppose that nis of classC1and bounded away\nfrom zero. Set ℓ= logn.\nIfφis a solution of (1.1), then the function ζdefined by\nζ=2\nnφz\nis a solution of the equation:\n(3.1) aζz+bζz+cζz+dζz= 0,\nwhere\n(3.2) a= 1+ζℓζ, b=−1\nζℓζ, c=1\nζ2(1−ζℓζ). d=ζℓζ.\nProof.Notice that (1.1) becomes:\n4φzφz=n2=e2ℓ.\nNext, we have that\nφz=eℓ\n2ζ, φz=eℓ\n2ζ.SOLUTIONS OF THE COMPLEX EIKONAL EQUATION 7\nThus, by Schwartz’s theorem on mixed second derivatives, we infer that\n1\nζℓz−1\nζ2ζz=ζℓz+ζz.\nTherefore, we get:\nζz+1\nζ2ζz=−ζℓz+1\nζℓz=−ζ(ℓζζz+ℓζζz)+1\nζ(ℓζζz+ℓζζz).\nWe can rearrange this equation and obtain (3.1), when (3.2) is in forc e. /square\nRemark 3.3. Similarly to what observed in Section 2, we have that the jacobian\n|ζz|2−|ζz|2/ne}ationslash= 0 if and only if ∇v/ne}ationslash= 0. In fact, in this case, (2.5) changes into\nnζ(|ζz|2−|ζz|2) =−2i|ζz|2(|ζ|2+1)(vx+ivy).\nSinceζis non-constant, we can assume that ζz/ne}ationslash= 0.\nProposition 3.4. Suppose that ζis a solution of (3.1)such that |ζz|2−|ζz|2/ne}ationslash= 0.\nThen, the function z(ζ)such thatz(ζ(z)) =zandζ(z(ζ)) =ζsatisfies the equation:\n(3.3) bzζ+azζ−czζ−dzζ= 0.\nProof.In order to obtain (3.3), it is sufficient to apply the first two equation s in\n(2.4) to (3.1). /square\nThe following result is the crucial step for the construction of para metrized\nsolutions of (1.1) with non-constant index of refraction.\nTheorem 3.5. Leta,b,c, anddbe given by (3.2), set\n(3.4)A11= 2Im/bracketleftbig\n(a+b)(c+d)/bracketrightbig\n|a+c|2−|b+d|2, A 12=|a+d|2−|b+c|2\n|a+c|2−|b+d|2,\nA21=|a−d|2−|b−c|2\n|a+c|2−|b+d|2, A 22= 2Im/bracketleftbig\n(a−b)(c−d)/bracketrightbig\n|a+c|2−|b+d|2,\nand suppose that\n(3.5) 4 A12A21−(A11+A22)2>0, A12>0.\nLetzbe a solution of (3.3). Then there exist a quasi-conformal homeomorphism\nχ=χ(ζ), an analytic function f(χ), and two complex-valued functions κ(χ)and\ns(χ)such that\n(3.6) z(ζ) =Z(χ(ζ)),\nwith\n(3.7) Z(χ) =es(χ)f(χ)+κ(χ)f(χ)es(χ)\n1−|κ(χ)|2.\nProof.Letz=x+iyandζ=ξ+iη. Lengthy computations give that (3.3) is\nequivalent to the system:\n(3.8) yξ=A11xξ+A12xη,−yη=A21xξ+A22xη.\nThe coefficients of the system are given by (3.4).\nNext, we follow the arguments of [2, Section 12] on pseudo-analytic functions.\nSince (3.5) holds, the system (3.8) is declaredelliptic and z(ζ) satisfiesthe equation:\n(3.9) zζ=µzζ+νzζ,8 R. MAGNANINI\nwhere\nµ=A11+A22+i(A12−A21)\n2−(A12+A21)+A12A21−A11A22,\nν=1+A11A22−A12A21−i(A22−A11)\n2−(A12+A21)+A12A21−A11A22.\nIt turns out that |µ|+|ν|<1, under the assumption (3.5).\nEquation (3.9) can be conveniently transformed into a canonical fo rm. In fact,\nsince|µ|+|ν|<1, it exists a homeomorphism χ=χ(ζ) with respect to the metric\nassociated to the system (3.8), i.e. a univalent solution of the Beltrami equation\n(3.10) χζ=σχζ,\nsuch that the function Zdefined by z(ζ) =Z(χ(ζ)) is a solution of the canonical\nequation\n(3.11) Zχ=κZχ.\nIn terms of the coefficients in the system (3.8), we get:\n(3.12) σ=A12−A21−i(A11+A22)\nA12+A21+/radicalbig\n4A12A21−(A11+A22)2.\nUnder the assumption (3.5), it holds that |σ|<1. Also, we compute that\n(3.13) κ=ν\n1−µσ,\nand again |κ|<1.\nWe can further transform the equation for Zby settingW=Z−κZ. It turns\nout thatWis a solution of the equation:\nWχ=BW+CW,\nwhere\nB=−κκχ\n1−|κ|2, C=−κχ\n1−|κ|2.\nThis means that Wis a pseudo-analytic function of the first kind in the χ-plane.\nIt is well known that pseudo-analytic functions of the first kind sat isfy the simi-\nlarity principle (see [2, Section 10]). Thus, we can infer that there ex ist an analytic\nfunctionfand a complex-valued function ssuch that\nW(χ) =es(χ)f(χ).\nThis last formula and the fact that\nZ=W+κW\n1−|κ|2\ngive the desired conclusion. /square\nCorollary 3.6. Under the assumptions of Theorem 3.5, any solution φof(1.1)\ncan be parametrized as in (1.9),\nz=Z(χ(ζ)), φ= Φ(ζ,χ(ζ)),\nwhere\nφζ=N(χ)\n2ζ/bracketleftBig\n(1+ζ2κ)Zχχζ+σ(κ+ζ2)Zχχζ/bracketrightBig\n,\nφζ=N(χ)\n2ζ/bracketleftBig\nσ(1+ζ2κ)Zχχζ+(κ+ζ2)Zχχζ/bracketrightBig\n,\nwhereN=n◦Z.SOLUTIONS OF THE COMPLEX EIKONAL EQUATION 9\nProof.We proceed as in the proof of Theorem 2.2. In fact, we still observe that\nφζ=φzzζ+φzzζandφζ=φzzζ+φzzζ, and hence we obtain that\nφζ=n\n2ζzζ+nζ\n2zζ, φζ=n\n2ζzζ+nζ\n2zζ.\nThus, we find the desired two formulas for φζandφζ, by using (3.10) and (3.11).\nByintegratingthe obtainedtwoformulasonefor φζandφζwecan finallyrecover\nφ= Φ(ζ,χ(ζ)). /square\n4.On caustics and light and shadow regions\nAs already observed in the previous sections, real-valued solutions of (1.1) de-\nscribe the propagation of light along rays. Thus, the region swept b y rays — the\nlight region — corresponds to the set in which the eikonal φ=u+ivis real-valued\nor, up to some suitable normalization, to the set of critical points of v. The enve-\nlope of rays create a so-called caustic that bounds the light region. Outside that\nregion, i.e. beyond the caustic, we have that ∇v/ne}ationslash= 0 and, up to normalizing an\nadditive imaginary constant, we can assume that v <0, so that the exponentials\nin (1.3) or (1.4) rapidly decay to zero as the wave number kbecomes large. This is\nthe so-called shadow region.\nIn this section, we shall thus investigateon the regionin which ∇v/ne}ationslash= 0, by means\nof our parametrization.\n4.1.The case of constant index of refraction n.As noticed in Section 2, (2.5)\ninforms us that the map z/ma√sto→ζ(z) is locally invertible, and hence (2.6) holds as\nwell.\nAn inspection of (2.8) tells us that the parametrizationis not defined on the unit\ndisk on which |ζ|2= 1. By (2.5), we see that this holds exactly if ∇v= 0. Next,\nobserve that equation (2.7) defines a complex ray for each fixed ζ. Sinceζ=ξ+iη,\n(2.7) can be re-written as\n(1−ξ2+η2)x−2ξηy= Re[f(ξ+iη)],\n−2ξηx+(1+ξ2−η2)y= Im[f(ξ+iη)].\nIf we use polar coordinates reiθforζ, the system becomes:\n[1−r2cos(2θ)]x−r2sin(2θ)y= Re[f(reiθ)],\n−r2sin(2θ)x+[1+r2cos(2θ)]y= Im[f(reiθ)].\nIt is easy to compute that the determinant of this linear system equ als 1−r4. Thus,\nwhen∇v/ne}ationslash= 0, we have that |ζ|2/ne}ationslash= 1, and hence each point ζ=reiθdetermines a\nunique point z=x+iy. In other words, the parametrization uniquely detemines\nall the regular points of v, which correspond to the points in the shadow region.\nIn the light region, i.e. when ∇v= 0, the system degenerates into the two\nequations:\n(4.1)[1−cos(2θ)]x−sin(2θ)y= Re[f(eiθ)],\n−sin(2θ)x+[1+cos(2 θ)]y= Im[f(eiθ)].\nSince in this case the determinant of this system is always zero, the s ystem has\nsolutions if and only if\n(4.2) cos θRe[f(eiθ)]+sinθIm[f(eiθ)] = 0\nor, in complex notation if and only if\n(4.3) Re/bracketleftbig\nf(eiθ)e−iθ] = 0.\nIf this condition holds for some θ, then the equation ∇v= 0 is satisfied at all\npoints of the line defined by either one equation in (4.1). If (4.2) does not hold10 R. MAGNANINI\nfor someθ, the lines defined in (4.1) are parallel, and hence no points on them is a\ncritical point of v. We can summarize this analysis into the following result.\nProposition 4.1. It holds that the parametrization (2.8)maps:\n(i) each point ζoutside the unit circle onto a point of the shadow region;\n(ii) each point ζon the unitcircle that satisfies Re/bracketleftbig\nf(eiθ)e−iθ] = 0ontoa segment\nwithin the light region, whose equation is given in (4.1)(either one will do);\n(iii) any other point on the unit circle to ∞.\nRemark 4.2. Letφ=u+ivbe a solution of (1.1) in an open domain Ω ⊆C.\nProposition 4.1 and (2.5) inform us that the set of critical points of v,\nCv={z∈Ω :∇v(z) = 0},\ndoes not contain isolated points. In fact, Cvis mapped by the parametrization for\nzin (2.8) from the set Sφdefined by\nSφ={ζ=eiθ: Re[f(eiθ)e−iθ] = 0}.\nThus, discrete points in Sφare mapped to segments in Cv(as it happens, for\ninstance, for (1.6)). If Sφcontains an arc, then a corresponding pencil of segments\nsweepsasubsetof Cvwithnon-emptyinterior. Thisanalysisconfirmssimilarresults\npreviously obtained in [21, 4], by other means and different viewpoint s.\nNotice that, because of the condition (4.3), the parametrization f orzin (2.8)\ndoes converge as ζconverges to points in Sφ. In fact, in the limit as |ζ| →1, we\nobtain from (2.8) that z(ζ) is a point in the shadow region that converges to the\npoint:\nz=−1\n4eiθ[f′(eiθ)+f′(eiθ)]−1\n2e2iθf(eiθ).\nWhenSφcontains an arc, by using (4.3) and its derivative, we then can infer t hat\nz=f(eiθ)−1\n2eiθf′(eiθ),\nforeiθbelonging to that arc. The last equation provides a parametrization of a\ncaustic, which separates the shadow region from the light region Cv.\nBy means of (2.8), we can also compute the values of φonCv.\nExample 4.3. One can concretely construct an analytic function f(ζ) in a neigh-\nborhood of the unit circle such that Svcontains an arc. It is sufficient to construct\nRe[f(ζ)/ζ] as a harmonic function which is zero on an arc of the unit circle, and\nthen extend it analytically on a neighborhood.\nIn fact, by the classical Poisson’s formula (see [26]),\ng(ζ) =1\n2π/integraldisplayπ\n−πeit+ζ\neit−ζReg(eit)dt\nrepresents the analytic function in the unit disk with real part assig ned on the unit\ncircle. We thus can choose, for instance, that\nReg(eit) = Re/bracketleftbig\nf(eit)/eit/bracketrightbig\n= max(|t|−τ,0),\nwith 0<τ <π, to obtain the formula:\nf(ζ) =1\nπ/integraldisplayπ−τ\n0ζ(1−ζ2)tdt\n1+ζ2−2ζcos(σ+t).\nThis formula gives an analytic function outside the arc γ={eiθ:τ≤ |θ| ≤π},\ncontinuous up to the boundary, and such that Re[ f(eiθ)/eiθ] = 0 for 0 ≤ |θ|< τ.\nThis choice of f(ζ) will thus generatea caustic, thanks to the parametrization(1.8) ,\nand a set Cvwith non-empty interior.SOLUTIONS OF THE COMPLEX EIKONAL EQUATION 11\nOf course, the values of Re[ f(eiθ)/eiθ] onγcan be arbitrarily changed, in order\nto obtain other examples.\n4.2.The case of non-constant index of refraction. The situation is more\ncomplicated, and slightly different since we do not directly work with ζ(z), but\nrather with z(ζ). However, the geometric information we derive is quite similar.\nOnce the coefficients in (3.4) are determined in terms of ℓ, we can compute σand\nκ, thanksto(3.12)and(3.13). After ahomeomorphism χ(ζ) satisfyingtheBeltrami\nequation (3.10) is determined, we can examine the parametrization ( 3.6)-(3.7).\nRecall that the functions ζ(z) andz(ζ) are the inverse of one another. Thus,\nfrom (3.6) we obtain that\nz=z(ζ(z)) =Z(χ(ζ(z))),\ni.e. the composition ω=χ◦ζtakespoints ofthe z-planetopoints ofthe Z-plane, so\nthat any curve in the Z-plane is deformed by the inverse ω−1onto a corresponding\ncurve in the z-plane.\nNow, notice that (3.6) and (3.7) give that\n(4.4) Z−κ(χ)Z=es(χ)f(χ).\nThus, any point in the χ-plane determines a complex ray in the Z-plane, and this\nis deformed by ω−1to obtain in the z-plane a complex ray in a metric induced by\nthe index of refraction n.\nAn analysis similar to that of Proposition 4.1 informs us that the points χnot\nbelonging to the set κ−1(S1) are mapped onto the shadow region. Each point χ\nsatisfying\n(4.5) |κ(χ)|2= 1 andes(χ)f(χ)+κ(χ)f(χ)es(χ)= 0\nis mapped toa segmentwith equation (4.4) in the Z-plane, andhence to adeformed\nsegment in the z-plane, by means of the mapping ω−1. When (4.5) is satisfied on\na curve in the χ-plane, these deformed segments sweep the light region and their\nenvelope creates a caustic separating the light from the shadow re gion.\nAcknowledgements\nThe author wish to thank Giulio Ciraolo for some useful discussions. T he paper\nis partially supported by the Gruppo Nazionale per l’Analisi Matematica , la Proba-\nbilit` a e le loro Applicazioni (GNAMPA) dell’Istituto Nazionale di Alta Mate matica\n(INdAM).\nReferences\n[1] G. Alessandrini, R. Magnanini, Elliptic equations in divergence form, geometric critical\npoints of solutions, and Stekloff eigenfunctions , SIAM Jour. Math. Anal. 25 (1994),\n1259–1268.\n[2] L. Bers, An outline of the theory of pseudoanalytic functions , Bull. Amer. Math. Soc. 62\n(1956), 291–331.\n[3] D. Bouche and F. Molinet, M´ ethodes Asymptotiques en ´Electromagn´ etisme, Springer-\nVerlag, Berlin, 1994.\n[4] S. Cecchini and R. Magnanini, Critical points of solutions of degenerate elliptic equati ons\nin the plane , Calc. Var. PDE 39 (2010), 121–138.\n[5] S. Choudhary and L. B. Felsen, Asymptotic theory for inhomogeneous waves , IEEE\nTrans. Antennas and Propagation, 21 (1973) 827–842.\n[6] S. Choudhary and L. B. Felsen, Analysis of Gaussian beam propagation and diffraction\nby inhomogeneous wave tracking , Proc. IEEE, 62 (1974) 1530–1541.\n[7] S.J. Chapman, J.M.H. Lawry, J.R. Ockendon and R.H. Tew, On the theory of complex\nrays,SIAM Review 41 (1999), 417–509.\n[8] P. Einzinger and L.B. Felsen, Evanescent waves and complex rays, IEEE Trans. Ant.\nProp. AP-30, 4 (1982), 594–605.12 R. MAGNANINI\n[9] P. Einzinger and S. Raz, On the asymptotic theory of inhomogeneous wave tracking,\nRadio Science 15 (1980) 763–771.\n[10] L.B. Felsen, Evanescent waves, J. Opt. Soc. Amer. 66 (1976) 751–760.\n[11] L.B. Felsen, Complex-source-point solutions of the field equations and t heir relation to\nthe propagation and scattering of Gaussian beams, Symposia Mathematica 18 (1976)\n39–56.\n[12] E. Heyman and L.B. Felsen, Evanescent waves and complex rays for modal propagation\nin curved open waveguides, SIAM J. Appl. Math. 43 (1983) 855–884.\n[13] Yu.A. Kravtsov, A modification of the geometrical optics method, Radiofizika 7 (1964)\n664–673.\n[14] Yu.A. Kravtsov, Asymptotic solutions of Maxwell’s equations near a caustic ,Radiofizika\n7 (1964) 1049-1056.\n[15] S. Kr¨ omer, Global invertibility for orientation-preserving Sobolev maps via invertibility\non or near the boundary , Arch. Rat. Mech. Anal. 238 (2020), 1113–1155.\n[16] R.M. Lewis, N. Bleistein and D. Ludwig, Uniform asymptotic theory of creeping waves,\nComm. Pure Appl. Math. 20 (1967) 295–328.\n[17] R.M. Lewis and J.B. Keller, Asymptotic methods for partial differential equations: the\nreduced wave equation and Maxwell’s equation. New York University, Courant Institute\nof Mathematical Sciences (1964).\n[18] R.K. Lunenburg, Mathematical Theory of Optics. University of California Press (1964).\n[19] D. Ludwig, Uniform asymptotic expansions at a caustic, Comm. Pure Appl. Math. 19\n(1966), 215-250.\n[20] D. Ludwig, Uniform asymptotic expansion of the field scattered by a conv ex object at\nhigh frequencies, Comm. Pure Appl. Math. 20 (1967), 103–138.\n[21] R. Magnanini and G. Talenti, On complex-valued solutions to a 2D eikonal equation.\nPart one: qualitative properties , in Nonlinear Partial Differential Equations, Gui-Qiang\nChen and Emmanuele Di Benedetto eds., Cont. Math. Amer. Math . Soc. 1999.\n[22] R. Magnanini and G. Talenti, On complex-valued solutions to a 2D eikonal equation.\nPart two: existence theorems , SIAM J. Math. Anal. 34 (2003), 805–835.\n[23] R.Magnanini and G.Talenti, On complex-valued solutions to a 2D eikonal equation. Part\nthree: analysis of a B¨ acklund transformation , Appl. Anal. 85, no. 1-3 (2006), 249–276.\n[24] R. Magnanini and G. Talenti, On complex-valued solutions to a 2D eikonal equation.\nPart four: continuation past a caustic , Milan Jour. Math. 77 (2009), 1–66.\n[25] R. Magnanini and G. Talenti, Approaching a partial differential equation of mixed\nelliptic-hyperbolic type , in Ill-posed and Inverse Problems, V.G. Romanov, S.I. Ka-\nbanikhin, Yu.E. Anikonov, A.L. Bukhgeim eds., VSP 2002, pp. 263–276.\n[26] W. Rudin, Real and Complex Analysis. Second edition. Mc Graw-Hill Book Co., New\nYork, D¨ usseldorf, Johannesburg, 1974.\n[27] I. N. Vekua, Generalized analytic functions. Pergamon Press, London-Paris-Frankfurt;\nAddison-Wesley Publishing Co., Inc., Reading, Mass. 1962.\nDipartimento di Matematica ed Informatica “U. Dini”, Univers it`a di Firenze, viale\nMorgagni 67/A, 50134 Firenze, Italy.\nEmail address :rolando.magnanini@unifi.it\nURL:http://web.math.unifi.it/users/magnanin"    },    {        "title": "2111.11797v2.Resonant_dynamics_of_skyrmion_lattices_in_thin_film_multilayers__Localised_modes_and_spin_wave_emission.pdf",        "content": "Resonant dynamics of skyrmion lattices in thin film multilayers: Localised modes and spin wave\nemission\nTitiksha Srivastava,1, 2,∗Yanis Sassi,1Fernando Ajejas,1Aymeric Vecchiola,1Igor Ngouagnia,2Herv ´e Hurdequint,2Karim\nBouzehouane,1Nicolas Reyren,1Vincent Cros,1Thibaut Devolder,3Joo-V on Kim,3,†and Gr ´egoire de Loubens2,‡\n1Unit´ e Mixte de Physique, CNRS, Thales, Universit´ e Paris-Saclay, 91767, Palaiseau, France\n2SPEC, CEA, CNRS, Universit´ e Paris-Saclay, 91191, Gif-sur-Yvette, France\n3Centre de Nanosciences et de Nanotechnologies, CNRS, Universit´ e Paris-Saclay, 91120, Palaiseau, France\nThe spectral signatures of magnetic skyrmions under microwave field excitation are of fundamental interest\nand can be an asset for high frequency applications. These topological solitons can be tailored in multilayered\nthin films, but the experimental observation of their spin wave dynamics remains elusive, in particular due to\nlarge damping. Here, we study Pt /FeCoB /AlO xmultilayers hosting dense and robust skyrmion lattices at room\ntemperature with Gilbert damping of ∼0.02. We use magnetic force microscopy to characterise their static mag-\nnetic phases and broadband ferromagnetic resonance to probe their high frequency response. Micromagnetic\nsimulations reproduce the experiments with accuracy and allow us to identify distinct resonant modes detected\nin the skyrmion lattice phase. Low ( <2 GHz) and intermediate frequency (2 −8 GHz) modes involve excitations\nlocalised to skyrmion edges in conjunction with precession of the uniform background magnetisation, while a\nhigh frequency ( >12 GHz) mode corresponds to in-phase skyrmion core precession emitting spin waves into\nuniform background with wavelengths in the 50–80 nm range commensurate with the lattice structure. These\nfindings could be instrumental in the investigation of room temperature wave scattering and the implementation\nof novel microwave processing schemes in reconfigurable arrays of solitons.\nThe dynamic response of magnetic materials at microwave\nfrequencies represents a rich field of research for its funda-\nmental interest and applications in information processing.\nLinear excitations in the form of spin waves (SWs) underpin\nthe field of magnonics, which describes the paradigm of trans-\nmitting and processing information with such waves1–3. SW-\nbased devices may allow for fast and energy-e fficient logic\napplications4, and a growing number of proposals have shown\npotential uses for computing5and spectral analysis6by SW\ninterference. Magnonic circuit elements such as transistors7,\ndiodes8and filters9have also been demonstrated in experi-\nments based on the manipulation of dipole-dominated SWs\nwith micrometer wavelengths. The generation and detection\nof shorter wavelength SWs, a prerequisite for miniaturiza-\ntion, is challenging, owing to the limitations of nanoscale fab-\nrication. Recent studies have shown that short wavelength\nemission can be achieved by broadband antennae10, grating\neffects11–13and spin torques14.\nIn this light, nonuniform magnetic textures have been ex-\nplored for generating and manipulating spin waves, and gen-\nerally o ffer an alternative route to expand the range of useful\nphenomena for applications15. Such textures can appear spon-\ntaneously at the micro- and nano-scale in magnetic materials\nas a result of the competing interactions, namely, exchange,\ndipolar, and anisotropy. Magnetic textures like bubbles16,17,\nstripes18–20, and vortices21,22have been shown to exhibit a\nrich diversity of magnetisation dynamics. For example, it has\nbeen demonstrated that magnetic domain walls can serve as\nnanoscale waveguides23,24, while the cores of magnetic vor-\ntices can be used to generate omnidirectional dipole-exchange\nSWs with sub-100 nm wavelengths25,26.\nRecently, topologically non-trivial chiral magnetic con-\nfigurations called skyrmions have generated much interest\nowing to their robust and particle-like nature27. They can\nbe stabilised at room temperature in thin films28,29with\nperpendicular magnetic anisotropy (PMA) and interfacialDzyaloshinskii-Moriya interaction (iDMI), which provides\ndifferent handles for tuning the desired magnetic parameters,\nboth statically and dynamically30. On one hand, their dc\ncurrent-driven dynamics31,32could be exploited for racetrack\nmemory and logic devices27, and on the other, their unique\nmicrowave response opens up the possibilities of skyrmion-\nbased spin-torque oscillators33, rf detectors34, and recon-\nfigurable magnonic crystals for microwave processing35–38.\nSkyrmions exhibit a rich variety of eigenmodes39–43, among\nwhich azimuthal bound states and breathing modes have been\nexperimentally observed at low temperatures in bulk crys-\ntals44–46, which have low damping parameters. Only very\nrecently, resonant dynamics with specific spectroscopic sig-\nnatures of thin-film multilayers hosting skyrmions has been\nevidenced47,48. However, much remains to be explored and\nunderstood concerning their individual and collective resonant\nresponse to microwave excitations.\nIn this work, we report a study of the resonant dynam-\nics of ultrathin film multilayers with perpendicular magnetic\nanisotropy, which host stable skyrmion lattices under ambi-\nent conditions with typical periods of 250 nm and skyrmion\ndiameters of 100 nm, while exhibiting Gilbert damping in\nthe range of α/similarequal0.02. By combining magnetic force mi-\ncroscopy (MFM) and ferromagnetic resonance (FMR) exper-\niments with micromagnetic simulations, we can identify dis-\ntinct SW modes associated with the skyrmion lattice phase.\nAt low frequency ( /lessorsimilar2 GHz), we observe a number of modes\nrelated to the precession of the uniform background state of\nindividual layers close to or at the surfaces of the stack, along\nwith eigenmodes localised to the skyrmion edges. At inter-\nmediate frequencies (2 −8 GHz), the precession of the uni-\nform background near the centre of the stack dominates the\nresponse. Similar modes were previously described and re-\nported in bulk crystals45,46, and lately in thin films47,48. In-\ntriguingly, we also observe a well-defined mode at high fre-\nquency (>12 GHz), which corresponds to the in-phase pre-arXiv:2111.11797v2  [cond-mat.mes-hall]  28 Jun 20222\ncession of the magnetisation within the skyrmion cores. The\ncores possess a distinct three-dimensional structure due to\nthe competition between all the existing magnetic interactions\nin these multilayers, notably the interlayer dipolar e ffects49.\nStrikingly, this precession is accompanied by the emission of\nspin waves, with wavelengths in the range of 50 to 80 nm,\ninto the uniformly magnetised background. These SWs in-\nterfere with those generated at neighbouring skyrmion cores,\nyielding a collective dynamical state governed by the subtle\ninterplay between the skyrmion diameter, the wavelength of\nthe emitted SWs, and the skyrmion lattice periodicity.\nRESULTS\nMultilayer composition\nThe basic element of the multilayer film studied is the\nPt(1.6) /Fe0.7Co0.1B0.2(1.2)/AlO x(1.0) trilayer (hereafter re-\nferred to as Pt /FeCoB /AlO x), where the figures in parenthe-\nses indicate the nominal film thickness in nm. This trilayer\nlacks inversion symmetry along the film thickness direction\nwhereby the Dzyaloshinskii-Moriya interaction is promoted\nat the interfaces of the ferromagnetic FeCoB film with Pt (and\npossibly AlO x50). The trilayer is repeated 20 times to form our\nmultilayer sample (see Methods), as shown in Fig. 1(a). The\nchoice of Fe-rich FeCoB and the aforementioned optimised\nthicknesses of Pt and FeCoB allow having su fficient DMI and\nPMA51–53to stabilise skyrmions, while limiting spin pumping\neffects which would otherwise lead to an increased damping\ncoefficient54. We estimate by FMR a Gilbert damping con-\nstant ofα=0.022 and an inhomogeneous broadening of ∼\n18 mT (see Supplementary Figure 1) for our samples, which\nare relatively low for such multilayer systems47,54. The over-\nall magnetic volume is enhanced with the 20 repetitions of the\ntrilayer, which not only increases the thermal stability of the\nskyrmions28but also provides a larger signal-to-noise ratio for\ninductive measurements.\nStatic characterization\nFigure 1(b) shows the out-of-plane hysteresis curve of the\nsample measured by alternating gradient field magnetome-\nter (AGFM), which is characteristic of thin films hosting\nskyrmions. The quasi-static magnetisation configuration of\nthe sample is probed by MFM using a low moment tip by\nsweeping the out-of-plane (OP) field from negative ( −z) to\npositive ( +z) saturation as illustrated in Fig. 1(d). On decreas-\ning the field from negative saturation, we observe nucleation\nof skyrmions with an average diameter around 100 nm (zone\n(i)), which develops into a dense lattice structure upon further\nreduction of the field (zone (ii)). In zone (iii) some of the\nskyrmions elongate and /or coalesce to form stripes leading to\na mixture of skyrmions and stripe domains. At small positive\nfields, labyrinthine domains (zone (iv)) become energetically\nstable and, upon increasing the field further, lead to the forma-\ntion of a dense skyrmion lattice once again (zone (vi)). From\nFIG. 1. Static characterization of the magnetic phases. (a)\nSchematics of the sample stack. (b) The out-of-plane (OP) mag-\nnetisation curve on sweeping the field from negative to positive val-\nues. (c) The corresponding skyrmion density and apparent skyrmion\ndiameter as a function of the field. The dashed red line indicates\nthe possible extrapolation of the observed skyrmion density at higher\npositive field values in accordance with the OP hysteresis curve. The\nskyrmion diameter is extracted only for the field ranges where the\nskyrmion density is close to maximum (see Methods). (d) Mag-\nnetic domain configurations of the sample: (i) randomly distributed\nskyrmions, (ii and vi) skyrmion lattice, (iii and v) skyrmion-stripe\nmix, and (iv) labyrinthine domains, measured by MFM as a function\nof OP magnetic field (corresponding to the values marked in (b))\nswept from negative to positive values.\nthe MFM images we extract the density of skyrmions as a\nfunction of the applied field µ0Hzand apparent skyrmion di-\nameter for the ranges of magnetic field where the skyrmion\ndensity is close to maximum, as shown in Fig. 1(c).\nIt is interesting to note that the observed skyrmion lattice is\nalmost quasi periodic hexagonal (see Supplementary Figure\n2) whereas more amorphous states were observed in previ-\nous studies47,48in such kind of multilayered films, indicating\nonly minor inhomogeneities and defects induced during the\nfilm growth in our samples. The skyrmion lattice phase is\nalso remarkably stable at positive field, as shown by the con-\nstant skyrmion density of about 21 per square micron between\n200 mT and 350 mT.\nBy using the period of the labyrinthine domain pattern in\nFig. 1(d)(iv) and the measured values of saturation magneti-3\nsation ( Ms=1.2 MA /m) and uniaxial anisotropy ( Ku=\n0.7 MJ/m3), we estimate the DMI to be D=1.2 mJ/m2with\nan exchange constant of A=15 pJ /m for our sample (see\nMethods), which are in good agreement with direct experi-\nmental determination of these parameters in Pt /FeCoB /MgO\nstacks55.\nDynamic characterization\nThe resonant dynamics of the sample was probed by broad-\nband FMR using a coplanar waveguide [Fig. 2(d)] and a vector\nnetwork analyser (VNA). The dc out-of-plane field is swept\nfrom negative to positive values while the frequency of the in-\nplane ac field is scanned over a range of 0.1–20 GHz for each\ndc field step. The real and the imaginary parts of the transmis-\nsion signal S21are recorded and processed to remove a back-\nground signal that is independent of the dc field, which im-\nproves the contrast (see Methods and Supplementary Figure\n3). The corresponding frequency versus field map is shown\nin Fig. 2(a). In the saturated state, we observe the high in-\ntensity Kittel mode (KM) along with two additional low in-\ntensity secondary modes at higher field (see also Supplemen-\ntary Figure 1). The latter are attributed to localised modes\nin the multilayer thickness that result from inhomogeneous\ninterfacial couplings of our multilayer system56. In addition\nto the KM observed above saturation fields, three groups of\nmodes with lower intensities appear when the magnetisation\nenters a non-uniform state. On ramping the field from nega-\ntive saturation towards zero, the KM softens close to µ0Hz=\n−335 mT where skyrmions start to nucleate (see MFM im-\nages in Fig. 1(d)). It then evolves into a mode with negative\nfield dispersion ( i.e.,∂f/∂|Hz|<0) as the skyrmion lattice\ngrows denser, which we call the intermediate frequency mode\n(IFM). At this point, a weak amplitude mode also emerges\nat high frequency ( >12 GHz) which has a positive field dis-\npersion ( i.e.,∂f/∂|Hz|>0) denoted as the high frequency\nmode (HFM). The IFM and HFM fade away close to zero\nfield, where the static magnetisation configuration consists of\nlabyrinthine domains. While the IFM reappears as the field\nis increased towards positive values corresponding to a mix-\nture of skyrmions and stripes, the HFM is visible only above\n200 mT when the magnetisation profile consists of a dense\nskyrmion lattice network (see Fig. 1(c)), and the IFM disper-\nsion nearly flattens. We also notice that at the very same field\na mode appears at low frequency ( <2 GHz) followed by an-\nother at even lower frequencies, which we refer to as low fre-\nquency modes (LFM). The HFM mode fades away at fields\nwhen the skyrmion lattice transforms into isolated skyrmions.\nThe IFM continues beyond this point however with an abrupt\nchange of slope until the magnetisation becomes uniform be-\nyond 390 mT. The line cuts along fixed fields ( −295 mT and\n295 mT) and fixed frequencies (14 GHz and 4 GHz) are shown\nin Figs. 2(b) and 2(c) respectively. The modes at negative and\npositive fields are not symmetric but instead depend on the\nmagnetic field history (see Supplementary Figure 4), as does\nthe static magnetisation profile.Micromagnetic simulations\nWe performed simulations with the finite-di fference mi-\ncromagnetic code M umax 357in order to gain better insight\ninto the static and dynamic properties of our sample (see\nMethods). We modelled the full 20-layer repetition of the\nPt/FeCoB /AlO xtrilayer (with periodic boundary conditions\nin the film plane) in order to account for dipolar e ffects as\naccurately as possible, since it is known that the skyrmion\ncore deviates from the usual tubular structure to complex\nthree-dimensional configurations due to inhomogeneous dipo-\nlar fields along the multilayer thickness49. Figures 3(a) and\n3(b) show the simulated hysteresis loop and the skyrmion den-\nsity and apparent diameter variation as a function of the OP\nfield swept from negative to positive values. The MFM im-\nages calculated from the simulated magnetisation profiles (see\nMethods) are shown in Fig. 3(c). The field evolution of the\nsimulated static characteristics presented in Figs. 3(a)–(c) is\nfound to be in good agreement with the experiments.\nA striking feature of the labyrinthine domain and skyrmion\nstructures found is that their micromagnetic configuration ex-\nhibits strong variations along the multilayer thickness direc-\ntion. An example of such complex structures is shown in\nFig. 3(d), where the mzcomponent is shown for a single\nskyrmion core in the lattice phase at µ0Hz=285 mT. The\nfigure shows the contours for mz=0.9 (red), mz=0 (yel-\nlow), and mz=−0.9 (blue) as surfaces where cubic interpo-\nlation has been used across the nonmagnetic layers. We note\nthat the skyrmion core, in particular the region of reversed\nmagnetisation mz≤−0.9, does not extend across the entire\nthickness of the multilayer. Recall that the FeCoB layers are\nonly coupled together through dipolar interactions, which are\nsufficiently large to maintain an alignment of the core cen-\ntre but too weak to promote a coherent magnetisation profile\nacross the di fferent layers. We can also observe that the mag-\nnetisation in the uppermost layers at the core centre is not re-\nversed at all, but slightly tilted away from the film normal as\nindicated by the presence of the inverted cone. The in-plane\ncomponents of the magnetic texture shown in Fig. 3(d) are\npresented in Fig. 3(e). Here, each cube represents the mag-\nnetic state of a finite-di fference cell, where the colour repre-\nsents the orientation of the magnetisation in the cell. In order\nto highlight the role of the in-plane components to comple-\nment the data shown in Fig. 3(d), the relative size and opacity\nof each cube is scaled with the function 1 −m2\nz; this renders\nthe regions of the uniform background and the reversed mag-\nnetisation transparent. A clear skyrmion profile can be seen\nfor the nine bottom layers, where the same left-handed ( i.e.\ncounterclockwise) N ´eel chirality is found in each layer. In\nlayers 10 to 14, on the other hand, the in-plane component\nof the magnetisation at the skyrmion boundary becomes more\nuniform and does not exhibit the same winding as in the bot-\ntom half of the stack. This means that the reversed domain\nstructure is non-topological. A skyrmion profile reappears in\nlayers 15, 16 and 17 but with a reversed chirality, where the\nin-plane magnetisation components of the right-handed ( i.e.\nclockwise) N ´eel structure are rotated by 180 degrees in the\nplane with respect to their left-handed counterparts. Finally in4\nFIG. 2. Broadband ferromagnetic resonance. (a) Frequency-field dispersion map measured by VNA-FMR over a frequency range of\n0.1 GHz to 20 GHz with the applied OP field swept from -500 mT to 500 mT. Apart from the Kittel mode (KM) above saturation, several\ndistinct modes appear in the non-saturated state which are labelled as HFM: high frequency mode ( f>12 GHz), IFM: intermediate frequency\nmode ( f<8 GHz) and LFM: low frequency modes ( f<2 GHz). (b) Line cuts for fixed OP field values: -295 mT and 295 mT. (c) Line cuts\nfor fixed frequency values: 4 GHz and 14 GHz. (d) Schematic of the VNA-FMR setup.\nlayers 18 to 20, we observe another type of non-topological\ntexture where the core magnetisation is closely aligned with\nthe background magnetisation, which corresponds to the in-\nverted cone at the top of the stack in Fig. 3(d). Figure 3(f)\nshows the variation of the topological charge density per layer\nas a function of the layer number, which shows that a similar\nthickness dependence is observed across the skyrmion lattice.\nThe skyrmions in the bottom half of the stack remain topo-\nlogical, while the top half comprises largely non-topological\nbubbles. Finally, Fig. 3(g) illustrates the magnetic configura-\ntion of layer 12 across the entire region simulated, where we\ncan observe that the mainly uniformly-magnetised regions of\nthe magnetic bubble walls can vary greatly from one bubble\nto the next, with no discernible spatial order. We have ver-\nified that these features persist for finite di fference cell sizes\ndown to∼2 nm, which indicates that the complex magneti-\nsation structure does not arise from discretization e ffects (see\nSupplementary Figure 5).\nWe next discuss the dynamical response of the system,\nwhere the frequency-dependent susceptibility is computed un-\nder di ���erent applied fields as in Fig. 2(a) (see Methods). The\nsimulated susceptibility map is shown in Fig. 4(a), which is\ndetermined from response of the static configurations com-\nputed in Fig. 3 to sinusoidal in-plane fields in the frequencyrange of 0.1 to 20 GHz. The Kittel mode (KM) is eas-\nily identified for the uniform state for fields above the sat-\nuration field. In the regime in which the magnetisation is\nnonuniform,−330< µ 0Hz<365 mT, three distinct types\nof modes can be identified, as illustrated by the line cut at\n285 mT shown in Fig. 4(b). As in experiments, the KM trans-\nforms into a negative field dispersion mode, an intermediate\nfrequency mode (IFM), at fields where skyrmion nucleation\nbegins. The IFM dispersion is rather rugged on the negative\nfield side, where the skyrmion density rapidly evolves with\nfield, and a faint splitting of IFM is seen around −260 mT.\nThe IFM is asymmetric with respect to zero field and exhibits\nsmoother variations on the positive field side. Similar to the\nnegative field side, another branch of the IFM is seen for pos-\nitive fields appearing at around 200 mT, which is relatively\nflat around 4 GHz in the range of positive magnetic fields\n240≤µ0Hz≤345 mT, where a dense skyrmion lattice is\nstable. It then merges to a single IFM and then varies sharply\nuntil saturation. Several low frequency modes (LFMs), com-\nprising several closely spaced branches in the frequency range\nof 0.5<f<2.5 GHz, extend over the entire magnetic field\nrange in which the magnetisation is non-uniform. Finally, a\nhigh frequency mode (HFM) is also observed above 10 GHz\nwith a positive field dispersion. It exhibits a sawtooth-like5\nFIG. 3. Micromagnetic simulations of equilibrium magnetic configuration. (a) Simulated magnetisation curve, (b) skyrmion density and\ndiameter evolution and (c) the corresponding simulated MFM images for the OP field swept from negative to positive values. The grey shaded\narea in (b) results from uncertainties in determining the skyrmion diameter, as discussed in the Methods section. (d) mzcontours and (e)\nin-plane components of the magnetisation texture of a single skyrmion in the lattice phase at 285 mT. (f) Topological charge density Qper\nlayer 285 mT. (g) Magnetic configuration of layer 12 at 285 mT.\nvariation at negative fields and a smoother variation for posi-\ntive fields. It is attenuated in the field range where magneti-\nsation profile consists of randomly oriented stripes (region\n(iv) in Fig. 3). The simulated susceptibility map reproduces\nwell the three families of modes observed experimentally in\nthe non-saturated state and gives good quantitative agreement\nfor the mode frequencies. Note that in experiments, the LFM\nbranches could not be resolved for the negative field side in the\nS21transmission data of Fig. 2(a) (as opposed to the S11and\nS22reflection data, presented in Supplementary Figure 6). It\nis because these modes have a nearly flat frequency–field dis-\npersion, and hence get diluted in the signal processing which\nis necessary to remove the field independent o ffset. Only two\nof the LFM branches become resolvable when they acquire\na certain slope on the positive field side, where the skyrmion\ndensity is constant and the skyrmion size strongly varies with\nthe field.\nThe spatial profiles of the excited modes are found by driv-\ning the system at the mode frequencies identified in Fig. 4(a)\nand by recording the resonant response. In Fig. 4(b), we\npresent the frequency response for µ0Hz=285 mT at which\nthe equilibrium ground state is a skyrmion lattice, where wecan identify well-defined peaks corresponding to the LFM,\nIFM, and HFM. Fig. 4(c) gives layer-resolved susceptibility\nwhose sum over all layers results in the curve in Fig. 4(b).\nFor the LFM, we observe that the peaks in Fig. 4(b) corre-\nspond to two kinds of resonances. For LFM 1 at f=0.94\nGHz, the primary response arises from the central part of the\nstack in layers 13 and 14, and also from the top of the stack in\nlayer 20. A similar behaviour is seen for LFM 2 at 1.50 GHz,\nwhere layers 1 and 19, close to the stack surfaces, resonate\nalong with the central part of the multilayer, as with LFM 3\nat 1.92 GHz, where a strong response is seen in layers 2 and\n18. This trend continues as the frequency increases, where\nsuccessive layers farther away from the surfaces exhibit local\nresonances in conjunction with the central part. These mode\nprofiles are visualised in Fig. 4(d), where we show the res-\nonant fluctuations in the xcomponent of the magnetisation,\n/angbracketleftδmx/angbracketright(see Methods). Blue regions give the main response to\nthe susceptibility, while the sum of equally intense blue and\nred regions cancel each other out. Similarly to Fig. 3(e), the\nopacity of the cells increase with the strength of the response\nin order to provide a clearer picture of their spatial profiles.\nLFM 1 mainly corresponds to the uniform precession of the6\nFIG. 4. Micromagnetic simulations of the dynamic response. (a) FMR spectra simulated by calculating the frequency-dependent suscepti-\nbility of multilayer to uniform excitation fields. (b) Imaginary part of the magnetic susceptibility at 285 mT (indicated by a dashed line in (a)),\nshowing three LFM (0.94, 1.50, and 1.92 GHz), two IFM (3.90 and 4.24 GHz) and one HFM (15.10 GHz). (c) Thickness-resolved susceptibil-\nity at 285 mT. (d, e, f) Three-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightfor (d) LFM, (e) IFM, and (f) HFM. (g, h, i) Two-dimensional\nview of the magnetic response /angbracketleftδmx/angbracketrightin layer 10 of the modes in (d,e,f) for (g) LFM, (h) IFM, and (i) IFM.\nbackground magnetisation at the top of the multilayer stack,\nwhile the skyrmion edges are also excited in the central part\nof the stack with a small overall contribution to the suscepti-\nbility. This can be seen more clearly in Fig. 3(g), where the\nfluctuations are shown for layer 10. For LFM 2 and LFM 3, a\nsimilar behaviour is seen where individual layers precess uni-\nformly close to the surfaces, while excitations in the central\npart are localised to skyrmion edges.\nFor the IFM, two broad resonances are identified at µ0Hz=\n285 mT and correspond to excitations that are mainly lo-\ncalised to the central part of the multilayer [Fig. 4(c,e)]. The\nmain contribution comes from the uniform precession of thebackground magnetisation between the skyrmion cores, i.e.,\nthe inter-skyrmion region which is aligned parallel to the ap-\nplied field, which can been seen from the predominantly blue\nregions in Fig. 4(h) that appear outside the cores. In addition,\nsome SW channelling is also observed at the skyrmion edges,\nas shown by alternating red and blue patches encircling the\nskyrmions.\nThe HFM involves the in-phase precession of the reversed\nmagnetisation within the skyrmion cores and extends across\nmost of the multilayer [Fig. 4(c,f)], with much of the power\nconcentrated in the central part of the stack. The coherence\nof the precession can be seen in Fig. 4(i) by the uniform blue7\nzones of/angbracketleftδmx/angbracketrightwithin the mz=0 contours denoted by the con-\ntinuous black circles. Moreover, this precession results in the\ngeneration of short wavelength SWs in the inter-skyrmion re-\ngion, which are almost as intense as the core precession. How-\never, only the volume occupied by the skyrmion cores pro-\nvides a coherent response to the susceptibility for the HFM,\nwhich explains its much weaker intensity compared to the\nIFM and KM.\nThese features are also present at other applied field val-\nues at which the equilibrium state comprises nonuniform\nmagnetisation. Supplementary Figure 7 shows an analo-\ngous behaviour at µ0Hz=−285 mT, where the lattice pos-\nsesses a lower skyrmion density with deformed core profiles.\nAtµ0Hz=140 mT where the equilibrium state comprises\nlabyrinthine domains, only the LFM and IFM are present,\nwhich further highlights the link between the HFM and the\nprecessing skyrmion cores (Supplementary Figure 8). At zero\nfield, the equilibrium state comprises a mixture of bubbles\nand elongated stripes in which no clear background orienta-\ntion can be defined. Here, LFM persist while the IFM and\nHFM are absent. Instead, we can observe the existence of cav-\nity modes that represent confined spin waves (Supplementary\nFigure 9).\nSpin wave emission from the high frequency mode\nThe short wavelength SWs associated with the HFM can-\nnot be excited directly with the uniform driving fields used\nin FMR, so their presence indicates the skyrmion cores act as\nnanoscale transducers that convert a spatially-uniform drive\ninto a spatially-nonuniform excitation. To obtain deeper in-\nsight into this process, we focus on the HFM in the simulation\ndata at two values of the applied fields, 300 and 364 mT, as\npresented in Fig. 5. The equilibrium state at 300 mT com-\nprises a skyrmion lattice, with a skyrmion density of 20 µm−2\nand an average core-to-core distance of L=262 nm, while\nonly one metastable skyrmion is present at 364 mT within the\nsame area. The skyrmion diameters are roughly equivalent at\nthese field values, which are 95 and 94 nm, respectively. The\nsusceptibility at these two fields, shown in Fig. 5(a) for the\nrange of 12–17 GHz, display a clear resonance peak around\n15.5 GHz, which is over an order of magnitude larger at\n300 mT than at 364 mT due to the di fference in the skyrmion\ndensity.\nAt both field values, the coherent precession of the reversed\nmagnetisation within the skyrmion cores emit spin waves into\nthe ferromagnetic background. Their wavelengths are simi-\nlar at both fields and are estimated from the /angbracketleftδmx/angbracketrightprofiles in\nlayer 10 shown in Figs. 5(b) and 5(c), yielding λ/similarequal60 nm and\nλ/similarequal54 nm atµ0Hz=300 mT and µ0Hz=364 mT, respec-\ntively. Despite the strong variations in the static core profile\nacross the multilayer thickness, the emitted spin waves are re-\nmarkably coherent across the thickness. This can be seen in\nFig. 5(d) for 300 mT and Fig. 5(e) for 364 mT where coher-\nent wavefronts, driven in the central part of the multilayer at\nwhich the core diameter is greatest, extend across much of the\nmultilayer thickness. The left inset in Figs. 5(d,e) indicate thenumber of topological charges per layer, which shows that the\nspin wave emission is just as e ffective for the topological bub-\nbles residing primarily in the bottom of the stack as for the\nnon-topological bubbles residing primarily in the top.\nThe overall coherence of these spin wave excitations\ncan also be seen from the layer-averaged spatial discrete\nFourier transform of the dynamical magnetisation shown in\nFigs. 5(b,c). These Fourier maps, shown in Figs. 5(f,g), dis-\nplay a ring pattern with wave vectors 100 <kSW<150µm−1\ncorresponding to wavelengths 40 < λ < 60 nm. At 300 mT,\nthe pattern in the Fourier transformed data also manifests the\nperiodic structure of the skyrmion lattice, from which the lat-\ntice periodicity can be estimated as shown in Fig. 5(f), using\n2π/k=262 nm, where k=24µm−1.\nThe spin wave dynamics can be understood from the lin-\nearised equations of motion for the dynamic magnetisation\nδm,∂tδm=−|γ0|(δm×Heff,0+m0×δHeff), where m0is\nthe static magnetisation configuration and Heff,0is the static\neffective (internal) field associated with m0, whileδHeffis the\ndynamic e ffective (internal) field associated with δm. The full\n3D structure of the core and the surrounding configuration de-\ntermines the spatial profile of the static internal field and the\nlocalisation of the mode. The average value of this static in-\nternal field Heff,0(r) inside the skyrmion core can be extracted\nfrom micromagnetic simulations and is about −200 mT, which\nleads to resonances at 5.8 GHz without any confinement ef-\nfects. To this, one needs to add the dynamic dipolar and ex-\nchange contributions δHeff(r,t) due to the localisation of mag-\nnetisation dynamics in a volume of ∼100×100×50 nm3,\nwhich increases the resonant frequency to about 15 GHz (see\nSupplementary Note 2). Similarly, we can estimate the wave-\nlength of the emitted SWs by using the dispersion relation\nof forward-volume SWs in the inter-skyrmion region. At\n300 mT, the average internal field in this region is around\n110 mT, which results in a wavelength of 60 nm at 15.5 GHz, a\ngroup velocity of 960 m /s and an attenuation length of 490 nm\n(see Supplementary Figure 10). These estimations are in good\nagreement with the SW characteristics observed in Fig. 5.\nThe di fference between the mode profiles shown in\nFigs. 5(b) and 5(c) is due to interference e ffects. At low\nskyrmion densities, the SW emitted at one core attenuates be-\nfore reaching another, resulting in the familiar concentric rip-\nples around each excitation source. At higher skyrmion densi-\nties, on the other hand, the inter-skyrmion distance is smaller\nthan the attenuation length and interference patterns can arise.\nThe cores can emit SWs in a broad spectral range, which de-\npends on the excitation frequency and the skyrmion size (see\nSupplementary Figure 11). As such, the overall collective be-\nhavior is governed by a subtle interplay between the charac-\nteristic length scales of the system. The skyrmion diameter,\nwhich varies with the applied magnetic field, determines the\nprecession frequency of the core thereby the wavelength of\nthe emitted SWs, whereas constructive or destructive SW in-\nterference patterns are determined by the commensurability of\nSW wavelength and lattice parameters.8\nFIG. 5. Emission and interference of spin waves in skyrmion lattice. (a) Simulated susceptibility curves at 300 mT and 364 mT with the\ndynamic magnetisation profiles /angbracketleftδmx/angbracketrightin (b) and (c) of layer 10, respectively, corresponding to the resonance at 15.5 GHz. Cross-sections\nof/angbracketleftδmx/angbracketrightmarked by dotted lines in (b) and (c) are also shown respectively in panels (d) and (e), highlighting the in-phase skyrmions core\nprecession (in blue) and spin wave emission in the surrounding medium (alternate blue and red zones) (f,g) Power spectral density (PSD) of\nthe layer averaged spatial discrete Fourier transform of the dynamic magnetisation profiles indicating the average k S W-vector of the emitted\nSWs at 300 mT and 364 mT respectively. At 300 mT, the skyrmion lattice parameter can also be deduced from the k=24µm−1periodic\npattern seen in the spatial Fourier transform.\nDISCUSSION\nSW emission related to the HFM can have advantages over\nother reported methods of sub-100 nm wavelength genera-\ntion. For example, magnetic vortex cores, which can act as\nSW emitters, require confined geometries such as dots for\ntheir existence and do not spontaneously form lattice struc-\ntures in continuous films, which precludes the appearance of\nSW interference phenomena seen here. The precession fre-\nquency of the skyrmions in our samples is also above the FMR\ngap of the surrounding domain background and hence SWs\ncan be excited at this frequency in the same medium, unlike\nin other hybrid structures such as Pt /Co/YIG58. Moreover,\nthe SWs generated in PMA materials are forward-volume\nSWs, which propagate isotropically59compared to backward-\nvolume and Damon-Eshbach modes in in-plane magnetised\nsystems, while maintaining high group velocities.\nAn interesting analogy of our skyrmion lattice as SW emit-\nters can be made with arrays of spin-torque nano-oscillators,\nwhich can generate propagating SWs and get synchronised via\nthem60. These structures have been proposed for implement-\ning unconventional computing schemes5. We have shown\nhere that SWs can be emitted in the uniform background by\nthe skyrmion dynamics, but one also expects that in return,\nSWs can influence the skyrmion dynamics39, which could en-\nlarge the processing capabilities of this type of devices.\nOur study also paves the way toward creating reconfig-\nurable magnonic crystals, since the skyrmion lattices we ob-serve are quasi-periodic and dynamically tunable37. We can\nget a glimpse of the role of this lattice in the observation\nof SW interference patterns in the simulated HFM. How-\never, the properties of the skyrmion lattice as a magnonic\ncrystal also needs to be explored using broadband propagat-\ning SW spectroscopy10or advanced SW imaging methods9,25.\nLastly, our system could serve as a platform to study magnon-\nskyrmion scattering39, emergent magnon motifs61, or chiral\nmagnonic edge states62.\nMETHODS\nExperimental details\nTa(10) /Pt(8)/[FeCoB(1.2) /AlO x(1.0)/Pt(1.6)] 19/FeCoB(1.2)\n/AlO x(1.0)/Pt(3) multilayer stacks (where FeCoB ≡\nFe70Co10B20, and thicknesses given in nm) were deposited by\ndc magnetron sputtering at room temperature on thermally-\noxidised silicon substrates, under Ar gas flow at a pressure\nof 0.25 Pa. The base pressure of the sputtering equipment\nwas 5×10−6Pa. magnetisation cycles were measured by\nalternating gradient force magnetometry on 3 ×3 mm2films.\nMFM images were obtained at room temperature and ambient\npressure using a double-pass lift mode, detecting the phase\nshift (MFM signal) of the second pass after a topographic\nmeasurement. The employed low moment tips were home\nmade49. Small permanent magnets with tunable gap and9\ncalibrated values of field were used to bias the samples while\nimaging them. VNA-FMR was conducted on 2 ×4 mm2films\nfacing the 50 µm wide central conductor of a 50 Ωmatched\ncoplanar waveguide. The OP magnetic field was ramped up\nand down between −550 mT and +550 mT. For each of the\n400 field steps, the VNA frequency was swept from 0.1 GHz\nto 20 GHz with a number of points set to 1491 and a sweep\ntime of 45 s. To improve the signal to noise ratio, the data\nwere averaged over 30 magnetic field cycles. To remove the\nfield independent background and normalise the transmission\nsignal data S21, we calculate d DS21using the derivative\ndivide method63. To increase the contrast further, we apply an\nadditional frequency di fferentiation, and plot the imaginary\npart of the obtained quantity in Fig. 2(a).\nExtraction of magnetic parameters\nThe magnetisation of the studied multilayers was deter-\nmined by SQUID magnetometry. The 1.2 nm thick FeCoB\nis found to have a saturation magnetisation Ms=1.20±\n0.1 MA /m. Using angle dependent cavity-FMR, the gyro-\nmagnetic ratio γ0/(2π)=28.75±0.10 GHz /T and the total\nuniaxial anisotropy field including demagnetising and PMA\ncontributions µ0(Ms−Hk)=354.7±0.5 mT are determined\n(see Supplementary Note 1). In the case of uniaxial anisotropy\nµ0Hk=2Ku/Ms, hence Ku=0.69±0.11 MJ /m3. By as-\nsuming Ms=1.2 MA /m and Ku=0.7 MJ/m3, the DMI and\nexchange parameters were iteratively determined by adjusting\nthe period of the simulated labyrinthine domains to the one\n(/similarequal170 nm) observed by MFM at remanence49. We find the\nDMI constant to be D=1.2 mJ/m2for an exchange constant\nA=15 pJ /m.\nDetermination of skyrmion diameter\nFrom the MFM measurements of the quasi static mag-\nnetic configuration as a function of the applied field, we ex-\ntract skyrmion density by using a MATLAB program which\nuses binarisation. However, as the image resolution is not\ngood enough to estimate the skyrmion diameters directly from\nthe MFM images (which would anyway require to know the\ntransfer function of the tip64, and might be complicated by\nthe influence of the tip’s stray field), we extract the appar-\nent skyrmion diameters (2 Rsk) indirectly, for the field values\nwhere the skyrmion density ( dsk) is close to maximum, us-\ning the out-of-plane experimental magnetisation curve: Rsk=/radicalbig\n(1−mz)/(2πdsk). In Fig. 1(c) the error bars constitute the er-\nrors on the skyrmion density (proportional to the density and\ninversely proportional to the image size: ∼8%) and magnetic\nfield values to account for the slight misalignment of the mag-\nnetic pole and the magnetic tip in the MFM setup (10%). In\nthe case of simulated MFM images, we used the same MAT-\nLAB program that binarises the simulated MFM images, de-\ntects and finds the diameters of all the skyrmions, and calcu-\nlates the mean and the standard deviation. For the simula-\ntions, we also compared the skyrmion diameter i /calculatedby the program and ii /estimated from skyrmion density and\nmzcurve as a function of field when the skyrmion density is\nclose to maximum. The small di fference in the skyrmion di-\nameters for the two methods lies within the standard deviation\nof the former method, indicated by the shaded area in grey in\nFig. 3(b).\nMicromagnetic simulations\nSimulations were carried out with the open-source finite-\ndifference micromagnetics code M umax 357. We used values\nofMs=1.3375 MA /m and Ku=0.9 MJ /m3, which are\nslightly larger than the experimentally-determined values but\nwere chosen to reproduce quantitatively the mode frequencies\n(see Supplementary Figure 12). The geometry consists of a\nrectangular slab of volume 1 µm×1µm×72 nm discretised\nusing 128×128×60 rectangular cells for the susceptibility\ncalculations and a volume of 2 µm×2µm×72 nm with\n256×256×60 cells for the quasi-static hysteresis loop calcu-\nlations. The simulated stack consists of 20 ferromagnetic lay-\ners with thickness 1.2 nm separated by 2.4 nm empty spacers,\nwhich accurately mimics the experimental stack, in which the\ntotal thickness of non-magnetic Pt and AlO xlayers separating\nadjacent FeCoB layers is 2.6 nm. Periodic boundary condi-\ntions are used in xandydirections in the film plane. The code\nperforms a numerical time integration of the Landau-Lifshitz-\nGilbert equation for the magnetisation dynamics,\ndm\ndt=−|γ0|m×Heff+αm×dm\ndt, (1)\nwhere/bardblm(r,t)/bardbl=1 is a unit vector representing the ori-\nentation of the magnetisation field, γ0=µ0gµB//planckover2pi1is the\ngyromagnetic constant, and αis the Gilbert damping con-\nstant. The e ffective field, Heff=−(1/µ0Ms)δU/δm, repre-\nsents a variational derivative of the total magnetic energy U\nwith respect to the magnetisation, where Ucontains contri-\nbutions from the Zeeman, nearest-neighbour Heisenberg ex-\nchange, interfacial Dzyaloshinskii-Moriya, and dipole-dipole\ninteractions along with a uniaxial anisotropy with an easy axis\nalong ˆz, perpendicular to the film plane. Note that while the\nlateral cell size of 7.8125 nm used for the simulations pre-\nsented here is larger than the characteristic exchange length,\nlex=/radicalbig\n2A/µ0M2s/similarequal3.65 nm, we have verified that using\nsmaller discretisation cells do not a ffect the salient features\npresented here (see Supplementary Figure 13).\nTo simulate quasi-static processes such as hysteresis loops,\nwe begin with a uniform magnetic state oriented along −z\nunder an applied field of µ0Hz=−500 mT. A Langevin-\ndynamics simulation with T=300 K is then performed over 1\nns to mimic experimental conditions in which thermal fluctu-\nations help to sample di fferent metastable states in the energy\nlandscape. The system is then relaxed without the preces-\nsional term in Eq. (1) toward an energy minimum. This proce-\ndure of running Langevin dynamics, followed by relaxation, is\nthen repeated two more times. After this, the micromagnetic\nconfiguration is recorded and assigned as the equilibrium state\nfor the applied field value. This equilibrium state then serves10\nas the initial state for the next value of the applied field, which\nis increased to µ0Hz=500 mT in increments of 1 mT. The\nsimulated MFM images are computed from the equilibrium\nstate by assuming a sample-to-tip distance of 50 nm.\nTo simulate the microwave susceptibility of the FeCoB\nmultilayer, we compute the response of the magnetisation to\nharmonic excitation fields. While it is common practice (and\nusually more e fficient computationally) to calculate the eigen-\nmode spectra from the Fourier transform of the transient re-\nsponse to pulsed fields, such an approach for the present sys-\ntem results in strong artifacts arising from non-resonant drift\nof the background magnetic state, such as the displacement\nof the skyrmion lattice or labyrinthine domain structure. In-\nstead, we adopt an approach that more closely resembles the\nexperimental protocol. For each value of the external field\nHz, we compute the response of the precomputed static con-\nfiguration, m0(r), to sinusoidal in-plane fields with frequency\nf,hx=hrfsin(2πf t), over 20 periods τ=1/f, where the\nspatial average of the resulting component mx(t) is recorded\nat intervals of 0.1 τ. In this procedure, the Gilbert damping\nparameter is set to α=0.02 and the excitation amplitude to\nµ0hrf=0.1 mT (1 mT in Fig. 5 and Supplementary Figure 12).\nWe estimate the imaginary part of the susceptibility χat this\nfrequency, Im[ χ(f)], by projecting/angbracketleftmx(t)/angbracketrightonto−cos(2πf t),\nIm/bracketleftbigχ(f)/bracketrightbig/similarequal−1\n20τ/integraldisplay20τ\n0/angbracketleftmx(t)/angbracketrightcos(2πf t)dt, (2)\nwhich is the quantity shown in Figs. 4(a) and 4(b). Im[ χ(f)]\nexhibits peaks at resonance when mxis in perfect quadrature\nwith respect to hx. This procedure is repeated as a function of\nfranging from 0.1 GHz to 20 GHz in steps of 0.1 GHz for\nthe map in Fig. 4(a) and in steps of 0.02 GHz for the data in\nFigs. 4(b,c).\nThe spatial mode profile at a given mode resonance fn\nis extracted by driving the system harmonically at fnwith\nhx=hrfsin(2πfnt)and by recording, cell-by-cell, the compo-\nnents of magnetisation at intervals of 0.1 τn, as for the calcula-\ntion of the susceptibility. The mode profiles are characterised\nby the fluctuation amplitude /angbracketleftδmx(r)/angbracketright, which is computed in\nan analogous way to the susceptibility, where we drive the\nsystem over 100 periods and then compute, after transients\nare washed out,\n/angbracketleftδmx(r)/angbracketright=−1\n2τn/integraldisplay2τn\n0[mx(r,t)−mx,0(r)] cos (2πfnt)dt,(3)\nover two additional periods. Visually, dark blue regions con-\ntribute most to the overall Im[ χ(f)], while equally dark redand blue regions cancel each other out because they represent\nfluctuations with opposite sign.\nDATA A V AILABILITY\nData are available from the corresponding authors upon rea-\nsonable request.\nACKNOWLEDGEMENTS\nThis work was partially supported by the French National\nResearch Agency (ANR) under grant no. ANR-17-CE24-\n0025 (TOPSKY) and the Horizon2020 Research Framework\nProgramme of the European Commission under grant nos.\n824123 (SKYTOP) and 899646 (k-NET). It is also supported\nby a public grant overseen by the ANR as part of the “In-\nvestissements d’Avenir” program (Labex NanoSaclay, refer-\nence: ANR-10-LABX-0035).\nAUTHOR CONTRIBUTIONS\nN.R., V .C., T.D., J.-V .K and G.d.L. conceived the project.\nThe samples were designed, grown and characterised by T.S.,\nY .S., F.A., N.R. and V .C. T.S. performed the MFM experi-\nments with the help of A.V . and K.B. H.H. conducted the\ncavity-FMR measurements. T.S. and I.N. performed initial\nFMR characterizations. T.D. conducted the VNA-FMR ex-\nperiments and analyzed the results with T.S. and G.d.L. J.-\nV .K. performed the micromagnetic simulations and analyzed\nthem with T.S. and G.d.L. T.S., J.-V .K. and G.d.L. wrote the\nmanuscript. All the authors discussed the data and commented\non the manuscript.\nCOMPETING INTERESTS\nThe authors declare no competing interests.\nADDITIONAL INFORMATION\nSupplementary information The online version contains\nsupplementary material available at https: //doi.org /xxxxx.\nCorrespondence and requests for materials should be ad-\ndressed to T.S., J.-V .K. or G.d.L.\n∗titiksha.srivastava@cea.fr\n†joo-von.kim@c2n.upsaclay.fr\n‡gregoire.deloubens@cea.fr\n1V . V . Kruglyak, S. O. Demokritov, and D. Grundler, “Magnonics,”\nJ. Phys. D Appl. Phys. 43, 264001 (2010).\n2M. Krawczyk and D. 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Mater. 551, 169073 (2022).Supplementary Information —\n“Resonant dynamics of skyrmion lattices in thin film multilayers: Localised modes and spin\nwave emission”\nTitiksha Srivastava,1, 2,∗Yanis Sassi,1Fernando Ajejas,1Aymeric Vecchiola,1\nIgor Ngouagnia,2Herv ´e Hurdequint,2Karim Bouzehouane,1Nicolas Reyren,1\nVincent Cros,1Thibaut Devolder,3Joo-V on Kim,3,†and Gr ´egoire de Loubens2,‡\n1Unit´ e Mixte de Physique, CNRS, Thales, Universit´ e Paris-Saclay, 91767, Palaiseau, France\n2SPEC, CEA, CNRS, Universit´ e Paris-Saclay, 91191, Gif-sur-Yvette, France\n3Centre de Nanosciences et de Nanotechnologies, CNRS, Universit´ e Paris-Saclay, 91120, Palaiseau, France\n∗titiksha.srivastava@cea.fr\n†joo-von.kim@c2n.upsaclay.fr\n‡gregoire.deloubens@cea.frarXiv:2111.11797v2  [cond-mat.mes-hall]  28 Jun 20222\nCONTENTS\nSupplementary Note 1. Additional FMR measurements in the saturated state 3\nAngle dependent cavity-FMR 3\nDamping parameter 3\nSupplementary Note 2. Analysis of the high frequency mode 4\nInternal field in skyrmion lattice phase 4\nContributions to the eigenfrequency 4\nCharacteristics of the spin waves emitted by the skyrmion cores 5\nSupplementary Figures 6\nSupplementary References 193\nSUPPLEMENTARY NOTE 1. ADDITIONAL FMR MEASUREMENTS IN THE SATURATED STATE\nAngle dependent cavity-FMR\nCavity-FMR measurements are performed at a fixed frequency of 9.725 GHz to probe the FMR of\nthe multilayer sample as a function of the angle θHbetween the applied field and the film plane. Out-\nof-plane (OP) and in-plane (IP) configurations correspond to θH=0◦andθH=90◦, respectively. The\nresonant field measured at θH=0◦is higher than at θH=90◦, meaning that the overall anisotropy is in-\nplane. From the values of the resonant field in these two configurations, one can extract the gyromagnetic\nratioγ0/(2π)=28.75 GHz /T and the total uniaxial anisotropy field including demagnetising and PMA\ncontributions µ0(Ms−Hk)=354.7 mT. Using these values, the full angular variation of the resonance field is\naccurately reproduced by the theoretical expectation [1], as shown by the continuous line in Supplementary\nFigure 1(a). The angular dependence of the full width ∆Hat half maximum of the resonance peak is\ndisplayed in Supplementary Figure 1(b). The minimal value of 33.2 mT is observed in the IP configuration,\nwhile the linewidth is only slightly larger in the OP configuration, ∆H=35.5 mT; it is rather low for this\ntype of multilayer film, which evidences its good dynamical quality.\nDamping parameter\nTo separate the intrinsic and extrinsic contributions to the FMR linewidth, broadband-FMR measure-\nments are performed in the IP configuration. At 10 GHz, the extracted linewidth is 32.5 mT, in good\nagreement with the cavity-FMR results. The frequency dependence of ∆His plotted in Supplementary Fig-\nure 1(c) together with its linear fit. The latter yields the frequency independent contribution to the linewidth,\nfound to be about 18 mT, which indicates a fair amount of inhomogeneities in the film. The frequency linear\npart of the linewidth yields the Gilbert damping parameter of the film, α=0.022±0.003.4\nSUPPLEMENTARY NOTE 2. ANALYSIS OF THE HIGH FREQUENCY MODE\nInternal field in skyrmion lattice phase\nThe spatial profile of the total internal field ( i.e., the sum of the applied, anisotropy, and dipolar fields)\ncan be obtained from micromagnetic simulations. The map obtained at 300 mT is shown for layer 12 in\nSupplementary Figure 10(a). It shows that su fficiently far away from the skyrmions, the static internal field\nis rather homogeneous in the background domains, around 110 to 120 mT. Due to the strong dipolar fields\nin the vicinity of the oppositely magnetised skyrmion cores, the internal field substantially increases while\napproaching the skyrmion boundaries, as shown by the closely spaced isocontours. Inside the skyrmion\ncores it becomes negative, about −200 mT.\nContributions to the eigenfrequency\nAs shown by Fig. 5 of the main text the high frequency mode (HFM) corresponds to a precession of\nthe magnetisation localized inside the skyrmion cores. Its frequency can be calculated as the convolution\nof its spatial profile with a frequency operator taking into account all the contributions to the magnetic\nenergy (Eq. 8 in ref. [2]): Zeeman, anisotropy, dipolar and exchange energies. Even for homogeneously\nmagnetised non-ellipsoidal bodies, it is di fficult to evaluate the magnetic self-interactions (magneto-dipolar\nand inhomogeneous exchange) contributing to the eigenfrequency. As shown in Fig. 3 of the main text,\nthe static configuration in the 20 dipolarly coupled magnetic layers is quite complex in the skyrmion phase.\nHowever, to roughly estimate the HFM frequency, one can consider the averaged value of the static internal\nfield inside the skyrmion core where the precession is localized, to which the dynamic magneto-dipolar\nself-interaction ( i.e.the depolarization energy of the mode profile on itself) and the exchange contributions\nshould be added. The averaged value µ0Heff,0≈−200 mT of the internal field in the core translates into a\nfrequency of 5 .8 GHz, which is less than the HFM frequency of 15.5 GHz determined by micromagnetic\nsimulations. The exchange contribution to the frequency is given by ( γ0Ak2\nHFM)/(πMs) where kHFM is the\neffective wavevector of the mode. Assuming an e ffective wavelength of the order of the skyrmion diameter\n≈100 nm, one obtains an exchange contribution of 2.6 GHz. The evaluation of the contribution due to the\ndynamic magneto-dipolar self-interaction is more di fficult, because it requires the knowledge of static and\ndynamic magnetisation profiles to perform the required integration. However, the missing 7.1 GHz fall in\nthe frequency range of this contribution in submicron samples [2].5\nCharacteristics of the spin waves emitted by the skyrmion cores\nTo find the characteristics of the spin waves (SWs) emitted by the precession of the skyrmion cores in\nthe surrounding background domains, one should consider that the latter are homogeneously magnetised\nOP along the applied field and use the dispersion relation of forward-volume waves [3]:\nω(k)=/radicalBigg/parenleftBigg\nωHeff,0+2Aγ0\nMsk2/parenrightBigg/parenleftBigg\nωHeff,0+2Aγ0\nMsk2+ωM/parenleftBigg\n1−1−e−kt\nkt/parenrightBigg/parenrightBigg\n, (S1)\nwhereωHeff=γ0Heff,0andωM=γ0M. To take into account the multilayer nature of the film, the dynamic\ndipolar term is evaluated using an e ffective film medium of total thickness t=20×(tFeCoB +tPt+tAlOx)=\n76 nm and magnetisation M=tFeCoB/(tFeCoB +tPt+tAlOx)Ms=0.42 MA /m. This dispersion relation is\nplotted in Supplementary Figure 10(b) using µ0Heff,0=110 mT, which corresponds to the mean value of\nthe static internal field in the interskyrmion region at the bias field of 300 mT shown in Supplementary\nFigure 10(a). The skyrmion core precession frequency at this field is equal to 15.5 GHz, as indicated by the\nhorizontal dashed line that crosses the dispersion curve at k=105µm−1, corresponding to a wavelength of\nλ=59.8 nm. This value is in excellent agreement with the one directly determined from the SW radiation\npattern of Fig. 5(c) in the main text. Moreover, it is also possible to compute the group velocity Vg=∂ω/∂ k\nof the forward-volume wave in the e ffective film, which is plotted in Supplementary Figure 10(c). The\ngroup velocity of the SWs propagating at 15.5 GHz is found to be Vg=960 m /s. Based on Eq. (S1),\nforward volume SWs exhibit almost circular precession in our material, therefore one can use τ=1/(αω)\nto evaluate their lifetime. At ω/(2π)=15.5 GHz,τ=1/(αω)=0.513 ns, which allows to estimate the\nattenuation length of this forward-volume SW, Latt=Vgτ=493 nm.6\nSUPPLEMENTARY FIGURES\nSupplementary Figure 1 |Ferromagnetic resonance in the saturated state . (a) Evolution of the resonance field as\na function of the angle θHbetween the applied field and the film plane, determined by cavity-FMR at 9.725 GHz. The\ncontinuous line is the theoretical expectation using the values of gyromagnetic ratio and total uniaxial anisotropy field\nextracted from out-of-plane (OP) and in-plane (IP) configurations given in the text. (b) Corresponding dependence of\nthe FMR linewidth on θH. The dashed line is a guide to the eye. Inset: spectrum measured in the OP configuration.\nThe resonance field of the uniform Kittel mode is 691.2 mT, additional modes are observed at 718.3 mT and 737.2 mT.\n(c) Dependence of the in-plane FMR linewidth on frequency measured by broadband FMR. The red line is the linear\nfit yielding the damping parameter and the inhomogeneous contribution.7\nSupplementary Figure 2 |Skyrmion lattice. (a) MFM image of skyrmions in disordered state at -280 mT, and\n(b) nearest neighbour (NN) probability distribution. (c) Skyrmion lattice at -190 mT, and (d) its nearest neighbour\nprobability distribution, demonstrating its almost quasi periodic hexagonal structure.8\nSupplementary Figure 3 |VNA-FMR data processing . (a) Intensity map of the 30-cycle averaged transmission\nsignal S21in the field–frequency coordinates, the field being swept upwards. (b) Map of the S21data processed\nusing the derivative method. (c) Map of the S21data processed using the derivative method followed by a frequency\ndifferentiation. (d–f) same as (a–c) for the magnetic field being swept downwards. In addition to the narrow signals\nascribed to ferromagnetic resonances in the multilayer sample, there is a very broad background signal coming from\nthe measurement cell which is symmetric between negative and positive field values, as can be seen from the distorted\n”V” shape in panels (b) and (e), and independent of the sweep direction – in strong contrast with the resonances\noriginating from the multilayer sample (see also Suppl. Fig. 4). The frequency di fferentiation attenuates the broad\nbackground signal and improves the signal-to-noise ratio of the narrow resonances from the sample, as seen from\npanels (c) and (f).9\nSupplementary Figure 4 |Hysteretic character of the FMR modes . (a) OP magnetisation curves determined using\nalternating gradient force magnetometry by sweeping the field up and down. (b) Corresponding dependence of FMR\nmodes determined by broadband VNA-FMR.10\nSupplementary Figure 5 |Mesh size dependence of static states . The simulated static properties, as discussed in\nFig. 3 of the main text, are shown for three di fferent discretisation schemes for 1 µm of the simulated geometry: 128 (as\ndiscussed in the main text), 256, and 512 cells. (a–c), Hysteresis loops. (d–f), Density plots of the topological charge\ndensity per layer as a function of perpendicular applied field Bz. (g–i), In-plane components of the magnetisation\nconfiguration for a skyrmion at Bz=285 mT. (j–l), Topological charge density as a function of layer number at\nBz=285 mT. (m–o), Magnetisation configuration of layer 12 at Bz=285 mT, where the dashed boxes indicate the\nstate shown in (g–i), respectively.11\nSupplementary Figure 6 |VNA-FMR reflection parameters . Real parts of the (a) S11and (b) S22parameters\nprocessed using the derivative divide method. A broad signal between 0.5 GHz and 1.5 GHz is observed on the full\nfield range between −330 mT and 390 mT, which corresponds to the simulated low frequency modes (LFM).12\nSupplementary Figure 7 |Micromagnetic simulations of the dynamic response at µ0Hz=−285mT. (a) Field-\ndependent susceptibility map, reproduced from Fig. 4(a). (b) Imaginary part of the magnetic susceptibility at -285 mT,\nshowing six LFM (0.6, 1.06, 1.16, 1.4, 1.6, and 2.08 GHz), two IFM (2.92 and 3.1 GHz) and one HFM (13.52 GHz).\n(c) Thickness-resolved susceptibility at -285 mT. (d, f, g) Three-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightfor\n(d) LFM, (f) IFM, and (g) HFM. (e, h, i) Two-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightin layer 10 of the\nmodes in (d,f,g) for (e) LFM, (h) IFM, and (i) IFM, respectively.13\nSupplementary Figure 8 |Micromagnetic simulations of the dynamic response at µ0Hz=140mT. (a) Field-\ndependent susceptibility map, reproduced from Fig. 4(a). (b) Imaginary part of the magnetic susceptibility at 140 mT,\nshowing five LFM (1.02, 1.2, 1.36, 1.96, and 3 GHz) and one IFM (6.54 GHz). No HFM is observed because\nno skyrmions are present. (c) Thickness-resolved susceptibility at 140 mT. (d, e) Three-dimensional view of the\nmagnetic response /angbracketleftδmx/angbracketrightfor (d) LFM and (e) IFM. (f, g) Two-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightin\nlayer 10 of the modes in (d,e) for (f) LFM and (g) IFM, respectively.14\nSupplementary Figure 9 |Micromagnetic simulations of the dynamic response at µ0Hz=0mT. (a) Field-\ndependent susceptibility map, reproduced from Fig. 4(a). (b) Imaginary part of the magnetic susceptibility at 0 mT,\nshowing four LFM (0.7, 1.06, 1.56, and 2.1 GHz) and three cavity modes (CM) (8.02, 10.62, and 13.14 GHz), where\nthe latter represent spin wave excitations confined within the bubble or elongated stripe domains. (c) Thickness-\nresolved susceptibility at 0 mT. (d, e) Three-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightfor (d) LFM and (e)\nCM. (f, g) Two-dimensional view of the magnetic response /angbracketleftδmx/angbracketrightin layer 10 of the modes in (d,e) for (f) LFM and\n(g) CM, respectively.15\nSupplementary Figure 10 |Internal field and spin wave dispersion . (a) Map of the internal field extracted from\nmicromagnetic simulations at 300 mT in layer 12. Values are given in units of mT. (b) Dispersion relation of forward-\nvolume SWs calculated using Eq. S1 for a uniform internal field of 110 mT. (c) Corresponding group velocity. The\ndashed lines in panels (b) and (c) highlight the SWs excited at 15.5 GHz, whose wavelength and group velocity are\nrespectively found to be λ=59.8 nm and Vg=960 m /s.16\nSupplementary Figure 11 |Frequency dependence of the SW interference pattern . Dynamic magnetisation\nexcited in layer 10 at 300 mT (top) and 364 mT (bottom) by an in-plane microwave field of frequency ranging\nfrom 10 GHz to 18 GHz, and associated layer averaged spatial FFT. The most e fficient frequency to drive the core\nprecession corresponds to the resonance frequency of 15.5 GHz. When using a lower or higher frequency to drive the\nsystem, its overall response has a lower amplitude. In addition, following the trend of the dispersion relation shown\nin Supplementary Figure 10(b), the wave vector of the emitted SWs increases with the excitation frequency of the\nskyrmion core, as seen from the FFT maps. As a result, the resulting interference patterns also change, in a way\nthat depends on the skyrmion lattice, as shown by the comparison between 300 mT and 364 mT. More generally, the\ninterference patterns depend on the characteristic lengths of the system, which can be tuned by the applied OP field.\nBoth the density and the diameter of the skyrmions indeed depend on the field (see Fig. 3(b) of the main text), which\ndetermines the travel length of the SWs between the boundaries of neighbouring skyrmions.17\nSupplementary Figure 12 |Optimisation of magnetic parameters used in micromagnetic simulations . Simulated\nimaginary part of the dynamic susceptibility, Im[ χ(f)], as a function of applied perpendicular field, µ0Hz, for devia-\ntions from experimentally-determined values of the micromagnetic parameters, (a) Ms,0=1.2 MA /m,A0=15 pJ /m,\nKu,0=0.70 MJ /m3, and D0=1.2 mJ /m2. (b) Ku=Ku,0+0.04 MJ /m3, (c) Ku=Ku,0−0.04 MJ /m3, (d) A=A0+5\npJ/m, (e) A=A0−5 pJ/m, (f) D=D0+0.03 mJ /m2, (g) Experimental values with applied field tilted 2◦from z-axis\nalong the xdirection, (h) Ms=Ms,0+0.1 MA /m and Ku=Ku,0+0.14 MJ /m3, (i)Ms=Ms,0+0.175 MA /m and\nKu=Ku,0+0.26 MJ /m3, and (j) Ms=Ms,0+0.1375 MA /m and Ku=Ku,0+0.20 MJ /m3, which represents the best\nfit. The blue dots represent the experimental data.18\n-500 -400 -300 -200 -100 0 100 200 300 400 50002468101214161820\n-500 -400 -300 -200 -100 0 100 200 300 400 50002468101214161820\n-500 -400 -300 -200 -100 0 100 200 300 400 50002468101214161820\nSupplementary Figure 13 |Mesh size dependence of dynamic susceptibility . Simulated imaginary part of the\ndynamic susceptibility, Im[ χ(f)], as a function of applied perpendicular field, µ0Hz, for three finite-di fference dis-\ncretization schemes of the 1000 ×1000×72 nm simulation box: (a) 128 ×128×60, (b) 256×256×60, and (c) 512\n×512×60 finite di fference cells. The discretisation scheme and corresponding cell size is indicated on each figure.\nThe micromagnetic parameters used correspond to the experimental values [see Supplementary Figure 12(a).].19\nSUPPLEMENTARY REFERENCES\n[1] Hurdequint, H. FMR studies of ultrathin permalloy layers sandwiched by Al 2O3.J. Magn. Magn. Mater. 242-245 ,\n521–524 (2002).\n[2] Naletov, V . V . et al. Identification and selection rules of the spin-wave eigenmodes in a normally magnetized\nnanopillar. Phys. Rev. B 84, 224423 (2011).\n[3] Kalinikos, B. A. & Slavin, A. N. Theory of dipole-exchange spin wave spectrum for ferromagnetic films with\nmixed exchange boundary conditions. J. Phys. C 19, 7013 (1986)."    },    {        "title": "2111.15142v1.First_and_second_order_magnetic_anisotropy_and_damping_of_europium_iron_garnet_under_high_strain.pdf",        "content": "1 \n First and second order magnetic anisotropy and damping of \neuropium iron garnet under high strain  \n \nVíctor H. Ortiz1, Bassim Arkook1, Junxue Li1, Mohammed Aldosary1, Mason Biggerstaff1, Wei \nYuan1, Chad Warren2, Yasuhiro Kodera2, Javier E. Garay2, Igor Barsukov1*, and Jing Shi1* \n \n1Department of Physics and Astronomy, University of California, Riverside, CA 92521, \nUSA  \n2 Department of Mechanical and Aerospace Engineering, University of California, San \nDiego, CA 92093, USA  \n \n \nUnderstanding and tailoring static and dynamic properties of magnetic insulator thin films \nis important for spintronic device applications. Here, we grow atomically flat epitaxial europium \niron garnet (EuIG) thin films by pulsed laser deposition on (111) -oriented garnet sub strates with a \nrange of  lattice parameters. By controlling the lattice mismatch between EuIG and the substrates, \nwe tune the strain in EuIG films from compressive to tensile regime, which is characterized by X -\nray diffraction. Using ferromagnetic resonance , we find that in addition to the first -order \nperpendicular magnetic anisotropy which depends linearly on the strain, there is a significant \nsecond -order one that has a quadratic strain dependence. Inhomogeneous linewidth of the \nferromagnetic resonance inc reases notably with increasing strain, while the Gilbert damping \nparameter remains nearly constant (≈ 2× 10-2). The se results provide valuable insight into the  spin \ndynamics in ferrimagnetic insulators and useful guidance for material synthesis and engineer ing \nof next -generation spintronics applications.  \n \n \n*: Corresponding authors: Igor Barsukov ( igorb@ucr.edu ) and Jing Shi ( jing.shi@ucr.edu )   2 \n Ferrimagnetic insulators (FMIs) have played an important role in uncovering a series of \nnovel spintronic effects such as spin Seebeck effect (SSE) and spin Hall magnetoresistance (SMR). \nIn addition, FMI thin films have proved to be an excellent source of proximity -induced \nferromagnetism in adjacent layers (e.g., heavy metals [1], graphene  [1] and topological \ninsulators  [2]) and of pure spin currents  [3–6]. FMIs have also been shown to be a superb medium \nfor magnon spin currents with a long decay length  [7,8]. Among FMIs, rare earth iron  garnets \n(REIGs) in particular have a plethora of desirable properties for practical applications: high Curie \ntemperature (T c > 550  K), strong chemical stability, and relatively large band gaps (~ 2.8 eV).  \nCompared to other magnetic materials, REIGs are distinct owing to their magnetoelastic \neffect with the magnetostriction coefficient ranging from -8.5×106  to +21 ×106 at room \ntemperature  [9] and up to two orders of magnitude increases at low temperatures  [10]. This  unique  \nfeature allows for tailoring ma gnetic anisotropy in REIG thin films via growth, for example, by \nmeans of controlling lattice mismatch with substrates, film thickness, oxygen pressure, and \nchemical substitution. In thin films, the magnetization usually prefers to be in the film plane due  \nto magnetic shape anisotropy; however, the competing perpendicular magnetic anisotropy (PMA) \ncan be introduced by utilizing magneto -crystalline anisotropy or interfacial strain, both of which \nhave been demonstrated through epitaxial growth  [11–14]. In the  study of Tb 3Fe5O12 (TbIG) and \nEu3Fe5O12 (EuIG) thin films, the PMA field H2ꓕ was found to be as high as 7 T under interfacial \nstrain  [11], much stronger than the demagnetizing field. While using strain is proven to be an \neffective way of manipulating magn etic anisotropy, it often comes at a cost of increasing magnetic \ninhomogeneity and damping of thin films  [15,16].  \nIn this work, we investigate the effect of strain on magnetic properties of (111) -oriented \nEuIG thin films  for the following reasons: (1) The  spin dynamics in EuIG bulk crystals is \nparticularly interesting but has not been studied thoroughly in the thin film form. Compared to \nother REIGs, the Eu3+ ions occupying the dodecahedral sites (c -site) should have the J = 0 ground \nstate according to the Hund’s rules, which do not contribute to the total magnetic moment; \ntherefore, EuIG thin films can potentially have a ferromagnetic resonance (FMR) linewidt h as \nnarrow as that of Y 3Fe5O12 (YIG)  [17,18] or Lu 3Fe5O12 (LuIG)  [19]. In EuIG crystals, a very \nnarrow linewidth (< 1 Oe)  [20] was indeed observed at low temperatures, but it showed a nearly \ntwo orders of magnitude increase at high temperatures, which ra ises fundamental questions \nregarding the damping mechanism responsible for this precipitous change. (2) Although it has 3 \n been shown that the uniaxial anisotropy can be controlled by moderate strain for different substrate \norientations and even in polycrysta lline form  [21], the emergence of the higher -order anisotropy \nat larger strain, despite its technological significance, has remained elusive.  \nWe grow EuIG films by pulsed laser deposition (PLD) from a target densified by powders \nsynthesized using the meth od described previously  [22]. The films are deposited on (111) -oriented \nGd3Sc2Ga3O12 (GSGG), Nd 3Ga5O12 (NGG), Gd 2.6Ca0.4Ga4.1Mg 0.25Zr0.65O12 (SGGG), \nY3Sc2Ga3O12 (YSGG), Gd3Ga5O12 (GGG), Tb 3Ga5O12 (TGG) and Y 3Al5O12 (YAG) single crystal \nsubstrates, with the lattice mismatch 𝜂=𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 −𝑎𝐸𝑢𝐼𝐺\n𝑎𝐸𝑢𝐼𝐺 (where 𝑎 represents the lattice parameter \nof the referred material) ranging from +0.45% (GSGG) to -3.95% (YAG) in the decreasing order \n(see Table I). After the standard solvent cleaning process, the substrates are annealed at 220 °C \ninside the PLD chamber with the base pressure lower than 10-6 Torr for 5 hours prior to deposition . \nThen the temperature is increased to ~ 600 °C in the atmosphere of 1.5 mT orr oxygen mixed with \n12% (wt.) ozone for 30 minutes. A 248 nm KrF excimer pulsed laser is used to ablate the target \nwith a power of 156 mJ and a repetition rate of 1 Hz. We crystalize the films by ex situ   annealing \nat 800 °C for 200 s in a steady flow of  oxygen using rapid thermal annealing  (RTA) .  \nReflection high energy electron diffraction (RHEED) is used to evaluate the crystalline \nstructural properties of the EuIG films grown on various substrates (Fig. 1a). Immediately after \nthe deposition, RHEED dis plays the absence of any crystalline order. After ex situ  rapid thermal \nannealing, all EuIG films turn into single crystals. We carry out atomic force microscopy (AFM) \non all samples and find that they show atomic flatness and good uniformity with root -mean-square \n(RMS) roughness < 2 Å (Fig. 1b). In addition, we perform X -ray diffraction (XRD) on all samples \nusing a Rigaku SmartLab with Cu K α radiation with a Ni filter and Ge(220) mirror as \nmonochromators, at room temperature in 0.002° steps over the 2  range from 10° to 90°  [23]. In \na representative XRD spectrum (Fig. 1c), two (444) Bragg peaks are present, one from the 50 nm \nthick EuIG film and the other from the YSGG substrate, which confirms the epitaxial growth and \nsingle crystal structure of the fi lm without evidence of any secondary phases. Other REIG films \ngrown under similar conditions , i.e., by PLD in oxygen mixed with ozone at ~600 °C during \nfollowed by RTA, have shown no observable interdiffusion across the interface from high \nresolution trans mission electron microscopy  and energy dispersive X -ray spectroscopy  (Fig. S1 , \n[24]). The EuIG Bragg peak ( a0 = 12.497 Å) is shifted with respect to the expected peak position \nof unstrained bulk crystal, indicating a change in the EuIG lattice parameter pe rpendicular to the 4 \n surface ( aꓕ). For the example shown in Fig. 1c, the EuIG (444) peak shifts to left with respect to \nits bulk value, indicating an out -of-plane tensile strain and therefore an in -plane compressive strain \nin the EuIG lattice.  \n \nA common app roach for inferring the in -plane strain ε|| of thin films from the standard −2 \nXRD measurements involves the following equation  [23],  \n \n𝜀∥= −𝑐11+2 𝑐12+4 𝑐44\n2𝑐11+4 𝑐12−4 𝑐44 𝜀⊥,  with 𝜀⊥=𝑎⊥−𝑎𝑜\n𝑎𝑜,    (1) \n \nwhere a0 is the lattice parameter of the bulk material, and aꓕ can be calculated using 𝑎⊥=\n𝑑ℎ𝑘𝑙√ℎ2+𝑘2+𝑙2 from the interplanar distance 𝑑ℎ𝑘𝑙  obtained from the XRD data (Fig. S 2, [25]), \nand cij are the elastic stiffness constants of the crystal which in most  cases can be found in the \nliterature  [9]. However, due to the wide range of strain values studied in this work and the \npossibility that the films may contain different amounts of crystalline defects, we perform \nreciprocal space mapping (RSM) measurements on a subset  of our EuIG samples (Fig. S 3, [26]) \nand compared the measured in -plane lattice parameters with the calculated ones using Eq. 1. We \nobserve that the average in -plane strain s measured by RSM has a  systematic difference of 40% \nfrom  the calculated values based on the  elastic properties (Fig. S 4, [26]). Given this nearly constant \nfactor  for all measured films, we find that the elastic stiffness constants of our EuIG films may \ndeviate  from the literature  reported bulk values , possibly due to stochiometric deviations  or slight \nunit cell distortion  in thin films . Here we adopt the reported lattice parameter value ( a0 = 12.497 \nÅ) as the reference due to the difficulty of  grow ing sufficiently thick, unstrained EuIG films usin g \nPLD .  \nIn the thickness -tuned magnetic anisotropy study [11], the anisotropy field in REIG films is \nfound to be proportional to η/(t+t o), which was attributed to the relaxation of strain as the film \nthickness  t increases. Here in EuIG samples with small lattice mismatch η (e.g., NGG/EuIG), the \nstrain is mostly preserved in 50 nm thick films (pseudomorphic regime), whereas for larger η (e.g., \nYAG/EuIG ), the lattice parameter of EuIG films shows nearly complete structural relaxation to \nthe bulk value. For this reason, in the samples with larger η (YAG = -3.95 %, GSGG = 0.45%), \nwe grow thinner EuIG films (20 nm) in order to retain a larger in -plane strain (compressive for 5 \n YAG, tensile for GSGG).  For EuIG films gr own on TGG and GGG substrates, the paramagnetic \nbackground of the substrates is too large to obtain a reliable magnetic moment measurement of the \nEuIG films; therefore, the results of thinner films on these two substrates are not included in this \nstudy.  \nRoom-temperature magnetic hysteresis curves for YSGG/EuIG sample are shown in Fig. 1d \nwith the magnetic field applied parallel and perpendicular to the film  [26]. The saturation field for \nthe out -of-plane loop (~1100 Oe) is clearly larger than that for the i n-plane loop, indicating that \nthe magnetization prefers to lie in the film plane. Moreover, since the demagnetizing field 4π Ms \n(≈ 920 Oe) is less than the saturation field in the out -of-plane loop (Fig. S 5, [27]), it suggests the \npresence of additional easy -plane anisotropy result ing from the magnetoelastic effect due to \ninterfacial strain. As shown in this example, we can qualitatively track the evolution of the \nmagnetic anisotropy in samples with different strains. However, this approach cannot provide a \nquantitative description when high -order anisotropy contributions are involved.  \nTo quantitatively determine magnetic anisotropy in all EuIG films, we perform polar angle  \n(H)-dependent FMR measurements using an X -band microwave cavity with f requency f = 9.32 \nGHz and field modulation. The samples are rotated from H = 0° to H = 180° in 10° steps, where \nH = 90° corresponds to the field parallel to the sample plane (Fig. 2a). The spectra at Η = 0° for \nall samples are displayed in Fig. 2b and show a single resonance peak which can be  well fitted by \na Lorentzian derivative. Despite different strains in all s amples, the resonance field Hres is lower \nfor the in -plane direction ( H = 90°) than for the out -of-plane direction ( H = 0°). A quick \ninspection reveals that the out -of-plane Hres shifts to larger values as η increases in the positive \ndirection (e.g., fro m YAG/EuIG to GSGG/EuIG), corresponding to stronger easy -plane \nanisotropy. Furthermore, the Hres values at θΗ = 0° show a large spread among the samples. Fig. 2c \nshows a comparison of FMR spectra at different polar angles between two representative samples: \nNGG/EuIG (small η) and YAG/EuIG (large η). \nFigs. 3a -c show  Hres vs. θH for three representative EuIG films . To evaluate  magnetic \nanisotropy, we fit the data using the Smit -Beljers  formalism by considering the first -order  \n−𝐾1cos2𝜃  and the second -order −1\n2𝐾2cos4𝜃  uniaxial anisotropy energy terms  [28]. From this \nfitting, we extract the parameters 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠− 2𝐾1\n𝑀𝑠= 4𝜋𝑀𝑠- 𝐻2⊥ and 𝐻4⊥= 2𝐾2\n𝑀𝑠 (see Table \nI), her e 𝐻2⊥ and 𝐻4⊥ being the  first- and second - order anisotropy fields, respectively , and favoring  6 \n out-of-plane (in -plane) orientation of magnetization when they are positive (negative). The \nspectroscopic  g-factor is treated as a fitted parameter  which is found as a nearly constant , g = \n1.40 (Fig. S 6, [28]), in accordance to the previous results obtained by Miyadai  [31]. In Figs. 3d \nand 3e, we present 𝐻2⊥ and 𝐻4⊥ as functions of the measured out -of-plane strain 𝜀⊥ and in -plane \nstrain 𝜀∥. Clear ly, the magnitude of  4𝜋𝑀𝑒𝑓𝑓 is greater than the demagnetizing field for EuIG \n4𝜋𝑀𝑠=920  𝑂𝑒; therefore,  𝐻2⊥ is negative for all samples, i.e., favoring the in -plane orientation. \nAs shown in  Fig. 3d, |𝐻2⊥| increases linearly with increasing in -plane strain η. This is consistent \nwith the magnetoelastic effect in (111) -oriented EuIG films  [9]. As briefly discussed earlier,  due \nto the constant scaling factor between the calculated and measured 𝜀∥, we rewrite t he \nmagnetoelastic contribution to the first -order perpendicular anisotropy as −9𝛯\n3𝑀𝑠𝜀⊥, with the \nparameter 𝛯 containing the information related to the magnetoelastic constant λ111 and elastic \nstiffness cii. We fit the magnetoelastic equation in Ref.  [11] using the parameter 𝛯 and obtain 𝛯=\n−(7.06±0.95)×104 𝑑𝑦𝑛𝑒\n𝑐𝑚2 from the slope. On the other hand, based on  the reported literature \nvalues ( 𝜆111=+1.8×10−6, c11 = 25.10 ×1011 dyne/cm2, c12 = 10.70 ×1011 𝑑𝑦𝑛𝑒\n𝑐𝑚2, c44 = 7.62 ×1011 \n𝑑𝑦𝑛𝑒\n𝑐𝑚2) [10], we obtain  𝛯𝑙𝑖𝑡=−6.12×104 𝑑𝑦𝑛𝑒\n𝑐𝑚2. This result suggests  that even though  the actual \nelastic properties of our EuIG films may be different from the ones reported in for EuIG crystals  \ndue to  the thin film unit cell distortion (Table S1 , [32]), the pertaining parameter 𝛯 appears to be \nrelatively insensitive to variations of stoichiometry . The intercept of the  straight -line fit  should \ngive the magneto -crystalline anisotropy coefficient  of EuIG  Kc.  We find Kc = (+62.76 ± 0.18 ) × \n103 erg/cm3, which is differ ent from the previously reported values for EuIG bulk crystals in both \nthe magnitude and sign ( Kc = -38 × 103 erg/cm3) [31]. Similar growth -modified magneto -\ncrystalline anisotropy was observed in EuIG  films grown with relatively lo w temperatures \n(requiring post -deposition annealing to crystalize)  [10]. In the absence of  interfacial interdiffusion, \nthe anomalous anisotropy may be related to partial deviation from the chemical ordering of the \ngarnet structure  [31].  \n By comparing the first - and second -order anisotropy fields 𝐻2⊥ and 𝐻4⊥ vs. 𝜀∥ plotted in \nFigs. 3d and 3e, we find that the former dominates over the entire range of 𝜀∥ (except for \nYAG/EuIG). In contrast to the linear dependence for 𝐻2⊥, 𝐻4⊥ can be fitted well with a quadratic \n𝜀∥ dependence , which is not surprising for materials with large magnetostriction constants (such 7 \n as EuIG) under large strains. For relatively small 𝜀∥, the linear strain term in the magnetic \nanisotropy energy  dictates . For large 𝜀∥, higher -order strain terms may not be neglected. By \nincluding the ( 𝜀∥cos2θ)2 term, we obtain excellent fitting to the FMR data, indicating that the \nsecond -order expansion in 𝜀∥ is adequate. In contrast to  𝐻2⊥, 𝐻4⊥ is always positive, thus favoring \nout-of-plane magnetization orientation. It is worth pointing out that for YAG and TGG, the \nmagnitude of the 𝐻2⊥ becomes comparable with that of the 𝐻4⊥, but the sign differ s. Comparison \nof 𝐻4⊥ with 4𝜋𝑀𝑒𝑓𝑓 reveals that a coexistence (bi -stable) magnetic state can be realized when \n𝐻4⊥>4𝜋𝑀𝑒𝑓𝑓 [31, 33 -35]. The results are summarized in Table I.  \nThe above magnetic anisotropy energy analysis only deals with  the polar angle  dependence , \nbut in principle, it can also vary in the film plane and  therefore depend on the azimuthal angle.  To \nunderstand the latter, w e perform azimuthal angle dependent FMR measurements on all samples. \nWe indeed observe a six -fold in -plane anisotropy in Hres due to the crystalline symmetry  of EuIG  \n(111). However, the amplitude of  the six -fold Hres variation is less than 15 Oe, about two orders \nof magnitude smaller than the average value of Hres for most samples, thus we omit the in-plane \nanisotropy in our analysis.  \nBesides the Hres information, t he FMR spectra  in Fig. 2 c reveal s significant variations in \nFMR linewidth, which contains information  of magnetic inhomogeneity and Gilbert damping. To \ninvestigate these properties systematically, we perform broad -band (up to 15  GHz) FMR \nmeasurements with m agnetic field applied in the film plane, using a coplanar waveguide setup. \nFrom the frequency dependence of Hres, we obtain 4𝜋𝑀𝑒𝑓𝑓 and g independently via fitting the data \nwith the Kittel equation. These values agree very well with those previously found from the polar \nangle dependence. We plot the half width at half maximum, ∆𝐻, as a function of frequency  f in \nFig. 4a. While ∆𝐻 varies significantly across the samples, the data for  each sample fall \napproximately on a straight line and the slope of ∆𝐻 vs. 𝑓 appears to be visibly close  to each other. \nFor a quantitative evaluation of ∆𝐻, we consider the following contributions: the Gilbert damping \n∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 , two -magnon scatt ering ∆𝐻𝑇𝑀𝑆, and the inhomogeneous linewidth ∆𝐻0 [36], \n \n∆𝐻=∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 +∆𝐻𝑇𝑀𝑆 +∆𝐻0 .     (3) \n 8 \n The Gilbert term, ∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 =2𝜋𝛼𝑓\n|𝛾|, depends linearly on f, where α is the Gilbert damping \nparameter; the two -magnon term is described through ∆𝐻𝑇𝑀𝑆 =𝛤0𝑎𝑟𝑐𝑠𝑖𝑛  √√𝑓2+(𝑓𝑜\n2)2\n−𝑓𝑜\n2\n√𝑓2+(𝑓𝑜\n2)2\n+𝑓𝑜\n2  [37], \nwhere  𝛤0 denotes the magnitude of the two -magnon scattering, f0 = 2γMeff; and ∆𝐻0, the \ninhomogeneous linewidth w hich is frequency independent.  \nBy fitting Eq. (3) to the linewidth data, we obtain quantitative information on magnetic \ndamping through the Gilbert parameter and two -magnon scattering magnitude as well as the \nmagnetic inhomogeneity  [39–40]. In Fig. 4a, the overall linear behavior for all samples is an \nindication of a relatively small two -magnon scattering contribution ∆𝐻𝑇𝑀𝑆 which therefore may \nbe disregarded in the fitting process. Figs. 4b and 4c  show both  ∆𝐻0 and α vs. 𝜀∥. It is cl ear that \nfour of the samples with the smallest ∆𝐻0 (~ 10 Oe) are those with relatively low in -plane strain \n(|𝜀∥|<0.30% ). In the meantime, the XRD spectra  of these samples show  fringes characteristic of \nwell conformed crystal planes (Fig. S 2), and moreover,  the RSM plots (Fig. S 3) reveal a uniform \nstrain distribution in the films  [41]. On the compressive strain side, ∆𝐻0 increases steeply to 400 \nOe at 𝜀∥ ~ -0.40 %, and their XRD spectra  show no fringes and the RSM graphs indicate  non-\nuniform strain relaxation in the samples (Figs. S2 and S3 ). In sharp contrast to the ∆𝐻0 trend, the \nGilbert damping α remains about 2 ×10-2 over the entire range of 𝜀∥, sugges ting that the intrinsic \nmagneti c damping of EuIG films is nearly unaffected by the inhomogeneity. In fact, the magnitude \nof α is significantly larger than that of YIG  [17,18] or LuIG films  [19], which is somewhat \nunexpected for Eu3+ in EuIG with J = 0. A possible reason for this enhance d damping is that other \nvalence states of Eu such as Eu2+ (J =7/2) may be present, which leads to non -zero magnetic \nmoments of Eu ions in the EuIG  lattice and thus results in a larger damping constant, common to \nother REIG with non -zero 4f -moments  [42]. The X -ray photoelectron spectroscopy data taken on \nYSGG(111)/EuIG(50 nm) (Fig. S7 , [43]) indicates such a possibility. While the FMR linewidth \npresents large variations across the sample set, we have identified that the non-uniform strain \nrelaxation process caused by large lattice mismatch with the substrate is a main source of the \ninhomogeneity linewidth ∆𝐻0, but it does not affect the Gilbert damp ing α. The results raise \ninteresting questions on the mechanisms of intrinsic damping and the origin of magnetic \ninhomogeneity in EuIG thin films , both of which  warrant further investigations.  9 \n In summary, we find that uniaxial magnetic anisotropy in PLD -grown EuIG(111) thin films \ncan be tuned over a wide range via magnetostriction and lattice -mismatch induced strain. The first -\norder anisotropy field depends linearly on the strain and the second order anisotropy field has a \nquadratic dependence. While non -uniform strain relaxation significantly increases the magnetic \ninhomogeneity, the Gilbert damping remains nearly constant over a wide range of in -plane strain. \nThe results demonstrate broad tunab ility of magnetic properties in REIG films and provide \nguidance for implementation of EuIG for spintronic applications. Further studies to elucidate the \nrole of Eu2+ sites in magnetic damping  are called upon.  \n \nWe thank Dong Yan and Daniel Borchardt for the ir technical assistance. This work was supported \nas part of the SHINES, an Energy Frontier Research Center funded by the US Department of \nEnergy, Office of Science, Basic Energy Sciences under Award No. SC0012670. J.S. \nacknowledges support by DOE BES Award  No. DE -FG02 -07ER46351 and I.B. acknowledges \nsupport by the National Science Foundation under grant number NSF -ECCS -1810541.  \n  10 \n References  \n \n[1] Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi, Proximity -Induced Ferromagnetism in \nGraphene Revealed by the Anomalous Hall Effect , Phys. Rev. Lett. 114, 016603 (2015).  \n[2] Z. Jiang, C. -Z. Chang, C. Tang, P. Wei, J. S. Moodera, and J. Shi, Independent Tuning of \nElectronic Properties and Induced Ferromagnetism in Topological Insulators with \nHeterostructure Approach , Nano Lett. 15, 5835 (2015).  \n[3] K. 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Interfaces 13, 20288 (2021).  \n[40] I. Barsukov, H. K. Lee, A. A. Jara, Y. J. Chen, A. M. Gonçalves, C. Sha, J. A. Katine, R. \nE. Arias, B. A. Ivanov, and I. N. Krivorotov, Giant Nonlinear Damping in Nanoscale \nFerromagnets , Sci. Adv. 5, 1 (2019).  \n[41] X. Guo, A. H. Tavakoli, S. Sutton, R. K. Kukkadapu, L. Qi, A. Lanzirotti, M. Newville, \nM. Asta, and A. Navrotsky, Cerium Substitution in Yttrium Iron Garnet: Valence State, \nStructure, and Energetics , (2013).  \n[42] C. Tang, P. Sellappan, Y. Liu, Y. Xu, J. E. Garay, and J. Shi, Anomalous Hall Hysteresis \nin Tm 3Fe5O12/Pt with Strain -Induced Perpendicular Magnetic Anisotropy , Phys. Rev. B \n94, 140403 (2016).  \n[43]    See Supplemental Information figure S7 at http://placeholder.html for the XPS spectra and \nanalysis.  \n \n 14 \n Figures  \n \n \n \n \n \n \n \n \nFigure  SEQ Figure \\* ARABIC 1 : Structural and magnetic property characterization of \nEuIG 50 nm film grown on YSGG(111) substrate. (a) Reflection high energy electron \ndiffraction (RHEED) pattern along the direction, displaying single crystal structure after \nrapid thermal anneal ing process. (b) 2 mm  2 mm atomic force microscope (AFM) surface \nmorphology scan, demonstrating a root -mean -square (RMS) roughness of 1.7 Å. (c) \nIntensity semi -log plot of \n  - 2\n XRD scan. The dashed line corresponds to the XRD peak \nfor bulk EuIG. (d) Mag netization hysteresis loops for field out -of-plane and in -plane \ndirections.  Figure 1: Structural and magnetic property characterization of EuIG 50 nm film grown on \nTGG(111)  substrate . (a) Reflection high  energy electron diffraction  (RHEED ) pattern along the \n⟨112⟩ direction, displaying single crystal structure after rapid thermal annealing process. (b) 5 mm \n 5 mm atomic force microscope ( AFM ) surface morphology scan, demonstrating  a root-mean -\nsquare (RMS) roughness of 1.8 Å. (c) Intensity semi -log plot of  - 2 XRD scan. The dashed line \ncorresponds to the XRD peak for bulk EuIG. (d) Magnetization hysteresis loops for field out -of-\nplane and in -plane directions.  15 \n  \n \nFigure 2  Polar angle dependent ferromagnetic resonance (FMR). (a) Coordinate system used for \nthe FMR measurement. (b) Room temperature FMR derivative absorption spectra for θH = 0° (out -\nof-plane configuration) for EuIG on different (111) substrates. (c) FMR derivative absorption \nspectra for 50 nm EuIG grown on NGG(111) ( 𝜀∥ ≈ 0)  and 20 nm EuIG on YAG(111) (𝜀∥< 0) with \npolar angle θH ranging from 0° (out -of-plane) to 90° ( in-plane) at 300 K, where 𝜀∥ is in-plane strain \nbetween the EuIG film and substrate.  \n \n \n \n16 \n  \nFigure 3  Polar angle dependent ferromagnetic resonance field Hres for (a) tensile in -plane strain \n(𝜀∥ > 0), (b) in -plane strain close to zero ( 𝜀∥ ≈ 0), and (c) compressive in -plane strain ( 𝜀∥ < 0). Solid \ncurves represent the best fitting results. In -plane strain dependence of the anisotropy fields H 2ꓕ (d) \nand  H 4ꓕ (e). \n \n \n \n \n \n17 \n  \nFigure 4  FMR linewidth and magnetic damping of EuIG films as a function of in -plane strain. (a) \nHalf width at half maximum ∆𝐻 vs. frequency f for EuIG films grown on different substrates, with \nthe corresponding fitting according to Eq. (3).  In -plane strain depen dence of inhomogeneous \nlinewidth ΔH0 (b) and Gilbert parameter α (c).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n18 \n  \n \nSubstrate  asubstrate  \n(Å) η \n(%) t \n(nm)  𝜀∥ (%) 𝜀⊥ (%) g H2ꓕ \n(Oe) H4ꓕ \n(Oe) α (×10-2) ΔHo \n(Oe) Γo (Oe) \nGSGG  12.554  0.45 50 0.34 -0.16 1.40 -1394.2 \n± 44.9  339.79 \n± 6.59  2.46 ± \n0.03 21.4 ± \n1.3 2.61 \n   25 0.46 -0.21 1.41 -1543.6 \n± 39.7  709.47 \n± 27.5  1.58 ± \n0.06 10.2 ± \n1.7 6.05 \nNGG  12.508  0.06 50 0.12 -0.06 1.38 -1224.4 \n± 5.7  18.34 ± \n0.05 2.41 \n±0.01  8.9 ± \n0.7 0.20 \nSGGG  12.480  -\n0.14 50 -0.13 0.06 1.40 -909.6 \n± 15.2  164.8 ± \n1.36 2.13 ± \n0.01 5.6 ± \n0.4 0.50 \nYSGG  12.426  -\n0.57 50 -0.27 0.12 1.37 -709.4 \n± 22.0  377.3 ± \n5.09 2.47 ± \n0.03 9.9 ± \n1.8 2.47 \nGGG  12.383  -\n0.92 50 -0.45 0.21 1.38 -1015.0 \n± 81.3  887.2 ± \n37.27  2.20 ± \n0.14 412.2 \n± 8.4  3.35 \nTGG  12.355  -\n1.14 50 -0.38 0.18 1.38 -393.4 \n± 53.6  245.0 ± \n10.00  2.29 ± \n0.20 253.4 \n± 11.8  0.20 \nYAG  12.004  -\n3.95 20 -0.42 0.20 1.37 -36.8 ± \n47.1 424.8 ± \n20.91  1.86 ± \n0.20 217.0 \n± 22.6  0.20 \n \nTable 1  Structural and magnetic parameters for the EuIG thin films grown on different substrates.  "    },    {        "title": "2112.11274v2.Exponential_decay_of_intersection_volume_with_applications_on_list_decodability_and_Gilbert_Varshamov_type_bound.pdf",        "content": "arXiv:2112.11274v2  [math.CO]  3 Jan 2022Exponential decay of intersection volume with application s on\nlist–decodability and Gilbert–Varshamov type bound\nJaehoon Kim∗Hong Liu†Tuan Tran‡\nAbstract\nWe give some natural sufficient conditions for balls in a metric space to have small intersection.\nRoughly speaking, this happens when the metric space is (i) expandin g and (ii) well-spread, and\n(iii) a certain random variable on the boundary of a ball has a small tail. As applications, we show\nthat the volume of intersection of balls in Hamming, Johnson spaces a nd symmetric groups decay\nexponentially as their centers drift apart. To verify condition (iii), w e prove some large deviation\ninequalities ‘on a slice’ for functions with Lipschitz conditions.\nWe then use these estimates on intersection volumes to\n•obtain a sharp lower bound on list-decodability of random q-ary codes, confirming a conjec-\nture of Li and Wootters; and\n•improve the classical bound of Levenshtein from 1971 on constant weight codes by a factor\nlinear in dimension, resolving a problem raised by Jiang and Vardy.\nOur probabilistic point of view also offers a unified framework to obtain improvements on\nother Gilbert–Varshamov type bounds, giving conceptually simple an d calculation-free proofs for\nq-ary codes, permutation codes, and spherical codes. Another c onsequence is a counting result on\nthe number of codes, showing ampleness of large codes.\n1 Introduction\nA well-known fact in convex geometry states that the volume o f the intersection of two Euclidean\nballs of the same radius in Rnis exponentially (in n) smaller than the two given balls. It can be\nproved by observing that the intersection is contained in a b all of smaller radius centered at the mid-\npoint of the centers of the two original balls. This simple pr oof, however, does not extend to some\ndiscrete settings, as the intersection might no longer be en closed by a ball of smaller radius. One\nsuch example is that of the Hamming space over a finite alphabe t, one of the most studied space in\ntheoretical computer science and information theory. Inde ed, take the discrete cube t0,1unendowed\nwith the Hamming metric and let k,rPNwith 2kďr. Consider the two radius- rHamming balls\nAandBcentered at a“0nandb“12k0n´2krespectively. Take a mid-point cofaandb, say by\nsymmetry c“1k0n´k. Then the point x“0k1r0n´r´klies in the intersection AXB, but it is of\nHamming distance r`kfrom the chosen mid-point c.\nThe expression of the intersection volume in such discrete m etric spaces can usually be written\nout explicitly. The problem is that such expression is often cumbersome and it is a grueling task\n∗Department of Mathematical Sciences, KAIST, South Korea. E -mail:jaehoon.kim@kaist.ac.kr . Supported by\nthe POSCO Science Fellowship of POSCO TJ Park Foundation, an d by the KAIX Challenge program of KAIST\nAdvanced Institute for Science-X.\n†Mathematics Institute, University of Warwick, UK. E-mail: h.liu.9@warwick.ac.uk . Supported by the UK Re-\nsearch and Innovation Future Leaders Fellowship MR/S01632 5/1.\n‡Discrete Mathematics Group, Institute for Basic Science (I BS), South Korea. E-mail: tuantran@ibs.re.kr . Sup-\nported by the Institute for Basic Science (IBS-R029-Y1).\n1to estimate. To illustrate, let us consider the q-ary Hamming space t0,1,...,q ´1un. Denote\nby volqpn,rqthe volume of a radius- r q-ary Hamming ball, and by vol qpn,r;kqthe volume of the\nintersections of two radius- rballs whose centers are distance kapart. It is not hard to show that the\nintersection volume is\nvolqpn,r;kq “ÿ\ni`jďkk!\ni!j!pk´i´jq!pq´2qk´i´jÿ\ntďtmaxˆn´k\nt˙\npq´1qt, (1)\nwheretmax:“minpn´k,r´k`i,r´k`jq.\nEstimating the asymptotics of the right hand side above is no t at all a straightforward task.\nIndeed, when kandrare linear in the dimension n, Jiang and Vardy [25] studied the binary case\nq“2 with the help of computer. Later, Vu and Wu [43] estimated th e general q-ary case for all\nqě2 using a discrete analog of Lagrange’s multiplier and some i nequalities on entropy functions;\ntheir proof, though computer-free and much cleaner, is stil l rather involved.\nConsider the following alternative probabilistic approac h to estimate the intersection volume. Let\nA,Bbe two radius- rEuclidean balls centered at a,bPRnrespectively. Let xbe a uniform random\npoint drawn from A, then the ratio of the volume of the intersection AXBand the volume of the\nradius-rball is precisely the probability that xlies inAXB, that is,volpAXBq\nvolpAq“PpxPAXBq. We\ncan then bound the probability PpxPAXBqusing for instance Talagrand’s celebrated deviation\ninequality [38] for functions with Lipschitz condition wit h respect to both ℓn\n1andℓn\n2norms. We refer\nthe readers to [4, 30] for related results on concentration o f measure.\nWe use this probabilistic approach to give some natural suffic ient conditions that guarantee small\nintersection of balls in a metric space. The advantage of thi s approach is that it can be implemented\nin the discrete settings, provided that appropriate concen tration inequalities can be proved.\n1.1 Sufficient conditions for small intersection\nTo state our result, we need some notations. Let pX,dqbe a finite metric space with dtaking values\ninNY t0u. ForaPXandrPN, we write Bpa,rqfor the ball of radius raroundaand write Spa,rq\nfor the shell of all points of distance exactly rfroma. We say the metric space pX,dqhasexponential\ngrowth at radius rwith rate cif for every aPXand every tăr,\nvolpBpa,r´tqq\nvolpBpa,rqqď2e´ct.\nFora,bPX, letℓa,b:XÑRbe given by\nℓa,bpxq “dpx,bq ´dpx,aq. (2)\nGivenr,kPNandαą0, we say that the metric space pX,dqispr,kq-dispersed with constant αif\nfor anya,bPXwithdpa,bq “kand any 0 ďiďαk,\nEx„Spa,r´iq”\nℓa,bpxqı\ně2αk,\nwherexis a uniform random point of Spa,r´iq.\nA real-valued random variable XisK-subgaussian if for any tě0,\nPp|X| ětq ď2exp`\n´t2{K˘\n.\nOur result reads as follows.\nTheorem 1.1. LetpX,dqbe a finite metric space with dtaking values in NY t0uand letk,rPN.\nSuppose\n2(A1) pX,dqhas exponential growth at radius rwith rate cą0;\n(A2) pX,dqispr,kq-dispersed with constant αą0;\n(A3)For any a,bPXwithdpa,bq “kand any 0ďiďαk,ℓa,bpxq ´Eℓa,bpxqisK-subgaussian,\nwhereℓa,bis as in(2)andxis drawn uniformly from Spa,r´iq.\nThen, for any a,bPXwithdpa,bq “k,\nvolpBpa,rq XBpb,rqq\nvolpBpa,rqq“2e´Ωc,αp1q¨pk`k2{Kq.\nThemerit of Theorem 1.1 is its formulation. Theconditions ( A1)–(A3) are quitenatural; they are\ninspired by properties of the Euclidean space. By our result , showing that the intersection volume\nis small then amounts to verifying these conditions, which a re more manageable. For instance,\nusing Theorem 1.1, we can get a conceptually simple and calculation-free proof that the intersection\nvolume in (1) is exponentially small, i.e.volqpn,pn;kq\nvolqpn,pn qďe´Ωpkq, for the optimal range 0 ăpă1´1{q\nand all 1 ďkďn(Lemma 4.2). It is important that the exponential bound hold s for not just when\nk“Ωpnq, but for all k, which is needed in some applications, e.g. the tightness on list-decoding\ncapacity theorem (Theorem 2.3).\nTo illustrate the power of Theorem 1.1, apart from the Hammin g cube example above, we shall\napply it to Johnson space (Lemma 4.3) and permutation group ( Lemma 4.4). Such estimates on the\nintersection volume of balls are useful for various problem s. We will use them in Section 2 to obtain\nresults on list-decodability of q-ary random codes with rate just below the limiting rate, and improve-\nments on Gilbert–Varshamov type bounds for constant weight codes,q-ary codes, permutation codes\nand spherical codes, and the corresponding counting result s.\nIn order to apply Theorem 1.1, it is not hard to check that the d iscrete metric spaces we con-\nsider have the exponential growth and they are well-dispers ed. To verify the third condition that\nthe centered random variable ℓa,bpxq ´Eℓa,bpxqis subgaussian in our applications, we prove some\nconcentration inequalities for Lipschitz functions define d on ‘slices’ of the space, see Lemmas 3.2\nand 3.5.\nNotations. Before discussing the applications in details, let us revie w the terminology that will be\nused throughout the paper. A codeover a finite alphabet Σ is simply a subset of Σn; the number n\nis referred to as the lengthof the code. The elements of the code are called codewords . If |Σ| “q,\nthe code is called q-ary code, with the term binary used for the case q“2. We say that the code has\nrateRif the number of codewords is |Σ|Rn. Given two words x“ px1,...,x nqandy“ py1,...,ynq\nin Σn, theHamming distance ∆px,yqbetween xandyis the number of coordinates iin which xi\nandyidiffer. For a word xwe denote by xithe value of its i-th coordinate. For xP t0,1,...,q ´1un,\nwe denote its weight, which is the number of non-zero entries in x, by wt pxq. TheJohnson distance\nbetween two binary words x,yP t0,1unof the same weight is half of their Hamming distance. The\nq-ary entropy function hq:r0,1s ÑRis\nhqpxq “xlogqpq´1q ´xlogqx´ p1´xqlogqp1´xq.\nWe generally use boldface letters for random variables. Giv en a finite set A, we write x„Afor a\ndiscrete random variable xchosen uniformly from A.\n2 Applications\n2.1 List decoding of random codes\nOne of the main goals of the theory of error-correcting codes is to understand the trade-off between\nthe rate of a code and the fraction of errors the code can toler ate during transmission over a noisy\n3channel. There are two natural error models: Hamming’s adve rsarial noise model, and Shannon’s\nstochastic noise model. Channels in Shannon’s world can flip each transmitted bit with certain\nprobability, independentlyofotherbits, whilechannelsi nHamming’sworldcancorruptthecodeword\narbitrarily, subject only to a bound on the total number of er rors.\nThere is a gap between Hamming and Shannon’s world: one can co rrect twice as many errors in\nShannon’s world. We refer the reader to [21] for a thorough co mparison. List decoding, which was\nintroduced by Elias [14] and Wonzencraft [40] in the late 195 0’s, can be used to bridge the gap. In\nlist decoding we give up unique decoding, allowing decoder t o output a list of all codewords that are\nwithin Hamming distance pnfrom the received word. Thus, if at most pnerrors occur, the list will\ninclude the correct codeword. Formally, we say that a q-ary code CĂΣnispp,Lq-list decodable if\nany Hamming ball of radius pnin Σncontains at most Lcodewords.\nList decoding has three important parameters: the rate Rof the code, the error fraction p, and\nthe list size L. A fundamental question in list decoding is to determine the feasible region of pR,p,L q.\nDespite significant efforts, afull description remains elusi ve. In1981, Zyablov and Pinsker[48] proved\nthe list-decoding capacity theorem, thus giving a partial s olution to the above question.\nTheorem 2.1 (Zyablov and Pinsker) .Letqě2,0ăpăq´1\nq, andεą0.\n1. There exist q-ary codes of rate 1´hqppq ´εthat are`\np,r1\nεs˘\n-list decodable.\n2. Anyq-ary code of rate 1´hqppq `εthat is pp,Lq-list decodable must have LěqΩpεnq.\nTheorem 2.1 establishes the optimal trade-off between the ra te and the error fraction for list\ndecoding. In particular, it shows that the list decoding cap acity is 1 ´hqppq, which matches the\ncapacity of Shannon’s model.\nThe existential part of Theorem 2.1 was achieved by demonstr ating that a random code of rate\n1´hqppq ´εispp,r1\nεsq-list decodable with high probability. Rudra [34] proved th at this result is\nbest possible up to a constant factor, in the sense that a rand om code of rate 1 ´hqppq ´εrequires\nL“Ωp,qp1{εq. In[20], GuruswamiandNarayanan providedamoredirectpro ofofRudra’sresult. For\nbinary codes, Li and Wootters [29] recently sharpened the ar gument of Guruswami and Narayanan\nto show that the list size of 1 {εin Theorem 2.1 is tight even in the leading constant factor:\nTheorem 2.2 (Li and Wootters) .For any pP p0,1{2qandεą0, there exist γp,ε“exp`\n´Ωp`1\nε˘˘\nandnp,εPNsuch that for all něnp,ε, a random code CĎ t0,1unof rateR“1´hppq ´εis with\nprobability 1´expp´Ωp,εpnqqnotpp,1´γp,ε\nε´1q-list decodable.\nLi and Wootters [29] conjectured that Theorem 2.2 generaliz es toq-ary codes, for any qě3. To\nquote their words, “our arguments only work for binary codes and do not extend to larger alphabets.”\nOur first application, making use of the intersection volume estimate, confirms their conjecture,\nshowing that the list size 1 {εin list decoding capacity theorem is optimal for all qě2.\nTheorem 2.3. Letqě2. Then for any pP p0,q´1\nqqandεą0, there exist γp,q,ε “exp`\n´Ωp,q`1\nε˘˘\nandnp,q,ε PNsuch that for all něnp,q,ε, a random code CĎ rqsnof rateR“1´hqppq ´εis with\nprobability 1´expp´Ωp,q,εpnqqnotpp,1´γp,q,ε\nε´1q-list decodable.\n2.2 The sphere-covering bounds\nOur second group of applications concern codes over metric s paces. Consider a metric space pX,dq\nand a real number rą0. We say a subset CofXis an pX,d,rq-codeifdpc,c1q ąrfor any distinct\ncodewords c,c1PC. A simple covering argument shows the existence of such a cod eCwith\n|C| ěinf\naPXmpXq\nmpBpa,rqq(3)\n4foranyfinite measure mon the Borel σ-algebra of X. To see why (3) holds, one can assume Cis a\nmaximal pX,d,rq-code of finite size. From the maximality of C, we deduce that X“Ť\naPCBpa,rq.\nBy the subadditivity of measures, we then get mpXq ďř\naPCmpBpa,rqq ď |C| ¨supaPXmpBpa,rqq,\nresulting in (3).\nImproving upon the sphere-covering bound (3) is a notorious ly difficult problem; more on this\nlater. Our next result improves the bound, assuming some mil d conditions on the metric space.\nTheorem 2.4. LetpX,dqbe a finite metric space, and let rą0. Suppose\n(P1)(Homogeneous) For every sPR, all the balls of radius shave the same volume volpsq.\nSuppose further that there exist tP p0,rqandKą0such that\n(P2)(Exponential growth)volpr´tq\nvolprqďe´K; and\n(P3)(Small intersection volume) for any a,bPXwithr´tădpa,bq ďr,volpBpa,rqXBpb,rqq\nvolprqďe´K.\nThen there is an pX,d,rq-code of size p1´oKÑ8p1qqK¨|X|\nvolprq, and the number of pX,d,rq-codes is\nat leastexp´\np1\n8`oKÑ8p1qqK2¨|X|\nvolprq¯\n.\nTheorem 2.4 can be used in conjunction with Theorem 1.1 (for v erifying condition (P3)) to\ngive a unified proof of improvements on Gilbert-Varshamov ty pe bounds on various models of error\ncorrection codes, which we now discuss in details. Theorem 2 .4 builds on recent developments on\nsome graph theoretic results; such approach was pioneered b y the work of Jiang and Vardy [25] and\nby Krivelevich, Litsyn and Vardy [26].\nq-ary codes\nAq-ary code Cis said to have minimum distance at least dif any two codewords in Chave distance\nat leastd. Given three parameters q,nandd, what is the largest possible size Aqpn,dqof aq-ary\nlength-ncode with minimum distance at least d? This question has been studied extensively for\nalmost seven decades, and remains one of the most important q uestions in coding theory.\nFor a word xP rqsn, the Hamming ball of radius dcentered at xis the collection of words in rqsn\nwith distance at most dfromx. The volume of this ball does not depend on the location of xand\ncan be expressed as\nvolqpn,dq “dÿ\ni“0ˆn\ni˙\npq´1qi.\nThe sphere-covering bound (3), applied to the the Hamming sp aceprqsn,∆q, gives\nAqpn,d`1q ěqn\nvolqpn,dq.\nThis is known in the literature as the famous Gilbert–Varsha mov bound [19, 39] from the 1950’s. For\nfive decades this was the best asymptotic lower bound for Aqpn,d`1q(see for example [22, page\n95]).\nThe case when dis proportional to n, that is, d{nis a positive constant, is of special interest in\ncoding theory. It is an easy exercise to see that for d{ně pq´1q{q, the fraction qn{volqpn,dqis\nless than 2. In this case, the Gilbert–Varshamov bound gives no useful information. Thus, the value\npq´1q{qis a natural threshold for the ratio d{n.\nIn a breakthrough, Jiang and Vardy [25] improved the Gilbert –Varshamov bound, for the binary\ncase, for dď0.4994n. Extending the work of Jiang and Vardy, Vu and Wu [43] proved t hat ifd{n\n5is less than pq´1q{q, then one can improve the Gilbert–Varshamov bound by a facto r linear in n.\nWe give a short proof of the following strengthening of Vu-Wu ’s result, showing ampleness of large\ncodes.\nTheorem 2.5. Letqě2and let0ăpăq´1\nqandd“pn. Then there exists a positive constant\nc“cp,qsuch that the number of q-ary length- ncodes with minimum distance at least d`1is at least\nexpˆ\ncn2¨qn\nvolqpn,dq˙\n.\nAs the number of subsets of rqsnof sizeop,qp1qn¨qn\nvolqpn,dqis exp´\nop,qp1qn2¨qn\nvolqpn,dq¯\n, Theorem 2.5\nrecovers the bound Aqpn,d `1q ěΩp,qp1qn¨qn\nvolqpn,dqof Vu and Wu. The original proof of Vu-\nWu’s bound was quite complicated, and involved heavy calcul ations. Our proof of Theorem 2.5 is\nconceptual and reflects, in a clean way, the necessity of the a ssumption d{nă pq´1q{q.\nConstant-weight codes\nGiven positive integers n,dandw, we denote by Apn,d,w qthe size of a largest constant-weight code\nof length nand minimum Johnson distance dall of whose codewords are in t0,1unwith weight w.\nEstimating Apn,d,w qaccurately is the central problem regarding constant-weig ht codes. With the\nexceptions of a few particular small cases [5] and the fixed wcase [7], it remains open in general.\nThanks to symmetry, all Johnson ball of radius din`rns\nw˘\nhave the same volume\nVwpn,dq:“dÿ\ni“0ˆw\ni˙ˆn´w\ni˙\n.\nThus, the sphere-covering bound, specialized to the Johnso n space, gives\nApn,d`1,wq ě`n\nw˘\nVwpn,dq.\nThis lower bound was obtained by Levenshtein back in 1971 [28 ].\nOur next result provides an improvement on this 50-year-old bound of Levenshtein by a factor\nlinear in the dimension. This resolves a problem posed by Jia ng and Vardy [25].\nTheorem 2.6. Letαandλbe constants satisfying 0ăαăλp1´λq. There is a positive constant\nc“cα,λsuch that for d“αnandw“λn\nApn,d`1,wq ěcn¨`n\nw˘\nVwpn,dq.\nPermutation codes\nLetSnbethe symmetric group of permutations on rns. Considera permutation σPSnas a codeword\npσp1q,...,σ pnqq P rnsn, thenSnis a subset of rnsn. With this view, the Hamming distance between\ntwo permutations σ,τPSnis naturally defined as\n∆pσ,τq “ˇˇtiP rns:σpiq ‰τpiquˇˇ.\nA codeCis called a permutation code ifCĎSn. It is said to have minimum distance at least dif\nany two codewords in Chave the Hamming distance at least d.\nPermutation codes have been extensively studied, see for ex ample [1, 2, 12, 13, 36]. It also has\nvarious applications including data transmission over pow er lines [8, 9, 16, 32, 42], and design of\n6block ciphers [10]. From an extremal perspective, the most n atural question for permutation codes\nis that for given nandd, what is the largest possible size Aperpn,dqof a length- npermutation code\nwith minimum distance at least d? Let volperpn,dqbe the volume of a radius- dHamming ball in Sn.\nOnce again, the sphere-covering bound (3) yields\nAperpn,d`1q ěn!\nvolperpn,dq.\nTait-Vardy-Verstra¨ ete [37], Yang-Chen-Yuan [47] and Wan g-Zhang-Yang-Ge [44] further improved\nthis to\nAperpn,d`1q ěΩpnq ¨n!\nvolperpn,dqfor Ω pnq ďdăn{2.\nWe prove the following strengthening which recovers this bo und for a larger range of distance d.\nTheorem 2.7. For given εP p0,1{2q, there exists a positive constant c“cεsuch that the following\nholds. For εnădă p1´εqn,Aperpn,d`1q ěcn¨n!\nvolperpn,dq. Furthermore, the number of length- n\npermutation codes with minimum distance at least dis at least\nexpˆ\ncn2¨n!\nvolperpn,dq˙\n.\nSpherical codes\nAspherical code of angleθin dimension nis a collection of vectors x1,...,x kin the unit sphere Sn´1\nsuch that xxi,xjy ďcosθfor every i‰j, that is, any two distinct vectors form an angle at least θ.\nLetApn,θqbe the size of the largest spherical code of angle θin dimension n.\nForθěπ{2, Rankin [33] determined Apn,θqexactly, so from now on we will assume that\nθP p0,π{2q. ForxPSn´1, we write\nCθpxq “ tyPSn´1:xx,yy ěcosθu\nfor the spherical cap of angular radius θaroundx, and let snpθqdenote the normalized surface area\nofCθpxq.\nThe sphere-covering bound (3) (observed by Chabauty [6], Sh annon [35], and Wyner [46]) implies\nApn,θq ě1\nsnpθq“ p1`op1qq?\n2πn¨cosθ\nsinn´1θ.\nFor over six decades there have been no improvements to this e asy lower bound. By estimating\nthe expected size of a random spherical code drawn from a Gibb s point process, Jenssen, Joos and\nPerkins [24] recently improved the lower bound by a linear fa ctor in dimension.\nTheorem 2.8 (Jenssen, Joos and Perkins) .ForθP p0,π{2q, letcθ“logsin2θ ?\np1´cosθq2p1`2cosθq. Then,\nApn,θq ě p1`op1qqcθ¨n\nsnpθq.\nThis bound was very recently further improved by Gil Fern´ an dez, Kim, Liu and Pikhurko [18].\nTheorem 2.9 (Gil Fern´ andez, Kim, Liu and Pikhurko) .LetθP p0,π{2qbe fixed. Then,\nApn,θq ě p1`op1qqlogsinθ?\n2sinθ\n2¨n\nsnpθq,asnÑ 8.\n7Although Theorem 2.4 is not directly applicable to the conti nuous setting of spherical codes,\nwe use discretization and the graph theoretic idea in Theore m 2.4 to give a short proof of the\nimprovement of Jenssen, Joos and Perkins [24] in Theorem 2.8 . This answers another question of\nJiang and Vardy [25], who asked whether discretization appr oach would work for spherical codes. A\nclosely related topic in continuous setting is the sphere pa cking problem, where a similar approach\nusing integer lattice instead was utilized by Krivelevich, Litsyn and Vardy [26].\nWe remark that the best lower bound by Gil Fern´ andez, Kim, Li u and Pikhurko [18] in Theo-\nrem 2.9, however, seems not attainable via discretization a nd requires to work directly with intrinsic\nproperties of spherical geometry.\nOrganization. The rest of the paper is organized as follows. In Section 3, we prove Theorem 1.1\nand concentration inequalities for Lipschitz functions ov er slices of Hamming spaces and symmetric\ngroup, see Lemmas 3.2 and 3.5. We then use these concentratio n inequalities in Section 4 to deduce\nboundsonthevolumeofintersectionsofHamming/Johnson/p ermutationballs, seeLemmas4.2to4.4.\nSection 5 comtains some graph theoretic tools, which will be used in Section 6 to prove Theorems 2.5\nto 2.8 on improvements on sphere-covering bounds. The proof of Theorem 2.3 is given in Section 7.\n3 Proof of Theorem 1.1 and concentration on the slice\nIn this section we will prove Theorem 1.1, and establish some new concentration inequalities that will\nbe used to verify (A3) when applying Theorem 1.1. Concentrat ion inequalities are fundamental tools\ninprobabilisticcombinatorics andtheoretical computers cienceforprovingthat nicerandomvariables\nare near their means. The main principle is that a random func tion that smoothly depends on many\nindependent random variables should be sharply concentrat ed. The new concentration inequalities\nwe need are for functions of dependent random variables. Our proofs use coupling techniques.\nProof of Theorem 1.1. LetT“Bpa,rq XBpb,rq, and let η„Bpa,rq. Then\nvolpBpa,rq XBpb,rqq\nvolpBpa,rqq“PpηPTq.\nBy definition, ηlies inTif and only if it is of distance at most rfromb, i.e.\nPpηPTq “Ppdpη,bq ďrq.\nAs the metric space has exponential growth at radius r,Ppdpη,aq ďr´αkq ď2e´Ωpkq. Thus,\nPpηPTq ďPpηPTˇˇdpη,aq ąr´αkq ¨Ppdpη,aq ąr´αkq `Ppdpη,aq ďr´αkq\nďαkÿ\ni“0P`\ndpη,bq ďrˇˇdpη,aq “r´i˘\n¨Ppdpη,aq “r´iq `2e´Ωpkq\nďmax\n0ďiďαkP`\ndpη,bq ďrˇˇdpη,aq “r´i˘\n`2e´Ωpkq.\nFix an arbitrary 0 ďiďα, and let x„Spa,r´iq. Note that, conditioning on dpη,aq “r´i,η\nandxare identically distributed. We thus have\nP`\ndpη,bq ďrˇˇdpη,aq “r´i˘\n“P`\ndpη,bq ´dpη,aq ďiˇˇdpη,aq “r´i˘\n“Ppdpx,bq ´dpx,aq ďiq\n“Ppℓa,bpxq ďiq.\n8Using that pX,dqispr,kq-dispersed with constant α, we see that Eℓa,bpxq ě2αk. Consequently,\ni´Eℓa,bpxq ďi´2αkď ´αk. Thus, since ℓa,bpxq ´Eℓa,bpxqisK-subgaussian, we get\nPpℓa,bpxq ďiq “Ppℓa,bpxq ´Eℓa,bpxq ďi´Eℓa,bpxqq\nďPpℓa,bpxq ´Eℓa,bpxq ď ´αkq\nď2e´Ωpk2{Kq,\nas desired.\n3.1 Slices of the q-ary cube\nOne of the most natural and easy-to-verify smoothness assum ptions that one may consider is the\nso-called bounded differences condition.\nDefinition 3.1 (Boundeddifferences condition) .A function f: ΩnÑRis said to satisfy the bounded\ndifferences condition with parameters pc1,...,cnq PRnif for every x,x1PΩn\n|fpxq ´fpx1q| ďnÿ\ni“1ci\n/BDtxi‰x1\niu.\nIn the proof of Theorems 2.3, 2.5 and 2.6 we will use the follow ing “non-uniform” concentration\ninequality.\nLemma 3.2. Suppose f:t0,1,...,q ´1unÑRsatisfies the bounded differences condition with\nparameters pc1,...,cnqand that ηis drawn uniformly at random from t0,1,...,q ´1unsubject to\nwtpηq “k. Then\nPp|fpηq ´Efpηq| ětq ď2expˆ\n´t2\n68řn\ni“1c2\ni˙\nfor alltě0.\nThe binary case above is Lemma 2.1 from [27]. For completenes s, we include its short proof.\nLemma 3.3 ([27]).Suppose g:t0,1unÑRsatisfies the bounded differences condition with parame-\nters pc1,...,cnqand that ξP t0,1unis a random vector uniformly distributed in`rns\nk˘\n. Then\nPp|gpξq ´Egpξq| ětq ď2expˆ\n´t2\n8řn\ni“1c2\ni˙\nfor alltě0.\nProof.We may assume without loss of generality that c1ě ¨ ¨ ¨ ěcn. Consider the Doob martingale\nZi“Ergpξq|ξ1,...ξis, soZ0“EgpξqandZn“Zn´1“gpξq. LetLpx1,...,x iqbe the conditional\ndistribution of ξgivenξ1“x1,...,ξi“xi.\nWe want to show that\n|ErgpLpx1,...,x i´1,0qqs ´ErgpLpx1,...,x i´1,1qqs| ď2ci\nfor all feasible x1,...,x i´1P t0,1u; this will imply that |Zi´Zi´1|is uniformly bounded by 2 ci, so\nthe desired result will follow from the Azuma–Hoeffding bound (see for example [17, Theorem 22.16]).\nIfξis distributed as Lpx1,...,x i´1,0q, we can change ξito 1 and then randomly choose one of\nthe ones among ξi`1,...,ξnand change it to 0; we thereby obtain the distribution Lpx1,...,x i´1,1q.\nThis provides a coupling between Lpx1,...,x i´1,0qandLpx1,...,x i´1,1qthat differs in only two\ncoordinates iandjąi, and since cjďcithis implies the required bound.\nWe also require some standard facts about subgaussian rando m variables (see for instance [41,\nProposition 2.5.2]).\n9Lemma 3.4 (Subgaussian properties) .LetXbe a random variable with mean zero. Then the\nfollowing properties are equivalent.\n(i) There exists K1ą0such that the tails of Xsatisfy\nPp|X| ětq ď2exp`\n´t2{K1˘\nfor alltě0.\n(ii) There exists K2ą0such that the moment generating function of Xsatisfies\nEexppλXq ďexp`\nK2λ2˘\nfor allλě0.\nIn particular, for piq ù ñ piiq, we can take K2“2K1and for piiq ù ñ piq, we can take K1“4K2.\nWe now have all the tools to prove Lemma 3.2.\nProof of Lemma 3.2. LetξP t0,1unbe a random vector uniformly distributed in`rns\nk˘\n. Letube\ndrawn uniformly from rq´1sn, independently from ξ. Then the distribution of ηcoincides with the\ndistribution of\nu‹ξ:“ pu1ξ1,...,u nξnq.\nWriting }c}2“řn\ni“1c2\ni, by Lemma 3.4, it suffices to show that\nEuEξeλpfpu‹ξq´Eu,ξfpu‹ξqqďe17}c}2λ2. (4)\nFix an instance of u. Note that, as fp¨q,fpu‹ ¨qalso satisfies the bounded differences condition\nwith parameters c“ pc1,...,cnq. Then, by Lemma 3.3 with fpu‹ ¨qplaying the role of gp¨qand\nLemma 3.4, we get that\nEξeλpfpu‹ξq´Eξfpu‹ξqqďe16}c}2λ2.\nThus,\nEuEξeλpfpu‹ξq´Eu,ξfpu‹ξqq“e´λEu,ξfpu‹ξq¨EueλEξfpu‹ξqEξeλpfpu‹ξq´Eξfpu‹ξqq\nďe16}c}2λ2¨EueλpEξfpu‹ξq´EuEξfpu‹ξqq. (5)\nIt is easy to check that gp¨q:“Eξfp¨ ‹ξqalso has the bounded differences condition with parameters\nc. Thus by McDiarmid’s inequality (see for example [17, Theor em 22.17]),\nPp|gpuq ´Eugpuq| ětq ď2e´2t2\n}c}2\nand so by Lemma 3.4,\nEueλpgpuq´Eugpuqqďe}c}2λ2.\nThis, together with (5), implies (4) and completes the proof .\n3.2 Slices of the symmetric group\nThe proof of Theorem 2.7 relies on the following concentrati on inequality for functions over slices of\nthe symmetric group. We define the weightof a permutation σinSnto be the Hamming distance\nbetween σand the identity.\nLemma 3.5. LetSn,kbe the set of all permutations in Snwith weight k. Suppose f:Sn,kÑR\nsatisfies\n|fpσq ´fpτq| ď∆pσ,τqfor allσ,τPSn,k. (6)\nLetσbe drawn uniformly at random from Sn,k. Then\nPp|fpσq ´Efpσq| ětq ď2exp p´t2{72kqfor alltě0.\n10To prove Lemma 3.5 we will use a coupling argument together wi th two well-known concentration\ninequalities. The first is a simple consequence of the Azuma– Hoeffding bound, obtained by Wormald\n[45, Theorem 2.19].\nTheorem 3.6 (Wormald [45]) .LetDnĂSnbe the set of derangements, that is, σPDnif and only\nifσpiq ‰ifor alliP rns. Suppose f:DnÑRsatisfies\n|fpσq ´fpτq| ď∆pσ,τqfor allσ,τPDn.\nLetσbe a uniformly random element of Dn. Then\nPp|fpσq ´Efpσq| ětq ď2exp p´t2{4nqfor alltě0.\nWe also need aGaussian boundfor Lipshitzfunctionson slice s of thediscrete cube, dueto Bobkov\n[3, Theorem 2.1].\nTheorem 3.7 (Bobkov [3]) .Letg:`rns\nk˘\nÑRbe a function such that\n|gpJq ´gpJ1q| ď1.\nfor anyJ,J1P`rns\nk˘\nwith |JXJ1| “k´1. LetIbe a uniformly random element of`rns\nk˘\n. Then\nPp|gpIq ´EgpIq| ětq ď2expˆ\n´t2\nmintk,n´ku˙\nfor alltě0.\nProof of Lemma 3.5. ForJĎ rns, letDJbe the set of all permutations which has exactly rnszJas\nthe set of fixed points, in other words,\nDJ“ tσPSn:σpiq ‰iif and only if iPJu. (7)\nIt is not difficult to see that the following two-step random pr ocess yields the uniform distribution\non the set of all permutations in Snwith weight k:\n1. Choose a set Iuniformly at random from`rns\nk˘\n, and\n2. Choose a permutation σuniformly at random from DI.\nFor a set JP`rns\nk˘\n, letgpJqbe the average of foverDJ, that is,\ngpJq “EI,σrfpσq |I“Js “Eσ„DJfpσq.\nClaim 3.8. For any J,J1P`rns\nk˘\nwith |JXJ1| “k´1, we have\n|gpJq ´gpJ1q| ď3.\nProof of claim. Letjbe the element in JzJ1andj1be the element in J1zJ. For each permutation\nσPDJ, we define a permutation rσPDJ1as follows:\n•rσpiq “ifor alliR tj,j1,σ´1pjqu,\n•rσpjq “j,\n•rσpj1q “σpjq, and\n•rσpσ´1pjqq “j1.\n11Sinceσpjq ‰jandσpj1q “j1, we see that σ´1pjq R tj,j1u,σpjq ‰j1, andj1‰σ´1pjq. Thusrσis\na permutation in DJ1. Also it is easy to see that the map σÞÑrσis a bijection from DJtoDJ1. Asσ\nandrσdiffer only at three places, by the hypothesis we have |fpσq ´fprσq| ď∆pσ,rσq “3. Therefore,\n|gpJq ´gpJ1q| “ˇˇEσ„DJrfpσq ´fprσqsˇˇď3,\nas desired. /squaresolid\nLetµbe the mean of f. Then note that\nµ“EgpIq.\nBy the triangle inequality,\nPp|fpσq ´µ| ětq ďPp|gpIq ´µ| ět{2q `Pp|fpσq ´gpIq| ět{2q.\nFor the first term, recalling Claim 3.8 and applying Theorem 3 .7 to1\n3g, we get\nPt|gpIq ´µ| ět{2u ď2exp p´t2{36kq.\nFor the second term, note that gpIq “Eσ„DIfpσqfor each instance of I. OnceIis fixed, for\nσ„DI, we can view fpσqas a function from DItoR. Then, by Eq. (6), we can apply Theorem 3.6\ntofand get\nPp|fpσq ´gpIq| ět{2q ď2exp p´t2{16kq.\nTherefore,\nPp|fpσq ´µ| ětq ď4exp p´t2{36kq.\nAs the left side is at most one, we get Pp|fpσq ´µ| ětq ď2exp p´t2{72kq.\n4 Small intersection\nIn this section, we will verify the conditions of Theorem 1.1 for Hamming/Johnson/permutation\nspaces, using the concentration inequalities proved in pre vious section, to show that the intersection\nof balls in these spaces has small volume.\nAs these metric spaces pX,dqhave the property that the balls of the same radius have the sa me\nvolume independent of the center point, we will use vol prqthroughout this section to denote the\nvolume of a radius- rball inX.\nWe start with the Hamming space. We will need the following st andard estimate on the volume\nof a Hamming ball.\nLemma 4.1. Suppose that 0ăpă1´1{qand that 1ďαnďpn. Then\nvolqpn,αn q “Θp,qp1q ¨qhqpαqn\n?αn.\nThe Hamming space satisfies the conditions of Theorem 1.1 as f ollows.\nLemma 4.2. Let0ăpăq´1\nq, and let kbe any positive integer. Consider X“ t0,1,...,q ´\n1unendowed with the Hamming distance ∆. Then pX,∆qsatisfies the conditions (A1)–(A3)of\nTheorem 1.1 as follows.\n(A1) pX,∆qhas exponential growth at radius pnwith rate c“Ωp,qp1q.\n(A2) pX,∆qisppn,k q-dispersed with constant α“1\n2p1´pq\nq´1q ą0.\n12(A3)For any a,bPXwith∆pa,bq “kand any 0ďiďαk,ℓa,bpxq ´Eℓa,bpxqis400k-subgaussian,\nwhereℓa,bis as in(2)andxis drawn uniformly from Spa,pn ´iq.\nConsequently, for every a,bPX,\nvolpBpa,rq XBpb,rqq\nvolpBpa,rqq“2e´Ωp,qp1q¨∆pa,bq. (8)\nProof.(A1) Consider tăpn. By the mean value theorem, hqppq ´hqpp´t{nq “h1\nqpxqt{nfor some\nxP pp´t{n,pq. Together with Lemma 4.1, this yields\nvolppnq\nvolppn´tq“Ωp,qp1q ¨?pn´t?pn¨qphqppq´hqpp´t{nqqn“Ωp,qp1q ¨?pn´t?pn¨qh1\nqpxqt.\nAsxďpă1´1{q, we have h1\nqpxq “logqpq´1q ´logqx\n1´xělogqpq´1q ´logqp\n1´pą0. Letting\nε“logqpq´1q ´logqp\n1´p, we thus get\nvolppnq\nvolppn´tqěΩp,qp1q ¨?pn´t?pn¨qεt.\nIftďpn{2, then?pn´t?pně1{2; while?pn´t?pn¨qεtěqεt{2ifpn{2ďtďpn´1 andpnis sufficiently\nlarge. Hencevolppn´tq\nvolppnqďOp,qp1q ¨q´εt{2in either case. As the left side is at most one, we conclude\nthat there exists c“Ωp,qp1qsuch thatvolppn´tq\nvolppnqď2e´ctfor alltăpn.\n(A2) Consider any two points a,bPXwith ∆ pa,bq “k. Let 0 ďiďαk, and let x„Spa,pn ´iq.\nWe can assume a“0nandb“1k0n´k. Writeγ“Ppx1‰1q “ ¨ ¨ ¨ “ Ppx1‰q´1qandδ“Ppx1‰0q.\nThenδ“pn´i\nnďp. Moreover, note that pq´1qγ`δ“q´1, and so γ“1´δ\nq´1ě1´p\nq´1. By the\nlinearity of expectation we have\nEℓa,bpxq “kÿ\ni“1pPpxi‰1q ´Ppxi‰0qq\n“kpγ´δq\něk`\n1´p\nq´1´p˘\n“2αk,\nwhere the second equality follows from the symmetry.\n(A3) Assume a“0nandb“1k0n´k. It is easy to see that the function ℓa,bsatisfies the bounded\ndifferenceconditionwithparameters p2,...,2,0,...,0qwhereonlythefirst kcoordinatesarenon-zero.\nLet 0 ďiďαk, and let x„Spa,pn ´iq. By Lemma 3.2, ℓa,bpxq ´Eℓa,bpxqis 400k-subgaussian.\nOur next result justifies the conditions of Theorem 1.1 for th e Johnson space.\nLemma 4.3. Letβ,λandεbe real numbers with 0ăεă1{10and0ăβă p1´εqλp1´λq. Let\nkbe any positive integer. Consider the slice X“`rns\nλn˘\nendowed with the Johnson distance d. Then\npX,dqsatisfies the conditions (A1)–(A3)of Theorem 1.1 as follows.\n(A1) pX,dqhas exponential growth at radius βnwith rate ε2;\n(A2) pX,dqispβn,k q-dispersed with constant ε{2;\n(A3)For any a,bPXwith∆pa,bq “kand any 0ďiďεk,ℓa,bpxq ´Eℓa,bpxqis8βn-subgaussian,\nwherex„Spa,βn ´iq.\n13Consequently, for every a,bPX,\nvolpBpa,βn q XBpb,βn qq\nvolpBpa,βn qq“2e´Ωεp1q¨`\ndpa,bq`dpa,bq2{pβnq˘\n. (9)\nProof.(A1) We wish to show that vol pβn´tq{volpβnq ď2e´ε2tfor alltďβn. Since the left side is\nat most one, we can assume tě1{p2ε2q. Recall that vol pdq “řd\ni“0`λn\ni˘`p1´λqn\ni˘\nfor all non-negative\nintegerd. For 1 ďiďβn, we have\n`λn\ni˘`p1´λqn\ni˘\n`λn\ni´1˘`p1´λqn\ni´1˘“pλn´i`1qpp1´λqn´i`1q\ni2\něpλ´βqpp1´λq ´βq\nβ2\n“1`λp1´λq ´β\nβ2ě1`4ε. (10)\nIt follows that\nvolpβn´tq ďˆλn\nβn´t˙ˆp1´λqn\nβn´t˙\n¨βn´tÿ\ni“0p1`4εq´iďˆλn\nβn´t˙ˆp1´λqn\nβn´t˙\n¨1`4ε\n4ε.\nFurthermore, (10) implies vol pβnq ě`λn\nβn˘`p1´λqn\nβn˘\ně`λn\nβn´t˘`p1´λqn\nβn´t˘\n¨ p1`4εqt. Therefore, we have\nvolpβn´tq\nvolpβnqď1`4ε\n4ε¨ p1`4εq´tď2e´ε2tassuming 0 ăεď1{10 andtě1{p2ε2q.\n(A2) Consider any two points a,bPXwithdpa,bq “k. Let 0 ďiďεk{2, and let x„Spa,βn ´iq.\nWe can assume a“1λn0p1´λqnandb“0k1λn0p1´λqn´k. SincexP`rns\nλn˘\nanddpx,aq “βn´i, we findřλn\nj“1xj“ pλ´βqn`iandřn\nj“λn`1xj“βn´i. We thus get Ex1“ ¨ ¨ ¨ “ Exλn“pλ´βqn`i\nλnand\nExλn`1“ ¨ ¨ ¨ “Exn“βn´i\np1´λqn, by the symmetry. Furthermore, notice that\nℓa,bpxq “dpx,bq ´dpx,aq “1\n2kÿ\nj“1p2xj´1q `1\n2λn`kÿ\nj“λn`1p1´2xjq “kÿ\nj“1xj´λn`kÿ\nj“λn`1xj.\nTherefore, by linearity of expectation, we obtain\nEℓa,bpxq “k¨ˆpλ´βqn`i\nλn´βn´i\np1´λqn˙\něk¨λp1´λq ´β\nλp1´λqěεk,\nas desired.\n(A3) Without loss of generality we can assume a“1λn0p1´λqnandb“0k1λn0p1´λqn´k. We\nwish to show Pp|ℓa,bpxq ´Eℓa,bpxq| ětq ď2e´t2{p8βnqfor alltě0. As the left side is at most\none, we may assume 2 e´t2{p8βnqď1. Observe that x„Spa,βn ´iqis a concatenation of two\nindependent random vectors px1,...,x λnq „`rλns\npλ´βqn`i˘\nand pxλn`1,...,x nq „`rnszrλns\nβn´i˘\n. Moreover,\nwe can decompose ℓa,bpxq “fpx1,...,x λnq `gpxλn`1,...,x nq, wherefpx1,...,x λnq “řk\nj“1xjand\ngpxλn`1,...,x nq “ ´řλn`k\nj“λn`1xj. Applying Theorem 3.7 to fandg, we therefore get\nPp|ℓa,bpxq ´Eℓa,bpxq| ětq ďPp|f´Ef| ět{2q `Pp|g´Eg| ět{2q\nď4exp´\n´t2\n4pβn´iq¯\nď4e´t2{p4βnqď2e´t2{p8βnq,\nwhere the last inequality holds as 2 e´t2{p8βnqď1. This completes our proof.\n14The last result of this section confirms the conditions of The orem 1.1 for the permutation space.\nLemma 4.4. Let0ăεă0.01,1ďrď p1´εqnandkě6{ε. Consider the symmetric group\nSnendowed with the Hamming distance ∆. Then pSn,∆qsatisfies the conditions (A1)–(A3)of\nTheorem 1.1 as follows.\n(A1) pSn,∆qhas exponential growth at radius rwith rate ε;\n(A2) pSn,∆qispr,kq-dispersed with constant ε{4;\n(A3)For any a,bPMwith∆pa,bq “kand any 0ďiďεk{4,ℓa,bpxq ´Eℓa,bpxqis72r-subgaussian,\nwherex„Spa,r´iq.\nConsequently, for every a,bPSnwith∆pa,bq ě6{ε,\nvolpBpa,rq XBpb,rqq\nvolpBpa,rqqď2e´Ωεp1q¨p∆pa,bq`∆pa,bq2{rq. (11)\nProof.(A1) We wish to show vol pr´tq{volprq ď2e´εtfor alltăr. As the left side is at most one,\nwe may assume 2 e´εtď1. It is well known that1\n3|I|!ď |D|I|| ď1\n2|I|! for |I| ě2 (where DIis as\ndefined in (7)). Hence,\nvolpr´tq\nvolprq“ř\nIPprns\nďr´tq|DI|\nř\nIPprns\nďrq|DI|ď1`1\n2řr´t\ni“2`n\ni˘\ni!\n1`1\n3řr\ni“2`n\ni˘\ni!\nď3\n2¨3npn´1q ¨ ¨ ¨ pn´r`t`1q\nnpn´1q ¨ ¨ ¨ pn´r`1qď5\nt!ď2e´εt,\nwhere the last inequality holds as εď0.01 and 2 e´εtď1.\n(A2) Consider any a,bPSnwith ∆ pa,bq “k. Let 0 ďiďεk{4, and let x„Spa,r´iq. We can\nassumeaPSnis the identity permutation and bis a permutation in Drks. To compute the mean of\nℓa,bpxq, we generate xby first drawing I„`rns\nr´i˘\nand then choosing x„DI.\nNote that for all iP rkszIandjP tk`1,...,n uzI, we have xpiq “i‰bpiqandxpjq “j“bpjq.\nHence, by the linearity of expectation, we have\nEr∆px,bq:I“Is “ÿ\niPIPrxpiq ‰bpiqs ` |rkszI|\ně |I|ˆ\n1´p|I| ´1q!\n|DI|˙\n` |rkszI|\ně pr´i´3q ` |rkszI|.\nHere the penultimate inequality holds as there are at most p|I| ´1q! permutations fixing one value,\nand the final inequality follows from the facts that |DI| ě1\n3|I|! and that |I| “r´i. From this we\nget\nEr∆px,bqs “ÿ\nIPprns\nr´iqEr∆px,bq:I“Is ¨PpI“Iq\ně pr´i´3q `E|rkszI|\n“ pr´i´3q `kpn´r`iq\nn.\n15Asℓa,bpxq “∆px,bq ´∆px,aq “∆px,bq ´ pr´iq, we obtain\nErℓa,bpxqs “Er∆px,bqs ´ pr´iq\někpn´r`iq\nn´3\něεk´3ěεk{2,\nassuming rď p1´εqnandkě6{ε.\n(A3) For all x,x1PBpa,r´iq, we have |ℓa,bpxq ´ℓa,bpx1q| “ |∆px,bq ´∆px1,bq| ď∆px,x1q. Hence\nLemma 3.5 implies ℓa,bpxq ´Eℓa,bpxqis 72r-subgaussian.\n5 Graph theoretic tools\nWe will reduce the lower bound on various codes to lower bound on independence number of some\nauxiliary graphs. We then show that all the auxiliary graphs are locally sparse. We can then use\nknown bound on independence number of locally sparse graphs . We will use the following variant\nwhich is tailored to our needs.\nTheorem 5.1. LetGbe anN-vertex with maximum degree Dand minimum degree at least D{2. Let\nKP r1,Ds, and let ΓĎGbe a subgraph induced by the neighborhood of an arbitrary ver tex. Suppose\nthere is a partition VpΓq “BYIsuch that\n•every vertex uPBhas degree degΓpuq ďD{K; and\n•|I| ďD{K.\nThen the independence number of Gis at least`\n1´oKÑ8p1q˘N\nDlogK, and the number of independent\nsets inGis at least exp`\np1\n8`oKÑ8p1qqN\nDlog2K˘\n.\nRemark. In some of our applications we have K“DΘp1q, in which case the second conclusion implies\nthat the average size of an independent set in Gis at least Ω p1q ¨N\nDlogD.\nProof of Theorem 5.1. As|Γ| ďD, we get\n2epΓq “ÿ\nvPBdegΓpvq `ÿ\nvPIdegΓpvq ď |B| ¨ pD{Kq ` |I| ¨ |Γ| ď2D2{K.\nHence Γ has average degree at most 4 D{K. By a result of Hurley and Pirot [23, Theorem 2], Ghas\nchromatic number at most p1`oKÑ8p1qqD\nlogK. It follows that the independence number of Gis at\nleast`\n1´oKÑ8p1q˘N\nDlogK, as desired.\nFor the second statement, we need to introduce some notation . LetIpGqbe the collection of\nindependent sets of G. Thehard-core model on Gat fugacity λą0 is a probability distribution on\nIpGq, where each IPIpGqoccurs with probability proportional to λ|I|. In other words,\nPrIs “λ|I|\nř\nJPIpGqλ|J|.\nThe denominator, PGpλq “ř\nJPIpGqλ|J|, is thepartition function of the hard-core model on G. Note\nthatPGpλqis an increasing function with PGp0q “1 andPGp1q “ |IpGq|.\nThe expected size of an independent set drawn from the hard-c ore model on Gat fugacity λis\nthe scaled logarithmic derivative of the partition functio n:\nsαGpλq “ÿ\nIPIpGq|I| ¨PrIs “ř\nIPIpGq|I|λ|I|\nPGpλq“λP1\nGpλq\nPGpλq“λ¨ plogPGpλqq1. (12)\n16We need a lower bound on sαGpλqfor certain range of λ, due to Davies et al. [11]. The lower\nbound is written in terms of the Lambert W function: for zą0,Wpzqis the unique positive real\nsatisfying WpzqeWpzq“z. Note that Wpzq “ p1`op1qqlogzaszÑ 8.\nConsider a graph Gthat satisfies the assumptions of Theorem 5.1. Let λ0“logK\nDandλ1“?\nK\nD.\nAsepΓq ďD2{K, Theorem 5 in [11] shows that for all λP rλ0,λ1swe have\n1\nNsαGpλq ě p1`op1qqλ\n1`λWpDlogp1`λqq\nDlogp1`λq.\nCombining this with (12) and letting ui“WpDlogp1`λiqq, we find\nlogPGpλ1q ´logPGpλ0q ěN\nDżλ1\nλ0WpDlogp1`tqq\np1`tqlogp1`tqdt\n“N\nDżWpDlogp1`λ1qq\nWpDlogp1`λ0qqp1`uqdu\n“N\n2Dru2\n1`2u1´u2\n0´2u0s,\nwherethefirstequality followsfromchangeofvariable u“WpDlogp1`tqq. Usingtheapproximations\nDlogp1`λ0q “ p1`op1qqlogK,Dlogp1`λ1q “ p1`op1qq?\nK, andWpzq “ p1`op1qqlogz, we have\nu0“ p1`op1qqloglogKandu1“ p1\n2`op1qqlogK. Therefore, we get\nlogPGpλ1q ´logPGpλ0q ě`1\n8`op1q˘N\nDlog2K.\nSince 1 ďPGpλ0q ďPGpλ1q ď |IpGq|, this gives log |IpGq| ě p1\n8`op1qqN\nDlog2K, as desired.\n6 Improvement on Gilbert–Varshamov bounds\nWe present in this section a unified short proofs of improveme nts on sphere-covering bounds on\nvarious codes by reducing it to lower bound on independence n umber of an auxiliary graph. In order\nto use Theorem 5.1, we need to show that the graph is locally sp arse. Our strategy is to split the edge\ncount in the subgraph induced by the neighbourhood of a verte x into two parts, one from vertices\nfrom the boundary of the Hamming/Johnson/Euclidean ball, a nd the other from interior vertices of\nthe ball. The contribution from boundary vertices is expone ntially small because the volume of the\nintersection of balls that are far apart is small as we have sh own usingTheorem 1.1 and concentration\nof measure. On the other hand, the contribution from the inte rior vertices is also small as there are\nnegligible amount of interior vertices using the growth of t he balls in such spaces.\nProof of Theorem 2.4. Define a graph Gwhose vertices are points in the metric space pX,dqand\ntwo points are adjacent if their distance is at most r. It is easy to see that Ghas|X|vertices, the\ndegree of every vertex is vol prq ´1, and the maximum size of an pX,d,rq-code is the independence\nnumberαpGqofG. Let Γ be a subgraph induced by the neighborhood of an arbitra ry vertex xPX.\nWe partition VpΓq “BYI, whereIis the punctured ball of radius r´tcentered at x. By the\nassumption,|I|\nvolprq“volpr´tq´1\nvolprqďe´K. Consider any vertex uPB. Asr´tădpx,uq ďr, we obtain\ndegΓpuq\nvolprq“volpBpx,rq XBpu,rqq\nvolprqďe´K.\nTherefore, Theorem 2.4 is a realization of Theorem 5.1.\nProof of Theorems 2.5 to 2.7. Each of Lemmas 4.2 to 4.4 verifies the conditions for each of q-ary\ncodes, constant-weight codes and permutation codes for app lying Theorem 2.4, respectively. Hence,\nTheorems 2.5 to 2.7 all follow from Theorem 2.4.\n176.1 Spherical codes\nWe need two lemmas for the short proof of Theorem 2.8. The first one is a folklore result that\npartitions the sphere into small pieces of equal measure (se e e.g. [15, Lemma 21]).\nLemma 6.1. For each δP p0,1qthe sphere Sn´1can be partitioned into N“ pOp1q{δqnpieces of\nequal measure, each of diameter at most δ.\nThe second one is an Euclidean version of results from Sectio n 4. For a measurable set AĂSn´1,\nletspAqdenote the normalized surface area of A. Recall that snpθqis the normalized surface area of\na spherical cap of angular radius θ. It is well known that for fixed angle θP p0,π{2q\nsnpθq “1`op1q?\n2πn¨sinn´1θ\ncosθ. (13)\nWeneedaparameter qθ, whichistheangular radiusofthesmallestcapcontaining t heintersection\nof twospherical caps ofangular radius θwhosecenters areat angle θ. Itisstraightforward tocompute\nthat\nqθ“arcsin´a\npcosθ´1q2p1`2cosθq\nsinθ¯\n. (14)\nLemma 6.2 ([24, Lemma 6]) .LetxPSn´1andAĂCθpxqbe measurable with spAq ą0. Then\nE\nu„ArspCθpuq XAqs ď2¨snpqθq,\nwhereqθis as in(14).\nProof of Theorem 2.8. Chooseδ!θ,n1, that is, δis less than a suitable function of θandn. Apply\nLemma 6.1 to partition the unit sphere into N“ pOp1q{δqnpiecesP1,...,P Nof equal measure, each\nwith diameter at most δ. For each iP rNs, pick an arbitrary point vifromPi. LetGbe a graph with\nvertex set being these Nchosen points, and two vertices form an edge if the angle betw een them\nis less than θ. Then by definition, Apn,θq ěαpGq. We first use a packing/covering argument to\nshow that every vertex in Ghas degree p1`op1qqsnpθqN. WriteNrxs:“Npxq Y txufor the closed\nneighborhood of x.\nClaim 6.3. For every xPVpGq,\nCθ´2δpxq Ăď\nviPNrxsPiĂCθ`2δpxq.\nProof of claim. We only prove the first inclusion. Let ybe any point in Cθ´2δpxq, that is, the angle\nbetween yandxis at most θ´2δ. As the Pi’s cover the sphere, there exists an index isuch that\nyPPi. By the assumption on Pi, we have ||y´vi|| ďδ. Thus, the angle between viandyis\n2arcsin p||y´vi||{2q ď2arcsin pδ{2q ă2δ. It follows from the triangle inequality that the angle\nbetween viandxis less than 2 δ` pθ´2δq “θ, implying viPNrxs. Therefore, for every yPCθ´2δpxq\nwe must have yPŤ\nviPNrxsPi, as desired. /squaresolid\nLetxbe an arbitrary vertex of G. Since the Pi’s are disjoint subsets of Sn´1of normalized\nsurface area 1 {N, Claim 6.3 gives snpθ´2δqNď |Nrxs| ďsnpθ`2δqN. Moreover, it follows from\n(13) that snpθ˘2δq “ p1`Opδqqnsnpθq “ p1`op1qqsnpθq. Therefore, every vertex in Ghas degree\nD:“ p1`op1qqsnpθqN.\nLetK“snpθq\n4snpqθq. By (13), we obtain log K“ p1`op1qqlogsinθ\nsinqθ¨n“ p1`op1qqcθ¨n. It suffices\nto show that we can apply Theorem 5.1 with this choice of K. This amounts to proving that for any\n18xPVpGq, the average degree of GrNpxqsis at most D{K. For this, we view the average degree of\nGrNpxqsprobabilistically as the expected degree of a uniform rando m vertex in Npxq.\nWe partition Nrxs “BYI, whereI“ tvi:PiĂCθpxqu. From Claim 6.3, we know thatŤ\nviPBPi\nis contained in Cθ`2δpxqzCθ´2δpxq. Thus,δ!θ,n1, the number of boundary point is\n|B| ď`\nsnpθ`2δq ´snpθ´2δq˘\nN“OpδnqsnpθqN“opD{Kq,\nwhich is negligible. So it suffices to estimate the average deg ree ofGrIs.\nLetA“Ť\nviPIPi, andletubeauniformrandompointin A. Now, aseach vertexin Gcorresponds\nto a piece of the sphere with the same measure, we can generate vi„Iby rounding uto the vertex\nvisuch that uPPi. Thus, we have by Lemma 6.2 that\nE\nvi„IrdegGrIspviqs “E\nu„ArspCθpuq XAqs ¨Nď2snpqθqNďD{K,\nas desired.\n7 List-decodability of random codes\nIn this section, we prove Theorem 2.3, which states that a uni formly chosen random code of rate\n1´hqppq ´εis with high probability notpp,p1´op1qq{εq-list decodable. In large part we follow the\nproof of Guruswami and Narayanan [20, Theorem 20]. As in [20] we define a random variable W\nthat counts the number of witnesses that certify the violati on of the pp,Lq-list decodability property.\nThus the code is pp,Lq-list decodable if and only if W“0. So our job becomes to bound the\nprobability of the event that W“0. For this we employ the Chebyshev’s inequality\nPpW“0q ďVarrWs\nErWs2.\nWe then show that Var rWs{ErWs2is exponentially small, which would finish the proof. To boun d\nthe variance, we introduce a new ingredient (Lemma 7.1), who se proof relies crucially on our bound\non intersection volume from Lemma 4.2.\nNotation. For the rest of this section, we shall employ the following no tation. Given aP rqsnand\nrPN, we write Bqpa,rqfor the Hamming ball of radius rcentered at a. Recall that vol qpn,rqis the\nvolume of a radius- rHamming ball in rqsn, and vol qpn,r;kqstands for the volume of the intersections\nof two radius- rballs whose centers are distance kapart.\nLemma 7.1. Let0ăpă1´1{q,1ďℓďLandµ:“q´nvolqpn,pn q. There exists a constant\nc“cp,qą0such that the following holds. Let\na,b,x1,...,xℓ,yℓ`1,...,yL,zℓ`1,...,zL\nbe chosen independently and uniformly at random from rqsn. Denote by Eℓthe event\n!\nx1,...,xℓPBqpa,pnq XBqpb,pnq,yℓ`1,...,yLPBqpa,pnq,zℓ`1,...,zLPBqpb,pnq)\n.\nThen\nPpEℓq ďmin!\nµ2L´ℓ`1,q´nµ2L´ℓ´\n1`2pq´1qq´cℓ¯n)\n.\nRemark. A version of Lemma 7.1, for the case q“2, appeared as [29, Lemma A.5]. The proof of [29,\nLemma A.5], however, does not extend to larger q.\n19Proof of Lemma 7.1. We first show that the probability of Eℓis at most µ2L´ℓ`1. For the event\nEℓto occur, one must have (i) a,bPBqpx1,pnq, (ii)x2,...,xℓ,yℓ`1,...,yLPBqpa,pnq, and (iii)\nzℓ`1,...,zLPBqpb,pnq. Note that the events (i), (ii), (iii) are independent. Cond itioned on the\nposition of x1, (i) occurs with probability µ2. Givenaandb, (ii) and (iii) happen with probability\nµL´1andµL´ℓ, respectively. It follows that PpEℓq ďµ2L´ℓ`1.\nFor the other bound, we first apply the law of total probabilit y to get\nPpEℓq “nÿ\nk“0Pp∆pa,bq “kq ¨PpEℓˇˇ∆pa,bq “kq.\nSince there are`n\nk˘\npq´1qkcodewords bP rqsnwhich are at distance kfromaP rqsn, the prob-\nability that ∆ pa,bq “kis exactly q´n`n\nk˘\npq´1qk. Conditioned on the positions of aandbbe-\ning distance kapart, the probability that x1,...,xℓPBqpa,pnq XBqpb,pnqis´\nvolqpn,pn;kq\nqn¯ℓ\n“\n´\nvolqpn,pn;kq\nvolqpn,pn q¯ℓ\nµℓ. The probability that yℓ`1,...,yLPBqpa,pnqisµL´ℓ, and the probability that\nzℓ`1,...,zLPBqpa,pnqisµL´ℓ. Thus, we have\nPpEℓˇˇ∆pa,bq “kq “ˆvolqpn,pn;kq\nvolqpn,pn q˙ℓ\nµℓ¨µL´ℓ¨µL´ℓ\n“ˆvolqpn,pn;kq\nvolqpn,pn q˙ℓ\nµ2L´ℓ.\nTherefore, we get the following for some c“cp,qas in Lemma 4.2.\nPpEℓq “nÿ\nk“0q´nˆn\nk˙\npq´1qk¨ˆvolqpn,pn;kq\nvolqpn,pn q˙ℓ\nµ2L´ℓ\npby Lemma 4.2 q ďq´nµ2L´ℓnÿ\nk“0ˆn\nk˙\npq´1qk¨ p2q´ckqℓ\n“q´nµ2L´ℓ´\n1`2pq´1qq´cℓ¯n\n,\nas desired.\nWe are now ready to prove Theorem 2.3.\nProof of Theorem 2.3. Letcbe the positive constant given by (8). Let\nµ:“q´nvolqpn,pn q, ℓ0:“1´hqppq\n2ε, γ:“4pq´1q\nlnq¨q´cℓ0,andL“1´γ\nε.\nFrom Lemma 4.1, and recalling that R“1´hqppq ´ε, we get\nµ“Θp1q?n¨q´p1´hqppqqnandqRnµ“Θp1q?n¨q´εn. (15)\nNotice that a random q-ary code of rate Ris simply a random map C:rqsRnÑ rqsnwhere, for each\nxP rqsRn, its image Cpxqis chosen independently and uniformly at random from rqsn. For any center\naP rqsnand any ordered list of Ldistinct messages X“ px1,...,x Lq P prqsRnqL, we define Ipa,Xq\nto be the indicator random variable for the event that Cpx1q,...,CpxLqall fall in Bqpa,pn q, and let\nW“ř\na,XIpa,Xq. ThenCispp,L´1q-list decodable if and only if W“0.\n20We have ErIpa,Xqs “P/visualspace\nCpx1q,...,CpxLq PBqpa,pn q(\n“µLand the number of pairs pa,Xqis\nqn¨L´1ś\ni“0pqRn´iq ěqn¨1\n2qRnL. Thus, by linearity of expectation,\nErWs ě1\n2µLqRnL `n. (16)\nObserve that if XandYare two disjoint lists (viewed as sets), then the events Ipa,XqandIpb,Yq\nare independent for any pair of centers a,b. It follows that\nVarrWs “ÿ\nXXY‰∅ÿ\na,b´\nErIpa,XqIpb,Yqs ´ErIpa,Xqs ¨ErIpb,Yqs¯\nďÿ\nXXY‰∅ÿ\na,bErIpa,XqIpb,Yqs\n“Lÿ\nℓ“1ÿ\n|XXY|“ℓÿ\na,bP/visualspace\nIpa,Xq “1 andIpb,Yq “1(\n“Lÿ\nℓ“1ÿ\n|XXY|“ℓq2n¨Pa,b,C/visualspace\nIpa,Xq “1 andIpb,Yq “1(\n,\nwhere in the last equality we converted the inner summation i nto an expectation by randomizing\nover the centers aandb.\nFixapair pX,Y qwith |XXY| “ℓ,andsupposethattheelementsof CpXqarex1,...,xℓ,yℓ`1,...,yL\nwhile the elements of CpYqarex1,...,xℓ,zℓ`1,...,zL. Then the event/visualspace\nIpa,Xq “1 andIpb,Yq “\n1(\nis exactly the event Eℓin Lemma 7.1. Thus, we can bound the variance of Was\nVarrWs ďLÿ\nℓ“1ÿ\n|XXY|“ℓq2n¨PpEℓq\nďLÿ\nℓ“1L2LqRnp2L´ℓq`2n¨PpEℓq,\nwhere the second inequality stems from the fact that the numb er of pairs pX,Y qwith |XXY| “ℓis\nat mostL2LqRnp2L´ℓq. We split the summation into ℓďℓ0andℓąℓ0, and get Var rWs ďVďℓ0`Vąℓ0.\nFrom Lemma 7.1 and (16), we find\nVďℓ0\nErWs2ď4\nµ2Lq2RnL `2nℓ0ÿ\nℓ“1L2LqRnp2L´ℓq`2n¨µ2L´ℓ`1\n“4L2Lℓ0ÿ\nℓ“1pqRnµq´ℓ¨µ\npby (15) q “Θp1q ¨`?nqεn˘ℓ0¨Θp1q?nq´p1´hqppqqn\npasℓ0“1´hqppq\n2εq “q´Ωpnq.\n21Again by appealing to Lemma 7.1 and (16), we see that\nVąℓ0\nErWs2ď4\nµ2Lq2RnL `2nÿ\nℓ0ăℓďLL2LqRnp2L´ℓq`2n¨q´nµ2L´ℓ´\n1`2pq´1qq´cℓ¯n\n“4L2Lÿ\nℓ0ăℓďLpqRnµq´ℓ¨ˆ1`2pq´1qq´cℓ\nq˙n\npby the choice of γq ď4L2Lÿ\nℓ0ăℓďLpqRnµq´ℓ¨q´p1´γ{2qn\npby (15) q “Θp1q ¨`?nqεn˘L¨q´p1´γ{2qn\npsinceL“1´γ\nεq “q´Ωpnq.\nPutting everything together, we get from Chebyshev’s inequ ality that\nPpW“0q ďVarrWs\nErWs2ďVďℓ0`Vąℓ0\nErWs2ďq´Ωpnq.\nSinceCispp,L ´1q-list decodable if and only if W“0, we conclude that Cis with probability\n1´q´Ωpnqnotpp,L´1q-list decodable.\nAcknowledgement\nWe would like to thank Benny Sudakov for bringing [26] to our a ttention.\nReferences\n[1] I.F. Blake, Permutation codes for discrete channels. IEEE Trans. Inform. 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Inform.\nTheory51(2005), 3200–3208.\n[44] X. Wang, Y. Zhang, Y. Yang and G. Ge, New bounds of permuta tion codes under Hamming\nmetric and Kendall’s τ-metric. Des. Codes Cryptogr. 85(2017) 533–545.\n[45] N.C. Wormald, Models of random regular graphs. In Surve ys in combinatorics, 1999 (Canter-\nbury), volume 267 of London Math. Soc. Lecture Note Ser., pag es 239–298. Cambridge Univ.\nPress, Cambridge, 1999.\n[46] A.D. Wyner, Capabilities of bounded discrepancy decod ing.Bell System Tech. J. 44(1965),\n1061–1122.\n[47] L. Yang, K. Chen and L. Yuan, New lower boundson sizes of p ermutation arrays. arXiv preprint\narXiv:0801.3986.\n24[48] V.V. Zyablov and M.S. Pinsker, List concatenated decod ing.Problemy Peredachi Informatsii\n17(1981), 29–33.\n25"    },    {        "title": "2112.11348v1.Fast_long_wavelength_exchange_spin_waves_in_partially_compensated_Ga_YIG.pdf",        "content": "Fast long-wavelength  exchange  spin waves  in partially -compensated  Ga:YIG  \nT. Böttcher,1, 2 M. Ruhwedel,1 K. O. Levchenko,3 Q. Wang,3  H. L. Chumak,4 M. A. Popov,4 I. V. Zavislyak,4 \nC. Dubs,5 O. Surzhenko,5 B. Hillebrands,1 A. V. Chumak,3 and P. Pirro1 \n \n1)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern,  \nGottlieb -Daimler -Straße  46, 67663  Kaiserslautern, Germany  \n2)MAINZ  Graduate School  of Excellence, Staudingerweg  9, 55128 Mainz, Germany  \n3)Faculty  of Physics, University  of Vienna,  A-1090  Wien,  Austria  \n4)Faculty of Radiophysics, Electronics and Computer Systems, Taras Shevchenko National University of Kyiv,  \nKyiv, 01601,  Ukraine  \n5)INNOVENT  e.V. Technologieentwicklung, Prüssingstrasse 27B,  07745  Jena,  Germany  \n (Dated:  20 December  2021)  \nSpin waves in yttrium iron garnet (YIG) nano -structures attract increasing attention from the perspective of novel  \nmagnon -based data processing applications. For  short wavelengths needed in small -scale devices, the group \nvelocity  is directly proportional to the spin -wave exchange stiffness constant λex. Using wave vector resolved \nBrillouin Light  Scattering (BLS) spectroscopy, we directly measure λex in Ga -substituted YIG thin films and \nshow that it is about  three times larger than for pure YIG. Consequently, the spin -wave group velocity overcomes \nthe one in pure YIG for  wavenumbers 𝑘 > 4 rad/ µm, and the ratio between the velocities reaches a constan t value of \naround 3.4 for all 𝑘 > 20 rad/µm. As revealed by vibrating -sample magnetometry (VSM) and ferromagnetic \nresonance (FMR) spectroscopy,  Ga:YIG films with thicknesses down to 59 nm have a low Gilbert damping ( α < \n10−3 ), a decreased saturation magn etization 𝜇0𝑀S ≈ 20 mT and a pronounced out -of-plane uniaxial anisotropy \nof about  𝜇0𝐻u1 ≈ 95 mT which leads to  an out -of-plane easy axis. Thus, Ga:YIG opens access to fast and \nisotropic spin -wave transport for all wavelengths in  nano -scale  systems  independently  of dipolar  effects.  \n \n \nWave -based logic concepts1-3 are expected to come \nalong  with major advantages over information processing \nbased on  current CMOS -based information  technology4,5. \nIn particular, coherent spin waves6  are envisaged to allow \nfor the realization of efficient wave -based logic devices3, 7-\n12. However, progress in this field places high demands on \nthe materials  used. Specifically,  spin-wave  elements  \noperating  at large  clock frequencies demand materials \nwhich exhibit a small  Gilbert damping constant13, large \nspin-wave velocities and  good  processing  properties.  \nYttrium  Iron Garnet  (YIG)  has a very small damping \nconstant14 and nano -structures of 50  nm lateral sizes have \nbeen demonstrated recently15-17. However, dipolar spin \nwaves in YIG waveguides feature velocities that are \nsignificantly reduced compared to plane YIG  films16 which \nis caused by the flattening of the dispersion  curve. The \nfastest dipolar waves in these structures are magnetostatic  \nsurface  waves  which  offer  a group  velocity  of around  \n0.2 µm/ns17. Exchange  spin waves  with wavelengths  in \nthe range of 100 nm or shorter are faster18 but the \nexcitation of such short wavelengths is a separate \nchallenge16. It can be addressed  by the utilization  of: \nnanoscopic  antennas with increased Ohmic loss; strongly \nnon-uniform ma gnetic  patterns19; hybrid  nanostructures  \nutilizing  different  magnetic materials20-22; or more complex \nphysical phenomena  like magnon  Cherenkov radiation23. \nTo operate  with waves  of maximized  speed  in nano - \nstructures, materials with large spin -wave exchange   \nstiffness  λex=2𝐴/(𝜇0𝑀S) are mandatory since the  \nexchange contribution  to the group  velocity  is directly  \nproportional to λex. Here,  𝐴 is the Heisenberg  exchange  \nconstant,  MS the saturation  magnetization and 𝜇0 is the \npermeability of the vacuum. In addition,  the possibility  to \noperate  with fast exchange -dominated  spin waves of larger \nwavelengths would not only allow for  the operations  with \n\"standard\"  micro -scaled  antennas11,16, but  also would give \nthe freedom required for the engineering of  data-processing \nunits1,24 since the exchange -dominated dispersion  relation  is \nhighly  isotropic.  \n In this context, it is promising to study ferrimagnetic in - \nsulators which are close to the magnetic compensation since  \nlow 𝑀S tends to increase λex. For this, Liquid Phase Epitaxy \n(LPE) based14 growth of Ga -substituted YIG25-28, which             has \nbeen adopted for the deposition of Y 3Fe5-xGaxO12 single  \ncrystalline  films  of sub-100 nm thicknesses,  is very \ninteresting.  In these films, non -magnetic Ga3+ ions \npreferentially  substitute the magnetic Fe3+ ions in the \ntetrahedral coordinated  magnetic  sub-lattice  (0 < x < 1.5), \ndecreasing  the MS of this ferrimagnetic material down to \nthe fully compensated   antiferromagnetic  state (for the single  \ncrystals  grown  from  the high-temperature  solutions  at \ncontents  x of about  1.27 formula   units at room temperature \nT = 295K)25,26. Besides, the Ga  substitution  induces  a strong  \nout-of-plane  uniaxial  anisotropy  enabling the easy axis of \nthe thin film perpendicular to the film  surface.  \nHere, we report on the investigation of the spin waves  in  2  \n \nFIG. 1. (a) Ferromagnetic resonance frequency fFMR as a function \nof the in -plane magnetic field H || [112̅]. Experimental data (blue  \ndots) is fitted by a modified Kittel equation (solid red line).  \n(b) Recalculated full width at half maximum (FWHM) field \nlinewidth µ0∆H as a function of fFMR. Measured data (dark red \nsquares) is fitted  by a linear regression (solid red line). A  full \nevaluation (see Supplementary Materials) provides the values \nshown in the Table  (c). (c) Parameters obtained from FMR -VNA \ndata of 59 nm Ga:YIG and  97 nm YIG films grown on GGG(111).  \n \nGa:YIG films of sub -100 nm thickness. Ga substitution \nx ≈ 1       was chosen so that the magnetization of the sample \nis decreased to one tenth of the pure YIG value. This was \ndone in   order to change the easy magnetization orientation \nfrom the  in-plane to out -of-plane, simultaneously \navoiding an excessive  increase  in the magnetic  damping.  \nFirst,  a thorough  characterization is performed using \nferromagnetic resonance spectroscopy (FMR) in \ncombination with vibrating -sample magnetometry (VSM) \nto obtain the saturation magnetization, the  anisotropy  \nconstants  and the damping  parameters.  Afterward,  wave \nvector resolved29 Brillouin light scattering (BLS) \nspectroscopy  is used to measure  the dispersion  relation  of \nthermal  spin waves  ω(k) and the spin-wave  exchange  \nstiffness  λex directly.  The resulting  group  velocity  dω/dk \nis compared  to the velocities of spin waves in pure YIG \nfilms.  Our presented  results are complementary and in \ngood agreement with recent results from an indirect \nmeasurement of the dispersion  relation at low wave \nvectors using electrical spectroscopy in  Ga:YIG  reported \nby Carmiggelt et  al.30 \nIn the following, we present exemplary Broadband Fer - \nromagnetic Resonance – Vector Network Analyser (FMR  \n- VNA) spectroscopy data for a LPE -grown 59 nm thick \nfilm of Ga:YIG/GGG(111). Data for the othe r films \nthicknesses of 105 nm -thick Ga:YIG/GGG(111), 95 nm -\nthick Ga:YIG/GGG(001), and a reference film of 97 nm -\nthick YIG/GGG(111) are provided in the Supplementary \nMaterials. The films grown on   GGG(111)  were cut  into the square specimens with the edges o riented along [112̅] \nand [11̅0] crystallographic directions, and samples on \nGGG (001) – along  [100] and [010] (see Figs.  S1 and S5 \nin the Supplementary Materials). For magnetic \ncharacterization, FMR -VNA was performed in the \nfrequency range up to 20  GHz and at the rf power of \n0 dBm. To define the crystallographic parameters  of the \nsamples, theoretical model of Bobkov and Zavislyak  was \nused31. The model differentiates three m ain anisotropy  \nfields – a cubic field 𝐻c, a uniaxial anisotropy field of the  \nfirst order 𝐻u1, and a uniaxial anisotropy field of the \nsecond  order  𝐻u2. The  cubic  anisotropy  originates  from  \nthe magnetization along the preferred crystallographic \ndirections  in the garnet lattice14,31, while the uniaxial \nanisotropy consists of cubic, growth -induced and strain -\ninduced contributions resulting in an effective uniaxial \nanisotropy14,32,33. In addition to the direction of a magnetic \nfield H, the FMR frequency  also depends on the \ncrystallographic orientation of the GGG  substrate31. To \ndefine all the anisotro py fields experimentally,  a magnetic \nfield was applied in -plane (IP) along the two orthogonal \ncrystallographic axes [112̅] and [11̅0] (corresponding to \nthe sample’s edges), and out -of-plane (OOP) along the  \n[111] direction normal to the film plane. The satura tion \nmagnetization 𝑀S was measured by VSM. Detailed \ndescriptions of  the employed theoretical model, the \nmeasurement procedure,  and the FMR analyses for the film \nunder the investigation are  given in the Supplementary \nMaterials.  \nThe dependence of the FMR frequency on the in -plane \nmagnetic field is shown in Fig. 1(a). The Gilbert damping \nparameter α and the inhomogeneous linewidth broadening  \n∆𝐻(0) were found according to the standard approach \ndescribed in Ref.34 (see Fig.  1(b)), taking into consideration  \nthat 𝐻u1 > 𝑀S. All obtained parameters are summarized \nin Fig. 1(c) along  with the results  of a reference  97 nm-\nthick  YIG/GGG(111)  film.  A comparison  of these  values  \ndemonstrates  that the Ga doping  led to a ~ 9 times  \nreduction  in the       saturation  magnetization  𝑀S, a ~ 27 times  \nincrease  in the uniaxial  anisotropy  𝐻u1, and a ~ 4.7 times  \nincrease  in the Gilbert       damping  α. A strong  increase  of the \nuniaxial  anisotropy  in dicates  the out-of-plane  easy axis of \nGa:YIG  thin film.  While   an increase  in α was observed,  \nit is still significantly  lower  compared  to metallic  \ncompounds2. The obtained  values  are in good  agreement \nwith those  reported in Ref 30. \nAfter the characterization via FMR, the dispersion \nrelation of thermally excited, magnetostatic surface spin \nwaves  propagating perpendicularly to the applied field is \nprobed  by          Brillouin light scattering spectroscopy (BLS)29 \nto obtain the  exchange  stiffness  λex. An external  field of \n𝜇0𝐻= 300 mT is applied  in the film plane  along  the \nsample  edge  which  ensures an in -plane magnetization. For \nthe spectral analysis of the scattered light, a 6 -pass tandem \nFabry -Pérot interferometer is used35. In all measurements \npresented here, a  laser with a wavelength  of 𝜆Laser =  \n3  \n \nFIG. 2. (a) Schematic  BLS setup.  (b) Three  exemplary  BLS spectra  obtained  for the different  in-plane  wave  vectors  from  a 59 nm \nthick         Ga:YIG  film at an applied  field of 300 mT. Two spin-wave  modes  as well as a phonon  mode  can be observed.  (c) Dispersion  \nrelations  for the        spin-wave  modes  extracted  from  all the measured  BLS spectra.  The solid  lines  are fits according  to the model  given  in \nEqn.1. The dashed  lines  are linear  fits of the phonon  mode.  \n \n \n491 nm is used.  The in -plane  component  𝑘𝜁 of the wave  \nvector  of the probed  spin wave is  varied  by changing  the \nangle  of light incidence  Θ,   𝑘𝜁=4𝜋 sin(Θ) / 𝜆Laser  – see \nFig. 2(a). The results  for the 59 nm thick  Ga:YIG  film \nare presented  in Fig. 2. Fig. 2(b) shows three exemplary \nBLS spectra (anti -stokes part).  Data  for the three different \nin-plane wave vectors are presented.  Besides the quite \nstrong phonon signal, one can distinguish  the fundamental \nspin-wave mode and the first perpendicular  standing  spin-\nwave  mode  (PSSW).  For wave  vectors  between  \n12 rad µm−1 and 20 rad µm−1 no fundamental mode  could  \n \n be observed  because  of the strong  signal  of the phonon  \nmode  that  crosses  the fundamental  spin-wave  mode  in \nthis area (see, e.g., the  spectrum  for 𝑘𝜁 = 14.7 rad µm−1). \n  The corresponding analytical description of the spin -\nwave  dispersion  relation  for the case of a \nferromagnetic  film in (111) orientation having uniaxial \nand cubic anisotropy  with unpinned  surface  spins  has \nbeen  obtained by Kalinikos et al.36. The wave  vector  \nquantization  along  the film normal, which  results  in the \nappearance  of the PSSWs37, is described by  the index  n \nsuch that the dispersion  relation  is given  by:  \n \n \n𝑓n(𝐤)=𝛾𝜇0\n2𝜋√(𝐻+λex𝑘𝑛2+𝑀S−𝑀S𝑃𝑛𝑛(𝑘𝜁𝑡)−𝐻c−𝐻u1)(𝐻+λex𝑘𝑛2+𝑀S𝑃𝑛𝑛(𝑘𝜁𝑡)sin2𝜙)−2𝐻c2cos23𝜙𝑀       (1) \n \n \nwhere 𝑘𝑛=√𝑘𝜁2+ĸ𝑛2 is the total spin wave vector \nconsisting of the in -plane spin wave vector 𝑘𝜁 and the out -\nof-plane  spin wave  vector  𝜅𝑛 with 𝜅𝑛=𝑛𝜋\n𝑡,𝑛 =\n 0,1,2,… Here,  t  is the thickness  of the film,  𝜙 is the \nangle  between  the static magnetization  and the in-plane  \nwave  vector  𝑘𝜁, 𝜙𝑀 is the angle  between the static \nmagnetization and the [11̅0] axis, H is the  applied  \nmagnetic  field and 𝛾 is the gyromagnetic  ratio which we  \ntake from  the FMR  measurements.  The matrix  element  𝑃𝑛𝑛 \nis a function of 𝑘𝜁𝑡 (0≤𝑃𝑛𝑛< 1 if 0≤𝑘𝜁𝑡<∞). In the \nlong wavelength limit  (𝑘𝜁𝑡≪ 1) and for unpinned  surface  \nspins  the following approximations have been obtained: \n𝑃00 =𝑘𝜁𝑡\n2  for 𝑛=0 and (𝑘𝜁𝑡\n𝑛𝜋)2\n for 𝑛≠038.  \nThe dispersion  relations  of the respective  modes  \nextracted                 from  the BLS spectra  are shown  in Fig. 2(c)   \ntogether with fit curves according to Eqn.  1. FMR  \nmeasurements  show  that 𝐻c is about  one order  of \nmagnitude  smaller  than 𝐻u1  (compare  Fig. 1(c)).   \nConsequently,  the last term in Eqn.1  that is quadratic  \nin 𝐻c can be safely  neglected.  For the fits  we have fixed \nthe saturation  magnetization  to 𝜇0𝑀S= 20.2  mT as \nobtained  from  VSM,  and the gyromagnetic  ratio     to 𝛾=\n179  rad T−1ns−1 as obtained from the FMR measurements.  \nThe extracted values from the simultaneous fits of the  \nfundamental  mode and the first PSSW  mode are:  exchange  \nstiffness λex= (13.54 ± 0.07) × 10−11 Tm2, exchange \nconstant 𝐴 = (1.37 ± 0.01) pJm−1, respectively. The sum \nof the anisotropy fields is 𝜇0(𝐻u + 𝐻c) = \n(91.3 ± 0.4) mT, in very  good agree ment  with the values  \nobtained  from  FMR  (compare to Fig. 1(c)). \n    The exchange stiffness in the film under investigation in  \nthis  work is  about  three times as large as the one for pure   \n4 YIG39. This results in a much higher group velocity than  \nin pure YIG as can be seen in Fig . 3. There the fitted \ndispersion relation of the fundamental mode for the  \ninvestigated Ga:YIG film and the corresponding \ndispersion relation for a pure YIG film of the same \nthickness of 59 nm and at the same applied field of \n300 mT is shown. Here the standard parameters of  \nYIG14,39 have been used:  𝛾= 177 rad T−1ns−1 (from \nFMR), λex= 4.03·10−11 Tm2, 𝜇0𝑀S= 177.2 mT  (all \nanisotropy contributions are neglected). The \ncorresponding group velocities calculated by 𝑣𝑔𝑟=\n 2𝜋𝜕𝑓𝑛(𝒌)/𝜕𝑘𝑛 are plotted in the lower   part of Fig. 3(b).  \nThe exchange  dominated  region  is characterized  by a \nlinear  dependence  of the group  velocity  on the wave \nvector. Thus, spin waves in the Ga:YIG can be considered \nas exchange dominated down to very low wave vectors  as \nit is directly visible from Fig.  3(b).  For wave vectors \nabove   𝑘 > 4 rad/µm, spin -waves in Ga:YIG are faster \ncompared to  pure YIG.  \n \nFIG. 3. (a) Dispersion fitted to the measured data from the \ninvestigated 59 nm thick Ga:YIG film at an applied field of 300 mT \naccording to Eqn. 1 (red) and a theoretical dispersion calculated \naccording to Eqn. 1 for a pure YIG film (green) of t he same \nthickness at the same ap plied field using standard YIG parameters \n(see text)14,39. (b) Group velocity calculated from the dispersion \nrelation in (a) for Ga:YIG (red) and pure YIG (green). The ratio r \nof the group velocities for Ga:YIG and pure YIG is plotted by a grey \ndashed line .    For wave vectors 𝑘 > 30 rad/ µm, both dispersion  \nrelations are dominated and the ratio r of the group  \nvelocities is converging to the ratio of the exchange \nstiffness  constants  𝑟 ≈ λex(Ga:YIG)/λex(YIG) ≈ 3.4. \nIn conclusion, we have investigated spin-wave \nproperties  in Ga-substituted  YIG with significantly  \ndecreased  saturation   magnetization  𝜇0𝑀S ≈20.2 mT and \nincreased  exchange  stiffness λex = (13.54 ± 0.07)·10−11 \nTm2. The saturation magnetization  MS was measured  using  \nVSM,  the three anisotropy constants  𝐻c, 𝐻u1,   𝐻u2 and the \ngyromagnetic  ratio 𝛾 were  determined using FMR, and the \nexchange stiffness  λex was determined from BLS \nmeasurements of the dispersion relation  of the fundamental \nand the first PSSW mode.  We find that  even  spin waves  of \nrelatively  small  wave  vector  𝑘 ≈ 4 rad/µm   exhibit an \nexchange nature, and their velocities are higher  than in \npure YIG, reaching a ratio of approximately 3.4 as  defined \nby the ratio of the individual exchange stiffness constants.  \nAs a further consequenc e, waves in Ga:YIG have a  \nsignificantly  more  isotropic  dispersion  relation  than waves  \nof the same  wavelength  in YIG.  Thus,  for magnonic  \nwaveguides  structured from Ga:YIG, only a weak \ndependence of important  parameters,  such as the wave  \nvelocity  and the wave  phase               accumulation,  on the structure  \nsizes  and on the magnetization   orientation can be expected.  \nThe small saturation magnetization and the uniaxial \nanisotropy lead to an out -of-plane easy  axis which  \nfacilitates  also the use of the entirely  isotropic  Forward \nVolume waves.  Since the relative  drop of the  exchange \nconstant 𝐴 with Ga substitution x is weaker than the drop \nof the saturation magnetization 𝑀S, one can expect that a \nfurther reduction of 𝑀S by an increased Ga substitution will \nlead to even faster and more isotropic spin waves. \nEventually, a fully compensated Ga:YIG film might serve \nas a model system for antiferromagnetic magnonics. Thus, \nGa:YIG opens access to the operation with fast and \nisotropic exchange spin waves of variable wavelengths in \nfuture magnonics networks.  \n \n \nACKNOWLEDGMENTS  \n \nThis research has been funded by the Deutsche Forschungs - \ngemeinschaft (DFG, German Research Foundation) - \n271741898, by the DFG Collaborative Research Center \nSFB/TRR 173 -268565370 (Projects B01 and B11), by the \nAustrian Science Fund (FWF) through the project I 4696 -N, \nand by the European Research Council project ERC Starting \nGrant 678309 MagnonCircuits. The authors thank Volodymyr \nGolub (Institute of Mag netism, National Academy of Sci - \nences of Ukraine) for support and valuable discussions, as \nwell as M. Lindner and T. Reimann (INNOVENT e.V.) for \nthe production of the YIG reference sample and R. Meyer for \nthe technical assistance.  \n  \n5  DATA  AVAILABILITY  \n \nThe data that support  the findings  of this study  are available  \nfrom  the corresponding  author  upon  reasonable  request.  \n \n REFERENCES  \n \n1 A. Mahmoud, F.  Ciubotaru, F.  Vanderveken, A.  V. Chumak, S.  Hamdioui,  \n  C. Adelmann,  and S. Cotofana,  J. Appl . Phys . 128, 161101  (2020) . \n2 A. Barman,  G. Gubbiotti,  S. Ladak,  A. O. Adeyeye,  M. Krawczyk,  J. Gräfe,  \n  C. Adelmann,  S. Cotofana,  A. Naeemi, V. I. Vasyuchka,  B. Hillebrands,  \n  et al., J. Phys .: Condens . Matter  33 (2021) .  \n3 A. V. Chumak,  P. Kabos,  M. Wu, C. Abert,  C. Adelmann,  A. Adeyeye,  \n  J. Åkerman, F. G. Aliev, A. Anane, A. Awad, C. H. 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Pirro1 \n \n1)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern,  Gottlieb -\nDaimler -Straße  46, 67663  Kaiserslautern, Germany  \n2)MAINZ  Graduate School  of Excellence, Staudingerweg  9, 55128 Mainz, Germany  \n3)Faculty  of Physics, University  of Vienna,  A-1090  Wien,  Austria  \n4)Faculty of Radiophysics, Electronics and Computer Systems, Taras Shevchenko National University of Kyiv,  \nKyiv, 01601,  Ukraine  \n5)INNOVENT  e.V. Technologieentwicklung, Prüssingstrasse 27B,  07745  Jena,  Germany  \n \n \nFerromagnetic Resonance – Vector Network Analyzer (FMR -VNA) spectroscopy is a fast and non -destructive \ntechnique that provides access to the fundamental magnetic properties of a material. This spectroscopy proved to be \nespecially useful in the study of the  magnetization in quaternary compounds of a partially compensated ferrimagnet \nGa:YIG/GGG. By changing the concentration of GaS1-S4, it is possible to tune the saturation magnetization and the \ndemagnetizing fields, opening a materials perspective towards magnonic logic devicesS5-S7 with isotropic spin -wave \npropagationS8. Considered as a milestone towards fully compensated antiferromagnetic Ga:YIG, i t was discovered that the \ncurrent samples already exhibit advantageous properties such as an induced perpendicular anisotropy.  \nHence, the primary aim of this section is to establish a concise and precise interpretation of the ferromagnetic resonance \ndata obtained for the thin epitaxial films of Ga:YIG/GGG and for the reference YIG/GGG film cut into 5 x 5 x 0.5 mm3 \npieces. To give our interpretation a better degree of flexibility, in the present analysis we have included samples grown by \nliquid phase epitaxy (LPE) on the substrates with different crystallographic directions – GGG (111) and GGG (001). The \nresults pre sented in the sections below were obtained for several films: a 59 nm thick Ga:YIG/GGG(111) film, which was \nin the focus of interest for the BLS investigations in the main manuscript, a 105 nm thick Ga:YIG/GGG(111) film, a 95 \nnm thick Ga:YIG/GGG(001) film,  and a reference 97 nm thick YIG/GGG(111) film. The Ga concentration in the \ninvestigated films is approximately xGa≈1.0, corresponding to saturation magnetization values of about 20 mTS1, although \nthe precise value is challenging to determine for such thin films on GGG substrate due to technical limitations.  \nFMR -VNA measurements were carried out in the frequency range of up to 20 GHz. To avoid non -linear contributions \nfrom ma gnon -magnon scattering processes that would contribute to the FMR linewidth broadening, we kept the RF power \nat 0 dBm. The measurement set -up consists of a VNA (Anritsu MS4642B) connected to an H -frame electromagnet GMW \n3473 -70 with an 8  cm air gap for var ious measurement configurations and magnet poles of 15 cm diameter  to induce a \nsufficiently uniform biasing magnetic field. The electromagnet is powered by a bipolar power supply BPS -85-70-EC, \nallowing to generate ≈0.9 T at 8 cm air gap. The calibrated VNA signal was transferred via SMA cables/non -magnetic \nSMA end -launch connectors to a straight Southwest Microwave RO4003.8mil microstrip with the sample mounted on top. \nAll the measurements were performed at 𝑇r≈295  𝐾. The measurements of the Ga:YIG samples proved to be challenging \nregarding lower applied magnetic fields. Therefore, to enhance the precision and the reliability of the results, an averaging  \nprocedure of the measured data was applied.  \nFor the films grown on G GG(111), in -plane measurements (IP) were carried out with the magnetic field 𝐻 applied along \nthe two orthogonal crystallographic axes [112̅] and [11̅0] corresponding to the sides of the samples, while for the film \ngrown on GGG(001), 𝐻 was oriented along [100] and [11̅0] (along the side and diagonal of the rectangular samples). Out -\nof-plane (OOP) measurements were performed by applying the magnetic field normal to the surface of the films.  \nA model of the magnetic permeability tensor for the ferritesS9 was adopted in order to determine the fundamental \nmagnetic properties, such as the gyromagnetic ratio 𝛾, the anisotropy fields 𝐻c, 𝐻u1,𝐻u2, and the effective magnetization \n𝑀eff.  Furthermore, the Gilbert damping parameter 𝛼 and the inhomogeneous linewidth broadening ∆𝐻(0) were extracted \nfrom the FMR -VNA measurements.  \n  S2  1. MODEL USED TO ANALYZE THE DATA  \n \nFor YIG films, a detailed description of the magnetocrystalline anisotropy (MCA) energy 𝑈A for different values of the \nmagnetization together with a derivation of the specific anisotropy fields ansatz is found in the work of Bobkov and \nZavislyakS9.  \nThe authors made an important conclusion regarding the influence of the substrate’s crystallographic orientation and \nthe direction of the applied magnetic field 𝐻. The work is mainly focused on a magnetostatic wave (MSW) analysis \ndescribing the frequency -dependent component s of the dynamic permeability tensor μ. The permeability tensor is derived \nin the linear approximation regarding the magnetization variables and includes the anisotropy constants up to the fourth -\norder terms.  \nWe can adopt this approach to the IP FMR analysis usingS9: \n \n𝑓2=(𝛾𝜇0\n2𝜋)2\n[(𝐻i+𝐻A1+𝑀s)(𝐻i+𝐻A2)−𝐻A32] ,   (Eq. S1)  \n \nwhere 𝐻i=𝐻0 – the internal magnetic field which is equal to the applied external field 𝐻0 in this case, 𝛾=𝑔𝜇B\nℏ − the \ngyromagnetic ratio with 𝑔 the g-factor, 𝜇B the Bohr magneton, and  ℏ the reduced Planck constant, 𝑀s− the saturation \nmagentization. 𝐻A1, 𝐻A2,𝐻A3 are the combinations of the anisotropy terms derived from the magnetocrystalline (and \nmagnetostrictive) energy densityS10 dependent on the magnetic field dir ection and the substrate’s crystallographic \norientations – see Tables S1, S2 . For practical reasons, the original model was adjusted from the CGS to SI system of units, \nand the circular frequency 𝜔 was converted to the linear frequency 𝑓= 𝜔/2π. \nThe main  mode for the OOP FMR is expressed as:  \n \n𝑓2=(𝛾𝜇0\n2𝜋)2\n[(𝐻i+𝐻A1)(𝐻𝑖+𝐻A2)−𝐻A32] ,            (Eq. S2)  \n \nwhere 𝜇0𝐻i=𝜇0𝐻0−𝜇0Ms.  \n \n \n Table S1. Expressions  for the 𝐻A1,𝐻A2, 𝐻A3 for ferrites on GGG(111) (adopted from [ S9]). \n 𝐻A1 𝐻A2 HA3 \n𝑰𝑷: 𝐻‖ [110 ] −𝐾c\n𝜇0Ms−2𝐾u1\n𝜇0Ms 0 −√2𝐾c\n𝜇0Ms \n𝑶𝑶𝑷 : 𝐻‖ [001 ] −4\n3𝐾c\n𝜇0Ms+2𝐾u1\n𝜇0Ms+4𝐾u2\n𝜇0Ms 0 \n \n  S3   Table S2.  Expressions  for the 𝐻A1,𝐻A2,𝐻A3 for ferrites on GGG(001) (adopted from [ S9]). \n \n \n \n \n \n \n \n \n \n \n \nThe MCA of the YIG -based epitaxial films consists of the cubic ( 𝑈Acubic), and the non-cubic part (𝑈Anon  cubic). For the \nequilibrium magnetization under the linear approximation assumption, this leads to three major anisotropy fields \ncontributing to an effective magnetization 𝜇0𝑀eff – a cubic field (𝐻c=𝐾c\n𝜇0Ms), a first order uniaxial field (𝐻u1=2𝐾u1\n𝜇0𝑀s) , \nand a second order uniaxial field (𝐻u2=4𝐾u2\n𝜇0𝑀s)S9,S11. The latter term, 𝐻u2 appears only for the out -of-plane applied \nmagnetic field, while the former two, 𝐻u1 and 𝐻c, contribute to the sample’s magnetization in both IP and OOP \nconfigurations.  \n𝐾c is the cubic anisotropy constant (denoted as 𝐾4 in other sourcesS12,S13). 𝐾u1 and 𝐾u2 are the uniaxial out -of-plane \nanisotropies of first and second order, respectively. 𝐾u1 can al so be found as 𝐾2⊥ in the literatureS12,S13, while 𝐾u2 (and, \nconsequently, 𝐻u2) is a term, distinguished separately from 𝐾u1 within the framework of the  currently discussed model. \nHere, we define the cubic anisotropy as 𝐻c=𝐾c\n𝜇0Ms, following the approach of Bobkov and ZavislyakS9, however, in other \narticlesS12,S13 it is defined as 𝐻c∗=𝟐 𝐾c\n𝜇0Ms. However, the dependencies of the ferromagnetic resonance frequency on the cubic \nanisotropy are identical.  \nThe cubic anisotropy originates from the preferred orientation of the magnetization along the crystallographic  axis in \nthe garnet latticeS9, S12. In the case of YIG, the negative 𝐾c has a positive contribution to the anisotropy field in (111) and a \nnegative contribution in (001) films. For stoichiometric garnets grown under near -equilibrium conditions via the LPE \ntechnique, uniaxial anisotropy is mainly caused by the misfit strain be tween the lattice constants of the film and the substrate \n(see, e.g.  S12 and Supplemental Materials of S12 for the strain calculations). This leads to a tensile stress and a stress -\ninduced out -of-plane anisotropy contribution in the case of GGG substrates, dominating in Ga:YIG films over the weaker \ncubic and shape anisotropies.  \nIt is also worth exploring the 𝐻A3 anisotropy term more closely in the case of iron garnet/GGG (111) ferromagnetic \nresonance under IP magnetic field . For these films, as the [100 ] axis lies outside the film’s plane, complementary \nmeasurements should be pe rformed along the [ 112̅] direction. Based on Baselgia et al.S15 2𝐻c2= 2(𝐾4\n𝜇0Ms)2\ncos2(3𝜑), \nwhere 𝜑 is the angle between the [11̅0] crystallographic axis, corresponding to the sample’s side (see Fig. S1), and the \nmagnetization 𝑀. For the 𝐻‖ [11̅0] case 𝜑=0°, cos2(3𝜑)=1 → 2(𝐾4\n𝜇0Ms)2\ncos2(3𝜑)=2𝐻c2, which holds true with a \ncurrent model. However, for the 𝐻‖ [112̅] case 𝜑=90°, cos2(3𝜑)=0 → 2(𝐾4\n𝑀)2\ncos2(3𝜑)=0 and the 2𝐻c2 \ncontribution vanishes. This simple dependency allows to quickly express the cubic field 𝐻c for iron garnet/GGG(111) \nthrough the measurements of two well -defined directions, corresponding in our case to the sample’s side s, opposing to the \nmore demanding angle -resolved measurements.  \n \n  𝐻A1 𝐻A2 HA3 \n𝑰𝑷: 𝐻‖ [100 ] 2𝐾c\n𝜇0𝑀s−2𝐾u1\n𝜇0Ms 2𝐾c\n𝜇0𝑀s 0 \n𝑰𝑷: 𝐻‖ [110 ] 𝐾c\n𝜇0𝑀s−2𝐾u1\n𝜇0𝑀s −2𝐾c\n𝜇0𝑀s 0 \n𝑶𝑶𝑷 : 𝐻‖ [001 ] 2𝐾c\n𝜇0𝑀s+2𝐾u1\n𝜇0𝑀s+4𝐾u2\n𝜇0𝑀s 0 S4  2. IRON GARNET/GGG(111)  \n \nLet us consider thin YIG and GaYIG films grown \non a GGG(111) substrate ( Fig. S1). Then, based on \n(Eq. S1, S2) with substituted corresponding \nanisotropy terms from Table  S1, S2 we obtain \nequations for the FMR frequency 𝑓|| under the IP \nmagnetic field 𝐻||: \n \n \n \nFigure  S1. Crystallographic orientations of YIG/GGG(111) \nand Ga:YIG/GGG(111).  \n \n \n𝑓|| [11̅0]=𝛾𝜇0\n2π√𝐻||(𝐻||−𝐻c−𝐻u1+𝑀s)−2𝐻c2        (Eq. S3) \n𝑓|| [112̅]=𝛾𝜇0\n2π√𝐻||(𝐻||−𝐻c−𝐻u1+𝑀s)         (Eq. S4) \n𝑓⊥=𝛾𝜇0\n2π(𝐻⊥−4\n3𝐻c+𝐻u1+𝐻u2−𝑀s),      (Eq. S5) \n \nThe 2𝐻c2 term is usually very small compared to the product before it. For currently investigated Ga:YIG film, it \nintroduces an approximate shift in the FMR frequency (rounded to the highest value to incorporate a measurement error) \nof ∆𝑓||≈110  MHz  for µ0𝐻||≈ 110 mT (@𝑓||≈2 GHz ) and ∆𝑓||≈3 MHz  for µ0𝐻||≈ 700  mT (@𝑓||≈19.5 GHz ). \nTherefore, the term 2𝐻c2 can be neglected for frequencies above 5 GHz as it is done in the analysis of the BLS data in the \ncurrent manuscript.  \nSimilar results with slightly different approximations are presented in a recent work of Dubs et  al.S12, where sub -100 \nnm YIG films were analyzed via the angle -resolved broadband FMR -VNA. There, only results for  samples grown on (111) \nsubstrates are shown and the main focus was made on the influence of the YIG film thickness on static and dynamic \nmagnetic properties. The main difference between the theoretical model used in the aforementioned publication and our \napproach stems from a slightly different interpretation of the uniaxial / stress -induced anisotropy. However, assuming that  \n𝐻c=𝐾c\n𝜇0𝑀s=𝐾4\n𝜇0𝑀s, 𝐻u1=2𝐾u1\n𝜇0𝑀s=𝐻2⊥−𝐻c , adding the second -order uniaxial anisotropy, and omitting the relatively weak \nin-plane anisotropy field 2𝐾u||\n𝜇0𝑀s, we will get to the same set of equations describing the ferromagnetic resonance in both the \nIP and OOP configurations.  \nA thorough investigation of the static and dynamic magnetic properties of Ga:YIG (111) was performed in a recent work \nof Joris J. Carmiggelt et al.S13. The authors have also underlined the role of anisotropy in switching the easy magnetization \naxis to OOP, as seen in the FMR measurements performed on their 45 nm thick sample. Precise measurements in the IP \nand OOP FMR configurations were performed, and the accumulated data was fit with the respective Kittel equations, \nmodified to incorporate both the cu bic and the uniaxial anisotropy. Similar to the earlier discussed work of Dubs et al.S12, \nthe authors do not consider the uniaxial anisotropy of the second order for OOP resonance, but analytically determine the \nuniaxial out -of-plane anisotropy 2𝐾2⊥\n𝜇0𝑀s=𝐻2⊥, which includes both the cubic and the stress -induced anisotropy \ncontributions. Considering different thicknesses and slightly different xGa concentrations in Carmiggelt’s work, the \nexpected slight change in cubic ( −4.1 mT) and uniaxial anisotropy ( 104 .7 mT) is observedS13.  \nOtherwise, both models are in good agreement and lead  to similar results.  \n \n \n \n  \nS5  2.1.   Determining 𝑯𝐜 from FMR  \n \nConsidering the many unknown parameters 𝛾,𝑀s,𝐻c,𝐻u1, and 𝐻u2 in the Equations S3 – S5, the fitting will yield \nhigh error margins and poor convergence. Hence, a more elaborated treatment is required to specify the terms.  \nTo determine the cubic anisotropy, samples are measured while magnetized along the [11̅0] and th e [112̅] \ncrystallographic directions. The applied in -plane magnetic field was selected in a specific range higher than the saturation \nfield 𝐻s (at which the sample is getting homogenously magnetized in -plane) but still low enough to deduce the 2𝐻c2 term. \nThe saturation field is discussed in the next section and can also be extracted from Fig. S2 . \nSubtracting Equations S4 and S3, we obtain:  \n \n𝐻c=√1\n2(2π𝑓|| [112̅]\n𝛾 𝜇0)2\n−1\n2(2π𝑓|| [11̅0]\n𝛾 𝜇0)2\n     (Eq. S6) \n \nUsually, the cubic anisotropy field 𝜇0𝐻c varies from ≈−4..5 mT for micrometer -thick and bulk YIGS16 or about \n−4.2 mT for sub -40 nm LPE YIG filmsS12. Evaluation of our experimental data using (Eq.  S6), gives the values 𝜇0𝐻c ≈\n−5.05 ±0.5 mT for 97 nm thick YIG film, −4.2 ±0.7 mT and −6.5 ±1.3 mT for 105 nm and 59 nm thick Ga:YIG \nsamples correspondingly. This is in good agreement with −4.1 mT found for the 45 nm Ga:YIG filmS13.  \n \n \nFigure  S2. Cubic anisotropy 𝐻c fitting for the 97 nm YIG (a) and 105 nm Ga:YIG (b) films grown on GGG (111). For Ga:YIG @ 𝜇0𝐻≈\n50 mT, a kink in the low field regime in (b) indicates a transition from unsaturated (multidomain state) to a homogeneous magnetization, while \nfor YIG such a transition is barely visible (in -plane bias field).  \n \n \n2.2.   Determining 𝑯𝐮𝟏 from FMR  \n \nThe uniaxial anisotropy of the first order can be fitted directly to the measured data for the in -plane resonance (Eq. S3 -\nS4) if the saturation magnetization 𝜇0𝑀s is known. In the absence of this value, it is better to use complementary equations \nto incr ease the fitting reliability and specify the range in which to search for 𝐻u1. Here, we propose an equationS9,S11,S17, \nthat includes anisotropy fields relevant for the in -plane configuration:  \n \n𝜇02𝐻s(𝐻s−𝐻c−𝐻u1)−2(𝜇0𝐻c)2=0     (Eq. S7) \n \nwhere 𝜇0𝐻𝑠 is the saturation field required to rotate the magnetization of the film in plane by suppressing the strong \nperpendicular anisotropy in Ga:YIGS13. Equation  S7, was derived  for iron garnet/GGG(111), 𝑓@ 𝐻 || [11̅0] assuming \n𝜇0𝐻=𝜇0𝐻s and 𝑓=0S9. \n  \n(a) \n (b) S6  The unsaturated region is identified on the 𝑓FMR(𝐻||) plot as an inverse dependence of the FMR frequency on the \napplied magnetic field. The critical field 𝐻s is indicated by the lowest extremum on this dependenceS13. The signal below \nthe field 𝐻s is hardly distinguishable from the noise background. Typical values for the micrometer thick YIG are around \n5 mT. For the 45 -nm thick Ga:YIG film, the magnetization reac hes saturation at about 87  mTS13.  \nIn our measurements, the saturation field 𝜇0𝐻s of Ga:YIG/GGG(111) was about 50±5 mT for the 105 nm film (see \nFig. S2 b ) and about 81±2 mT for the 59 nm film. An alternative Ga:YIG sample of a relatively similar thickness (56  nm), \nmatching 𝑀s (20.2 mT), and grown via the LPE under the same conditions clearly shows ( Fig. S3 a ) a negative derivative \nof the frequency with respect to the ma gnetic field, 𝜕𝑓\n𝜕𝐻<0, until 𝜇0𝐻s≅78±2 mT. Above this field, the resonances are \nmore pronounced and the derivative is positive, 𝜕𝑓\n𝜕𝐻>0 (Fig. S3 a ). The 59 nm thick Ga:YIG film had shown the same \nbehavior and reached saturation at a similar field 𝜇0 𝐻s ≅81±2 mT, corresponding to the resonance in a slightly lower \nfrequency 𝑓FMR  ≅1.0 GHz. For YIG/GGG(111) the transition was below 2  mT, and was hard to define . \nHaving both 𝐻s and 𝐻c fields, we may derive 𝐻u1 from (Eq.  S7) and increase precision of the obtained value through \nthe fit (Eq. S3 -S4). Uniaxial anisotropy was estimated to be around  74.8±1 mT for the 105  nm thick Ga:YIG and around \n94.1±0.5 mT for 59 nm thick Ga:YIG films.  \n \nFigure S3.  Magnetization saturation of the 56 -nm Ga:YIG/GGG(111) thick film. (a) Color map of the data collected from the S 21 VNA trace \nas a function of the in -plane magnetic field, 𝐻|| || [112̅], increasing with a step ∆𝜇0𝐻=1mT . (b) Ferromagnetic resonance 𝑓FMR as a function \nof the applied magnetic field obtained from the fit of the corresponding resonance curves in (a) with a Lorentzian.  \n \n \n2.3.   Determining 𝜸, 𝑴𝐬, and 𝑯𝐮𝟐 from FMR  \n \nThe saturation magnetization 𝑀s can be derived from the FMR measurements using (Eq.  S3 – S4) if the fields 𝐻c and \n𝐻u1 are known.  \nIn the case of pure YIG, anisotropy contributions 𝐻u1 and 𝐻c are usually two orders of magnitude smaller than the \nsaturation magnetization 𝑀s. Hence, th ese two terms are often either neglected or combined in one termS18. However, they \nstart to play a crucial role in Ga:YIG films, leading to the negative effective magnetization 𝜇0𝑀eff=𝜇0(𝑀s−𝐻c−𝐻u1) \n(Fig. S4 ) as determined from the direct Kittel fitting of 𝑓FMR(𝐻||). One of the reasons behind the negative values of 𝑀eff \nin Ga:YIG is a substitution o f the magnetic Fe3+ ions with the non -magnetic Ga3+ ions in a tetrahedral sub -lattice leading \nto a decreased 𝑀s. The second reason is the pronounced uniaxial anisot ropy 𝐻u1 of the Ga:YIG films due to a large lattice \nmisfit strain between the film and the GGG substrate.  \n \n(a) \n (b) S7   Figure  S4. Simplified Kittel equation fit (red and green dashed lines) to \n105 nm thick Ga:YIG /GGG  (111) experimental FMR -VNA data (blue \ncircles). The red line corresponds to the fitted gyromagnetic ratio 𝛾, while the \ngreen dashed line – to the fixed value 𝛾=176  rad\nns∙T. In both assumptions, the \nterm 𝜇0𝑀eff yields a negative value hinting a dec reased saturation \nmagnetization and a strong anisotropy contribution.  \n \n \nDomination of the uniaxial anisotropy over the saturation magnetization in Ga -substituted YIG samples suggests a \nperpendicular magnetic anisotropy with the easy magnetization pointing  out of plane.  \nThe saturation magnetization 𝑀s determined using FMR was compared to the values obtained using vibrational sample \nmagnetometry (VSM). The values differ only by 2% for the 105 nm -thick and by 19.8 % for the 59 nm -thick Ga:YIG films.  \nMoreover , the described methodology does not allow for the determination of 𝑀s for the films grown on (001) GGG \nsubstrates. In the following, we use the VSM values to determine the saturation magnetization 𝑀s, and the FMR -VNA \nspectroscopy to define all the aniso tropy fields contributions.  \nThe gyromagnetic ratio  𝛾 could be fixed to the free -electron value 176  rad\nT ∙ ns, as it is typically done for the YIG films, \nor could be fitted experimentally. The difference in the results for the effective magnetization 𝑀eff was reaching quite a \nsubstantial value of 20 % – see Fig. S4 . The obtained 𝛾 values are given in Table 3 , and are around 179  rad\nT ∙ ns for Ga:YIG \nand 177  rad\nT ∙ ns for YIG.  \nThe uniaxial anisotropy of the second order, 𝜇0𝐻u2, is obtained through the OOP FMR measurements and the \nsubsequent fitting with Equation  S5.   \nThe parameters obtained following the described procedure are given in Tables S3 .  \n \n \n3. IRON GARNET/GGG(001)  \n \nThe crystallographic orientation ( Fig. S5) of the \nsubstrate influences the anisotropy ansatz (see \nTable S2 ). For Ga:YIG/GGG(001), the in -plane \nand out -of-plane ferromagnetic resonances are \ndescribed through a system of Equations S8 – S10. \n \n \n \n \n \nFigure  S5. Crystallographic orientations of Ga:YIG /GGG(001).  \n \n \n \n \nS8  𝑓|| [11̅0]=𝛾𝜇0\n2π√(𝐻||−2𝐻c)( 𝐻||+𝐻c−𝐻u1+𝑀s)       (Eq. S8) \n𝑓|| [100 ]=𝛾𝜇0\n2π√(𝐻||+2𝐻c)(𝐻||+2𝐻c−𝐻u1+𝑀s)          (Eq. S9) \n𝑓⊥=𝛾𝜇0\n2π(𝐻⊥+𝐻u1+2𝐻c+𝐻u2−𝑀s)        (Eq. S10) \n \nThe case of (001) crystallographic anisotropy is more complex to analyze compared to the (111) case since (1) it is not \npossible to introduce the same effective magnetization 𝑀eff for IP and OOP configurations, and (2) the algorithm developed \nfor the FMR extraction of the saturation magnetization 𝑀s is not applicable. The VSM values for 𝑀s are used in the \nfollowing.  \n \n \n3.1 Determining 𝛾,𝑯𝐜,𝑯𝐮𝟏,𝑯𝐮𝟐 from FMR  \n \nSince the cubic anisotropy field 𝐻c could not be separately expressed like in the case of (111) films, 𝐻c was fitted \nsimultaneously with 𝐻u1 and 𝛾 for IP measurements with (Eq.  S8-S10). If there is a need to increase the precision of the \nfitting, the c omplementary equations derived from the saturation magnetization under the assumptions 𝐻||=𝐻s,𝑓=0 \ncan be used:  \n \n{𝐻|| [11̅0]: 𝜇02(𝐻s−2𝐻c)(𝐻s+𝐻c−𝐻u1)=0  \n 𝐻|| [100 ]:  𝜇02(𝐻s+2𝐻c)(𝐻s+2𝐻c−𝐻u1)=0             (Eq. S11)  \n \nAn important difference in the measurement approach for the sample on GGG(001) is based upon a different set of IP \ncrystallographic axes, [11̅0] (diagonal) and [100 ] (side), along which the magnetic field 𝐻|| is applied.  \nConclusively, the parameters obtai ned according to the procedures described above are summarized in Table S3  for all \nthe samples under the investigation.  \n \nTable S3 . The parameters obtained from FMR -VNA analyses for YIG/GGG (111), Ga:YIG/GGG(111)  and \nGa:YIG/GGG(001) films. 𝜇0𝑀s (Ga:YIG) obtained from VSM.  \nGarnet/  \nsubstrate  Thickness, \n𝒕 Gyromagnetic \nratio, 𝜸 𝝁𝟎𝑴𝐬 𝝁𝟎𝑯𝐜 𝝁𝟎𝑯𝐮𝟏 𝝁𝟎𝑯𝐮𝟐 \nrad/ns·T  mT mT mT mT \nYIG \nGGG (111)  97 nm  Fit: 177   Fit: 182 .4  \n± 1.8 −5.1 \n± 0.5 −3.5 \n± 0.5 3.6 \n± 0.5 \nGa:YIG  \nGGG (111)  59 nm  Fit: 179  Fix: 20.2 −6.5 \n± 1.3 94.1 \n± 0.5 2.4 \n± 0.1 \nGa:YIG  \nGGG (111)  105 nm  Fit: 179  Fix: 24.4 −4.2 \n± 0.7 74.8 \n± 1.0 2.1 \n± 0.1 \nGa:YIG  \nGGG (001)  96 nm  Fit: 179 Fix: 21.7 −5.3  \n± 0.6 92.4  \n± 1.7 5.2 \n± 0.2 \n \n  S9  4. DETERMINING  α, ∆𝑯(𝟎) FROM FMR  \n \nThe FMR linewidth ∆𝐻 depends on the ferromagnetic resonance 𝑓FMR(𝐻||) frequency according toS19,: \n \n𝜇0 ∆𝐻= 𝜇0 ∆𝐻(0)+𝛼 4π 𝑓FMR\n𝛾,                (Eq. S12) \n \nwhere 𝜇0 ∆𝐻 is the FMR full width at half maximum (FWHM), 𝜇0 ∆𝐻(0) – the inhomogeneous linewidth broadening, 𝛼 \n– the Gilbert damping parameter, and 𝜇0 – the permeability of free space.  \nIn order to recalculate ∆𝑓, obtained from the broadband frequency FMR -VNA meas urements, into ∆𝐻 in (Eq.  S12), \none can use the approach introduced by Kalarickal et al. : \n \n2𝜋 ∆𝑓=∆𝐻 𝜕 𝑓Kittel  (𝐻||)\n𝜕𝐻|| |\n𝐻|| = 𝐻Kittel (𝑓FMR )=𝜇0 ∆𝐻 𝛾 𝑃A(𝑓FMR)        (Eq. S13) \n \n𝜇0 ∆𝐻= 2π ∆𝑓\n𝛾 𝑃A(𝑓FMR ),          (Eq. S14) \n \nwhere 𝑃A(𝑓FMR)= √1+(𝛾 𝜇0 𝑀s\n4π 𝑓FMR)2\n. The term 2π was included in the Equation S14 to recalculate the linear frequency 𝑓 \nfrom the angular frequency 𝜔. \nThe original formulas (Eq.  S12-S14) were derived for YIG/GGG(111) considering 𝑀s≫|𝐻u1|,|𝐻c|. Hence, in the \ndifferentiated ferromagnetic resonance equation 𝜕 𝑓Kittel  (𝐻||)\n𝜕𝐻||, the effective magnetization 𝑀eff was substituted with the \nsaturation magnetization 𝑀s. However, for the Ga:YIG films, as shown earlier in this section, 𝑀s<|𝐻u1|, and the direction \nof the applied magnetic field 𝐻|| with respect to the specific GGG substrate influences the resonance equa tion. Therefore, \nto obtain an appropriate field swept linewidth, each of the specific ferromagnetic resonance equations (Eq.  S3-S4, Eq.  S8-\nS9) should be differentiated separately with the anisotropy fields included ( Table S4 ).  \nHere, we discuss the results  only for the in -plane configuration, as the damping constant 𝛼 is enhanced in the out -of-\nplane measurements. This is attributed to the influence of a magnetically inhomogeneous transient layer near the substrate \ninterfaceS12. \n \n \nTable S4.  The expressions for the recalculated field linewidth 𝜇0∆𝐻 from the frequency linewidth ∆𝑓 based upon the \nmagnetic field orientation with respect to the crystallographic axis of the GGG substrate.  \nSubstrate  Direction of 𝝁𝟎𝑯|| 𝝁𝟎 ∆𝑯  (recalculated from ∆𝑓) \nGGG (111)  𝜇0𝐻|| || [11̅0] 8π2 ∆𝑓∙𝑓FMR [11̅0]\n𝜇0𝛾2 (2𝐻||−𝐻c−𝐻u1 +𝑀s) \n𝜇0𝐻|| || [112̅] 8π2 ∆𝑓∙𝑓FMR [112̅]\n𝜇0𝛾2 (2𝐻||−𝐻c−𝐻u1 +𝑀s) \nGGG (001)  𝜇0𝐻|| || [11̅0] 8π2 ∆𝑓∙𝑓FMR [11̅0]\n𝜇0𝛾2 (2𝐻||−𝐻c−𝐻u1 +𝑀s) \n𝜇0𝐻|| || [100] 8π2 ∆𝑓∙𝑓FMR [100 ]\n𝜇0𝛾2 (2𝐻||+4𝐻c−𝐻u1+𝑀s) \n \n \n S10  In a specific case for the iron garnet/GGG (111) films under the in -plane magnetic field 𝜇0𝐻|| || [112̅] or under \n𝜇0𝐻|| || [11̅0]  field in the frequency range above 5 GHz, it is possible to derive 𝜇0 ∆𝐻 similar to Kalarickal et al. . Because \nthe anisotropy fields ensemble in the ferromagnetic resonance equation (Eq.  S4) assumes plain form 𝜇0(−𝐻c−𝐻u1+\n𝑀s)= 𝜇0 𝑀eff, the expression given in the second row in Table S4  could be re -written as:  \n \n2π ∆𝑓\n𝛾 √1+(𝛾 𝜇0 𝑀eff\n4π 𝑓FMR [112̅])2                (Eq. S15) \n \nBased on the (Eq. S12) and expressions from Table S4 , the Gilbert damping constant 𝛼 and the inhomogeneous \nlinewidth broadening 𝜇0∆𝐻(0) were calculated, and, subsequently, summarized in Table S5 . The errors were calculated \nbased on the corresponding fits convergences.  \n \n \nTable S5.  The damping parameters obtained from the FMR -VNA analyses for the thin films.  \nIron garnet/  \nsubstrate  Thickness , \n𝒕 (nm)  𝜶,   𝟏𝟎−𝟒 𝝁𝟎∆𝑯(𝟎) 𝝁𝟎∆𝑯 (mT)  \n @ 𝑓≈10.5 GHz ∆𝒇 (MHz)  \n @ 𝑓≈10.5 GHz \nGGG (111)   [112̅] [11̅0] [112̅] [11̅0] [112̅] [11̅0] [112̅] [11̅0] \nYIG 97 nm  1.3 \n±0.15 0.6 \n±0.16 0.1 \n±0.01 0.2 \n±0.01 0.195 \n±0.003 0.313 \n±0.008 5.7 \n±0.1 9.1 \n±0.23 \nGa:YIG  59 nm  6.1 \n±0.62 4.3 \n±1.02 0.4 \n±0.05 0.7 \n±0.08 0.786 \n±0.017 0.934 \n±0.024 22.5 \n±0.49 26.9 \n±0.69 \nGa:YIG  105 nm  4.6 \n±0.28 4.9 \n±0.52 0.4 \n±0.02 0.4 \n±0.04 0.637 \n±0.009 0.727 \n±0.017 18.2 \n±0.27 20.8 \n±0.50 \nGGG(001)   [100 ] [11̅0] [100 ] [11̅0] [100 ] [11̅0] [100 ] [11̅0] \nGa:YIG  96 nm  8.4 \n±0.85 6.7 \n±0.68 0.4 \n±0.06 0.4 \n±0.05 1.066 \n±0.055 0.804 \n±0.037 30.9 \n±1.60 23.4 \n±1.06 \n \n \nDivergence between the Gilbert damping  constant 𝛼 along the different crystallographic directions hints a pronounced \ninfluence of the inhomogeneous linewidth broadening  𝜇0∆𝐻(0), but might be also associated with a relatively large error \nbar. Therefore, a more detailed investigation is required to verify the origins if this phenomenon.  \nTo compare these values with the literature, it is worth to mention that the typical values of Gilbert damping parameter \n𝛼 of discs made from the bulk crystalsS21 are 𝛼=0.4∙10−4 for YIG and 𝛼=1.25 ∙10−4..  2.44∙10−4 for Ga:YIG \n(𝑥Ga=0.78..0.88) [respectivelyS22,S21]. Epitaxially -grown micrometer -thick YIG LPE films have slightly higher \ndamping. Their Gilbert parameters range  from 𝛼=0.4∙10−4 (𝑡= 23 μm)S21 to 0.5∙10−4 (𝑡= 3 μm)S14. High -quality \nLPE YIG films with thickness down to hundreds of nanometers are reported to possess 𝛼=1.0..2.0∙10−4 (𝑡=\n 200  nm)S23,S24, 𝛼=1.7∙10−4 (𝑡= 100  nm)S14. A new dimensionality milestone was achieved with the high -quality \nsub-100 nm LPE -grown  YIG filmsS12, that were shown to exhibit low ferromagnetic losses and relatively low Gilbert \ndamping 𝛼=1.0∙10−4..  1.2∙10−4 (𝑡= 42..11 nm)S12. Just recently, a 45 nm thick Ga:YIG film was reported to have \n𝛼=1.0∙10−3 S13, which is higher compared to the 𝛼=6.1∙10−4 presented in this study. However, considering thinner \nsample with a slightly lower saturation magnetization in the work of Carmiggelt et al.S13, both damping constants are in \nrelatively good agreement.   S11   LITERATURE : \n \n[S1] P. Hansen, P. Röschmann, W. Tolksdorf “Saturation magnetization of gallium -substituted yttrium iron garnet”, J. Appl. Phys. 45, 2728 -27-32 (1974). \nDOI:   10.1063/1.1663657  \n[S2] P. Görnert and C. d’Ambly “Investigations of the growth and the saturation magnetization of garnet single crystals Y 3Fe5-xGaxO12 and Y 3Fe5-xAlxO12”, PSS \n(a) 29 (1975). DOI:  10.1002/pssa.2210290111  \n[S3] J. Guigay, J. Baruchel, D. Challeton, J. Daval and F. Mezei “Local measurement of magnetization in two Ga -YIG single crystals grown by di fferent \nmethods”, J. Magn. Magn. Mater  51, 342 (1985). DOI:  10.1016/0304 -8853(85)90034 -4 \n[S4] P. Röschmann “Annealing effects on FMR linewidth in Ga substituted YIG”, IEEE Transactions on Magnetics  17(6), 2973 (1981). \nDOI:  10.1109/TMAG.1981.1061632  \n[S5] A. Mahmoud, F. Ciubotaru, F. Vanderveken, A. V. Chumak, S. Hamdioui, C. Adelmann and S. C otofana “Introduction to spin wave computing (Tutorial \nArticle)”, J. of Appl. Phys.  128, 161101 (2020). DOI:  10.1063/5.0019328   \n[S6] P. Pirro, V. I. Vasyuchka, A. A. Serga, et al., “Advances in coherent magnonics”,  Nat. Rev. Mater.  (2021). DOI : 10.1038/s41578 -021-00332 -w  \n[S7] A. V. Chumak, et al., “ Roadmap on spin -wave computing ”, (2021). arXiv:2111.00365   \n[S8] S. Klingler, P. Pirro, T. Brächer, B. Leven, B. Hillebrands, A. V. Chumak “Spin -wave logic devices based on isotropic forward volume magnetostatic \nwaves”, Appl. Phys. Lett ., 106(2), 2124 06 (2015). DOI: 10.1063/1.4921850  \n[S9] V. B. Bobkov, I. V. Zavislyak “Equilibrium State and Magnetic Permeability Tensor of the Epitaxial Ferrite Films”, Phys. Stat. Sol . (a) 164, 791 (1997). \nDOI:  10.1002/1521 -396X(199712)164:2<791::AID -PSSA791>3.0.CO;2 -7 \n[S10] H. Szymczak and N. Tsuya “Phenomenological Theory of Magnetostriction a nd Growth -Induced Anisotropy in Garnet Films”, Phys. Stat. Sol.  (a) 54, 117 \n(1979). DOI: 10.1002/pssa.2210540115   \n[S11] I.  V. Zavislyak and M.  A. Popov “Yttrium: Compounds, Production and Applications ” edited by B.  D. Volkerts, Chapter 3 , Nova Science Publishers, \n(2009).  \n[S12] C. Dubs, O. Surzhenko, R. Thomas, J. Osten, T. Schneider, K. Lenz, J. Grenzer, R. Hübner, and E. Wendler “Low damping a nd microstructural perfection \nof sub -40nm -thin yttrium iron  garnet films grown by liquid phase epitaxy”, Phys. Rev. Materials  4, 024416 (2020). \nDOI: 10.1103/PhysRevMaterials.4.024416   \n[S13] J. J. Carmiggelt, O.C. Dreijer, C. Dubs, O.Surzhenko , T. van der Sar “Electrical spectroscopy of the spin -wave dispersion and bistability in gallium -doped \nyttrium iron garnet”, Appl. Phys. Lett.  119, 202403 (2021). DOI: 10.1063/5.0070796  \n[S14 ] C. Dubs, O. Surzhenko, R. Linke, A. Danilewsky, U. Brckner, and J. Dellith “Sub -micrometer yttrium iron garnet LPE films with low ferromagnetic \nresonance losses”, J. Phys. D: Appl. Phys. , 50(20), 204005 (2017). DOI: 10.1088/1361 -6463/aa6b1c   \n[S15]  L. Baselgia, M. Warden, F. Waldner, Stuart L. Hutton, John E. Drumheller, Y. Q. He, P.  E. Wigen, and M. Maryško, “Derivation of the resonance frequency \nfrom the free energy of ferromagnets”,  Phys. Rev. B  38, 2237 (1988). DOI : 10.1103/PhysRevB.38.2237  \n[S16] S. A. Manuilov, S. I. Khartsev, and A. M.  Grishin \"Pulsed laser deposited Y 3Fe5O12 films: Nature of magnetic anisotropy I\", J. Appl. Phys.  106, 123917  \n(2009) DOI:  10.1063/1.3272731  \n[S17] I. V. Zavislyak, M. A. Popov, G. Sreenivasulu , and G. Srinivasan “Electric field tuning of domain magnetic resonances in yttrium iron garnet films”, Appl. \nPhys. Lett.  102, 222407 (2013). DOI: 10.1063/1.4809580   \n[S18] M. C. Onbasli, A. Kehlberger, D. H . Kim, G. Jakob, M. Kläui, A. V. Chumak, B. Hillebrands, and C. A. Ross “Pulsed laser deposition of epitaxial yttrium \niron garnet films with low Gilbert  damping  and bulk-like magnetization”,  APL Mater.  2, (2014). DOI : 10.1063/1.4896936  \n[S19] P. Pirro, T.Brächer, A. V. Chumak, B. Lägel, C. Dubs, O. Surzhenko, P. Görnert, B. Leven, and B. Hillebrands “Spin -wave excitation and propagation in \nmicrostructured wave guides of yttrium iron garnet/Pt bilayers”, Appl. Phys. Lett.  104, 012402 (2014). DOI : 10.1063/1.4861343   \n[S20] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, an d J. P. Nibarger “Ferromagnetic resonance linewidth in metallic \nthin films: Comparison of measurement methods”, J. Appl. Phys.  99, 093909 (2006).  DOI: 10.1063/1.2197087   \n[S21] P. Röschmann and W. Tolksdorf  “Epitaxial growth and annealing control of FMR properties of thick homogeneous Ga substituted yttrium iron garnet \nfilms”, Mat. Res. Bull.  18, 449 (1983). DOI: 10.1016/0025 -5408(83)9013 7-X  \n[S22] P. R öschmann \"Annealing effects on FMR linewidth in Ga substituted YIG\", in  IEEE Transactions on Magnetics , 17 (6), 2973, (1981). \nDOI: 10.1109/TMAG.1981.1061632 . \n[S23]  S. Maendl , I. Stasinopoulos, D. Grundler “Spin waves with large decay length and few 100  nm wavelengths in thin yttrium iron garnet grown at the wafer \nscale”, APL 111, 012403 (2017). DOI:  10.1063/1.4991520  \n[S24] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef “Comparative measurements of inverse spin  Hall effects and \nmagnetoresistance in YIG/Pt and YIG/Ta”, Phys. Rev. B  87(17), 174417 (2013). DOI: 10.1103/PhysRevB.87.174417   "    },    {        "title": "2112.12967v2.Skyrmion_nucleation_on_the_surface_of_a_topological_insulator.pdf",        "content": "Skyrmion nucleation on the surface of a topological insulator\nDaichi Kurebayashi1, 2,\u0003and Oleg A. Tretiakov1,y\n1School of Physics, The University of New South Wales, Sydney 2052, Australia\n2Center for Emergent Matter Science, RIKEN, Wako 351-0198, Japan\nSkyrmion nucleation induced by spin-transfer torques at an interface of a topological insulator and a ferro-\nmagnetic insulator is investigated. Due to strong spin-orbit coupling on a surface of topological insulators, which\nenhances the e \u000bect of spin torques, e \u000ecient manipulation of skyrmions is expected, and therefore, topological\ninsulators could provide the ideal platform to achieve high-performance skyrmionic devices. Using micromag-\nnetic simulations and energetics, we evaluate properties of the skyrmion nucleation on a surface of topological\ninsulators, such as nucleation time, critical electric field, and skyrmion numbers. We show that the nucleation\ntime is inversely proportional to the applied electric field. We also identify the Gilbert damping and temperature\ndependencies of the critical field. Furthermore, we analytically evaluate the e \u000bect of the Dzyaloshinskii-Moriya\ninteraction and demonstrate that the temperature dependence can be explained by the reduction of a magnon\nexcitation gap due to the self-energy corrections.\nA magnetic skyrmion is a real-space topological object de-\nfined by its non-coplanar spin texture [1–7]. Because of their\ntopological spin structure, skyrmions exhibit unique dynam-\nics [8, 9] and transport properties associated with a nontriv-\nial quantum Berry phase resulting in emergent electromag-\nnetic fields [10–12]. On the one hand, a topological insulator\n(TI) is a momentum-space topological object characterized\nby its nontrivial band structure [13, 14]. Due to this topo-\nlogically nontrivial structure, transport phenomena related to\nthe momentum-space Berry curvature, such as the quantum\nspin Hall e \u000bect [15–17] and the quantum anomalous Hall ef-\nfect [18–20], are realized and experimentally observed. An-\nother consequence of the nontrivial bands is the appearance\nof metallic gapless surface states. The surface Dirac elec-\ntrons mediate strong correlations between spin and current as\ntheir spin and momentum have a one-to-one correspondence,\nknown as the spin-momentum locking. Although both TIs\nand skyrmions separately have been recent emergent topics in\ncondensed matter physics, a combination of them could be an\nideal platform to study the interplay of real- and momentum-\nspace topology.\nRecently, in a heterostructure consisting of a TI and a mag-\nnetic insulator, the ferromagnetic (FM) skyrmion formation\nhas been observed by transport measurements [21–23]. In\naddition to the conventional anomalous Hall e \u000bect, an extra\nHall signal has been observed, which was attributed to the\ntopological Hall e \u000bect arising from the emergent electromag-\nnetic field of skyrmions. Beyond that, a real-space observation\nby a scanning transmission X-ray microscopy has been made\nand confirmed the formation of antiferromagnetically coupled\nN´eel-type skyrmions at a TI interface with a ferrimagnet [24].\nSkyrmions on a TI surface are exhilarating not only from\nthe viewpoint of emerging physics arising from the interplay\nof real- and momentum-space topologies but also due to their\nprospects for spintronic nanodevices [6, 7]. It has been theo-\nretically proposed that a skyrmion on a TI surface is accom-\npanied by a nonzero charge density due to the chiral edge\n\u0003d.kurebayashi@unsw.edu.au\nyo.tretiakov@unsw.edu.austates [25–28]. Because of this additional charge attached to\nit, a skyrmion can be manipulated by external electric fields\nwithout Ohmic losses from currents. Another mechanism\nhas been proposed to manipulate skyrmions by utilizing spin-\ntransfer torques, which are greatly enhanced due to the spin-\nmomentum locking on the TI surface [29, 30]. Consequently,\nthe dynamics of skyrmions is expected to be faster, which is\nalso highly favorable for memory applications.\nAlthough the skyrmion dynamics on a TI surface has been\nintensively investigated, the missing ingredient for success-\nful applications of skyrmions in TIs is their nucleation stud-\nies. In conventional FMs, skyrmion nucleation has been ex-\nplored [31], for example, by employing geometric structures\nand local magnetic fluctuations, such as notches, edges, or\nimpurity sites [32–38] as well as by utilizing local injection\nof charge and spin currents [39–42]. However, the e \u000bect of\ndissipation on the nucleation process, including the influence\nof the Gilbert damping and thermal fluctuations, has not been\nunderstood well and therefore requires further investigation.\nMoreover, skyrmion nucleation on a TI surface might be sig-\nnificantly di \u000berent from the one in conventional FMs because\nof the spin-momentum locking. With increasing interest in\nskyrmionics with TIs, detailed studies on the skyrmion nu-\ncleation process are highly demanded and therefore are the\nsubject of this paper. We investigate the general properties of\nskyrmion nucleation on a TI surface, such as nucleation time\nand critical field. We perform micromagnetic simulations at\na finite temperature and demonstrate the Gilbert damping and\ntemperature dependences of the nucleation process. To give\nphysical understanding, we describe analytically the e \u000bect of\ntemperature based on a self-energy renormalization. By treat-\ning the Dzyaloshinskii-Moriya interaction (DMI) as a pertur-\nbation within the random-phase approximation, we succeed in\nreproducing the temperature dependence obtained in our mi-\ncromagnetic simulations.\nStochastic magnetization dynamics model.– Magnetization\ndynamics at finite temperature is analyzed with the stochastic\nLandau-Lifshitz-Gilbert (sLLG) equation [43, 44],\n@n\n@t=\u0000\r\n1+\u000b2\u0002n\u0002Be+\u000bn\u0002(n\u0002Be)\u0003+T; (1)\nwhere\ris the gyromagnetic ratio, \u000bis the Gilbert damp-arXiv:2112.12967v2  [cond-mat.mes-hall]  15 Nov 20222\ning constant, n(r)=M(r)=Msis the normalized mag-\nnetization, Msis the saturation magnetization, Be(r)=\n\u0000(1=Ms)\u000eFM=\u000en(r)+Bthis the e \u000bective magnetic field, Bth\nis the thermal field, FMis the magnetic free energy, and T\nis the spin-transfer torque. On a surface of TIs, the mag-\nnetic free energy FM=Fex+Fani+FZ+FDMI, where\nFex=(2J0S2=la)R\ndV(rn)2,Fani=\u0000KR\ndV(nz)2,FZ=\n\u0000MsR\ndV B znz, and FDMIare the exchange, anisotropy, Zee-\nman, and DMI energies, respectively. Here J0is the ex-\nchange constant between local magnetic moments, lais the\nmagnetic lattice constant, Sis the amplitude of the local\nmoments, Kis the easy-axis anisotropy constant along z-\naxis, and Bzis the magnetic field perpendicular to the film\n[45]. On a TI surface, electrons mediate anisotropic ex-\nchange interaction, namely the DMI, reflecting an inversion\nsymmetry breaking. Then the DMI takes the form FDMI=\n(D=\u0018l)R\ndV[nz(@xnx+@yny)\u0000nx@xnz\u0000ny@ynz], where\u0018lis the\npenetration length of a TI surface state into the magnetic in-\nsulator, D=\u0000J2[\u0002(jJj+EF)\u0000\u0002(jJj\u0000EF)]=(8\u0019vF) is the\nDMI constant mediated by the surface Dirac electrons [46],\nwhere Jis the s-dcoupling constant between electron’s spins\nand local moments, vFis the Fermi velocity, EFis the Fermi\nenergy, and \u0002(x) is the Heaviside function. The e \u000bective mag-\nnetic field is then given by\nBe\nx;y=4J0S2\nMsla\u0001nx;y+2D\nMs\u0018l@x;ynz+Bth\nx;y; (2)\nBe\nz=4J0S2\nMsla\u0001nz+Bz\u00002D\nMs\u0018lr\u0001n+2K\nMsnz+Bth\nz:(3)\nIt is known that the spin-transfer torques on a TI surface that\ncouples to local magnetic moments are substantially modified\ndue to the strong spin-orbit coupling:\nT=\u0000e\r\nMs\u0018l(\u000beˆx+\feˆy) (r\u0001n)Ex; (4)\nwhere\u000be=\u001cJ3sgn(EF)\u0002(E2\nF\u0000J2)=(8\u0019E2\nF) and\fe=\n\u001c2J2(E2\nF\u0000J2)sgn( EF)\u0002(E2\nF\u0000J2)=(8\u0019E2\nF) are dimensionless\ncoe\u000ecients [29, 30]. These specific spin-transfer torques oc-\ncur at the TI /magnetic insulator interface or in magnetic TI\nthin films. A detailed derivation of the spin-transfer torques\non the TI surface is given in the Appendix. Note that we ne-\nglected the e \u000bect of the spin-orbit torques (SOT) on the mag-\nnetization dynamics in Eq. (1). On the TI surface, the SOT\nplays an important role in the magnetic dynamics because of\nthe spin-momentum locking, however, for the skyrmion nu-\ncleation, which requires spatial inhomogeneity, the SOT does\nnot give rise to any qualitative di \u000berences as the SOT are uni-\nform contribution. Instead, the SOT under the DC current is\nequivalent to a static uniform in-plane magnetic field, there-\nfore reducing the energy barrier between the uniform FM and\nskyrmion states and thus contributing to the reduction of the\ncritical field.\nFor micromagnetic simulations, we discretize a space into\na square lattice by using relations @in(r)jrj\u0019(nj+ei\u0000\nnj\u0000ei)=(2la),@2\nin(r)\f\f\frj\u0019(nj+ei\u00002nj+nj\u0000ei)=l2\na, and impos-\ning periodic boundary conditions in x- and y-directions. To\nnumerically simulate finite temperature, the thermal field is\n050100150200y/la(a)t= 0 ns\n (b)t= 0.2 ns\n0 50 100 150 200\nx/la050100150200y/la(c)t= 1 ns\n0 50 100 150 200\nx/la(d)t= 2 ns\n−1.0−0.50.00.51.0\nnz\n−1.0−0.50.00.51.0\nnzFIG. 1. Magnetic profile at (a) t=0 s when no current is applied,\n(b)t=0:2 ns after current pulse is applied, (c) t=1 ns after current\npulse is applied, and (d) after the current pulse is switched o \u000b. The\ncolor code and arrows show the z- and in-plane components of mag-\nnetization, respectively. The parameters are \u000b=0:04,T=0:2K, and\nEx=6:5\u0002104V/m.\ndefined by Bth=\u0011p\n2\u000bkBT=(Ms\rV\u0001t), where \u0011is the ran-\ndom vector drawn from a standard normal distribution, Vis\nthe average magnetic-ion volume, and \u0001tis the simulation\ntime-step. As an integration scheme, we have employed the\nHeun’s method with the time-step \u0001t\u00183 fs. Since these\nsimulations are all stochastic, all numerical data are obtained\nas the statistical average over 50 independent simulations.\nFor typical parameters of a magnetic TI, we have used val-\nues estimated from the first-principle calculations and exper-\niments: vF=2:55 eV Å, J=0:15 eV , J0=1:38\u000210\u000023\nJ,K=7:25\u000210\u000027J/Å3,Ms=1:16\u0002104J/(Tm3), and\nla=\u0018l=8:1 Å [47, 48].\nSkyrmion nucleation.– We first examine the skyrmion nu-\ncleation with uniform currents. Figure 1 shows the magnetic\nprofiles under a uniform current pulse. Here, we chose the\nparameters as \u000b=0:04,T=0:2 K, and Ex=6:5\u0002104\nV/m. At t=0 ns, when no current is applied, the magne-\ntization is along the z-axis. After application of the current\npulse, it first develops particle-like small fluctuations as shown\nin Fig. 1(b). Eventually, these particle-like fluctuations grow\ninto N ´eel skyrmions once their radius reaches the critical one\ndetermined by J,D, and Bz[49, 50], otherwise they collapse\nback into a uniform state. Note that, during an early stage of\nthe nucleation, a skyrmion - antiskyrmion pair is created due\nto the topological number conservation [11, 51]. However, be-\ncause our DMI stabilizes only skyrmions, the antiskyrmions\nquickly decay into the uniform state [52]. After the current\npulse is switched o \u000b, only the skyrmions survive, see Fig. 1\n(d). These results clearly show that skyrmions can be nucle-3\nFIG. 2. (a) The nucleation time and (b) the skyrmion number as a\nfunction of electric field Exare plotted for various Gilbert damping\n\u000bandT=1 K. The dots are numerical data and solid lines are\nfitting functions. The inset of panel (a) shows the Gilbert damping\ndependence of the critical field Ecin the units of 104V/m.\nated by uniform current pulses at TI /FM interfaces [53].\nE\u000bect of Gilbert damping.– There is always a delay be-\nfore the first skyrmion is nucleated. To investigate this nu-\ncleation time, we first study its Gilbert damping \u000bdepen-\ndence. Since \u000bdepends on various factors such as disor-\nder, it is important to understand how it a \u000bects the nucle-\nation process. The skyrmion nucleation time tnfor various\n\u000bis shown in Fig. 2 (a) as a function of applied field Ex.\nWe define the tnas a time before the total skyrmion num-\nber,Nsk=\f\f\f!\ndxdy n\u0001(@xn\u0002@yn)\f\f\f=(4\u00192);exceeds one. The\nskyrmion nucleation is absent for small Ex, i.e., there is a\ncritical field Ecfor the nucleation process. In this regime,\nthe energy dissipation caused by the Gilbert damping exceeds\nthe energy influx due to the spin-transfer torque [54], such\nthat the total accumulated energy is insu \u000ecient to nucleate a\nskyrmion.\nIn terms of the nucleation time, Fig. 2 (a) shows diverg-\ning behavior at Ecand monotonically decreases with Ex. This\ncan be explained based on the energy considerations. Since\nthe energy influx per unit time to the system is linearly pro-\nportional to Exas the coupling to electrons is treated within\nthe linear response theory, the total accumulated energy is\nFtot/Ex. The nucleation rate 1 =tnis, then, proportional to\nthe energy di \u000berence between the total accumulated energy\nFtotand the nucleation energy of a single skyrmion Fsk, i.e.,\n1=tn/Ftot\u0000Fsk\u0018Ex\u0000Ec. Thus, the nucleation time scales\nastn/(Ex\u0000Ec)\u00001. Indeed, the numerical data is in excellent\nagreement with it, see Fig. 2 (a), when fitted by,\ntn(\u000b)=A[jExj\u0000Ec(\u000b)]\u00001; (5)\nwhere Ais a coe \u000ecient and Ec(\u000b) is the critical field at the\ngiven\u000b. As shown in the inset of Fig. 2 (a), the critical field Ec\nis linear in\u000b. This is because the energy dissipation is linear in\n\u000band the energy influx is /Ex. Thus, as the total accumulated\nenergy is determined as the di \u000berence of the energy influx\ndue to spin-transfer torques and the Gilbert dissipation, the\nrequired energy influx to nucleate a skyrmion should linearly\nincrease with \u000b.\nWe also examined the total skyrmion number nucleated af-\nter 1 ns pulse as a function of applied field Exfor several\n\u000b. As shown in Fig. 2 (b), the nucleated skyrmion number\nNsklinearly increases with Exin the vicinity of the critical\n0 5 10 15 20 25\nEx[104V/m]0.00.20.40.60.81.0t[ns]\n(a)\nSteady\nnucleationTurbulence\nregime\n05101520Nsk\n(b)\n(c)FIG. 3. (a) The skyrmion number Nskas a function of time and elec-\ntric field Ex, for\u000b=0:04 and T=0:2 K. (b), (c) Magnetic profile at\nt=0:5 ns and Ex=7\u0002104V/m, and (c) t=0:5 ns and Ex=22\u0002104\nV/m.\nfield. However, as Exincreases further, Nskdeviates from\na linear slope and saturates. The saturation occurs because\nskyrmions start overlapping and merging into large domains\nas the skyrmion density increases. By further increasing Ex,\nskyrmion states are almost destroyed by strong magnon exci-\ntations. Time evolution of Nskat each applied field and corre-\nsponding magnetic profiles are shown in Fig. 3. As seen from\nFig. 3 (a), Nskstays constant after reaching a steady state for\nthe fields below Ex\u00181:5\u0002105m/V . In this regime, all nucle-\nated skyrmions are well separated, as shown in Fig. 3 (b). On\nthe other hand, for larger Excorresponding to the turbulence\nregime in Fig. 3 (a), Nskdecreases as Exincreases and oscil-\nlates in time. Figure 3 (c) shows a typical magnetic profile in\nthe strong field regime, swirling magnetic structures no longer\nsurvive [53], and larger magnetic domains are formed because\nof strong magnon excitations. Note that it is a crossover, not a\nphase transition, between the steady nucleation and turbulence\nregimes.\nE\u000bect of temperature.– Next, we examine the temperature\ne\u000bects on the nucleation phenomenon. The nucleation time\nfor various temperatures is presented in Fig. 4. One can notice\nthat the temperature Tonly a \u000bects the critical field, see Ec(\u000b)\nin Eq. (5), whereas the functional form of Exis hardly mod-\nified. The critical field linearly decreases with T, as shown\nin the inset of Fig. 4. Phenomenologically, the linear depen-\ndence on Tcan be understood as follows. The thermal fluc-\ntuations supply the energy \u0018kBTto the system, where kBis\nthe Boltzmann constant. Due to this additional contribution,\nthe energy required to create a skyrmion reduces linearly with\nT. We note that the critical field vanishes around T\u00188 K;\nabove this temperature, skyrmions are nucleated even without\nthe spin-transfer torques due to the thermal fluctuations.\nAlthough phenomenological energy considerations explain\nthe temperature e \u000bect on the nucleation process, we move one\nstep further and try to explain this phenomenon in terms of\nferromagnetic magnon excitations. Introducing the Holstein-\nPrimako \u000brepresentation, the free energy of the system can be4\nFIG. 4. The nucleation time as a function of electric field Exfor\nvarious temperatures and \u000b=0:04. The dots are numerical data,\nwhile the solid curves are fitting functions given by Eq. (5). The\ninset shows the temperature dependence of the critical field Ec.\ntransformed to the magnonic Hamiltonian as\nˆHm=X\nkay\nk\u0010\nJk+˜K+˜B\u0000tk\u0011\nak\n+X\nk;q\u0010\nDqay\nqay\nk\u0000q=2ak+q=2+D\u0003\nqay\nk+q=2ak\u0000q=2aq\u0011\n;(6)\nwhere akis the magnon annihilation operator with the wave\nnumber k,Jk=8J0S2P\ni=x;y(1\u0000coskj) is the exchange en-\nergy, ˜K=2Kl3\nais the easy-axis anisotropy, ˜B=MsBzl3\nais\nthe Zeeman energy, Dk=2iDla(sinkx+isinky) is the DMI,\nandtk=4eExla(\fesinky+\u000besinkx) describes the e \u000bect of\nthe spin-transfer torque. Note that we have retained the lin-\near in Exterm and neglected the higher-order terms. The first\nterm gives a single-particle magnon dispersion as it already\nhas the bilinear form in magnon operators. On the other hand,\nthe DMI in the second line of Eq. (6) contains three-magnon\noperators and has to be treated perturbatively.\nThe full Green’s function is given by Gk(D)=h\u001ek¯\u001eki=R\nD(¯\u001e;\u001e)\u001ek¯\u001eke\u0000S[\u001e;¯\u001e]\nR\nD(¯\u001e;\u001e)e\u0000S[\u001e;¯\u001e], where S[\u001e;¯\u001e]=P\nk¯\u001ek(\u0000i!n)\u001ek+Hm[\u001e;¯\u001e]\nis the imaginary-time action, \u001ek=(!n;k)is the eigenvalue of\nthe magnon operator ak, andHmis the Hamiltonian in the\nmagnon coherent states basis. Within the random phase ap-\nproximation, the Green’s function Gk\u0019(\u0000i!n+Jk+˜K+˜B\u0000\ntk\u0000\u0006k)\u00001, where \u0006kis the self-energy induced by the DMI.\nThe real part of the self-energy modifies the magnon disper-\nsion, while the imaginary part gives a finite lifetime. There-\nfore, the e \u000bective magnon dispersion including the e \u000bect of\nDMI takes the form\n!e\u000b(k)=Jk+˜K+˜B\u0000tk\u0000Re[\u0006k]: (7)\nAs depicted in Fig. 5, the self-energy has two contributions\nFIG. 5. The Feynman diagrams contributing to the magnon self-\nenergy. (a) and (b) show the density-density and the pair correlations,\nrespectively.\n\u0006k= \u0006 k;d+ \u0006 k;pwith\n\u0006k;d=\u00004jDkj2X\nqfB(!q)\u0000fB(!k+q)\n!q\u0000!k+q+i0+; (8)\n\u0006k;p=4X\nqjDqj21+fB(!q)+fB(!k\u0000q)\n!q+!k\u0000q\u0000i0+; (9)\nwhere fBis the Bose-Einstein distribution and !k=Jk+˜K+\n˜B\u0000tkis the bare magnon dispersion. Equations (8) and (9) cor-\nrespond to the magnon density-density response function, \u0006d,\nand pair-correlation function, \u0006p. From Eqs. (8) and (9), one\nnotices that the real part of the self-energy is always positive,\nnamely, the magnon correlations always reduce the magnon\nexcitation gap min( !e\u000b). As the Bose-Einstein distribution\nfunction can be expanded as fB(!)\u0019kBT=!, the reduction of\nthe gap min( !e\u000b) due to the self-energy contributions is lin-\near in temperature. Then, because the magnon instability is a\nprecursor of the skyrmion nucleation and it occurs when the\nmagnon excitation gap collapses, we conclude that the criti-\ncal field linearly decreases with T. Note that this expansion\nis valid when a single-particle magnon excitation gap, !gap,\nis smaller than kBT. In typical magnetically-doped TIs, the\nperpendicular magnetic anisotropy is \u001810\u00006eV [48], which\ncorresponds to\u00180:1 K. Because the condition !gap<kBTis\nalways met within the temperature range of our simulations,\nthe linear- Tdependence observed in the inset of Fig. 4 can\nbe explained analytically based on the magnon excitations.\nNote that the instability and the nucleation of magnetic texture\nis a highly nonequilibrium process, and equilibrium analysis\nbased on the self-energy corrections may fail to completely\nexplain dynamical properties. We also neglected the magnon\nexcitation due to the electric field, which may a \u000bect the be-\nhavior in a low-temperature regime. However, it is still capa-\nble of qualitatively describing statistical properties such as the\naverage nucleation time and critical field.\nDiscussion.– Let us review potential candidate materials\nfor magnetic TIs. In this study, we used material param-\neters for a typical magnetic TI, such as Cr- and V-doped\n(Bi;Sb) 2Te3[47, 48]. The Curie temperature of this group\nof materials, however, is Tc\u001915\u001830K, and therefore the\nskyrmion state is only stable at the cryogenic temperatures,\nthus hindering some device applications. In order to overcome\nthis limitation, achieving high Tcis particularly desirable. In\na similar group of materials, such as MnSb 2Te4[55], a higher\nCurie temperature Tc\u001850K is observed. Furthermore, the\nincrease of Tcby 60K due to the exchange bias e \u000bect is ob-\nserved in a superlattice structure of Dy-doped Bi 2Te3and Cr-\ndoped Sb 2Te3[56]. As another approach to realizing high Tc5\nmagnetic TIs, bilayer systems consisting of a TI and magnetic\ninsulator with a higher Tcare expected to be more commer-\ncially realizable candidates. It is reported that a TI grown on\nYIG by molecular-beam epitaxy exhibits the anomalous Hall\ne\u000bect, i.e., a signature of magnetically-coupled surface states,\nup to 300K [57, 58]. Thus, with recent advances in nanotech-\nnology, there is a real prospect that room-temperature mag-\nnetic TIs will be within the reach in the near future.\nIn conclusion, we have comprehensively studied the\ncurrent-induced skyrmion nucleation on a surface of a topo-\nlogical insulator. As a system, we have considered a het-\nerostructure consisting of a TI and an ultrathin ferromag-\nnetic insulator (applicable to the recent experimental realiza-\ntions [21–24]) and employed the specific spin-transfer torque\nrealized on a TI surface coupling to magnetic local moments.\nBy solving the stochastic LLG equation, we have determined\nthe critical field Ecfor skyrmion nucleation and found that the\nnucleation time tn/(Ex\u0000Ec)\u00001, where Echas been found to\nbe proportional to the Gilbert damping \u000b. These results sug-\ngest that with the advances in nanotechnology, when much\ncleaner samples with lower damping will be available, faster\nskyrmion nucleation with lower fields may be achieved. Fur-\nthermore, we have investigated the temperature dependence\nof the nucleation and have observed the temperature e \u000bects on\nthe critical field, which linearly decreases as temperature rises.\nWe have given a quantum microscopic description for this lin-\near dependence based on magnon excitations. The self-energy\ncorrections due to the Dzyaloshinskii-Moriya interaction pro-\nvide linear in temperature reduction of the magnon excitation\ngap and the critical field. Our results give a phenomenologi-\ncal understanding of the skyrmion nucleation on a surface of a\ntopological insulator and open doors for developing TI-based\nskyrmionic memory and logic nanodevices.\nACKNOWLEDGMENTS\nThe authors are grateful to N. Nagaosa for insightful discus-\nsions. O.A.T. acknowledges the support from the Australian\nResearch Council (Grant No. DP200101027), the Cooperative\nResearch Project Program at the Research Institute of Electri-\ncal Communication, Tohoku University (Japan), and the NC-\nMAS grant.\nAppendix: Spin-transfer torque on the surface of a TI\nIn the appendix, we present a detailed derivation of spin-\ntransfer torques on the surface of a topological insulator. TheHamiltonian describing the surface states of a topological in-\nsulator is given by\nH=H0\u0000evFX\nk y\nk[A(t)\u0002\u001b]z k\u0000JX\nrn(r)\u0001 y\nr\u001b r;\n(A.1)\nwhere H0=P\nk y\nkh\n\u0000vF\u0010\nkx\u001by\u0000ky\u001bx\u0011\n\u0000J\u001bzi\n kdescribes a\nmassive Dirac Fermions with a helical spin texture [47],  kis\nthe Fermionic annihilation operator at momentum k,vFis the\nFermi velocity, \u001bis the Pauli matrix describing real spin de-\ngrees of freedom, A=(Ax;Ay) is the electromagnetic vector\npotential, Jis as-dexchange constant, and n(r)=ˆz+\u000en(r) is\nmagnetization of local moments. Here we take the saturated\nmagnetization direction along ˆzaxis and assume \u000en(r) is a\nsmall magnetic fluctuation around ˆzaxis, namely ˆz\u0001\u000en(r)=0\nandj\u000en(r)j<<1.\nBy treating the electromagnetic field and the magnetic fluc-\ntuation,\u000en(r), as perturbations, a non-equilibrium spin ac-\ncumulation induced by an electric field applied along the x-\ndirection is evaluated as\nh\u001b(r;t)i=\u0000iTr\u0002\u001bG<(r;r0;t;t)\u0003\f\f\fr0!r; (A.2)\nwhere G<(r;r;t;t)=ih y\nr0(t0) r(t)iis the lesser Green’s func-\ntion on the Keldysh contour and the trace is over the spin in-\ndices [59]. Turning into momentum space, the contour or-\ndered Green’s function obeys the Dyson equation given as\nGk;k0(t;t0)=gk(t\u0000t0)\u000ek;k0\n\u0000evFX\nkZ\nCdt1gk(t\u0000t1)Ax(t1)\u001byGk;k0(t1;t0)\n\u0000JX\nk;qZ\nCdt1gk(t\u0000t1)\u000en(q)\u0001\u001bGk\u0000q;k0(t1;t0);\n(A.3)\nwhere Gk;k0=P\nk;k0exp(\u0000ik\u0001r+ik0\u0001r0)G(r;r0) and gk=\n[i@t\u0000H0]\u00001is an unperturbed Green’s function. By expanding\nthe contour-ordered Green’s function up to the first order in\nthe external field and the magnetic fluctuation, \u000en(r), one can\nobtain that\nGk;k0(t;t0)\u0019\u0000evFJ\u000ek\u0000q;k0X\nk;qZ\nCdt1Z\nCdt2\u000en(q)\u0001h\ngk(t\u0000t1)Ax(t1)\u001bygk(t1\u0000t2)\u001bgk\u0000q(t2\u0000t0)\n+gk+q(t\u0000t1)\u001bgk(t1\u0000t2)Ax(t2)\u001bygk(t2\u0000t0)i\n: (A.4)\nNote that, in Eq. (A.4), we have neglected the terms irrele- vant to the non-equilibrium spin accumulation. By substitut-6\ning Eq. (A.4) into Eq. (A.2), the spin accumulation is given by\nh\u001b\u000b(r;t)i=ievFJX\nk;qeiq\u0001rAx(t1)\u000en\f(q)Tr\"Z\nCdt1Z\nCdt2\u001b\u000bgk(t\u0000t1)\u001bygk(t1\u0000t2)\u001b\fgk\u0000q(t2\u0000t)\n+Z\nCdt1Z\nCdt2\u001b\u000bgk+q(t\u0000t2)\u001b\fgk(t2\u0000t1)\u001bygk(t1\u0000t)#<\n: (A.5)\nWith use of the Langreth theorem,\n\"Z\nCdt1Z\nCdt2A(t\u0000t1)B(t1\u0000t2)C(t2\u0000t0)#<\n=Z1\n\u00001dt1Z1\n\u00001dt2\u0002Ar(t\u0000t1)Br(t1\u0000t2)C<(t2\u0000t0)\n+Ar(t\u0000t1)B<(t1\u0000t2)Ca(t2\u0000t0)+A<(t\u0000t1)Ba(t1\u0000t2)Ca(t2\u0000t0)\u0003;(A.6)\nwhere superscripts ( a;r) denote the advanced and retarded\nfunctions, the non-equilibrium spin accumulation is obtainedas\nh\u001b\u000b(r;t)i=X\nqZd\n2\u0019eiq\u0001r\u0000i\nt\u001f\u000b\f(q;\n)Ax(\n)\u000en\f(q):(A.7)\nHere\u001f\u000b\f(q;\n) is a spin susceptibility defined by\n\u001f\u000b\f(q;\n)=ievFJZd!\n2\u0019X\nkTrh\n\u001b\u000bgk(!+ \n)\u001bygk(!)\u001b\fgk\u0000q(!)+\u001b\u000bgk+q(!)\u001b\fgk(!)\u001bygk(!\u0000\n)i<\n\u0019i\n2\u0019evFJX\nkh\n\u001b\u000bgr\nk\u001byga\nk\u001b\fga\nk\u0000q+\u001b\u000bgr\nk+q\u001b\fgr\nk\u001byga\nki\n; (A.8)\nwhere gr;a\nk\u0011gr;a\nk(!=0)=\u0002\u0000H0\u0006i\u0011\u0003\u00001,\u0011is the damping\nrate induced by impurity scattering, and we have expanded\nthe function with respect to the frequency up to the first order.\nHere we have considered the small scattering regime ( \u0011 <<\nEF;J) and only retained the dominant contributions which\ninclude both the retarded and advanced Green’s functions. By\nexpanding the Green’s function with respect to the external\nmomentum, q, asgk+q\u0019gk\u0000vF\u000f\r\u001aq\rgk\u001b\u001agk+O(q2), one\ncan obtain that\n\u001f\u000b\f(q;\n)\u0019i\n2\u0019ev2\nFJX\nk\u000f\r\u001ah\n\u001b\u000bgr\nk\u001byga\nk\u001b\fga\nk\u001b\u001aga\nk\n\u0000\u001b\u000bgr\nk\u001b\u001agr\nk\u001b\fgr\nk\u001byga\nki\nq\r: (A.9)\nNote that we have dropped the O(q0)-terms because they are\nthe uniform contributions, namely corresponding to the spin-\norbit torque, which is well known and there is no need to\nreproduce in this Appendix. After momentum integration in\nEq. (A.9), the non-equilibrium spin accumulation correspond-\ning to the spin-transfer torques is obtained by\nh\u001b?(r;t)i=C(EF)r\u0001\u000en(r)Ex; (A.10)\nwhere C(EF)=eJ\n8\u0019E2\nFh\n\u001c2(E2\nF\u0000J2) ˆx\u0000\u001cJM zˆyi\nsgn(E F)\u0002(E2\nF\u0000J2),\u001c=1\n2\u0011is the electron scattering lifetime, sgn( x) is the\nsign function, and \u0002(x) is the Heaviside function. Finally, the\nspin-transfer torques are given by\nT=\u0000\r0J\nMS\u0018lˆz\u0002h\u001b?(r;t)i; (A.11)\nTx=\u0000e\r0\u000be\nMS\u0018l(r\u0001\u000en)Ex; (A.12)\nTy=\u0000e\r0\fe\nMS\u0018l(r\u0001\u000en)Ex; (A.13)\nwhere\u000be=\u001cJ3\n8\u0019E2\nFsgn(EF)\u0002(E2\nF\u0000J2) and\fe=\n\u001c2J2(E2\nF\u0000J2)\n8\u0019E2\nFsgn(EF)\u0002(E2\nF\u0000J2) are dimensionless coe \u000ecients\ncharacterizing the spin-transfer torque mediated by the Dirac\nelectrons [29, 30]. Note that Eq. 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Jpn. 75,\n113706 (2006)."    },    {        "title": "2201.04932v1.Enhanced_Planar_Antenna_Efficiency_Through_Magnetic_Thin_Films.pdf",        "content": "> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  1 \n  Abstract— This work proposes to use magnetic material as the substrate of planar antennas to overcome the platform effect caused by the conducting ground plane. The upper bound of the radiation efficiency of an electric-current-driven low-profile antenna is theoretically derived, which is inversely proportional to the Gilbert damping factor of the magnetic material. Meanwhile, the improvement of radiation due to the use of magnetic material is demonstrated by a three-dimensional (3D) multiphysics and multiscale time-domain model. The simulation results match the theoretical derivation, showing 25% radiation efficiency from a planar antenna backed by a FeGaB thin film with 2.56 µm thickness. Furthermore, for conductive ferromagnetic materials, it is shown that the eddy current loss can be well suppressed by laminating the thin film into multiple layers. The radiation efficiency of the modeled antenna with a conductive ferromagnetic substrate is improved from 2.2% to 11.8% by dividing the substrate into 10 layers, with a ferromagnetic material fill factor of 93%.   Index Terms— ADI, eddy current loss, electromagnetics, antenna, FDTD, ferromagnetic resonance, lamination, magnetic thin films, numerical computation, planar structures, radiation, radiation efficiency, solver  I. INTRODUCTION caling down of circuitry has been a growing trend in modern electronics, enabling miniaturized and interconnected systems. Specifically, conformal devices with very small thicknesses are popular in applications such as wearable devices for law enforcement, military, and civilian emergency services [1], [2]. However, a major challenge that stands in the way of realizing these new technologies is scaling  Manuscript received **, 2021. The work was supported by NSF Nanosystems Engineering Research Center for Translational Applications of Nanoscale Multiferroic Systems (TANMS) Cooperative Agreement Award (No. EEC-1160504), and the Defense Advanced Research Projects Agency (DARPA) Magnetic Miniaturized and Monolithically Integrated Components (M3IC) Program under award W911NF-17-1-0100. Zhi Yao is with Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA (email: jackie_zhiyao@lbl.gov). Sidhant Tiwari is with Sandia National Laboratory, Albuquerque, NM 87123, USA. Joseph Schneider is with Lawrence Livermore National Laboratory, Livermore, CA 94550, USA. Robert N. Candler and Yuanxun Ethan Wang are with the Department of Electrical and Computer Engineering, University of California, Los Angeles, CA 90095, USA. Robert N. Candler is jointly with the California NanoSystems Institute (CNSI), Los Angeles, CA 90095, USA. Gregory P. Carman is with the Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA 90095, USA. down the antenna. Planar dimensions of traditional antennas must be on par with the electromagnetic (EM) wavelength to transmit efficiently [3]. Moreover, space-saving low-profile antennas require a ground plane for operation, causing poor radiation due to the platform effect. The platform effect is the major issue that conventional current-based antennas suffer when placed at a short distance above a conducting plane. The radiation becomes inefficient because the image current flows in the opposite direction and cancels the original current source. Therefore, antenna scaling is prevented by the excessive reactive energy stored between the radiating element and the conducting plane, thus raising the radiation quality factor (Q factor) and making the antenna difficult to match. In order to alleviate the platform effect and to allow for miniaturization, new materials and technologies need to be implemented.  One of the effective ways to increase efficiency is to replace the regular dielectric substrates with magnetodielectric materials, which provide high values of both relative permittivity 𝜖! and relative permeability 𝜇!\t. In patch antennas with such substrates, the effective EM wavelength is reduced by approximately √𝜖!𝜇! times, leading to a reduction of antenna characteristic length by approximately the same scale factor [4]. Efforts have been made on implementing both natural ferrites [5]–[7] and artificial magnetic materials, such as metamaterials [8]. When the material loss is considered, modeled with a the complex permeability 𝜇!=𝜇!\"−𝑗𝜇!\"\", the common understanding in these works is to avoid large values of 𝜇!\"\", which in turn limits the value of 𝜇!\" to be below a hundred and the operation frequency of the natural magnetic materials to be below hundreds of megahertz [9], i.e. below domain wall resonance frequencies. However, as the frequency increases to the gigahertz range, magnetic materials now exhibit ferromagnetic resonance (FMR), manifesting itself as dramatically high values of  𝜇!\"\"  resulting in large values for the magnitude of 𝜇!. Moreover, there is a causality relation between 𝜇!\" and 𝜇!\"\", meaning these two values cannot be tuned independently of each other. The ferrites/ferromagnets perform as imperfect magnetic conductors, converting the electric current image [10], [11] into one that is parallel to the original source current, enhancing the radiation from the source rather than canceling it. The strong magnetic flux existing in the antenna structure enables the antenna to switch from being electric field dominated to magnetic field dominated. Following this strategy, new types of radiating mechanisms targeting antenna miniaturization have been proposed. These Enhanced Planar Antenna Efficiency Through Magnetic Thin-Films  Zhi Yao, Member, IEEE, Sidhant Tiwari, Member, IEEE, Joseph Schneider, Member, IEEE, Robert N. Candler, Senior Member, IEEE, Gregory P. Carman, and Yuanxun Ethan Wang, Fellow, IEEE \nS > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  2 include a strain-mediated antenna composed of composite multiferroic materials [12], [13] and mechanical antennas based on physically oscillating magnets [14], [15]. By utilizing the time-varying magnetic flux as the radiating source, these new types of antennas could potentially be immune to the conductive loss and the platform effect. The recent work on electrically small loop antenna with a ferrite substrate has experimentally demonstrated the enhancement of the antenna radiation performance by FMR of the ferrite [16]. It is worth pointing out that the multiferroic antenna and mechanical antenna both rely on high permeability magnetic material to lower the radiation quality factor, which is consistent with the approaches attempted by merely using magnetodielectric substrates [5]–[8].  The magnetic material used in magnetic antennas should possess an FMR frequency in the gigahertz range, large value of permeability, as well as low eddy current loss, i.e. low electric conductive loss. To achieve gigahertz FMR frequency and large permeability at the same time, in-plane biased magnetic thin films should be used. The reason is that, compared to that of bulk materials, the FMR frequency of an in-plane biased thin film is increased by the factor of (𝑀#/𝐻$ , where 𝑀# is the saturation magnetization and 𝐻$ is the magnetic DC bias. As the FMR frequency is increased in the in-plane biased case, lower bias magnetic fields can be used to achieve the same FMR frequencies, easing the requirements on the strength of the electromagnets used to provide the bias magnetic fields. Moreover, rather than ferrites, ferromagnetic materials are preferred. The reason is that in saturated ferromagnetic materials, all the spins are aligned to the bias direction, leading to a large spontaneous magnetization and a higher permeability near FMR. This is in contrast to the saturated ferrites, where adjacent spins are opposite to each other and different in magnitude, leading to a low saturation magnetization. Such spin orientation results in a smaller net magnetization thus lowering the magnitude of the permeability near the FMR frequency. However, even though ferromagnetic materials provide large permeability, they are typically conductive, leading to severe eddy current loss. Therefore, one also needs to resolve the dilemma between having the large magnetic permeability from ferromagnetic materials and requiring low conductive losses. It is proposed in this work to suppress the eddy current loss by segmenting the ferromagnetic film into thin layers, so that the giant eddy current loops are broken, and the conductive loss is reduced. In summary, this work proposes that by inserting in-plane biased, multi-layered, ferromagnetic thin films between the radiating source and the conducting ground plane, one can drastically improve the radiation performance, such as radiation efficiency.  In this manuscript, the elimination of the platform effect is demonstrated by studying a planar antenna. The antenna is composed of a planar electric current backed by a ferromagnetic thin film. At the bottom of the thin film, a perfect electric conducting (PEC) ground plane is assigned. The radiation efficiency of this idealized radiator is derived theoretically, based on the plane wave assumption. Furthermore, a three-dimensional (3D) finite difference time domain (FDTD) algorithm is developed [17], [18], to demonstrate the theory. The algorithm is based on the alternating directional implicit (ADI) method to achieve unconditional stability. In the model, an electric current source on top of an FeGaB substrate is used. The modeling results show that even if the substrate is only several-micrometer-thick, it can boost up the radiation power by six orders of magnitude. Additionally, the simulation results demonstrate the suppression of eddy current loss by laminating the continuous thin film into multiple layers. With laminated ferromagnetic substrate, the radiation efficiency of the current source with the ferromagnetic substrate can be improved from 2.2% to 12% by dividing the substrate into 10 layers. II. THEORY Consider an infinite, uniform current sheet 𝑖%\tthat radiates EM waves into free space, as shown in Fig. 1(a). The current source is placed over an infinite PEC ground plane, with a ferromagnet substrate inserted between them. The PEC-backed current source is the model of the antenna. The thickness of the substrate is electrically small such that 𝑘ℎ≪1, where k is the wave number in the substrate. The ferromagnetic substrate is biased to saturation by an in-plane magnetic DC field 𝐻$ that is parallel to the current source 𝑖%. EM waves are directly radiated into the free space and reflected by the PEC plane, resulting in destructive interference.  Therefore, according to the classical EM theory, the amplitude of the time-varying EM waves in the different regions are: Stored field: 1𝐸&=𝐸$sin(𝑘𝑧)𝐻'=−𝐸$/𝑗𝜂cos(𝑘𝑧),  Radiated field: 1𝐸&=𝐸$sin(𝑘ℎ)exp(−𝑗𝑘$𝑧)𝐻'=−𝐸$/𝜂$sin(𝑘ℎ)exp(−𝑗𝑘$𝑧), (1) where 𝐸$ is the aperture electric field amplitude at the interface between the free space and the substrate. The radiated power into the free space is thus calculated as 𝑃!()=12𝜂$A|𝐸|*#𝑑𝑠≈12𝜂$𝐸$*(𝑘ℎ)*𝑆 =+*,!𝐸$*ℎ*𝑆𝜔*|(𝜇\"−𝑗𝜇′′)𝜖|,    (2) Note that in Equation (2), the off-diagonal permeability terms in the Polder tensor are ignored for mathematical simplicity. In Equation (2), the approximation of a linear distribution along the z-direction for the electric field is applied since the tangential electric field on the PEC is zero. Similarly, an approximation of a uniform magnetic field distribution along the z-direction leads to the stored magnetic energy and magnetic power loss as in Equations (3) and (4), respectively.  𝑊-=+*∭𝜇′|𝐻|*./0𝑑𝑣≈+*𝜇′|𝐻|*ℎ𝑆=+*𝜇′1!\"23#$%&'$%%()\t2\"ℎ𝑆,   (3) > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  3 𝑃5=+*𝜔∭𝜇′′|𝐻|*./0𝑑𝑣≈+*𝜔𝜇′′1!\"23#$%&'$%%()\t2\"ℎ,    (4) \n Fig. 1.  Radiation from electric current source. The coordinate system is chosen such that the surface current is in the y direction. The thickness of the substrate satisfies the condition 𝑘ℎ≪1. (a) An infinite, uniform current sheet 𝑖!\tthat radiates EM waves into free space. The current source is grounded by an infinite perfect electrically conducting (PEC) plane, with a ferromagnet substrate inserted between them. The thickness of the substrate is electrically small such that 𝑘ℎ≪1, where k is the wave number in the substrate. The ferromagnetic substrate is biased to saturation by an in-plane magnetic DC field 𝐻\" that is aligned with the current source 𝑖!. Note that in the numerical model, the actual size of the structure is set to be 5×5×10 µm in the x, y, and z directions, respectively. The infinite planar size is realized by periodic boundary conditions applied at the four side walls. (b) The y-z cross section of (a). (c) Analytical permeability spectrum of a ferromagnetic material, FeGaB, under a DC magnetic bias of 60 Oe, with the saturation magnetization 4𝜋𝑀# being 1.2×10$\tGauss. (d) Circuit model of the structure in (a) and (b). The parallel RLC resonator represents the ferromagnetic resonance, and the shunt resistor 𝜂\" represents the intrinsic resistance of the fee space.  In ferromagnetic materials, the electric energy stored in the structure is negligible compared to the magnetic stored energy, thus the total amount of stored energy is approximately equal to the magnetic stored energy, or in the mathematical form, 𝑊67689≈𝑊-. Therefore, the total quality factor of the system is given by: 𝑄:;:(<≈𝜔=*>+,-?>.=+/0\t2!∙$%\"3$%%\"$%?$%%$%=A+\"0B!CA+%\"?A+%%\"D?A+\"\" .  (5) Note that in Equation (5), the electric power loss is neglected by assuming that the conductivity of the material is zero. Similarly, the radiation quality factor is 𝑄!()≈𝜔=*>+,-=+/0\t2!$+%\"3$+%%\"$%=A+\"0B!CA+%\"?A+%%\"D .  (6) Hence, for an antenna working around 2 GHz, 𝑄E8F is on the order of 104 when the thickness of the substrate is 1.5 µm if the material is non-magnetic, or in mathematical form, 𝜇!\"=1 and 𝜇!\"\"=0. Since traditional antennas are mostly made of conductors and rely on conductive current to radiate, the platform effect is an inevitable shortcoming of traditional low-profile antennas. The platform effect results in more energy being stored in the structure instead of being radiated away into free space, raising the antenna quality factor. On the other hand, a magnetic substrate with high relative permeability offers the capability of significantly lowering 𝑄E8F and improving the radiation performance of low-profile antennas. According to the definition, the radiation efficiency can be calculated as  𝜉!()=G454,6G+,-=++?$+%%$+%\"3$+%%\"∙78!/.     (7) In saturated ferromagnetic materials that is biased in-plane, the relative permeability is calculated by (8) [19]. 𝜇!=H9H!H+\"H+\"IH\"?JKH(*H!?H9)+1.     (8) In (8), 𝜔$ is the Larmor frequency, defined as 𝜔$=𝜇$𝛾𝐻$. Similarly, 𝜔N is defined as 𝜇$𝛾𝑀#, where 𝑀# is the saturation magnetization of the ferromagnet. The term 𝜔! stands for ferromagnetic resonance frequency, which is calculated according to Kittel’s equation as 𝜔!=𝜇$𝛾(𝐻$(𝐻$+𝑀#) for in-plane biased films. The term 𝛼 is the Gilbert damping constant of the ferromagnetic material, which is related to the FMR linewidth by the formula Δ𝐻=2𝛼𝜔/𝜇$𝛾. Substituting (8) into (7) yields [16] \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  4 𝜉!()=++?KH∙0!09∙\"0!3090+\"∙78!/ ,    (7’) Note that in the derivation of Equation (9), the latter term of Equation (8) (i.e. the constant number 1) is neglected. This is a valid approximation as near FMR, the value of 𝜇!\"\"* is very large such that 𝜇!\"*+𝜇!\"\"*≈𝜒!\"*+𝜇!\"\"*.  However, as mentioned previously, ferromagnetic materials are electrically conductive. Therefore, heat dissipation will be generated by the oscillating electric field in such materials. With a conductivity 𝜎 in the ferromagnetic film, the electric power dissipation can be calculated as in 𝑃1=12U𝜎|𝐸|*./0𝑑𝑣≈12U𝜎𝐸$*|𝑘𝑧|*./0𝑑𝑣 =+O𝜎𝐸$*𝜔*V((𝜇\"−𝑗𝜇\"\")𝜖V*ℎP𝑆,   (9) Similarly, 𝜉!()=G454,6G+,-=>+,->+,-?>.?>:\t=++?$+%%$+%\"3$+%%\"78!/?;/0$!<8!  . (10) Substituting Equation (8) into Equation (10) yields: 𝜉!()=++?KH∙0!09∙\"0!3090+\"78!/?;/0$!<8! ,  (11) Equation (11) again reveals the previously mentioned conclusion that the electric conductivity of the substrate decreases the radiation performance of the structure. III. MODEL In order to accurately simulate the performance of the low-profile antenna considered in this work, an algorithm that describes both the EM wave propagation and the micromagnetic dynamics is applied [20]. Mathematically, this algorithm simultaneously solves the Maxwell’s Equations (12) and the Landau-Lifshitz-Gilbert (LLG) Equation (13) using finite-difference time-domain (FDTD) method in a coupled fashion simultaneously. 𝛻×𝑯=𝜖Q𝑬Q:+𝑱+𝜎𝑬,\t𝛻×𝑬=−Q𝑩Q: ,  (12) Q𝑴Q:=𝜇$𝛾]𝑴×𝑯UVV_+K|5|𝑴×Q𝑴Q:.   (13) In Equation (12), E represents the electric field, J represents the electric current volume density, H represents the magnetic field intensity, and B represents the magnetic flux density. The material properties are also involved in the Maxwell’s equations, where 𝜎 is the electric conductivity and 𝜖 is the relative permittivity. In Equation (13),  𝜇$ represents the vacuum permeability. 𝛾 is the gyromagnetic ratio, possessing a value of −1.759×10++𝐶/𝑘𝑔. The term 𝛼 is the Gilbert magnetic damping constant defined as 𝛼=𝜇$𝛾𝛥𝐻/4𝜋𝑓:, with 𝛥𝐻 being the FMR linewidth and 𝑓: being the frequency at which the linewidth is measured. As the governing law of micromagnetics, the LLG Equation (13) describes the evolution of magnetization 𝑴, with 𝑯XYY being the total effective magnetic field that drives the magnetic spins [21]. The low-profile antennas considered in this work consist of structures with characteristic dimensions much smaller than the EM wavelength. Conventional FDTD operates under the limit of Courant–Friedrichs–Lewy (CFL) stability condition, which incurs a tremendous amount of calculation, especially with such a small-scale structure. To overcome the stability constraint and reduce the time consumption of the simulation, alternating direction implicit (ADI) methods are used to obtain unconditional stability. The size of the entire simulation space shown in Fig. 1(a) is 5×5×10 µm in the x, y and z directions, respectively. The lower boundary, as mentioned, is set to be PEC. The ferromagnetic film is placed on the PEC, with the thickness being h = 2.56 µm, and the planar dimensions 5×5 µm. Periodic boundary conditions are applied at the four side walls to realize the infinite dimensions of the magnetic film in the planar directions. The upper surface of the space is terminated with absorbing boundary. A uniform electric current excitation is applied on the top surface of the ferromagnetic thin film. The field components are defined such that all the electric field components are along the edges of the spatial cell, and all the magnetic field components are face-centered on the cell surfaces. The spatial resolution is Δ𝑥=1 𝜇𝑚, Δ𝑦=1 𝜇𝑚, Δ𝑧=0.01 𝜇𝑚, and the time step is set as Δ𝑡=2.31´10-13 s, which is 104 times of the CFL limit. The surface current excitation center frequency is 2.4 GHz, and it is in the form of a modified Gaussian pulse, with a bandwidth of ±500 MHz. In this work, FeGaB is used due to its attractive ferromagnetic properties, e.g., low electric conductivity and high saturation magnetization. In the model, the saturation magnetization of FeGaB is 4𝜋𝑀#=12000\tGauss, the FMR linewidth is Δ𝐻=30\tOersted, and the electric conductivity is 𝜎=5×10Z\tSiemens/meter\t[22]. The magnetic DC bias applied in-plane is 60 Oersted so that the FMR frequency and the input signal frequency overlap. Fig. 2 shows the simulated results with the geometry and material properties specified in the last paragraph. To explore the effect of conductive dissipation on the radiation performance, two additional cases have been simulated:  1. Artificial nonconductive ferromagnetic substrate. The setup of this control case is identical to the one previously introduced, except that the conductivity of FeGaB is artificially set to be zero. Therefore, this case is the optimal circumstance with a nonconductive ferromagnetic material.  2. No substrate under the current. In this case, the space between the current source and the PEC ground is filled with air, which is the original platformed antenna with the current source close to the PEC.   (a) \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  5 \n (b) \n Fig. 2.  Simulated radiation efficiency from the electric current source. (a) Antenna structure. The current is placed on the top surface of the ferromagnetic substrate. The size of the entire simulation space is 5×5×10 µm in the x, y and z directions, respectively. The spatial resolution is Δ𝑥=1 𝜇𝑚, Δ𝑦=1 𝜇𝑚, Δ𝑧=0.01 𝜇𝑚. (b) Simulated permeability of the FeGaB thin film, with conductivity of 5×10%\tSiemens/meter, and an artificial conductivity of zero. (c) Simulated radiation efficiency with conductive and nonconductive materials, as well as without substrate.   As can be seen in Fig. 2(b), the simulated permeability is almost independent from the electric conductivity, and it matches the analytical results. At FMR, the imaginary permeability 𝜇′′ is as large as 3000, and close to FMR the real permeability 𝜇′ is as large as 1500. Fig. 2(c) shows the comparison between the radiation efficiency of different substrate materials. Without the ferromagnetic substrate, the simulated radiation efficiency is on the order of 10-7 (as shown by the yellow curve with triangle marks), indicating that almost the entire radiated field is cancelled out by the PEC platform. Compared to the air-filled antenna, the artificial nonconductive ferromagnetic substrate improves the radiation efficiency by 106 times, leading to a radiation efficiency of 25%, shown by the black solid curve with circle marks. The red solid line represents the analytical radiation efficiency calculated by Equation (7’), showing a good match to the simulation. However, this is the ideal case with no eddy current loss. Practically, the conductive FeGaB results in the radiation efficiency being only 2.2%, as shown by the black dashed curve with circle marks in Fig. 2(c). The analytical radiation efficiency corresponding to the FeGaB material is calculated with Equation (11), and plotted as the red dashed curve in Fig. 2(c). Therefore, in order to achieve the full advantage of using a ferromagnetic substrate to improve the radiation performance, modified geometries need to be explored to suppress the eddy current loss. IV. EDDY CURRENT SUPPRESSION It is evident that magnetic materials with high relative permeability help overcome the platform effect. However, unfortunately, most of the materials that have such high permeability are ferromagnetic materials, which are highly conductive and suffer significant eddy current loss. By briefly analyzing Faraday’s law in the integral form ∮𝑬[∙𝑑𝒍=−(𝜕/𝜕𝑡)∬𝑩∙𝑑𝑨#, one can quickly conclude that the eddy current loss can be well suppressed by reducing the magnetic flux by laminating the thin film into multiple layers, as shown in the inset of Fig. 3. Note that the thickness of the laminates should be at least comparable to the skin depth, so that the eddy current loop could be broken into smaller loops and the conductive loss will be reduced. Fig. 3 shows the radiation efficiency of antennas with laminated substrates of various numbers of layers and different layer thicknesses. The gap between the PEC ground and the current source is approximately 2.56 µm for each lamination geometry. Since the planar dimensions of the antenna structure are constant, the thickness ratio represents the volume ratio of the material. The skin depth of FeGaB close to FMR is approximately 0.3 µm. Therefore, laminates with thicknesses smaller than 0.3 µm are effective for the eddy current suppression, such as the 8-layer, 10-layer and 12-layer structures, leading to the peak radiation efficiency of 9.27%, 11.8% and 10.31%, respectively. It is noticed that a dispersive radiation efficiency spectrum is formed, in contrast to the single-layer cases simulated in Fig. 2, where the radiation efficiency is constant over the frequency band under observation. Peaks in the dispersive spectrum 𝜉E8F,]8^(𝑓) are formed around 2.4 GHz, due to the FMR effect influenced by the inductive-capacitive coupling between the laminates. \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  6 \n Fig. 3.  Radiation efficiency of antennas with laminated substrate with various numbers of layers and different layer thicknesses. Inset: geometry of the laminated substrate. tm: thickness of the ferromagnetic layers; tair: gap between adjacent layers. N: number of ferromagnetic layers.   Intuitively, the more layers the ferromagnetic substrate is cut into, the more effectively the eddy current loops will be broken down, and the better suppression effect will be achieved. However, different from the intuition, it is observed from Fig. 3 that a maximum radiation efficiency is achieved with N=10 (N is the number of ferromagnetic layers), instead of the maximum number N=24. This optimum number of laminate layers is attributed to the balance between the interlayer coupling effects and the ferromagnetic material volume fraction. One type of possible interlayer coupling is capacitive coupling between adjacent ferromagnetic layers through the air gap, which is elaborated in Appendix A. To be brief, the electric fields couple to each other through the air gap, leading to an equivalent continuous, giant dielectric eddy current loop, which degrades the radiation efficiency. This effect grows stronger for thinner air gaps. Another type of interlayer coupling captured by the FDTD model is coupling via dipolar magnetic fields generated by the magnetization in each layer. Therefore, competing effects exist between the interlayer coupling, the ferromagnet/air volume ratio, and the change in the eddy current suppression effectiveness as the ferromagnetic layer thickness changes. This multi-factor scenario is captured by the numerical model proposed in this work, while is too complicated to be captured by theoretical analysis. In summary, the numerical model proposed in this work serves as a tool for optimizing the antenna structure for the purpose of suppressing the eddy current loss to the largest extent.  V. CONCLUSION In this work a layered ferromagnetic material is explored as a method to increase radiation efficiency in planar electrically small antennas. Air-filled planar electrically small antennas radiate almost no power into the space due to an image current that is anti-parallel to the source current. Furthermore, the radiation quality factor (Qrad) is large due to the reactive energy associated with this antenna geometry. Inserting a ferromagnetic material between the source and ground plane enhances the radiation efficiency by reversing the direction of the image current in addition to reducing Qrad through the high relative permeability of the material. Radiation efficiency is further improved by dividing the ferromagnetic material into several layers reducing eddy current losses, thus, lowering the negative effects caused by the materials conductivity. An ADI-FDTD model is developed coupling the LLG Equation with Maxwell’s Equations to study the complicated dynamics of a planar antenna with a laminated ferromagnetic substrate. Numerical results show that inserting FeGaB into the airgap increases efficiency from lower than 0.00001% to greater than 2%. Furthermore, dividing the FeGaB into 10 layers further improves the radiation efficiency to approximately 2% to 11.8%. These results show that the radiation efficiency of planar electrically small antennas can be dramatically increased by using a ferromagnetic material inserted between the source plane and the ground plane.  ACKNOWLEDGMENT The work was supported by NSF Nanosystems Engineering Research Center for Translational Applications of Nanoscale Multiferroic Systems (TANMS) Cooperative Agreement Award (No. EEC-1160504), and the Defense Advanced Research Projects Agency (DARPA) Magnetic Miniaturized and Monolithically Integrated Components (M3IC) Program under award W911NF-17-1-0100. \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  7 APPENDIX A: CIRCUIT MODELLING OF DISPLACEMENT EDDY CURRENT FLOW A. Analysis Setup As discussed in the main body of this work, a common method to reduce eddy currents in a ferromagnetic material is to laminate the material into several thin sheets separated by insulators. This has the effect of restricting the possible paths of the eddy currents, limiting the amount of current that flows and thus, limiting the power dissipated. However, as the layers get too thin, interlayer coupling can grow as the layer boundaries become closer together. One type of interlayer coupling is the capacitive coupling between the ferromagnetic laminations through the dielectric gaps. \n Fig. A1. Structure used to study the effect of displacement eddy currents on power dissipation: Two conductive ferromagnetic layers of thickness 𝑡& separated by an air gap of 𝑡'().  A RF magnetic field is applied along the width of the ferromagnetic layers.  To study the effect of the air gap thickness on power loss due to eddy currents, a simple structure of two conductive ferromagnetic layers separated by an air gap (Fig. A1) is used. By Faraday’s Law, the RF magnetic field applied to the sample will generate an electromotive force, which will cause eddy currents to circulate in the ferromagnetic layers. When the gap between the ferromagnetic layers is small, the two layers become strongly capacitively coupled and displacement currents travel between them, leading to an increase in power dissipation as new eddy current paths are created [23]–[26]. If the air gap is too thin, the eddy current suppression effects of the laminations becomes practically nonexistent.   \n Fig. A2. Equivalent circuit model used to intuitively study eddy currents generated in the structure depicted in Fig. A1.  To analyze the structure in Fig. A1, the approach of [27] is used and the structure is modelled by an equivalent circuit (Fig. A2). Here, the resistive elements 𝑅+ and 𝑅*\trepresents the power dissipation by eddy currents traveling along the length and thickness, respectively. The capacitive elements are used to account for the coupling between the layers, and the voltage sources account for the electromotive force generated by the RF magnetic field through Faraday’s Law. The values for the lumped elements and voltage sources used in the circuit model are shown in Table 1. Additional interlayer coupling effects (such as dipolar magnetic fields between the layers), dynamic magnetization effects due to ferromagnetic resonance, or incorporating a large number of layers would be difficult to capture using this approach and is a task better suited for the ADI-FDTD model discussed in the main body of this work.  Variable Formula Description 𝛿 A2/(𝜎𝜇)𝜇\"𝜔) Skin depth in ferromagnetic layer 𝑑* \t𝛿⋅{1−exp[−𝑡&/(2𝛿)\t]} Effective depth of eddy currents traveling along the length of the structure 𝑑+ 𝛿⋅{1−exp[−𝑙/(2𝛿)\t]} Effective depth of eddy currents traveling along the thickness of the structure 𝐶 𝜖\"𝑙𝑤/(2𝑡+) Capacitance coupling the eddy currents between the two ferromagnetic layers 𝑅* 𝑙/(𝜎𝑤𝑑*) Approximate resistance seen by eddy currents traveling along the length of the structure 𝑅+ 𝑡*/(2𝜎𝑤𝑑+) Approximate resistance seen by eddy currents traveling along the thickness of the structure 𝑍, 1/(𝑗𝜔∙𝐶) Impedance due to capacitive coupling of eddy currents 𝑉* 𝜔∙𝜇)𝜇\"∙𝑙∙(𝑡*/2)∙𝐻-. Electromotive force acting on eddy current loops contained solely in the ferromagnetic layer 𝑉+ 𝑉*+(𝜔∙𝜇\"∙𝑙∙𝑡+∙𝐻-.) Electromotive force acting on eddy current loops crossing between the adjacent two layers  Table 1: Formulas for variables used to calculate equivalent circuit parameters. Note that in the formulas for 𝑅* and 𝑅+, crowding of the current due to the skin effect is considered through the parameters 𝑑* and 𝑑+ [27].   B. Determining Unknown Currents  There are six unknown currents in the circuit shown in Fig. A2 that must be solved for to calculate the total power dissipated by the eddy currents. Using Kirchhoff’s voltage and current laws, six equations can be derived to solve for the six \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  8 unknowns. 𝑉+−2𝐼+𝑅*−(𝐼++𝐼P)𝑅+=0 ,   (A1a) 𝑉*−(𝐼*+𝐼_)∙(𝑅*+𝑍`)+(𝐼P+𝐼Z)𝑅+=0,   (A1b) 𝑉+−2𝐼O𝑅*−(𝐼O+𝐼Z)𝑅+=0,   (A1c) 𝐼+−𝐼P=𝐼* ,   (A1d) 𝐼*+𝐼Z=𝐼O ,   (A1e) 𝐼_+𝐼P=𝐼+ ,   (A1f) By adding Equations (A1d) and (A1f), we can solve for 𝐼_.  (𝐼+−𝐼P)+(𝐼_+𝐼P)=𝐼*+𝐼+   𝐼_=𝐼* (A2a) Using Equation (A2a), Equations (A1e) and (A1f) can be subtracted from each other to get the following relation.  (𝐼*+𝐼Z)−(𝐼*+𝐼P)=𝐼O−𝐼+   𝐼Z−𝐼P=𝐼O−𝐼+ (A2b) Rewriting Equations (A1a) and (A1c), solutions for 𝐼P and 𝐼Z can be found.  𝐼P=𝑉+𝑅+−1+2𝑅*𝑅+𝐼+ (A2c)  𝐼Z=𝑉+𝑅+−1+2𝑅*𝑅+𝐼O (A2d) Combining Equations (A2b), (A2c), and (A2d), 𝐼+ and 𝐼O can be solved for.  −1+2𝑅*𝑅+𝐼O+1+2𝑅*𝑅+𝐼+=𝐼O−𝐼+   1+2𝑅*𝑅+(𝐼O−𝐼+)=𝐼O−𝐼+ (A2e) The only way Equation (A2e) can be true for any value of a\"a7 is if 𝐼O−𝐼+ is zero.  Combining this with Equation (A2b), gives the following two equations for 𝐼Z and 𝐼O.  𝐼Z=𝐼P (A2f)  𝐼O=𝐼+ (A2g) With Equations (A2a), (A2f), and (A2g) in mind, the currents in Fig. A2 can be relabeled to reduce the number of unknowns down to three. \n Fig. A3. Equivalent circuit model shown in Fig. A2, but with Equations (A2a), (A2f), and (A2g) applied to reduce the number of unknowns.  Rewriting Equations (A1a) and (A1b), gives the following two equations for the remaining three currents.  𝑉+𝑅+−1+2𝑅*𝑅+𝐼+−𝐼P=0 (A3a)  𝑉*2𝑅+−𝑅*𝑅++𝑍[𝑅+𝐼*+𝐼P=0 (A3b) Combining Equation (A1d), (A3a), and (A3b), 𝐼P can be eliminated and two equations for 𝐼+ and 𝐼* can be found.  𝑉+𝑅+−1+2𝑅*𝑅+𝐼+−𝐼++𝐼*=0   𝑉+𝑅+−21+𝑅*𝑅+𝐼++𝐼*=0 (A3c)  𝑉*2𝑅+−𝑅*𝑅++𝑍[𝑅+𝐼*+𝐼+−𝐼*=0   𝑉*2𝑅+−1+𝑅*𝑅++𝑍[𝑅+𝐼*+𝐼+=0 (A3d) Using Equations (A1d), (A3c), and (A3d), 𝐼+, 𝐼*, and 𝐼P can readily be solved.   𝐷=21+𝑅*𝑅+1+𝑅*𝑅++𝑍`𝑅+−1 (A4a)  𝐼+=𝑉+𝑅+1+𝑅*𝑅++𝑍`𝑅++𝑉*2𝑅+𝐷\t (A4b)  𝐼*=𝑉+𝑅++1+𝑅*𝑅+𝑉*𝑅+𝐷\t (A4c)  𝐼P=𝑉+𝑅+𝑅*𝑅++𝑍`𝑅+−𝑉*𝑅+12+𝑅*𝑅+𝐷 (A4d)  C. Perfect Insulation Limit Assuming 𝑅+≫𝑅*, in the limit of a perfectly insulating air gap (𝑍`→∞), these equations reduce down to the following equations.  𝐼+=𝑉+2𝑅+\t (A5a)  𝐼*=0\t (A5b)  𝐼P=𝑉+2𝑅+ (A5c) In this limit, there is no coupling between the layers and the entirety of the eddy currents are contained within the individual ferromagnetic layers (Fig. A4a). Power dissipation in this scenario will be the minimum possible for the structure shown in Fig. A1. D. No Air Gap Limit Again assuming 𝑅+≫𝑅*,\tin the limit where the air gap goes to zero (𝑡(b!→0, 𝑉*→𝑉+, 𝑍`→0), Equations (A4b), (A4c), and (A4d) reduce down to the following equations.  𝐼+=32𝑉+𝑅+\t (A6a) \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  9  𝐼*=2𝑉+𝑅+\t (A6b)  𝐼P=−𝑉+2𝑅+ (A6c) In this limit, 𝐼P now reverses direction as additional eddy currents are allowed to flow throughout the structure since the two ferromagnetic layers are now shorted together (Fig. A4(b)). Power dissipation in this scenario is now the maximum possible for the structure shown in Fig. A1. \n  (a) (b) Fig. A4. (a) Current flow in the perfect insulation limit. (b) Current flow in the no air gap limit. E. Power Dissipation vs Air Gap Thickness The total power dissipated due to eddy currents is simply the sum of the power dissipated in each of the resistors in the equivalent circuit. Note that because of the reactive contribution from capacitive coupling between the layers, the currents found through Equations (A4b), (A4c), and (A4d) are generally complex.  𝑃)bccbd(:U)=2×12𝐼+𝐼+∗∙(𝑅++2𝑅*)+12𝐼*𝐼*∗∙𝑅*+12𝐼P𝐼P∗∙𝑅+ (A7) Equation (A7) is a strong function of 𝑡(b!, as it not only tunes the capacitive coupling between the layers (𝑍`), but also the electromotive force felt by the central current loop (𝑉*). For small thickness, the power dissipation will increase as it approaches the limit of completely shorted ferromagnetic layers. As the thickness increases, the impedance of the air gap also increases and eventually approaches the limit of the perfectly insulating air gap. Shown below is Equation (A7) plotted as a function of air gap thickness. Here, the structure has in-plane dimensions of 300 μm × 300 μm and the two ferromagnetic layers are 0.24\tµm thick. The ferromagnetic material is FeGaB at ferromagnetic resonance (2.4 GHz for a DC bias of 60 Oe). 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"    },    {        "title": "2201.06060v2.Ferromagnetic_resonance_modulation_in__d__wave_superconductor_ferromagnetic_insulator_bilayer_systems.pdf",        "content": "Ferromagnetic resonance modulation in d-wave superconductor/ferromagnetic\ninsulator bilayer systems\nYuya Ominato,1Ai Yamakage,2Takeo Kato,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China.\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: May 6, 2022)\nWe investigate ferromagnetic resonance (FMR) modulation in d-wave superconductor\n(SC)/ferromagnetic insulator (FI) bilayer systems theoretically. The modulation of the Gilbert\ndamping in these systems re\rects the existence of nodes in the d-wave SC and shows power-law\ndecay characteristics within the low-temperature and low-frequency limit. Our results indicate the\ne\u000bectiveness of use of spin pumping as a probe technique to determine the symmetry of unconven-\ntional SCs with high sensitivity for nanoscale thin \flms.\nI. INTRODUCTION\nSpin pumping (SP)1,2is a versatile method that can\nbe used to generate spin currents at magnetic junctions.\nWhile SP has been used for spin accumulation in vari-\nous materials in the \feld of spintronics3,4, it has recently\nbeen recognized that SP can also be used to detect spin\nexcitation in nanostructured materials5, including mag-\nnetic thin \flms6, two-dimensional electron systems7{9,\nand magnetic impurities on metal surfaces10. Notably,\nspin excitation detection using SP is sensitive even for\nsuch nanoscale thin \flms for which detection by con-\nventional bulk measurement techniques such as nuclear\nmagnetic resonance and neutron scattering experiment is\ndi\u000ecult.\nRecently, spin injection into s-wave superconductors\n(SCs) has been a subject of intensive study both theoret-\nically11{20and experimentally21{34. While the research\ninto spin transport in s-wave SC/magnet junctions is ex-\npected to see rapid development, expansion of the devel-\nopment targets toward unconventional SCs represents a\nfascinating research direction. Nevertheless, SP into un-\nconventional SCs has only been considered in a few recent\nworks35,36. In particular, SP into a d-wave SC, which is\none of the simplest unconventional SCs that can be real-\nized in cuprate SCs37, has not been studied theoretically\nto the best of our knowledge, although experimental SP\nin ad-wave SC has been reported recently38.\nIn this work, we investigate SP theoretically in a bi-\nlayer magnetic junction composed of a d-wave SC and\na ferromagnetic insulator (FI), as shown in Fig. 1. We\napply a static magnetic \feld along the xdirection and\nconsider the ferromagnetic resonance (FMR) experiment\nof the FI induced by microwave irradiation. In this setup,\nthe FMR linewidth is determined by the sum of the in-\ntrinsic contribution made by the Gilbert damping of the\nbulk FI and the interface contribution, which originates\nfrom the spin transfer caused by exchange coupling be-\nMicrowavex yz\nSpin current\nFerromagnetic resonanceInteractionFIG. 1. Schematic of the d-wave SC/FI bilayer system. The\ntwo-dimensional d-wave SC is placed on the FI. Precessional\nmotion of the magnetization is induced by microwave irradia-\ntion. The spins are injected and the magnetization dynamics\nare modulated because of the interface magnetic interaction.\ntween thed-wave SC and the FI. We then calculate the\ninterface contribution to the FMR linewidth, which is\ncalled the modulation of the Gilbert damping hereafter,\nusing microscopic theory based on the second-order per-\nturbation39{41. We show that the temperature depen-\ndence of the modulation of the Gilbert damping exhibits\na coherent peak below the transition temperature that\nis weaker than that of s-wave SCs11,13{15. We also show\nthat because of the existence of nodes in the d-wave SCs,\nthe FMR linewidth enhancement due to SP remains even\nat zero temperature.\nThe paper is organized as follows. In Sec. II, we in-\ntroduce the model Hamiltonian of the SC/FI bilayer sys-\ntem. In Sec. III, we present the formalism to calculate\nthe modulation of the Gilbert damping. In Sec. IV, we\npresent the numerical results and explain the detailed\nbehavior of the modulation of the Gilbert damping. In\nSec. V, we brie\ry discuss the relation to other SC sym-\nmetries, the proximity e\u000bect, and the di\u000berence between\nd-wave SC/FI junctions and d-wave SC/ferromagnetic\nmetal junctions. We also discuss the e\u000bect of an e\u000bectivearXiv:2201.06060v2  [cond-mat.mes-hall]  5 May 20222\nZeeman \feld due to the exchange coupling. In Sec. VI,\nwe present our conclusion and future perspectives.\nII. MODEL\nThe model Hamiltonian of the SC/FI bilayer system\nHis given by\nH=HFI+HdSC+HT: (1)\nThe \frst term HFIis the ferromagnetic Heisenberg\nmodel, which is given by\nHFI=\u0000JX\nhi;jiSi\u0001Sj\u0000~\rhdcX\njSx\nj; (2)\nwhereJ>0 is the exchange coupling constant, hi;ji\nrepresents summation over all the nearest-neighbor sites,\nSjis the localized spin at site jin the FI,\ris the gy-\nromagnetic ratio, and hdcis the static magnetic \feld.\nThe localized spin Sjis described as shown using the\nbosonic operators bjandby\njof the Holstein-Primako\u000b\ntransformation42\nS+\nj=Sy\nj+iSz\nj=\u0010\n2S\u0000by\njbj\u00111=2\nbj; (3)\nS\u0000\nj=Sy\nj\u0000iSz\nj=by\nj\u0010\n2S\u0000by\njbj\u00111=2\n; (4)\nSx\nj=S\u0000by\njbj; (5)\nwhere we require [ bi;by\nj] =\u000ei;jto ensure that S+\nj,S\u0000\nj,\nandSx\njsatisfy the commutation relation of angular mo-\nmentum. The deviation of Sx\njfrom its maximum value S\nis quanti\fed using the boson particle number. It is conve-\nnient to represent the bosonic operators in the reciprocal\nspace as follows\nbk=1p\nNX\nje\u0000ik\u0001rjbj; by\nk=1p\nNX\njeik\u0001rjby\nj;(6)\nwhereNis the number of sites. The magnon opera-\ntors with wave vector k= (kx;ky;kz) satisfy [bk;by\nk0] =\n\u000ek;k0. Assuming that the deviation is small, i.e., that\nhby\njbji=S\u001c1, the ladder operators S\u0006\njcan be approx-\nimated asS+\nj\u0019(2S)1=2bjandS\u0000\nj\u0019(2S)1=2by\nj, which\nis called the spin-wave approximation. The Hamiltonian\nHFIis then written as\nHFI\u0019X\nk~!kby\nkbk; (7)\nwhere we assume a parabolic dispersion ~!k=Dk2+\n~\rhdcwith a spin sti\u000bness constant Dand the constant\nterms are omitted.\nThe second term HdSCis the mean-\feld Hamiltonian\nfor the two-dimensional d-wave SC, and is given by\nHdSC=X\nk(cy\nk\";c\u0000k#)\u0012\n\u0018k \u0001k\n\u0001k\u0000\u0018k\u0013\u0012ck\"\ncy\n\u0000k#\u0013\n;(8)wherecy\nk\u001bandck\u001bdenote the creation and annihilation\noperators, respectively, of the electrons with the wave\nvectork= (kx;ky) and thexcomponent of the spin\n\u001b=\";#, and\u0018k=~2k2=2m\u0000\u0016is the energy of conduc-\ntion electrons measured from the chemical potential \u0016.\nWe assume that the d-wave pair potential has the form\n\u0001k= \u0001 cos 2\u001ekwith the phenomenological temperature\ndependence\n\u0001 = 1:76kBTctanh \n1:74r\nTc\nT\u00001!\n; (9)\nwhere\u001ek= arctan(ky=kx) denotes the azimuth angle of\nk. Using the Bogoliubov transformation given by\n\u0012ck\"\ncy\n\u0000k#\u0013\n=\u0012\nuk\u0000vk\nvkuk\u0013\u0012\rk\"\n\ry\n\u0000k#\u0013\n; (10)\nwhere\ry\nk\u001band\rk\u001bdenote the creation and annihilation\noperators of the Bogoliubov quasiparticles, respectively,\nandukandvkare given by\nuk=r\nEk+\u0018k\n2Ek; vk=r\nEk\u0000\u0018k\n2Ek; (11)\nwith the quasiparticle energy Ek=p\n\u00182\nk+ \u00012\nk, the mean-\n\feld Hamiltonian can be diagonalized as\nHdSC=X\nk(\ry\nk\";\r\u0000k#)\u0012\nEk 0\n0\u0000Ek\u0013\u0012\rk\"\n\ry\n\u0000k#\u0013\n:(12)\nThe density of states of the d-wave SC is given by43\nD(E)=Dn= Re\u00142\n\u0019K\u0012\u00012\nE2\u0013\u0015\n; (13)\nwhereDn=Am= 2\u0019~2is the density of states per spin of\nthe normal state, Ais the system area, and K(x) is the\ncomplete elliptic integral of the \frst kind in terms of the\nparameterx, where\nK(x) =Z\u0019=2\n0d\u001ep\n1\u0000xcos2\u001e: (14)\nD(E) diverges at E=\u0001 = 1 and decreases linearly when\nE=\u0001\u001c1 because of the nodal structure of \u0001 k. The\ndensity of states for an s-wave SC, in contrast, has a\ngap forjEj<\u0001. This di\u000berence leads to distinct FMR\nmodulation behaviors, as shown below.\nThe third term HTdescribes the spin transfer between\nthe SC and the FI at the interface\nHT=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+J\u0003\nq;k\u001b\u0000\n\u0000qS+\n\u0000k\u0001\n; (15)\nwhereJq;kis the matrix element of the spin transfer pro-\ncesses, and \u001b\u0006\nq= (\u001by\nq\u0006i\u001bz\nq)=2 andS\u0006\nk=Sy\nk\u0006iSz\nkare3\n(a) Spin transfer process (b) Self-energy\nJq,kJ*q,k\np/uni2191p+q/uni2193\np/uni2191p+q/uni2193\n−k −k\n/uni03A3R\nk(/uni03C9)=\nFIG. 2. (a) Diagrams of the bare vertices of the spin transfer\nprocesses at the interface. (b) Self-energy within the second-\norder perturbation.\nthe Fourier components of the ladder operators and are\ngiven by\n\u001b+\nq=X\npcy\np\"cp+q#; \u001b\u0000\n\u0000q=X\npcy\np+q#cp\"; (16)\nS\u0000\n\u0000k\u0019(2S)1=2by\nk; S+\nk\u0019(2S)1=2bk: (17)\nUsing the expressions above, HTcan be written as\nHT\u0019p\n2SX\np;q;k\u0010\nJq;kcy\np\"cp+q#by\n\u0000k+J\u0003\nq;kcy\np+q#cp\"b\u0000k\u0011\n:\n(18)\nThe \frst (second) term describes a magnon emission\n(absorption) process accompanying an electron spin-\rip\nfrom down to up (from up to down). A diagrammatic\nrepresentation of the interface interactions is shown in\nFig. 2 (a).\nIn this work, we drop a diagonal exchange coupling at\nthe interface, whose Hamiltonian is given as\nHZ=X\nq;kJq;k\u001bx\nqSx\nk: (19)\nThis term does not change the number of magnons in\nthe FI and induces an e\u000bective Zeeman \feld on electrons\nin the two-dimensional d-wave SC. We expect that this\nterm does not a\u000bect our main result because the coupling\nstrength is expected to be much smaller than the super-\nconducting gap and the microwave photon energy. We\nwill discuss this e\u000bect in Sec. V brie\ry.\nIII. FORMULATION\nThe coupling between the localized spin and the mi-\ncrowave is given by\nV(t) =\u0000~\rhacX\ni(Sy\nicos!t\u0000Sz\nisin!t); (20)wherehacis the amplitude of the transverse oscillating\nmagnetic \feld with frequency !. The microwave irra-\ndiation induces the precessional motion of the localized\nspin. The Gilbert damping constant can be read from\nthe retarded magnon propagator de\fned by\nGR\nk(t) =1\ni~\u0012(t)h[S+\nk(t);S\u0000\n\u0000k(0)]i; (21)\nwhere\u0012(t) is a step function. Second-order perturbation\ncalculation of the magnon propagator with respect to the\ninterface interaction was performed and the expression of\nthe self-energy was derived in the study of SP39{41. Fol-\nlowing calculation of the second-order perturbation with\nrespect to Jq;k, the Fourier transform of the retarded\nmagnon propagator is given by\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!); (22)\nwhere\u000bis the intrinsic Gilbert damping constant that\nwas introduced phenomenologically44{46. The diagram\nof the self-energy \u0006R\nk(!) is shown in Fig. 2 (b). From the\nexpressions given above, the modulation of the Gilbert\ndamping constant is given by\n\u000e\u000b=\u00002SIm \u0006R\nk=0(!)\n~!: (23)\nWithin the second-order perturbation, the self-energy\nis given by\n\u0006R\nk(!) =\u0000X\nqjJq;kj2\u001fR\nq(!); (24)\nwhere\u001fR\nq(!) represents the dynamic spin susceptibility\nof thed-wave SC de\fned by\n\u001fR\nq(!) =\u00001\ni~Z\ndtei(!+i0)t\u0012(t)h[\u001b+\nq(t);\u001b\u0000\n\u0000q(0)]i:(25)\nSubstituting the ladder operators in terms of the Bogoli-\nubov quasiparticle operators into the above expression\nand performing a straightforward calculation, we then\nobtain434\n\u001fR\nq(!) =\u0000X\npX\n\u0015=\u00061X\n\u00150=\u00061\u0012(\u0018p+\u0015Ep)(\u0018p+q+\u00150Ep+q) + \u0001 p\u0001p+q\n4\u0015Ep\u00150Ep+q\u0013f(\u0015Ep)\u0000f(\u00150Ep+q)\n\u0015Ep\u0000\u00150Ep+q+~!+i0; (26)\nwheref(E) = 1=(eE=kBT+ 1) is the Fermi distribution\nfunction.\nIn this paper, we focus on a rough interface modeled in\nterms of the mean J1and variance J22of the distribution\nofJq;k(see Appendix A for detail). The con\fgurationally\naveraged coupling constant is given by\njJq;k=0j2=J12\u000eq;0+J22: (27)\nIn this case, \u000e\u000bis written as\n\u000e\u000b=2SJ12\n~!Im\u001fR\nq=0(!) +2SJ22\n~!X\nqIm\u001fR\nq(!):(28)\nThe \frst term represents the momentum-conserved spin-\ntransfer processes, which vanish as directly veri\fed from\nEq. (26). This vanishment always occurs in spin-singlet\nSCs, including sandd-wave SCs, since the spin is\nconserved43. Consequently, the enhanced Gilbert damp-\ning is contributed from spin-transfer processes induced\nby the roughness proportional to the variance J22\n\u000e\u000b=2SJ22\n~!X\nqIm\u001fR\nq(!): (29)\nThe wave number summation can be replaced as\nX\nq(\u0001\u0001\u0001)!Dn\n2\u0019Z1\n\u00001d\u0018Z2\u0019\n0d\u001e(\u0001\u0001\u0001): (30)\nChanging the integral variable from \u0018toEand substi-\ntuting Eq. (26) into Eq. (29), we \fnally obtain\n\u000e\u000b=2\u0019SJ 22D2\nn\n~!Z1\n\u00001dE[f(E)\u0000f(E+~!)]\n\u0002Re\u00142\n\u0019K\u0012\u00012\nE2\u0013\u0015\nRe\u00142\n\u0019K\u0012\u00012\n(E+~!)2\u0013\u0015\n:\n(31)\nNote that the coherence factor vanishes in the above ex-\npression by performing the angular integral. The en-\nhanced Gilbert damping in the normal state is given by\n\u000e\u000bn= 2\u0019SJ 22D2\nn; (32)\nfor the lowest order of !. This expression means that \u000e\u000b\nis proportional to the product of the spin-up and spin-\ndown densities of states at the Fermi level7.IV. GILBERT DAMPING MODULATION\nFigure 3 shows the enhanced Gilbert damping constant\n\u000e\u000bas a function of temperature for several FMR frequen-\ncies, where \u000e\u000bis normalized with respect to its value in\nthe normal state. We compare \u000e\u000bin thed-wave SC shown\nin Figs. 3 (a) and (c) to that in the s-wave SC shown in\nFigs. 3 (b) and (d). The enhanced Gilbert damping for\nthes-wave SC is given by13\n\u000e\u000b=2\u0019SJ 22D2\nn\n~!Z1\n\u00001dE[f(E)\u0000f(E+~!)]\n\u0002\u0012\n1 +\u00012\nE(E+~!)\u0013\n\u0002Re\u0014jEjp\nE2\u0000\u00012\u0015\nRe\"\njE+~!jp\n(E+~!)2\u0000\u00012#\n;\n(33)\nwhere the temperature dependence of \u0001 is the same as\nthat for the d-wave SC, given by Eq. (9). Note that\nthe BCS theory we are based on, which is valid when\nthe Fermi energy is much larger than \u0001, is described by\nonly some universal parameters, including Tc, and inde-\npendent of the detail of the system in the normal state.\nWhen ~!=k BTc= 0:1,\u000e\u000bshows a coherence peak just\nbelow the transition temperature Tc. However, the co-\nherence peak of the d-wave SC is smaller than that of\nthes-wave SC. Within the low temperature limit, \u000e\u000bin\nthed-wave SC shows power-law decay behavior described\nby\u000e\u000b/T2. This is in contrast to \u000e\u000bin thes-wave SC,\nwhich shows exponential decay. The di\u000berence in the low\ntemperature region originates from the densities of states\nin thed-wave ands-wave SCs, which have gapless and full\ngap structures, respectively. When the FMR frequency\nincreases, the coherence peak is suppressed, and \u000e\u000bde-\ncays monotonically with decreasing temperature. \u000e\u000bhas\na kink structure at ~!= 2\u0001, where the FMR frequency\ncorresponds to the superconducting gap.\nFigure 4 shows \u000e\u000batT= 0 as a function of !. In\nthed-wave SC,\u000e\u000bgrows from zero with increasing !as\n\u000e\u000b/!2. When the value of \u000e\u000bbecomes comparable to\nthe normal state value, the increase in \u000e\u000bis suppressed,\nand\u000e\u000bthen approaches the value in the normal state.\nIn contrast, \u000e\u000bin thes-wave SC vanishes as long as the\ncondition that ~! < 2\u0001 is satis\fed. When ~!exceeds\n2\u0001,\u000e\u000bthen increases with increasing !and approaches\nthe normal state value. This di\u000berence also originates\nfrom the distinct spectral functions of the d-wave and\ns-wave SCs. Under the low temperature condition that\nT= 0:1Tc, the frequency dependence of \u000e\u000bdoes not5\n0.1 5.0\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2\n0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n0.1\n0.5\n1.0\n1.5\n2.03.04.05.0/uni210F/uni03C9/kBTc\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0\n0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n(a)  d-wave (b)  s-wave\n(c)  d-wave (d)  s-wave\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2\n0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0\n0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n0.1\n0.5\n1.0\n1.5\n2.03.04.05.0\nFIG. 3. Enhanced Gilbert damping \u000e\u000bas a function of tem-\nperatureT. The left panels (a) and (c) show \u000e\u000bin thed-\nwave SC in the low and high frequency cases, respectively.\nThe right panels (b) and (d) show \u000e\u000bin thes-wave SC in the\nlow and high frequency cases, respectively. \u000e\u000bnis the normal\nstate value.\nchange for the s-wave SC, and it only changes in the\nlow-frequency region where ~!.kBTfor thed-wave SC\n(see the inset in Fig. 4).\nV. DISCUSSION\nWe discuss the modulation of the Gilbert damping\nin SCs with nodes other than the d-wave SC consid-\nered in this work. Other SCs with nodes are expected\nto exhibit the power-law decay behavior within the low-\ntemperature and low-frequency limit as the d-wave SCs.\nHowever, the exponent of the power can di\u000ber due to\nthe di\u000berence of the quasiparticle density of states. Fur-\nthermore, in the p-wave states, two signi\fcant di\u000berences\narise due to spin-triplet Cooper pairs. First, the uni-\nform spin susceptibility \u001fR\nq=0(!) can be \fnite in the spin-\ntriplet SCs because the spin is not conserved. Second,\nthe enhanced Gilbert damping exhibits anisotropy and\nthe value changes by changing the relative angle between\nthe Cooper pair spin and localized spin35.\nIn our work, proximity e\u000bect between FIs and SCs\nwas not taken into account because the FMR modula-\ntion was calculated by second-order perturbation based\non the tunnel Hamiltonian. Reduction of superconduct-\n/uni210F/uni03C9=2/uni0394(T =0)\ns-waved-wave\n00.2\n2 4 6 81.2\n0.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n\n/uni210F/uni03C9/kBTc0.4\n0.0\n100 10.05\n0.000.1T/Tc=0.0FIG. 4. Enhanced Gilbert damping \u000e\u000bas a function of fre-\nquency!. The vertical dotted line indicates the resonance\nfrequency ~!= 2\u0001(T= 0). The inset shows an enlarged\nview in the low-frequency region.\ning gap due to the proximity e\u000bect15and e\u000bect of the\nsubgap Andreev bound states that appear in the ab-axis\njunction47would also be an important problem left for\nfuture works.\nPhysics of the FMR modulation for d-wave\nSC/ferromagnetic metal junctions is rather di\u000ber-\nent from that for d-wave SC/FI junctions. For d-wave\nSC/ferromagnetic metal junctions, spin transport is\ndescribed by electron hopping across a junction and\nthe FMR modulation is determined by the product\nof the density of states of electrons for a d-wave SC\nand a ferromagnetic metal. (We note that the FMR\nmodulation is determined by a spin susceptibility of\nd-wave SC, which in general includes di\u000berent informa-\ntion from the density of states of electrons.) While the\nFMR modulation is expected to be reduced below a SC\ntransition temperature due to opening an energy gap, its\ntemperature dependence would be di\u000berent from results\nobtained in our work.\nFinally, let us discuss e\u000bect of the diagonal exchange\ncoupling given in Eq. (19) (see also the last part of\nSec. II). This term causes an exchange bias, i.e., an e\u000bec-\ntive Zeeman \feld on conduction electrons in the d-wave\nSC, which is derived as follows. First, the x-component\nof the localized spin is approximated as hSx\nji \u0019S,\nwhich gives Sx\nk\u0019Sp\nN\u000ek;0. Next, the matrix element\nJq;k=0is replaced by the con\fgurationally averaged value\nJq;k=0=J1\u000eq;0. Consequently, the e\u000bective Zeeman\n\feld term is given by\nHZ\u0019EZX\np(cy\np\"cp\"\u0000cy\np#cp#); (34)\nwhere we introduced a Zeeman energy as EZ=J1Sp\nN.\nThis term induces spin splitting of conduction electrons6\nin thed-wave SC and changes the spin susceptibility of\nthe SC. The spin-splitting e\u000bect causes a spin excitation\ngap and modi\fes the frequency dependence in Fig. 4, that\nwill provide additional information on the exchange cou-\npling at the interface. In actual experimental setup for\nthed-wave SC, however, the Zeeman energy, that is less\nthan the exchange bias between a magnetic insulator and\na metal, is estimated to be of the order of 0 :1 erg=cm2.\nThis leads to the exchange coupling that is much less\nthanJ\u00180:1 meV for YIG48. Therefore, we expect that\nthe interfacial exchange coupling is much smaller than\nthe superconducting gap and the microwave photon en-\nergy though it has not been measured so far. A detailed\nanalysis for this spin-splitting e\u000bect is left for a future\nproblem.\nVI. CONCLUSION\nIn this work, we have investigated Gilbert damping\nmodulation in the d-wave SC/FI bilayer system. The\nenhanced Gilbert damping constant in this case is pro-\nportional to the imaginary part of the dynamic spin sus-\nceptibility of the d-wave SC. We found that the Gilbert\ndamping modulation re\rects the gapless excitation that\nis inherent in d-wave SCs. The coherence peak is sup-\npressed in the d-wave SC when compared with that in\nthes-wave SC. In addition, the di\u000berences in the spec-\ntral functions for the d-wave ands-wave SCs with gap-\nless and full-gap structures lead to power-law and ex-\nponential decays within the low-temperature limit, re-\nspectively. Within the low-temperature limit, \u000e\u000bin the\nd-wave SC increases with increasing !, while\u000e\u000bin the\ns-wave SC remains almost zero as long as the excitation\nenergy ~!remains smaller than the superconducting gap\n2\u0001.\nOur results illustrate the usefulness of measurement of\nthe FMR modulation of unconventional SCs for determi-\nnation of their symmetry through spin excitation. We\nhope that this fascinating feature will be veri\fed exper-\nimentally in d-wave SC/FI junctions in the near future.\nTo date, one interesting result of FMR modulation in\nd-wave SC/ferromagnetic metal structures has been re-\nported38. This modulation can be dependent on metallic\nstates, which are outside the scope of the theory pre-\nsented here. The FMR modulation caused by ferromag-\nnetic metals is another subject that will have to be clar-\ni\fed theoretically in future work.\nFurthermore, our work provides the most fundamental\nbasis for application to analysis of junctions with vari-\nous anisotropic SCs. For example, some anisotropic SCs\nare topological and have an intrinsic gapless surface state.\nSP can be accessible and can control the spin excitation of\nthe surface states because of its interface sensitivity. The\nextension of SP to anisotropic and topological supercon-\nductivity represents one of the most attractive directions\nfor further development of superconducting spintronics.\nAcknowledgments.| This work is partially supportedby the Priority Program of Chinese Academy of Sciences,\nGrant No. XDB28000000. We acknowledge JSPS KAK-\nENHI for Grants (No. JP20H01863, No. JP20K03835,\nNo. JP20K03831, No. JP20H04635, and No.21H04565).\nAppendix A: Magnon self-energy induced by a\nrough interface\nThe roughness of the interface is taken into account\nas an uncorrelated (white noise) distribution of the ex-\nchange couplings35, as shown below. We start with an\nexchange model in the real space\nHex=X\njZ\nd2rJ(r;rj)\u001b(r)\u0001Sj\n=X\nq;kJq;k\u001bq\u0001Sk: (A1)\nThe spin density \u001b(r) in the SC and the spin Sjin the\nFI are represented in the momentum space as\n\u001b(r) =1\nAX\nqeiq\u0001r\u001bq; (A2)\nSj=1p\nNX\nkeik\u0001rjSk; (A3)\nwhereAdenotes the area of the system and Nis the\nnumber of sites. The exchange coupling constant is also\nobtained to be\nJq;k=1\nAp\nNX\njZ\nd2rei(q\u0001r+k\u0001rj)J(r;rj): (A4)\nThe exchange model Hexis decomposed into the spin\ntransfer term HTand the e\u000bective Zeeman \feld term HZ\nasHex=HT+HZ.\nNow we consider the roughness e\u000bect of the interface.\nUncorrelated roughness is expressed by the mean J1and\nvarianceJ22as\n1p\nNX\njJ(r;rj) =J1; (A5)\n1\nNX\njj0J(r;rj)J(r0;rj0)\u0000J12=J22A\u000e2(r\u0000r0);(A6)\nwhereOis the con\fgurational average of Oover the\nroughness. 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Kato1,\n1Institute for Solid State Physics,\nThe University of Tokyo, Kashiwa, Japan\n2Kavli Institute for Theoretical Sciences,\nUniversity of Chinese Academy of Sciences, Beijing, China\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing, China\n4Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai, Japan\n5RIKEN Center for Emergent Matter Science (CEMS),\nWako, Saitama, Japan\n(Dated: May 15, 2023)\nWe theoretically consider the e\u000bect of vertex correction on spin pumping from a ferromagnetic\ninsulator (FI) into a two-dimensional electron gas (2DEG) in which the Rashba and Dresselhaus\nspin-orbit interactions coexist. The Gilbert damping in the FI is enhanced by elastic spin-\ripping\nor magnon absorption. We show that the Gilbert damping due to elastic spin-\ripping is strongly\nenhanced by the vertex correction when the ratio of the two spin-orbit interactions is near a special\nvalue at which the spin relaxation time diverges while that due to magnon absorption shows only\nsmall modi\fcation. We also show that the shift in the resonant frequency due to elastic spin-\ripping\nis strongly enhanced in a similar way as the Gilbert damping.\nI. INTRODUCTION\nIn the \feld of spintronics1,2, spin pumping has long\nbeen used as a method of injecting spins into various\nmaterials3{5. Spin pumping was \frst employed to inject\nspins from a ferromagnetic metal into an adjacent normal\nmetal (NM)6{9. Subsequently, it was used on ferromag-\nnetic insulator (FI)/NM junctions10. Because spin injec-\ntion is generally related to the loss of the magnetization\nin ferromagnets, it a\u000bects the Gilbert damping measured\nin ferromagnetic resonance (FMR) experiments11. When\nwe employ spin injection from the FI, the modulation of\nthe Gilbert damping re\rects the properties of the spin ex-\ncitation in the adjacent materials, such as magnetic thin\n\flms12, magnetic impurities on metal surfaces13, and su-\nperconductors14{17. This is in clear contrast with the\nGilbert damping of a bulk FI, which re\rects properties\nof electrons and phonons18{20.\nAn attractive strategy is to combine spin pumping\nwith spin-related transport phenomena in semiconduc-\ntor microstructures1,21. A two-dimensional electron gas\n(2DEG) in a semiconductor heterostructure is an easily\ncontrolled physical system that has been used in spin-\ntronics devices22{25. A 2DEG system has two types of\nspin-orbit interaction, i.e., Rashba26,27and Dresselhaus\nspin-orbit interactions28,29.\nIn our previous work30, we theoretically studied spin\npumping into a 2DEG in semiconductor heterostructures\nwith both Rashba and Dresselhaus spin-orbit interac-\ntions, which can be regarded as a prototype for a 2DEG\nwith a complex spin-texture near the Fermi surface [see\nFig. 1 (a)]. In that study, we formulated the modu-\nlation of the Gilbert damping in the FI by using the\n2DEG\nmicrowave\nFI\n(a) (b)\ntotFIG. 1. (a) Schematic picture of junction composed of a fer-\nromagnetic insulator (FI) and a two-dimensional electron gas\n(2DEG) realized in a semiconductor heterostructure. Stotin-\ndicates the total spin of the FI. We consider a uniform spin\nprecession of the FI induced by microwave irradiation. (b)\nLaboratory coordinates ( x;y;z ) and the magnetization-\fxed\ncoordinates ( x0;y0;z0). The red arrow indicates the expec-\ntation value of the spontaneous spin polarization of the FI,\nhSi.\nsecond-order perturbation with respect to the interfa-\ncial coupling15,31{35and related it to the dynamic spin\nsusceptibility of the 2DEG. We further calculated the\nspin susceptibility and obtained characteristic features\nof the Gilbert damping modulation. This modulation\ncontains two contributions: elastic spin-\ripping, which\ndominates at low resonant frequencies, and magnon ab-\nsorption, which dominates at high resonant frequencies.\nIn addition, we clari\fed that these contributions have\ndi\u000berent dependence on the in-plane azimuth angle \u0012of\nthe ordered spin in the FI [see Fig. 1 (b)].\nWhen the Rashba and Dresselhaus spin-orbit interac-\ntions have almost equal magnitudes, spin relaxation byarXiv:2201.11498v3  [cond-mat.mes-hall]  12 May 20232\nnonmagnetic impurity scattering is strongly suppressed\nbecause the direction of the e\u000bective Zeeman \feld gen-\nerated by the spin-orbit interactions is unchanged along\nthe Fermi surface. Due to this substantial suppression of\nspin relaxation, there emerge characteristic physical phe-\nnomena such as the persistent spin helix state36{39. In\ngeneral, the vertex corrections have to be taken into ac-\ncount to treat various conservation laws, i.e., the charge,\nspin, momentum, and energy conservation laws in cal-\nculation of the response functions40{43. Therefore, for\nbetter description of realistic systems, we need to con-\nsider vertex correction, which captures e\u000bect of impurity\nmore accurately by re\recting conservation laws. How-\never, the vertex corrections were neglected in our pre-\nvious work30. This means that our previous calculation\nshould fail when the Rashba and Dresselhaus spin-orbit\ninteractions compete.\nIn this study, we consider the same setting, i.e., a\njunction composed of an FI and a 2DEG as shown in\nFig. 1 (a), and examine e\u000bect of the spin conservation law\nby taking the vertex correction into account. We theo-\nretically calculate the modulation of the Gilbert damping\nand the shift in the FMR frequency by solving the Bethe-\nSalpeter equation within the ladder approximation. We\nshow that the vertex correction substantially changes the\nresults, in particular, when the strengths of the Rashba-\nand Dresselhaus-type spin-orbit interactions are chosen\nto be almost equal but slightly di\u000berent; Speci\fcally,\nboth the Gilbert damping and the FMR frequency shift\nare largely enhanced at low resonant frequencies re\rect-\ning strong suppression of spin relaxation. This remark-\nable feature should be able to be observed experimen-\ntally. In contrast, the vertex correction changes their\nmagnitude only slightly at high resonant frequencies.\nBefore describing our calculation, we brie\ry comment\non study of the vertex corrections in a di\u000berent context.\nIn early studies of the spin Hall e\u000bect, there was a de-\nbate on the existence of intrinsic spin Hall e\u000bect44{46. By\nconsidering the vertex corrections, the spin Hall conduc-\ntivity, which is calculated from the correlation function\nbetween the current and spin current, vanishes in the\npresence of short-range disorder for simple models even if\nits strength is in\fnitesimally small47{49. This seemingly\ncontradictory result stimulated theoretical researches on\nrealistic modi\fed models50,51as well as de\fnition of the\nspin current52{56. However, we stress that the vertex\ncorrections for the dynamic spin susceptibility, which is\ncalculated from the spin-spin correlation function, have\nno such subtle problem57because it does not include the\nspin current.\nThe rest of this work is organized as follows. In Sec. II,\nwe brie\ry summarize our model of the FI/2DEG junc-\ntion and describe a general formulation for the magnon\nself-energy following Ref. 30. In Sec. III, we formulate\nthe vertex correction that corresponds to the self-energy\nin the Born approximation. We show the modulation of\nthe Gilbert damping and the shift in the FMR frequency\nin Secs. IV and V, respectively, and discuss the e\u000bect\nky\nkx\n(a) (b)\nFIG. 2. Schematic picture of the spin-splitting energy bands\nof 2DEG for (a) \f=\u000b= 0 and (b) \f=\u000b= 1. The red and blue\narrows represent spin polarization of each band. In the case\nof (b), the spin component in the direction of the azimuth\nangle 3\u0019=4 is conserved.\nof the vertex correction in detail. Finally, we summa-\nrize our results in Sec. VI. The six Appendices detail the\ncalculation in Sec. III.\nII. FORMULATION\nHere, we describe a model for the FI/2DEG junction\nshown in Fig. 1 (a) and formulate the spin relaxation\nrate in an FMR experiment. Because we have already\ngiven a detailed formulation on this model in our previous\npaper30, we will brie\ry summarize it here.\nA. Two-dimensional electron gas\nWe consider a 2DEG whose Hamiltonian is given as\nHNM=Hkin+Himp, whereHkinandHimpdescribe the\nkinetic energy and the impurity, respectively. The kinetic\nenergy is given as\nHkin=X\nk(cy\nk\"cy\nk#)^hk\u0010ck\"\nck#\u0011\n; (1)\n^hk=\u0018k^I\u0000he\u000b(k)\u0001\u001b; (2)\nwhereck\u001bis the annihilation operator of conduction elec-\ntrons with wave number k= (kx;ky) andzcomponent\nof the spin, \u001b(=\";#),^Iis a 2\u00022 identity matrix, \u001ba\n(a=x;y;z ) are the Pauli matrices, \u0018k=~2k2=2m\u0003\u0000\u0016\nis the kinetic energy measured from the chemical poten-\ntial, andm\u0003is an e\u000bective mass. Hereafter, we assume\nthat the Fermi energy is much larger than the other en-\nergy scales such as the spin-orbit interactions, the tem-\nperature, and the ferromagnetic resonance energy. Then,\nthe low-energy part of the spin susceptibility depends on\nthe chemical potential \u0016and the e\u000bective mass m\u0003only\nthrough the density of states at the Fermi energy, D(\u000fF).\nThe spin-orbit interaction is described by the e\u000bective3\nZeeman \feld,\nhe\u000b(k) =jkj(\u0000\u000bsin'\u0000\fcos';\u000bcos'+\fsin';0)\n'kF(\u0000\u000bsin'\u0000\fcos';\u000bcos'+\fsin';0); (3)\nwhere\u000band\frespectively denote the amplitudes of the\nRashba- and Dresselhaus-type spin-orbit interactions and\nthe electron wave number is expressed by polar coordi-\nnates as (kx;ky) = (jkjcos';jkjsin'). In the second\nequation of Eq. (3), we have approximated jkjwith the\nFermi wave number kFassuming that the spin-orbit in-\nteraction energies, kF\u000bandkF\f, are much smaller than\nthe Fermi energy58. When only the Rashba spin-orbit in-\nteraction exists ( \f= 0), the energy band is spin-splitted\nas shown in Fig. 2 (a). The spin polarization of each band\ndepends on the azimuth angle 'because it is determined\nby the e\u000bective Zeeman \feld he\u000bwhich is a function of\n'as seen in Eq. (3). In the special case of \f=\u000b= 1, the\nspin polarization always becomes parallel to the direction\nof the azimuth angle 3 \u0019=4 in thexyplane as shown in\nFig. 2 (b). Then, the spin component in this direction is\nconserved. This observation indicates that e\u000bect of the\nspin conservation may become important when the two\nspin-orbit interactions compete ( \u000b'\f).\nThe Hamiltonian of the impurity potential is given as\nHimp=uX\ni2impX\n\u001b\ty\n\u001b(ri)\t\u001b(ri); (4)\nwhere \t\u001b(r) =A\u00001=2P\nkck\u001beik\u0001r,Ais the area of the\njunction,uis the strength of the impurity potential, and\nriis the position of the impurity site.\nThe \fnite-temperature Green's function for the con-\nduction electrons is de\fned by a 2 \u00022 matrix ^g(k;i!m)\nwhose elements are\ng\u001b\u001b0(k;i!m) =Z~\f\n0d\u001cei!m\u001cg\u001b\u001b0(k;\u001c); (5)\ng\u001b\u001b0(k;\u001c) =\u0000~\u00001hck\u001b(\u001c)cy\nk\u001b0i; (6)\nwhereck\u001b(\u001c) =eHNM\u001c=~ck\u001be\u0000HNM\u001c=~,HNM=Hkin+\nHimp,!m=\u0019(2m+ 1)=~\fis the fermionic Matsubara\nfrequency, and \fis the inverse temperature. By em-\nploying the Born approximation, the \fnite-temperature\nGreen's function can be expressed as\n^g(k;i!m) =(i~!m\u0000\u0018k+i\u0000sgn(!m)=2)^I\u0000he\u000b\u0001\u001bQ\n\u0017=\u0006(i~!m\u0000E\u0017\nk+i\u0000sgn(!m)=2);\n(7)\nwhereE\u0006\nk=\u0018k\u0006jhe\u000b(')jis the spin-dependent electron\ndispersion,\n\u0000 = 2\u0019niu2D(\u000fF) (8)\nis level broadening, and niis the impurity concentration\n(see Appendix A and Ref. 30 for detailed derivation).\nAs already mentioned, the case of \f=\u000b = 1 is special\nbecause the spin component parallel to the direction ofthe azimuth angle 3 \u0019=4 in thexyplane is conserved (see\nFig. 2 (b)). By de\fning the spin component in this di-\nrection as\ns3\u0019=4\ntot\u00111\n2X\nk(cy\nk+ck+\u0000cy\nk\u0000ck\u0000); (9)\n\u0012\nck+\nck\u0000\u0013\n=\u0012\n1=p\n2e\u0000i3\u0019=4=p\n2\n\u0000ei3\u0019=4=p\n2 1=p\n2\u0013\u0012\nck\"\nck#\u0013\n;(10)\nwe can prove [ Hkin+Himp;s3\u0019=4\ntot] = 0. When the value of\n\f=\u000b is slightly shifted from 1, the spin conservation law\nis broken slightly and this leads to a slow spin relaxation.\nAs will be discussed in Secs. IV and V, this slow spin re-\nlaxation, which is a remnant of the spin conservation at\n\f=\u000b = 1, strongly a\u000bects the spin injection from the FI\ninto the 2DEG. To describe this feature, we need to con-\nsider the vertex correction to take the conservation law\ninto account in our calculation as explained in Sec. III.\nB. Ferromagnetic insulator\nWe consider the quantum Heisenberg model for the FI\nand employ the spin-wave approximation assuming that\nthe temperature is much lower than the magnetic tran-\nsition temperature and the magnitude of the localized\nspins,S0, is su\u000eciently large. We write the expectation\nvalue of the localized spins in the FI as hSi, whose direc-\ntion is (cos \u0012;sin\u0012;0) as shown in the Fig. 1 (b). Using\nthe Holstein-Primakov transformation, the Hamiltonian\nin the spin-wave approximation is obtained as\nHFI=X\nk~!kby\nkbk; (11)\nwherebkis the magnon annihilation operator with wave\nnumberk,~!k=Dk2+~\rhdcis the energy dispersion of\na magnon,Dis the spin sti\u000bness, \ris the gyromagnetic\nratio, andhdcis the externally applied DC magnetic \feld.\nWe note that the external DC magnetic \feld controls the\ndirection of the ordered spins. We introduce new coor-\ndinates (x0;y0;z0) \fxed on the ordered spins by rotating\nthe original coordinates ( x;y;z ) as shown in Fig. 1 (b).\nThen, the magnon annihilation operator is related to the\nspin ladder operator by the Holstein-Primakov transfor-\nmation asSx0+\nk\u0011Sy0\nk+iSz0\nk= (2S0)1=2bk. The spin\ncorrelation function is de\fned as\nG0(k;i!n) =Z~\f\n0d\u001cei!n\u001cG0(k;\u001c); (12)\nG0(k;\u001c) =\u00001\n~hSx0+\nk(\u001c)Sx0\u0000\nk(0)i; (13)\nwhere!n= 2n\u0019=~\fis the bosonic Matsubara fre-\nquency. The spin correlation function is calculated from\nthe Hamiltonian (11), as\nG0(k;i!n) =2S0=~\ni!n\u0000!k\u0000\u000bGj!nj; (14)4\nwhere\u000bG>0 is a phenomenological dimensionless pa-\nrameter that describes the strength of the Gilbert damp-\ning in the bulk FI.\nC. E\u000bect of the FI/2DEG interface\nThe coupling between the FI and 2DEG can be ac-\ncounted for by the Hamiltonian,\nHint=X\nk(TkSx0+\nksx0\u0000\nk+T\u0003\nksx0+\nkSx0\u0000\nk); (15)\nwhereTkis an exchange interaction at a clean interface,\nfor which the momentum of spin excitation is conserved.\nThe spin ladder operators for conduction electrons, sx0\u0006\nk,\nare obtained using a coordinate rotation as30\nsx0\u0006\nk=1\n2X\n\u001b;\u001b0X\nk0cy\nk0\u001b(^\u001bx0\u0006)\u001b\u001b0ck0\u0006k\u001b0; (16)\n^\u001bx0\u0006=\u0000sin\u0012\u001bx+ cos\u0012\u001by\u0006i\u001bz; (17)\nwhere ^\u001bx0\u0006\u0011^\u001by0\u0006i^\u001bz0and\n0\n@^\u001bx0\n^\u001by0\n^\u001bz01\nA=0\n@cos\u0012sin\u00120\n\u0000sin\u0012cos\u00120\n0 0 11\nA0\n@\u001bx\n\u001by\n\u001bz1\nA:\nAssuming that the interfacial exchange interaction is\nmuch smaller than the spin-orbit interactions, kF\u000band\nkF\f59,60, we perform a second-order perturbation theory\nwith respect to the interfacial exchange interaction Hint.\nAccordingly, the spin correlation function of the FI is\ncalculated as\nG(k;i!n) =1\n(G0(k;i!n))\u00001\u0000\u0006(k;i!n); (18)\n\u0006(k;i!n) =jTkj2A\u001f(k;i!n); (19)\nwhere \u0006(k;i!n) is the self-energy due to the interfacial\nexchange coupling and \u001f(k;i!n) is the spin susceptibility\nfor conduction electrons per unit area, de\fned as\n\u001f(k;i!n) =Z~\f\n0d\u001cei!n\u001c\u001f(k;\u001c); (20)\n\u001f(k;\u001c) =\u00001\n~Ahsx0+\nk(\u001c)sx0\u0000\nk(0)i; (21)\nwheresx0\u0006\nk(\u001c) =eHNM\u001c=~sx0\u0006\nke\u0000HNM\u001c=~. Within the\nsecond-order perturbation, we only need to calculate the\nspin susceptibility for pure 2DEG without considering\nthe junction because the interfacial coupling is already\ntaken into account in the prefactor of the self-energy in\nEq. (19). The uniform component of the retarded spin\ncorrelation function is obtained by analytic continuation\n(a)(b)\nFIG. 3. Feynman diagrams of (a) the uniform spin susceptibil-\nity and (b) the Bethe-Salpeter equation for the ladder-type\nvertex function derived from the Born approximation. The\ncross with two dashed lines indicates interaction between an\nelectron and an impurity.\ni!n!!+i\u000e, as\nGR(0;!) =2S0=~\n!\u0000(!0+\u000e!0) +i(\u000bG+\u000e\u000bG)!;(22)\n\u000e!0\n!0'2S0jT0j2A\n~!0Re\u001fR(0;!0); (23)\n\u000e\u000bG'\u00002S0jT0j2A\n~!0Im\u001fR(0;!0); (24)\nwhere the superscript Rindicates the retarded compo-\nnent,!0=!k=0(=\rhdc) is the FMR frequency, and\n\u000e!0and\u000e\u000bGare respectively the changes in the FMR\nfrequency and Gilbert damping due to the FI/2DEG in-\nterface. We note that in contrast with the bulk Gilbert\ndamping\u000bG, the increase of the Gilbert damping, \u000e\u000bG,\ncan be related directly to the spin susceptibility of 2DEG\nas shown by Eq. (24). In fact, measurement of \u000e\u000bG\nhas been utilized as a qualitative indicator of spin cur-\nrent through a junction61,62. In Eqs. (23) and (24), we\nmade an approximation by replacing !with the FMR\nfrequency!0by assuming that the FMR peak is su\u000e-\nciently sharp ( \u000bG+\u000e\u000bG\u001c1). Thus, both the FMR\nfrequency shift and the modulation of the Gilbert damp-\ning are determined by the uniform spin susceptibility of\nthe conduction electrons, \u001f(0;!). In what follows, we\ninclude the vertex correction for calculation of \u001f(0;!),\nwhich was not taken into account in our previous work30.\nIII. VERTEX CORRECTION\nWe calculate the spin susceptibility in the ladder\napproximation42,43that obeys the Ward-Takahashi re-\nlation with the self-energy in the Born approximation57.\nThe Feynman diagrams for the corresponding spin sus-\nceptibility and the Bethe-Salpeter equation for the vertex\nfunction are shown in Figs. 3 (a) and 3 (b), respectively.5\nThe spin susceptibility of 2DEG is written as\n\u001f(0;i!n) =1\n4\fAX\nk;i!mTrh\n^g(k;i!m)^\u0000(k;i!m;i!n)\n^g(k;i!m+i!n)^\u001bx0\u0000i\n; (25)\nwhere the vertex function ^\u0000(k;i!m;i!n) is a 2\u00022 matrix\nwhose components are determined by the Bethe-Salpeter\nequation [see Fig. 3 (b)],\n\u0000\u001b0\u001b(k;i!m;i!n)\n= (^\u001bx0+)\u001b0\u001b+u2ni\nAX\nqX\n\u001b1\u001b2g\u001b0\u001b2(q;i!m)\n\u0002\u0000\u001b2\u001b1(q;i!m;i!n)g\u001b1\u001b(q;i!m+i!n):(26)\nSince the right-hand side of this equation is indepen-\ndent ofk, the vertex function can simply be described\nas^\u0000(i!m;i!n). We express the vertex function with the\nPauli matrices as\n^\u0000(i!m;i!n)\u0011E^I+X^\u001bx0+Y^\u001by0+Z^\u001bz0; (27)\nwhereE,X,Y, andZwill be determined self-\nconsistently later. The Green's function for the conduc-\ntion electrons can be rewritten as\n^g(q;i!m) =A^I+B^\u001bx0+C^\u001by0\nD; (28)\nA(i!m) =i~!m\u0000\u0018q+i\u0000\n2sgn(!m); (29)\nB=\u0000he\u000bcos(\u001e\u0000\u0012); (30)\nC=\u0000he\u000bsin(\u001e\u0000\u0012); (31)\nD(i!m) =Y\n\u0017=\u0006[i~!m\u0000E\u0017\nq+i\u0000\n2sgn(!m)]; (32)\nwhere\u001eis the azimuth angle by which the e\u000bective Zee-\nman \feld is written as he\u000b= (he\u000bcos\u001e;he\u000bsin\u001e;0).\nThishe\u000bis written as he\u000b'kFp\n\u000b2+\f2+ 2\u000b\fsin 2'\nusing the Fermi wave number kF. By substituting\nEqs. (27) and (28) into the second term of Eq. (26) and\nby the algebra of Pauli matrices, we obtain\nu2ni\nAX\nq^g(q;i!m)^\u0000(q;i!m;i!n)^g(q;i!m+i!n)\n=E0^I+X0^\u001bx0+Y0^\u001by0+Z0^\u001bz0; (33)\nwhere\n0\nB@E0\nX0\nY0\nZ01\nCA=0\nB@\u00030+ \u00031 0 0 0\n0 \u0003 0+ \u00032 \u00033 0\n0 \u0003 3 \u00030\u0000\u00032 0\n0 0 0 \u0003 0\u0000\u000311\nCA0\nB@E\nX\nY\nZ1\nCA;\n(34)and \u0003j(i!m;i!n) (j= 0;1;2;3) are expressed as\n\u00030(i!m;i!n) =u2ni\nAX\nqAA0\nDD0; (35)\n\u00031(i!m;i!n) =u2ni\nAX\nqh2\ne\u000b\nDD0; (36)\n\u00032(i!m;i!n) =u2ni\nAX\nqh2\ne\u000bcos 2(\u001e\u0000\u0012)\nDD0; (37)\n\u00033(i!m;i!n) =u2ni\nAX\nqh2\ne\u000bsin 2(\u001e\u0000\u0012)\nDD0; (38)\nusing the abbreviated symbols, A=A(i!m),A0=\nA(i!m+i!n),D=D(i!m), andD0=D(i!m+i!n).\nHere, we have used the fact that the contributions of the\n\frst-order terms of BandCbecome zero after replacing\nthe sum with the integral with respect to qand perform-\ning the azimuth integration. We can solve for E,X,\nY, andZby combining Eq. (34) and the Bethe-Salpeter\nequation (26), which we rewrite as\nE^I+X^\u001bx0+Y^\u001by0+Z^\u001bz0\n= ^\u001bx0++E0^I+X0^\u001bx0+Y0^\u001by0+Z0^\u001bz0; (39)\nwith ^\u001bx0+= ^\u001by0+i^\u001bz0. The solution is\nE= 0; (40)\nX=\u00033\n(1\u0000\u00030)2\u0000\u00032\n2\u0000\u00032\n3; (41)\nY=1\u0000\u00030\u0000\u00032\n(1\u0000\u00030)2\u0000\u00032\n2\u0000\u00032\n3; (42)\nZ=i\n1\u0000\u00030+ \u00031: (43)\nBy replacing the sum with an integral as \u0018\u0011\u0018q,\n1\nAX\nq(\u0001\u0001\u0001)'D(\u000fF)Z1\n\u00001d\u0018Z2\u0019\n0d'\n2\u0019(\u0001\u0001\u0001); (44)\nEqs. (35)-(38) can be rewritten as\n\u0003j(i!m;i!n) =\u0012(\u0000!m)\u0012(!m+!n)~\u0003j(i!n); (45)\n~\u0003j(i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019\n\u0002X\n\u0017;\u00170=\u0006fj(\u0017;\u00170;')\ni~!n+ (\u0017\u0000\u00170)he\u000b(') +i\u0000;\n(46)\nwhere we have used Eq. (8), \u0012(x) is a step function, and\nf0(\u0017;\u00170;') = 1; (47)\nf1(\u0017;\u00170;') =\u0017\u00170; (48)\nf2(\u0017;\u00170;') =\u0017\u00170cos 2(\u001e(')\u0000\u0012); (49)\nf3(\u0017;\u00170;') =\u0017\u00170sin 2(\u001e(')\u0000\u0012): (50)6\nFor detailed derivation, see Appendix B. Substituting the\nGreen's function and the vertex function into Eq. (25),\nwe obtain\n\u001f(0;i!n) =1\n4\fAX\nk;i!m2\nDD0h\n2BCX\n+ (AA0\u0000B2+C2)Y\u0000i(AA0\u0000B2\u0000C2)Zi\n:(51)\nBy summing over kand!mand by analytical continu-\nation,i!n!!+i\u000e, the retarded spin susceptibility is\nobtained as63\n\u001fR(0;!)\n=D(\u000fF)~!\n2i\u0000\u0014~\u0003R\n0(1\u0000~\u0003R\n0)\u0000~\u0003R\n2(1\u0000~\u0003R\n2) + ( ~\u0003R\n3)2\n(1\u0000~\u0003R\n0)2\u0000(~\u0003R\n2)2\u0000(~\u0003R\n3)2\n+~\u0003R\n0\u0000~\u0003R\n1\n1\u0000~\u0003R\n0+~\u0003R\n1\u0015\n\u0000D(\u000fF); (52)\nwhere\n~\u0003R\nj=~\u0003R\nj(!) =~\u0003j(i!n!!+i\u000e)\n=i\u0000\n4\u00010Z2\u0019\n0d'\n2\u0019\n\u0002X\n\u0017\u00170fj(\u0017;\u00170;')\n~!=\u00010+ (\u0017\u0000\u00170)he\u000b=\u00010+i\u0000=\u00010:(53)\nA detailed derivation is given in Appendix C. Here, we\nhave introduced a unit of energy, \u0001 0=kF\u000b, for the con-\nvenience of making the physical quantities dimensionless.\nUsing Eqs. (23) and (24), we \fnally obtain the shift in\nthe FMR frequency and the modulation of the Gilbert\ndamping as\n\u000e!0\n!0=\u000bG;0ReF(!0); (54)\n\u000e\u000bG=\u0000\u000bG;0ImF(!0); (55)\nF(!) =\u00010\n2\u0019i\u0000\u0014~\u0003R\n0(1\u0000~\u0003R\n0)\u0000~\u0003R\n2(1\u0000~\u0003R\n2) + ( ~\u0003R\n3)2\n(1\u0000~\u0003R\n0)2\u0000(~\u0003R\n2)2\u0000(~\u0003R\n3)2\n+~\u0003R\n0\u0000~\u0003R\n1\n1\u0000~\u0003R\n0+~\u0003R\n1\u0015\n\u0000\u00010\n\u0019~!; (56)\nwhere\u000bG;0= 2\u0019S0jT0j2AD(\u000fF)=\u00010is a dimensionless\nparameter that describes the coupling strength at the\ninterface. This is our main result.\nThe spin susceptibility without the vertex correction\ncan be obtained by taking the \frst-order term with re-spect to ~\u0003R\nj:\n\u001fR(0;!)'~!D(\u000fF)\n2i\u0000\u0002\n2~\u0003R\n0\u0000~\u0003R\n1\u0000~\u0003R\n2\u0003\n\u0000D(\u000fF)\n=~!D(\u000fF)Zd'\n2\u0019h1\n~!+i\u00001\u0000cos2(\u001e(')\u0000\u0012)\n2\n+1\n~!\u00002he\u000b(') +i\u00001 + cos2(\u001e(')\u0000\u0012)\n4\n+1\n~!+ 2he\u000b(') +i\u00001 + cos2(\u001e(')\u0000\u0012)\n4i\n\u0000D(\u000fF):\n(57)\nThe imaginary part of \u001fR(0;!) reproduces the result of\nRef. 30. Using this expression, the shift in the FMR\nfrequency and the modulation of the Gilbert damping\nwithout the vertex correction are obtained as\n\u000e!nv\n0\n!0=\u000bG;0ReFnv(!0); (58)\n\u000e\u000bnv\nG=\u0000\u000bG;0ImFnv(!0); (59)\nFnv(!) =\u00010\n2\u0019i\u0000\u0014\n2~\u0003R\n0\u0000~\u0003R\n1\u0000~\u0003R\n2\u0015\n\u0000\u00010\n\u0019~!; (60)\nIV. MODULATION OF THE GILBERT\nDAMPING\nFirst, we show the result for the modulation of the\nGilbert damping, \u000e\u000bG, for\f=\u000b = 0, 1, and 3 and dis-\ncuss the e\u000bect of the vertex correction by comparing it\nwith the result without the vertex correction in Sec. IV A.\nNext, we discuss the strong enhancement of the Gilbert\ndamping near \f=\u000b= 1 in Sec. IV B.\nA. E\u000bect of vertex corrections\nFirst, let us discuss the case of \f=\u000b = 0, i.e., the\ncase when only the Rashba spin-orbit interaction exists64.\nFigure 4 (a) shows the e\u000bective Zeeman \feld he\u000balong\nthe Fermi surface. Figures 4 (b) and 4 (c) show the\nmodulations of the Gilbert damping without and with\nthe vertex correction. The horizontal axes of Figs. 4 (b)\nand 4 (c) denote the resonant frequency !0=\rhdcin\nthe FMR experiment. Note that the modulation of the\nGilbert damping, \u000e\u000bG, is independent of \u0012, i.e., the az-\nimuth angle ofhSi. The four curves in Figs. 4 (b) and\n4 (c) correspond to \u0000 =\u00010= 0:1, 0:2, 0:5, and 1:065. We\n\fnd that these two graphs have a common qualitative\nfeature; the modulation of the Gilbert damping has two\npeaks at!0= 0 and!0= 2\u0001 0and their widths become\nlarger as \u0000 increases. The peak at !0= 0 corresponds\nto elastic spin-\ripping of conduction electrons induced\nby the transverse magnetic \feld via the exchange bias\nof the FI, while the peak at ~!0= 2\u0001 0is induced by\nspin excitation of conduction electrons due to magnon7\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\n\u0001\nWithout vertex corrections With vertex corrections With vertex corrections\nFIG. 4. (Left panels) E\u000bective Zeeman \feld he\u000bon the Fermi surface. (Middle panels) Modulation of the Gilbert damping,\n\u000e\u000bnv\nG, without vertex correction. (Right panels) Modulation of the Gilbert damping with vertex correction, \u000e\u000bG. In the middle\nand right panels, the modulation of the Gilbert damping is plotted as a function of the FMR frequency, !0=\rhdc. The\nspin-orbit interactions are as follows. (a), (b), (c): \f=\u000b= 0. (d), (e), (f): \f=\u000b= 1. (g), (h), (i): \f=\u000b= 3. We note that (b),\n(e), (h) are essentially the same result as Ref. 30.\nabsorption30. In the case of \f=\u000b = 0, the vertex cor-\nrection changes the modulation of the Gilbert damping\nmoderately [compare Figs. 4 (c) with 4 (b)]. The widths\nof the two peaks at !0= 0 and!0= 2\u0001 0become nar-\nrower when the vertex correction is taken into account\n(see Appendix D for the analytic expressions).\nThe case of \f=\u000b = 1 is special because the e\u000bective\nZeeman \feld he\u000balways points in the direction of ( \u00001;1)\nor (1;\u00001), as shown in Fig. 4 (d). The amplitude of\nhe\u000bdepends on the angle of the wave number of the\nconduction electrons, ',\nhe\u000b(') = 2\u0001 0jsin('+\u0019=4)j; (61)\nand varies in the range of 0 \u00142he\u000b\u00144\u00010. Figures 4 (e)\nand 4 (f) show the modulation of the Gilbert dampingwithout and with the vertex correction for \u0000 =\u00010= 0:5.\nThe \fve curves correspond to \fve di\u000berent angles of hSi,\n\u0012=\u0000\u0019=4;\u0000\u0019=8;0;\u0019=8, and\u0019=4. The most remarkable\nfeature revealed by comparing Figs. 4 (f) with 4 (e) is that\nthe peak at !0= 0 disappears if the vertex correction\nis taken into account (see Appendix E for the analytic\nexpressions). In the subsequent section, we will show\nthat\u000e\u000bG(!0) has a\u000e-function-like singularity at !0= 0\nfor\f=\u000b = 1 due to the spin conservation law along the\ndirection ofhe\u000b.\nIn the case of \f=\u000b = 3, the direction of the e\u000bec-\ntive Zeeman \feld he\u000bvaries along the Fermi surface\n[Fig. 4 (g)]. Figures 4 (h) and 4 (i) show the modula-\ntion of the Gilbert damping without and with the ver-\ntex correction for \u0000 =\u00010= 0:5. For\f=\u000b = 3, a peak8\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\n\u0001\nWithout vertex corrections With vertex corrections With vertex corrections\nFIG. 5. Modulation of the Gilbert damping calculated for \f=\u000b= 1:1 (a) without the vertex correction and (b) with the vertex\ncorrection. The horizontal axis is the FMR frequency !0and the \fve curves correspond to \fve di\u000berent angles of hSi, i.e.,\n\u0012=\u0000\u0019=4;\u0000\u0019=8;0;\u0019=8, and\u0019=4. (c) Enlarged plot of the modulations of the Gilbert damping as a function of the FMR\nfrequency!0. The angle ofhSiis \fxed as\u0012=\u0019=4 and the three curves correspond to \f=\u000b = 1:03, 1:05, and 1:1. In all the\nplots, we have chosen \u0000 =\u00010= 0:5.\nat!0= 0 appears even when the vertex correction is\ntaken into account. The broad structure in the range of\n4\u00010\u0014~!0\u00148\u00010is caused by the magnon absorption\nprocess where its range re\rects the distribution of the\nspin-splitting energy 2 he\u000balong the Fermi surface. By\ncomparing Figs. 4 (h) and 4 (i), we \fnd that the vertex\ncorrection changes the result only moderately as in the\ncase of\f=\u000b = 0; the peak structure at !0= 0 becomes\nsharper when the vertex correction is taken into account\nwhile the broad structure is slightly enhanced.\nB. Strong enhancement of the Gilbert damping\nHere, we examine the strong enhancement of the\nGilbert damping for \f=\u000b'1. As explained in Sec. II A,\nthe spin component in the direction of the azimuth angle\n3\u0019=4 in thexyplane is exactly conserved at \f=\u000b = 1\n[see also Fig. 4 (d)]. When the value of \f=\u000b is shifted\nslightly from 1, the spin conservation law is broken but\nthe spin relaxation becomes remarkably slow. To see this\ne\u000bect, we show the modulation of the Gilbert damping\nwithout and with the vertex correction for \f=\u000b = 1:1\nin Figs. 5 (a) and 5 (b), respectively. The \fve curves\ncorrespond to \fve di\u000berent azimuth angles of hSi, and\nthe energy broadening is set as \u0000 =\u00010= 0:5. Figs. 5 (a)\nand 5 (b) indicate that the Gilbert damping is strongly\nenhanced at !0= 0 only when the vertex correction is\ntaken into account. This is the main result of our work.\nFigure 5 (c) plots the modulation of the Gilbert damp-\ning with the vertex correction for \u0000 =\u00010= 0:5 and\n\u0012=\u0019=4, the latter of which corresponds to the case of\nthe strongest enhancement at !0= 0. The three curves\ncorrespond to \f=\u000b= 1:03, 1:05, and 1:1. As the ratio of\n\f=\u000bapproaches 1, the peak height at !0= 0 gets larger.\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\nWithout vertex corrections With vertex corrections With vertex corrections\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001FIG. 6. Modulation of the Gilbert damping as a function\nof\f=\u000b. The \fve curves correspond to ~!0=\u00010= 0;0:005,\n0:01, 0:02, and 0:05. We have taken the vertex correction into\naccount and have chosen \u0000 =\u00010= 0:5. The inset illustrates\nmaximum values of the modulation of the Gilbert damping,\n\u000e\u000bG;max, in varying \f=\u000b for a \fxed value of ~!0=\u00010.\nFor\f=\u000b'1,\u000e\u000bGis calculated approximately as\n\u000e\u000bG\n\u000bG;0'\u00010\n2\u0019\u0000s\n(~!0)2+ \u00002ssin2\u0010\n\u0012+\u0019\n4\u0011\n; (62)\n\u0000s\u00112\n\u0000Z2\u0019\n0d'\n2\u0019(hx+hy)2\n1 + (2he\u000b=\u0000)2; (63)\nwhere \u0000sgives the peak width in Figs. 5 (b) and 5 (c) (see\nAppendix F for a detailed derivation). For \f=\u000b= 1 +\u000e\n(\u000e\u001c1), \u0000sis proportional to \u000e2and approaches zero in\nthe limit of \u000e!0. This indicates that \u0000 scorresponds\nto the spin relaxation rate due to a small breakdown of\nthe spin conservation law away from the special point of\n\f=\u000b = 1. Note that the peak height of \u000e\u000bGat!0= 0\ndiverges at \f=\u000b = 1. This indicates that for \f=\u000b =\n1,\u000e\u000bG(!0) has a\u000e-function-like singularity at !0= 0,\nwhich is not drawn in Fig. 4 (f).9\nFIG. 7. (Upper panels) Modulations of the Gilbert damping, \u000e\u000bG=\u000bG;0for (a)\f=\u000b= 0, (b)\f=\u000b= 1, and (c) \f=\u000b= 3. (Lower\npanels) Shifts in the FMR frequency, \u000e!0=(\u000bG;0!0), for (d)\f=\u000b = 0, (e)\f=\u000b = 1, and (f) \f=\u000b = 3. The horizontal axes are\nthe FMR frequency, !0=\rhdc, while the vertical axes show the azimuth angle of the spontaneous spin polarization, \u0012, in the\nFI. In all the plots, we have considered vertex corrections and have chosen \u0000 =\u00010= 0:5. In (a), (c), and (e) there are regions\nin which the values exceed the upper limits of the color bar located in the right side of each plot; the maximum value is about\n0:45 in (a), 0 :65 in (c), and about 10 in (e) (see also Fig. 8). In addition, (b) cannot express a \u000e-function-like singularity at\n!0= 0 (see the main text).\nFigure 6 plots the modulation of the Gilbert damping\nfor \u0000=\u00010= 0:5 and\u0012=\u0019=4 as a function of \f=\u000b. The\n\fve curves correspond to ~!0=\u00010= 0;0:005;0:01;0:02,\nand 0:05, respectively. This \fgure indicates that when\nwe \fx the resonant frequency !0and vary the ratio of\n\f=\u000b, the Gilbert damping is strongly enhanced when \f=\u000b\nis slightly smaller or larger than 1. We expect that this\nenhancement of the Gilbert damping is strong enough to\nbe observed experimentally. We note that \u000e\u000bG=\u000bG;0ap-\nproaches 0:378 (0:318) for\f=\u000b!0 (\f=\u000b!1 ). The\ninset in Fig. 6 plots maximum values of \u000e\u000bG=\u000bG;0when\n\f=\u000bis varied for a \fxed value of ~!0=\u00010. In other words,\nthe vertical axis of the inset corresponds to the peak\nheight in the main panel for each value of ~!0=\u00010. We\n\fnd that the maximum value of \u000e\u000bG=\u000bG;0diverges as !0\napproaches zero.\nV. SHIFT IN THE FMR FREQUENCY\nNext, we discuss the shift in the FMR frequency when\nthe vertex correction is taken into account. The den-\nsity plots in Figs. 7 (a), 7 (b), and 7 (c) for \f=\u000b= 0, 1,\nand 3 summarize the modulation of the Gilbert damping,\n\u000e\u000bG. These plots have the same features as in Figs. 4 (c),\n4 (f), and 4 (i). Figures. 7 (d), 7 (e), and 7 (f) plot the\nshift in the FMR frequency \u000e!0=!0with density plots\nfor\f=\u000b = 0, 1, and 3. By comparing Figs. 7 (a), 7 (b),\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\nWithout vertex corrections With vertex corrections With vertex corrections\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001FIG. 8. Shift in FMR frequency, \u000e!0=(\u000bG;0!0), as a func-\ntion of the resonance frequency !0for\f=\u000b = 1:1. The in-\nset shows the same quantities in the low-frequency range of\n0\u0014~!0=\u00010\u00140:05 with a larger scale on the vertical axis.\nWe have taken the vertex correction into account and have\nchosen \u0000=\u00010= 0:5.\nand 7 (c) with 7 (d), 7 (e), and 7 (f), we \fnd that some\nof the qualitative features of the FMR frequency shift\nare common to those of the modulation of the Gilbert\ndamping,\u000e\u000bG; (i) they depend on \u0012for\f=\u000b > 0, while\nthey do not depend on \u0012for\f=\u000b = 0, (ii) the structure10\nat!0= 0 due to elastic spin-\ripping appears, and (iii)\nthe structure within a \fnite range of frequencies due to\nmagnon absorption appears. We can also see a few dif-\nferences between \u000e\u000bGand\u000e!0=!0. For example, \u000e!0=!0\nhas a dip-and-peak structure at ~!0=\u00010= 2 where\u000e\u000bG\nhas only a peak. Related to this feature, \u000e!0=!0has a\ntail that decays more slowly than that for \u000e\u000bG. The most\nremarkable di\u000berence is that \u000e!0=!0diverges at !0= 0\nfor\f=\u000b = 1 except for \u0012= 3\u0019=4;7\u0019=4, re\recting the \u000e-\nfunction-like singularity of \u000e\u000bGat!0= 0. These features\nare reasonable because \u000e!0=!0and\u000e\u000bG, which are de-\ntermined by the real and imaginary parts of the retarded\nspin susceptibility, are related to each other through the\nKramers-Kronig conversion.\nThe main panel of Fig. 8 shows the frequency shift\n\u000e!0=!0for\f=\u000b = 1:1 as a function of the resonant\nfrequency!0. The \fve curves correspond to \u0012=\n\u0000\u0019=4;\u0000\u0019=8;0;\u0019=8, and\u0019=4. Although the frequency\nshift appears to diverge in the limit of !0!0 in the\nscale of the main panel, it actually grows to a \fnite value\nand then goes to zero as !0approaches zero (see the inset\nof Fig. 8). For \f=\u000b= 1 +\u000e(\u000e\u001c1), the frequency shift\nis calculated approximately as\n\u000e!0\n\u000bG;0!0'\u00010\n2\u0019~!0\n(~!0)2+ \u00002ssin2\u0010\n\u0012+\u0019\n4\u0011\n; (64)\nwhere \u0000sis the spin relaxation rate de\fned in Eq. (63)\n(see Appendix F for the detailed derivation). We expect\nthat this strong enhancement of the frequency shift near\n\f=\u000b= 1 can be observed experimentally.\nVI. SUMMARY\nWe theoretically investigated spin pumping into a two-\ndimensional electron gas (2DEG) with a textured e\u000bec-\ntive Zeeman \feld caused by Rashba- and Dresselhaus-\ntype spin-orbit interactions. We expressed the change\nin the peak position and the linewidth in a ferromag-\nnetic resonance (FMR) experiment that is induced by\nthe 2DEG within a second-order perturbation with re-\nspect to the interfacial exchange coupling by taking the\nvertex correction into account. The FMR frequency\nand linewidth are modulated by elastic spin-\ripping or\nmagnon absorption. We found that, for almost all of the\nparameters, the vertex correction modi\fes the modula-\ntion of the Gilbert damping only moderately and does not\nchange the qualitative features obtained in our previous\npaper30. However, we found that the Gilbert damping at\nlow frequencies, which is caused by elastic spin-\ripping,\nis strongly enhanced when the Rashba- and Dresselhaus-\ntype spin-orbit interactions are chosen to be almost equal\nbut slightly di\u000berent. Even in this situation, the Gilbert\ndamping at high frequencies, which is caused by magnon\nabsorption, shows small modi\fcation. This strong en-\nhancement of the Gilbert damping at low frequencies ap-\npears only when the vertex correction is taken into ac-\ncount and is considered to originate from the slow spinrelaxation related to the spin conservation law that holds\nwhen the two spin-orbit interactions completely match.\nA similar enhancement was found for the frequency shift\nof the FMR due to elastic spin-\ripping. We expect that\nthis remarkable enhancement can be observed experi-\nmentally.\nOur work provides a theoretical foundation for spin\npumping into two-dimensional electrons with a spin-\ntextured Zeeman \feld on the Fermi surface. Although\nwe have treated a speci\fc model for two-dimensional\nelectron systems with both the Rashba and Dresselhaus\nspin-orbit interactions, our formulation and results will\nbe helpful for describing spin pumping into general two-\ndimensional electron systems such as surface/interface\nstates66{68and atomic layer compounds69,70.\nACKNOWLEDGEMENTS\nThe authors thank Y. Suzuki, Y. Kato, and A. Shi-\ntade for helpful discussion. T. K. acknowledges sup-\nport from the Japan Society for the Promotion of Sci-\nence (JSPS KAKENHI Grant No. JP20K03831). M. M.\nis \fnancially supported by a Grant-in-Aid for Scienti\fc\nResearch B (Grants No. JP20H01863, No. JP21H04565,\nand No. JP21H01800) from MEXT, Japan. M. Y. is sup-\nported by JST SPRING (Grant No. JPMJSP2108).\nAppendix A: Calculation of Green's function\nIn our work, Green's function of conduction electrons\nis calculated by taking e\u000bect of impurity scattering into\naccount. In general, the \fnite-temperature Green's func-\ntion ^g(k;i!m) after the impurity average is described\nby the Dyson equation with the impurity self-energy\n^\u0000(k;!m) as\n^g(k;i!m) =1\n^g0(k;i!m)\u00001\u0000^\u0000(k;i!m); (A1)\nwhere ^g0(k;i!m)\u00001is Green's function of electrons in the\nabsence of impurities. In our work, we employ the Born\napproximation in which the self-energy is approximated\nby second-order perturbation with respect to an impurity\npotential. In the Born approximation, the self-energy is\ngiven as\n^\u0000(k;i!m) =niu2Zd2k\n(2\u0019)2^g0(k;i!m); (A2)\nwhereniis the impurity concentration. The correspond-\ning Feynman diagram of the Dyson equation is shown in\nFig. 9. By straightforward calculation, Eq. (7) can be\nderived. For a detailed derivation, see Ref. 30.11\nFIG. 9. The Feynman diagram for Green's function within\nthe Born approximation.\nAppendix B: Derivation of Equations. (45)-(50)\nEqs. (35)-(38) can be rewritten with \u0000 = 2 \u0019niu2D(\u000fF)\nas\n\u00030(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019X\n\u0017;\u00170I\u0017\u00170; (B1)\n\u00031(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019X\n\u0017;\u00170\u0017\u00170I\u0017\u00170; (B2)\n\u00032(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019cos 2('\u0000\u0012)X\n\u0017;\u00170\u0017\u00170I\u0017\u00170;\n(B3)\n\u00033(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019sin 2('\u0000\u0012)X\n\u0017;\u00170\u0017\u00170I\u0017\u00170;\n(B4)\nwhere\nI\u0017\u00170=Z1\n\u00001d\u0018\n2\u0019i1\ni~!m\u0000\u0018\u0000\u0017he\u000b+i(\u0000=2)sgn(!m)\n\u00021\ni~(!m+!n)\u0000\u0018\u0000\u00170he\u000b+i(\u0000=2)sgn(!m+!n):\n(B5)\nWe note that one needs to calculate this integral only for\n!n>0 to obtain the retarded component by analytic\ncontinuation. Then, we can easily prove by the residue\nintegral that I\u0017\u00170= 0 for!m>0 and!m+!n>0\n(!m<0 and!m+!n<0) because both of the two poles\nin the integrand are located only in the upper (lower) half\nof the complex plane of \u0018. For!m<0 and!m+!n>0,\nthe integral is evaluated by the residue integral as\nI\u0017\u00170=1\ni~!n+ (\u0017\u0000\u00170)he\u000b+i\u0000: (B6)\nBy combining these results, Eqs. (45)-(50) can be de-\nrived.\n(a) (b)\nFIG. 10. Schematic picture of the change in the contour in-\ntegral. (a) The original contour. (b) The modi\fed contour.\nAppendix C: Derivation of Eq. (52)\nIn this Appendix, we give a detailed derivation of\nEq. (52) from Eq. (51). First, we modify Eq. (51) as\n\u001f(0;i!n) =1\n8AX\nkX\n\u0017;\u00170\"\n\u0017\u00170sin 2(\u001e\u0000\u0012)I\u0017\u00170;1\n+n\n1\u0000\u0017\u00170cos 2(\u001e\u0000\u0012)o\nI\u0017\u00170;2\n\u0000i(1\u0000\u0017\u00170)I\u0017\u00170;3#\n; (C1)\nwhere\nI\u0017\u00170;j\u00111\n\fX\ni!mXj\ni~!m\u0000E\u0017\nk+i\u0000=2 sgn(!m)\n\u00021\ni~!m+i~!n\u0000E\u00170\nk+i\u0000=2 sgn(!m+!n);(C2)\nand (X1;X2;X3) = (X;Y;Z ). A standard procedure\nbased on the residue integral enables us to express the\nsumI\u0017\u00170;jfor!n>0 as a complex integral on the con-\ntour C shown in Fig. 10 (a). This contour can be modi\fed\ninto a sum of the four contours, C l(l= 1;2;3;4), shown\nin Fig. 10 (b). Accordingly, I\u0017\u00170;jis written as\nI\u0017\u00170;j=4X\nl=1ICl\n\u0017\u00170;j; (C3)\nICl\n\u0017\u00170;j=\u0000Z\nCldz\n2\u0019if(z)Xj(z;i!n)\nz\u0000E\u0017\nk+i\u0000=2 sgn(Imz)\n\u00021\nz+i~!n\u0000E\u00170\nk+i\u0000=2 sgn(Imz+!n);(C4)\nwheref(z) = 1=(e\fz+ 1) is the Fermi distribution func-\ntion. The sum of the contributions from the two contours,12\nC2and C 3, is calculated as\nIC2\n\u0017\u00170;j+IC3\n\u0017\u00170;j\n=\u0000~Xj(i!n)ZdE\n2\u0019if(E)\n\u0002\"\n\u00001\nE\u0000E\u0017\nk\u0000i\u0000=21\nE+i~!n\u0000E\u00170\nk+i\u0000=2\n+1\nE\u0000i~!n\u0000E\u0017\nk\u0000i\u0000=21\nE\u0000E\u00170\nk+i\u0000=2#\n:(C5)\nHere, we have used the fact that Xj(z;i!n) is indepen-\ndent ofzfor 0<Imz < !nfrom Eq. (45) and have\nde\fned its value as ~Xj(i!n) (j= 1;2;3). From Eqs. (41)-\n(43), ~Xj(i!n) are calculated as\n~X1(i!n) =~\u00033(i!n)\n(1\u0000~\u00030(i!n))2\u0000~\u00032(i!n)2\u0000~\u00033(i!n)2;(C6)\n~X2(i!n) =1\u0000~\u00030(i!n)\u0000~\u00032(i!n)\n(1\u0000~\u00030(i!n))2\u0000~\u00032(i!n)2\u0000~\u00033(i!n)2;(C7)\n~X3(i!n) =i\n1\u0000~\u00030(i!n) +~\u00031(i!n): (C8)\nBy changing the integral variable to E0=E\u0000E\u0017\nkin the\n\frst term and to E0=\u0000(E\u0000E\u00170\nk) in the second term in\nEq. (C5), we obtain\nIC2\n\u0017\u00170;j+IC3\n\u0017\u00170;j=\u0000~Xj(i!n)ZdE0\n2\u0019i1\nE0\u0000i\u0000=2\n\u0002\"\nf(\u0000E0+E\u00170\nk)\u0000f(E0+E\u0017\nk)\nE0+i~!n+E\u0017\nk\u0000E\u00170\nk+i\u0000=2#\n:(C9)\nUsing formula (44), we replace the sum over kin Eq. (C1)\nby the integral with respect to \u0018and'. We can perform\nthe\u0018-integral by using\n\u0000Z1\n\u00001d\u0018[f(E0+E\u0017\nk)\u0000f(\u0000E0+E\u00170\nk)]\n= 2E0+E\u0017\nk\u0000E\u00170\nk: (C10)\nThen, by performing the E0-integral, we obtain\nZ1\n\u00001d\u0018(IC2\n\u0017\u00170;j+IC3\n\u0017\u00170;j) =i~!n~Xj(i!n)\nE\u0017\nk\u0000E\u00170\nk+i~!n+i\u0000:\n(C11)\nNext, let us consider the contribution from C 1and C 4.\nOn these two contours, Xj(z;i!n) is independent of z\nand its value is de\fned by ~X0\nj(i!n) (j= 1;2;3). Because\n\u0003j(z;i!n) (j= 0;1;2;3) becomes zero for Im z < 0 or\n!n<Imzfrom Eq. (45), ~X0\nj(i!n) are given as\n~X0\n1(i!n) = 0;~X0\n2(i!n) = 1;~X0\n3(i!n) =i:(C12)A similar calculation to that of C 2and C 3yields\nZ1\n\u00001d\u0018(IC1\n\u0017\u00170;j+IC4\n\u0017\u00170;j) =\u0000~X0\nj(i!n): (C13)\nBy substituting these results into Eq. (C1), we obtain\n\u001f(0;i!n) =D(\u000fF)\n8X\n\u0017;\u00170Z2\u0019\n0d'\n2\u0019\"\n\u0017\u00170sin 2(\u001e\u0000\u0012) ~x1(i!n)\n+ [1\u0000\u0017\u00170cos 2(\u001e\u0000\u0012)][\u00001 + ~x2(i!n)]\n\u0000i(1\u0000\u0017\u00170)[\u0000i+ ~x3(i!n)]#\n: (C14)\nwhere\n~xj(i!n) =i~!n~Xj(i!n)\nE\u0017\nk\u0000E\u00170\nk+i~!n+i\u0000: (C15)\nFinally, Eq. (52) is derived by substituting the expres-\nsions for ~Xj(i!n) and by analytic continuation i!n!\n!+i\u000e.\nAppendix D: Analytic Expression for \f=\u000b= 0\nIn this appendix, we derive analytic expressions of the\nmodulation of the Gilbert damping when \f=\u000b= 0, only\nthe Rashba spin-orbit interaction exists, to see the quan-\ntitative e\u000bect of taking the vertex correction into account.\nFor\f=\u000b= 0, the spin-splitting energy 2 he\u000b= 2\u0001 0is con-\nstant along the Fermi surface, and ~\u0003R\nj(!) (j= 0;1;2;3)\nis simpli\fed as\n~\u0003R\n0(!) =i\u0000\n4\u00010X\n\u0017\u001701\n~!=\u00010+ (\u0017\u0000\u00170) +i\u0000=\u00010;(D1)\n~\u0003R\n1(!) =i\u0000\n4\u00010X\n\u0017\u00170\u0017\u00170\n~!=\u00010+ (\u0017\u0000\u00170) +i\u0000=\u00010;(D2)\n~\u0003R\n2(!) =~\u0003R\n3(!) = 0: (D3)\nThen, we obtain the modulation of the Gilbert damping\nwith the vertex corrections,\n\u000e\u000bG\n\u000bG;0'\u00010\n2\u0019\u0000Re\"~\u0003R\n0(!0)\n1\u0000~\u0003R\n0(!0)+~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)\n1\u0000~\u0003R\n0(!0) +~\u0003R\n1(!0)#\n:\n(D4)\nThe modulation of the Gilbert damping without the ver-\ntex correction is obtained by considering only the \frst-\norder term with respect to ~\u0003R\nj(!0),\n\u000e\u000bnv\nG\n\u000bG;0'\u00010\n2\u0019\u0000Re\"\n2~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)#\n: (D5)\nWhen \u0000\u001c\u00010, the contribution of \u0017=\u00170is dominant\nfor the peak at !0= 0 and the modulation of the Gilbert13\ndamping can be analytically calculated as\n\u000e\u000bG\n\u000bG;0'\u00010\n4\u0019\u0001\u0000=2\n(~!0)2+ (\u0000=2)2; (D6)\n\u000e\u000bnv\nG\n\u000bG;0'\u00010\n4\u0019\u0001\u0000\n(~!0)2+ \u00002: (D7)\nThis indicates that the peak width is halved by taking\nthe vertex correction into account, which is consistent\nwith the results shown in Figs. 4 (b) and 4 (c).\nIn a similar way, we can evaluate the modulation of\nthe Gilbert damping near the peak at !0= 2\u0001 0=~as\n\u000e\u000bG\n\u000bG;0'\u00010\n4\u0019\u0001\u00141\n23\u0000=4\n(~!0\u00002\u00010)2+ (3\u0000=4)2\n+\u0000=2\n(~!0\u00002\u00010)2+ (\u0000=2)2\u0015\n; (D8)\n\u000e\u000bnv\nG\n\u000bG;0'\u00010\n4\u0019\u00013\u0000=2\n(~!0\u00002\u00010)2+ \u00002: (D9)\nAs well, for the peak at !0= 2\u0001 0=~, the peak width\nbecomes smaller when the vertex correction is taken into\naccount. This observation is consistent with the results\nshown in Figs. 4 (b) and 4 (c). For a \fnite value of \u0000, a\nsum of Eqs. (D6) and (D8) [Eqs. (D7) and (D9)] gives a\nbetter analytic form which \fts the numerical result with\n(without) the vertex correction. Note that \u000e\u000bGand\u000e\u000bnv\nG\ndepend on the impurity potential strength, u, and im-\npurity concentration, ni, through \u0000 = 2 \u0019niu2D(\u000fF) [see\nEq. (8)]. As shown in Eqs. (D6)-(D9), the peak widths\nof the Lorentzian functions in \u000e\u000bGand\u000e\u000bnv\nGare deter-\nmined by \u0000 [see Figs. 4 (b) and 4 (c)]. It is remarkable\nthat the peak width in \u000e\u000bGis reduced from \u0000 to \u0000 =2 by\ntaking the vertex correction into account. To summarize\nthe e\u000bect of the vertex correction, we show \u000e\u000bG\u0000\u000e\u000bnv\nG\nand\u000e!0\u0000\u000e!nv\n0in Figs. 11 (a) and 11 (d), respectively.\nWe \fnd that the vertex correction modi\fes mainly the\npeak width around ~!0=\u00010= 0 and 2, in consistent with\nthe above analytic expressions.\nFinally, we note that the same analytical expressions\nfor\u000e\u000bGand\u000e\u000bnv\nGcan be obtained for the case of \u000b=\f=\n0, i.e., when only the Dresselhaus spin-orbit interaction\nexists. We also note that for general values of \f=\u000b,\u000e\u000bG\nand\u000e\u000bnv\nGdepend on \u0000 in a more complicated way.\nAppendix E: Analytic Expression for \f=\u000b= 1\nIn this Appendix, we derive analytic expressions of the\nmodulation of the Gilbert damping when \f=\u000b = 1. In\nthis case, the e\u000bective Zeeman \feld is parallel to the\n(\u00001;1;0) direction and its amplitude is given as\nhe\u000b(') = 2\u0001 0jsin('+\u0019=4)j: (E1)Then, ~\u0003R\nj(!) (j= 0;1;2;3) becomes\n~\u0003R\n0(!) =i\u0000\n4\u00010X\n\u0017\u00170J\u0017\u00170 (E2)\n~\u0003R\n1(!) =i\u0000\n4\u00010X\n\u0017\u00170\u0017\u00170J\u0017\u00170; (E3)\n~\u0003R\n2(!) =\u0000sin 2\u0012~\u0003R\n1(!); (E4)\n~\u0003R\n3(!) =\u0000cos 2\u0012~\u0003R\n1(!) (E5)\nwhere\nJ\u0017\u00170(!)\u0011Z2\u0019\n0d'\n2\u0019\u00010\n~!+ (\u0017\u0000\u00170)he\u000b(') +i\u0000:(E6)\nIn the case of \u0012=\u0019=4, the modulation of the Gilbert\ndamping with the vertex correction is expressed as\n\u000e\u000bG\n\u000bG;0=\u00010\n2\u0019\u0000Re\"\n\u00002 +1\n1\u0000~\u0003R\n0(!0) +~\u0003R\n1(!0)\n+1\n1\u0000~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)#\n: (E7)\nThe third term of the above equation is calculated as\n1\n1\u0000~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)=1\n1\u0000i\u0000\n~!0+i\u0000=~!0+i\u0000\n~!0:(E8)\nThis indicates that the expansion with respect to ~\u0003R\njcan\nnot be allowed for !0\u001c\u0000. This is why the modula-\ntion without the vertex correction, which is obtained by\ntaking from the \frst-order term of ~\u0003R\njin Eq. (E7) as\n\u000e\u000bnv\nG\n\u000bG;0=\u00010\n2\u0019\u0000Re\u0014\n2~\u0003R\n0(!0)\u0015\n; (E9)\ngives a di\u000berent result near !0'0. Actually, for \u0012=\u0019=4,\n\u000e\u000bGand\u000e\u000bnv\nGare calculated as\n\u000e\u000bG\n\u000bG;0=\u00010\n2\u0019\u0000Re\"\ni\u0000\n2\u00010(J+\u0000+J\u0000+)\n1\u0000i\u0000\n2\u00010(J+\u0000+J\u0000+)#\n; (E10)\n\u000e\u000bnv\nG\n\u000bG;0=1\n4\u0019Re\"\ni\u0012\nJ+\u0000+J\u0000++J+++J\u0000\u0000\u0013#\n:(E11)\nNote that Eq.(E10) is not valid for !0= 0. As indicated\nfrom the absence of J++andJ\u0000\u0000, the graph of \u000e\u000bG(!0)\nhas no peak at zero frequency even though \u000e\u000bnv\nG(!0) has a\npeak there. This observation is consistent with Figs. 4 (e)\nand 4 (f).\nIn the case of \u0012=\u0000\u0019=4, the modulations of the Gilbert\ndamping with and without the vertex correction are\n\u000e\u000bG\n\u000bG;0=\u00010\n\u0019\u0000Re\"\ni\u0000\n2\u00010(J+\u0000+J\u0000+)\n1\u0000i\u0000\n2\u00010(J+\u0000+J\u0000+)#\n; (E12)\n\u000e\u000bnv\nG\n\u000bG;0=1\n2\u0019Re\"\ni(J+\u0000+J\u0000+)#\n: (E13)14\nFIG. 11. (Upper panels) Change of the Gilbert damping due to the vertex correction, \u000e\u000bG\u0000\u000e\u000bnv\nG, for (a)\f=\u000b= 0, (b)\f=\u000b= 1,\nand (c)\f=\u000b= 3. (Lower panels) Change of the FMR frequency due to the vertex correction, \u000e!0\u0000\u000e!nv\n0, for (d)\f=\u000b= 0, (e)\n\f=\u000b= 1, and (f) \f=\u000b= 3. The horizontal axes are the FMR frequency, !0=\rhdc, whereas the vertical axes are the azimuth\nangle of the spontaneous spin polarization, \u0012, in the FI. In all the plots, we have set \u0000 =\u00010= 0:5.\nNote that\u000e\u000bnv\nGis obtained by taking the \frst-order term\nin Eq. (E12). As indicated by the absence of the terms,\nJ++andJ\u0000\u0000, neither\u000e\u000bGnor\u000e\u000bnv\nGhas any structure\naround!0= 0. It can be checked that these two expres-\nsions give almost the same result when \u0000 .\u00010, which is\nconsistent with Figs. 4 (e) and 4 (f). Note as well that\n\u000e\u000bGis just doubled compared with the result for \u0012=\u0019=4\nin Eq. (E10).\nTo summarize the e\u000bect of the vertex correction, we\nshow\u000e\u000bG\u0000\u000e\u000bnv\nGand\u000e!0\u0000\u000e!nv\n0in Figs. 11 (b) and\n11 (e), respectively. We \fnd that the vertex correction\nmodi\fes mainly the peak width around ~!0=\u00010= 0.\nIn addition, the broad peak in the range of 0 <~!0<\n2\u00010is enhanced or suppressed depending on the azimuth\nangle of the ordered spin. These features are consistent\nwith the above analytic expressions. We note that similar\nfeatures are observed for \f=\u000b= 3 as seen in Figs. 11 (c)\nand 11 (f).\nAppendix F: Approximate Expressions near \f=\u000b= 1\nIn this Appendix, we derive the approximate expres-\nsions Eqs. (62) and (64) for \f=\u000b = 1 +\u000e(\u000e\u001c1) and\n!'0. For\f=\u000b= 1 +\u000e(\u000e\u001c1), we can use the approx-\nimation,\ncos 2(\u001e\u0000\u0012)'sin 2\u0012\u0010\n\u00001 +(hx+hy)2\nh2\ne\u000b\u0011\n; (F1)\nsin 2(\u001e\u0000\u0012)'cos 2\u0012\u0010\n\u00001 +(hx+hy)2\nh2\ne\u000b\u0011\n: (F2)Then, we obtain\n~\u0003R\n2'Xsin 2\u0012; (F3)\n~\u0003R\n3'Xcos 2\u0012; (F4)\nX\u0011i\u0000\n4Z2\u0019\n0d'\n2\u0019X\n\u0017\u00170\u0017\u00170\u0010\n\u00001 +(hx+hy)2\nh2\neff\u0011\n~!+ (\u0017\u0000\u00170)he\u000b+i\u0000(F5)\nin the low-frequency region. Here, the contribution of\nthe second term of the bracket in Eq. (56) does not have\na singularity at !0= 0 because ~\u0003R\n0and ~\u0003R\n1do not de-\npend on the e\u000bective Zeeman \feld he\u000b. Therefore, the\nsingularity comes from the \frst term of the bracket in\nEq. 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International Center for Materials Nanoarchitectonics,  Nationa l Institute for Materials Science,  1-1 \nNamiki, Tsukuba 305 -0044, Japan  \n \n \n \n \n \n \n \n \n \n \n \n*wu.2314@osu.edu     \n \n2 \n Abstract:  \nEngineering magnetic anisotropy in a ferro - or ferrimagnetic (FM) thin film is crucial in \nspintronic device. One way to modify the magnetic anisotropy is through the surface of the FM \nthin film. Here, we report the emergence of a perpendicular magnetic anisotropy (PMA) induced \nby interfacial interactions in a heterostructure comprised of a garnet ferrimagnet, Y 3Fe5O12 \n(YIG), and the low -symmetry, high spin orbit coupling (SOC) transition metal dichalcogenide, \nWTe 2.  At the same time, we also observed an enhancement in Gilbert damping in the WTe 2 \ncovered YIG area. Both the magnitude of interface -induced PMA and the Gilb ert damping \nenhancement have no observable WTe 2 thickness dependence down to single quadruple -layer, \nindicating that the interfacial interaction plays a critical role. The ability of WTe 2 to enhance the \nPMA in FM thin film, combined with its previously rep orted capability to generate out -of-plane \ndamping like spin torque, makes it desirable for magnetic memory applications.   \n \nKey words:  perpendicular magnetic anisotropy, magnetic resonance force microscope, transition metal \ndichalcogenides, ferrimagnetic i nsulator      \n \n3 \n Perpendicular magnetic anisotropy  (PMA)  in a ferromagnetic thin film is of great interest in \nspintronics research and application s. Ferromagnetic nano -element s with PMA overcome  their shape \nanisotropy , greatly ease the memory cell size reduction and improves  memory retention . These \nexceptional properties, improving the performance  of magnetic devices , make PMA highly desirable for \nmagnetic memory  application s.  PMA becomes even more important in the recent development of solid \nstate magnetic random -access memory (MRAM) since it allows MRAM to have lower switching current \nand faster switching speed compare d to in-plane magnetized materials 1, 2.  \nMagnetic storage devices generally rely on metallic magnetic material s due to their robust \nelectrical response . Interfacial magnetic anisotropy plays a critical role in generat ing PMA in metallic \nferromagnet s. When interfacing with a nonmagnetic material (NM), electron orbital angular momentum \nof the magnetic ions at the ferromagnet surface will be modified, in some cases  enabling strong covalent \nbonding, resulting in distinct magnetic properties compare d to the single layer 3-6. However, spintronics \ndevices made of metallic magnetic materials are inherently energy consumptive due to resistive losses. \nRecently, complex oxide ferro - or ferrimagnet  insulator s (FMI)  have  attracted substantial  interest  due to \ntheir ability to transport spin excitation s with low dissipation  7. Inducing PMA in FMIs naturally \nbecomes an important topic both for scientific and technologic al reasons . Several successful route s to \nachiev ing PMA in FMIs ha ve been reported using bulk intrinsic anisotropy 8 or lattice strain 9-12. But in \nmost experiments,  the sign of the resulting interfacial anisotropy in FMI/NM heterostructures is such as \nto enhance  the easy-plane anisotropy  13-15. Only one recent experiment has shown the possibility of \ngenerating interfacial PMA, and this was attributed to topological surface states  16. Nevertheless, t hese \nresults demonstrate  the possibility of controlling magnetic anisotropy through interfac ial interaction s in    \n \n4 \n FMI/NM heterostructures . Here, we report a study on YIG/WTe 2/hBN heterostructures, which shows  \nthat when interfacing with a low symmetry nonmagnetic  van der Waals  material , WTe 2, an additional  \ninterfac e-induced  PMA  (iPMA) term  emerges  in the magnetic anisotropy  of the YIG thin film . The \nabsence of topological surface states at room temperature  in WTe 2 17, 18 forces us to seek an explanation \nfor our observation of enhanced PMA that is distinct from that proposed for top ological insulator/YIG \nbilayers 16. We therefore turn to an analysis of the broken symmetries in WTe 2. We point out that low \nsymmetry WTe 2 has recently shown the capability of generating both in -plane and out -of-plane spin \npolarization in charge -spin conversion experiments 19-22. It also enables field-free switching of PMA \nmagnet ic material , which ease s the application of PMA material s in MRAM application s 23-25.   \nFerrimagnetic insulator  YIG is of significant research interest in spintronics  due to its \nexceptionally  low Gilbert damping  26, which describes the  relaxation rate  of magnetization precession . \nAnd 1T’-WTe 2 is a semi -metallic transition metal dichalcogenide (TMD) layered material with strong \nSOC  27, 28. The crystal structure of 1T’-WTe 2 lacks twofold rotational symmetry about the c -axis (Fig. \n1a). The only symmetry in the  WTe 2 crystal lattice  ab plane is the mirror symmetry about  the bc plane  \n29. This u nique symmetry breaking allows  out-of-plane damping -like torque  to be generated 30, 31, \nenabling efficient switching of the  out-of-plane magnetization  of the adjacent magnetic material  24.  \nA 20nm thick YIG thin film used in our experiment is epitaxially grown on (111) -oriented  \nGd3Ga5O12 (GGG) substrate  by off -axis sputtering  32. WTe 2 flakes are then mechanically exfoliated \nfrom a flux-grown crystal, and dry transferred on to the clean top YIG surface without touching any \nother substances. This whole process is carried out in an Ar -filled glove box with <0.1 ppm of H 2O and    \n \n5 \n O2 to protect the flakes from degradation and  ensure  the clean liness  of the YIG/WTe 2 interface. We \nemploy hexagonal boron nitride ( hBN ) encapsulation to protect the WTe 2 flakes from oxidation after \nbeing removed from the glove box. We make two samples  and focus on the  data taken from sample 1 in \nthe main text. The raw data taken from sample 2 can be found  in Supporting Information  Fig. S 2.  \n \nFig. 1 Crystal structure of WTe 2 and sample schematic . a) Crystal lattice structure of WTe 2 viewed  \nfrom the top along  the c-axis and looking from the side along the a-axis. The black dashed box in the \nside view indicates a monolayer of WTe 2. b) Schematic of the ferromagnetic resonance force \nmicroscope. RF excitation is generated by a stripline underneath the sample , where the hBN \nencapsulation is not shown . The  region of  localized mode is shown as a yellow dot adjacent to the  WTe 2 \nflake, and the probe magnetic moment is shown as a yellow arrow on the particle. The cantilever \noscillation is detected by a fiber laser interferometer.  \nFigure 2a shows an optical image  of the sample 1. Due to the small  lateral size of the exfoliate d \nWTe 2 and hBN  flake s having  length  scale s of 10 μm, we use a home -built ferromagnetic resonance \nforce microscope (FMRFM) to measure  the local ferromagnetic resonance (FMR) signal. FMRFM is a \nsensitive technique to detect  the local magnetic properties with high spatial and spectral resolution  33. In \nour FMRFM, the external magnetic field 𝐻⃗⃗ ext is aligned perpendicular to  the sample plane. The \ncantilever tip holds a  high coercivity SmCo 5 magnetic particle , whose  moment  is magnetized in the  \n   \n \n6 \n direction opposite to 𝐻⃗⃗ ext to create a magnetic field well . The field well supports a set of  localized \nstanding spin wave modes (LMs). During the measurement, we excite spin precession uniformly by a \nstripline underneath the sample at a fixed RF frequency  (2 GHz)  and sweep the magnetic field. The \nresonance of each LM  generate s a stray field, whic h can then  be detected by the SmCo 5 magnetic \nparticle attached on the cantilever through their magnetic dipole -dipole interaction (Fig. 1b). During the \nmeasurement, we keep the probe -to-sample separation around 4 μm. The operation  of FMRFM is \ndescribed in detail in Ref s. 34-36. For reference, w e separate  a region of  YIG that does not contain  \nWTe 2/hBN heterostructures and measur e its Gilbert damping using  broadband FMR. To eliminate two -\nmagnon scattering, w e perform broadband FMR in the out -of-plane  field geometry. The FMR linewidth \nas a function of frequency measured on bare YIG (sample 1) shows a linear dependence (Fig. 2b), from \nwhich we can extract the Gilbert damping  of bare YIG  𝛼YIG=1.05×10−3. We also confirm that the \nWTe 2 used in the experiment is indeed  the 1T’ phase  through polarized Raman measurements. The \npolarization angle dependence of the Raman peak at 212 cm-1 (spectrum is shown in Fig. S4) exhibits \nminimum intensity when the excitation laser polarization is along the crystallographic a axis of WTe 2 37 \nas shown by the polar plot in Fig. 2c and Raman intensity plot in Fig. 2d .  \nWe find t he position of the YIG/WTe 2/hBN heterostructure with the assist ance of magnetic \nalignment markers  (Fig. 2a) . Figure 2e shows t wo raw FMRFM scans taken in the region  of YIG/hBN \nand YIG/WTe 2/hBN , indicated by the blue and the red dot in Fig. 2a, respectively , which  reveals  the \nchange in FMRFM spectra  at two different location s. Here we focus on the 𝑛=1 LM because it has the \nmode radius of around 1 μm and gives the highest spatial resolution. Higher order modes  have \nincreasing  mode radius and therefore, detect  less local ized magnetic  properties.  This is the reason why    \n \n7 \n the quasi -uniform mode at ~  3325 Oe does not show obvious change in resonance field or signal \namplitude.  We further take a  line scan across the edge of WTe 2 flake (Fig. 2 f) to resolve the spatial \nevolution of FMRFM spectra . The line scan in Fig. 2f (along the dashed line shown in Fig. 2a) shows \nthree main features : first, the magnitude of the LM resonance signal is reduced in the YIG/WTe 2/hBN \nregion compare d to the YIG/hBN region; second, the LM resonance field for all LMs is decreased by \n~40 Oe in the YIG/WTe 2/hBN region; third , the LMs  show complex  splitting and crossing when the \nprobe is close to the boundary (−5 μm<𝑋<10 μm).  \n \nFig. 2 FMRFM and Raman measurement data . a) An optical micrograph of the YIG/WTe 2/hBN \nheterostructure under study . WTe 2 crystal a and b axis are labeled.  b) Broadband FMR measurement of \nthe frequency -dependent linewidth of the YIG thin film. The measurement is done on the same piece of \nYIG used to make sample shown in Fig. 1b. c) Polar plot of the 21 2 cm-1 peak  Raman  intensity. Angle \ndenote s the relative angle between the measurement laser polarization and the WTe 2 a axis. d) 2D \nintensity plot showing Raman peak intensities versus polarization angle . e) FMRFM spectra, one over \nthe YIG/hBN  region (blue line)  and the second over the YIG/WTe 2/hBN  region (red line); these \nlocations are  indicated by the  blue and red dot s in Fig. 2a respectively . f) Color plot of field -dependence \n   \n \n8 \n FMRFM scans as a function of position along the trace indicated by the black dashed line in Fig. 2a. A \nconstant background is subtracted to show only the signal from the several LM resonance s. \nIn the following, we will explain the origin of the three observed effects using spin pumping and \nmagnetic anisotropy. The first effect , i.e. signal reduction in  the YIG/WTe 2/hBN area relative to the \nYIG/hBN  area, is the result of enhanced relaxation due to spin pumping from YIG to WTe 2 38. The 𝑛=\n1 LM resonance signal amplitude ∆𝐴 is inversely proportional to the square of Gilbert damping , 𝛼2. We \ndetermine  the Gilbert damping  constant  𝛼 for YIG/WTe 2/hBN  using 𝛼YIG/WTe2/hBN=\n𝛼YIG/hBN×√∆𝐴YIG/hBN∆𝐴YIG/WTe2/hBN ⁄  (see Ref. 39), where 𝛼YIG/hBN is assumed to be the same as \n𝛼YIG=1.05×10−3 due to  the low SOC and insulating character of hBN . The second  effect is  the \ndecrease of 𝑛=1 LM resonance field 𝐻r,1 by ~40 Oe . And the third effect is  splitting and crossing  of \ncomplex modes  in the region −5 μm<𝑋<10 μm. The second and the third effects  are due  to an \nabrupt change of uniaxial anisotropy across the boundary  separating the YIG/WTe 2/hBN and YIG/hBN \nregions  15. Here, the uniaxial anisotropy refers to the magnetic free energy  depends on the angle between \nmagnetization and sample normal  ℱu=−𝐾u𝒎z2, where 𝒎z is the component  of magnetization unit \nvector in the direction normal to sample plane  and 𝐾u is the uniaxial anisotropy constant specific to \nsample  and depends on the total interaction in the sample . When 𝐾u is positive, ℱu is called to be of \nPMA type, on the other hand, if 𝐾u is negative, ℱu is called to be of easy -plane  type.  This uniaxial \nanisotropy will lead to an effective uniaxial magnetic field 𝑯u=−𝜕ℱu𝜕𝑴⁄ , where 𝑴 is the \nmagnetization . And therefore, a change in 𝐾u can modify the resonance field in a FMR measurement . In \nFMRFM spatial mapping, a n abrupt change in 𝐾u spatially could disturb the LM and lead to mode \nsplitting and crossing as described in Ref. 15. Moreover, i n striking contrast to the previously studied    \n \n9 \n YIG/Au interface 15, which result s in a 32 Oe increase of 𝐻r,1 due to  the enhanced easy -plane \nanisotropy, the observed decrease of 𝐻r,1 indicates that the WTe 2 overlayer induces  an iPMA  in YIG. \nWe note that the  magnitude of the shift in  𝐻r,1 is comparable to the easy -plane anisotropy induced by a \nheavy metal 15, 40 or the iPMA generated by topological surface state 16 on garnet ferrimagnetic material .  \nIn order to probe  the global effect of  a WTe 2 overlayer on YIG, we spatial ly map 𝐻r,1 using  the \n𝑛=1 LM. Figure 3a present s an optical image of WTe 2 flakes on a Si/SiO 2 (285nm ) substrate , where \ndifferent c olors of WTe 2 flakes indicat e different WTe 2 thickness es. Figure 3b and 3 c show spatial  maps \nof magnetic properties  in the region enclosed by the black dashed rectangle  in Fig. 3a .  We acquire the \nmaps using the procedure described in Ref. 39, i.e., simultaneously  measuring spatial variation of  the \nmagnetic anisotropy and Gilbert damping using the  𝑛=1 LM resonance field 𝐻r,1 and signal amplitude  \n∆𝐴. The  entire  WTe 2-covered area show s uniformly lower ed 𝐻r,1 and increased Gilbert damping relative \nto the area without WTe 2. In Fig. 3c, despite the not great signal to noise ratio in damping imaging, the re \nis a clear  Gilbert damping enhancement in WTe 2-covered area . The averaged Gilbert damping of YIG in \nWTe 2-covered area is 𝛼̅YIG/WTe2/hBN≈1.30×10−3, about 24% higher than  𝛼YIG. We note that due to \nthe slight relative tilting of the scan plane and the sample plane, there is a color shift in Fig. 3b  that \nmight conceal the contrast difference in different WTe 2 thickness region . Therefore, to study the WTe 2 \nthickness dependence, we will show fine line scans across edge s of flakes  having  different WTe 2 \nthickness es.     \n \n10 \n  \nFig. 3 Two -dimensional FMRFM scan resolving the spatial variation of magnetic anisotropy and \nGilbert damping. a ) Optical micrograph showing the color contrast of different thickness WTe 2 flake s \n(ranging from 4.7 nm to 44.8 nm) on Si/SiO 2(300 nm). Black dashed box outlines the FMRFM scanned \narea for 2D mapping. b) 2D map of the 𝑛=1 LM resonance field . The d ashed line s labeled 1 -4 \ncorrespond to the  four line-scans shown in Fig. S1a-S1d. c) 2D mapping of the Gilbert damping \nextracted from  the 𝑛=1 LM resonance peak amplitude.  \n  \n   \n \n11 \n Next, we want to understand what gives rise to the PMA in WTe 2/YIG. We rule out the effect \ninduced by a modification of the gyromagnetic ratio by showing the resonance field shift across the \nWTe 2 edge does not depend on RF excitation frequency (See Fig. S3). We also exclude a strain induced \neffect given the absence of an epitaxial relation and the weakness of the van der Waals interaction \nbetween YIG and WTe 2. We further note that we can ignore the role of topological surface states  16 in \nour analysis; they are not relevant for our room temperature experiment since WTe 2 is a topological \nWeyl semimetal only below 100 K  17, 18. \nWe show how an analysis based on symmetry and the nature of the interfacial SOC , generalizing \nthe theory in Ref. 41, gives insight into the PMA observed in our experiment. This will also help us \nunderstand why the easy-axis anisotropy we observe in WTe 2/YIG is so different from the results of \nRef. 13, 15  on YIG interfaces with a dozen different metallic and semiconducting materials, all of which \nexhibit interface -induced easy-plane  anisotropy, as is predicted by theory  41.  \nYIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via \nantiferromagnetic (AFM) superexchange interactions. We focus on how interfacial SOC impacts AFM \nsuperexchange in YIG and show that it leads to a very specific form of the mag netic anisotropy that is \ngoverned by the direction of the effective B-field (see Supporting Information for a details).  \nBefore turning to WTe 2/YIG, it is useful to first consider the simpler case when the only broken \nsymmetry is the mirror plane defined b y the interface.  The abrupt change in lattice potential then results \nin an effective electric field that points normal to the interface, which in turn leads to an effective \nmagnetic field in the rest frame of the electron that couples to its spin. Since t he E-field points normal to    \n \n12 \n the interfacial plane in which the electron moves, the resulting B-field arising from SOC lies within  the \ninterfacial plane. As we show in the SI, this leads to a SOC -induced correction to AFM superexchange \nthat necessarily lead s to an easy-plane  anisotropy.  \nIn the case of WTe 2/YIG, however, when there are additional broken symmetries. Not only does \nthe interface break inversion  symmetry , but the crystal structure of WTe 2 itself breaks in -plane inversion  \nsymmetry . The electric field is now no longer normal to the interface, and the effective B-field arising \nfrom SOC necessarily has an out -of-plane component, as shown in Fig S5b in SI. Thus, we see why the \nlower symmetry of WTe 2/YIG can naturally result in an easy-axis or perpendicular magnetic anisotropy  \n(PMA);  see Supporting Information for details.  \nWe note that the lack of two -fold rotational symmetry in the ab plane in WTe 2 that plays a \ncritical role in our understanding of PMA in WTe 2/YIG, has also been pointed  out be crucial for the out -\nof-plane damping -like torque in WTe 2/Permalloy30. We note, however, that the out -of-plane damping -\nlike torque necessarily involves current flow in WTe 2, while the PMA is an equilibrium property of the \nsystem independent of current flow.  \nWe further demonstrate the interfacial  origin of the observed effect by studying the influence of \nWTe 2 thickness.  We show four  line-scans , labeled in Fig. 3b, across the edges of WTe 2 with different \nthickness es, ranging from 4. 7 nm to 44.8 nm . From these four line -scans, we extract  the 𝑛=1 LM \nresonance field 𝐻r,1 and the 𝑛=1 LM resonance signal amplitude  ∆𝐴. Figures S1a-d in the Supporting \nInformation  show the evolution of 𝐻r,1 and ∆𝐴 along the traces labeled correspondingly . The thickness \nof WTe 2 at each measurement location is later measured using atomic force microscop y. From these    \n \n13 \n line-scans , we choose the region s where the probe is far away from the edge of WTe 2 so that the \nmagnetic propert ies are uniform,  to obtain  spatial average s of 𝐻r,1 and ∆𝐴, which are denoted  \n𝐻̅r,1,YIG/hBN and ∆𝐴̅̅̅̅YIG/hBN in the YIG/hBN region , and 𝐻̅r,1,YIG/WTe2/hBN and ∆𝐴̅̅̅̅YIG/WTe2/hBN in the \nYIG/WTe 2/hBN region , respectively . We further extract the 𝑛=1 LM resonance field difference \nbetween two regions using ∆𝐻r,1=𝐻̅r,1,YIG/hBN−𝐻̅r,1,YIG/WTe2/hBN, as well as the Gilbert damping \ndifference using ∆𝛼=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ −1) as a function of the WTe 2 \nthickness . We note  that the hBN overlayer does not change the Gilbert damping in YIG . The \nsummarized results containing the data from both sample 1 and sample 2  are shown in Fig s. 4a and 4 b. \nRaw data from sample 2 can be found in Supporting Information  Fig. S 2. The thinnest WTe 2 acquired in \nthe experiment is 3.2nm from sample 2, which is approximately the thickness of  a quadruple -layer \nWTe 2.  \nFigures 4a and 4 b indicate  that both ∆𝐻r,1 and ∆𝛼 have  almost no WTe 2 thickness  dependence . \nThere is a  small sample -to-sample variation possibly  due to different YIG/WTe 2 interfacial quality . The \nchange of 𝑛=1 LM resonance field,  ∆𝐻r,1, is as large as ~38 Oe even when the WTe 2 thickness \napproaches the quadruple -layer thickness . This indicates that the modification of magnetic anisotropy is \ndue to the YIG/WTe 2 interfac ial interaction , with no bulk contribution. For the increase of Gilbert \ndamping ∆𝛼, no obvious thickness dependence is observed when comparing the data from the same \nsample. In sample 2, the Gilbert damping enhancement  due to the quadruple -layer WTe 2 has almost the \nsame value as the 50 nm thick WTe 2 flake, indicating that no thickness dependence of spin pumping  can \nbe resolved from our measurement. There are two possible interpretations of these results . First, if the    \n \n14 \n spin current injected into WTe 2 is mainly relaxed due to spin relaxation in the bulk, then the \nexperimental result is a demonstration of ultra -short spin diffusion length along the c axis38, smaller or \ncomparable to the thinnest WTe 2 flake (3.2 nm), employed in this experiment . It is much smaller than \nthe 8nm spin diffusio n length  in the  in-plane direction measured using inverse spin Hall effect  22. Note \nthat due to the chang e in mo bility and the metal -insulator transition  in few layer WTe 2 when its \nthickness reduces  42, the spin diffusion length approximated here could be inaccurate . Alternatively , it is \npossible that the spin relaxation is primarily due to the interfacial SOC induced by inversion symmetry \nbreaking at the interface and in the WTe 2 crystal lattice. In this case, the Gilbert damping enhancement \nwill have no WTe 2 thickness dependence.     \n \n15 \n  \nFig. 4  WTe 2 thickness dependence  of resonance field and damping enhancement . a) 𝐻r,1 in the \nYIG/hBN and YIG/WTe 2/hBN regions  are averaged  respectively to get 𝐻̅r,1,YIG and 𝐻̅r,1,YIG/WTe2, and \n∆𝐻r,1=𝐻̅r,1,YIG−𝐻̅r,1,YIG/WTe2. b) ∆𝛼 as a function of WTe 2 thickness , and  ∆𝛼=𝛼YIG/WTe2−𝛼YIG \nwhere 𝛼YIG is the Gilbert damping of bare YIG measured using broadband FMR for each sample , and \n𝛼YIG/WTe2=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ ).   \nIn conclusion, we have shown that the YIG/WTe 2 interface plays a critical  role in both interfacial \nmagnetic anisotropy and spin relaxation , making WTe 2 a promising material  in magnetic memory \n   \n \n16 \n application s. Combining the iPMA  created by WTe 2 with the out-of-plane spin orbit torque  generated by \nflowing a charge current along  the a axis of WTe 2, one can possibly achieve field -free switching of a \nPMA magnetic cell  for magnetic memory application s. It will improv e the  scalability , reduc e the  power \nconsumption  and increas e operation speed  of magnetic solid -state devices . Our result reveals new \npossibilities in selecting materials and designing spintronic devices. For example, one can consider other \nmaterials with low lattice symmetry and strong SOC to induce larger PMA type interfacial ani sotropy in \nFMIs. To achieve a fully PMA material, one could utilize thinner FMIs to magnify the role of iPMA. \nMoreover, interfacial SOC also plays an important role in  generat ing topologically protected magnetic \ntextures in the FMIs  43. These findings will motivate further research to reveal the fundamental  physics \narising at the  interface between FMIs and nonmagnetic materials.  \n \nData availability:  \nThe data generated by the present study are available from the corresponding author on request.  \nSupporting Information:  \nA description of raw data on WTe 2 thickness dependence, a FMRFM measurement on a second sample, \na FMRFM measurement at different RF frequency, a description of polarized Raman measurement \nresult, and a detailed illustration of impact of broken  mirror reflection symmetries on the magnetic \nanisotropy.  \nAckno wledgements:     \n \n17 \n This work was primarily supported by the Center for Emergent Materials: an NSF MRSEC under award \nnumber DMR -2011876 (GW, NV , YC, SG, FY , MR and PCH) . KW and TT acknowledge support from \nthe Elemental Strategy Initiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001) \nand JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).  DW, GC, CNL, and MB \nare supported by NSF under award DMR -2004801.  We gr atefully acknowledge N. Trivedi for insightful \ndiscussions. Fabrication and some characterization were performed in the Ohio State University \nNanoSystems Laboratory.  \n     \n \n18 \n Methods:  \nSample fabrication  \nOur YIG/WTe 2/hBN heterostructure was prepared by means of dry transfer and stacking 44. hBN crystals \nwere mechanically exfoliated under ambient conditions onto SiO 2/Si substrates (285 nm thick SiO 2). 20-\n40 nm thick hBN flakes were identified under an optical microscope and used for the capping lay er for \nthe stack. The hBN was picked up using a polymer -based dry transfer technique and then moved into an \nAr-filled glove box with oxygen and water level below 0.1 ppm. Flux -grown WTe 2 crystals 45 were \nexfoliated inside the glove box and flakes with different thicknesses were optically identified and \nquickly picked up with the capping hBN layer then transferred to the YIG substrate. Finally, we removed \nthe fully encapsulated sample from the glove b ox and performed the e -beam lithography and \nmetallization (Ni/Au) step for alignment in our ferromagnetic resonance force microscope (FMRFM).  \n \nPolarized Raman measurement  \nPolarized Raman spectra from the WTe 2 sample were collected using 633 nm excitation w avelength in \nan inVia Renishaw Raman microscope.  The sample was loaded onto the microscope stage and \npositioned in such a way that the long edge of the flake was aligned parallel to the laser polarization ( θ = \n0°). In this configuration, the incident illu mination is polarized vertically coming out of the laser and is \naligned with the long axis of the WTe 2 flake. The polarization of the incident laser was rotated from 0 to \n360° by 10° increments using a polarization rotator, while an analyzer was set to onl y allow vertically \npolarized light to enter the spectrometer.  Raman spectra were collected at each polarization for 3 \nacquisitions with a 20 s time per acquisition.  The laser power was set to 0.5 mW at the sample to avoid \nany damage by heating.  Followin g spectral collection, the (baseline corrected) integrated intensities \nunder each peak were calculated to make the contour plots and polar plots in Fig. 2c and 2d.  \n \nFMRFM measurement and signal fitting  \nOur FMRFM perform s local ly measures  FMR at room temper ature in vacuum. The cantilever has \nnatural frequency of ~18 KHz, spring constant of 0.2 N/m and Q factor of ~20000, resulting in force \ndetection sensitivity of  10-15 N/Hz1/2. The SmCo 5 magnetic particle attached on the cantilever has a \nmagnetic moment of ~4 nemu.  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B 2015,  92, (4), 041104.  \n \n     \n \n23 \n Enhancing  Perpendicular Magnetic Anisotropy in Garnet  Ferrimagnet \nby Interfacing with Few-Layer  WTe 2 \nGuanzhong Wu1*, Dongying Wang1, Nish chhal  Verma1, Rahul Rao2, Yang Cheng1, Side Guo1, \nGuixin Cao1, Kenji Watanabe3, Takashi Taniguchi4, Chun  Ning Lau1, Fengyuan Yang1,  \nMohit Randeria1, Marc Bockrath1, and P. Chris Hammel1 \n1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA  \n2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air \nForce Base, Dayton, OH, 45433, USA  \n3. Research Center for Functional Materials,  National Institute for Materials Science, 1 -1 Namiki, \nTsukuba 305 -0044, Japan  \n4. International Center for Materials Nanoarchitectonics , National Institute for Materials Science,  1-1 \nNamiki, Tsukuba 305 -0044, Japan  \n*wu.2314@osu.edu  \n     \n \n24 \n FMRFM  line-scan across edge of WTe 2 with different thickness  \n \nFig. S1 a-d, FMRFM line-scans along the traces 1~4 indicated in Fig. 3b respectively.  The gray shaded \narea in four figures are outlining the location of  WTe2 flake . 𝐻r,1 and ∆𝐴 at each position are derived by \nfitting the 𝑛=1 LM to a Lorentzian line shape. The thickness of WTe2 flake at each location are \nmeasured by atomic force microscope.  \n  \n   \n \n25 \n FMRFM measurement on Sample 2  \n \nFig. S 2 FMRFM measurement on sample 2. a, The optical picture of the YIG/WTe 2/hBN \nheterostructure. b, 2D mapping of the 𝑛=1 LM resonance field in the black dash line circled area.  c, \n2D mapping of the Gilbert damping extracted from 𝑛=1 LM resonance peak amplitude in the black \ndash line circled area. d, FMRFM line scan along the trace indicated by the solid black line in Fig. S 2a. \nA constant background is subtracted to show only the signal from the LMs resonance. e, Fine scan \nzoomed in o n the quadruple layer WTe 2 stripe area  \n  \n   \n \n26 \n FMRFMR measurement at 4 GHz  \n \nFig. S 3 FMRFM measurement across WTe 2 edge at 4 GHz. FMRFM line scan is measured at 4 GHz \nacross the WTe 2 flake edge. The shift of the resonance field 𝐻r,1 is 36 Oe, similar to the 𝐻r,1 shift \nmeasured at 2 GHz. This result excludes the possibility that the resonance field shift arises from \nmodification of the gyromagnetic ratio.  \n \n  \n   \n \n27 \n Polarized Raman measurement  \n \nFig. S 4 Polarized Raman measurement. As shown by the red curve, the Raman  spectrum taken on \nGGG/YIG/WTe 2/hBN heterostructure contains more peaks than WTe 2. The Raman spectrum taken in \nthe GGG/YIG/hBN area identifies the peaks arising from the  substrate GGG/YIG or top hBN \nencapsulation layer. By subtracting the Raman spectrum in the GGG/YIG/hBN area, the Raman spectra \nfrom WTe 2 layer are extracted and plotted in Fig. 2d. The black dash line are the markers indicating the \nRaman peaks of WTe 2  \n   \n \n28 \n Impact of  broken mirror reflection  symmetr ies on the  magnetic anisotropy  \nWe describe theo retical constraints on the interface -induced  magnetic anisotropy in the WTe 2/YIG \nbilayer. We first show that symmetry arguments alone do not provide strong constraints on the anisotropy \ntensor, given that we are dealing with an interface between two crystalline materials at an arbitrary \norientation with respect to each other . We then present qualitative arguments, based on the interfacia l spin -\norbit coupling, that give insight into the magnetic anisotropy in WTe 2/YIG. This helps us understand why \nthe easy-axis anisotropy that we observe  in WTe 2/YIG differ s from the results  of Lee et al. [1] on YIG \ninterfaces with a dozen different metallic and semiconducting materials , all of which exhibit interface -\ninduced easy-plane anisotropy  as predicted  by theory  [2]. \nOn general grounds, the  anisotropy (free) energy can be written as   \nℱ𝑎𝑛𝑖𝑠= ∑ 𝐾𝑎𝑏𝑎,𝑏 𝑚𝑎 𝑚𝑏,           (S1) \nwhere a and b take on values x,y,z. We focus here on the leading term, quadratic in the magnetization, and \nignore higher order anisotropy terms like (mx4+my4+mz4) or (mx2my2+my2mz2+mz 2mx2). The form of \n 𝐾𝑎𝑏=  𝐾𝑏𝑎 is constrained by symmetry. Let us consider three cases , going from the most symmetric to \nthe least .  \nCase I: The only broken symmetry is interfacial inversion (z →  - z), which  is relevant for the \nexperiments of Ref. [1]. The magnetization is an axial vector (or pseudovector) that transforms under  \nrotation s like a vector but is unchanged  under inversion . Thus (𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧)→ (𝑚𝑥 ,− 𝑚𝑦 ,−𝑚𝑧) under \nreflection in  a mirror plane with normal 𝑥̂. Using reflection symmetry in mirror planes normal  to 𝑥̂ and to    \n \n29 \n 𝑦̂ , we can see that all off -diagonal components of  𝐾𝑎𝑏  vanish. Further, f our-fold rotational symmetry \nabout the 𝑧̂ axis shows that  𝐾𝑥𝑥=  𝐾𝑦𝑦. Using mx2+my2+mz2=1, we write  𝐾𝑥𝑥(m𝐱2+m𝐲2) in terms \nof m𝑧2, and d efining 𝐾𝑢=  (𝐾𝑥𝑥−  𝐾𝑧𝑧), we obtain  \nℱ𝑎𝑛𝑖𝑠= −  𝐾𝑢 m𝑧2.         (S2) \nThis symmetry analysis only constrains the form  of the anisotropy energy, but not the sign of  𝐾𝑢. We will \ngive below a simple microscopic argument [2] that shows that   𝐾𝑢<0  (easy plane)  for Case I.  \nCase II: In addition to broken interfacial inversion (z → - z), let u s also break reflection symmetry \nin the plane normal to 𝑥̂. This would be the case if the crystalline axes of WTe 2 were aligned with YIG.  \nThis also breaks four-fold rotational symmetry about 𝑧̂, so that   𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. However, we  can still use \nreflection symmetry in the plane normal to 𝑦̂ to conclude that  𝐾𝑥𝑦=  𝐾𝑦𝑧=0. Thus  we find that  \n      K = (𝐾𝑥𝑥0 𝐾𝑥𝑧\n0 𝐾𝑦𝑦0\n 𝐾𝑥𝑧0 𝐾𝑧𝑧)                                                               (S3) \nCase III: When the crystalline axes of WTe 2 are not aligned with YIG, which is the experimentally \nrelevant case, all mirror reflection and rotation symmetries are broken. Then there are no symmetr y \nconstrain ts on   𝐾𝑎𝑏 and all six components of this symmetric tensor are in general non -zero.      \nLet us now see how, despite the lack of general symmetry -based constrai nts, we can still get some \nqualitative insight about the form of the anisotropy from simple microscopic considerations informed by \nsymmetry. YIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via    \n \n30 \n antiferromagnetic (AFM) superexch ange interactions. We thus focus on how interfacial spin -orbit \ncoupling (SOC) impacts AFM superexchange.   \nThe broken symmetry at the interface leads to an electric field ℇ=−𝛁𝑉(𝒓), whose direction will \nbe discussed in detail below for three cases. This in turn produces a magnetic field in the rest frame of the \nelectron  which underlies SOC. As the electron moves along  𝐫̂ij from site i to j, it experiences an  SOC field  \nin the direction 𝐝̂ij  which is determined by  ℇ ×𝐫̂ij . The SOC Hamiltonian  is thus given by \n−𝑖𝜆∑𝑐𝐢𝛼†(𝐝̂ij∙𝝈𝛼𝛽)𝑐𝐣𝛃 𝛼𝛽 . Including the effect of this term in addition to the usual hopping t and Hubbard \nU in the standard strong coupling expansion calculation leads to the Hamiltonian  \n                 ℋex=J∑𝐒i∙𝐒j <𝐢,𝐣>+D∑𝐝̂ij∙𝐒i×𝐒j <𝐢,𝐣> +K0∑(𝐝̂ij∙𝐒i)(𝐝̂ij∙𝐒j). <𝐢,𝐣>                  (S4) \nHere  the spin  𝐒i   at site i is coupled to its neighbors via  the AFM superexchange  𝐽 ~𝑡2\n𝑈  and the \nDzyaloshinskii -Moriya interaction (DMI)  𝐷 ~𝑡𝜆\n𝑈. The K0 term will be the focus of our attention belo w as \nit leads  to magnetic anisotropy.  We note that the general form of ℋex is in fact substantially independent \n[2] of the microscopic mechanism and very similar results are obtained not only for superexchange but \nalso for Zener double exchange and RKKY interactions.  \nCase I: Let us again return to the simplest case with broken interfacial inversion (z → - z). This \nleads to an electric field ℇ=−𝛁𝑉(𝒓)  along ẑ , the normal to the interface.  The SOC magnetic field \ndirection is then given by  𝐝̂ij= ẑ ×𝐫̂ij ; see Figure S4(a). This is the well -known Rashba SOC at \ninterfaces . We note in passing that 𝐝̂ij is antis ymmetric under the interchange of i and j, and thus  leads to \na DMI term where 𝐒i×𝐒j is also antisymmetric.     \n \n31 \n  \n \nFig. S 5 Symmetry based selection of magneto -crystalline anisotropy. Interfacial SOC originates from \nan effective Electric field ℇ=−𝛁𝑉(𝒓) whose direction is determined by the broken mirror planes in the \nsystem. This electric field leads to  spin-orbit coupling (SOC), with the 𝐝̂ij= ℇ̂ ×𝐫̂ij, the direction of the \nSOC magnetic  field. Note that the direction of the electron hop 𝐫̂ij lies in the xy plane of the interface. As \nshown in the text 𝐝̂ij controls the interface -induced magnetic anisotropy. (a) When only surface inversion \nis broken, 𝐝𝐢𝐣 is constrained to lie in the int erface and interfacial SOC leads to easy -plane anisotropy. (b) \nIf there are other broken mirror planes, the 𝐝𝐢𝐣 must lie outside the interfacial plane. This can lead to a \nperpendicular magnetic anisotropy in systems like YIG/WTe 2 bilayers.  \n \nWe see that in Case I,  𝐝̂ij lies in the plane  of the interface, and the third term in eq. (S 4) then takes \nthe form K0∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+���𝑦) 𝐢  for a square lattice . To make the connection with  magnetic anisotropy, \nwe look at a continuum approximation with a s lowly  varying magnetization 𝐦(𝐫). We make a Taylor \nexpan sion of  𝐒r in terms of its value at 𝒓, denoted by  𝐦(𝐫), and its spatial derivatives . The exchange and \nDMI terms involve gradients of 𝐦(𝐫), but we focus here on local terms that  do not involve derivatives  to \n   \n \n32 \n understand the magnetic anisotropy . The leading term is + K0(m𝐱2+m𝐲2)  which can be rewritten as \n– K0 m𝑧2 using the fact that mx2+my2+mz2=1 at each 𝒓. Thus, we may identify K0 with the anisotropy  \nK𝑢 defined in eq. (S2).  \nThe microscopic analysis leads to the result K0= − 𝜆2\n𝑈   < 0 and this explains the easy-plane \nanisotropy  arising Rashba SOC at the interface . The easy-plane nature of the anisotropy  is in fact a general \nfeature of various microscopic models as emphasized  in Ref. [2]. We note however that these author s use d \nthe opposite sign convention for anisotropies  from the one we use here . The easy plane vs. easy -axis \ncharacter is , of course, independent of sign conventions.  The FMR experiments of Ref. [1] have seen the \ninterface -induced easy-plane anisotropy  predicted by the theory in a YIG interfaces with several  metallic \nand semiconducting materials . \nThe key difference between the YIG/WTe 2 bilayer studied here and systems  studied earlier [1] is \nthat WTe 2 has a broken mirror plane  (the ac plane ) as shown in  Fig. 1(a)  of the paper . We now  look at the \neffect of this lower symmetry on the microscopic analysis.  \nCase II: Let us break reflection symmetry in the plane normal to 𝑥̂ in addition to broken interfacial \ninversion. We choose x̂ parallel to the b axis, ŷ parallel to a, and ẑ parallel to c. Reflection symmetry in \nthe ŷ mirror plane constrains the  electric field ℇ =−𝛁𝑉(𝒓) to lie in the xz plane, at an angle 𝜃 from the \nz-axis as shown in Fig. S 5(b). Thus  \n𝐝ij=(sin𝜃𝑥̂+cos𝜃𝑧̂)×𝐫̂ij     (S5)    \n \n33 \n where 𝐫̂ij is a vector in the interface  (xy plane ) and 0≤𝜃≤𝜋. Using eq. (S5), we may rewrite  the last \nterm in the  Hamiltonian (S4) as \n K0 sin2𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑧)\n𝐢+K0 cos2𝜃∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦)\n𝐢\n−K0sin𝜃cos𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑥+S𝐢𝑥𝑆𝐢+𝑦𝑧)\n𝐢  \nAs before, we  make  a continuum approximation with a smoothly varying 𝐦(𝐫) and focus only on the \nlocal terms, without gradients, to obtain the magnetic anisotropy . We find that the leading order \ncontribution to anisotropy is −K0cos2𝜃m𝑧2+K0sin2𝜃 mzmx. This analysis correctly captures the non -\nzero K𝑥𝑧 expected on general grounds; see eq. (S3). We did not include here , for simplicity, the effects of \nbroken four -fold rotation that would have led to  𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. \nCase III: When we lose all mirror symmetries, the case relevant to the YIG/WTe 2 experiment, the \nelectric field ℇ =−𝛁𝑉(𝒓) will point in a general direction specified by 0≤𝜃≤𝜋 and 0≤𝜑≤2𝜋, and \nthere will be no symmetry constraints on the anisotropy tensor  𝐾𝑎𝑏. \n Let us conclude by highlighting the  key qualitative difference  between Case I  on the one hand and \nCases II and III  on the other . In Case I, the only broken symmetry is interfaci al inversion (z → - z). Then \nsymmetry constrains the  𝐝̂ij, the direction of the SOC B-field,  to lie in the plane  of the interface  and this \nleads to easy -plane anisotropy as described above. In Cases II and III, there are other additional broken \nmirror planes, and this leads to the 𝐝̂ij  vector being pulled out of the plane of the interface. This \nimmediately leads to the possibility of an easy -axis like character to the anisotropy, although in the general \ncase one has a non -trivial anisotropy tensor  𝐾𝑎𝑏.     \n \n34 \n Reference  \n[1] Lee, A. J.; Ahmed, A. S.; McCullian, B. A.; Guo, S. D.; Zhu, M. L.; Yu, S. S.; Woodward, P. M.; \nHwang, J.; Hammel, P. C.; Yang, F. Y . Interfacial Rashba -Effect-Induced Anisotropy in Nonmagnetic -\nMaterial -Ferrimagnetic -Insulator Bilayers. Phys. Rev. Lett. 2020,  124, (25), 257202.  \n[2] Banerjee, S.; Rowland, J.; Erten, O.; Randeria, M. Enhanced Stability of Skyrmions in Two -\nDimensional Chiral Magnets with Rashba Spin -Orbit Coupling. Physical Review X 2014,  4, (3), 031045.  \n \n "    },    {        "title": "2202.05295v1.Non_stationary_Anderson_acceleration_with_optimized_damping.pdf",        "content": "arXiv:2202.05295v1  [math.NA]  10 Feb 2022Non-stationary Anderson acceleration with optimized\ndamping⋆\nKewang Chena,b,∗, Cornelis Vuikb\naCollege of Mathematics and Statistics, Nanjing University of Information Science and\nTechnology, Nanjing, 210044, China.\nbDelft Institute of Applied Mathematics, Delft University o f Technology, Delft, 2628XE, the\nNetherlands.\nAbstract\nAnderson acceleration (AA) has a long history of use and a strong r ecent inter-\nest due to its potential ability to dramatically improve the linear conve rgence\nof the fixed-point iteration. Most authors are simply using and analy zing the\nstationary version of Anderson acceleration (sAA) with a constan t damping\nfactor or without damping. Little attention has been paid to nonsta tionary\nalgorithms. However, damping can be useful and is sometimes crucia l for simu-\nlations in which the underlying fixed-point operator is not globally cont ractive.\nThe role of this damping factor has not been fully understood. In th e present\nwork, we consider the non-stationary Anderson acceleration algo rithm with op-\ntimized damping (AAoptD) in each iteration to further speed up linear and\nnonlinear iterations by applying one extra inexpensive optimization. W e an-\nalyze this procedure and develop an efficient and inexpensive implemen tation\nscheme. We also show that, compared with the stationary Anderso n accelera-\ntion with fixed window size sAA(m), optimizing the damping factors is related\nto dynamically packaging sAA(m) andsAA(1) in each iteration (alternating\nwindow size mis another direction of producing non-stationary AA). More-\n⋆Funding: This work was partially supported by the National Natural Sc ience Foundation\nof China [grant number 12001287]; the Startup Foundation fo r Introducing Talent of Nanjing\nUniversity of Information Science and Technology [grant nu mber 2019r106]\n∗Corresponding author\nEmail addresses: kwchen@nuist.edu.cn (Kewang Chen), c.vuik@tudelft.nl (Cornelis\nVuik)\nURL:https://homepage.tudelft.nl/d2b4e/ (Cornelis Vuik)\nPreprint submitted to Journal of Computational and Applied Mathematics.February 14, 2022over, we show by extensive numerical experiments that, in the cas e a larger\nwindow size is needed, the proposed non-stationary Anderson acc eleration with\noptimized damping procedure often converges much faster than s tationary AA\nwith constant damping or without damping. When the window size is ver y\nsmall (m≤3 was typically used, especially in the early days of application),\nAAoptD and AA are comparable. Lastly, we observed that when the system is\noverdamped (i.e. the damping factor is close to the lower bound zero ), incon-\nsistency may occur. So there is some trade-off between stability an d speed of\nconvergence. We successfully solve this problem by further restr icting damping\nfactors bound away from zero.\nKeywords: Anderson acceleration, fixed-point iteration, optimal damping.\n2010 MSC: 65H10, 65F10\n1. Introduction\nIn this part, we first give a literature review on Anderson Accelerat ion\nmethod. Thenwediscussourmainmotivationsandthe structurefo rthe present\npaper. To begin with, let us consider the nonlinearaccelerationfor t he following\ngeneral fixed-point problem\nx=g(x), g:Rn→Rn\nor its related nonlinear equations problem\nf(x) =x−g(x) = 0.\nThe associated basical fixed-point iteration is given in Algorithm 1.\nAlgorithm 1 Picard iteration\nGiven:x0.\nfork= 0,1,2,···do\nSetxk+1=g(xk).\nend for\n2The main concern related to this basic fixed-point iteration is that th e it-\nerates may not converge or may converge extremely slowly (only line ar conver-\ngent). Therefore, various acceleration methods are proposed t o alleviate this\nslow convergence problem. Among these algorithms, one popular ac celeration\nprocedure is called the Anderson acceleration method [1]. For the ab ove basic\nPicard iteration, the usual general form of Anderson acceleratio n with damping\nis given in Algorithm 2. In the above algorithm, fkis the residual for the kth\nAlgorithm 2 Anderson acceleration: AA(m)\nGiven:x0andm≥1.\nSet:x1=g(x0).\nfork= 0,1,2,···do\nSet:mk= min{m,k}.\nSet:Fk= (fk−mk,···,fk), where fi=g(xi)−xi.\nDetermine: α(k)=/parenleftig\nα(k)\n0,···,α(k)\nmk/parenrightigT\nthat solves\nmin\nα=(α0,···,αmk)T/bardblFkα/bardbl2s. t.mk/summationdisplay\ni=0αi= 1.\nSet:xk+1= (1−βk)mk/summationdisplay\ni=0α(k)\nixk−mk+i+βkmk/summationdisplay\ni=0α(k)\nig(xk−mk+i).\nend for\niteration; mis the window size which indicates how many history residuals will\nbe used in the algorithm. The value of mis typically no larger than 3 in the\nearly days of applications and now this value could be as large as up to 1 00,\nsee [2]. It is usually a fixed number during the procedure, varying mcan also\nmake the algorithm to be non-stationary. We will come back to this po int in\nsection Section 2; βk∈(0,1] is a damping factor (or a relaxation parameter) at\nkth iteration. We have, for a fixed window size m:\nβk=\n\n1, no damping,\nβ,(a constant independent of k) stationary AA,\nβk,(depending on k) non-stationary AA.\nThe constrained optimization problem can also be formulated as an eq uivalent\n3unconstrained least-squares problem [3, 4]:\nmin\n(ω1,···,ωmk)T/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublefk+mk/summationdisplay\ni=1ωi(fk−i−fk)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2(1)\nOne can easily recover the original problem by setting\nω0= 1−mk/summationdisplay\ni=1ωi.\nThis formulation of the linear least-squares problem is not optimal fo r imple-\nmentation, we will discuss this in more detail in Section 3.\nAnderson acceleration method dates back to the 1960s. In 1962, Anderson\n[1] developeda techniqueforacceleratingthe convergenceofthe Picarditeration\nassociated with a fixed-point problem which is called Extrapolation Algo rithm.\nThis technique is now called Anderson Acceleration (AA) in the applied m athe-\nmatics community and Anderson Mixing in the physics and chemistry co mmu-\nnities. This method is “essentially” (or nearly) similar to the nonlinear G MRES\nmethod or Krylov acceleration [5, 6, 7, 8] and the direct inversion on the itera-\ntive subspace method (DIIS) [9, 10, 11]. And it is also in a broad categ ory with\nmethods based on quasi-Newton updating [12, 13, 14, 15, 16]. Howe ver, unlike\nNewton-like methods, AA does not require the computation or appr oximation\nof Jacobians or Jacobian-vector products which could be an advan tage.\nAlthough the Anderson acceleration method has been around for d ecades,\nconvergence analysis has been reported in the literature only rece ntly. Fang\nand Saad [14] had clarified a remarkable relationship of AA to quasi-Ne wton\nmethods and extended it to define a broader Anderson family metho d. Later,\nWalker and Ni [17] showed that, on linear problems, AA without trunc ation\nis “essentially equivalent” in a certain sense to the GMRES method. Fo r the\nlinear case, Toth and Kelley [3] first proved the stationary version o f AA (sAA)\nwithout damping is locally r-linearly convergent if the fixed point map is a\ncontractionand the coefficients in the linear combination remain boun ded. This\nwork was later extended by Evens et al. [18] to AA with damping and t he\nauthors proved the new convergence rate is θk((1−βk−1)+βk−1κ), where κis\n4the Lipschitz constant for the function g(x) andθkis the ratio quantifying the\nconvergence gain provided by AA in step k. However, it is not clear how θk\nmay be evaluated or bounded in practice and how it may translate to im proved\nasymptotic convergence behavior in general. In 2019, Pollock et al. [19] applied\nsAA to the Picard iteration for solving steady incompressible Navier– Stokes\nequations(NSE) andprovedthat the accelerationimprovesthe co nvergencerate\nofthe Picarditeration. Then, De Sterck[20] extended the resultt o moregeneral\nfixed-point iteration x=g(x), given knowledge of the spectrum of g′(x) at\nfixed-point x∗and Wang et al. [21] extended the result to study the asymptotic\nlinear convergence speed of sAA applied to Alternating Direction Met hod of\nMultipliers (ADMM) method. Sharper local convergence results of A A remain\na hot research topic in this area. More recently, Zhang et al. [22] pr oved a\nglobal convergent result of type-I Anderson acceleration for no nsmooth fixed-\npointiterationswithoutresortingtolinesearchoranyfurtherass umptionsother\nthan nonexpansiveness. For more related results about Anderso n acceleration\nand its applications, we refer the interested readers to [2, 23, 24, 25, 26, 27, 28]\nand references therein.\nAs mentioned above, the local convergence rate θk((1−βk−1) +βk−1κ) at\nstagekis closely related to the damping factor βk−1. However, questions like\nhow to choose those damping values in each iteration [2] and how it will a ffect\nthe global convergence of the algorithm have not been deeply stud ied. Besides,\nAA is often combined with globalization methods to safeguard against erratic\nconvergence away from a fixed point by using damping. One similar idea in\nthe optimization context for nonlinear GMRES is to use line search str ategies\n[29]. This is an important strategy but not yet fully explored in the liter ature.\nMoreover, the early days of Anderson Mixing method (the 1980s, f or electronic\nstructure calculations) initially dictated the window size m≤3 due to the\nstorage limitations and costly gevaluations involving large N. However, in\nrecent years and a broad range of contexts, the window size mranging from\n20 to 100 has also been considered by many authors. For example, W alker\nand Ni [17] used m= 50 in solving the nonlinear Bratu problem. A natural\n5question will be should we try to further steep up Anderson acceler ationmethod\nor try to use a larger size of the window? No such comparison results have been\nreported. Motivated by the above works, in this paper, we propos e, analyze\nand numerically study non-stationary Anderson acceleration with o ptimized\ndamping to solve fixed-point problems. The goal of this paper is to ex plore the\nrole of damping factors in non-stationary Anderson acceleration.\nThe paper is organized as follows. Our new algorithms and analysis are in\nSection 2, the implementation ofthe new algorithmis in Section 3, expe rimental\nresults and discussion are in Section 4. Conclusions follow in Section 5.\n2. Anderson acceleration with optimized dampings\nIn this section, we focus on developing the algorithm for Anderson a cceler-\nation with optimized dampings at each iteration and studying its conve rgence\nrate explicitly.\nxk+1= (1−βk)mk/summationdisplay\ni=0α(k)\nixk−mk+i+βkmk/summationdisplay\ni=0α(k)\nig(xk−mk+i)\n=mk/summationdisplay\ni=0α(k)\nixk−mk+i+βk/parenleftiggmk/summationdisplay\ni=0α(k)\nig(xk−mk+i)−mk/summationdisplay\ni=0α(k)\nixk−mk+i/parenrightigg\n.(2)\nDefine the following averagesgiven by the solution αkto the optimization prob-\nlem by\nxα\nk=mk/summationdisplay\ni=0α(k)\nixk−mk+i,˜xα\nk=mk/summationdisplay\ni=0α(k)\nig(xk−mk+i). (3)\nThen (2) becomes\nxk+1=xα\nk+βk(˜xα\nk−xα\nk). (4)\nA natural way to choose “best” βkat this stage is that choosing βksuch that\nxk+1gives a minimal residual. This is similar to the original idea of Anderson\naccelerationwithwindowsizeequaltoone. Sowejustneedtosolvet hefollowing\nunconstrained optimization problem:\nmin\nβk/bardblxk+1−g(xk+1)/bardbl2= min\nβk/bardblxα\nk+βk(˜xα\nk−xα\nk)−g(xα\nk+βk(˜xα\nk−xα\nk))/bardbl2.(5)\n6Noting the fact that\ng(xα\nk+βk(˜xα\nk−xα\nk))≈g(xα\nk)+βk∂g\n∂x/vextendsingle/vextendsingle/vextendsingle\nxα\nk(˜xα\nk−xα\nk)\n≈g(xα\nk)+βk(g(˜xα\nk)−g(xα\nk)). (6)\nTherefore, (5) becomes\nmin\nβk/bardblxk+1−g(xk+1)/bardbl2\n= min\nβk/bardblxα\nk+βk(˜xα\nk−xα\nk)−g(xα\nk+βk(˜xα\nk−xα\nk))/bardbl2\n≈min\nβk/bardblxα\nk+βk(˜xα\nk−xα\nk)−[g(xα\nk)+βk(g(˜xα\nk)−g(xα\nk))]/bardbl2\n≈min\nβk/bardbl(xα\nk−g(xα\nk))−βk[(g(˜xα\nk)−g(xα\nk))−(˜xα\nk−xα\nk)]/bardbl2.(7)\nThus, we just need to calculate the projection\nβk=/vextendsingle/vextendsingle/vextendsingle(xα\nk−g(xα\nk))·[(xα\nk−g(xα\nk))−(˜xα\nk−g(˜xα\nk))]\n/bardbl[(xα\nk−g(xα\nk))−(˜xα\nk−g(˜xα\nk))]/bardbl2/vextendsingle/vextendsingle/vextendsingle. (8)\nSet\nrp= (xα\nk−g(xα\nk)), rq= (˜xα\nk−g(˜xα\nk)),\nwe have\nβk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(rp−rq)Trp\n/bardblrp−rq/bardbl2/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (9)\nWe will discuss how much work is needed to calculate this βkin Section 3.\nFinally, our analysisleads to the followingnon-stationaryAnderson a cceleration\nalgorithm with optimized damping: AAoptD(m).\nRemark 2.1. As mentioned in Section 1, changing the window size mat each\niteration can also make a stationary Anderson acceleration to be non-stationary.\nComparing withthe stationary Anderson acceleration with fi xedwindow sAA(m),\nour proposed nonstationary procedure ( AAoptD(m)) of choosing optimal βkis\nsomewhat related to packaging sAA(m)andsAA(1)in each iteration in a cheap\nway. Combining sAA(m)withsAA(1)can provide really good outcomes, espe-\ncially in the case when larger mis needed. We will discuss this in detail for the\nnumerical results in Section 4.\n7Algorithm 3 Anderson acceleration with optimized dampings: AAoptD(m)\nGiven:x0andm≥1.\nSet:x1=g(x0).\nfork= 0,1,2,···do\nSet:mk= min{m,k}.\nSet:Fk= (fk−mk,···,fk), where fi=g(xi)−xi.\nDetermine: α(k)=/parenleftig\nα(k\n0,···,α(k)\nmk/parenrightigT\nthat solves\nmin\nα=(α0,···,αmk)T/bardblFkα/bardbl2s. t.mk/summationdisplay\ni=0αi= 1.\nSet:xα\nk=mk/summationdisplay\ni=0α(k)\nixk−mk+i,˜xα\nk=mk/summationdisplay\ni=0α(k)\nig(xk−mk+i).\nSet:rp= (xα\nk−g(xα\nk)), rq= (˜xα\nk−g(˜xα\nk)).\nSet: βk=(rp−rq)Trp\n/bardblrp−rq/bardbl2.\nSet:xk+1=xα\nk+βk(˜xα\nk−xα\nk).\nend for\nRemark 2.2. Here this optimized damping step is a “local optimal” strate gy at\nkth iteration. It usually will speed up the convergence rate c ompared with an\nundamped one, but not always. Because in (k+1)th iteration, it uses a combi-\nnation of all previous m history information. Moreover, whe nβkis very close\nto zero, the system is over-damped, which, sometimes, may al so slow down the\nconvergence speed. We may need to further modify our βk. See more discussion\nin our numerical results in Section 4.\nLastly, we summarize the convergence results with damping in Theor em 2.1.\nThe proof of this theorem can be found in [18].\nTheorem 2.1. [18] Assume that g:Rn→Rnis uniformly Lipschitz con-\ntinuously differentiable and there exists κ∈(0,1)such that/bardblg(y)−g(x)/bardbl2≤\nκ/bardbly−x/bardbl2for allx,y∈Rn. Suppose also that ∃Mandǫ >0such that for all\nk > m,/summationtextm−1\ni=0|αi|< Mand|αm|≥ǫ. Then\n/bardblf(xk+1)/bardbl2≤θk+1[(1−βk)+κβk]/bardblf(xk)/bardbl2+m/summationdisplay\ni=0O(/bardblf(xk−m+i)/bardbl2\n2),(10)\n8where\nθk+1=/bardbl/summationtextm\ni=0αif(xk−m+i)/bardbl2\n/bardblf(xk)/bardbl2.\n3. Implementation\nForimplementation, wemainlyfollowthepathin[4]andmodifyitasneed ed.\nWe first briefly review the implementation of AA without damping. Then we\nfocus on how to implement the optimized damping problem efficiently and ac-\ncurately.\nThe constrained linear least-squares problem in Algorithm 2 can be so lved\nin many ways. Here we rewrite it into an equivalent unconstrained for m which\ncan be solved efficiently by using QR factorizations. We define ∆ fi=fi+1−fi\nfor each iand setFk= (∆fk−mk,···,∆fk−1), then the least-squares problem\nis equivalent to\nmin\nγ=(γ0,···,γmk−1)T/bardblfk−Fkγ/bardbl2,\nwhereαandγare related by α0=γ0,αi=γi−γi−1for 1≤i≤mk−1, and\nαmk= 1−γmk−1.We assumeFhas a thin QRdecomposition i.e., Fk=QkRk\nwithQk∈Rn×mkandRk∈Rmk×mk, forwhichthesolutionoftheleast-squares\nproblem is obtained by solving the mk×mktriangular system Rkγ=QT\nkfk.\nAs the algorithm proceeds, the successive least-squares problem s can be solved\nefficiently by updating the factors in the decomposition.\nAssume that γk= (γk\n0,···,γk\nmk−1)Tis the solution to the above modified\nform of Anderson acceleration, we have\nxk+1=g(xk)−mk−1/summationdisplay\ni=0γk\ni[g(xk−mk+i+1)−g(xk−mk+i)] =g(xk)−Gkγk,\nwhereGk= (∆ggk−mk,···,∆gk−1) with ∆ gi=g(xi+1−g(xi)) for each i. For\nAnderson acceleration with damping\nxk+1= (1−βk)mk/summationdisplay\ni=0α(k)\nixk−mk+i+βkmk/summationdisplay\ni=0α(k)\nig(xk−mk+i)\n=mk/summationdisplay\ni=0α(k)\nixk−mk+i+βk/parenleftiggmk/summationdisplay\ni=0α(k)\nig(xk−mk+i)−mk/summationdisplay\ni=0α(k)\nixk−mk+i/parenrightigg\n.\n9Follow the idea in [4], we have\nmk/summationdisplay\ni=0α(k)\nig(xk−mk+i) =g(xk)−Gkγk, (11)\nmk/summationdisplay\ni=0α(k)\nixk−mk+i=/parenleftbig\ng(xk)−Gkγk/parenrightbig\n−/parenleftbig\nfk−Fkγk/parenrightbig\n. (12)\nThen this can be achieved equivalently using the following strategy:\nStep 1: Compute the undamped iterate xk+1=g(xk)−Gkγk.\nStep 2: Update xk+1again by\nxk+1←xk+1−(1−βk)/parenleftbig\nfk−QRγk/parenrightbig\n.\nNow we talk about how to efficiently calculate βkas described in Algorithm 3.\nTaking benefit of the QR decomposition in the first optimization proble m and\nnoting (11) and (12), we have\n˜xα\nk=mk/summationdisplay\ni=0α(k)\nig(xk−mk+i) =g(xk)−Gkγk,\nxα\nk=mk/summationdisplay\ni=0α(k)\nixk−mk+i= ˜xα\nk−/parenleftbig\nfk−Fkγk/parenrightbig\n.\nThen we could calculate optimized βkby doing two extra function evaluations\nand two dot products, which are not very expensive:\nrp= (xα\nk−g(xα\nk)), rq= (˜xα\nk−g(˜xα\nk)), βk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(rp−rq)Trp\n/bardblrp−rq/bardbl2/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nIn practice, when xkis very close to the fixed-point x∗, scientific computing\nerrors may arise in calculating these two high dimension vectors rpandrp−rq.\nThus we normalize these two vectors first, then calculate βkby simply doing a\ndot product.\n4. Experimental results and discussion\nInthissection,wenumericallycomparetheperformanceofthisnon -stationary\nAAoptD with sAA (with constant damping or without damping). The fir st part\n10containsexamples where largerwindow sizes mareneeded in orderto accelerate\nthe iteration. The second part consists of some examples where sm all window\nsizes are working very well. All these experiments are done in MATLAB 2021b\nenvironment. MATLAB codes are available upon request to the auth ors.\nThis first example is from Walker and Ni’s [17] paper, where a stationar y\nAnderson acceleration with window size m= 50 is used to solve the Bratu\nproblem. This problem has a long history, we refer the reader to Glow inski et\nal. [30] and Pernice and Walker [31], and the references in those pape rs. It is\nnot a difficult problem for Newton-like solvers.\nProblem 4.1. The Bratu problem. The Bratu problem is a nonlinear PDE\nboundary value problem as follows:\n∆u+λ eu= 0, in D = [0,1]×[0,1],\nu= 0, on ∂D.\nIn this experiment, we used a centered-difference discretization o n a 32×32,\n64×64and 128×128grid, respectively. We take λ= 6in the Bratuproblemand\nuse the zero initial approximate solution in all cases. We also applied pr econ-\nditioning such that the basic Picard iteration still works. The precon ditioning\nmatrix that we used here is the diagonal inverse of the matrix A, whereAis a\nmatrix for the discrete Laplace operator.\nThe resultsareshowninthe followingfigures. InFig.1, weplot there sultsof\napplying AA(m) andAAoptD(m) to acceleratePicarditeration with m= 5 and\nm= 10 on a grid of 32 ×32. As we see from the picture, AA(5) andAA(10) does\nnot accelerate the convergence speed very much. AAoptD(5) andAAoptD(10)\nperform much better than AA(5) andAA(10). However, we also notice that\nthere are some inconsistencies and stagnations in AAoptD(m). Thus we go\nfurther to plot the βkvalues that are used in each iteration, see Fig. 2. From\nFig. 2 we see that: for AAoptD(10), some optimized damping factors are below\n0.3(see the dashedline). As we know, the damping factor βk∈(0,1] andβk= 1\nmeans no damping. Thus small βkmay cause an over-damping phenomenon,\n110 20 40 60 80 100 120 140 160 180 200\niteration10-910-810-710-610-510-410-310-210-1residual\nAA(5)\nAA(10)\nAAoptD(5)\nAAoptD(10)\nFigure 1: Compare AA and AoptD for solving nonlinear Bratu pr oblems.\nwhich might be the reasonfor small inconsistencies observedin Fig. 1 ; Similarly,\nwe see that the residual of AAoptD(10) in Fig. 1 is not decreasing consistently\naround 10th iteration (see the read dashed square region in Fig. 1) , where the\ncorresponding βkvalues are super close to zero as shown in Fig. 2.\nTo balance the over-damping effect, we bound these βkaway from zero. The\nfirst strategy we propose is to use\nˆβk= max{βk,η}, (13)\nwhereηis a small positive number such that 0 < η <0.5. For example, to\nreduce the over-damping effect, we take η= 0.3 in (13) as a lower bound. We\nplot the new βkvalues at each iteration in Fig. 3. There are no βkvalues less\nthan 0.3 anymore. The corresponding results are in Fig. 4. Compared with t he\nresults in Fig. 1, we see that there is less stagnation (see the red da shed square\nregion in Fig. 4) and onvergence is also faster. We also note that the βkvalues\nin Fig. 3 differs a lot from the values of βkin Fig. 2. Because changing βkin\n120 10 20 30 40 50 60 70 80 90\niteration00.10.20.30.40.50.60.70.8k\nFigure 2: Optimal damping factors in each iteration for m= 10.\nprevious iterations will affect the later ones.\n0 10 20 30 40 50 60 70 80\niteration00.10.20.30.40.50.60.70.8k\nFigure 3: Modified optimal damping factors: ˆβk= max{βk,η}withη= 0.3\nAlthough the results in Fig. 4 are better than those in Fig. 1, we notic e that\nthere are still some inconsistencies in the red dashed square region . To further\nsmooth out these inconsistencies, we change these “bad” βkvalues further away\nfrom zero. Therefore, we propose our second strategy:\nˆβk=\n\nβkifβk≥η,\n1−βkifβk< η.(14)\nWe note here that there is some trade-off between stability and spe ed of conver-\ngence. This does not mean that larger βkwork better, since larger βkmay not\nspeed up the convergence if it is not appropriate. Therefore, dam ping is good,\n130 20 40 60 80 100 120 140 160 180 200\niteration10-910-810-710-610-510-410-310-210-1residual\nAA(5)\nAA(10)\nAAoptD(5)\nAAoptD(10)\nFigure 4: Solving nonlinear Bratu problems: ˆβk= max{βk,η}withη= 0.3\nbut over-damping may cause inconsistencies and stagnation. In ou r numerical\nexperiment, we take η= 0.3 in (14) as an example. The results are in Fig. 5.\nCompared with the results in Fig. 1 and Fig. 4, it becomes better. We s ee that\nthere are almost no inconsistencies and there is faster convergen ce. We also plot\nthe new βkin Fig. 5.\nTo compare with the results provided in [17], we go further to increas e the\nwindows until m= 50. Again, without bounding away from zero, there are\nsome stagnations and inconsistencies. To avoid strong over-damp ing, we apply\n(14) again with η= 0.3 and obtain our new results in Fig. 7. We easily see that\nAAoptD(20) works as well as AA(50). Moreover, to test its scaling properties,\nwe also solve the Bratu problem on larger grids. In Fig. 8, for a grid siz e 64×64,\nwe see that AAoptD(10) is already comparable with AA(60) and AAoptD(30)\nperforms better than AA(60). Similarly, for a grid size 128 ×128, Fig. 9 shows\nthatAAoptD(40) performs much better than AA(80).\n140 20 40 60 80 100 120 140 160 180 200\niteration10-910-810-710-610-510-410-310-210-1residual\nAA(5)\nAA(10)\nAAoptD(5)\nAAoptD(10)\nFigure 5: Solving nonlinear Bratu problems: ˆβk= 1−βkwhenβk<0.3.\n0 10 20 30 40 50 60 70 80 90 100\niteration0.10.20.30.40.50.60.70.8k\nFigure 6: Modified optimal damping factors: ˆβk= 1−βkwhenβk<0.3.\n150 20 40 60 80 100 120 140 160 180 200\niteration10-910-810-710-610-510-410-310-210-1residual\nPicard\nAA(10)\nAA(20)\nAA(30)\nAA(40)\nAA(50)\nAAoptD(10)\nAAoptD(20)\nFigure 7: Using larger size windows and bounding the damping factor away from zero.\n0 50 100 150 200 250 300\niteration10-910-810-710-610-510-410-310-210-1residual64x64\nPicard\nAA(10)\nAA(20)\nAA(30)\nAA(50)\nAA(60)\nAAoptD(10)\nAAoptD(20)\nAAoptD(30)\nFigure 8: Scaling: solve the Bratu problem on a 64 ×64 gird.\n160 50 100 150 200 250 300 350 400 450 500\niteration10-910-810-710-610-510-410-310-210-1residual128x128\nPicard\nAA(20)\nAA(40)\nAA(60)\nAA(80)\nAAoptD(20)\nAAoptD(40)\nFigure 9: Scaling: solve the Bratu problem on a 128 ×128 grid.\nProblem 4.2. The nonlinear convection-diffusion problem. Use AAand\nAAoptD to solve the following 2D nonlinear convection-diffu sion equation in a\nsquare region:\n(−uxx−uyy)+(ux+uy)+ku2=f(x,y),(x,y)∈D= [0,1]×[0,1]\nwith the source term\nf(x,y) = 2π2sin(πx)sin(πy)\nand zero boundary conditions: u(x,y) = 0on∂D.\nIn this numerical experiment, we use a centered-difference discre tization on\n32×32 and 64×64 grids, respectively. We take k= 3 in the above problem\nand use u0= (1,1,···,1)Tas an initial approximate solution in all cases. As\nin solving the Bratu problem, the same preconditioning strategy is us ed here so\nthat the basic Picard iteration still works. To bound βkaway from zero, we use\n(14) with η= 0.25. The results are shown in Fig. 10 and Fig. 11 for n= 32×32\n17andn= 64×64, respectively. From Fig. 10, we see that AAoptD(5) is already\nbetter than AA(15); From Fig. 11, we also observe that AAoptD(20) is better\nthanAA(50). In both cases, AAoptD(m) does a much better job than AA(m),\nwhich is consistent with our previous example.\n0 100 200 300 400 500\niteration10-1010-810-610-410-2100102residual32  32\nPicard\nAA(5)\nAA(10)\nAA(15)\nAAoptD(5)\nAAoptD(10)\nAAoptD(15)\nFigure 10: Solving the nonlinear convection-diffusion prob lem on a 32 ×32 gird.\nOur next example is about solving a linear system Ax=b. As proved by\nWalker and Ni in [17], AA without truncation is “essentially equivalent” in a\ncertain sense to the GMRES method for linear problems.\nProblem 4.3. The linear equations. Apply AA and AAoptD to solve the\nfollowing linear system Ax=b, whereAis\nA=\n2−1···0 0\n−1 2···0 0\n...............\n0 0···2−1\n0 0··· −1 2\n, A∈Rn×n\n180 100 200 300 400 500 600\niteration10-1010-810-610-410-2100102residual64  64\nPicard\nAA(10)\nAA(20)\nAA(30)\nAA(50)\nAAoptD(10)\nAAoptD(20)\nAAoptD(30)\nFigure 11: Solving the nonlinear convection-diffusion prob lem on a 64 ×64 gird.\nand\nb= (1,···,1)T.\nChoosen= 10andn= 100, respectively. Here, we choose a large nso that\na large window size mis needed in Anderson Acceleration. We also note that\nthe Picard iteration does not work for this problem.\nThe initial guess is x0= (0,···,0)T. Without bounding βkaway from zero,\nthe results are shown in Fig. 12 and Fig. 13. For small m,AA(1) does not work,\nbutAAoptD(1) works. Moreover, we obtain from Fig. 12 that AAoptD(m) still\ndoes better than AA(m). When n= 100, we need larger mvalues. In this case,\nas shown in Fig. 13, AAoptD(5) already performs much better than AA(25).\nThis example shows that AAoptD can also be used to solve linear problems.\nFinally, we consider cases where very small mworks. Our example is from\nToth and Kelley’s paper [3], where AA is applied to solve the Chandrase khar\nH-equation.\n190 50 100 150 200 250 300\niteration10-910-810-710-610-510-410-310-210-1100101residualn=10\nAA(2)\nAA(3)\nAA(4)\nAAoptD(1)\nAAoptD(2)\nAAoptD(3)\nAAoptD(4)\nFigure 12: Small m: solving a linear problem Ax=bwithn= 10.\n0 50 100 150 200 250 300 350 400 450 500\niteration10-810-710-610-510-410-310-210-1100101residualn=100\nAA(5)\nAA(10)\nAA(15)\nAA(20)\nAA(50)\nAAoptD(5)\nAAoptD(10)\nAAoptD(15)\nAAoptD(20)\nFigure 13: Large m: solving a linear problem Ax=bwithn= 100.\n20Problem 4.4. the Chandrasekhar H-equation, arising in Radiative Heat Tr ans-\nfer theory, is a nonlinear integral equation:\nH(µ) =G(H) =/parenleftbigg\n1−c\n2/integraldisplay1\n0µ\nµ+vH(v)dv/parenrightbigg−1\n,\nwherec∈[0,1)is a physical parameter.\nWe will discretize the equation with the composite midpoint rule. Here we\napproximate integrals on [0,1]by\n/integraldisplay1\n0f(µ)dµ≈1\nNN/summationdisplay\nj=1f(µj)\nwhereµj= (i−1/2)/Nfor1≤i≤N. The resulting discrete problem is\nF(x)i=xi−\n1−c\n2NN/summationdisplay\nj=1µixj\nµi+µj\n−1\n,\nwhich is a fully nonlinear system.\nIt is known [32] both for the continuous problem and its follo wing midpoint\nrule discretization, that if c <1\nρ(G′(H∗))≤1−√\n1−c <1,\nwhereρdenotes spectral radius. Hence the local convergence theor y and Picard\niteration works.\nIn our numerical experiment, we choose N= 500,c= 0.5,c= 0.99 and\nc= 1. The case c= 1 is a critical value (Picard does not work in this case, but\nAA does). The numerical results are in Fig. 14 to Fig. 16. Firstly, AA(m) and\nAAoptD(m), with very small m( ≤3) values, work for all cases including the\ncritical case c= 1 and their performances are comparable. Secondly, increasing\nmdoes not always increase the performance. Thirdly, AAoptD may no t always\nhave advantages over AA for small window size m. This result is reasonable\nsince AAoptD(m) is kind of like packaging AA(m) andAA(1). Ifmis small,\nthere is almost no difference between AA(m) andAA(1), thus packaging them\n(varying window sizes) may not give better results.\n210 1 2 3 4 5 6 7 8 9 10\niteration10-810-710-610-510-410-310-210-1100101102residualPicard\nAA(1)\nAA(2)\nAA(3)\nAAoptD(1)\nAAoptD(2)\nAAoptD(3)\nFigure 14: Solving Chandrasekhar H-equation with AA and AAo ptD:c= 0.5\n0 5 10 15 20 25 30 35 40 45 50\niteration10-1010-810-610-410-2100102residualPicard\nAA(1)\nAA(2)\nAA(3)\nAAopt(1)\nAAoptD(2)\nAAoptD(3)\nFigure 15: Solving Chandrasekhar H-equation with AA and AAo ptD:c= 0.99\n220 5 10 15 20 25 30 35 40 45 50\niteration10-810-710-610-510-410-310-210-1100101102residualPicard\nAA(1)\nAA(2)\nAAoptD(1)\nAAoptD(2)\nFigure 16: Solving Chandrasekhar H-equation with AA and AAo ptD:c= 1\n5. Conclusions\nWe proposedanon-stationaryAndersonaccelerationalgorithmwit h anopti-\nmized damping factor in each iteration to further speed up linear and nonlinear\niterations by applying one extra optimization. This procedure has a s trong con-\nnection toanotherperspective ofgeneratingnon-stationaryAA (i.e. varyingthe\nwindow size mat different iterations). It turns out that choosing optimal βk\nis somewhat similar to packaging sAA(m) and sAA(1) within a single itera tion\nin a cheap way. Moreover, by taking benefit of the QR decomposition in the\nfirst optimization problem, the calculation of optimized βkat each iteration is\ncheap if two extra function evaluations are relatively inexpensive. O ur numer-\nical results show that the gain of doing this extra optimized step on βkcould\nbe large. Moreover, damping is good but over damping is not good bec ause it\nmay slow down the convergence rate. Therefore, when the statio nary AA is not\nworking well or a larger size of the window is needed in AA, we recommen d to\n23use AAoptD proposed in the present work.\nAcknowledgments\nThis work was partially supported by the National Natural Science F ounda-\ntion of China [grant number 12001287]; the Startup Foundation for Introduc-\ning Talent of Nanjing University of Information Science and Technolo gy [grant\nnumber 2019r106]; The first author Kewang Chen also gratefully ac knowledge\nthe financial support for his doctoral study provided by the China Scholarship\nCouncil (No. 202008320191).\nReferences\n[1] D. G. Anderson, Iterative procedures for nonlinear integral e quations, J.\nAssoc. Comput. Mach. 12 (1965) 547–560. doi:10.1145/321296.321305 .\n[2] D. G. M. Anderson, Comments on “Anderson acceleration, mix-\ning and extrapolation”, Numer. Algorithms 80 (1) (2019) 135–234.\ndoi:10.1007/s11075-018-0549-4 .\n[3] A. Toth, C. T. 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Mathematical Phys. 19 (2) (1978) 500–501.\ndoi:10.1063/1.523673 .\n27"    },    {        "title": "2202.06154v1.Generalization_of_the_Landau_Lifshitz_Gilbert_equation_by_multi_body_contributions_to_Gilbert_damping_for_non_collinear_magnets.pdf",        "content": "Generalization of the Landau-Lifshitz-Gilbert equation by multi-body contributions to\nGilbert damping for non-collinear magnets\nSascha Brinker,1Manuel dos Santos Dias,2, 1,\u0003and Samir Lounis1, 2,y\n1Peter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, 52425 J ulich, Germany\n2Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg, Germany\n(Dated: February 15, 2022)\nWe propose a systematic and sequential expansion of the Landau-Lifshitz-Gilbert equation utilizing\nthe dependence of the Gilbert damping tensor on the angle between magnetic moments, which arises\nfrom multi-body scattering processes. The tensor consists of a damping-like term and a correction\nto the gyromagnetic ratio. Based on electronic structure theory, both terms are shown to depend\non e.g. the scalar, anisotropic, vector-chiral and scalar-chiral products of magnetic moments: ei\u0001ej,\n(nij\u0001ei)(nij\u0001ej),nij\u0001(ei\u0002ej), (ei\u0001ej)2,ei\u0001(ej\u0002ek)..., where some terms are subjected to the\nspin-orbit \feld nijin \frst and second order. We explore the magnitude of the di\u000berent contributions\nusing both the Alexander-Anderson model and time-dependent density functional theory in magnetic\nadatoms and dimers deposited on Au(111) surface.arXiv:2202.06154v1  [cond-mat.mtrl-sci]  12 Feb 20222\nI. INTRODUCTION\nIn the last decades non-collinear magnetic textures have been at the forefront in the \feld of spintronics due to the\npromising applications and perspectives tied to them1,2. Highly non-collinear particle-like topological swirls, like\nskyrmions3,4and hop\fons5, but also domain walls6can potentially be utilized in data storage and processing devices\nwith superior properties compared to conventional devices. Any manipulation, writing and nucleation of these various\nmagnetic states involve magnetization dynamical processes, which are crucial to understand for the design of future\nspintronic devices.\nIn this context, the Landau-Lifshitz-Gilbert (LLG) model7,8is widely used to describe spin dynamics of materials\nranging from 3-dimensional bulk magnets down to the 0-dimensional case of single atoms, see e.g. Refs.9{12. The\nLLG model has two important ingredients: (i) the Gilbert damping being in general a tensorial quantity13, which can\noriginate from the presence of spin-orbit coupling (SOC)14and/or from spin currents pumped into a reservoir15,16;\n(ii) the e\u000bective magnetic \feld acting on a given magnetic moment and rising from internal and external interactions.\nOften a generalized Heisenberg model, including magnetic anisotropies and magnetic exchange interactions, is utilized\nto explore the ground state and magnetization dynamics characterizing a material of interest. Instead of the con-\nventional bilinear form, the magnetic interactions can eventually be of higher-order type, see e.g.17{23. Similarly to\nmagnetic interactions, the Gilbert damping, as we demonstrate in this paper, can host higher-order non-local contri-\nbutions. Previously, signatures of giant anisotropic damping were found24, while chiral damping and renormalization\nof the gyromagnetic ratio were revealed through measurements executed on chiral domain wall creep motion24{28.\nMost \frst-principles studies of the Gilbert damping were either focusing on collinear systems or were case-by-case\nstudies on speci\fc non-collinear structures lacking a general understanding of the fundamental behaviour of the Gilbert\ndamping as function of the non-collinear state of the system. In this paper, we discuss the Gilbert damping tensor\nand its dependencies on the alignment of spin moments as they occur in arbitrary non-collinear state. Utilizing linear\nresponse theory, we extract the dynamical magnetic susceptibility and identify the Gilbert damping tensor pertaining\nto the generalized LLG equation that we map to that obtained from electronic structure models such as the single\norbital Alexander-Anderson model29or time-dependent density functional theory applied to realistic systems10,30,31.\nApplying systematic perturbative expansions, we \fnd the allowed dependencies of the Gilbert damping tensor on the\ndirection of the magnetic moments. We identify terms that are a\u000bected by SOC in \frst and second order. We generalize\nthe LLG equation by a simple form where the Gilbert damping tensor is amended with terms proportional to scalar,\nanisotropic, vector-chiral and scalar-chiral products of magnetic moments, i.e. terms like ei\u0001ej, (nij\u0001ei)(nij\u0001ej),\nnij\u0001(ei\u0002ej), (ei\u0001ej)2,ei\u0001(ej\u0002ek)..., where we use unit vectors, ei=mi=jmij, to describe the directional dependence\nof the damping parameters and nijrepresents the spin-orbit \feld.\nThe knowledge gained from the Alexander-Anderson model is applied to realistic systems obtained from \frst-principles\ncalculations. As prototypical test system we use 3 dtransition metal adatoms and dimers deposited on the Au(111)\nsurface. Besides the intra-site contribution to the Gilbert damping, we also shed light on the inter-site contribution,\nusually referred to as the non-local contribution.\nII. MAPPING THE GILBERT DAMPING FROM THE DYNAMICAL MAGNETIC SUSCEPTIBILITY\nHere we extract the dynamical transverse magnetic response of a magnetic moment from both the Landau-Lifshitz-\nGilbert model and electronic structure theory in order to identify the Gilbert damping tensor Gij10,11,32,33. In linear\nresponse theory, the response of the magnetization mat siteito a transverse magnetic \feld bapplied at sites jand\noscillating at frequency !reads\nm\u000b\ni(!) =X\nj\f\u001f\u000b\f\nij(!)b\f\nj(!); (1)\nwith the magnetic susceptibility \u001f\u000b\f\nij(!) and\u000b;\fare thex;ycoordinates de\fned in the local spin frame of reference\npertaining to sites iandj.\nIn a general form13the LLG equation is given by\ndmi\ndt=\u0000\rmi\u00020\n@Be\u000b\ni+X\njGij\u0001dmj\ndt1\nA; (2)3\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)°2°101234Energy [U]°1.00°0.75°0.50°0.250.000.250.500.751.00DOS [#states/U]m=0.2m=0.3m=0.4m=0.5m=0.6m=0.7m=0.8\n°2°101234Energy [U]°1.0°0.50.00.51.0DOS [#states/U]m=0.2m=0.3m=0.4m=0.5m=0.6m=0.7m=0.8dI/dVVexcitationa)b)\nabc\nFIG. 1. Illustration of the Landau-Lifshitz-Gilbert model and local density of states within the Alexander-Anderson model.\n(a) A magnetic moment (red arrow) precesses in the the presence of an external \feld. The blue arrow indicates the direction\nof a damping term, while the green arrow shows the direction of the precession term. (b) Density of states for di\u000berent\nmagnetizations in the range from 0 :2 to 0:8. Density of states of dimers described within the Alexander-Anderson model for\ndi\u000berent magnetizations in the range from 0 :2 to 0:8. Shown is the ferromagnetic reference state. The magnetizations are\nself-consistently constrained using a longitudinal magnetic \feld, which is shown in the inset. Model parameters: U= 1:0 eV,\nEd= 1:0 eV; t= 0:2 eV;\u0000 = 0:2 eV; 'R= 0 °.\nwhere\r= 2 is the gyromagnetic ratio, Be\u000b\ni=\u0000dHspin=dmiis the e\u000bective magnetic \feld containing the contributions\nfrom an external magnetic \feld Bext\ni, as well as internal magnetic \felds originating from the interaction of the\nmoment with its surrounding. In an atomistic spin model described by e.g. the generalized Heisenberg hamiltonian,\nHspin=P\nimiKimi+1\n2P\nijmiJijmj, containing the on-site magnetic anisotropy Kiand the exchange tensor Jij,\nthe e\u000bective \feld is given by Be\u000b\ni=Bext\ni\u0000Kimi\u0000P\njJijmj(green arrow in Fig. 1a). The Gilbert damping tensor\ncan be separated into two contributions { a damping-like term, which is the symmetric part of the tensor, S, (blue\narrow in Fig. 1a), and a precession-like term A, which is the anti-symmetric part of the tensor. In Appendix A we\nshow how the antisymmetric intra-site part of the tensor contributes to a renormalization of the gyromagnetic ratio.\nTo extract the magnetic susceptibility, we express the magnetic moments in their respective local spin frame of\nreferences and use Rotation matrices that ensure rotation from local to globa spin frame of reference (see Appendix B).\nThe magnetic moment is assumed to be perturbed around its equilibrium value Mi,mloc\ni=Miez\ni+mx\niex\ni+my\niey\ni,\nwhere e\u000b\niis the unit vector in direction \u000bin the local frame of site i. Using the ground-state condition of vanishing\nmagnetic torques, Miez\ni\u0002\u0000\nBext\ni+Bint\ni\u0001\n= 0 and the inverse of the transverse magnetic susceptibility can be identi\fed\nas\n\u001f\u00001\ni\u000bj\f(!) =\u000eij\u0012\n\u000e\u000b\fBe\u000b\niz\nMi+i!\n\rMi\u000f\u000b\f\u0016\u0013\n+1\nMiMj(RiJijRT\nj)\u000b\f+ i!(RiGijRT\nj)\u000b\f; (3)\nfrom which it follows that the Gilbert damping is directly related to the linear in frequency imaginary part of the\ninverse susceptibility\nd\nd!=[\u001f\u00001]\u000b\f\nij=\u000eij\u00121\n\rMi\u000f\u000b\f\u0016\u0013\n+ (RiGijRT\nj)\u000b\f: (4)\nNote thatRiandRjare rotation matrices rotating to the local frames of site iandj, respectively, which de\fne the\ncoordinates \u000b;\f=fx;yg(see Appendix B).\nBased on electronic structure theory, the transverse dynamical susceptibility can be extracted from a Dyson-like\nequation:\u001f\u00001(!) =\u001f\u00001\n0(!)\u0000U, where\u001f0is the susceptibility of non-interacting electron while Uis a many-body\ninteraction Kernel, called exchange-correlation Kernel in the context of time-dependent density functional theory30.\nThe Kernel is generally assumed to be adiabatic, which enables the evaluation of the Gilbert damping directly from4\nthe non-interacting susceptibility. Obviously:d\nd!\u001f\u00001(!) =d\nd!\u001f\u00001\n0(!). For small frequencies !,\u001f0has a simple\n!-dependence11:\n\u001f0(!)\u0019<\u001f0(0) +i!=d\nd!\u001f0j!=0 (5)\nand as shown in Ref.33\nd\nd!\u001f\u00001\n0(!)\u0019[<\u001f0(0)]\u00002=d\nd!\u001f0j!=0: (6)\nStarting from the electronic Hamiltonian Hand the corresponding Green functions G(E\u0006i\u0011) = (E\u0000H\u0006i\u0011)\u00001, one\ncan show that the non-interacting magnetic susceptibility can be de\fned via\n\u001f\u000b\f\n0;ij(!+ i\u0011) =\u00001\n\u0019TrZEF\ndE\u0002\n\u001b\u000bGij(E+!+ i\u0011)\u001b\fImGji(E) +\u001b\u000bImGij(E)\u001b\fGji(E\u0000!\u0000i\u0011)\u0003\n;(7)\nwith\u001bbeing the vector of Pauli matrices. Obviously to identify the Gilbert damping and how it reacts to magnetic\nnon-collinearity, we have to inspect the dependence of the susceptibility, and therefore the Green function, on the\nmisalignment of the magnetic moments.\nIII. MULTI-SITE EXPANSION OF THE GILBERT DAMPING\nAssuming the hamiltonian Hconsisting of an on-site contribution H0and an inter-site term encoded in a hopping\ntermt, which can be spin-dependent, one can proceed with a perturbative expansion of the corresponding Green\nfunction utilizing the Dyson equation\nGij=G0\ni\u000eij+G0\nitijG0\nj+G0\nitikG0\nktkjG0\nj+::: : (8)\nWithin the Alexander-Anderson single-orbital impurity model29,H0\ni=Ed\u0000i \u0000\u0000Uimi\u0001\u001b\u0000Bi\u0001\u001b, whereEd\nis the energy of the localized orbitals, \u0000 is the hybridization in the wide band limit, Uiis the local interaction\nresponsible for the formation of a magnetic moment and Biis an constraining or external magnetic \feld. SOC can be\nincorportated as tsoc\nij=i\u0015ijnij\u0001\u001b, where\u0015ijandnij=\u0000njirepresent respectively the strength and direction of the\nanisotropy \feld. It can be parameterized as a spin-dependent hopping using the Rashba-like spin-momentum locking\ntij=t(cos'R\u001b0\u0000i sin'Rnij\u0001\u001b)34.\nDepending on whether the considered Green function is an on-site Green function Giior an inter-site Green function\nGijdi\u000berent orders in the hopping are relevant. On-site Green functions require an even number of hopping processes,\nwhile inter-site Green functions require at least one hopping process.\nThe on-site Green function G0\nican be separated into a spin-less part Niand a spin dependent part Mi,\nG0\ni=Ni\u001b0+Mi\u0001\u001b ; (9)\nwhere the spin dependent part is parallel to the magnetic moment of site i,Mikmi(note that SOC is added later on\nto the hoppings). Using the perturbative expansion, eq. (8), and the separated Green function, eq. (9), to calculate\nthe magnetic susceptibility, eq. (7), one can systematically classify the allowed dependencies of the susceptibility with\nrespect to the directions of the magnetic moments, e.g. by using diagrammatic techniques as shown in Ref.18for a\nrelated model in the context of higher-order magnetic exchange interactions.\nSince our interest is in the form of the Gilbert damping, and therefore also in the form of the magnetic susceptibility, the\nperturbative expansion can be applied to the magnetic susceptibility. The general form of the magnetic susceptibility\nin terms of the Green function, eq. (7), depends on a combination of two Green functions with di\u000berent energy\narguments, which are labeled as !and 0 in the following. The relevant structure is then identi\fed as33,\n\u001f\u000b\f\nij(!)\u0018Tr\u001b\u000b\niGij(!)\u001b\f\njGji(0): (10)\nThe sake of the perturbative expansion is to gather insights in the possible forms and dependencies on the magnetic\nmoments of the Gilbert damping, and not to calculate explicitly the strength of the Gilbert damping from this5\nexpansion. Therefore, we focus on the structure of eq. (10), even though the susceptibility has more ingredients,\nwhich are of a similar form.\nInstead of writing all the perturbations explicitly, we set up a diagrammatic approach, which has the following\ningredients and rules:\n1. Each diagram contains the operators NandM, which are \u001b\u000band\u001b\ffor the magnetic susceptibility. The\noperators are represented by a white circle with the site and spin index: i\u000b\n2. Hoppings are represented by grey circles indicating the hopping from site itoj:ij. The vertex corresponds\ntotij.\n3. SOC is described as a spin-dependent hopping from site itojand represented by: ij;\u000b. The vertex\ncorresponds to tsoc\nij=i\u0015ij^n\u000b\nij\u001b\u000b.\n4. The bare spin-independent (on-site) Green functions are represented by directional lines with an energy at-\ntributed to it: !. The Green function connects operators and hoppings. The line corresponds to Ni(!).\n5. The spin-dependent part of the bare Green function is represented by: !;\u000b .\u000bindicates the spin direction.\nThe direction ensures the right order within the trace (due to the Pauli matrices, the di\u000berent objects in the\ndiagram do not commute). The line corresponds to Mi(!)m\u000b\ni\u001b\u000b.\nNote that the diagrammatic rules might be counter-intuitive, since local quantities (the Green function) are represented\nby lines, while non-local quantities (the hopping from itoj) are represented by vertices. However, these diagrammatic\nrules allow a much simpli\fed description and identi\fcation of all the possible forms of the Gilbert damping, without\nhaving to write lengthy perturbative expansions.\nSpin-orbit coupling independent contributions.\nTo get a feeling for the diagrammatic approach, we start with the simplest example: the on-site susceptibility without\nany hoppings to a di\u000berent site, which describes both the single atom and the lowest order term for interacting atoms.\nThe possible forms are,\n\u001fii\n\u000b\f(!)/\n!0\ni\u000b i\f+\n!;\r0\ni\u000b i\f\n+\n!0;\r\ni\u000b i\f+\n!;\u000e0;\r\ni\u000b i\f; (11)6\nwhich evaluate to,\n!0\ni\u000b i\f= Tr\u001b\u000b\u001b\fNi!)Ni(0) =\u000e\u000b\fNi(!)Ni(0) (12)\n!;\r0\ni\u000b i\f= Tr\u001b\u000b\u001b\r\u001b\fMi(!)Ni(0)m\r\ni= i\u000f\u000b\r\fMi(!)Ni(0)m\r\ni (13)\n!0;\r\ni\u000b i\f= Tr\u001b\u000b\u001b\f\u001b\rNi(!)Mi(0)m\r\ni= i\u000f\u000b\f\rMi(!)Mi(0)m\r\ni (14)\n!;\u000e0;\r\ni\u000b i\f= Tr\u001b\u000b\u001b\u000e\u001b\f\u001b\rMi(!)Mi(0)m\u000e\nim\r\ni\n= (\u000e\u000b\u000e\u000e\f\r+\u000e\u000b\r\u000e\f\u000e\u0000\u000e\u000b\f\u000e\r\u000e)Mi(!)Mi(0)m\u000e\nim\r\ni: (15)\nThe \frst diagram yields an isotropic contribution, the second and third diagrams yield an anti-symmetric contribution,\nwhich is linear in the magnetic moment, and the last diagram yields a symmetric contribution being quadratic in the\nmagnetic moment. Note that the energy dependence of the Green functions is crucial, since otherwise the sum of\neqs. (13) and (14) vanishes. In particular this means that the static susceptibility has no dependence linear in the\nmagnetic moment, while the the slope of the susceptibility with respect to energy can have a dependence linear in\nthe magnetic moment. The static part of the susceptibility maps to the magnetic exchange interactions, which are\nknown to be even in the magnetic moment due to time reversal symmetry.\nCombining all the functional forms of the diagrams, we \fnd the following possible dependencies of the on-site Gilbert\ndamping on the magnetic moments,\nG\u000b\f\nii(fmg)/f\u000e\u000b\f;\u000f\u000b\f\rm\r\ni;m\u000b\nim\f\nig: (16)\nSince we work in the local frames, mi= (0;0;mz\ni), the last dependence is a purely longitudinal term, which is not\nrelevant for the transversal dynamics discussed in this work.\nIf we still focus on the on-site term, but allow for two hoppings to another atom and back, we \fnd the following new7\ndiagrams,\n!00\n0\ni\u000b i\fij ji\n+\n!;\r00\n0\ni\u000b i\fij ji\n+:::+\n!;\r0;\u000e0\n0\ni\u000b i\fij ji\n+:::\n+\n!;\r0;\u000e0;\u0011\n0\ni\u000b i\fij ji\n+:::+\n!;\r0;\u000e0;\u0011\n0;\u0017\ni\u000b i\fij ji\n: (17)\nThe dashed line in the second diagram can be inserted in any of the four sides of the square, with the other possibilities\nomitted. Likewise for the diagrams with two or three dashed lines, the di\u000berent possible assignments have to be\nconsidered. The additional hopping to the site jyields a dependence of the on-site magnetic susceptibility and\ntherefore also the on-site Gilbert damping tensor on the magnetic moment of site j.\nAnother contribution to the Gilbert damping originates from the inter-site part, thus encoding the dependence of the\nmoment site ion the dynamics of the moment of site jviaGij. This contribution is often neglected in the literature,\nsince for many systems it is believed to have no signi\fcant impact. Using the microscopic model, a di\u000berent class\nof diagrams is responsible for the inter-site damping. In the lowest order in t=Um the diagrams contain already two\nhopping events,\n! !0 0\ni\u000b j\f\nijij\n+\n!;\r !0 0\ni\u000b j\f\nijij\n+:::+\n!;\r !;\u000e0 0\ni\u000b j\f\nijij\n+:::\n+\n!;\r !;\u000e0;\u0011 0\ni\u000b j\f\nijij\n+:::+\n!;\r !;\u000e0;\u0011 0;\u0010\ni\u000b j\f\nijij\n: (18)\nIn total, we \fnd that the spin-orbit independent intra-site and inter-site Gilbert damping tensors can be respectively\nwritten as\nGii=\u0010\nSi+Sij;(1)\ni (ei\u0001ej) +Sij;(2)\ni (ei\u0001ej)2\u0011\nI\n+\u0010\nAi+Aij\ni(ei\u0001ej)\u0011\nE(ei);(19)8\nand\nG\u000b\f\nij=\u0000\nSij+Sdot\nij(ei\u0001ej)\u0001\n\u000e\u000b\f\n+\u0000\nAij+Adot\nij(ei\u0001ej)\u0001\n(E(ei) +E(ej))\u000b\f\n+Scross\nij(ei\u0002ej)\u000b(ei\u0002ej)\f+Sba\nije\f\nie\u000b\nj; (20)\nwhere as mentioned earlier SandArepresent symmetric and asymmetric contributions, Iis the 3\u00023 identity while\nE(ei) =0\n@0ez\ni\u0000ey\ni\n\u0000ez\ni0ex\ni\ney\ni\u0000ex\ni01\nA.\nRemarkably, we \fnd that both the symmetric and anti-symmetric parts of the Gilbert damping tensor have a rich\ndependence with the opening angle of the magnetic moments. We identify, for example, the dot and the square\nof the dot products of the magnetic moments to possibly play a crucial role in modifying the damping, similarly to\nbilinear and biquadratic magnetic interactions. It is worth noting that even though the intra-site Gilbert damping can\nexplicitly depend on other magnetic moments, its meaning remains unchanged. The anti-symmetric precession-like\nterm describes a precession of the moment around its own e\u000bective magnetic \feld, while the diagonal damping-like\nterm describes a damping towards its own e\u000bective magnetic \feld. The dependence on other magnetic moments\nrenormalizes the intensity of those two processes. The inter-site Gilbert damping describes similar processes, but with\nrespect to the e\u000bective \feld of the other involved magnetic moment. On the basis of the LLG equation, eq. (2), it can be\nshown that the term related to Sba\nijwith a functional form of e\f\nie\u000b\njdescribes a precession of the i-th moment around\nthej-th moment with a time- and directional-dependent amplitude, @tmi/(mi\u0002mj) (mi\u0001@tmj). The double\ncross product term yields a time dependence of @tmi/(mi\u0002(mi\u0002mj)) ((mi\u0002mj)\u0001@tmj). Both contributions\nare neither pure precession-like nor pure damping-like, but show complex time- and directional-dependent dynamics.\nSpin-orbit coupling contributions. The spin-orbit interaction gives rise to new possible dependencies of the\ndamping on the magnetic structure. In particular, the so-called chiral damping, which in general is the di\u000berence\nof the damping between a right-handed and a left-handed opening, rises from SOC and broken inversion symmetry.\nUsing our perturbative model, we can identify all possible dependencies up to second order in SOC and third order\nin the magnetic moments.\nIn the diagramms SOC is added by replacing one spin-independent hopping vertex by a spin-dependent one,\n!00\n0\ni\u000b i\fij ji\n!\n!00\n0\ni\u000b i\fij\r ij\n: (21)\nUp to \frst-order in SOC, we \fnd the the following dependencies were found for the on-site Gilbert damping\nGii(fmg)/f\u000f\u000b\f\r^n\r\nij;^n\u000b\nij^n\f\nji;^n\f\nijm\u000b\ni;^n\u000b\nijm\f\ni;\u000e\u000b\f(^nij\u0001mi);\u000e\u000b\f(^nij\u0001mj);\n^n\f\nijm\u000b\nj;^n\u000b\nijm\f\nj;m\u000b\ni(^nij\u0002mi)\f;m\f\ni(^nij\u0002mi)\u000b;\n\u000e\u000b\f^nij\u0001(mi\u0002mj);m\u000b\ni(^nij\u0002mj)\f;m\f\ni(^nij\u0002mj)\u000b;(^nij\u0001mj)\u000f\u000b\f\rm\r\ni;\nm\u000b\nim\f\ni(^nij\u0001mj);(m\u000b\nim\f\nj\u0000m\f\nim\u000b\nj)(^nij\u0001mj);^n\f\nijm\u000b\ni(mi\u0001mj);^n\u000b\nijm\f\ni(mi\u0001mj)g: (22)\nWe identi\fed the following contributions for the on-site and intersite damping to be the most relevant one after the\nnumerical evaluation discussed in the next sections:\nGsoc\nii=Ssoc;ij\ni nij\u0001(ei\u0002ej)I\n+Ssoc;ij;(2)\ni (nij\u0001ei)(nij\u0001ej)I\n+Asoc;ij\ni nij\u0001(ei\u0002ej)E(ei)\n+Asoc;ij;(2)\ni (nij\u0001ej)E(nij); (23)9\nand\nGsoc;\u000b\f\nij =Ssoc\nijnij\u0001(ei\u0002ej)\u000e\u000b\f+Ssoc;ba\nijn\f\nij(ei\u0002ej)\u000b\n+Asoc\nijE\u000b\f(nij): (24)\nThe contributions being \frst-order in SOC are obviously chiral since they depend on the cross product, ei\u0002ej. Thus,\nsimilar to the magnetic Dzyaloshinskii-Moriya interaction, SOC gives rise to a dependence of the Gilbert damping\non the vector chirality, ei\u0002ej. The term chiral damping used in literature refers to the dependence of the Gilbert\ndamping on the chirality, but to our knowledge it was not shown so far how this dependence evolves from a microscopic\nmodel, and how it looks like in an atomistic model.\nExtension to three sites. Including three di\u000berent sites i,j, andkin the expansions allows for a ring exchange\ni!j!k!iinvolving three hopping processes, which gives rise to new dependencies of the Gilbert damping on\nthe directions of the moments.\nAn example of a diagram showing up for the on-site Gilbert damping is given below for the on-site Gilbert damping\nthe diagram,\n!00 0;\r\n0\ni\u000b i\fijjk\nki(25)\nApart from the natural extensions of the previously discussed 2-site quantities, the intra-site Gilbert damping of site i\ncan depend on the angle between the sites jandk,ej\u0001ek, or in higher-order on the product of the angles between site\niandjwithiandk, (ei\u0001ej)(ei\u0001ek). In sixth-order in the magnetic moments the term ( ei\u0001ej)(ej\u0001ek)(ek\u0001ei) yields\nto a dependence on the square of the scalar spin chirality of the three sites, [ ei\u0001(ej\u0002ek)]2. Including SOC, there are\ntwo interesting dependencies on the scalar spin chirality. In \frst-order one \fnds similarly to the recently discovered\nchiral biquadratic interaction18and its 3-site generalization19, e.g. ( nij\u0001ei) (ei\u0001(ej\u0002ek)), while in second order a\ndirect dependence on the scalar spin chirality is allowed, e.g. n\u000b\nijn\f\nki(ei\u0001(ej\u0002ek)). The scalar spin chirality directly\nrelates to the topological orbital moment35{37and therefore the physical origin of those dependencies lies in the\ntopological orbital moment. Even though these terms might not be the most important ones in our model, for speci\fc\nnon-collinear con\fgurations or for some realistic elements with a large topological orbital moment, e.g. MnGe20, they\nmight be important and even dominant yielding interesting new physics.\nIV. APPLICATION TO THE ALEXANDER-ANDERSON MODEL\nMagnetic dimers. Based on a 2-site Alexander-Anderson model, we investigated the dependence of the Gilbert\ndamping on the directions of the magnetic moments using the previously discussed possible terms (see more details\non the method in Appendix C). The spin splitting Ude\fnes the energy scale and all other parameters. The energy of\norbitals is set to Ed= 1:0. The magnetization is self-consistently constrained in a range of m= 0:2 tom= 0:8 using\nmagnetic constraining \felds. The corresponding spin-resolved local density of states is illustrated in Fig. 1b, where\nthe inter-site hopping is set to t= 0:2 and the hybridization to \u0000 = 0 :2. We performed two sets of calculations: one\nwithout spin-dependent hopping, 'R= 0 °, and one with a spin-dependent hopping, 'R= 20 °.\nThe di\u000berent damping parameters are shown in Fig. 2 as function of the magnetization. They are obtained from a\nleast-squares \ft to several non-collinear con\fgurations based on a Lebedev mesh for `= 238. The damping, which is\nindependent of the relative orientation of the two sites, is shown in Fig. 2a. The symmetric damping-like intra-site\ncontributionSidominates the damping tensor for most magnetizations and has a maximum at m= 0:3. The anti-\nsymmetric intra-site contribtuion Ai, which renormalizes the gyromagnetic ratio, approximately changes sign when\nthe Fermi level passes the peak of the minority spin channel at m\u00190:5 and has a signi\fcantly larger amplitude\nfor small magnetizations. Both contributions depend mainly on the broadening \u0000, which mimics the coupling to an10\nMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)abc\nab\nFIG. 2. Damping parameters as function of the magnetization for the dimers described within the Alexander-Anderson\nmodel including spin orbit coupling. A longitudinal magnetic \feld is used to self-consistently constrain the magnetization. The\nparameters are extracted from \ftting to the inverse of the transversal susceptibility for several non-collinear con\fgurations\nbased on a Lebedev mesh. Model parameters in units of U:Ed= 1:0; t= 0:2;\u0000 = 0:2; 'R= 20 °.\nelectron bath and is responsible for the absorption of spin currents, which in turn are responsible for the damping of\nthe magnetization dynamics15,16.\nThe directional dependencies of the intra-site damping are shown in Fig. 2b. With our choice of parameters, the\ncorrection to the damping-like symmetric Gilbert damping can reach half of the direction-independent term. This\nmeans that the damping can vary between \u00190:4\u00001:0 for a ferromagnetic and an antiferromagnetic state at m= 0:4.\nAlso for the renormalization of the gyromagnetic ratio a signi\fcant correction is found, which in the ferromagnetic case\nalways lowers and in the antiferromagnetic case enhances the amplitude. The most dominant contribution induced\nby SOC is the chiral one, which depends on the cross product of the moments iandj, which in terms of amplitude is\ncomparable to the isotropic dot product terms. Interestingly, while the inter-site damping term is in general known\nto be less relevant than the intra-site damping, we \fnd that this does not hold for the directional dependence of the\ndamping. The inter-site damping is shown in Fig. 2c. Even though the directional-independent term, Sij, is nearly\none order of magnitude smaller than the equivalent intra-site contribution, this is not necessarily the case for the\ndirectional-dependent terms, which are comparable to the intra-site equivalents.\nV. APPLICATION TO FIRST-PRINCIPLES SIMULATIONS\nTo investigate the importance of non-collinear e\u000bects for the Gilbert damping in realistic systems, we use DFT and\ntime-dependent DFT to explore the prototypical example of monoatomic 3 dtransition metal adatoms and dimers\ndeposited on a heavy metal surface hosting large SOC (see Fig. 3a for an illustration of the con\fguration). We\nconsider a Cr, Mn, Fe and Co atoms deposited on the fcc-Au(111) surface (details of the simulations are described in\nAppendix D). The parameters and the corresponding functional forms are \ftted to our \frst-principles data using 196\nnon-collinear states based on a Lebedev mesh for `= 238.\nAdatoms on Au(111). To illustrate the di\u000berent e\u000bects on the Gilbert damping, we start by exploring magnetic\nadatoms in the uniaxial symmetry of the Au(111) surface. For the adatoms no non-local e\u000bects can contribute to the\nGilbert damping.\nThe Gilbert damping tensor of a single adatom without SOC has the form shown in relation to eq. (16),\nG0\ni=SiI+AiE(ei): (26)\nNote that SOC can induce additional anisotropies, as shown in eq. (22). The most important ones for the case of a\nsingle adatom are f\u000f\u000b\f\r^n\r\nij;^n\u000b\nij^n\f\njig, which in the C3vsymmetry result in\nGi=G0\ni+Ssoc\ni0\n@0 0 0\n0 0 0\n0 0 11\nA+Asoc\ni0\n@0 1 0\n\u00001 0 0\n0 0 01\nA; (27)11\nDampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters\nSi 0:083 0 :014 0 :242 0 :472\nAi 0:204 0 :100 0 :200 0 :024\nSsoc\ni 0:000 0 :000 0 :116 0 :010\nAsoc\ni 0:000 0 :000\u00000:022 0 :012\n\rrenorm\nx=y 1:42 1 :67 1 :43 1 :91\n\rrenorm\nz 1:42 1 :67 1 :48 1 :87\nTABLE I. Gilbert damping parameters of Cr, Mn, Fe and Co adatoms deposited on the Au(111) surface as parametrized in\neqs. (26) and (27). The SOC \feld points in the z-direction due to the C3vsymmetry. The renormalized gyromagnetic ratio\n\rrenormis calculated according to eqs. (28) for an in-plane magnetic moment and an out-of-plane magnetic moment.\nsince the sum of all SOC vectors points in the out-of-plane direction with ^nij!ez. Thus, the Gilbert damping tensor\nof adatoms deposited on the Au(111) surface can be described by the four parameters shown in eqs. (26) and (27),\nwhich are reported in Table I for Cr, Mn, Fe and Co adatoms. Cr and Mn, being nearly half-\flled, are characterized\nby a small damping-like contribution Si, while Fe and Co having states at the Fermi level show a signi\fcant damping\nof up to 0:47 in the case of Co. The antisymmetric part Aiof the Gilbert damping tensor results in an e\u000bective\nrenormalization of the gyromagnetic ratio \r, as shown in relation to eq. (A5), which using the full LLG equation,\neq. (2), and approximating mi\u0001dmi\ndt= 0 is given by,\n\rrenorm=\r1\n1 +\r(ei\u0001Ai); (28)\nwhere Aidescribes the vector Ai=\u0000\nAi;Ai;Ai+Asoc\ni\u0001\n. For Cr and Fe there is a signi\fcant renormalization of the\ngyromagnetic ratio resulting in approximately 1 :4. In contrast, Co shows only a weak renormalization with 1 :9 being\nclose to the gyromagnetic ratio of 2. The SOC e\u000bects are negigible for most adatoms except for Fe, which shows a\nsmall anisotropy in the renormalized gyromagentic ratio ( \u001910 %) and a large anisotropy in the damping-like term of\nnearly 50 %.\nDimers on Au(111). In contrast to single adatoms, dimers can show non-local contributions and dependencies on\nthe relative orientation of the magnetic moments carried by the atoms. All quantities depending on the SOC vector\nare assumed to lie in the y-z-plane due to the mirror symmetry of the system. A sketch of the dimer and its nearest\nneighboring substrate atoms together with adatoms' local density of states are presented in Fig. 3.\nThe density of states originates mainly from the d-states of the dimer atoms. It can be seen that the dimers exhibit\na much more complicated hybridization pattern than the Alexander-Anderson model. In addition the crystal \feld\nsplits the di\u000berent d-states resulting in a rich and high complexity than assumed in the model. However, the main\nfeatures are comparable: For all dimers there is either a fully occupied majority channel (Mn, Fe, and Co) or a fully\nunoccupied minority channel (Cr). The other spin channel determines the magnetic moments of the dimer atoms\nf4:04;4:48;3:42;2:20g\u0016Bfor respectively Cr, Mn, Fe and Co. Using the maximal spin moment, which is according\nto Hund's rule 5 \u0016B, the \frst-principles results can be converted to the single-orbital Alexander-Anderson model\ncorresponding to approximately m=f0:81;0:90;0:68;0:44g\u0016Bfor the aforementioned sequence of atoms. Thus by\nthis comparison, we expect large non-collinear contributions for Fe and Co, while Cr and Mn should show only weak\nnon-local dependencies.\nThe obtained parametrization is given in Table II. The Cr and Mn dimers show a weak or nearly no directional\ndependence. While the overall damping for both nanostructures is rather small, there is a signi\fcant correction to\nthe gyromagnetic ratio.\nIn contrast, the Fe and Co dimers are characterized by a very strong directional dependence. Originating from the\nisotropic dependencies of the damping-like contributions, the damping of the Fe dimer can vary between 0 :21 in\nthe ferromagnetic state and 0 :99 in the antiferromagnetic state. For the Co dimer the inter-site damping is even\ndominated by the bilinear and biquadratic term, while the constant damping is negligible. In total, there is a very\ngood qualitative agreement between the expectations derived from studying the Alexander-Anderson model and the\n\frst-principles results.12\nDampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters\nSi 0:0911 0 :0210 0 :2307 0 :5235\nSij;(1)\ni 0:0376 0 :0006\u00000:3924\u00000:2662\nSij;(2)\ni 0:0133\u00000:0006 0 :3707 0 :3119\nAi 0:2135 0 :1158 0 :1472 0 :0915\nAij\ni 0:0521 0 :0028\u00000:0710\u00000:0305\nSij\u00000:0356 0 :0028 0 :2932 0 :0929\nSdot\nij\u00000:0344\u00000:0018\u00000:3396\u00000:4056\nSdot;(2)\nij 0:0100 0 :0001 0 :1579 0 :2468\nAij\u00000:0281\u00000:0044 0 :0103 0 :0011\nAdot\nij\u00000:0175 0 :0000\u00000:0234\u00000:0402\nScross\nij 0:0288 0 :0002\u00000:2857\u00000:0895\nSba\nij 0:0331 0 :0036 0 :2181 0 :2651\nSsoc;ij;y\ni 0:0034 0 :0000 0 :0143\u00000:0225\nSsoc;ij;z\ni 0:0011 0 :0000\u00000:0104 0 :0156\nAsoc;ij;y\ni 0:0024\u00000:0001\u00000:0036 0 :0022\nAsoc;ij;z\ni 0:0018\u00000:0005 0 :0039\u00000:0144\nSsoc;y\nij 0:0004 0 :0001 0 :0307 0 :0159\nSsoc;z\nij\u00000:0011 0 :0000\u00000:0233 0 :0206\nSba,soc ;y\nij\u00000:0027 0 :0000\u00000:0184\u00000:0270\nSba,soc ;z\nij 0:0005\u00000:0001 0 :0116\u00000:0411\nTABLE II. Damping parameters of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The possible forms of the\ndamping are taken from the analytic model. The SOC \feld is assumed to lie in the y-zplane and inverts under permutation\nof the two dimer atoms.\n\u00003\u00002\u000010123E\u0000EF[eV]\u00006\u0000303DOS [#states/eV]Cr dimerMn dimerFe dimerCo dimersurface\nab\nFIG. 3. aIllustration of a non-collinear magnetic dimer (red spheres) deposited on the (111) facets of Au (grey spheres).\nFrom the initial C3vspatial symmetry of the surface the dimers preserve the mirror plane (indicated grey) in the y-zplane. b\nLocal density of states of the Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The grey background indicates the\nsurface density of states. The dimers are collinear in the z-direction.\nVI. CONCLUSIONS\nIn this article, we presented a comprehensive analysis of magnetization dynamics in non-collinear system with a special\nfocus on the Gilbert damping tensor and its dependencies on the non-collinearity. Using a perturbative expansion\nof the two-site Alexander-Anderson model, we could identify that both, the intra-site and the inter-site part of the\nGilbert damping, depend isotropically on the environment via the e\u000bective angle between the two magnetic moments,\nei\u0001ej. SOC was identi\fed as the source of a chiral contribution to the Gilbert damping, which similarly to the\nDzyaloshinskii-Moriya and chiral biquadratic interactions depends linearly on the vector spin chirality, ei\u0002ej. We\nunveiled dependencies that are proportional to the three-spin scalar chirality ei\u0001(ej\u0002ek), i.e. to the chiral or\ntopological moment, and to its square. Using the Alexander-Anderson model, we investigated the importance of the13\ndi\u000berent contributions in terms of their magnitude as function of the magnetization. Using the prototypical test\nsystem of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface, we extracted the e\u000bects of the non-collinearity\non the Gilbert damping using time-dependent DFT. Overall, the \frst-principles results agree qualitatively well with\nthe Alexander-Anderson model, showing no dependence for the nearly half-\flled systems Cr and Mn and a strong\ndependence on the non-collinearity for Fe and Co having a half-\flled minority spin-channel. The realistic systems\nindicate an even stronger dependence on the magnetic texture than the model with the used parameters. The Fe and\nthe Co dimer show signi\fcant isotropic terms up to the biquadratic term, while the chiral contributions originating\nfrom SOC have only a weak impact on the total Gilbert damping. However, the chiral contributions can play the\ndeciding role for systems which are degenerate in the isotropic terms, like e.g. spin spirals of opposite chirality.\nWe expect the dependencies of the Gilbert damping on the magnetic texture to have a signi\fcant and non-trivial\nimpact on the spin dynamics of complex magnetic structures. Our \fndings are readily implementable in the LLG\nmodel, which can trivially be amended with the angular dependencies provided in the manuscript. Utilizing multiscale\nmapping approaches, it is rather straightforward to generalize the presented forms for an implementation of the\nmicromagnetic LLG and Thiele equations. The impact of the di\u000berent contributions to the Gilbert damping, e.g. the\nvector (and/or scalar) chiral and the isotropic contributions, can be analyzed on the basis of either free parameters\nor sophisticated parametrizations obtained from \frst principles as discussed in this manuscript. It remains to be\nexplored how the newly found dependencies of the Gilbert damping a\u000bect the excitations and motion of a plethora of\nhighly non-collinear magnetic quasi-particles such as magnetic skyrmions, bobbers, hop\fons, domain walls and spin\nspirals. Future studies using atomistic spin dynamics simulations could shed some light on this aspect and help for\nthe design of future devices based on spintronics.\nACKNOWLEDGMENTS\nThis work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research\nand innovation program (ERC-consolidator grant 681405 { DYNASORE) and from Deutsche Forschungsgemeinschaft\n(DFG) through SPP 2137 \\Skyrmionics\" (Project LO 1659/8-1). The authors gratefully acknowledge the computing\ntime granted through JARA-HPC on the supercomputer JURECA at the Forschungszentrum J ulich39.\nVII. METHODS\nAppendix A: Analysis of the Gilbert damping tensor\nThe Gilbert damping tensor Gcan be decomposed into a symmetric part Sand an anti-symmetric part A,\nA=G\u0000GT\n2andS=G+GT\n2: (A1)\nWhile the symmetric contribution can be referred to as the damping-like contribution including potential anisotropies,\nthe anti-symmetric Atypically renormalizes the gyromagnetic ratio as can be seen as follows: The three independent\ncomponents of an anti-symmetric tensor can be encoded in a vector Ayielding\nA\u000b\f=\u000f\u000b\f\rA\r; (A2)\nwhere\u000f\u000b\f\ris the Levi-Cevita symbol. Inserting this into the LLG equation yields\ndmi\ndt=\u0000\rmi\u00020\n@Be\u000b\ni+X\njAijdmj\ndt1\nA (A3)\n\u0019\u0000\rmi\u00020\n@Be\u000b\ni\u0000\rX\njAij\u0000\nmj\u0002Be\u000b\nj\u00011\nA: (A4)\nThe last term can be rewritten as\n(Aij\u0001mj)Be\u000b\nj\u0000\u0000\nAij\u0001Be\u000b\nj\u0001\nmj: (A5)14\nFor the local contribution, Aii, the correction is kmiandkBe\u000b\niyielding a renormalization of \rmi\u0002Be\u000b\ni. However,\nthe non-local parts of the anti-symmetric Gilbert damping tensor can be damping-like.\nAppendix B: Relation between the LLG and the magnetic susceptibility\nThe Fourier transform of the LLG equation is given by\n\u0000i!mi=\u0000\rmi\u00020\n@Bext\ni\u0000X\njJijmj\u0000i!X\njGijmj1\nA: (B1)\nTransforming this equation to the local frames of site iandjusing the rotation matrices RiandRjyields\ni!\n\rMimloc\ni=mloc\ni\nMi\u00020\n@RiBext\ni\u0000X\njRiJijRT\njmloc\nj\u0000i!X\njRiGijRT\njmloc\nj1\nA; (B2)\nwhere mloc\ni=Rimiandmloc\nj=Rjmj. The rotation matrices are written as R(#i;'i) = cos(#i=2)\u001b0+\ni sin(#i=2)\u0000\nsin('i)\u001bx\u0000cos('i)\u001by\u0001\n, with (#i;'i) being the polar and azimuthal angle pertaining to the moment\nmi. In the ground state the magnetic torque vanishes. Thus, denoting mloc\ni= (mx\ni; my\ni; Mi), wheremx=y\niare\nperturbations to the ground states, yields for the ground state\n0\n@(RiBext\ni)x\u0000P\nj(RiJijRT\njMjez)x\n(RiBext\ni)y\u0000P\nj(RiJijRT\njMjez)y\n(RiBext\ni)z\u0000P\nj(RiJijRT\njMjez)z1\nA=0\n@0\n0\n(RiBe\u000b\ni)z1\nA: (B3)\nLinearizing the LLG and using the previous result and limiting our expansion to transveral excitations yield\ni!\n\rMimx\ni=my\ni(RiBe\u000b\ni)z\nMi\u0000(RiBext\ni)y+X\nj(RiJijRT\njmloc\nj)y+ i!X\nj(RiGijRT\njmloc\nj)y(B4)\ni!\n\rMimy\ni=\u0000mx\ni(RiBe\u000b\ni)z\nMi+ (RiBext\ni)x\u0000X\nj(RiJijRT\njmloc\nj)x\u0000i!X\nj(RiGijRT\njmloc\nj)x; (B5)\nwhich in a compact form gives\nX\nj\n\f=x;y0\n@\u000eij\u0012\n\u000e\u000b\f(RiBe\u000b\ni)z\nMi+\u000f\u000b\f\u0016i!\n\rMi\u0013\n+X\nj(RiJijRT\nj)\u000b\f+ i!X\nj(RiGijRT\nj)\u000b\f1\nAm\f\nj= (RiBext\ni)\u000b; (B6)\nand can be related to the inverse of the magnetic susceptibility\nX\nj\n\f=x;y\u001f\u00001\ni\u000b;j\f(!)m\f\nj= (RiBext\ni)\u000b: (B7)\nThus, the magnetic susceptibility in the local frames of site iandjis given by\n\u001f\u00001\ni\u000b;j\f(!) =\u000eij\u0012\n\u000e\u000b\f(RiBe\u000b\ni)z\nMi+\u000f\u000b\f\u0016i!\n\rMi\u0013\n+X\nj(RiJijRT\nj)\u000b\f+ i!X\nj(RiGijRT\nj)\u000b\f(B8)\nAppendix C: Alexander-Anderson model{more details\nWe use a single orbital Alexander-Anderson model,\nH=X\nij[\u000eij(Ed\u0000i \u0000\u0000Uimi\u0001\u001b\u0000Bi\u0001\u001b)\u0000(1\u0000\u000eij)tij]; (C1)15\nwhereiandjsum over all n-sites,Edis the energy of the localized orbitals, \u0000 is the hybridization in the wide band\nlimit,Uiis the local interaction responsible for the formation of a magnetic moment, miis the magnetic moment of site\ni,Biis an constraining or external magnetic \feld, \u001bare the Pauli matrices, and tijis the hopping parameter between\nsiteiandj, which can be in general spin-dependent. SOC is added as spin-dependent hopping using a Rashba-like\nspin-momentum locking tij=t(cos'R\u001b0\u0000i sin'Rnij\u0001\u001b), where the spin-dependent hopping is characterized by its\nstrength de\fned by 'Rand its direction nij=\u0000nji34. The eigenenergies and eigenstates of the model are given by,\nHjni= (En\u0000i \u0000)jni: (C2)\nThe single particle Green function can be de\fned using the eigensystem,\nG(E+ i\u0011) =X\nnjnihnj\nE\u0000En+ i\u0011; (C3)\nwhere\u0011is an in\fnitesimal parameter de\fning the retarded ( \u0011!0+) and advanced ( \u0011!0\u0000) Green function. The\nmagnitude of the magnetic moment is determined self-consistently using\nmi=\u00001\n\u0019Im TrZ\ndE\u001bGii(E); (C4)\nwhereGii(E) is the local Green function of site idepending on the magnetic moment. Using the magnetic torque\nexerted on the moment of site i,\ndH\nd^ei=\u0000miBe\u000b\ni; (C5)\nmagnetic constraining \felds can be de\fned ensuring the stability of an arbitrary non-collinear con\fguration,\nBconstr=\u0000Pm\n?mi\njmijBe\u000b\ni) Hconstr=\u0000Bconstr\u0001\u001b ; (C6)\nwherePm\n?is the projection on the plane perpendicular to the moment m. The constraining \felds are added to the\nhamiltonian, eq. (C1), and determined self-consistently.\nAppendix D: Density functional theory{details\nThe density functional theory calculations were performed with the Korringa-Kohn-Rostoker (KKR) Green function\nmethod. We assume the atomic sphere approximation for the the potential and include full charge density in the\nself-consistent scheme40. Exchange and correlation e\u000bects are treated in the local spin density approximation (LSDA)\nas parametrized by Vosko, Wilk and Nusair41, and SOC is added to the scalar-relativistic approximation in a self-\nconsistent fashion42. We model the pristine surfaces utilizing a slab of 40 layers with the experimental lattice constant\nof Au assuming open boundary conditions in the stacking direction, and surrounded by two vacuum regions. No\nrelaxation of the surface layer is considered, as it was shown to be negligible43. We use 450\u0002450k-points in the\ntwo-dimensional Brillouin zone, and the angular momentum expansions for the scattering problem are carried out up\nto`max= 3. Each adatom is placed in the fcc-stacking position on the surface, using the embedding KKR method.\nPreviously reported relaxations towards the surface of 3 dadatoms deposited on the Au(111) surface44indicate a\nweak dependence of the relaxation on the chemical nature of the element. Therefore, we use a relaxation towards the\nsurface of 20 % of the inter-layer distance for all the considered dimers. The embedding region consists of a spherical\ncluster around each magnetic adatom, including the nearest-neighbor surface atoms. 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L upke1,x\n1Department of Applied Science, The College of William and Mary, Williamsburg, Virginia, 23187, USA\n2Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education),\nShanghai Ultra-precision Optical Manufacturing Engineering Research Center,\nDepartment of Optical Science and Engineering, Fudan University, Shanghai, 200433, China\n3State Key Laboratory of Supperlattices and Microstructures,\nInstitute of Semiconductors, Chinese Academy of Sciences, Beijing, 100083, China\n(Dated: May 10, 2022)\nSpin angular momentum transfer in magnetic bilayers o\u000bers the possibility of ultrafast and low-loss\noperation for next-generation spintronic devices. We report the \feld- and temperature- dependent\nmeasurements on the magnetization precessions in Co 2FeAl/(Ga,Mn)As by time-resolved magneto-\noptical Kerr e\u000bect (TRMOKE). Analysis of the e\u000bective Gilbert damping and phase shift indicates\na clear signature of an enhanced dynamic exchange coupling between the two ferromagnetic (FM)\nlayers due to the reinforced spin pumping at resonance. The temperature dependence of the dynamic\nexchange-coupling reveals a primary contribution from the ferromagnetism in (Ga,Mn)As.\nINTRODUCTION\nThere has been growing interest in the ultrafast op-\ntical manipulation of magnetic dynamics in ferromag-\nnetic heterostructures due to its potential applications\nin advanced functional spintronic devices. The spin-\npumping (SP) e\u000bect, in which a spin-precessing ferro-\nmagnetic layer transfers its angular momentum into an-\nother layer by a chargeless spin current, brings a new\nmechanism for spin controlling and hence plays an im-\nportant role in the design of future spintronic devices.[1]\nSince Heinrich, B. et al \frst reported the spin-pumping\ne\u000bect as increased damping of the source layer in fer-\nromagnetic resonance (FMR) experiments [2] and a few\nFMR experiments on SP e\u000bect have been performed on\ntransition-metal multilayers [3{6], topological insulators\n[7, 8] and semiconductors [9{11]. In addition, Danilov\nA. et al. demonstrated that the mutual SP e\u000bect modi-\n\fes the precession dynamics in a pseudo spin-valve where\nmagnetization precessions are excited simultaneously in\ntwo FM layers by femtosecond laser pulses.[12] However,\nno SP e\u000bect has ever been observed yet for the het-\nerostructure of a Heusler alloy and a ferromagnetic (FM)\nsemiconductor. Importantly, the hard and soft ferromag-\nnetic phases in such materials can potentially exhibit a\ndynamic exchange coupling that is completely indepen-\ndent of the static exchange coupling due to spin pumping.\nThis could o\u000ber a possibility of ultrafast low-power con-\ntrol of spin current for next-generation spintronic devices.\nIn this study, we investigate the magnetization preces-\nsion dynamics of the Heusler alloy Co 2FeAl/FM semi-\nconductor (Ga,Mn)As heterostructure as a function of\napplied \feld and temperature by time-resolved magneto-\noptical Kerr e\u000bect (TRMOKE). Analysis of the \feld-\ndependent e\u000bective Gilbert damping indicates a clear\nsignature of the enhanced dynamic exchange couplingbetween the two FM layers due to a reinforced spin\npumping. In addition, curvatures of the phase shift as a\nfunction of applied \feld elucidate the dynamic exchange-\ncoupling model where the counter-precessing precessions\nare damped signi\fcantly at the resonant frequency of\nthe two FM layers. The magnetization precession in the\nCo2FeAl layer transfers a pure spin current directly into\nthe ferromagnetic semiconductor (Ga,Mn)As layer with-\nout a nonmagnetic metal spacer. On the other hand,\nthe temperature-dependent results manifest a strong con-\ntribution from the ferromagnetism of (Ga,Mn)As to the\ndynamic exchange-coupling e\u000bect. These results provide\nvaluable insight into the topic of dynamic exchange cou-\npling and the detection of spin current. Furthermore,\nthey suggest a new pathway of ultrafast spin manipula-\ntion in metal/semiconductor bilayer systems at low power\nand therefore promote the development and design of fu-\nture spintronic devices.\nRESULTS AND DISCUSSION\nThe Co 2FeAl/(Ga,Mn)As bilayer sample is grown\non GaAs (001) substrates by molecular-beam epitaxy\n(MBE). The thickness of Co2FeAl and Ga1-xMnxAs\n(x=0.07) layer is 10 nm and 150 nm, respectively. The\nsample is capped with 2-nm thick Al layer to avoid ox-\nidation and contamination. The hard FM Co 2FeAl ex-\nhibits an in-plane uniaxial magnetic anisotropy with an\neasy axis along the [110] direction (Fig. 1(a)), whereas\nthe easy axis of the soft FM (Ga,Mn)As is along the [1-\n10] direction below its Curie temperature Tc= 50 K.[13]\nRe\rection high-energy electron di\u000braction (RHEED)\npatterns, high-resolution double-crystal x-ray di\u000brac-\ntion (DCXRD) measurements, and high-resolution cross-\nsectional transmission electron microscopy (HRTEM) re-arXiv:2203.03225v2  [cond-mat.mtrl-sci]  7 May 20222\nFIG. 1. (a) Schematic of TRMOKE measurement geom-\netry, depicting the structure of the sample and the mag-\nnetization M precessing around the e\u000bective \feld He\u000b in\nCo2FeAl/(Ga,Mn)As bilayer in a canted magnetization con-\n\fguration with H applied along hard axis [1-10]. (b)\nTRMOKE data from Co 2FeAl/(Ga,Mn)As bilayer under the\ndi\u000berent intensity of the applied \feld. (c) FFT analysis for\nmagnetization precession frequency under the di\u000berent inten-\nsities of the applied \feld, where the solid lines represent the\nFFT peaks \ftted by Lorentz functions.\nveal high-quality, single-crystalline, epitaxial growth of\nthe Co2FeAl and (Ga,Mn)As thin \flms.[13] At low tem-\nperatures ( T < T c), a ferromagnetic alignment of lo-\ncal Mn moments in the (Ga,Mn)As layer is expected,\nwhereas at high temperatures ( T >T c) the Mn ions ex-\ntending a few nanometers from the interface remain spin-\npolarized due to the ferromagnetic proximity e\u000bect.[13]\nFigure 1(a) shows the experimental geometry\nof TRMOKE measurements. Field-dependent\nTRMOKE measurements are performed on the\nCo2FeAl/(Ga,Mn)As bilayer sample from 7 K to\n300 K utilizing 100-fs pump and probe pulses at 800\nnm wavelength with a repetition rate of 80 MHz. The\nexternal magnetic \feld is set along the easy-hard axis\n[100] of the Co 2FeAl layer and the pump \ruence is\nset atI= 5\u0016J/cm2. The probe pulses utilize the\nbalanced detection technique with a half-wave plate and\nWollaston prism to investigate the transient magnetic\nstate change along longitudinal and polar directions.\nFigure 1b displays the TRMOKE data at 10 K with\nan in-plane magnetic \feld scanning from 366 Oe to\n550 Oe. The precession signals can be well \ftted by a\ndamped-harmonic function with a linear background:\n\u0012k=a0+b0t+Aexp (\u0000t=\u001c) sin (2\u0019ft+'0) , where\na0+b0trepresents the linear background, Ais the\nprecession amplitude, \u001cis the relaxation time, f is the\nprecession frequency and '0is the phase. The mag-\nnetization precession decays with di\u000berent relaxation\ntimes, with the fastest decay at 458 Oe. This indicates\nthat a dynamic exchange coupling may occur between\nthe magnetization precession in the Co 2FeAl layer and\nthe (Ga,Mn)As layer. Here, only one frequency can\nbe extracted from FFT analyses, as seen in Fig. 1(c),\ncoherent spin precession of (Ga,Mn)As decays fast and\nthus vanishes very shortly.\nThe magnetization precession in the\nCo2FeAl/(Ga,Mn)As bilayer system is described bythe following modi\fed Landau-Lifshitz-Gilbert(LLG)\nequation with an additional spin-torque term:\ndM\ndt=\u0000\rM\u0002He\u000b+\u000b0M\u0002dM\ndt+\u000bsp\u0012\nM\u0002dM\ndt\u0000M0\u0002dM0\ndt\u0013\nwhere Mis the magnetization direction of the Co 2FeAl\nlayer,\ris the gyromagnetic ratio, \u000b0is the intrinsic\nGilbert damping constant, and He\u000bis the e\u000bective mag-\nnetic \feld in the Co 2FeAl layer including the external\nmagnetic \feld, the demagnetization \feld, the anisotropy\n\feld, and the exchange-coupling \feld. The last term de-\nscribes the spin torque which acts on both layers as a\nbidirectional e\u000bect, in which \u000bsprepresents the contribu-\ntion of spin pumping to the damping and M0denotes the\nmagnetization of (Ga,Mn)As. Then, the e\u000bective Gilbert\ndamping can be obtained from the relaxation time \u001c, us-\ning [14, 15]\n\u000b= 2=(\u001c\r(Ha+Hb))\nwhereHaandHbare determined in our previous anal-\nysis, which includes the out-of-plane, in-plane uniaxial,\ncrystalline cubic, unidirectional and rotatable magnetic\nanisotropies.[15]\nFigure 2(a) shows the temperature-dependent Gilbert\ndamping as a function of the external \feld. At T=\n10 K, the damping of the magnetization precession is\nmost pronounced with an external \feld H= 458 Oe. Be-\nlow the Curie temperature ( Tc= 50 K) of (Ga,Mn)As,\nthe damping peak \frst shifts from 450 Oe to 650 Oe\nwith the temperature increasing from 10 K to 35 K, as\nsummarized in Fig. 2(b) insert, and then gradually be-\ncomes inconspicuous and \fnally disappears at T=Tc.\nSuch a temperature dependency clearly shows that the\ndamping peak only exists when the ferromagnetism of\n(Ga,Mn)As is well-developed. Meanwhile, the strongest\nGilbert damping extracted across all the \felds as a func-\ntion of temperature (Fig. 2(b)) shows a transition tem-\nperature close to Tc. This manifests the crucial role of\nspontaneous (Ga,Mn)As magnetization in the damping\nof magnetization precession of Co 2FeAl.\nThe ultrafast pump excitation causes a transient en-\nhancement of exchange coupling,[15] which induces a dy-\nnamic exchange-coupling torque acting on both Fe(Co)\nspins and Mn spins. In such a case, as shown in Fig. 2(c),\nthe magnetizations of both FM layers are suddenly pulled\ntowards each other and start to precess with opposite an-\ngular momentum along their own equilibrium directions.\nAt the resonance, i.e., fCFA =fGMA , the precessing\nmagnetization of Co 2FeAl \\pumps\" a spin current Iidi-\nrectly into the (Ga,Mn)As layer, which exerts a torque\nonto the (Ga,Mn)As magnetization and thereby counter-\nacts its precession. Meanwhile, this spin current Iicar-\nries an out\row of angular momentum from the Co 2FeAl3\nFIG. 2. (a) E\u000bective Gilbert damping constant as a func-\ntion of externally applied \feld at di\u000berent temperatures from\n10 K to 50 K. (b) Red rectangular solids denote the peak\n(or strongest if not peak) Gilbert damping from 10 K to\n50 K. Blue circle solids are the corresponding \felds as a\nfunction of temperature. (c) Illustration of photo-excited\nexchange-coupling torque and spin-pumping generated dy-\nnamic exchange-coupling mode for damping.\nlayer and leads to damping to its magnetization preces-\nsion. In other words, the spin current reinforces the M\ndamping for both FM layers at the resonance. Techni-\ncally, there should also be a spin current Ijinjecting into\nthe Co 2FeAl layer from the magnetization precession of\n(Ga,Mn)As. [2] However, such a spin current should be\nmuch smaller than that from the Co 2FeAl layer.\nIn addition to the discussion on damping, the dynamic\nexchange coupling between the two FM layers can also be\nevinced by the \feld-dependency of the precession phase.\nFig. 3(a) shows that from 10 K to 35 K, the phase drops\ndown dramatically around certain \feld windows that\ncorrespond to the Gilbert damping peaks, which move\nto higher \feld ranges as temperature increases. When\nT > 30 K, the dramatic phase shift becomes less con-\ntrastive and then completely disappears when T=Tc.\nWe notice that similar features of phase-shifting are re-\nported in the FMR experiments on similar hard/soft FM\nsystems.[16, 17] The observed 30 °{ 40 °phase shift at 10\nK, as shown in Fig. 3(c), is comparable with those of the\ndynamic exchange-coupling spin-valve structures.[7, 16{\n19]\nFIG. 3. Phase of magnetization precession as a function of\nexternally applied \feld at di\u000berent temperatures from 10 K to\n50 K (a), and zoomed speci\fcally at 10 K (b). (c) TRMOKE\ndata as a function of normalized precession phase under dif-\nferent applied \felds. The red straight line is a guideline of\nthe phase shift.\nCONCLUSION\nIn summary, we have studied the dynamics of the\nmagnetization precession of Co 2FeAl/(Ga,Mn)As het-\nerostructure as a function of applied \feld and temper-\nature. The pronounced peaks in e\u000bective Gilbert damp-\ning of Co 2FeAl magnetization precession reveal the en-\nhanced dynamic exchange coupling between the two FM\nlayers due to spin pumping. The corresponding \feld-\ndependency of phase shift, which corresponds to that\nof the Gilbert damping, con\frms the counter-precessing\nexchange-coupled model where both precessions are\ndamped signi\fcantly at the resonance. In addition, the\ntemperature-dependent results manifest a strong contri-\nbution from the ferromagnetism in (Ga,Mn)As to the dy-\nnamic exchange-coupling e\u000bect. These results provide\nvaluable insight into the topic of dynamic exchange cou-\npling and the detection of spin current. Mover, they sug-\ngest a novel route of ultrafast low-power spin manipu-\nlation in metal/semiconductor bilayer system and hence\npromote the research of the future spintronic devices.4\nMETHODS\nMOKE experiments. The magnetization of the\nexchange-coupled Co 2FeAl/(Ga,Mn)As bilayer is mea-\nsured using a longitudinal MOKE setup. The sample\nis illuminated with p-polarized light and the re\rected s-\npolarized light is detected with a photodiode. The mag-\nnetic \feld is applied along the in-plane [110] or [-110]\ncrystallographic directions. The measurements are con-\nducted from 5 K to above room temperature.\nTRMOKE experiments. For the pump-probe\nTRMOKE measurements, a Ti:sapphire oscillator laser\nsystem is employed, which produces 150-fs pulses at 800-\nnm wavelength with a repetition rate of 80 MHz. The\nprobe(pump) \ruence is \fxed at \u00180.5(5)\u0016J/cm2. The\nprobe pulses ( \u0015= 800 nm) use the balanced detection\napproach with a half-wave plate and Wollaston prism to\ninvestigate the transient magnetic state change along lon-\ngitudinal and polar directions. The measurements are\nconducted from 5 K to above room temperature.\nDATA AVAILABILITY\nThe data that support the \fndings of this study are\navailable from the corresponding author upon reasonable\nrequest.\nACKNOWLEDGMENTS\nThe work at the College of William and Mary was\nsponsored by the DOE through Grant No. DEFG02-\n04ER46127. The work at the Department of Optical\nScience and Engineering, Fudan University, was sup-\nported by the National Natural Science Foundation of\nChina with Grant No. 11774064, National Key Re-\nsearch and Development Program of China (Grant No.\n2016YFA0300703), and National Key Basic Research\nProgram (No. 2015CB921403). The work at the State\nKey Laboratory of Superlattices and Microstructures, In-\nstitute of Semiconductors, Chinese Academy of Sciences,\nwas supported by National Natural Science Foundation\nof China with Grant No. U1632264.\nAUTHOR CONTRIBUTIONS\nX. L., P. L., H. C. Y, J. H. Z., H. B. Z., and G. L.\ndesigned and analyzed the experiments. H. L. W., S.\nH. N., J. Y. S., and X. Z. Y. prepared the samples and\ncarried out characterizations using MOKE, RHEED, and\nSQUID measurements. H. C. Y., J. Y. S., X. L., P. L.,\nand F. J. performed the TRMOKE experiments. X. L.,\nP. L., H. C. Y. and J. Y. S. conducted the data analysis\nand simulations. All authors discussed the results. X. L.,P. L., J. H. Z., H. B. Z., and G. L. wrote the manuscript\nwith contributions from all authors.\nCOMPETING INTERESTS\nThe authors declare no competing \fnancial interests.\n\u0003These authors contributed equally to the work.\nyemail: jhzhao@red.semi.ac.cn\nzemail: hbzhao@fudan.edu.cn\nxemail: gxluep@wm.edu\n[1] Tserkovnyak, Y. et al. Enhanced Gilbert damping in thin\nferromagnetic \flm. Phys. Rev. Lett. 88, 117601 (2002).\n[2] Heinrich, B. et al. Dynamic exchange coupling in mag-\nnetic bilayers. Phys. Rev. Lett. 90, 187601 (2003).\n[3] Stenning, G. B. G. et al. Magnetization dynamics in an\nexchange-coupled NiFe/CoFe bilayer studied by x-ray de-\ntected ferromagnetic resonance. New J. Phys. 17, 013019\n(2015).\n[4] van der Laan, G. et al. Time-resolved X-ray detected fer-\nromagnetic resonance of spin currents. J. Electron Spec-\ntrosc. 220, 137-146 (2017).\n[5] Saitoh, E. et al. Conversion of spin current into charge\ncurrent at room temperature: Inverse spin-Hall e\u000bect.\nAppl. Phys. Lett. 88, 182509 (2006).\n[6] Costache, M. V. et al. Electrical detection of spin pump-\ning due to the precessing magnetization of a single ferro-\nmagnet. Phys. Rev. Lett. 97, 216603 (2006).\n[7] Figueroa, A. I. et al. Spin pumping through a topo-\nlogical insulator probed by x-ray detected ferromagnetic\nresonance. J. Magn. Magn. Mater. 400, 178-183 (2016).\n[8] Shiomi, Y. et al. Spin-electricity conversion induced by\nspin injection into topological insulators. Phys. Rev. Lett.\n113, 196601 (2014).\n[9] Shikoh, E. et al. Spin-pump-induced spin transport in\np-type Si at room temperature. Phys. Rev. Lett. 110,\n127201 (2013).\n[10]\u0014Zuti\u0013 c, I. & Dery, H. Spintronics: Taming spin currents.\nNat. Mater. 10, 647 (2011).\n[11] Ando, K. et al. Electrically tunable spin injector free\nfrom the impedance mismatch problem. Nat. Mater. 10,\n655 (2011).\n[12] Danilov, A. P. et al. Optically excited spin pumping\nmediating collective magnetization dynamics in a spin\nvalve structure. Phys. Rev. B 98, 060406 (2018).\n[13] Nie, S. H. et al. Ferromagnetic interfacial interaction\nand the proximity e\u000bect in a Co2FeAl/(Ga,Mn)As bilaye.\nPhys. Rev. Lett. 111, 027203 (2013).\n[14] Zhao, H. B. et al. Ultrafast magnetization dynamics of\nepitaxial Fe \flms on AlGaAs (001). Appl. Phys. Lett. 86,\n152512 (2005).\n[15] Liu, X. et al. Ultrafast enhancement of interfa-\ncial exchange coupling in ferromagnetic bilayer. ,\narXiv:2203.00293 ().\n[16] Marcham, M. K. et al. Phase-resolved x-ray ferromag-\nnetic resonance measurements of spin pumping in spin\nvalve structures. Phys. Rev. B 87, 180403 (2013).\n[17] Baker, A. A. et al. Anisotropic absorption of pure spin\ncurrents. Phys. Rev. Lett 116, 047201 (2016).5\n[18] Li, J. et al. Direct detection of pure ac spin current by\nx-ray pump-probe measurements. Phys. Rev. Lett 117,\n076602 (2016).\n[19] Baker, A. A. et al. Spin pumping in magnetic trilayer\nstructures with an MgO barrier. Sci. Rep. 6, 35582(2016)."    },    {        "title": "2204.09923v1.Transport_theory_for_topological_Josephson_junctions_with_a_Majorana_qubit.pdf",        "content": "Transport theory for topological Josephson junctions with a Majorana qubit\nZhi Wang,1,\u0003Jia-Jin Feng,2,\u0003Zhao Huang,3,yand Qian Niu4\n1School of Physics, Sun Yat-sen University, Guangzhou 510275, China\n2International Center for Quantum Materials, Peking University, Beijing 100871, China\n3Texas Center for Superconductivity, University of Houston, Houston, Texas 77204, USA\n4School of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China\nWe construct a semiclassical theory for the transport of topological junctions starting from a microscopic\nHamiltonian that comprehensively includes the interplay among the Majorana qubit, the Josephson phase, and\nthedissipationprocess. Withthepathintegralapproach,wederiveasetofsemiclassicalequationsofmotionthat\ncan be used to calculate the time evolution of the Josephson phase and the Majorana qubit. In the equations we\nrevealrichdynamicalphenomenasuchasthequbitinducedchargepumping,theeffectivespin-orbittorque,and\nthe Gilbert damping. We demonstrate the influence of these dynamical phenomena on the transport signatures\nof the junction. We apply the theory to study the Shapiro steps of the junction, and find the suppression of the\nfirst Shapiro step due to the dynamical feedback of the Majorana qubit.\nIntroduction. –Josephsonphysicsreceivesrevivinginterests\nduetotherapidprogressofsuperconductingquantumcompu-\ntationinrecentyears[1,2]. Thedemandfortheminificationof\nthe superconducting quantum circuits pushes the limits of the\nsizeofJosephsonjunctions[3,4]. Forjunctionsthataresmall\nenough,asingleembeddedqubitmaysignificantlymodifythe\ntransport signatures[5]. While this effect has been discussed\nin a number of systems with various models[6–8], a compre-\nhensive theory that takes account the qubit dynamics is still\nabsent.\nThis issue is particularly relevant to topological Josephson\njunctionswithMajoranazeromodes[9–31]. ThetwoMajorana\nzero modes in the junction construct a Majorana qubit which\nresults in a 4𝜋-periodic Josephson current[32, 33]. Previous\ntheoretical studies take a variety of phenomenological mod-\nels which are different extensions of the standard resistively\nshunted junction model for conventional junctions [6, 7, 34–\n37]. However, these phenomenological models still have dif-\nficultiesinexplainingtheexperimentalreportedtransportfea-\ntures such as the suppression of the first Shapiro step[38–45].\nIt is highly desirable to construct a microscopic theory to ex-\namine the validity of the phenomenological models and to\nunderstand the experimental results.\nInthiswork,wedevelopasemiclassicaltheoryforstudying\nthe transport properties of the topological junctions. Our the-\nory starts from a microscopic Hamiltonian that characterizes\nthe coupling between the Josephson junction and the Majo-\nranaqubit. Wetakeapathintegralapproachtoincorporatethe\ndissipation process that is essential for studying the transport\nproperties,andderivethesemiclassicalequationsofmotionfor\ntheJosephsonphaseandtheMajoranaqubit. Intheequations\nof motion, we identify the effective spin-orbit torque and the\nGilbert damping in the qubit dynamics, and reveal the charge\npumping driven by the qubit rotation in the dynamics of the\nJosephson phase. Solving the equations of motion, we ob-\ntainthetimeevolutionoftheJosephsonphasewhichprovides\ntransport and spectroscopic signatures for the junction.\nAs an application of this theory, we calculate the Shapiro\nsteps of the topological junction. We find that the first step\nis strongly suppressed while higher odd-number steps are ro-bustlyvisibleforarangeofjunctionparameters. Weshowthat\nthis bizarre behavior is due to the feedback of the Majorana\nqubitdynamicstothetransportofthejunction. Atthevoltage\nof the first step, the Majorana qubit evolves to a stable state\nwhich supports a finite 4𝜋-period Josephson current, and this\n4𝜋-periodicity in the Josephson phase dynamics suppresses\nthefirstShapirostep. Atvoltagesofhigherodd-numbersteps,\nhowever, the Majorana qubit evolves to a stable state which\ncontributes a vanishing 4𝜋-period Josephson current, and the\nShapiro steps are naturally intact. Our theory provides an\nintrinsic mechanism for the reported Shapiro step missing in\ntopological junctions.\nMicroscopic Hamiltonian and equations of motion. –The\nlow-energy effective Hamiltonian for the junction with a Ma-\nTSC TSC\nMajorana qubit(a)\n(b)\n0 1 2-101(c)\n0 1 2-101\nFIG. 1. (a) Schematic illustration of the two tunneling processes\nthrough the Majorana qubit: The qubit assisted half-pair tunneling\nthat leads to the fractional Josephson effect, and the qubit rotation\ninduced charge pumping. (b) The energy levels of the Hamiltonian\nh\u0001ˆ𝜎. Landau-Zener transitions happen at the anti-crossing points\nat𝜃=¹2𝑛¸1º𝜋. (c) The energy levels of the Hamiltonian ¹h¸\n¤𝜃B𝑓º\u0001ˆ𝜎for high voltage of ¤𝜃𝐵𝑓\u001d𝐸M, where B𝑓=𝐵𝑓ˆ𝑥. The\nLandau-Zener transition at the anti-crossing points are significantly\nsuppressed and the qubit dynamics would follow one of the levels.arXiv:2204.09923v1  [cond-mat.supr-con]  21 Apr 20222\njorana qubit can be written as[46],\nHJ=ˆ𝑝2\n𝜃\n𝐶0\u0000𝐸Jcosˆ𝜃\u0000𝐼exˆ𝜃\u0000𝐸M𝜎𝑧cosˆ𝜃\n2¸𝐸0\nM𝜎𝑥(1)\nwhere ˆ𝜃is the Josephson phase with canonical momentum\nˆ𝑝𝜃=2𝑒𝜕𝜃,𝐸Jis the Josephson energy, 𝐶0=2𝐶~2¹2𝑒º2\nis a dimensionless constant determined by the effective ca-\npacitance𝐶of the junction, 𝐼exrepresents the experimentally\ncontrollable external current injected into the junction, 𝜎𝑥𝑧\nare Pauli matrices which represent the pseudo-spin direction\nfor the Majorana qubit, 𝐸Mand𝐸0\nMrepresent the energies\nof the Majorana qubit from various couplings between Ma-\njorana zero modes. The first three terms of the Hamiltonian\nhave been widely adopted for studying conventional Joseph-\nson junctions[5], while the last two terms come from the\nMajorana zero modes[32, 47] and can be derived from the\nBogoliubov- de Gennes Hamiltonian of a topological Joseph-\nson junction[48, 49].\nThis Hamiltonian can be understood as describing a spin-\none-half particlewith a mass of 𝐶0, moving under a po-\ntential energy 𝑈𝑝=\u0000𝐸Jcos𝜃\u0000𝐼ex𝜃, and a Zeeman en-\nergy𝑈𝑧=h\u0001ˆ𝜎where the direction of the Zeeman field\nh=\u0000𝐸0\nM0\u0000𝐸Mcos𝜃\n2\u0001varies along the path of the motion.\nThepotentialenergyisidenticaltothetilted-washboardpoten-\ntialthatwastakeninstudyingconventionaljunctions[5],while\nthe unique Zeeman energy comes from the coupling between\nthe Josephson phase and the Majorana qubit.\nThe time-evolution of the Josephson phase determines the\ntransport properties of the junction through the ac Josephson\nrelation[50]. To derive the equation of motion for this time-\nevolution, we rewrite the Hamiltonian of Eq. (1) into an\naction[46, 51, 52],\n𝑆J=∫\n𝑑𝑡\u0012𝐶0\n2¤𝜃2¸𝐸Jcos𝜃¸𝐼ex𝜃¸A𝑠\u0001¤s¸h\u0001s\u0013\n(2)\nwhere s=𝜓yˆ𝜎𝜓=¹sin𝜑sin𝜙sin𝜑cos𝜙cos𝜑ºrepre-\nsents the psuedo-spin state on the Bloch sphere with 𝜓=\n¹𝑒\u0000𝑖𝜙cos𝜑\n2sin𝜑\n2ºthe spinor wave function of the qubit,\nA𝑠=ˆe𝜙¹1\u0000cos𝜑ºsin𝜑represents the Berry connection\non the Bloch sphere[52], which provides a Berry curvature\nofr\u0002A𝑠=s. The extreme action path of Eq. (2) gives\nthesemiclassicalequationsofmotionfortheJosephsonphase\n𝐶0¥𝜃¸𝐼𝑐1sin𝜃¸𝐼𝑐2𝑠𝑧sin𝜃\n2\u0000𝐼ex=0, and the pseudo spin\n¤s=h\u0002s, where𝐼𝑐1=2𝑒𝐸J~is the supercurrent from the\nCooper pair tunneling and 𝐼𝑐2=𝑒𝐸M~is the supercurrent\nfrom the half-pair tunneling through the Majorana qubit. The\nequations of motion explicitly demonstrate the coupling be-\ntween the Josephson phase and the Majorana qubit through\nthe𝑠𝑧dependent Zeeman term in the first equation and the\n𝜃dependent effective magnetic field in the second equation.\nHowever,theseequationsareinadequateforstudyingthetrans-\nportpropertiesofthejunction. Themissingpieceisthedissi-\npation process.\nTo include the dissipation into the equations of motion, we\nfollowtheCaldeira-Leggettapproachandintroduceathermalbath of harmonic modes to characterize the environment[53,\n54]. Theenvironmentaldegreesoffreedomandtheircoupling\nwith the junction can be described with the action[46],\n𝑆en=∑︁\n𝑖∫\n𝑑𝑡\u00121\n2\u0010\n¤ℎ2\n𝑖\u0000Ω2\n𝑖ℎ2\n𝑖\u0011\n¸ℎ𝑖¹𝑔𝑖𝜃¸B𝑖\u0001sº\u0013\n(3)\nwhereℎ𝑖are the coordinates of the environmental modes and\nΩ𝑖are their oscillating energies, 𝑔𝑖represents the minimal\ncouplingbetweentheenvironmentalmodesandtheJosephson\nphase[54], and B𝑖represents the minimal coupling between\nthe environmental modes and the qubit[55]. The details of\n𝑔𝑖andB𝑖are determined by the coupling between each envi-\nronmental mode and the junction. For topological junctions\ndescribed by Eq. (1), the environmental modes that modulate\nthe tunneling barrier of the junction can be understood as a\nfluctuation on the amplitude of 𝐸Mand we have B𝑖=𝐵𝑖ˆ𝑧,\nwhile the environmental modes that modulate the coupling\nbetween Majorana zero modes in one side of the junction can\nbeunderstoodasafluctuationon 𝐸0\nMandwehave B𝑖=𝐵𝑖ˆ𝑥. In\ngeneral, the environment modes may originate from multiple\nsources, such as electromagnetic perturbations, lattice vibra-\ntions, as well as thermally activated electrons. These modes\ncould have much more complex couplings with the junction,\nwhile the Caldeira-Leggett model is the minimal model for\ndescribing the dissipation processes.\nThe dissipated evolutionof the junction can beobtained by\nintegratingouttheenvironmentaldegreesoffreedom. Thisis\nachievable since the environment is modeled with harmonic\nmodes. After the integration, we arrive at an effective action\nfor the junction variables[46],\n𝑆eff=∫\n𝑑𝑡\u0012𝐶0\n2¤𝜃2¸𝐸Jcos𝜃¸𝐼ex𝜃¸A𝑠\u0001¤s¸h\u0001s\u0013\n(4)\n¸1\n4∫\n𝑑𝑡𝑑𝑡0»𝜂𝛼¹𝑡º\u0000𝜂𝛼¹𝑡0º¼𝐺𝛼𝛽¹𝑡𝑡0º\u0002\n𝜂𝛽¹𝑡º\u0000𝜂𝛽¹𝑡0º\u0003\n\nwhere𝜂𝛼=¹𝜃𝑠𝑥𝑠𝑦𝑠𝑧ºrepresents the junction degree of\nfreedom,𝐺𝛼𝛽¹𝑡𝑡0º=\u0000𝑖˜𝑀\n𝜋1\nj𝑡\u0000𝑡0j2is the averaged Green func-\ntionfromtheenvironmentalmodes,with ˜𝑀theaveragedcou-\nplingmatrix[46]. Theleastactionpathforthiseffectiveaction\nprovides the full semiclassical equations of motion for the\njunction variables,\n𝐼ex=𝐶0¥𝜃¸¤𝜃\n𝑅¸𝐼𝑐1sin𝜃¸𝐼𝑐2𝑠𝑧sin𝜃\n2¸B𝑓\u0001¤s(5a)\n¤s=h\u0002s¸¤𝜃B𝑓\u0002s¸¹˜𝛾\u0001¤sº\u0002s (5b)\nwhere𝑅=1Í\n𝑖𝑔2\n𝑖is the effective resistance of the junction\nwhichcomesfromthecouplingbetweentheenvironmentand\nthe Josephson phase, B𝑓=Í\n𝑖𝑔𝑖B𝑖is the environment medi-\natedcouplingfieldbetweentheJosephsonphaseandthequbit,\nand˜𝛾𝛼𝛽=Í\n𝑖𝐵𝑖𝛼𝐵𝑖𝛽representstheenvironmentinduceddis-\nsipation to the qubit.\nThe equation (5) is the central result of this work. Solving\nthese two self-consistent equations we can obtain 𝜃¹𝑡ºwhich3\ndetermines the dc and ac voltage of the junction through the\nJosephson relation 𝑉¹𝑡º=~¤𝜃¹𝑡º2𝑒. Starting from a mi-\ncroscopic Hamiltonian, we provide a framework to study the\ntransportpropertiesofaJosephsonjunctionwithanembedded\nMajorana qubit.\nPhysical interpretation of the equations of motion. – Let us\nsetup a physicalpicture for interpreting the termsin the equa-\ntions of motion, particularly those terms coming from the\nembedded Majorana qubit. The equation (5a) is a current\nconservationequationwhichstatesthattheexternallyinjected\ncurrent𝐼exequals the current flowing through all the physical\nchannels in the junction. If the qubit is completely ignored,\nthenthelasttwotermsontherightsideoftheEq. (5a)should\nbe dropped out and the equation becomes a self-consistent\nequation for the Josephson phase. This is exactly the resis-\ntively and capacitively shunted junction model that has been\nwidely used for studying bulk Josephson junctions[5].\nThe qubit provides two additional terms in Eq. (5a) as\nillustratedinFig. 1a. Thefirsttermisthe 4𝜋-periodicJoseph-\nson current 𝐼𝑐2���𝑧sin𝜃\n2, which is linearly dependent on the\nz-component of the pseudo spin. This is the extensively dis-\ncussed fractional Josephson effect[32, 48, 56], which comes\nfromthequbitassistedhalf-pairtunnelinginthejunction. The\nother term is B𝑓\u0001¤swhich is non-vanishing only when the\npseudo spin rotates. Since the pseudo-spin state of the Ma-\njorana qubit is defined by the parity of the superconducting\nground state, this current represents the pumped current by\nthe parity flipping of the Majorana qubit. This qubit pumping\nhasneverbeenrevealedinpreviousmodels,andonlybecomes\napparent from the effective action of the microscopic theory.\nNow we take a closer look at the Eq. (5b). In the ab-\nsence of the environment, only the first term on the right\nside of equation survives. The residing equation, ¤s=h\u0002s,\ndescribes a qubit procession where the direction of the pro-\ncession hoscillates with 𝜃. When the oscillating component\nℎ𝑧is much larger than the stable component ℎ𝑥, the qubit\nwould evolve under an oscillating energy spectrum shown in\nFig. 1b. When 𝜃moves through the anti-crossing points\n𝜃=¹2𝑛¸1º𝜋,thequbitexperiencesLandau-Zenertransitions,\nandmultiplecoherentLandau-ZenertransitionsexhibitStück-\nelburg interference[57]. These effects have been thoroughly\nanalyzed in previous phenomenological models[6, 58].\nThe second term on the right side of Eq. (5b) is a unique\ndiscoveryofourtheory. Itresemblesaspin-orbittorquewhich\nlinearly depends on the velocity of the Josephson phase. This\nspin-orbittorquedominatesthequbitdynamicsathighvoltage\nof¤𝜃\u001d𝐸0\nM~,causingasignificantsuppressionoftheenergy\ncrossing and the Landau-Zener transition, as shown in Fig.\n1c. For this reason, the transport and spectroscopic signals\nof the junction are expected to exhibit qualitatively different\nbehaviors for different voltage regimes. This is useful for\nunderstandingthevoltage-dependentsignaturesthathavebeen\nwidelyreportedintheI-VcharacteristicscurvesandJosephson\nradiations of Josephson junctions constructed by topological\nsystems[10, 13, 19].\nThethirdtermontherightsideofEq. (5b)istheanisotropic\n01234\n104-101(a)\n01234\n104-101(b)FIG. 2. Typical time evolution of the Majorana qubit for the low\nvoltage regime (a) and the high voltage regime (b). The damped\noscillation are combined effect of the Landau-Zener-Stückelburg in-\nterference and the Gilbert damping. The two different stable values\nof the𝑠𝑧represent the different fixed points in the dynamics of the\nMajorana qubit.\nGilbertdampingwhichdeterminesthedissipationofthequbit\nfrom the coupling to the environment. For isotropic case\nwhere the matrix ˜𝛾becomes a number, this term turns into\nthe standard Gilbert damping which appeared in the Landau-\nLifshiz-Gilbert equation[59]. While the Gilbert damping has\nbeenwidelytakentostudythedynamicsofthemagnetization,\nourworkprovidesaderivationforitsmicroscopicorigininthe\nMajorana qubit. This damping process influences the dynam-\nics of the qubit and thereby modifies the transport properties\nof the junction.\nFinally, we hope to point out that if the environment me-\ndiated coupling B𝑓and the Gilbert damping ˜𝛾are ignored,\nthen Eq. (5) will reduce to the phenomenological quantum\nresistively and capacitively shunted junction model that has\nbeen taken to study the I-V characteristics and the Josephson\nradiationsofthetopologicaljunction[6,58]. Ourmicroscopic\ntheoryclarifiesthevalidityandlimitsofthephenomenological\nmodel.\nFixed point analysis. – The Eq. (5) are complicate nonlin-\near equations for which obtaining analytical solutions is im-\npossible. However, the fixed points of the equations can be\nanalyticalcalculatedwiththemethodofaveraging,whichisa\nmethod to decouple the nonlinear equations with the division\nofthedynamicalvariablestothe\"fastvariables\"andthe\"slow\nvariables\" based on their time scales[60]. In Eq. (5) we treat\nthe psuedo spin sas the slow variable since it has a larger\ntime scale. We take it as constant to solve the Eq. (5a) for the\nfastvariable 𝜃¹𝑡º,andthesolutionprovidesthetime-averaged\nJosephsonenergy∫\n𝑑𝑡𝐸 Mcos𝜃¹𝑡º2\u0019𝛼𝑠𝑧𝐸M. Pluggingthis\ninto the Eq. (5b), we obtain an approximated self-consistent\nequation for s, and the fixed points of this equation can be\ndetermined analytically. There are two sets of fixed points.\nThe first is the trivial fixed points at s0=\u0006¹100ºwhich are\nstable fixed points for all parameters. If the system evolves\ntowards these fixed points, the 4𝜋-periodic Josephson current\nin Eq. (5a) vanishes and all experimental 4𝜋-periodic signa-\ntureswoulddisappear. Wealsofindanothersetoffixedpoints4\n0123456789101112131401234567(a)\n0500010000-101(c)\n05000 10000-101(b)\nFIG. 3. (a) The Shapiro steps for the topological Josephson junction\nsimulated with Eq. (5). The first Shapiro step is strongly suppressed\nwhile all other steps are clearly visible. (b) The time evolution of 𝑠𝑧\nfor the voltage around the first Shapiro step. The pseudo spin oscil-\nlates with a non-zero averaging value, and the resulted 4𝜋-periodic\nsupercurrentstronglysuppressesthefirststep. (c)Thetimeevolution\nof𝑠𝑧for the voltage around the third Shapiro step. The pseudo spin\ngoes to the fixed point of 𝑠𝑧\u00190which effectively shuts down the\n4𝜋-periodic channel for the Josephson current.\nat[46]\ns1=\u0006\u0012\n𝐸0\nM¸𝑉0𝐵𝑓𝑅0√︃\n1\u0000¹𝐸0\nM¸𝑉0𝐵𝑓𝑅º2\u0013\n(6)\nwhicharestablefixedpointsonlywhentheinjectedcurrentis\nsmall so that the dc voltage is smaller than a critical value of\n𝑉𝑐=j𝐸M𝛼\u0000𝐸0\nMj𝑅𝐵𝑓. The existence of these fixed points\nis voltage-dependent, which is qualitatively different from the\ntrivial fixed points.\nTheseanalyticalresultsforfixedpointsprovideinsightinto\nthe experimentally reported voltage-dependent behaviors of\ntopological junctions[10, 13]. From the fixed point analysis,\nwe find two different voltage regimes. At dc voltage below\n𝑉𝑐, there are two sets of fixed points, and the system has a\nchance of evolving to either of them. If the system evolves to\nthe fixed point s1as shown in Fig. 2a, the final stable state\nwould have a non-vanishing 𝑠𝑧and therefore a non-vanishing\n4𝜋-periodic Josephson current shows up in the equation for\nthe Josephson phase. In this voltage regime, we should ex-\npect transport signatures for 4𝜋-periodicity. However, for the\nvoltage above 𝑉𝑐, there exits only the trivial fixed points at\ns0. When the system evolves towards it as shown in Fig. 2b,\nthe final stable state would have a vanishing 𝑠𝑧, and the 4𝜋-\nperiodic Josephson current vanishes. In this voltage regime,\nall the transport signatures for the 4𝜋-periodicity should dis-\nappear. Basedonthesefixedpointanalysis,wepredictthatthe\ntransportoftopologicaljunctionswouldexhibitnontrivial 4𝜋-\nperiodicsignaturesonlyatlowvoltage,whileathighvoltages\nit would look quite similar to the trivial junctions.Shapirosteps. –TheShapirostepsaretheplateausoftheI-V\ncurve at voltages 𝑉𝑛=𝑛~𝜔2𝑒under an injected ac current\nwith frequency 𝜔. It is a powerful tool for probing the dy-\nnamics of Josephson junctions since it reflects the resonance\nbetweenthedcandtheacJosephsonrelation. Fortopological\njunctions,itwasanticipatedthattheoddnumberShapirosteps\nwith𝑛=135should be suppressed by the 4𝜋-periodic\nsupercurrent. The experimental results, however, often show\nstrong suppression of low order odd number steps such as the\nonewith𝑛=1,whileotheroddnumberShapirostepsathigher\nvoltagesarerobust. Sincetheunderstandingoftheexperimen-\ntal results are crucial for detecting Majorana zero modes, it is\ntimely to implement the Eq. (5) to calculate the Shapiro steps\nof Majorana Josephson junctions.\nWe consider an injected current of 𝐼ex¹𝑡º=𝐼¸𝐼0cos𝜔𝑡\nand calculate the I-V curve of the junction, with the results\nfor a typical junction parameter shown in Fig. 3a. We find\nShapiro steps at 𝑉=𝑛~𝜔2𝑒, where𝑛labels the number of\nthestep. Intriguingly,itisclearthatthefirstShapirostepwith\n𝑛=1is strongly suppressed, while all other steps are clearly\nvisible. At first glance, the suppression of only one Shapiro\nstep seems mysterious. One would expect a suppression of\nall odd-number steps if the 4𝜋-periodic supercurrent carried\nby the Majorana qubit is significant, or no suppression to any\nof the steps if the 4𝜋-periodic supercurrent is irrelevant. For\nthis phenomenon, our theory provides a possible mechanism:\nthe feedback from the dynamics of the Majorana qubit. As\nwe have shown in the analytical results, the Majorana qubit\nevolves to different stable states for different voltages. We\nexamine the qubit dynamics at the voltages for the first step\nand the third step. As shown in Fig. 3b, at the voltage where\nthe first step should appear, the Majorana qubit evolves to the\nstable state with a finite 𝑠𝑧. Then the 4𝜋-periodic Josephson\ncurrent will dominate and the Shapiro step is suppressed. For\nthe higher voltage of the third step, however, the Majorana\nqubitevolvestoastablestatewith 𝑠𝑧\u00190,asshowninFig. 3c.\nThenthe 4𝜋-periodicsupercurrentisblocked,andthejunction\nwould behaves similar to a conventional junction presenting\nShapiro steps. This feed back of the qubit dynamics provides\na simple mechanism for the suppression of the first Shapiro\nstep, and gives a possible explanation to one of the puzzles in\nthe experimental findings of topological superconductors.\nWe emphasize that, while our theory is derived for topo-\nlogical junctions with Majorana qubit, it is actually valid for\nany junction with an embedded qubit that can be described\nby the low energy effective Hamiltonian Eq. (1). One such\nexample is the Josephson junction with quantum dots[8]. In\nthissense,ourcalculationofShapirostepsprovideasignalfor\nthefeedbackofembeddedqubit,insteadofauniquesignature\nof Majorana zero modes.\nConclusion. – In summary, we constructed a semiclassical\ntheory for the topological Josephson junctions with an em-\nbedded Majorana qubit. We revealed nontrivial qubit dynam-\nics such as the Landau-Zener transitions and the anisotropic\nGilbert damping. We found that the feedback of the qubit dy-\nnamicsstronglymodifiesthetransportfeaturesofthejunction.5\nWe applied the theory to study the Shapiro steps of the topo-\nlogicaljunctionsanddemonstratedthesuppressionofthefirst\nShapirostepwhichagreeswithrecentexperiments. Wereveal\nthat this phenomenon is due to the voltage-selective feedback\nfrom the dynamics of Majorana qubit.\nAcknowledgments.— We thank Zhongbo Yan, Peng Ye\nand Shuai Yin for valuable discussions. 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Murdock,\nAveraging methods in nonlinear dynamical systems, Vol. 59\n(Springer, 2007)."    },    {        "title": "2204.10596v2.A_short_circuited_coplanar_waveguide_for_low_temperature_single_port_ferromagnetic_resonance_spectroscopy_set_up_to_probe_the_magnetic_properties_of_ferromagnetic_thin_films.pdf",        "content": "arXiv:2204.10596v2  [cond-mat.mtrl-sci]  19 Jul 2022A short-circuited coplanar waveguide for low-temperature single-port ferromagnetic\nresonance spectroscopy set-up to probe the magnetic proper ties of ferromagnetic thin\nfilms\nSayani Pal, Soumik Aon, Subhadip Manna and Chiranjib Mitra∗\nIndian Institute of Science Education and Research Kolkata ,\nWest Bengal, India\nA coplanar waveguide shorted in one end is proposed, designe d, and implemented successfully to\nmeasure the properties of magnetic thin films as a part of the v ector network analyser ferromag-\nnetic resonance (VNA-FMR) spectroscopy set-up. Its simple structure, potential applications and\neasy installation inside the cryostat chamber made it advan tageous especially for low-temperature\nmeasurements. It provides a wide band of frequencies in the g igahertz range essential for FMR\nmeasurements. Our spectroscopy set-up with short-circuit ed coplanar waveguide has been used to\nextract Gilbert damping coefficient and effective magnetizat ion values for standard ferromagnetic\nthin films like Py and Co. The thickness and temperature depen dent studies of those magnetic\nparameters have also been done here for the afore mentioned m agnetic samples.\nINTRODUCTION\nIn recent years, extensive research on microwave mag-\nnetization dynamics in magnetic thin films[1–3], planar\nnanostructures[4–6] and multi-layers[7–9] havebeen per-\nformedduetotheirpotentialapplicationsinvariousfields\nof science and technology. Spintronics is one such emerg-\ning discipline that encompasses the interplay between\nmagnetization dynamics and spin transport. It also in-\ncludes fields like spin-transfer torque [10–13], direct and\ninversespin hall effect [14–18], spin pumping [19, 20] etc.,\nwhich are crucial in industrial applications for develop-\ning devices like magnetic recording head[21], magnetic\ntunnel junction(MTJ) sensors [22, 23], magnetic memory\ndevices[24, 25] andspin-torquedevices[26, 27]. Thus ex-\nploring more about the static and dynamic properties of\nmagnetic materials in itself is an interesting subject. Fer-\nromagnetic resonance spectroscopy(FMR) is a very ba-\nsic and well-understood technique that is used to study\nthe magnetization dynamics of ferromagnets[28, 29, 31].\nNowadays, most advanced FMR spectroscopy methods\nuse a vector network analyzer (VNA)[30, 31] as the mi-\ncrowave source and detector. We have used VNA in our\nset-up too.\nTo determine the magnetic parameters of the ferromag-\nnetic materials using the VNA-FMR spectroscopy, one\nneeds to carry out the measurements at a wide range of\nfrequencies. Since the microwave magnetic field in the\ncoplanar waveguide (CPW) is parallel to the plane, it\nservesthepurposeofexploringthemagneticpropertiesof\nthe concernedsystem overabroadfrequencyrangein the\nGHz region. The advantage of using CPW in the spec-\ntroscopy system lies in the fact that we no longer need\nto remount samples at different waveguides or cavities\nforeveryotherfrequency measurements, which consumes\n∗Corresponding author:chiranjib@iiserkol.ac.ina lot of time and effort in an experiment[32, 33]. Re-\nsearchers design and use different types of CPW for vari-\nous other purposes like micron-sized CPW in microwave-\nassisted magnetic recording; two-port CPW in antenna;\nshorted CPW in ultra-wideband bandpass-filter and per-\nmeability measurements [34–36]. However, in broadband\nFMR spectroscopy two-port CPW jigs have most com-\nmonly been used till date. Using two-port CPW in FMR\nspectroscopy, one gets absorption spectra in terms of\nthe transmissioncoefficient of scatteringparameters, and\nfrom there magnetic parameters of the samples can be\ndetermined. The use of two-port CPW in VNA-FMR\ncan be replaced by one-port CPW where the reflection\ncoefficient of scattering parameters of the FMR spectra\ncan be used to determine the magnetic parameters of\nthe sample. One port reflection geometry is a lot more\nconvenient in terms of easy design, calibration, installa-\ntion, and sample loading. This is especially true when\nthe whole CPW arrangement is kept inside the cryostat\nchamber for low-temperature measurements and the sys-\ntem becomes very sensitive to vibration and any kind\nof magnetic contacts, one port CPW seems very con-\nvenient to operate rather than the two-port one. Previ-\nously, manyhavedesignedandusedshort-circuitedCPW\njigs for other purposes but to the best our knowledge it\nhas not been used for low-temperature VNA-FMR spec-\ntroscopy measurements before.\nIn this work, we report the development of short-\ncircuited CPW based low-temperature broadband VNA-\nFMR spectroscopy set-up to study the magnetic param-\neters of standard ferromagnetic samples. For measure-\nments, we chose the permalloy(Py) thin films as ferro-\nmagnetic (FM) material which has greatly been used in\nresearchfields like spintronics and industrial applications\ndue to its interesting magnetic properties like high per-\nmeability, large anisotropy magnetoresistance, low coer-\ncivity, and low magnetic anisotropy. We have also con-\nsidered another standard magnetic thin film, Co of thick-\nness 30nm as a standard for ascertaining the measure-2\nment accuracy. In our system, we swept the magnetic\nfield keeping the frequencies constant, and got the FMR\nspectra for several frequencies. From there we found the\nvariation of resonance fields and field linewidths with\nthe resonance frequencies. We have used the linear fit\nfor resonance frequencies vs field line-widths data to\ncalculate the Gilbert damping coefficient( α). We fit-\nted the set of resonance frequencies vs resonance fields\ndata to the Kittel formula [59] to obtain the effec-\ntive magnetization(4 πMeff). Subsequently, we investi-\ngated the thickness and temperature-dependent studies\nof 4πMeffandαfor FM thin films of different thickness\ninthetemperaturerangeof7.5Kto300K.Tocharacterise\nthe measurement set-up using short-circuited CPW, we\ncompared the previous measurements in the literature\nwith ourresults and there wasa good agreementbetween\nthe two[36, 41].\nEXPERIMENTAL DETAILS\nA short-circuited CPW has been designed and fab-\nricated as a part of our low-temperature VNA-FMR\nspectroscopy set-up. To make the CPW we have used\nRogers AD1000, a laminated PCB substrate with copper\ncladding on both sides of the dielectric. The thickness of\nthe dielectric and the copper layer are 1.5 mm and 17.5\nmicrons respectively and the dielectric constant of the\nsubstrate is 10.7. The main concern about the design of\nthe CPW is to match its characteristic impedance with\nthe impedance of the microwave transmission line con-\nnected to it. We haveused the line calculatorto calculate\nthe dimensions of CPW. For a CPW with a characteris-\ntic impedance of 50 ohms, the line calculator calculated\nthe width of the signal line and the gap to be 900 mi-\ncrons and 500 microns respectively. The fabrication is\ndone using optical lithography which is described in de-\ntail in the literature[49]. Other components of our mea-\nCryostatVNA\nElectromagnetSample\nCPWCoaxial Transmission Line\nFIG. 1. The schematic diagram of measurement system and\nthe arrangement inside the cryostat with the sample on top\nof the CPW\nsurement system are a)Vector Network Analyser(VNA),\nwhich is a microwave source as well as a detector, b)theelectromagnet that generates the external magnetic field,\ni.e., Zeemanfieldand, c)optistatdrycryogen-freecooling\nsystem from Oxford instruments which is used for low-\ntemperature measurements. One end of the CPW signal\nline is shorted to the ground, and the other end is con-\nnected to the VNA through a SMA connector and coax-\nial cable (fig 3b). On top of the CPW, thin-film samples\nhave been placed face down after wrapping them with\nan insulating tape to electrically isolate them. For low-\ntemperature measurements, the sample has been glued\nto the CPW using a low-temperature adhesive to ensure\ncontact of sample and resonator at all times, in spite of\nthe vibration caused by the cryostat unit. This whole ar-\nrangementis then placed inside the twopole pieces of the\nelectromagnet as we can see from the diagram in fig 1.\nTherearetwostandardmethods ofgettingFMR spectra:\nsweeping the frequency keeping the field constant and\nsweeping the magnetic field while keeping the frequency\nconstant. We have adopted the second method. We have\nworked in the frequency range from 2.5GHz to 5.5GHz\nand in the magnetic field range from 0 Oe to roughly\naround 500 Oe. We have used 1mW of microwave power\nthroughout the experiment. From the FMR spectra, we\nhavedeterminedeffectivemagnetizationanddampingco-\nefficient of FM thin films and studied their variation with\ntemperature and sample thickness.\nSAMPLE PREPARATION AND\nCHARACTERIZATION\nPy (Ni80Fe20) and Co thin films were fabricated by\nthermal evaporation technique on Si/SiO 2substrates,\nfrom commercially available pellets (99 .995%pure) at\nroom temperature. The substrates were cleaned with\nacetone, IPA and DI water respectively in ultrasonica-\ntor and dried with a nitrogen gun. The chamber was\npumped down to 1 ×10−7torr using a combination of\na scroll pump and turbo pump. During the deposition,\npressure reached upto 1 ×10−6torr. Thin films were fab-\nricated at a rate of 1 .2˚A/swhere thickness can be con-\ntrolled by Inficon SQM 160 crystal monitor. For our\nexperiments a series of Py thin films of different thick-\nnesses were fabricated by keeping the other parameters\nlike base pressure, deposition pressure and growth rate\nconstant. Film thickness and morphology was measured\nby using atomic force microscopy technique as shown in\nfig 2(a). We have used Py films with thicknesses 10nm,\n15nm, 34nm, 50nm, and 90nm with a surface roughness\nof around 1nm and one Co film of thickness 30nm. X-ray\ndiffraction experiment confirms the polycrystalline struc-\nture of the samples as shown in fig 2b and fig 2c for Py\nand Co respectively.3\n2µm\n2µm\n(a)\n/s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53 /s54/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s52/s52/s46/s51/s54/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41\n/s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s80/s121/s32/s40/s49/s53/s110/s109/s41\n(b)\n/s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41\n/s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s67/s111/s32/s40/s51/s48/s110/s109/s41\n(c)\nFIG. 2. (a)Atomic force microscope (AFM) image of 30 nm\nthick Py thin film with a surface roughness of 1 nm . X-ray\ndiffraction peak of (b)15nm thick Py film and (c)30nm Co\nprepared by thermal evaporation.\nRESULTS AND DISCUSSION\nWe have calculated the dimensions of the short-\ncircuited CPW using the line calculator of the CST Stu-\ndio Suite software as mentioned in the experimental de-\ntails section. Using those dimensions we have also done\nthe full-waveelectromagneticsimulation in CST software\nto get the electric and magnetic field distribution of the\nCPW. One can see from the simulation result displayed\nin figure 3a that the farther it is from the gap, the weaker\nthe intensity of the magnetic field, and the magnitude of\nthe field in the gap area is one order of greater than that\non the signal line. When placing the thin film sample\non top of the CPW, the dimension of the sample shouldDielectricSampleSignal  Line\nGap\nMagnetic field lines\nElectric field lines\na) b)\nc)\nFIG. 3. (a) Schematic diagram of the cross-sectional view of\nCPW. (b) Top view of the short-circuited CPW after fabri-\ncation. (c)Intensity distribution of microwave magnetic fi eld\nin the one end shorted CPW at 5GHz (top view)\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s83\n/s49/s49/s40/s100/s66/s41\n/s72/s32/s40/s79/s101/s41/s32/s32/s32 /s102/s114/s101/s113/s117/s101/s110/s99/s121\n/s32/s50/s46/s53/s71/s72/s122\n/s32/s51/s46/s53/s71/s72/s122\n/s32/s52/s46/s53/s71/s72/s122\n/s32/s53/s46/s53/s71/s72/s122/s49/s53/s110/s109/s32/s80/s121\n/s84/s61/s51/s48/s48/s75\nFIG. 4. Ferromagnetic Resonance spectra of absorption at\nfrequencies 2.5 GHz, 3.5 GHz, 4.5 GHz, 5.5 GHz for 15nm Py\nthin films at room temperature after background subtraction\nbe such that it can cover the gap area on both sides of\nthe signal line of the CPW because the magnetic field is\nmost intense in that area. This microwave magnetic field\ncirculatingthe signal line ofthe CPW is perpendicular to4\n/s50 /s51 /s52 /s53 /s54 /s55/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s51/s48/s48/s75\n/s32/s67/s111/s32/s40/s51/s48/s110/s109/s41\n/s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s68 /s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41\n(a)/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/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/s97\n/s116/s32\n/s80/s121 /s32/s40/s110/s109/s41/s32/s84/s61/s51/s48/s48/s75\n(b)\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s50/s51/s52/s53/s54/s55\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s32/s51/s48/s48/s75\n/s32/s67/s111/s32/s40/s51/s48/s110/s109/s41\n/s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s79/s101/s41\n(c)/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48/s56/s57/s49/s48/s49/s49/s52 /s112 /s77\n/s101/s102/s102/s32/s40/s107/s71/s41\n/s116/s32/s45/s49\n/s32/s80/s121/s32/s40/s110/s109/s45/s49\n/s41/s32/s84/s61/s51/s48/s48/s75\n(d)\nFIG. 5. a)Field linewidth variation with resonance frequen cies at 300K for 34nm Py and 30nm Co thin films. Equation 1 has\nbeen used for fitting the curve and to determine the Gilbert da mping coefficient; b)thickness dependence of Gilbert dampin g\ncoefficient at room temperature for Py thin films; c)resonance field variation with resonance frequencies at 300K for 34 nm P y\nand 30 nm Co thin films. Kittel formula (eqn-3)has been used fo r fitting the curve and to determine the effective magnetizati on;\nd)thickness dependence of effective magnetization for Py th in films at room temperature.\nthe external magnetic field and both the magnetic fields\nare parallel to the film surface as can be seen from fig\n3a and 3b. On account of the static magnetic field, the\nmagnetic moment will undergo a precession around the\nstatic magnetic field at a frequency called the Larmor\nprecession frequency. Absorption of electromagnetic en-\nergy happens when the frequency of the transverse mag-\nnetic field (microwave) is equal to the Larmor frequency.\nFig4exhibitsthe absorptionspectrafor15nmbarePy\nfilm after subtraction of a constant background for four\ndifferent frequencies, 2.5 GHz, 3.5 GHz, 4.5 GHz and 5.5\nGHz at room temperature in terms of S-parameter re-\nflection coefficient ( S11) vs. external magnetic field. We\nfitted these experimental results to the Lorentz equation\n[56]. We extracted the field linewidth at half maxima\nfrom the FMR spectra at different frequencies and fitted\nthem using equation 1 to obtain αas one can see from\nfig 5a and fig 6a. The experimental values of the absorp-tion linewidth (∆ H) contains both the effect of intrinsic\nGilbert damping and the extrinsic contribution to the\ndamping. Linewidth due to Gilbert damping is directly\nproportional to the resonance frequency and follows the\nequation:\n∆H= (2π\nγ)αf+∆H0 (1)\nwhereγis the gyromagneticratio, αis the Gilbert damp-\ning coefficient and ∆ H0is the inhomogeneous linewidth.\nA number of extrinsic contributions to the damping coef-\nficient like magnetic inhomogeneities, surface roughness,\ndefects of the thin films bring about the inhomogeneous\nlinewidth broadening [55]. αhas been determined using\nthe above equation only. Damping coefficient values ob-\ntainedhereareintherangeofabout0 .005to0.009forPy\nsamplesofthicknessescoveringthe whole thin film region\ni.e., 10nm to 90nm at room temperature. These values5\n/s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48/s53/s53/s54/s48\n/s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121/s68 /s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s53/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121\n(a)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50\n/s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s116 /s32\n/s80/s121 \n/s32/s49/s53/s110/s109\n/s32/s49/s48/s110/s109\n(b)\n/s54/s48 /s49/s50/s48 /s49/s56/s48 /s50/s52/s48 /s51/s48/s48 /s51/s54/s48 /s52/s50/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53/s54/s46/s48\n/s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s53/s110/s109/s32/s84/s61/s32/s52/s53/s75/s80/s121/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s79/s101/s41/s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121\n(c)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s56/s46/s48/s56/s46/s50/s56/s46/s52/s56/s46/s54/s56/s46/s56/s57/s46/s48/s57/s46/s50/s57/s46/s52/s57/s46/s54/s52 /s77\n/s101/s102/s102/s32/s40/s107/s71/s41\n/s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s116/s32\n/s80/s121 \n/s32/s49/s53/s110/s109\n/s32/s49/s48/s110/s109\n(d)\nFIG. 6. a)Field linewidth variation with resonance frequen cies at 300K and 45K for 10 nm and 15nm Py films. Equation 1\nhas been used for fitting the curve and to determine the Gilber t damping coefficient; b)temperature dependence of damping\ncoefficient for 10nm and 15nm Py thin films; c)resonance field va riation with resonance frequencies at 300K and 45K for 10nm\nand 15nm Py thin films. Kittel formula (eqn-3) has been used fo r fitting the curve and to determine the 4 πMeff; d)temperature\ndependence of 4 πMefffor 10nm and 15nm Py thin films.\nare pretty close to the values previously reported in the\nliterature [39–41, 43, 44]. For the Co film of thickness 30\nnm we have obtained the value of αto be 0.008 ±0.0004.\nBaratiet al.measured the damping value of 30nm Co\nfilm to be 0.004 [37, 38]. There are other literature also\nwhere Co multilayers have been studied where damping\ncoefficient value increasesbecause ofspin pumping effect.\nαis a veryinterestingparameterto investigatebecause it\nis used in the phenomenological LLG equation [57], [58]\nto describe magnetization relaxation:\nd/vectorM\ndt=−γ/vectorM×/vectorHeff+α\nMS/vectorM×d/vectorM\ndt(2)\nwhere,µBisBohrmagneton, /vectorMisthemagnetizationvec-\ntor,MSis the saturation magnetization and Heffis the\neffectve magnetic field which includes the external field,\ndemagnetization and crystalline anisotropy field. The in-troduction of the Damping coefficient in LLG equation is\nphenomenological in nature and the question of whether\nit has a physical origin or not has not been fully under-\nstood till date. We have measured 4 πMeffalso from\nthe absorption spectra. We have fitted the Kittel for-\nmula (equation 3) into resonance field vs. the resonance\nfrequency ( fres) data as shown in fig 5c and fig 6c.\nfres= (γ\n2π)[(H+4πMeff)H]1\n2 (3)\nwhere,His the applied magnetic field, and Meffis the\neffective magnetization which contains saturation mag-\nnetization and other anisotropic contributions. We ob-\ntained the 4 πMeffvalue for 30nm thick Co and 34nm Py\nto be 17.4 ±0.2kG and 9.6 ±0.09kG respectively at room\ntemperature. These values also agree quite well with the\nliterature. For a 10nm Co film, Beaujour et al.measured\nthe value to be around 16 kG[45] and for a 30nm Py the6\nvalue is 10 .4kGas measured by Zhao et al[41].\nWe tried to address here the thickness and tempera-\nture dependence of αand 4πMeffusing our measure-\nment set-up. The variation of the αwith thickness is\nshown here in figure 5b. It increases smoothly as film\nthickness decreases and then shows a sudden jump below\n15nm. Increased surface scattering could be the reason\nbehind this enhanced damping for thinner films. It has\nbeen previously observed [60] that damping coefficient\nand electrical resistivity follows a linear relation at room\ntemperature for Py thin film. It suggests a strong corre-\nlation between magnetization relaxation( α) and electron\nscattering. Magnetization relaxation could be explained\nby electron scattering by phonons and magnons. In the\nformer case, αis proportional to the electron scatter-\ning rate, τ−1and in the later case, α∼τ. Theoretical\npredictions by Kambersky [61] suggests that at higher\ntemperature α∼τ−1as electron scattering by phonons\nare predominant there. So, here in our case we can elim-\ninate the possibility of electron scattering by magnons as\nthickness dependent study has only been done at room\ntemperature where phonon scattering is prevalent. Ing-\nvasson et.al in their paper[60] also suggests that the re-\nlaxation of magnetization is similar to bulk relaxation\nwhere phonon scattering in bulk is replaced by surface\nand defect scattering in thin films.\nThicknessdependent studyof4 πMeffalsohasbeen done\nfor Py thin films at room temperature. As we can see\nfrom fig 5d, Meffis linear for thinner films and becomes\nalmost independent of thickness for thicker films. The\nchange in Meffwith thickness mainly comes from the\nsurface anisotropy,\nµ0Meff=µ0Ms−2Ks\nMsd(4)\nwhereMsis the saturation magnetization and2Ks\nMsdis\nthe surface anisotropy field. Surface anisotropy is higher\nfor thinner films and the anisotropy reduces as one in-\ncreases the film thickness. We have obtained saturation\nmagnetization(4 πMs) value of Py to be 10 .86kGusing\nthe linear fit (equation 4). Previously Chen et al.has re-\nported the 4 πMeffvalue for a 30nm Py film to be 12 kG\n[54] which includes both 4 πMsand anisotropy field.\nTemperature dependence of αfor 15nm and 10nm Py\nfilm is represented in figure 6b. The αvalue decreases\nmonotonically from room temperature value and reaches\na minimum value at around 100K and then starts to in-\ncrease with further decrease of temperature and reaches\na maximum value at 45K. Zhao et al.have seen this\nkind of damping enhancement at around50Kin their low\ntemperature experiment with Py thin films with differ-\nent types of capping layers and Rio et al.observed the\ndamping anomaly at temperature 25K when they have\nusedPtas a capping layer on Py thin film.[39, 41]. We\ndid not use any capping layer on Py film in our mea-\nsurement. So there is no question of interface effect for\nthe enhanced damping at 45K. A possible reason for the\nstrong enhancement of damping at 45K could be the/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48\n/s32/s57/s71/s72/s122\n/s32/s52/s71/s72/s122/s51/s48/s110/s109/s32/s67/s111/s68 /s72 /s32/s40/s79/s101/s41\n/s84/s32/s40/s75/s41\nFIG. 7. Temperature induced linewidth variation of 30nm Co\nthin film at two different frequencies 4GHz and 9GHz\nspin reorientation transition(SRT) on the Py surface at\nthat particular temperature [41, 42]. Previously it has\nbeen established that the competition between different\nanisotropy energies: magnetocrystalline anisotropy, sur-\nface anisotropy, shape anisotropy decides the magnetiza-\ntion direction in magnetic films. For thin films, the vari-\nation of temperature, film thickness, strain can alter the\ncompetition between shape and surface anisotropy. In\nour case, temperature variation could be the reason for\nthe spin reorientation transition on Py surface at around\n45K.Foradeeperunderstandingofthespinreorientation\nwe investigated the temperature dependence of 4 πMeff\nfor 15nm and 10nm Py film as shown in fig 6d. There\nMeffis showing an anomaly at around 45K, otherwise it\nis increasing smoothly with the decrease of temperature.\nSince there is no reason of sudden change in saturation\nmagnetization at this temperature, the possible reason\nfor the anomaly in Meffshould come from any change\ninmagneticanisotropy. Thatchangeofanisotropycanbe\nrelated to a spin reorientation at that particular temper-\nature value. Sierra et.al., [42], have also argued that in\nthe temperature dependent spin re-orientation (T-SRT),\nthe central effect of temperature on the magnetic prop-\nerties of Py films was to increase the in-plane uniax-\nial anisotropy and to induce a surface anisotropy which\norients the magnetization out of plane in the Py sur-\nface. They have verified this using X-Ray diffraction\nexperiments and high resolution transmission electron\nmicroscopy images. This establishes reasonably enough\nthat it is a spin re-orientation transition around 45K.\nLastly, for a 30nm Co thin film we have studied the\ntemperature variation of FMR linewidth(∆ H) at mi-\ncrowave frequencies 9GHz and 4GHz. One can see from\nfig7 that the linewidth does not change much in the tem-\nperature range 100 <T<300 but below 100K, ∆ Hstarts\nto increase significantly. This behaviour of ∆ Hhas been7\nobserved previously by Bhagat et al.[62]. They sug-\ngested that the increase of relaxation frequency at low\ntemp might be related to the rapid variation in the elec-\ntronicmeanfreepath. Thiscomparisonsuggeststhatour\nset-up with a single port CPW has the same sensitivity\nof the previously reported set-up.\nCONCLUSIONS\nShort-circuited coplanar waveguide can successfully be\nused for low-temperature single-port broadband FMR\nspectroscopy measurements. The magnetic parameters\nobtained here for the standard ferromagnetic materials,\nPy and Co are in good agreement with other experimen-\ntal works and theoretical predictions. Temperature de-\npendentstudiesof αand4πMeffforPyfilmshereexhibit\nspin reorientation phenomenon at low temperatures. We\nbelieve that the findings with Py sample will help in bet-\nter understanding of magnetic phenomena for other fer-\nromagnetic materials at low temperature. Though this\nsetup has limitations for angle-dependent FMR measure-\nments, itcanreadilybe usedforstudiesonmagnetization\ndynamics for multi-layered films and planar nanostruc-\ntures and is currently under way. In future we aim to in-\ntegrate electrical measurement facilities with the existing\nset-up thereby extending the possibilities of inverse spinhall(ISHE) measurements and spin-torque ferromagnetic\nresonance(ST-FMR)spectroscopyexperimentswhich are\nvery relevant for current scientific interests. We believe\nthe short-circuited CPW will serve its purpose conve-\nniently there too.\nACKNOWLEDGEMENTS\nTheauthorssincerelyacknowledgesMinistryofEduca-\ntion, Government of India and Science and Engineering\nResearch Board (SERB) (grant no:EMR/2016/007950)\nand Department of Science and Technology (grant\nno. DST/ICPS/Quest/2019/22) for financial sup-\nport. S.P. acknowledges Department of Science and\nTechnology(DST)-INSPIRE fellowship India, S. A. ac-\nknowledges Ministry of Education of Government of In-\ndia and S.M. acknowledges Council Of Scientific and\nIndustrial Research(CSIR),India for research fellowship.\nThe authors would like to thank Dr. Partha Mitra of the\nDepartment of Physical Sciences, Indian Institute of Sci-\nence Education and Research Kolkata for providing the\nlab facilities for sample deposition. The authors would\nalso like to thank Mr. Subhadip Roy, IISER Kolkata for\nhis help in fabricating the CPW structure using home-\nbuilt optical lithography set-up.\n[1]Barman, A. and Haldar, A., 2014. Time-domain study\nof magnetization dynamics in magnetic thin films and\nmicro-and nanostructures. Solid State Physics, 65, pp.1-\n108.\n[2]Silva, T.J., Lee, C.S., Crawford, T.M. and Rogers, C.T.,\n1999. Inductive measurement of ultrafast magnetization\ndynamics in thin-film Py. 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Physical\nReview B. 1974 Jul 1;10(1):179."    },    {        "title": "2205.06399v1.Precession_dynamics_of_a_small_magnet_with_non_Markovian_damping__Theoretical_proposal_for_an_experiment_to_determine_the_correlation_time.pdf",        "content": "arXiv:2205.06399v1  [cond-mat.mes-hall]  13 May 2022Precession dynamics of a small magnet with non-Markovian da mping: Theoretical\nproposal for an experiment to determine the correlation tim e,✩✩\nHiroshi Imamura, Hiroko Arai, Rie Matsumoto, Toshiki Yamaj i, Hiroshi Tsukahara✩\nNational Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan\nAbstract\nRecent advances in experimental techniques have made it pos sible to manipulate and measure the magnetization dynamics on\nthe femtosecond time scale which is the same order as the corr elation time of the bath degrees of freedom. In the equations of\nmotion of magnetization, the correlation of the bath is repr esented by the non-Markovian damping. For development of th e science\nand technologies based on the ultrafast magnetization dyna mics it is important to understand how the magnetization dyn amics\ndepend on the correlation time. It is also important to deter mine the correlation time experimentally. Here we study the precession\ndynamics of a small magnet with the non-Markovian damping. E xtending the theoretical analysis of Miyazaki and Seki [J. C hem.\nPhys. 108, 7052 (1998)] we obtain analytical expressions of the prece ssion angular velocity and the e ffective damping constant for\nany values of the correlation time under assumption of small Gilbert damping constant. We also propose a possible experi ment for\ndetermination of the correlation time.\nKeywords: non-Markovian damping, generalized Langevin equation, LL G equation, ultrafast spin dynamics, correlation time\n1. Introduction\nDynamics of magnetization is the combination of precession\nand damping. The precession is caused by the torque due to\nthe internal and external magnetic fields. Typical time scal e\nof the precession around the external field and the anisotrop y\nfield is nanosecond. The damping is caused by the coupling\nwith the bath degrees of freedom such as conduction electron s\nand phonons. The typical time scale of the relaxation of con-\nduction electrons and phonons is picosecond or sub-picosec ond\nwhich is much faster than precession. In typical experiment al\nsituations such as ferromagnetic resonance and magnetizat ion\nprocess, the time correlation of the bath degrees of freedom\ncan be neglected and the magnetization dynamics is well repr e-\nsented by the Landau-Lifshitz-Gilbert (LLG) equation with the\nMarkovian damping term[1–3].\nRecent advances in experimental techniques such as fem-\ntosecond laser pulse and time-resolved magneto-optical Ke rr\neffect measurement have made it possible to manipulate and\nmeasure magnetization dynamics on the femtosecond time\nscale[4–11]. In 1996, Beaurepaire et al. observed the femto sec-\nond laser pulse induced sub-picosecond demagnetization of a\nNi thin film[4], which opens the field of ultrafast magnetiza-\ntion dynamics. The all-optical switching of magnetization in a\n✩Permanent address: High Energy Accelerator Research Organ ization\n(KEK), Institute of Materials Structure Science (IMSS), Ts ukuba, Ibaraki 305-\n0801, Japan\n✩✩This work is partly supported by JSPS KAKENHI Grant Numbers\nJP19H01108 and JP18H03787.\nEmail addresses: h-imamura@aist.go.jp (Hiroshi Imamura),\narai-h@aist.go.jp (Hiroko Arai)ferrimagnetic GdFeCo alloy was demonstrated by Stanciu et a l.\nusing a 40 femtosecond circularly polarized laser pulse[5] . The\nhelicity-dependent laser-induced domain wall motion in Co /Pt\nmultilayer thin films was reported by Quessab et al.[11].\nTo understand the physics behind the ultrafast magnetizati on\ndynamics it is necessary to take into account the time correl a-\ntion of bath in the equations of motion of magnetization. The\nfirst attempt was done by Kawabata in 1972[12]. He derived the\nBloch equation and the Fokker-Planck equation for a classic al\nspin interacting with the surroundings based on the Nakajim a-\nZwanzig-Mori formalism[13–15]. In 1998, Miyazaki and Seki\nconstructed a theory for the Brownian motion of a classical\nspin and derived the integro-di fferential form of the generalized\nLangevin equation with non-Markovian damping[16]. They\nalso showed that the generalized Langevin equation reduces to\nthe LLG equation with modified parameters in a certain limit.\nAtxitia et al. applied the theory of Miyazaki and Seki to the\natomistic model simulations and showed that materials with\nsmaller correlation time demagnetized faster[17].\nDespite the experimental and theoretical progresses to dat e\nlittle attention has been paid to how to determine the correl a-\ntion time experimentally. For development of the science an d\ntechnologies based on the ultrafast magnetization dynamic s it\nis important to determine the correlation time experimenta lly\nas well as to understand how the magnetization dynamics de-\npend on the correlation time.\nIn this paper the precession dynamics of a small magnet with\nnon-Markovian damping is theoretically studied based on th e\nmacrospin model. The magnet is assumed to have a uniaxial\nanisotropy and to be subjected to an external magnetic field\nparallel to the magnetization easy axis. The non-Markovian ity\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials May 16, 2022enhances the precession angular velocity and reduces the da mp-\ning. Assuming that the Gilbert damping constant is much\nsmaller than unity, the analytical expressions of the prece ssion\nangular velocity and the e ffective damping constant are derived\nfor any values of the correlation time by extending the analy sis\nof Miyazaki and Seki[16]. We also propose a possible exper-\niment for determination of the correlation time. The correl a-\ntion time can be determined by analyzing the external field at\nwhich the enhancement of the precession angular velocity is\nmaximized.\nThe paper is organized as follows. Section 2 explains the\ntheoretical model and the equations of motion. Section 3 giv es\nthe numerical and theoretical analysis of the precession dy nam-\nics in the absence of an anisotropy field. The e ffect of the\nanisotropy field is discussed in Sec. 4. A possible experimen t\nfor determination of the correlation time is proposed in Sec . 5.\nThe results are summarized in Sec. 6.\n2. Theoretical model\nWe calculate the magnetization dynamics in a small mag-\nnet with a uniaxial anisotropy under an external magnetic fie ld\nbased on the macrospin model. The magnetization easy axis\nis taken to be z-axis and the magnetic field is applied in the\npositive z-direction. In terms of the magnetization unit vector,\nm=(mx,my,mz), the energy density is given by\nE=K(1−m2\nz)−µ0MsH m z, (1)\nwhere Kis the effective anisotropy constant including the crys-\ntalline, interfacial, and shape anisotropies. µ0is the vacuum\npermeability, Msis the saturation magnetization, His the exter-\nnal magnetic field. The e ffective field is obtained as\nHeff=(Hkmz+H)ez, (2)\nwhere ezis the unit vector in the positive zdirection and Hk=\n2K/(µ0Ms) is the effective anisotropy field.\nThe magnetization precesses around the e ffective field with\ndamping. The energy and angular momentum are absorbed by\nthe bath degrees of freedom such as conduction electrons and\nphonons until the magnetization becomes parallel to the e ffec-\ntive field. The equations of motion of mcoupled with the bath\nis given by the Langevin equation with the stochastic field re p-\nresenting the bath degrees of freedom. If the time scale of th e\nbath is much smaller than the precession frequency the stoch as-\ntic field can be treated as the Wiener process[18] as shown by\nBrown[3].\nSince we are interested in the ultrafast magnetization dyna m-\nics of which time scale is the same order as the correlation ti me\nof the bath degrees of freedom, the stochastic field should be\ntreated as the Ornstein-Uhlenbeck process[18, 19]. As show n\nby Miyazaki and Seki [16] the equations of motion of mtakes\nthe following integro-di fferential form:\n˙m=−γm×(Heff+r)+αm×/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′,(3)whereγis the gyromagnetic ratio, αis the Gilbert damping con-\nstant, and ris the stochastic field. The first term represents the\nprecession around the sum of the e ffective field and the stochas-\ntic field, and the second term represents the non-Markovian\ndamping. The memory function in the non-Markovian damping\nterm is defined as\nν(t−t′)=1\nτcexp/parenleftigg\n−|t−t′|\nτc/parenrightigg\n, (4)\nwhereτcis the correlation time of the bath degrees of freedom.\nThe stochastic field, r, satisfies/angbracketleftri(t)/angbracketright=0 and\n/angbracketleftrj(t)rk(t′)/angbracketright=µ\n2δj,kν(t−t′), (5)\nwhere/angbracketleft/angbracketrightrepresents the statistical mean, and\nµ=2αkBT\nγMsV. (6)\nThe subscripts jandkstand for x,y, orz,kBis the Boltzmann\nconstant, Tis the temperature, Vis the volume of the mag-\nnet, andδj,kis Kronecker’s delta. The LLG equation with the\nMarkovian damping derived by Brown [3] is reproduced in the\nlimit ofτc→0 because lim τc→0ν(t−t′)=2δ(t−t′), where\nδ(t−t′) is Dirac’s delta function. Equation (3) is equivalent to\nthe following set of the first order di fferential equations,\n˙m=−γm×[Heff+δH] (7)\n˙δH=−1\nτcδH−α\nτ2cm−γ\nτcR, (8)\nwhere Rrepresents the stochastic field due to thermal agita-\ntion. Equations (7), (8) are used for numerical simulations . The\nstochastic field, R, satisfies/angbracketleftRj(t)/angbracketright=0 and\n/angbracketleftRj(t)Rk(t′)/angbracketright=µδj,kδ(t−t′). (9)\n3. Precession dynamics in the absence of an anisotropy field\nIn this section the precession dynamics in the absence of an\nanisotropy field, i.e. Hk=0, is considered. The initial di-\nrection of magnetization is assumed to be m=(1,0,0). The\nnumerical simulation shows that the non-Markovian damping\nenhances the precession angular velocity and reduces the da mp-\ning. The numerical results are theoretically analyzed assu ming\nthatα≪1. The analytical expressions of the precession an-\ngular velocity and the e ffective damping constant are obtained.\nThe case with Hk/nequal0 will be discussed in Sec. 4.\n3.1. numerical simulation results\nWe numerically solve Eqs. (7), (8) for H=10 T,α=0.05,\nandτc=1 ps. The temperature is assumed to be low enough\nto set R=0 in Eq. (8). Figure 1(a) shows the trajectory of m\non a unit sphere. The initial direction is indicated by the fil led\ncircle. The plot of the temporal evolutions of mx,my, and mzare\nshown in Fig. 1(b). The magnetization relaxes to the positiv ez\ndirection with precessing around the external field. The res ults\n2t [ps] 0.00 0.01 0.03 \n0.02 0.04 \n100 200 0\nt [ps] \na) b) \nc) d) z\nx yHφ [rad / ps] \n1.76 1.77 1.79 \n1.78 1.80 100 200 0\n100 200 0-1 1\n0mx, m y, m z\nt [ps] \nφ00.05 \nαmzmymx\nyαeff \nFigure 1: (a) Trajectory of mon a unit sphere. The external field of H=\n10 T is applied in the positive zdirection. The initial direction is assumed to\nbem=(1,0,0) as indicated by the filled circle. The other parameters are\nτc=1 ps, andα=0.05. (b) Temporal evolution of mx,my,mz. (c) Temporal\nevolution of the precession angular velocity, ˙φ. The solid red curve shows the\nsimulation result. The dotted black line indicates the resu lt of the Markovian\nLLG equation, i.e. ˙φ0=γH/(1+α2). (d) Temporal evolution of the e ffective\ndamping constant, αeff. The solid red curve shows the simulation result. The\ndotted black line indicates α=0.05.\nare quite similar to that of the Markovian LLG equation, whic h\nimplies that the non-Markovianity in damping causes renorm al-\nization of the gyromagnetic ratio and the Gilbert damping co n-\nstant in the Markovian LLG equation.\nThe renormalized value of the gyromagnetic ratio can be\nobserved as a variation of the precession angular velocity, ˙φ,\nwhere the polar and azimuthal angles are defined as m=\n(sinθcosφ,sinθsinφ,cosθ). Figure 1(c) shows that temporal\nevolution of ˙φ(solid red) together with the precession angular\nvelocity without non-Makovianity, ˙φ0=γH/(1+α2), (dotted\nblack). The precession angular velocity increases with inc rease\nof time and saturates to a certain value around 1.798. The sha pe\nof the time dependence of ˙φis quite similar to that of mzshown\nin Fig. 1(b), which suggests that the non-Markovian damping\nacts as an effective anisotropy field in the precession dynamics.\nThe renormalization of the Gilbert damping constant can be\nobserved as a variation of the temporal evolution of the pola r\nangle, ˙θ. Rearranging the LLG equation for ˙θ, the effective\ndamping constant can be defined as\nαeff=−˙θ/(γHsinθ). (10)\nIn Fig. 1(d)αeffis shown by the red solid curve as a function of\ntime. The effective damping constant is reduced to about one-\nfifth of the original value of α=0.05 (dotted black). Contrary\nto˙φ,αeffdoes not show clear correlation with the dynamics of\nm. During the precession, αeffis kept almost constant.\nThe enhancement of the precession angular velocity and the\nreduction of the Gilbert damping constant due to the non-\nMarkovian damping will be explained by deriving the e ffectiveLLG equation that is valid up to the first order of αin the next\nsubsection.\n3.2. Theoretical analysis\nSince the Gilbert damping constant, α, of a conventional\nmagnet is of the order of 0 .01∼0.1, it is natural to take the\nfirst order ofαto derive the effective equations of motion for\nm. The other parameters related to the motion of mareγ,H,\nandτc. Multiplying these parameters we can obtain the dimen-\nsionless parameter, ξ=γHτc, which represents the increment\nof the precession angle during the correlation time.\nIn the case ofξ<1 Miyazaki and Seki dereived the e ffective\nLLG equation using time derivative series expansion[16]. W e\nfirst briefly review their analysis. Then we derive the e ffective\nLLG equation forξ>1 using the time-integral series expansion\nand show that the e ffective LLG equation has the same form for\nbothξ< 1 andξ> 1. Therefore, it is natural to assume that\nthe derived effective LLG equation is valid for any values of ξ\nincludingξ=1.\n3.2.1. Brief review of Miyazaki and Seki’s derivation of the ef-\nfective LLG equation for ξ<1\nIn Ref. 16, Miyazaki and Seki derived the e ffective LLG\nequation with renormalized parameters using the time deriv a-\ntive series expansion. Similar analysis of the LLG equation\nwas also done by Shul in the study of the damping due to\nstrain[20, 21]. The following is the brief summary of the der iva-\ntion.\nSuccessive application of the integration by parts using ν(t−\nt′)=τc[dν(t−t′)/dt′] gives the following time derivative se-\nries expansion:\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay\nn=1(−τc)n−1dnm\ndtn. (11)\nThen the non-Markovian damping term in Eq. (3) is expressed\nas\nα∞/summationdisplay\nn=1(−τc)n−1/parenleftigg\nm×dnm\ndtn/parenrightigg\n. (12)\nThe first derivative, n=1, is given by\n˙m=−γHm×ez+O(α), (13)\nwhere Ois the Bachmann–Landau symbol. For n=2, substi-\ntution of Eq. (13) into the time derivative of Eq. (13) gives\n¨m=(−γH)2(m×ez)×ez+O(α). (14)\nThe n-th order time derivative is obtained by using the same\nalgebra as\ndn\ndtnm=(−γH)n/bracketleftbig(m×ez)×ez.../bracketrightbig+O(α), (15)\nwhere ezappears ntimes. Expanding the vector products we\nobtain for even order time derivatives\nd2nm\ndt2n=(−1)n(γH)2n/bracketleftbigm−mzez/bracketrightbig+O(α), (16)\n3and for odd order time derivatives\nd2n+1m\ndt2n+1=(−1)n(γH)2n˙m+O(α). (17)\nSubstituting Eqs. (16) and (17) into Eq. (12) the non-\nMarkovian damping term is expressed as\n−∞/summationdisplay\nn=1γ2nm×ez+∞/summationdisplay\nn=0α2n+1m×˙m, (18)\nwhere\nγ2n=αγH m z(−1)n−1ξ2n−1(19)\nα2n+1=α(−1)nξ2n. (20)\nThe sums in Eq. (18) converge for ξ<1. Introducing\n˜γ=γ/parenleftigg\n1+αmzξ\n1+ξ2/parenrightigg\n(21)\n˜α=α\n1+ξ2, (22)\nEq. (3) can be expressed as the following e ffective LLG equa-\ntion with renormalized gyromagnetic ratio, ˜ γ, and damping\nconstant, ˜α:\n˙m=−˜γm×(H+r)+˜αm×˙m+O(α2). (23)\n3.2.2. Derivation of the e ffective LLG equation for ξ>1\nForξ>1 we expand Eq. (3) in power series of 1 /ξusing the\ntime integral series expansion approach. Using the integra tion\nby parts with dν(t−t′)/dt′=ν(t−t′)/τcthe integral part of the\nnon-Markovian damping can be written as\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=1\nτc/integraldisplayt\n−∞˙m(t′)dt′\n−1\nτc/integraldisplayt\n−∞ν(t−t′)/bracketleftigg/integraldisplayt′\n−∞˙m(t′′)dt′′/bracketrightigg\ndt′. (24)\nSuccessive application of the integration by parts gives\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=−∞/summationdisplay\nn=1/parenleftigg\n−1\nτc/parenrightiggn\nJn, (25)\nwhere Jnis the nth order multiple integral defined as\nJn=/integraldisplayt\n−∞/integraldisplayt1\n−∞···/integraldisplaytn−1\n−∞˙m(tn)dtn···dt2dt1. (26)\nFrom Eq. (17), on the other hand, ˙ mis expressed as\n˙m=1\n(−1)n(γH)2nd2n\ndt2n˙m+O(α). (27)\nSubstituting Eq. (27) into Eq. (26) the multiple integrals a re\ncalculated as\nJ2n=1\n(−1)n(γH)2n˙m (28)\nJ2n−1=1\n(−1)n(γH)2n¨m. (29)Then Eq. (25) becomes\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay\nn=11\n(−1)n−1ξ2n˙m\n+∞/summationdisplay\nn=1τc\n(−1)nξ2n¨m. (30)\nSubstituting Eq. (30) into the second term of Eq. (3) the non-\nMarkovian damping term is expressed as\nα∞/summationdisplay\nn=11\n(−1)n−1ξ2nm×[ ˙m+τc¨m]+O(α2). (31)\nFrom Eq. (16) ¨ mis expressed as\n¨m=(−1)(γH)2/bracketleftbigm−mzez/bracketrightbig. (32)\nSubstituting Eqs. (31) and (32) into Eq. (3) we obtain\n˙m=−γH∞/summationdisplay\nn=1/bracketleftigg\n1+αmz\n(−1)n−1ξ2n−1/bracketrightigg\nm×ez−γm×r\n+α∞/summationdisplay\nn=11\n(−1)n−1ξ2nm×˙m+O(α2). (33)\nThe sums converge for ξ>1, and the effective LLG equation\nforξ>1 has the same form as ξ<1, i.e. Eq. (23). Since the\neffective LLG equation has the same form for both ξ< 1 and\nξ>1, it is natural to Eq. (23) is valid for any values of ξ.\nAs pointed out by Miyazaki and Seki, and independently by\nSuhl the effect of the non-Markovian damping on the precession\ncan be regarded as the renormalization of the e ffective field [16,\n20, 21]. Equation (23) can be expressed as\n˙m=−γm×/parenleftigg\nH+αHξ\n1+ξ2mz/parenrightigg\nez−γm×r\n+˜αm×˙m+O(α2). (34)\nThe second term in the bracket represents the fictitious unia xial\nanisotropy field originated from the non-Markovian damping .\nThe fictitious anisotropy field increases with increase of ξfor\nξ<1 and takes the maximum value of αH m z/2 atξ=1, i.e.\nγHτc=1. Forξ>1 the fictitious anisotropy field decreases\nwith increase ofξand vanishes in the limit of ξ→∞ because\nthe non-Markovian damping term vanishes in the limit of τc→\n∞. The precession angular velocity, ˙φ, is expected to have the\nsameξdependence as the fictitious anisotropy field and to have\nthe same temporal evolution as mzas shown in Figs. 1(b) and\n1(c).\n3.2.3. The Correlation time dependence of the precession an -\ngular velocity, and e ffective damping constant\nEquation (21) tells us that up to the first order of αthe pre-\ncession angular velocity can be approximated as\n˙φ≃˜γH=γH/bracketleftigg\n1+αmzγHτc\n1+(γHτc)2/bracketrightigg\n, (35)\n4τc’ =1/( γH) \n0.1 1 10 0.01 \nτc [ps] τc [ps] a) b) \nφ [rad / ps] \n1.76 1.77 1.79 \n1.78 1.80 \n0.1 1 10 0.01 \nα, αeff ~\nαeffα0.04 \n0.00 0.02 0.05 \n0.01 0.03 \nααα\neffeffeffαeffαeff~sim. approx. \nFigure 2: (a) The correlation time, τc, dependence of the precession angular\nvelocity,δ˙φ, atθ=5◦forH=10 T. The solid yellow curve shows the ap-\nproximation result, ˜ γH. The dotted black curve shows the simulation results\nobtained by numerically solving Eqs. (7) and (8). The thin ve rtical dotted line\nindicates the critical value of the correlation time, τ′\nc=1/(γH). (b)τcdepen-\ndence of ˜α(solid yellow) andαeff(dotted black). The parameters and the other\nsymbols are the same as panel (a).\nwhere the second term in the square bracket represents the en -\nhancement due to the fictitious anisotropy field.\nIn Fig. 2(a) the approximation result of Eq. (35) at θ=5◦\nwhere ˙φis almost saturated is plotted as a function of τcby the\nsolid yellow curve. The external field and the Gilbert damp-\ning constant are assumed to be H=10 T andα=0.05, re-\nspectively. The corresponding simulation results obtaine d by\nnumerically solving Eqs. (7) and (8) are shown by the dotted\nblack curve. Both curves agree well with each other because\nαis as small as 0.05. The precession angular velocity is maxi-\nmized at the critical value of the correlation time τ′\nc=1/(γH).\nFigure 2(b) shows the τcdependence of ˜α(solid yellow) and\nαeff(dotted black) for the same parameters as panel (a). Both\ncurves agree well with each other and are monotonic decreasi ng\nfunctions ofτc. They vanish in the limit of τc→∞ similar to\nthe non-Markovian damping term.\n4. Effect of an anisotropy field on precession dynamics\nThe theoretical analysis given in the previous section can\nbe applied to the case with Hk/nequal0 by replacingξwithξk=\nγ(H+Hkmz)τc. Following the same procedure as for Hk=0\nEq. (3) can be expressed as\n˙m=−γm×/parenleftig\nH+αHξk\n1+ξ2\nkmz+αHkξk\n1+ξ2\nkm2\nz/parenrightig\nez\n−γm×r+α\n1+ξ2\nkm×˙m+O(α2). (36)\nThe second and the third terms in the bracket can be regarded\nas the fictitious uniaxial and unidirectional anisotropy fie lds\ncaused by the non-Markovian damping. Similar to the re-\nsults for Hk=0 the precession angular velocity is maximized\natξk=1. The renormalized damping constant is given by\nα/(1+ξ2\nk) which is a monotonic decreasing function of ξkand\nvanishes in the limit of ξk→∞ .c) d) a) b) \n24 6 14 12 10 8 0\nH [T] δφ /φ0 [%] 2\n013\nδφ /φ0 [%] 2\n01\n24 6 14 12 10 8 0\nH [T] Hk = 0 H’=1/( γτ c) \nτc = 1 ps \nθ = 5 oθ = 5 oHk = 1 T \nτc = 1 ps H’=1/( γτ c) − HkmzH = 6, 7, 8, 9 T \nH = 2, 3, 4, 5 T \n = 1 ps \n = 5  = 0 \n = 1 ps τ\nθH\nτ = 1 ps  = 1 ps  = 1 ps τc = 1 ps τ = 1 ps τ = 1 ps \nθτH\nτ\n = 5  = 1 ps  = 1 T \n = 1 ps  = 1 ps  = 1 T \n = 1 ps  = 1 ps τc = 1 ps ττ = 1 ps  = 1 ps 3δφ /φ0 [%] 2\n013\nδφ /φ0 [%] 2\n013\nt [ps] 100 200 0\nt [ps] 100 200 0Hk = 0 \nτc = 1 ps Hk = 0 \nτc = 1 ps \nFigure 3: (a)τcdependence ofδ˙φ/˙φ0atθ=5◦. From top to bottom the\nexternal field is H=2,3,4,5 T. The parameters are Hk=0, andτc=1 ps. (b)\nThe same plot as panel (a) for H≥5 T. From top to bottom the external field is\nH=6,7,8,9 T. (c) Hdependence ofδ˙φ/˙φ0atθ=5◦obtained by solving Eqs.\n(7) and (8). The parameters are Hk=0, andτc=1 ps. The critical value of\nthe external field, H′=1/(γτc), is indicated by the thin vertical dotted line. (d)\nThe same plot as panel (c) for Hk=1 T. The thin vertical dotted line indicates\nthe critical value of the external field, H′=1/(γτc)−Hkmz.\n5. A possible experiment to determine the correlation time\nBased on the results shown in Secs. 3 and 4 we propose a\npossible experiment to determine the correlation time, τc. Sim-\nilar to the previous sections we first discuss the case withou t\nanisotropy field, i.e. Hk=0, and then extend the discussion to\nthe case with Hk/nequal0.\nIn Figs. 3(a) and 3(b) we show the temporal evolution of the\nenhancement of angular velocity, δ˙φ/˙φ0, obtained by the solv-\ning Eqs. (7) and (8) for various values of H. The increment\nof the precession angular velocity is defined as δ˙φ=˙φ−˙φ0.\nThe initial state and the correlation time are assumed to be\nm=(1,0,0) andτc=1 ps, respectively. As shown in Fig.\n3(a),δ˙φ/˙φ0increases with increase of HforH≤5T. Once\nthe external field exceeds the critical value of 1 /(γτc)=5.7 T,\nδ˙φ/˙φ0decreases with increase of Has shown in Fig. 3 (b). The\nresults suggest that correlation time can be determined by a na-\nlyzing the external field that maximizes the enhancement of t he\nprecession angular velocity.\nFigure 3(c) shows the Hdependence ofδ˙φ/˙φ0atθ=5◦\nwhereδ˙φ/˙φ0is almost saturated. The enhancement is maxi-\nmized at the critical value of the external field, H′=5.7 T. The\ncorrelation time is calculated as τc=1/(γH′)=1 ps.\nIf the system has a uniaxial anisotropy field, Hk, the en-\nhancement of the precession angular velocity is maximized a t\nH′=1/(γτc)−Hkmzas shown in Fig. 3(d). The correlation\ntime is obtained as τc=1/γ(H′+Hkmz).\nThe above analysis is expected to be performed experimen-\n5tally using the time resolved magneto optical Kerr e ffect mea-\nsurement technique. In the practical experiments the analy sis\ncan be simplified as follows. The polar angle of the initial st ate\nis not necessarily large. It can be small as far as the preces-\nsion angular velocity can be measured. Instead of analyzing\nδ˙φ/˙φ0, one can analyze ˙φ/Hor˙φ/(H+Hkmz) because they are\nmaximized at the same value of Hasδ˙φ/˙φ0. Since the required\nmagnetic field is as high as 10 T, a superconducting magnet [22 ]\nis required.\n6. Summary\nIn summary we theoretically analyze the ultrafast precessi on\ndynamics of a small magnet with non-Markovian damping. As-\nsumingα≪1, we derive the effective LLG equation valid for\nany values ofτc, which is a direct extension of Miyazaki and\nSeki’s work[16]. The derived e ffective LLG equation reveals\nthe condition for maximizing ˙φin terms of Handτc. Based on\nthe results we propose a possible experiment for determinat ion\nofτc, whereτccan be determined from the external field that\nmaximizesδ˙φ/˙φ0.\nReferences\n[1] L. Landau, E. Lifshits, ON THE THEORY OF THE DISPER-\nSION OF MAGNETIC PERMEABILITY IN FERROMAGNETIC\nBODIES, Physikalische Zeitschrift der Sowjetunion 8 (1935 ) 153.\ndoi:10.1016/B978-0-08-010586-4.50023-7 .\n[2] T. Gilbert, Classics in Magnetics A Phenomenological Th eory of Damp-\ning in Ferromagnetic Materials, IEEE Transactions on Magne tics 40 (6)\n(2004) 3443–3449. doi:10.1109/TMAG.2004.836740 .\n[3] W. F. Brown, Thermal Fluctuations of a Single-Domain Par ticle, Physical\nReview 130 (5) (1963) 1677–1686. doi:10.1103/PhysRev.130.1677 .\n[4] E. Beaurepaire, J.-C. Merle, A. Daunois, J.-Y . Bigot, Ul trafast Spin Dy-\nnamics in Ferromagnetic Nickel, Physical Review Letters 76 (22) (1996)\n4250–4253. doi:10.1103/PhysRevLett.76.4250 .\n[5] C. D. Stanciu, F. Hansteen, A. V . Kimel, A. Kirilyuk, A. Ts ukamoto,\nA. Itoh, T. Rasing, All-Optical Magnetic Recording with Cir cu-\nlarly Polarized Light, Physical Review Letters 99 (4) (2007 ) 047601.\ndoi:10.1103/PhysRevLett.99.047601 .\n[6] G. P. 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Chubykalo-Fesenko, R. W. Chantrell, U. N owak,\nA. Rebei, Ultrafast Spin Dynamics: The E ffect of Col-\nored Noise, Physical Review Letters 102 (5) (2009) 057203.\ndoi:10.1103/PhysRevLett.102.057203 .\n[18] C. W. Gardiner, Stochastic Methods: A Handbook for the N atural and\nSocial Sciences, 4th Edition, no. 13 in Springer Series in Sy nergetics,\nSpringer, Berlin Heidelberg, 2009.\n[19] G. E. Uhlenbeck, L. S. Ornstein, On the Theory of the\nBrownian Motion, Physical Review 36 (5) (1930) 823–841.\ndoi:10.1103/PhysRev.36.823 .\n[20] H. Suhl, Theory of the magnetic damping constant, IEEE T ransactions on\nMagnetics 34 (4) (1998) 1834–1838. doi:10.1109/20.706720 .\n[21] H. Suhl, Relaxation Processes in Micromagnetics, Oxfo rd University\nPress, 2007. doi:10.1093/acprof:oso/9780198528029.001.0001 .\n[22] H. W. Weijers, U. P. Trociewitz, W. D. Markiewicz, J. Jia ng, D. My-\ners, E. E. Hellstrom, A. Xu, J. Jaroszynski, P. Noyes, Y . Viou chkov,\nD. C. Larbalestier, High field magnets with HTS conductors, I EEE\nTransactions on Applied Superconductivity 20 (3) (2010) 57 6–582.\ndoi:10.1109/TASC.2010.2043080 .\n6"    },    {        "title": "2205.09913v1.Effects_of_Crystalline_Disorder_on_Interfacial_and_Magnetic_Properties_of_Sputtered_Topological_Insulator_Ferromagnet_Heterostructures.pdf",        "content": "Effect s of Crystal line Diso rder on Interfacial and \nMagnetic  Properties of Sputter ed Topological \nInsulator/Ferromagnet  Heterostructures  \n1Nirjhar Bhattacharjee,3Krishnamurthy Mahalingam , 2Adrian Fedorko, 1Alexandria Will -Cole,  \n1Jaehyeon Ryu, 3Michael Page, 3Michael McConney,  1†Hui Fang, 2Don Heiman, 1Nian Xiang \nSun* \n1 Northeastern University, Department of Electrical and Computer Engineering , Boston MA \n02115  \n2 Northeastern University, Department of Physics , Boston MA 02115  \n3Air Force Research Laboratory, Nano -electronic Materials Branch, Wright Patterson Air Force \nBase, OH 05433  \nKEYWORDS. Topological Insulator, Ferromagnet, Interface, Spin Pum ping, Spin Orbit Torque, \nAntiferromagnet  \nABSTRACT . Thin film s of Topological insulators  (TIs)  coupled with ferromagnets  (FMs) are \nexcellent  candidates for energy -efficient spintronics devices.  Here, the effect of crystalline \nstructural disorder of TI on interfac ial and magnetic properties of  sputter -deposited  TI/FM , \nBi2Te3/Ni 80Fe20, heterostructures is reported . Ni and a small er amount  of Fe from Py was found to diffuse  across the interface  and react with  Bi2Te3. For highly crystalline  c-axis oriented  Bi2Te3 \nfilms , a giant enhancement in Gilbert damping  is observed , accompanied by an effective  out-of-\nplane magnetic anisotropy  and enhanced damping -like spin -orbit torque  (DL-SOT) , possibly  due \nto the topological surface states  (TSS)  of Bi2Te3. Further more , a spontaneous exchange bias is \nobserved in hysteresis loop measurements at low temperatures . This is because of an  \nantiferromagnetic  topological interfac ial layer  formed by reaction of the diffused  Ni with Bi2Te3 \nwhich  couples with the FM , Ni80Fe20. For increas ing disorder of Bi 2Te3, a significant  weakening  \nof exchange interaction in the AFM interfacial layer  is found . These experimental results  Abstract \nlength is one paragraph.  \n \n1. INTRODUCTION  \nTopological insulators (TIs) of the (Bi,Sb) 2(Te,Se) 3 family of compounds are van der Waals (vdW) \nchalcogenide materials with tetradymite structures. TIs possess large spin -orbit coupling (SOC) \nresulting in  dissipationless surface conducting st ates – topological surface states ( TSS) [1-3]. \nIntroducing magnetic order in TIs leads to gap opening in the TSS bands and possibility of \ndissipationless quantum anomalous Hall (QAH) and axion insulator states [ 4-21]. Stimulated by \nthese remarkable  material properties, TIs are regarded as  promising candidates for realization of \nenergy efficient spintronic devices. TIs possess highly reactive surfaces , thus making them  \nsusceptible to formation of interfacial phases  when coupled with metalli c films [ 22-25]. Because \nof their composition, t hese interfacial layers  have the potential for hosting fascinating topological \nmagnetic phases [ 25]. The m ajority of reported experiment s have studied TIs grown from \nMolecular Beam Epitaxy (MBE) [22-24], whic h is a standard  technique for growing high -quality , \ncrystalline -ordered thin films . However, MBE  suffers from low throughput and is constrained by sample dimensions , making it incompatible for integration in industrial CMOS processes. \nMagnetron sputtering on the other hand is the semiconductor industry’s  accepted  thin film \ndeposition technique because of its advantage of high throughput  and large area film growth. \nSputtering also allows easy deposition of TIs with varying crystalline disorder [25 -29]. This \ncontrollability  opens up the possibility of exploration of their disorder -dependent electronic and \nmagnetic properties .  \nRecently, the topological antiferromagnetic (AFM) compound  NiBi 2Te4 was discovered in the \ninterface of highly c-axis-oriente d sputtered Bi2Te3/Ni80Fe20 heterostructures [25]. Ni from the \nNi80Fe20 (Py) layer diffuses and reacts with Bi 2Te3 layer, and the reaction is promoted by the \ndelocalized TSS electrons  [23-25]. Also, recent experiment s have  show n the  presence of TSS even \nin amorphous Bi2Se3 [30]. In this work , the effects of crystalline structural disorder on the interface \nand magnetic properties  of Bi 2Te3/Py heterostructures  are investigated . The m agnetic  species , \nlargely Ni and small er amounts of Fe , are found to diffuse across the interface into Bi 2Te3, resulting \nin a magnetic  interfacial layer. For increas ing c-axis-oriented texture of Bi 2Te3, increasing  amount s \nof diffused magnetic  species were found to react with Bi 2Te3, which also leads  to enhanced  \nmagnetic  propert ies. This phenomenon  was identified in  room tem perature  hysteresis loop \nmeasurements  of the magnetic moment versus applied magnetic field , m(H), for the Bi 2Te3/Py \nsamples compared to a Py control sample.  As a result of the diffusion  of the magnetic species (Ni, \nFe) and reaction with Bi 2Te3, the saturation magnetic moment  (m) is reduced by  Δm in the \nBi2Te3/Py compared to Py  samples  suggesting change in valence state of the magnetic species . \nThe values of Δm becomes smaller  for increasing  disorder  in Bi 2Te3 suggesting lesser reaction \nbetween diffused Ni, Fe and Bi 2Te3. Further, a giant enhancement in Gilbert damping , an out-of-\nplane canting of magnetization  and enhanced DL -SOT  were  observed  in samples with highly c-axis oriented TI . However, with  significantly reduced  crystallinity, surprisingly the granular \nBi2Te3 samples had a comparabl e enhanced spin -charge conversion efficiency as samples with \nhighly c-axis-oriented Bi 2Te3, possibly due to the  quantum confinement effect in smaller crystallite \ngrains [ 26,27 ]. Low-temperature m(H) and m(T) measurements revealed an AFM ordered phase \nin the predominantly Ni-diffused Bi2Te3 interface from the  formation of the topological AFM \ncompound  NiBi 2Te4 [25]. Interestingly, the strength of the exchange interaction of the  interfacial  \nAFM phase , as monitore d by the exchange bias,  was found to weaken significantly with increase \nin disorder of the Bi 2Te3 layer. These results indicate  strong topological property of TI s with high \ncrystalline c-axis-oriented growth , which weakens  considerably with in creasing crystalline \ndisorder . These experimental results  show the possibility of tailoring topological properties of TIs \nby control of crystalline structural disorder .  \n2. EXPERIMENTAL RESULTS AND DISCUSSIONS  \n \nFigure 1.  a) Schematic model of three  quintuple Bi2Te3 unit cell s.  Cross -sectional HRTEM images \nshowing structural disorder in b) GBT, c) DBT and d) CBT samples. e) XRD data for the GBT, \nDBT and CBT samples. Inset: normalized plots of symmetric XRD data. f) XRR plots and \ntheoretical fitting for GBT, DBT and CBT samples used for characterization of thickness and \nsurface roughness. The data for CBT samples are similar to the ones in ref [25] . \n2.1. Crystalline Structure Properties of Sputter -deposited Bi 2Te3. Samples of 30 nm  Bi2Te3 \nwith varying crystalline disorder , (1) granular  (GBT), (2) randomly oriented polycrystalline \ndisordered  (DBT) , and (3) highly c-axis-oriented crystalline  (CBT) were grown using RF \nmagnetron sputtering  on amorphous thermally oxidized Si substrates  (see Supporting Information  \nSection S1 for grain size characterization ). Crystalline structural propert y of the Bi 2Te3 samples  \nwere verified using  X-ray diffraction (XRD ) and high -resolution transmission electron microscope \n(HRTEM) i maging measurements , as shown in Fig. 1a,b,d. The GBT samples did not show  any \nsignificant diffraction peaks in the XRD  measurement , suggesting a high amorphous content. \nFurther, HRTEM images of the GBT, DBT and CBT samples shown in Fig. 1a verifies the \ngranular, randomly oriented vdW domains and high c-axis-oriented layered structure , respectively. \nThe thickness and surface roughness  of the samples  were characterized using  X-ray reflectometry \n(XRR ) measurements , as shown in Fig. 1e. From the fitting of XRR data , thickness of ~30 nm was \nobtained for all three samples. The fits to the XRR data also revealed surface roughness of  0.7 nm, \n1.7 nm and 1.0 nm for the GBT, DBT and CBT samples , respectiv ely, which are typical surface \nroughness values for sputter -deposited thin films , confirming growth of high -quality TI films .  \n \n \n  \nFigure 2.  Cross -sectional HRTEM images of a) highly amorphous GBT/Py, b) disordered DBT/Py  \nwith randomly oriented vdW domains  and c) highly c-axis oriented CBT/Py  (similar to ref [25]). \nThe yellow dashed lines mark the approximate interface between the Bi 2Te3 and Py layers.  Atomic \n% (At %) characterized using EDS for d) GBT/Py, e) DBT/Py and f) CBT/Py samples. The \ninterface layers are highlighted in green and blue colors for the Ni -Bi2Te3 and Py*, respectively. \ng) Schematic of the Bi 2Te3/Py samples with the layers marked. h) m(H) measurements for IP \norientation at room temperature showing loss of moments in the Bi 2Te3/Py samples due to \ninterfacial Ni and Fe diffusion and reaction with Bi 2Te3. Inset: comparison of % loss of moments \n(Δm%) in GBT/Py, DBT/Py and CBT/Py samples compared to the control Py sample.  The data  \npresented  for the CBT sample are similar to the ones in ref [25].  \n2.2. Morphology of Interfacial Layer formed by Ni Diffusion  into Bi 2Te3. Heterostructure \nsamples of GBT/Py, DBT/Py and CBT/Py were grown where the thickness of the layers was \nmaintained at 30 nm and 20 nm respectively for Bi 2Te3 and Py respectively. HRTEM imaging and \nenergy  dispersive X -ray spectroscopy (EDS) measurements were performed to characterize the \nmorphology and stoichiometric composition a long the cross section of the samples , as shown in \nFigures  2a-f (see Supporting Information  Table S1 for average atomic % ). The HRTEM images in \nFigures 2 a -c clearly show a highly amo rphous nature of GBT, randomly oriented vdW layered \ncrystalline domains in DBT and highly oriented vdW layers in the CBT layers. A closer \nexamination of the interfaces of the heterostructures also reveal a rougher interface in the \ndisordered GBT/Py and DBT /Py samples compared to the CBT/Py sample. The EDS cross -\nsection al profiles of atomic % of elements in Figures 2d -f show  a significant  diffusion of Ni (and \nsmaller amounts of Fe) across the TI/FM interface into the Bi 2Te3 layers , forming an inter facial \nlayer denoted as  Ni-Bi2Te3. In general, the Ni and Fe have a large gradient over a Bi2Te3 distance \nof 5 to 14 nm, where the Ni averages 30 to 40 %, while the Fe diffusion is much smaller  in the \nGBT /Py and CBT/Py samples.  The disordered GBT/Py and DBT/Py samples also have ~3% of Ni  \ndiffused  throughout the thickness of the Bi 2Te3 layer. However, t he predominantly Ni-Bi2Te3 layer  \nin the  highly  ordered CBT/Py sample  appears to  act as a barrier . This prevents the  diffusion of Ni further into the Bi 2Te3 bulk. The formation of the Ni -Bi2Te3 layer is likewise accompanied by a \nthin Fe -rich region in the intermediate Py layer (marked Py*). It is also noted that the moderately -\ndisordered DBT/Py sample which has randomly oriented vdW polycrystalline domains has \ndeveloped  a much higher Ni and Fe diffusion of ~47% and ~9%  at the interface , respectively. The \nFe diffusion, however, is only ~3 -4% at the interface in the GBT/Py and CBT/Py samples.  \nThe diffusion of Ni into high -quality CBT Bi 2Te3 was previously shown to result from  solid -\nstate reactions leading to the formation of Ni -Te bonds and formation of the topological AFM \ncompound, NiBi 2Te4 [25].  Similar to that study, t he room temperature m(H) measurements can \nbe used here as an indicator of the reaction of Ni with Bi2Te3 that is promoted  by the delocalized \nTSS electrons . As shown in Figure  2h, all the Bi2Te3/Py samples  show a clear decrease in \nsaturation magnetic moment for increasing disorder . This reduction in moments  result s from \nchange in valence state of the reacting magnetic species . This loss of saturation moment  is \ncompared to a control sample of Py  by Δm. The Δm% values were found to be 13%, 3 7% and 41% \nfor the GBT, DBT and CBT samples, respectively. This clear  enhancement  in the  loss of moment s \nwith crystalline order  and hence reactivity of Ni with Bi 2Te3 [25] is due to  strengthening  of TSS \nwith increas ing crystallinity  of Bi 2Te3.  \n2.3. Disorder Effects o n Room -Temperature Magnetic Properties of Bi2Te3/Py. To investigate \nthe effect s of disorder, m(H) hysteresis loop, and ferromagnetic resonance (FMR) were performed \non the three types of samples, highly disordered GBT/Py, moderately disordered DBT/Py and \nhighly ord ered CBT/Py. First, m(H) hysteresis loop measurements were performed on the samples , \nwith the magnetic field oriented in -plane (IP) and out -of-plane (OP) relative to  the film plane , as \nshown in Figures 3 a -c. For increasing c-axis-oriented growth of  the Bi2Te3 layer , the saturation \nfield, Hs measured in the IP and OP configurations show an increasing and decreasing trend , respectively . Also,  the ratio of remanence  to saturation  magnetization  (Mr/Ms) in IP m(H) loop \ndecreases  for increasing crystalline order of Bi 2Te3. These trends indicate a n increase  in effective \nOP magnetic easy-axis with increased c-axis-oriented texture of Bi 2Te3 in the Bi 2Te3/Py \nheterostructure samples . This enhanced OP magnetic anisotropy  is a characteristic of interaction \nof the magnetic moments with large  SOC  in the  interfaces  [31]. A large OP anisotropy has been  \npreviously  predicted and observed in other  TI/FM -based materials systems [32,33 ] (also see \nSupporting Information  Section S4 ). As shown in Figure  3b, the OP m(H) loops for the Bi 2Te3/Py \nsamples also exhibit a smaller hysteresis -loop in the low -field region s. These smaller components \nof m(H) loop are more prominent in the DBT/Py and CBT/Py samples , which otherwise exhibit a \nlower OP easy-axis of magnetic moments compared to the highly c-axis-oriented CBT/Py sample . \nThe DBT/Py sample also had an unusually large coercive field ( Hc) resulting from the randomly \noriented vdW layered crystalline domains , as observed in both the IP and OP m(H) loop \nmeasurements in Figures  3a,b. These effects in the samples with highly disordered TIs are possibly \npresent due to disordered magnetic texture that emerg e in the ir relatively  rougher interfaces with \na net OP component.  \nFurther  information was obtained using  ferromagnetic resonance (FMR) measurements to \nunderstand the changes in magnetization dynamic s with changes in TI disorder . The FMR \nlinewidth ( ΔH) and resonance field ( Hres) were extracted from the FMR sign al at different constant \nfrequencies  (fres) (Supporting Information  Section S3 ). The Gilbert damping parameter  (𝛼) was \nextracted  by fitting a straight line to the FMR linewidth versus frequency plot  using the equation , \n𝛥𝐻=𝛥𝐻0+𝛾\n2𝜋𝛼𝑓𝑟𝑒𝑠. Here , ΔH 0 is the inhomogeneous linewidth and 𝛾 is the gyromagnetic ratio . \nAs shown in Figure  3d, the values of  𝛼 extracted  for the Py, GBT/Py, DBT/Py and CBT/Py \nsamples  are 0.0053, 0.0076, 0.0089 and 0.0123 , respectively.  This shows a progressive increase in 𝛼 with increase in crystallite grain size of Bi 2Te3, and a giant enhancement  when  the Bi2Te3 layer \nis highly c-axis oriented  (summarized in Figure 3f) . This effect was also observed in other \nBi2Te3/FM heterostructure materials (Supporting Information  Section S 5), which signals  a large \nenhancement in SOC and presence of robust  TSS in highly c-axis-oriented Bi 2Te3. In addition , the \neffective magnetization, 4π Meff, and the perpendicular magnetic anisotropy (PMA) field, 𝐻⊥, were \nextracted by fitting the modified Kittel equation  to the fres versus Hres plots shown in Fig ures 3e,f , \n𝑓𝑟𝑒𝑠=𝛾\n2𝜋√(𝐻𝑟𝑒𝑠+𝐻𝑎)(𝐻𝑟𝑒𝑠+𝐻𝑎+4𝜋𝑀𝑒𝑓𝑓), where  4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻⊥ and 𝐻𝑎 is the \nuniaxial anisotropy field . The 4𝜋𝑀𝑠 values were found to be 15.2, 14 .7, 14. 3 and 14.1 kOe , \nrespectively , for Py, GBT /Py, DBT/Py and CBT/Py samples . The decrease  in 4𝜋𝑀𝑠 for increasing \ncrystalline order  demonstrates  the reduction  in saturation magnetization due to interfacial diffusion \nof Ni  and Fe from Py into Bi 2Te3 [25]. Also , the 𝐻⊥ values increased from 4.35, 5.07, 5.54 and \n5.79 kOe, for the Py, GBT/Py , DBT/Py and CBT/Py sample s, respectively. This enhancement  in \n𝐻⊥ supports the m(H) results  that show an increase in effective OP anisotropy  with increasing  c-\naxis-oriented texture of Bi 2Te3. The magnetic properties measured using m(H) loops and FMR are \nsummarized in Table 1. \nThe enhancement in α for increasing c-axis texture of Bi 2Te3 can be attributed to a large spin -\npumping  effect  modeled  by the spin -mixing conductance, 𝑔↑↓=4𝜋𝑀𝑠𝑡𝐹𝑀𝛥𝛼\nℏ𝛾, where tFM is the \nthickness of the FM layer  and 𝛥𝛼 is the enhancement in Gilbert damping  and ℏ is the reduced \nPlank’s constant . The resulting  𝑔↑↓ values increased with increasing crystalline c-axis orientation \nof Bi 2Te3 and, were 2.10×10-18, 2.38×10-18 and 4.08×10-18 m-2 for the GBT/Py, DBT/Py and \nCBT/Py samples , respectively . The 𝑔↑↓ values were calculated  assuming the Gilbert damping \nenhancement  in the Bi 2Te3/Py samples  are entirely due to sp in-pumping . The loss of magnetization \nfrom  interfacial diffusion  of Ni from Py  and spin -memory loss due to interfacial proximity -induced magnetization [ 13-19] may also play a role in the enhancement of α in Bi 2Te3/Py samples . But \nthese contributions to wards  enhancement in α could  not be isolated  because of complexity  in these \nheterostructure material systems. However, the large enhancement in 𝛼 with highly c-axis-oriented \nTIs is also observed in other TI/FM materials  systems  [27,33 -37], including those which do not  \nshow  interfacial diffusion  (see Supporting Information  Section S4) . This suggests that for highly  \ncrystalline  oriented TI s, the TI/FM heterostructures experience a giant  enhancement  in spin -\npumping  predominantly from  the presence of robust TSS. The reduction in magnetization of the \nPy layer because of diffusion of Ni and reacti on with Bi2Te3 was previously shown [ 25]. These \nresults provide strong evidence of enhancement  in SOC strength and topological properties in \nhighly c-axis-oriented TI samples  compared to disordered TIs .  \n \nFigure 3.  Normalized m(H) loops measured : a) in-plane and b) out-of-plane for the GBT/Py, \nDBT/Py and CBT/Py samples. Inset: Expanded low -field regions showing enhanced Hc for the \nBi2Te3/Py samples compared to the Py control sample. c) Comparison of the saturation fields (IP \nand OP) and Mr/Ms ratio clearly highlight ing an increase in OP anisotropy with crystalline c-axis \norientation of Bi 2Te3. d) FMR linewidth versus frequency for extraction of α. e) FMR resonance \nfrequency versus field for extracting 𝐻⊥ and 4πM eff. f) Visual comparison of α and 𝐻⊥ extracted \nfrom d and e respectively. All measurements here were performed at 300 K.  \nTable 1. Summar y of room -temperature magnetic properties of the GBT/Py, DBT/Py and CBT/Py \nheterostructure samples.   \nSample  Hs OOP  Hs IP Mr/Ms 4πMs 𝑯⊥-FMR  \nα \n(kOe)  (kOe)  (%) (kOe)  (kOe)  \nPy 11.9 0.95 96 1.52 4.35 0.0053  \nGBT/Py  11.2 1.22 96 1.47 5.07 0.0075  \nDBT/Py  9.7 2.01 90 1.43 5.54 0.0089  \nCBT/Py  8.1 3.05 84 1.41 5.78 0.0123  \n \n \nFigure 4 . a) Schematic for ST -FMR experimental setup  with LIA and phase -locked RF current \nsource . The brown and dark blue arrows signify up -spin and down -spin states , respectively. ST -\nFMR data and Lorentzian fitting for : b) GBT/Py, c) DBT/Py , and d) CBT/Py samples , measured \nat 4 GHz frequency. e) DL-SOT (blue) and Oersted  plus FL-SOT for the GBT/Py, CBT/Py and \nDBT/Py samples extracted from b -d. f) 𝜃𝑆𝐻 of the GBT/Py, CB T/Py and DBT/Py samples.  Inset: \n𝜎𝑆𝐻 of the GBT/Py, CBT/Py and DBT/Py samples . \n2.4. Spin -Orbit Torque Properties of CBT/Py, DBT/Py and GBT/Py Samples . The spin-orbit \ntorque ( SOT ) characteristics were extracted from the symmetric and antisymmetric components  \nof the fitted Lorentzian , as shown in Fig ures 4b-d, using the equations  [43], 𝑉𝑠=\n−𝐼𝑟𝑓𝛾cos 𝜃𝐻\n4𝑑𝑅\n𝑑𝜃𝐻[𝜏𝐷𝐿1\n𝛥𝐹𝑠𝑦𝑚] and 𝑉𝐴=−𝐼𝑟𝑓𝛾cos 𝜃𝐻\n4𝑑𝑅\n𝑑𝜃𝐻𝜏⊥(1+𝜇0𝑀𝑒𝑓𝑓\n𝐻𝑒𝑥𝑡)1\n2\n𝛥𝑓𝑟𝑒𝑞𝐹 𝑎𝑠𝑦𝑚 ] (see Supporting \nInformation  Section s 3-5). Here , 𝐼𝑟𝑓 is the RF current injected, 𝜃𝐻 is the in -plane angle of the \nexternal DC field relative to the  injected  RF current, 𝑑𝑅\n𝑑𝜃𝐻 is the derivative of the anisotropic \nmagnetoresistance (AMR) relative to 𝜃𝐻, 𝛥𝑓𝑟𝑒𝑞 is the linewidth in the frequency domain, 𝜏𝐷𝐿 is \nthe damping -like DL -SOT,  𝜏⊥, includes  a combination the Oersted field torque  and the field -like \nSOT (FL -SOT) , 𝛾 is the gyromagnetic rati o, 𝜇0 is the permeability of vacuum  and 𝛥𝐻 is the \nlinewidth of the FMR signal . The 𝜏DL correspond s to the symmetric  component of the Lorentzian, \nwhile the 𝜏⊥ correspond to the  antisymmetric components of the Lorentzian function  [43]. The \nDL-SOT was the largest in the CBT/Py sample with a value of 0.1 6 Oe,  compared to 0.1 1 Oe and \n0.10 Oe in GBT/Py and DBT/Py, respectively, as shown in Fig ure 4e. The spin-Hall conductivity, \n𝜎𝑆𝐻, which measures the spin current, 𝐽𝑠 generated from electric field , E across the STFMR device  \nis given by 𝜎𝑆𝐻=𝐽𝑠\n𝐸=𝜏𝐷𝐿𝑀𝑠𝑡𝐹𝑀\n𝐸. The average 𝜎𝑆𝐻 values  shown in Figure 4f were calculated to be 8.1×104ℏ\n2𝑒, 5.5×104ℏ\n2𝑒 and 8.1×104ℏ\n2𝑒Ω−1𝑚−1, for GBT/Py, DBT/Py, and CBT/Py samples, \nrespectively.  Assuming negligible FL -SOT, t he spin -charge current conversion efficiency \nmeasured by the spin -Hall angle  given by , 𝜃𝑆𝐻= =(𝑉𝑠\n𝑉𝑎)(𝑒𝜇0𝑀𝑠𝑡𝐹𝑀𝑡𝑇𝐼\nℏ)√1+(4𝜋𝑀𝑒𝑓𝑓\n𝐻𝑒𝑥𝑡)  (see \nSupporting Information  Section S5) . The values of 𝜃𝑆𝐻 are calculated as 0.69, 0.51 and 0.65 for \nthe GBT/Py, DBT/Py and CBT/Py samples , respectively . The 𝜎𝑆𝐻 and 𝜃𝑆𝐻 are presented in Figure \n4f which  follow similar trends as expected . The GBT/Py and  CBT/Py sample s have a much larger \n𝜎𝑆𝐻 and 𝜃𝑆𝐻 compared to the disordered polycrystalline DBT/Py sample. This  points towards a \nreduction in charge -spin current conversion efficiency with degradation in crystalline ordering  \npossibly due to scattering of spin current in the randomly oriented crystalline TI domains . The \nGBT/Py sample , however , regains the spin -charge conversion efficiency possibly because of \nquantum confinement effect in the smaller grain size of the GBT sample [26, 27] . The symmetric \nLorentzian in ST -FMR also includes a contribution from spin-pumping due the inverse spin -Hall \neffect ( ISHE ), which results in the 𝛼 enhancement  due to spin -pumping , as shown in the Figure \n3d of the main text . However, because of the complex ity of the interface in the Bi 2Te3/Py samples, \nISHE contribution to the symmetric component of FMR spectra could not be accurately isolated \nfrom the DL -SOT. Furthermore, t he contribution of spin -pumping in Py -based heterostructures \nhas been shown to be muc h smaller than the AMR component [44,45]. H ence, the ISHE \ncomponent s can be safely neglected  from the  calculations of DL -SOT  and 𝜃𝑆𝐻 for comparison of \nthe samples .  \nFigure 5.  a) Schematic of the AFM -FM heterostructure materials system and exchange bias in \nm(H) loops. b) m(H) loops measured at 6 K under ZFC condition of the GBT/Py, DBT/Py and \nCBT/Py  [25] samples showing spontaneous exchange bias.  The size of the arrows qualitative ly \nindicates the magnitude of shifts in the exchange bias.  c) m(T) measurements of the GBT/Py, \nDBT/Py and CBT/Py  samples and their derivatives for characterization of TN. d) Exchange bias \nand TN values of the GBT/Py, DBT/Py and CBT/Py samples extracted fro m b and c.  The m(H) and \nm(T) data presented for the CBT sample are similar to the ones in ref [25].  \n2.5. Effect of Crystalline Disorder in  Interfacial Topological AFM  Phase . Creation of an  \ninterfacial AFM -ordered layer was reported  in the interface of  highly c-axis-oriented Bi 2Te3/Py \nheterostructure s [25]. That AFM ordering in the interfacial layer was found to exist  because of  the \npresence of the topological AFM  compound, NiB i2Te4. Here, t he effect of crystalline  disorder of \nBi2Te3 on the AFM propert y of the Ni /Fe-diffused Bi2Te3 interface  was also  studied using  zero-\nfield-cooled (ZFC) , m(H) and m(T) measurements  at low temperatures  as shown in Figure 5 . \nWhereas  the m(H) measurements  performed at 300 K are well-centered along the H-field axis, the \nm(H) loops measured at 6 K shown in Figure 5 b are significantly shifted off -center. This shift in \nthe magnetic hysteresis loop is characteristic of spontaneous exchange bias that arises from an \ninterfacial AFM -FM interaction given by 𝐻𝐸𝐵=𝐽𝑒𝑥\n4𝜋𝑀𝑠𝑡𝐹𝑀 [38-42], as illustrated in Figure 5 a. Here , \nJex is the interfacial AFM -FM exchange energy and tFM is the thickness of the ferromagnetic layer. \nAs shown in Fig ures 4b,d, the CBT/Py with highly c-axis-oriented crystalline texture d Bi2Te3 has \nthe largest exchange bias of HEB = 83 Oe, while  the DBT/Py sample with randomly oriented \npolycrystalline grains o f Bi 2Te3 has a slightly reduced  exchange bias of HEB = 73 Oe. The exchange \ninteraction strength in AFM materials is related to the Néel  temperature , TN which were determined \nusing ZFC m(T) measurements [39,41 ] at a constant field of 50 Oe , as shown in Figure 5 c. The \nCBT/Py and DBT/Py samples also show high values of TN = 63 and 60 K , respectively. It must \nalso be noted that the large exchange bias and high TN in the disordered DBT/Py sample should  \nalso have significant contributions from th e larger Ni and Fe interface concentration of 46% and \n9% respectively, possibly causing  the HEB and TN to be comparable to the highly c-axis-oriented \nCBT /Py sample . The DBT/Py sample also shows a secondary magnetic phase at 38 K also as \nobserved from the smaller peak in the d m/dT plot in Figure 5c  which can also influence the HEB \nand Hc in the sample  at measurement temperature of 6 K. This possibly emerges due to the \nrandomly oriented vdW crystalline domains affecting exchange interaction between the interfacial \nmagnetic species. However, the highly disordered GBT/Py sample with granular Bi 2Te3 had a \nlarge  reduction in exchange bias to HEB = 12 Oe and TN = 20 K , clearly showing reduction in \nexchange in teraction strength in the AFM interfac ial layer . This follows from the much lower Δm and hence lesser reaction between the diffused Ni and Bi 2Te3 in the GBT/Py sample compared to \nthe DBT/Py and CBT/Py samples.  In addition to the spontaneous exchange bias, the m(H) \nmeasurements also show a characteristic enhancement in coercive field ( Hc) in all the Bi 2Te3/Py \nsamples , as shown in Figure 5 b, due to f rustrated magnetic moments at the interface [ 40]. These \nresults indicate the persistence of exchange properties even in highly disordered TIs.  \n \n3. CONCLUSION  \nInterfacial and magnetic properties of sputtered -deposited  TI/FM Bi 2Te3/Py heterostructures were \nstudied for varying crystalline structural disorder of the Bi2Te3. An interface layer was found to \nform because of diffusion of Ni and small er amounts of Fe into Bi 2Te3. The diffused Ni and Fe \nwere found to undergo solid -state che mical reactions with Bi 2Te3 promoted by the TSS electrons. \nWith increas ing crystall ine c-axis-oriented texture  of the Bi 2Te3, the strengthening  of topological \nproperty of Bi 2Te3 led to an enhanced  reaction between the diffused species and Bi 2Te3, which \nwas observed by a larger loss of Py magnetic moment. Increase in crystalline c-axis orientation of \nBi2Te3 also resulted in a notable  increase in OP magnetic anisotropy , Gilbert damping and spin-\norbit torque  as observed from m(H) loop and FMR measurements. Interestingly, polycrystalline \ndisordered Bi 2Te3 sample had a reduced charge -spin current conversion efficiency  possibly \nbecause of scattering of spins from polycrystalline grain boundaries . Whereas  the samples with \ngranular and  highly c -axis-oriented Bi 2Te3 had a comparable charge -spin current conversion \nefficiencies , which possibly resulted from quantum confinement effect  in smaller crystalline grains  \nand strong TSS , respectively.  As such, t his is expected to result in  reduced spin-charge conversion \nefficiency . Further more , low temperature magnetization measurements showed surprising \nresilience of the topological property of Bi 2Te3 as the AFM order persisted even in highly disordered granular Bi 2Te3/Py heterostructures. However,  for this sample t he exchange interaction \nstrength of the interfacial AFM phase was  found to weaken significantly with the increase in \nstructural disorder of Bi 2Te3. This was verified by degradation in HEB and TN with increase in \ndisorder of the Bi2Te3. These results open the path for further exploration of crystalline disorder \nin TIs and TI/FM interfaces . These thin film heterostructures  can be easily grown using a CMOS -\ncompatible sputtering process that can lead  to realization of energy efficient spintronic devices.  \n4. EXPERIMENTAL SECTION  \n4.1. Material Growth.  Bi2Te3 thin films of thickness 30 nm were grown by  co-sputtering a \ncomposite Bi 2Te3 target with Te target , using RF magnetron sputtering at 90  W and 20  W, \nrespectively , with 4  mTorr Ar pressure on thermally oxidized Si substrate s. The base pressure of \nthe sputtering chamber was ~ 8×10-8 Torr. The GBT, DBT and CBT samples were grown with \nsubstrate maintained at 20 °C , 160 °C and 250°C , respectively. The 160°C DBT and 250°C CBT \nsamples were further annealed at the growth temperatures inside the PVD process chamber in 45 \nmTorr pressure in Ar environment for 25 minutes.  The CBT samples were gown using the same \nmethod as ref [25].  The sampl es were capped with 2  nm Al at room temperature before breaking \nvacuum  which oxidizes to AlO x on exposure to atmosphere . For the magnetic and ST -FMR \nexperiments, 20 nm Py and 3 nm TiO x capping w ere deposited at room  temperature after deposition \nof Bi 2Te3. \n4.2. XRD Characterization.  X-ray diffraction was collected using a background -free, highly \ncollimated beam of Cu -Kα1 radiation (wavelength λ = 1.54056 Å) . The X -rays were captured  by \na 2D charged -coupled device (CCD). The Bragg reflections were  indexed acc ording to the Bi 2Te3 \nbulk hexagonal unit cell, as indicated by ( h, k, -(h+k), l) where h, k, and l are the Miller indices  \n[25].  4.3. TEM and XEDS Characterization.  Samples for TEM investigations were prepared by \nfocused ion beam milling (FIB) using a Ga+ ion source. Prior to TEM observation an additional \ncleaning procedure was performed by Ar -ion milling to reduce a surface amorphous layer and \nresidual Ga from  the FI B process. The TEM observations were performed using a Talos 200 -FX \n(ThermoFiszher Scientific Inc.) TEM operated at an acceleration voltage of 200 kV  [25]. EDS \nmeasurements were performed using a ChemiSTEM (ThermoFisher Scientific) and processing of \nthe sp ectra was performed using Esprit 1.9 (Brucker Inc.) software  [25].  \n4.4. FMR Measurements.  FMR measurements of α, 4𝜋𝑀𝑒𝑓𝑓, and Ha were performed using a \nspin-torque FMR (ST -FMR) experimental setup. The analysis of the experiment is explained in \nSupporting Information  Section S4. RF current is provided by a HP 8350  RF source. A SR830 \nlock-in amplified (LIA) provides reference low -frequ ency AC for modulation phase -locked with \nthe RF current. The LIA was used for detection of ST -FMR signal. The bias DC field is provided \nby an Fe -core electromagnet on a rotating stage with precise angular control. The reported ST -\nFMR experiment was perform ed at a 45° angle of the microstrip relative to  the DC bias field. \nControl of the experiment and data acquisition was done using NI LabVIEW. The S11, S12 and \nimpedance values were used to calculate the RF current,  and the E-field using vector network \nanaly zer (VNA) . The total power lost in the electrical components , such as wires and connectors , \nwere measured to be ~60% , hence, 40% of 8 dbm power was used for RF current calculations.  \nFMR  characteristics in the Bi 2Te3/Py heterostructure samples  were also extracted from the ST -\nFMR experiment spectra as shown in Figure S2 . The devices were patterned using  ion-milling \nprocess for the DBT/Py and C BT/Py samples  and using lift -off process for the Py and GBT/Py \nsamples . The FMR  characteristics were ext racted by fitting Lorentzian functions to the spectra  \n[43], as shown in Figures S2a-d, using the equation , 𝑉𝑚𝑖𝑥 =𝑉𝑆𝐹𝑠𝑦𝑚+ 𝑉𝐴𝐹𝑎𝑠𝑦𝑚 , which clearly shows broadening of FMR linewidth progressively from Py, GBT/Py, DBT/Py to CBT/Py \nsamples . Here,  𝑉𝑚𝑖𝑥 is the DC voltage output recorded in the LIA , 𝐹𝑠𝑦𝑚 = Δ𝐻2\n(𝜇0𝐻−𝜇0𝐻𝑟𝑒𝑠)2+Δ𝐻2 and \n𝐹𝑎𝑠𝑦𝑚 = 𝛥𝐻(𝜇0𝐻−𝜇0𝐻𝐹𝑀𝑅 )\n(𝜇0𝐻−𝜇0𝐻𝑟𝑒𝑠)2+𝛥𝐻2 are the symmetric and antisymmetric components of the Lorentzian \nfunction , 𝛥𝐻 is the linewidth of the FMR signal , and Hres is the FMR field.  The ST -FMR \nmeasurements were performed at 45° angle relative to the external applied magnetic field. T he \nvalues of 𝛥𝐻, 𝐻𝐹𝑀𝑅, 𝑉𝑆 and 𝑉𝐴 were extracted by fitting the ST -FMR signal using above equation  \nfor the analysis of  𝛼, 4𝜋𝑀𝑠, 𝐻⊥ and 𝜃𝑆𝐻 reported in the main text  [25]. \n4.5. Hysteresis Loop Measurements.  Magnetization m(H) and m(T) measurements were obtained \nusing a  Quantum Design MPMS XL -7 superconducting quantum interference device (SQUID) \nmagnetometer  [25]. Hysteresis loop m(H) measurements were carried out at various temperatures \nbetween 6 and 300 K. The ZFC m(T) measurements were obtained while increasing the \ntemperature in an applied f ield of 5 0 Oe . Room temperature m(H) measurements were taken using \na vibrating sample magnetometer (VSM).  \n \nPresent Addresses  \n†Dartmouth College , Thayer School of Engineering, Hanover, NH 03755  \nNotes  \nThe authors declare no competing financial interests.  \nAuthor Contributions  \nThe manuscript was written throug h contributions of all authors. All authors have given approval \nto the final version of the manuscript.   ACKNOWLEDGEMENT  \nWe thank Charles Settens  and MIT, Materials Research Laboratory for their help with XRD \nmeasurements.  We thank Neville Sun and Mehdi Nasrollahpourmotlaghzanjani for help with VNA \nmeasurements.  We also thank Ivan Lisenkov for his valuable input in understanding FMR \nexperiments.  Certain commercial equipment s are identified in this paper to foster understanding , \nbut such identification does not imply recommendation or endorsement by Northeastern University  \nand AFRL.  \nFunding Sources  \nThis work is partially supported by the U.S Army under grant no. W911NF20P0009, the NIH \nAward UF1NS107694 and by the NSF TANMS ERC Award 1160504. The work of DH and AF \nwas partially supported by the National Science Foundation grant DMR -1905662 and the Air  \nForce Office of Scientific Research award FA9550 -20-1-0247. The work of KM was supported by \nAir Force Research Laboratory under AFRL/NEMO contract: FA8650 -19-F-5403 TO3. Studies \nemploying the Titan 60 -300 TEM was performed at the Center for Electron Micro scopy and \nAnalysis (CEMAS) at The Ohio State University with support through Air Force contract FA8650 -\n18-2-5295.  \nABBREVIATIONS  \nAFM, Antiferromagnet; CBT, c -axis oriented Bi2Te3; DBT, Disordered Bi2Te3; DL -SOT, \nDampin -like spin orbit torque; EDS, Energy -dispersive X -ray spectroscopy; FM, Ferromagnet; \nFMR, Ferrommagnetic resonance; GBT, Granular Bi2Te3; HRTEM, High resolution \ntransmission electron microscopy; IP, In -plane; MBE, Molecular beam epitaxy; OP, Out -of-\nplane; QAH, Quantum anomalous hall; RF, Radio  frequency; SOC, Spin orbit coupling; TI, Topological insulator; TSS, Topological surface states; XRD, X -ray diffraction; XRR, X -ray \nreflectometry; ZFC, Zero field cooled . \nREFERENCES  \n1. H Zhang, C.X. Liu, X.L. Qi, X. Dai, Z. Fang and S.C. Zhang, Topologica l insulators in  \nBi2Se3, Bi2Te3 and Sb 2Te3 with a single Dirac cone on the surface. Nat . 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Valenzuela , Control of Spin –Orbit Torques by Interface Engineering in \nTopological In sulator Heterostructures , Nano Lett., 20, 8, 5893 (2020 ). \n \n \n Supplementary Materials : Supporting Information for Effects of Crystalline Disorder on \nInterfacial and Magnetic Properties of Sputtered Topological Insulator/Ferromagnet \nHeterostructures  \nS1. Grain size calculations of Bi 2Te3 Samples  \n \nFigure B1. a) GIXD plots for GBT, DBT and CBT samples. Representative Gaussian fitting of the \nGIXD data for b) GBT, c) DBT and d) CBT samples, used for extracting the FWHM.]  \nAs shown in Fig. S1a, grazing -angle XRD (GIXD) measured at a small incidence angle (~1º) \nclearly show the presence of comparable diffraction peak intens ities from ( hkl) = (00l), (10l) and \n(11l) orientations in both the GBT and DBT samples. The comparable intensities suggest the \nsamples are disordered polycrystalline in nature, where h, k and l are the miller indices. The CBT \nsample had clear ( 00l) orienta tion that was determined from both symmetric -XRD (main text Fig. \n1e) and GIXD measurements, showing that the CBT thin film is highly textured and oriented along \nthe crystalline c-axis. Crystallite grain sizes for the sputter -grown Bi 2Te3 samples were deter mined \nusing the Scherrer equation, 𝑑𝑔𝑟𝑎𝑖𝑛 =𝐾𝜆\n𝐹𝑊𝐻𝑀  𝑐𝑜𝑠𝜃 . Here, dgrain is the grain size, K is the shape factor \nassumed to be 0.9, λ=1.54 Å is the wavelength of the X -ray source, θ is the Bragg angle and \nFWHM is the full -width at half -maximum of the peaks. The FWHM parameters of the peaks are \nextracted by fitting a Gaussian function to the raw data for the GIXD peaks as shown in Fig. S1. \nThe average calculated grain sizes for the sa mples are 8.2 nm, 16.4 nm and 18.7 nm for GBT, DBT \nand CBT, respectively.  \nS2. Summary of Cross -Sectional Atomic% in GBT/Py, DBT/Py and CBT/Py Samples  \nTable S1. Average atomic % of elements along the cross section of the GBT/Py, DBT/Py and \nCBT/Py samples, measured using cross -sectional EDS.  \nSample  Element  Bi2Te3 Ni-Bi2Te3 Py* Py \nGBT/Py  Bi 39 20 2 0 \nTe 57 39 6 0 \nNi 4 38 68 81 \nFe 0 3 25 19 \nDBT/Py  Bi 39 18 2 0 \nTe 57 26 4 0 \nNi 3 47 74 81 \nFe 0 9 19 19 \nCBT/Py  Bi 39 22 3 0 \nTe 60 34 5 0 \nNi 0 39 70 81 \nFe 0 4 22 19 \n S3. FMR Spectra for Bi 2Te3/Py Samples  \n \nFigure B2. Representative FMR spectra for the a) Py, b) GBT/Py, c) DBT/Py and d) CBT/Py \nsamples, measured using ST -FMR experimental setup for extracting Gilbert damping and \ninterfacial magnetic anisotropy field.  \nFMR characteristics in the Bi 2Te3/Py heterostructure samples were extracted from fitting \nLorentzian functions to the ST -FMR experiment spectra as shown in Fig. S2. The devices were \npatterned using ion -milling process for the DBT/Py and CBT/Py samples and using lift -off process \nfor the Py and GBT/Py samples (see Methods for details).  \nS4. Calculation of Anisotropic Magnetoresistance  in Bi2Te3/Py Samples  \n \nFigure B3. Angular dependence of resistance of a) GBT/Py, b) DBT/Py and c) CBT/Py relative to \nthe in -plane external field of 200 Oe, which was used to calculate AMR. The blue dots represent \nraw data, and the blue dashed curve is the theoretical fit.  \nAnisotropic magnetoresistance (AMR), which is the angular dependenc e of the resistivity of the \nmaterial in an applied DC magnetic field, is the primary component of the ST -FMR signal. The \nAMR of a rectangular microstrip can be characterized by in -plane angular -dependent measurement \nof resistance and is given by, 𝑅=𝑅𝐴𝑀𝑅cos2𝜃𝐻+𝑅0. Here, 𝑅 is the total resistance of the \nmicrostrip, 𝑅𝐴𝑀𝑅 is the AMR component of the resistance and 𝑅0 is the resistance at 90° angle. \nFigure S2 shows the resistance measured for the Bi2Te3/Py samples GBT, DBT and CBT at various \nangles  in an in -plane magnetic field of 200 Oe. From fitting resistance measurements with the \nabove equation, we obtained the average AMR values for the samples as 0.93 Ω, 0.57 Ω and 0.72 \nΩ, respectively. Further, taking a derivative of the above relation, we ob tain the 𝑑𝑅/𝑑𝜃𝐻 plots that \nare used in Eq. (3) in the main text for calculation of SOT field values using ST -FMR.  \n \nS5. Spin -Hall Angle Analysis of Bi 2Te3/Py Samples  \n \nFigure S4. Trend of a) θ SH calculated using the Vs -only method, b) θ SH  calculated using the V s/Va \nmethod and c) 𝜎𝑆𝐻 for the GBT/Py, DBT/Py and CBT/Py samples.  \nSpin-Hall angle ( θSH) is the standard measure of spin -charge conversion efficiency in FM/normal \nmetal (NM) heterostructure systems (Wang et al., J. Phys. D: Appl. Phys. 51, 273002, 2018). In \nmost cases, the θSH can be accurately determined from ST -FMR experiments using the Vs -only \nmethod, 𝜃𝑆𝐻=𝜎𝑆𝐻\n𝜎𝑥𝑥, where 𝜎𝑥𝑥 is the longitudinal conductivity of the NM thin film. The average \n𝜎𝑆𝐻 values shown in Figure S4c are 8.1×104ℏ\n2𝑒, 5.5×104ℏ\n2𝑒 and 8.1×104ℏ\n2𝑒Ω−1𝑚−1, for \nGBT/Py, DBT/Py, and CBT/Py samples, respectively. T he determination of 𝜎𝑥𝑥 for the GBT, DBT \nand CBT thin films (with AlO x cap) without Py deposition was found to be 2.80×10−4Ω−1𝑚−1, \n1.72×10−4Ω−1𝑚−1, and 4.08×10−4Ω−1𝑚−1, respectively.  Using the above relation, average \nθSH  values obtained were 3.68, 3.18 and 1.98 for the GBT/Py, DBT/Py and CBT/Py, respectively, \nas shown in Fig. S4a. However, the Vs -only method assumes no interfacial diffusion, which leads \nto a non -physical result in th ese TI/FM materials systems. The resistance in the TI layer is certainly \naltered by the diffusion of Ni and Fe from Py and reaction with Bi 2Te3. Moreover, the trend of θSH  \nobtained from the Vs -only method does not follow the 𝜎𝑆𝐻 trend, which also means  the θSH values \nobtained using the Vs -only method is incorrect. For a better estimation of the θSH  parameter in \nthese TI/FM systems with a complex interfacial morphology, the Vs/Va method given by, 𝜃𝑆𝐻=\n(𝑉𝑠\n𝑉𝑎)(𝑒𝜇0𝑀𝑠𝑡𝐹𝑀𝑡𝑇𝐼\nℏ)√1+(4𝜋𝑀𝑒𝑓𝑓\n𝐻𝑒𝑥𝑡), should be a more reliable method. This method assumes the \nantisymmetric component is completely from the Oersted field. Given the comparable values of \n𝜏⊥ for the GBT/Py, DBT/Py and CBT/Py (see Main Text Figure 4e), the FL -SOT can be safely \nassume d to be negligibly small compared to Oersted torque.  Using the Vs/Va method, the average \nθSH  for the GBT/Py, DBT/Py and CBT/Py were 0.69, 0.51 and 0.65, respectively, as shown in Fig. \nS4b. As expected, values also follow the trend of 𝜎𝑆𝐻 for the sampl es. This clearly suggests \nreduction in spin -charge conversion efficiency for the polycrystalline disordered TI sample \nDBT/Py.  \nS6. Crystalline Disorder -Dependent Spin -Orbit Coupling Effects in TI/FM Heterostructures  \nS6.1. Crystalline Disorder -Dependent Magnetic Properties of Bi 2Te3/CoFeB  \nThe crystalline disorder -dependent, room -temperature magnetic properties of TI/FM were also \nstudied for Bi 2Te3/CoFeB (CFB) heterostructure samples, as shown in Fig. S4. The FMR spectra \nshow a giant enhancement in α and 𝐻⊥ for the CBT/CFB samples with high c-axis orientation. The \nmeasurements were performed using a broad -band FMR setup for comparison with the Bi 2Te3/Py \nsamples. The extracted α values from FMR measurements in Fig. S4b were found to be 0.015, \n0.026 and 0.283 for CFB, GBT/CFB and CBT/CFB, respectively. These increasing values for \nincreasing order confirm the giant enhancement in SOC and strong TSS eff ects in highly c-axis \noriented TI compared to highly disordered TI materials. Further, Kittel equation fits to the \nresonance frequency versus field in Fig. S4c give estimates for 𝐻⊥ of -13.2 Oe,  100.7 Oe and 1485 \nOe, for the respective sample revealing a  giant enhancement in PMA field for the highly c-axis \noriented Bi 2Te3 sample. This clearly shows a large enhancement in interfacial PMA field in highly c-axis oriented CBT/CFB samples compared to a higher in -plane anisotropy of the disordered \nGBT/CFB sampl es. The FMR measurements are also supported by the increase in Mr/Ms ratio in \nthe m(H) measurements, shown in Fig. S4d. The Mr/Ms values are 86%, 70% and 66% for the \nCFB, GBT/CFB and CBT/CFB samples, respectively, which shows an enhanced OP magnetic \nanisot ropy for the highly oriented CBT/CFB sample compared to the disordered GBT/CFB \nsample. Table S2 summarizes the magnetic properties of the Bi 2Te3/CoFeB samples obtained from \nthe FMR and m(H)  loop measurements.  \n \nFigure B5. a) Representative normalized FMR spectra for CFB, GBT/CFB and CBT/CFB at 6 \nGHz showing broadening of linewidth. b) FMR linewidth versus frequency plots and linear fit for \nextracting Gilber t damping in GBT/Py, CBT/Py and DBT/Py samples. c) FMR frequency versus \nfield used for extracting 4𝜋𝑀𝑠 and 𝐻⊥ in GBT/Py, CBT/Py and DBT/Py samples. d) m(H) \nmeasurement for the GBT/Py, CBT/Py and DBT/Py samples.  \nTable S2. Magnetic Properties of Bi 2Te3/CoFeB Samples  \nSample  𝜶 𝟒𝝅𝑴𝒔 \n(kOe)  𝑯⊥ \n(Oe)  Mr/Ms (%) \nCFB  0.01484  16.01  -13.17  86 \nGBT/CFB  0.02604  15.89  100.74  70 \nCBT/CFB  0.28336  14.51  1485.39  66 \n \nS6.2. Crystalline Disorder -Dependent Magnetic Properties of YIG/Bi 2Te3 \nThe effect of TSS for giant enhancement in α was verified using Y 3Fe5O12 (YIG) as the FM layer. \nDisordered GBT and highly c-axis oriented CBT thin films of thickness 30 nm were grown on \nGGG (111)/YIG (444) (70 nm) substrates using the same deposition condi tions mentioned in the \nmain text. YIG is a chemically stable rare -earth garnet material, and because it is an oxide \ncompound it is not expected to experience diffusion across the interface with TI (Chen et al., Appl. \nPhys. Lett. 114, 031601, 2019). This wa s verified by m(H) measurements of  YIG and YIG/GBT \nsamples that did not show any reduction in magnetic moment, as shown in Fig. S4c, unlike with \nPy. FMR measurements for the YIG/Bi 2Te3 samples revealed a similar trend as the Bi 2Te3/FM \nsamples and showed a  giant enhancement in FMR linewidth and α for YIG/GBT heterostructures, \nas shown in Figs. S5b,d. The α values extracted from the linear fitting of Δ𝐻 versus resonance \nfrequency plots in Figure S3d are 8.43×10−4, 2.83×10−3 and 7.09×10−3 for increasing \ncrystallinity. This clearly shows a giant enhancement in Gilbert damping in the YIG/CBT \nheterostructure sample as a result of the TSS in the highly c-axis oriented Bi 2Te3. Fitting the Kittel equation also revealed changes in 𝛾\n2𝜋 values fr om 2.81 for YIG to 2.83 and 2.84 for YIG/GBT and \nYIG/CBT samples, respectively. These changes in 𝛾\n2𝜋 values provide strong evidence of interaction \nbetween the TSS and magnetic moments in YIG that are enhanced in the YIG/CBT sample (Liu \net al., Phys. Rev.  Lett. 125, 017204, 2020). Table S3 summarizes the magnetic properties of the \nYIG/Bi 2Te3 samples obtained from FMR measurements.  \n \nFigure B6. a) XRD data for sputtered CBT grown on YIG showing strong c -axis oriented texture. \nb) FMR spectra of YIG/Bi 2Te3 samples measured at 6 GHz showing enhancement in linewidth \nafter deposition of GBT and CBT. c) FMR frequency versus resonance field plots and Kittel \nequation fitting of YIG, YIG/GBT and YIG/CBT samples. d) FMR linewidth versus frequency plots \nof YIG, YIG/ GBT and YIG/CBT samples and linear fitting.  \nTable S3. Magnetic Properties of YIG/Bi 2Te3 Samples.  \nSample  𝜸\n𝟐𝝅 𝜶 𝟒𝝅𝑴𝒔 (kOe)  \nYIG 2.81 0.000843  1.89 \nGBT/YIG  2.83 0.00283  1.89 \nCBT/YIG  2.84 0.00710  1.89 \n "    },    {        "title": "2205.14717v1.Generalized_Stochastic_Matching.pdf",        "content": "arXiv:2205.14717v1  [cs.DS]  29 May 2022Generalized Stochastic Matching\nAlireza Farhadi, Jacob Gilbert, MohammadTaghi Hajiaghayi\nUniversity of Maryland\nfarhadi@cs.umd.edu, jgilber8@umd.edu, hajiagha@cs.umd .edu\nAbstract\nIn this paper, we generalize the recently studied stochastic\nmatching problem to more accurately model a significant\nmedical process, kidney exchange, and several other applic a-\ntions. Up until now the stochastic matching problem that has\nbeen studied was as follows: given a graph G= (V,E), each\nedge is included in the realized sub-graph GofGmutually\nindependently with probability pe, and the goal is to find a\ndegree-bounded sub-graph QofGthat has an expected max-\nimum matching that approximates the expected maximum\nmatching of G. This model does not account for possibilities\nof vertex dropouts, which can be found in several applica-\ntions, e.g. in kidney exchange when donors or patients opt\nout of the exchange process as well as in online freelancing\nand online dating when online profiles are found to be faked.\nThus, we will study a more generalized model of stochastic\nmatching in which vertices and edges are both realized inde-\npendently with some probabilities pv,pe, respectively, which\nmore accurately fits important applications than the previ-\nously studied model.\nWe will discuss the first algorithms and analysis for this gen -\neralization of the stochastic matching model and prove that\nthey achieve good approximation ratios. In particular, we\nshow that the approximation factor of a natural algorithm fo r\nthis problem is at least 0.6568 in unweighted graphs, and\n1/2 +ǫin weighted graphs for some constant ǫ >0. We\nfurther improve our result for unweighted graphs to 2/3us-\ning edge degree constrained subgraphs (EDCS).\nIntroduction\nThe stochastic matching problem has been used to model\nkidney exchange in several research papers in recent years,\nand in this paper, we generalize this model to better suit\nthe needs of kidney exchange and other applications. Kid-\nney exchange is an important medical procedure that is uti-\nlized to increase the amount of possible successful kidney\ntransplants between patients and donors for hundreds of\ndonor-patient pairs in the U.S. each year. This medical pro-\ncess occurs when an incompatible kidney donor-patient pair\nmatches with another incompatible pair such that the donors\nare swapped to become compatible pairs with the patients.\nUnfortunately, compatibility medical testing can require pa-\ntients and donors to be hospitalized and are expensive. Thus ,\nCopyright © 2022, Association for the Advancement of Artific ial\nIntelligence (www.aaai.org). All rights reserved.minimizing the amount of compatibility tests while maxi-\nmizing compatible exchanges is an important problem in the\nmedical world. Moreover, in this paper we also account for\nthe possibility that a patient or donor may decide to drop\nout of the exchange due to health conditions or at their dis-\ncretion at any point throughout the months’ long process.\nTherefore, while prior papers only considered donor-patie nt\ncompatibility, we will also consider potential dropouts fr om\nthe exchange process on top of compatibility.\nIn our proposed stochastic matching model of kidney ex-\nchange, each donor-patient pair is represented by a vertex\nin the graph. Edges in this graph Grepresent donor-patient\npairs that may be compatible for exchange. Only a subset\nof the edges are found to be compatible through medical\nrecords and testing, and this subset forms a realized sub-\ngraphGof possible successful exchanges. We say these\nedges are realized , i.e. appear inG, with some probability\npe. Similarly, a vertex is realized with probability pvif the\npair does not dropout during the exchange process. In our\ngeneralized model, an edge can only be included in Gif both\nof its vertices are realized as well. A maximum matching al-\ngoritm seeks to pair vertices connected by an edge of the\ngraph together to create the maximum amount of matches.\nSo, a maximum matching of Grepresents maximized com-\npatible kidney exchanges. As mentioned, medical tests for\ncompatibility are expensive, and so querying the edges of G\nto see if they were realized should be kept to a minimum.\nWithout knowing the sub-graph G, the goal of the stochas-\ntic matching problem is to find some degree-bounded sub-\ngraphQwith an expected maximum matching of realized\nedges that has a size approximately that of the actual max-\nimum matching of G. We will state and prove the existence\nof the first bounds for the approximation ratio achieved with\nthis model of kidney exchange in which vertices and edges\nmay be dropped from the original graph.\nAdditional Applications\nWith our generalized stochastic matching , in addition to kid-\nney exchange we can model the freelancing industry com-\nprised of freelance workers and their potential employers.\nIn modern freelancing, workers may have online profiles on\nwebsites that businesses can look through to find freelancer s\nwith compatibility for a job or project. Unfortunately, a la rge\namount of fake profiles and fake job offerings plague thesewebsites. Finding out profiles and jobs are fake costs time\nand money from those who hired the fake profiles or the\nfreelancers who took up a fake job. Therefore, in the online\nfreelancing problem the goal is to maximize matchings be-\ntween jobs and freelancers while minimizing the amount of\nqueries to freelancers and employers needed to match real\nfreelancer profiles to real job offers.\nTo model online freelancing with stochastic matching ,\nprofiles and companies will make up the vertices of some\ngraphG, and there is an edge between profiles and compa-\nnies if a freelancer fits the qualifications for a company’s jo b\nopening. Edges may be weighted by the amount a company\nwill pay a freelancer for the job, or remain unweighted if all\njobs are nearly equally valuable. Each vertex is realized wi th\nsome probability pvas long as the online profile or company\nis real. Each edge (u,v)is realized as long as both of the\nverticesuandvare realized. As in kidney exchange, queries\nof edges/vertices are expensive since they require profile r e-\nviews and lengthy communications, but in this version of the\nproblem only vertices have a realization probability while\nedges are always realized if both of its vertices have been\nrealized. Online dating is a very similar scenario with po-\ntential fake vertices, but edges between dating profiles may\nalso drop out if a match does not lead to a relationship. In\nour results later in the paper, we will state and prove bounds\nto the approximation ratio achieved for weighted graphs for\nthe freelancing and dating model.\nBesides the aforementioned applications, the problem is\nsignificant from a computer science theory perspective as a\ndiscussion of graph sparsification. We will show that a sim-\nple, well-studied algorithm provides a sparse sub-graph wi th\na good approximation of the expected maximum matching\nof the original graph for our generalized version of stochas -\ntic matching. This sub-graph will conform to a tight restric -\ntion: any vertex has at most constant O(1)degree.\nGeneralized Stochastic Matching Model\nAs discussed, the kidney exchange problem may be modeled\nwith our proposed generalized stochastic matching model.\nIn the stochastic setting, we have a random sub-graph, the\nrealized sub-graph, of some given graph, and we want to\napproximate some property of the realized sub-graph.\nDefinition 1. Given fixed parameters pv,pe∈(0,1]and\nweighted or unweighted graph G= (V,E)with vertex set\nVand edge set E⊂V2, let graphG= (V,E)be a sub-\ngraph of Gsuch that any vertex v∈Vis inVmutually\nindependently randomly with probability pvand any edge\ne= (u,v)∈Eis inEmutually independently randomly\nwith probability peifu,v∈V. We callGtherealized sub-\ngraph ofG.\nDefinition 2. Given weighted graph G= (V,E,W)where\nWis a set of edge weights, let we∈Wbe the weight of edge\ne∈E. DefineM(G)to be the maximum weighted matching\nofG; furthermore, let µ(G) :=/summationtext\ne∈M(G)webe the weight\nof the the maximum matching of G.\nIn the stochastic matching problem, we want to find a\nsparse sub-graph of a given graph such that the realized\nportion of the sparse sub-graph approximates the maximumweighted matching of the realized sub-graph. More for-\nmally, given a graph Gwithnvertices, we want to find a\nsub-graph Q= (V,EQ)that satisfies the following two con-\nditions:\n1. LetQ=Q∩ G, then the approximation ratio\nE[µ(Q)]/E[µ(G)]is as large as possible.\n2. The degree of QisO(1). Specifically, the maximum de-\ngree of any vertex in Qmay be bounded by a constant\ndetermined by pv,pebut notn.\nSo, if we can find such a sub-graph Q, then we may query\ntheO(n)edges ofQinstead of doing expensive queries to all\nO(n2)edges ofGto find out which were realized. However,\nfinding such a sparse sub-graph and proving it has a large\napproximation ratio is non-trivial.\nRelated Work\nThe less generalized version of stochastic matching in\nwhich all vertices are realized with probability 1was first\nintroduced by (Blum et al. 2015) primarily to model the\nkidney exchange setting. In this paper, the authors showed\npositive empirical results on simulated and real data from\nthe United Network for Organ Sharing in which stochastic\nmatching algorithms resulted in a good approximation\nof the optimal solution. This problem has been exten-\nsively studied since then (Assadi, Khanna, and Li 2016;\nYamaguchi and Maehara 2018; Behnezhad and Reyhani\n2018; Behnezhad et al. 2019a). The first discussion of this\nless generalized problem by (Blum et al. 2015) achieved\nan approximation ratio of (1/2−ǫ)in unweighted graphs,\nand then (Assadi, Khanna, and Li 2017) broke the half\napproximation barrier with an approximation ratio of .5001 .\nThis bound was later improved by (Behnezhad et al. 2019b)\nto .6568 and by (Assadi and Bernstein 2019) to (2/3−ǫ).\nAfterwards, (Behnezhad, Derakhshan, and Hajiaghayi\n2020) and (Behnezhad and Derakhshan 2020) both built on\nthe analysis of the algorithm proposed by (Behnezhad et al.\n2019b) to further improve approximation ratios for un-\nweighted and weighted graphs to (1−ǫ), respectively. We\nadapt this same algorithm as Algorithm 1 below to fit our\nmodel.\nOur Results\nIn the Crucial Edges and Unweighted Approximation sec-\ntion, we will achieve and prove a .65approximation ra-\ntio for unweighted stochastic matching, i.e. the kidney\nexchange model. By adapting the analysis techniques of\n(Behnezhad et al. 2019b) for our new generalization of\nstochastic matching, we will prove the following theorem\nand lower bound for the unweighted case:\nTheorem 3. For unweighted graph G, constant ǫ >0, ver-\ntex and edge realization probabilities pv,pe∈(0,1], there\nis an algorithm to find an Oǫ,p(1)1-degree subgraph QofG\nsuch that E[µ(Q)]/E[µ(G)]≥.6568−ǫ.\nIn the Weighted Approximation section, we further con-\nsider the weighted stochastic matching problem, i.e. the fr ee-\nlancing model. Here, we prove the following bounds:\n1We useOǫ,p(.)to hide the dependency on poly(ǫ,pv,pe).Algorithm 1: An algorithm for the generalized stochastic\nmatching problem.\nInput: Input weighted graph G= (V,E)and realiza-\ntion probabilities pv,pe∈[0,1].\nParameter: R:=2000log(1 /ǫ)log(1/(ǫp2\nvpe))\nǫ4p2vpe\n1:Q←(V,∅)\n2:forr= 1,...,R do\n3: Construct a sample Gr= (Vr,Er)ofG, where any\nvertexv∈Vappears inVrindependently with proba-\nbilitypv, and each edge e∈Econnecting vertices u,v\nappears inErindependently with probability peif and\nonly ifu,v∈Vr.\n4: Add the edges in maximum weighted matching\nM(Er)ofGrtoQ.\n5:end for\n6:Query the edges in Qand report the maximum weighted\nmatching of it.\nTheorem 4. For weighted graph G, constant ǫ >0, vertex\nand edge realization probabilities pv,pe∈(0,1], there is an\nalgorithm to find an Oǫ,p(1)-degree subgraph QofGsuch\nthatE[µ(Q)]/E[µ(G)]≥.501−ǫ.\nOne important distinction between our generalizations\nfor freelancing and kidney exchange from prior work is\nthat some edges are no longer realized completely inde-\npendently. Specifically, if a vertex is not realized in our\nmodel, then every edge connected to it is also not real-\nized. This dependence sets our model apart from previous\nstochastic matching papers. Thus, our techniques will not\nutilize independent realizations of certain edges, a prope rty\nthat both (Behnezhad, Derakhshan, and Hajiaghayi 2020)\nand (Behnezhad and Derakhshan 2020) have relied on be-\nfore. We also improve our bound for unweighted graphs to\n(2/3−ǫ)in the EDCS 2/3Approximation section using\nedge degree constraint sub-graphs (EDCS).\nTheorem 5. For unweighted graph G, constant ǫ >0, ver-\ntex and edge realization probabilities pv,pe∈(0,1], there\nis an algorithm to find an Oǫ,p(1)-degree subgraph QofG\nsuch that E[µ(Q)]/E[µ(G)]≥2/3−ǫ.\nWhile the bound of Theorem 5currently dominates that\nof Theorem 3for unweighted graphs, future improvements\nto Algorithm 1 and its analysis will eventually most likely\novertake the 2/3approximation ratio provided by the EDCS\napproach.\nAlgorithm 1 Analysis\nBefore we can prove our main results, we must introduce the\nconcept of fractional matchings and the related procedures\nwe use to build these fractional matchings. Constructing an\nintegral matching directly on our sparse sub-graph Qis dif-\nficult since we want a good approximation ratio in expecta-\ntion without directly knowing G. Instead, we can relax our\nmatching requirements to allow assigning fractional value s\nto edges of our matching, and then later show that the frac-\ntional matching serves as proof of the existence of a integra l\nmatching of the same approximation ratio.Fractional Matchings\nIn order to prove Theorems 3 and 4, we will find a frac-\ntional matching xofQthat achieves a .6568−ǫapproxi-\nmation ratio and .501−ǫapproximation ratio, respectively.\nIn an integral matching, each vertex can only be matched\nto one other vertex. Alternatively, one can think of an inte-\ngral matching as assigning a value of either 1 or 0 to every\nedge such that no vertex has two incident edges with value\n1. A fractional matching xprovides more flexibility in anal-\nysis than an integral matching since it allows assigning fra c-\ntional values xe∈[0,1]to edgeesuch that for any vertex,\nxv:=/summationtext\nv∈exe≤1. Once we have our fractional matching\nand prove that it achieves our target approximation ratios, we\nwill use the following folklore lemma to claim the existence\nof an integral matching ythat achieves the same approxima-\ntion ratio to complete the proofs of Theorem 3and Theorem\n4. Note that in the following lemma, Lemma 6, given graph\nG= (V,E)and subset U⊆V, we useE(U)to refer to the\nedges of the induced sub-graph on GbyUwhich includes\nevery edge (u,v)∈Esuch that u,v∈U.\nLemma 6. Letxbe a fractional matching, ǫ >0be a con-\nstant, and G= (V,E)be an edge weighted graph where\nweis the weight of edge e∈E. If for all U⊆Vsuch\nthat|U|≤1/ǫit is true that/summationtext\ne∈E(U)xe≤⌊|U|/2⌋, then\nGhas an integral matching ysuch that/summationtext\ne∈Ewe·ye≥\n(1−ǫ)/summationtext\ne∈Ewe·xe.\nProof of Lemma 6 and further discussion about fractional\nmatchings can be found in (Behnezhad et al. 2019b) in sec-\ntion 2.2. Now we see that to prove Algorithm 1 provides a\ngood expectecd maximum matching, a fractional matching\nxonQwill need to satisfy the requirements of Lemma 6 in\naddition to achieving the target approximation ratio. To cr e-\nate such a matching, we will combine two smaller matchings\nover two disjoint sets of edges, a set of non-crucial edges\nand a set of crucial edges. Each edge will be classified as\nnon-crucial or crucial based on the probability that the edg e\nappears in the maximum matching of G.\nDefinition 7. For edge e∈E, we define qe:= Pr(e∈\nM(G))as the probability that eappears in the maximum\nweighted matching of realized sub-graph G, and we will re-\nfer toqeas the matching probability of edgee2. Additionally,\nfor vertex v∈V, letqv:=/summationtext\ne∋vqe. For a vertex vand sub-\nsetX⊆E,q(X) :=/summationtext\ne∈XqeandqX\nv:=/summationtext\ne:e∈X,v∈eqe.\nDefinition 8. Let threshold τ=ǫ3p2\nvpe\n20log(1 /ǫ), then edge eis\ncrucial ifqe≥τandnon-crucial ifqe< τ. We will use C\nto denote the set of crucial edges and Nto denote the set of\nnon-crucial edges.\nThe matchings over non-crucial and crucial edges will be\nconstructed with procedures analyzed below. When creating\nthese procedures, we will have to keep a few things in mind\nabout our new model. First, edges are only realized if their\nincident vertices are realized, and so they can be thought of\n2Given a realization, we can assume that edges belong to the\nmaximum weighted matching are unique. These edges can be the\nedges returned by an arbitrary deterministic algorithm.as having a realization probability of not just pebutp2\nvpe.\nWith this in mind, when we sort edges into non-crucial and\ncrucial sets, we make sure our threshold incorporates this r e-\nalization probability p2\nvpein Definition 8. Furthermore, note\nthat in Algorithm 1, the product p2\nvpemakes an appearance\nin the number of iterations. One of the main reasons is to\neasily relate the number of rounds of the algorithm to our\nmatching probability threshold τ.\nAnother quirk of the new model is that there is a correla-\ntion in realization probabilities of adjacent edges that sh are\nan incident vertex. In step 2 of the upcoming non-crucial\nedge procedure, we scale down our fractional matching by a\nfactor ofpvto account for this.\nThe first procedure we discuss will create a near-optimal\nmatching on the non-crucial edges using the following use-\nful observation.\nObservation 9. E[µ(G)] =/summationtext\ne∈Eweqe.\nEssentially, we will use matching probabilities as the as-\nsigned values in our fractional matching because the ex-\npected size of the maximum matching of Gis just the sum of\nedge weights times matching probabilities. Since we don’t\nactually know the matching probabilities exactly, we will a s-\nsign the value of fe, defined as the fraction of times edge e\nappears in a maximum matching of an iteration of Algorithm\n1out ofR, the total number of iterations.\nDefinition 10. For an edge e∈E, letkbe the number\nof timeseappeared in a maximum weighted matching of a\nsampled graphGifor1≤i≤Rduring Algorithm 1. We\ndefinefe:=k\nR.\nObserve that from the above definition, E[fe] =qe, which\nis what we wanted.\nNon-crucial Edge Procedure\nGivenQ= (V,EQ)from Algorithm 1 with realized sub-\ngraphQ= (V,EQ). Let˜xe= 0for all edges e∈E. Then,\n1. For any realized edge e∈EQ∩N, set˜xeto be\nmin{fe/(p2\nvpe),2τ/(p2\nvpe)}.\n2. Letsebe the scaling-factor ofewhere the default value\nisse= 1. For each vertex v∈Vand edge eincident to\nv, setse\nse= min/braceleftBig\nse,max{qN\nv,ǫ}/(pv·/summationdisplay\ne∋v˜xe)/bracerightBig\n.\nNote that this step may be done for each vertex in an\narbitrary order.\n3. Finally scale down the fractional matching with se. So\nfor all edges e, letxe:= ˜xe·se.\nBy definition of a fractional matching, it is required that\nxv≤1for any vertex v∈V. So, each vertex can be thought\nto have a “budget” of size 1. We want to make sure the non-\ncrucial edge procedure leaves some remaining budget for the\ncrucial edges later. To this end, we set the scaling-factors in\nstep 2 to have a factor of/summationtext\ne∋v˜xein the denominator. More-\nover, we place a pvin the denominator because vertices can\nonly be matched if they are first realized. Altogether sinceqv≤pv, in expectation the scaling-factor qN\nv/(pv·/summationtext\ne∋v˜xe)\nshould keep xebelow 1. Note that in the actual definition of\nthe scaling-factors we have a max over qN\nvandǫ, and so,\nwe must make sure it is small enough to stay within the ver-\ntex budget even with ǫ. So as discussed, the properties of\nthe following lemma prove an upper bound on the size of\nthe fractional matching per vertex, i.e. some budget remain s\nfor crucial edges. Additionally, the first property of the fo l-\nlowing lemma, Lemma 11, proves that the non-crucial edge\nmatching satisfies the requirements of Lemma 6, which as\nmentioned will be applied in our last step of this analysis to\ngo from our fractional matching to an integral matching. The\nproof of Lemma 11 and other missing proofs of this section\nare available in the Appendix.\nLemma 11. Given graph G= (V,E), constant ǫ∈(0,1],\nand fractional matching xfrom the non-crucial edge proce-\ndure:\n1.∀U⊆Vsuch that|U|≤1/ǫ,/summationtext\ne∈E(U)xe≤ǫ⌊|U|/2⌋.\n2.∀v∈V,xv≤max{qv,ǫ}/pv.\nNow, we have proven that the non-crucial edge procedure\nis not only a fractional matching, but leaves some room to\ngrow when we discuss crucial edges. However, as previously\ndiscussed, we must now also prove that the current matching\nis nearly optimal in size. The following definition will be\nuseful to see this.\nDefinition 12. ForX⊂E, defineϕ(X) :=/summationtext\ne∈Xwe·qe\nas the expected matching weight ofX. Also, for a vertex v\nand subset X⊂E, we define ϕX\nv:=/summationtext\ne∈X,v∈ewe·qe.\nBy Observation 9, the expected size of the maximum\nmathcing onGisϕ(E). So in Lemma 13, we will show that\nmatching xover non-crucial edges of our sparse sub-graph\nis within a (1−ǫ)factor ofϕ(N). To begin, we will bound\nthe size of ˜xefrom the first step of the non-crucial edge pro-\ncedure.\nLemma 13. E/bracketleftBigg\n/summationtext\ne∈EQ∩Nwe·˜xe/bracketrightBigg\n≥(1−ǫ)ϕ(N).\nWe then show that the scaling factor is at least (1−5ǫ)\nwith high probability. Therefore, we will know that the scal -\ning factor does not decrease the size of the fractional match -\ning too greatly.\nLemma 14. For vertex vof realized edge e∈EQ∩Nwith\nprobability at least 1−2ǫ, it is true that max{qN\nv,ǫ}/(pv·/summationtext\ne∋v˜xe)≥1−5ǫ.\nProof of Lemma 14. Note that the following proof is an up-\ndated proof from Claim A.5, (Behnezhad et al. 2019b) for\nthe new model. In order to prove this lemma, we will need to\nrelate the numerator of the scaling factor, qN\nv, to the denom-\ninator of the scaling factor,/summationtext\ne∋v˜xe. Let˜xv:=/summationtext\ne∋v˜xe.\nTo create this relation, we will analyze fesince it is related\ntoqeand˜xe. Note from the lemma’s statement, we have al-\nready specified some incident edge e= (u,v)∈EQ∩N. If\neis not realized, we know that xewill just be 0 and does not\ncontribute to the matching. Therefore, moving forward we\nassume that eis realized. We will use e1,e2,...,ek∈EQ∩Nto denote the rest of the edges incident to v, realized or un-\nrealized. Now, let fN\nv:=/summationtext\neife; we will begin the proof by\nshowing that fN\nvis a good approximation of qN\nv.\nTrivially, since E[fe] =qe,E[fN\nv]≤qN\nv. Intuitively,\nfN\nvis the fraction representing the number of matchings in\nwhichvis matched by some edge eiin a round of Algo-\nrithm1divided by R−1since we do not count the round v\nis matched by e. So, by Hoeffding’s inequality we have\nPr(fN\nv−qN\nv≥ǫ2)≤Pr(fN\nv−E[fN\nv]≥ǫ2)\n≤exp(−2(R−1)ǫ4)≤ǫ.\nSo far we have that with probability at least 1−ǫ,fN\nv−qN\nv≤\nǫ2, and this inequality implies that with probability at least\n1−ǫ,max{fN\nv,ǫ}−max{qN\nv,ǫ}≤ǫ2. Therefore, with\nprobability at least 1−ǫ,\nmax{fN\nv,ǫ}≤max{qN\nv,ǫ}+ǫ2≤(1+ǫ)max{qN\nv,ǫ}.\n(1)\nSo, we have shown that fN\nvis close to (1 +ǫ)qN\nv. Note\nthat in step 2 of the non-crucial edge procedure, seonly de-\ncreases when ˜xvis greater than max{qN\nv,ǫ}. Therefore, we\nwill show that the probability of this happening is small,\nbut we will use fN\nvin place of qN\nv. To do so, we define\nX1,X2,...,Xkto be the random variables conditioned on\nedgeealready being realized where we set Xito 0 ifeiis\nnot realized and (min{fei/(pvpe),2τ/(pvpe)})ifeiis re-\nalized. Note that we conditioned on the realization of edge\neto keep the set of events Ximutually independent. Since\nedgeeis realized, we know that the incident vertex vto each\nedgeeimust also be realized. Furthermore, the probability\nthat each edge eiis realized is now only pvpesince a factor\nofpvhas been removed for the incident vertex. Now, we de-\nfineX:=/summationtext\neiXi. Observe that X= ˜xv−pv·˜xesince step\n1 of the non-crucial edge procedure sets any realized edges\neito bemin{fei/(p2\nvpe),2τ/(p2\nvpe)}. Additionally by lin-\nearity of expectation, E[X] =/summationtext\niE[Xi]≤fv.\nNow we will consider two cases. First, assume E[X]≤\nǫ/(2pv), then by definition of Xwe also have that E[X]≤\nmax{fN\nv,ǫ}/(2pv). From here, we can achieve the follow-\ning probability bound:\nPr(X >max{fN\nv,ǫ})\n= Pr(X−max{fN\nv,ǫ}/2≥max{fN\nv,ǫ}/2)\n≤Pr(X−E[X]≥max{fN\nv,ǫ}/2)\n= Pr/parenleftbigg\nX≥/parenleftbigg\n1+max{fN\nv,ǫ}\n2·E[X]/parenrightbigg\nE[X]/parenrightbigg\n≤exp/parenleftbigg\n−max{fN\nv,ǫ}/2\n6τ/(pvpe)/parenrightbigg\n, by Chernoff bound.\n≤exp/parenleftbigg\n−ǫ\n12τ/(pvpe)/parenrightbigg\n≤exp(−1/ǫ2)≤ǫ.\nFor the second case, if E[X]> ǫ/(2pv):Pr(X >(1+ǫ)max{fN\nv,ǫ}/pv)≤Pr(X >(1+ǫ)E[X])\n≤exp/parenleftbigg\n−ǫ2E[X]\n6τ/(pvpe)/parenrightbigg\n, by Chernoff bound.\n≤exp/parenleftbiggǫ3\n12τ/pe/parenrightbigg\n, sinceE[X]> ǫ/(2pv).\n≤exp/parenleftbigg\n−1\nlog(1/ǫ)/parenrightbigg\n≤ǫ.\nThus, in either case we know that\nPr(X≥(1+ǫ)max{fN\nv,ǫ})≤ǫ.\nSinceX= ˜xv−˜xe, we also know that with probability at\nleast1−ǫ,\n˜xv≤(1+ǫ)(max{fN\nv,ǫ}/pv)+pv·˜xe\n≤(1+ǫ)(max{fN\nv,ǫ}/pv)+ǫ\n≤(1+2ǫ)(max{fN\nv,ǫ}/pv).\nNow, we can use Inequality 1to obtain our inequality to\nsubstitute in qN\nv. With probability at least 1−2ǫ,\n˜xv≤(1+2ǫ)(max{fN\nv,ǫ}/pv)\n≤(1+2ǫ)(1+ǫ)(max{qN\nv,ǫ}/pv)\n≤(1+5ǫ)(max{qN\nv,ǫ}/pv).\nFinally, with probability at least 1−2ǫ,\nmax{qN\nv,ǫ}/(˜xv·pv)≥1\n1+5ǫ≥1−5ǫ.\nLemma 15. Given a non-crucial edge matching xfrom Pro-\ncedure 1,\nE/bracketleftBigg/summationdisplay\ne∈Nwe·xe/bracketrightBigg\n≥(1−10ǫ)ϕ(N).\nCrucial Edges and Unweighted Approximation\nNow we will augment the previous fractional matching from\nthe non-crucial edge procedure with a second procedure\non crucial edges. The crucial edge procedure is different\nfor unweighted and weighted graphs, and we first give the\nslightly simpler procedure for unweighted graphs. The re-\nsulting fractional matching will give us our desired approx i-\nmation ratio to prove Theorem 3.\nIn our new stochastic matching model, it is important to\nemphasize that the realization of non-crucial edges actu-\nally gives some information about the realization of crucia l\nedges. In fact, because some non-crucial edges share an in-\ncident vertex with crucial edges, the probability of a cruci al\nedge being realized loses a factor of pvwhen we know an\nadjacent non-crucial edge already has been realized as that\nimplies the shared incident vertex has been realized as well .\nIn order to avoid this further complexity to the problem,\nthis crucial edge procedure will assign remaining budget toa crucial edge based on the incident vertices’ non-crucial\nedge budget such that no vertex contributes more than size\n1 to the whole matching. Specifically, we will set xeof edge\ne= (u,v)for crucial edges equal to approximately 1−qN\nv\nor1−qN\nu(with an additional (1−ǫ)factor) in the crucial\nedge procedure. Additionally, we utilize a probability dis tri-\nbution defined below.\nDefinition 16. Take any matching Mof the realized crucial\nedges in sub-graph Q= (V,EQ)from Algorithm 1. Given\nG, the appearance probability ofMis the probability that M\nis the exact set of crucial edges that is in both EQand the\nmaximum matching of G.\nCrucial Edge Procedure\n1. Draw a matching MCofEQ∩Cbased on the appearance\nprobabilities of matchings given G∩C.\n2. Fore= (u,v)∈MC, letxe= (1−ǫ)min{1−qN\nu,1−\nqN\nv}.\nTrivially, this procedure makes sure that all vertices v∈\nVdo not exceed their budget, i.e. xv≤1. So, first, we will\nprove that this procedure still allows fractional matching x\nto satisfy the constraints of Lemma 6.\nLemma 17.∀U⊆Vwith|U|≤1/ǫ,/summationtext\ne∈Exe≤⌊|U|/2⌋.\nSecond, we show that the size of the fractional matching\nafter the crucial edge procedure is the size we wanted. The\nproof of the following lemma largely relies on the optimal\nmatching on non-crucial edges from the non-crucial edge\nprocedures as well as a bit of algebra involving matching\nprobabilities and the additional value assignments from th e\ncrucial edge procedure.\nLemma 18. Given unweighted graph Gand fractional\nmatching xfrom the non-crucial and crucial edge proce-\ndures,E[/summationtext\ne∈EQxe]/E[µ(G)]≥(1−2ǫ)(4√\n2−5).\nFinally, we may prove our first theorem, Theorem 3.\nProof of Theorem 3. Take fractional matching xonQ=\n(V,EQ)as constructed by the non-crucial edge and crucial\nedge procedures. By Lemma 17, xsatisfies the condition for\nLemma 6. Thus, by Lemma 6, there exists an integral match-\ningyonQwith size at least (1−ǫ)times that of x. So, we\nhave that\nE[µ(Q)]≥E\n/summationdisplay\ne∈EQwe·ye\n\n≥(1−ǫ)E\n/summationdisplay\ne∈EQwe·xe\n By Lemma 6.\n≥(1−3ǫ)(4√\n2−5)E[µ(G)] By Lemma 18.\n≥(.6568−ǫ0)E[µ(G)]\nwhereǫ0= 3(.6568)ǫ.\nUnfortunately, it can be shown that this bound is tight for\nour analysis using the non-crucial and crucial edge proce-\ndures as there are examples of graphs where the best ap-\nproximation using these procedures has ratio (4√\n2−5)(see\n(Behnezhad et al. 2019b))Weighted Approximation\nFor weighted graphs, we will begin similarly to the un-\nweighted graph analysis. The end goal will be to cre-\nate a fractional matching of non-crucial edges and crucial\nedges to prove our desired .501 bound. We have slightly\nchanged and improved the results of this section compared\nto (Behnezhad et al. 2019b) by working with a smaller con-\nstantδin Definition 20. From this update, the approximation\nratio is a bit better for the following weighted case analysi s.\nDue the new model, we must again work around the cor-\nrelation between realization probabilities of adjacent ed ges\nin our new model. First, the non-crucial edge procedure will\ncreate a near-perfect fractional matching xon non-crucial\nedges as before. However, a new procedure for crucial edges\nwill be used that will not only assign fractional matching va l-\nues to crucial edges but may also modify the values in the\nfractional matching given to non-crucial edges. By lower-\ning the previously assigned non-crucial edge matching val-\nues, there will be more budget for crucial edges to take. In\nweighted graphs, it is possible that crucial edges have high\nweights and significantly contribute to the maximum match-\ning ofG. For this reason, increasing the remaining vertex\nbudget accordingly for crucial edges is necessary. So, the\nfollowing procedure considers matching probabilities and\nedge weights when assigning fractional matching values to\nthe crucial edges, and then it modifies the non-crucial frac-\ntional matching if needed.\nWeighted Crucial Edge Procedure\n1. As in the previous crucial edge procedure, draw some\nmatching MCofEQ∩Caccording to appearance-\nprobabilities. For vertex vand constant α, define\ng(v,α) :=min{qN\nv,1−α}\nqNvϕN\nv.\n2. Fore= (u,v)∈MC, set\nxe:= (1−ǫ)argmax\n0≤α≤1(g(u,α)+g(v,α)+α·we).\n3. For any vertex v, ifxv>1, scale down fractional match-\ning of incident non-crucial edges until xv≤1.\nBefore analyzing this new procedure, we want to bound\nϕ(N)andϕ(C)so that they are easier to work with. To do\nso, we utilize the following lemma discussing the relation o f\nϕ(C)to Algorithm 1’s expected matching size. The proof of\nthe lemma and other missing proofs of this section can be\nfound in Appendix.\nLemma 19. Given sub-graph Q= (V,EQ)from Algorithm\n1,E[µ(Q)]≥(1−ǫ)ϕ(C).\nRecall that from Procedure and Lemma 15, there is an\nexpected matching of Qof size at least (1−10ǫ)ϕ(N). From\nLemma19, we know that there is also an expected matching\nsize ofQwith size at least (1−ǫ)ϕ(C). Thus, if either ϕ(C)\norϕ(N)is at least .501·E[µ(G)], we have our approximation\nratio from Theorem 4. We may now focus on when this is not\nthe case. Since E[µ(G)] =ϕ(C) +ϕ(N), we will analyze\nthe case when\n.499·E[µ(G)]≤ϕ(C),ϕ(N)≤.501·E[µ(G)].For this analysis, we will utilize the matching constructed by\nthe weighted crucial edge procedure and further classify cr u-\ncial edges by edge weights to make claims about this match-\ning’s expected weight.\nDefinition 20. Letδ=.09, then a crucial edge e= (u,v)∈\nCisheavy ifwe≥(1 +δ)(ϕN\nu+ϕN\nv), and we denote the\nset of heavy edges with H.\nA crucial edge is semi-heavy ifeis not heavy, we≥2(1+\nδ)ϕN\nv, andqN\nu≤(1−δ)whereqN\nv≥qN\nu. We denote the set\nof semi-heavy edges with H∗.\nObserve that by definition, the weight of a heavy edge\ne= (u,v)∈His larger than the expected fractional match-\ning of adjacent non-crucial edges. As such, from step 2 of\nthe weighted crucial edge procedure, xewill be maximized\nwhenα= 0andxe= (1−ǫ). Furthermore, step 3 will re-\nduce the fractional matching of adjacent non-crucial edges\nto 0 so that xu,xv≤1. Similarly, for semi-heavy edge\ne= (u,v)∈H∗, by definition the weight of eis greater\nthan the expected non-crucial fractional matching of one of\nits vertices. Specifically, from step 2 of the procedure we\nhave that xe≥(1−ǫ)(1−qN\nu)≥(1−ǫ)δ. With these cru-\ncial edge bounds, we can prove that if a graph has enough\nheavy and semi-heavy edges, then will will reach at least the\ndesired .501 approximation ratio.\nLemma 21. Ifϕ(H) +ϕ(H∗)≥0.074ϕ(C), then\nE[µ(Q)]/E[µ(G)]≥.501−11ǫ.\nNow, we must look at when ϕ(H) +ϕ(H∗)<\n0.074ϕ(C), i.e. when the heavy and semi-heavy edges do\nnot constitute a large part of the expected maximum match-\ning on crucial edges. We will use C∗to denote the set of\ncrucial edges that are not heavy nor semi-heavy. We will di-\nrectedges ofC∗based on their adjacent non-crucial edges’\ncontribution to the maximum matching of G.\nDefinition 22. We will classify edges e= (u,v)∈C∗into\nthe following three types of edges:\n1. IfϕN\nv≥ϕN\nu, directetowardsv.\n2. IfϕN\nv< ϕN\nuandwe≤2(1+δ)ϕN\nv, directetowardsv.\n3. Else, direct etowardsu. Note, in this case ϕN\nv< ϕN\nuand\nwe>2(1+δ)ϕN\nv.\nNote that edge types 1-3 partition C∗. Thus, using these\ndefinitions and directed edge types, we may prove properties\nofC∗and our desired bound for the approximation ratio of\nsub-graph Qfrom Algorithm 1.\nTheorem 23. Given sub-graph Qfrom Algorithm 1,\nE[µ(Q)]/E[µ(G)]≥.501−11ǫ.\nEDCS 2/3 Approximation\nIn this section we improve our bound for unweighted graphs\nto2/3−ǫfor an arbitrary constant ǫ >0. Inspired by a\nwork of (Assadi and Bernstein 2019) we use edge-degree\nconstrained sub-graph (EDCS) for designing our algorithm.\nBefore stating our result we first give the definition of EDCS\nfrom (Bernstein and Stein 2015) and (Bernstein and Stein\n2016).Definition 24 ((Bernstein and Stein 2015, 2016)) .For any\ngraphG= (V,E), and integers β≥β−≥0, an edge-\ndegree constrained sub-graph(EDCS) (G,β,β−)is a sub-\ngraphH= (V,EH)with the following two properties.\n1. For every edge (v,u)∈EH:degH(v)+degH(u)≤β.\n2. For every edge (v,u)/∈EH:degH(v)+degH(u)≥β−.\nIt has been shown in (Bernstein and Stein 2015) and\n(Bernstein and Stein 2016) that for any graph G, and any\nparameters β > β−, an EDCS of Gexists. Also it is easy to\nsee that an EDCS of Gis degree-bounded and has a maxi-\nmum degree of β. An interesting property of EDCS is that\nfor a large enough βandβ−, it always preserves 2/3ap-\nproximation of maximum matching in G. Specifically, we\nhave the following.\nTheorem 25 ((Assadi and Bernstein 2019)) .LetG=\n(V,E)be any graph, ǫ <1/2,λ≤ǫ/32,β≥\n8λ−2log(1/λ),β−≥(1−λ)β, and Let Hbe an\nEDCS(G,β,β−). Thenµ(H)≥(2/3−ǫ)µ(G).\nA result by (Assadi and Bernstein 2019) shows that for\nany stochastic graph Gwhere each edge is realized with a\nprobability of pe, an EDCS (G,β,β−1)also preserves a\n2/3−ǫapproximation of the expected maximum match-\ning. We show a similar result for the generalized stochas-\ntic matching problem where both edges and vertices are\nstochastic. Specifically, let Qbe an EDCS (G,β,β−1)\nforβ≥Clog(1/(ǫ·pv·pe))\nǫ2pvpewhereCis a large constant.\nAlso, LetQbe the realized portion of Q. In the following\nlemma we show that using the EDCS approach, Qachieves\na2/3−O(ǫ)matching approximation ratio in expectation.\nThe proof of Lemma 26 can be seen in the final section of\nthe Appendix.\nLemma 26. E[µ(Q)]≥(2/3−O(ǫ))E[µ(G)].\nConclusion\nWe have now proven bounds on the approximation ratio\nof Algorithm 1 breaking a half-approximation. The natural\nnext step is to improve the ratio up to (1−ǫ). For the old\nmodel of stochastic matching, (Behnezhad and Derakhshan\n2020) achieved a (1−ǫ)approximation for weighted\nstochastic matching using 1 complemented by a greedy sub-\nalgorithm. Unfortunately, differences in the models bar us\nfrom adapting their new algorithm directly. Previous analy -\nsis techniques relied on the complete independence of edge\nrealization, and it seems to us to be non-trivial to overcome\nthis difference.\nAcknowledgements\nThis research was supported by the NSF BIGDATA Grant\nNo. 1546108, NSF SPX Grant No. 1822738, NSF AF Grant\nNo. 2114269, and an Amazon AWS award.\nReferences\nAssadi, S.; and Bernstein, A. 2019. Towards a Unified\nTheory of Sparsification for Matching Problems. In Fine-\nman, J. T.; and Mitzenmacher, M., eds., 2nd Symposium onSimplicity in Algorithms, SOSA@SODA 2019, January 8-9,\n2019 - San Diego, CA, USA , volume 69 of OASICS , 11:1–\n11:20. Schloss Dagstuhl - Leibniz-Zentrum f¨ ur Informatik .\nAssadi, S.; Khanna, S.; and Li, Y . 2016. The Stochastic\nMatching Problem with (Very) Few Queries. In Conitzer,\nV .; Bergemann, D.; and Chen, Y ., eds., Proceedings of the\n2016 ACM Conference on Economics and Computation, EC\n’16, Maastricht, The Netherlands, July 24-28, 2016 , 43–60.\nACM.\nAssadi, S.; Khanna, S.; and Li, Y . 2017. The Stochastic\nMatching Problem: Beating Half with a Non-Adaptive Al-\ngorithm. In Daskalakis, C.; Babaioff, M.; and Moulin, H.,\neds., Proceedings of the 2017 ACM Conference on Eco-\nnomics and Computation, EC ’17, Cambridge, MA, USA,\nJune 26-30, 2017 , 99–116. ACM.\nBehnezhad, S.; and Derakhshan, M. 2020. Stochas-\ntic Weighted Matching: $(1- ǫ)$ Approximation. CoRR ,\nabs/2004.08703.\nBehnezhad, S.; Derakhshan, M.; Farhadi, A.; Hajiaghayi,\nM.; and Reyhani, N. 2019a. Stochastic Matching on Uni-\nformly Sparse Graphs. In Fotakis, D.; and Markakis, E.,\neds., Algorithmic Game Theory - 12th International Sym-\nposium, SAGT 2019, Athens, Greece, September 30 - Octo-\nber 3, 2019, Proceedings , volume 11801 of Lecture Notes in\nComputer Science , 357–373. Springer.\nBehnezhad, S.; Derakhshan, M.; and Hajiaghayi, M. 2020.\nStochastic matching with few queries: (1- ǫ) approximation.\nIn Makarychev, K.; Makarychev, Y .; Tulsiani, M.; Kamath,\nG.; and Chuzhoy, J., eds., Proccedings of the 52nd Annual\nACM SIGACT Symposium on Theory of Computing, STOC\n2020, Chicago, IL, USA, June 22-26, 2020 , 1111–1124.\nACM.\nBehnezhad, S.; Farhadi, A.; Hajiaghayi, M.; and Reyhani,\nN. 2019b. Stochastic Matching with Few Queries: New Al-\ngorithms and Tools. In Chan, T. M., ed., Proceedings of the\nThirtieth Annual ACM-SIAM Symposium on Discrete Algo-\nrithms, SODA 2019, San Diego, California, USA, January\n6-9, 2019 , 2855–2874. SIAM.\nBehnezhad, S.; and Reyhani, N. 2018. Almost Optimal\nStochastic Weighted Matching with Few Queries. In Tardos,\n´E.; Elkind, E.; and V ohra, R., eds., Proceedings of the 2018\nACM Conference on Economics and Computation, Ithaca,\nNY, USA, June 18-22, 2018 , 235–249. ACM.\nBernstein, A.; and Stein, C. 2015. Fully Dynamic Match-\ning in Bipartite Graphs. In Halld´ orsson, M. M.; Iwama, K.;\nKobayashi, N.; and Speckmann, B., eds., Automata, Lan-\nguages, and Programming - 42nd International Colloquium,\nICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings,\nPart I , volume 9134 of Lecture Notes in Computer Science ,\n167–179. Springer.\nBernstein, A.; and Stein, C. 2016. Faster Fully Dy-\nnamic Matchings with Small Approximation Ratios. In\nKrauthgamer, R., ed., Proceedings of the Twenty-Seventh\nAnnual ACM-SIAM Symposium on Discrete Algorithms,\nSODA 2016, Arlington, VA, USA, January 10-12, 2016 ,\n692–711. SIAM.Blum, A.; Dickerson, J. P.; Haghtalab, N.; Procaccia, A. D.;\nSandholm, T.; and Sharma, A. 2015. Ignorance is Al-\nmost Bliss: Near-Optimal Stochastic Matching With Few\nQueries. In Roughgarden, T.; Feldman, M.; and Schwarz,\nM., eds., Proceedings of the Sixteenth ACM Conference on\nEconomics and Computation, EC ’15, Portland, OR, USA,\nJune 15-19, 2015 , 325–342. ACM.\nYamaguchi, Y .; and Maehara, T. 2018. Stochastic Packing\nInteger Programs with Few Queries. In Czumaj, A., ed.,\nProceedings of the Twenty-Ninth Annual ACM-SIAM Sym-\nposium on Discrete Algorithms, SODA 2018, New Orleans,\nLA, USA, January 7-10, 2018 , 293–310. SIAM.Missing Proofs of Algorithm 1 Analysis\nProof of Lemma 11. To prove the first property, note that\nfrom step 1 of the procedure we have that xe≤\n2τ/(p2\nvpe)≤ǫ3for at most/parenleftbig|U|\n2/parenrightbig\nedges. This implies that\nfor anyU⊆Vwith|U|≤1/ǫ,\n/summationdisplay\ne∈E(U)xe=ǫ3|U|(|U|−1)\n2≤ǫ31\nǫ(|U|−1)\n2\n≤ǫ2⌊|U|/2⌋≤ǫ⌊|U|/2⌋.\nTo prove the second property, we then observe the scaling\nfrom steps 2 and 3 for the edges incident to any vertex v∈\nV. Since for all edges se≤1, if˜xv≤max{qN\nv,ǫ}, then\ntriviallyxv=se·˜xv≤max{qN\nv,ǫ}.\nIf˜xv>max{qN\nv,ǫ}, thenmax{qN\nv,ǫ}/˜xv≤1and so, at\nthe end of step 2, se≤max{qN\nv,ǫ}/˜xv. Thus after step 3,\nxv= ˜xv·se≤˜xv·max{qN\nv,ǫ}/˜xv≤max{qN\nv,ǫ}.\nProof of Lemma 13. First, we will show a simple equality\nbetween the non-crucial edges in Qandϕ(N). LetµN\nide-\nnote the random variable representing the weight of the non-\ncrucial edges in the maximum matching from round iof Al-\ngorithm1. In Algorithm 1, the matching chosen at any round\nihas the same probability of being chosen as the matching\nchosen forG. In other words, E[µN\ni] =/summationtext\ne∈Nwe·qe=\nϕ(N). Additionally, let µN= (µN\n1+µN\n2+...+µN\nR)/R.\nThen, by definition of fe,µN=/summationtext\ne∈EQ∩Nwe·fe, and\nmoreover, by linearity of expectation, we have that E[µN] =\nϕ(N). Thus, we have the key equality between ϕ(N)and\nour non-crucial edges of Q:\nE\n/summationdisplay\ne∈EQ∩Nwe·fe\n=ϕ(N). (2)\nUnfortunately, this does not tell us enough about the frac-\ntional matching on the realization of Q, and so, we will need\nto analyze step 1 of our non-crucial edge procedure. As pre-\nviously discussed, E[fe] =qeis the observation that was\nthe basis of our choice of procedure. However, since xe=\nmin{fe/(p2\nvpe),2τ/(p2\nvpe)}, we will discuss the probability\nthat2τ < fe. We want to show that this probability is small,\nonly at most ǫ·qe, and thus, does not affect our matching\nmuch. To do so, we let X=X1+X2+...+XRbe the\nsum of random independent events Xi, which are 1 if edge\neis in the maximum matching of iteration iof Algorithm 1\nand 0 otherwise. We will use a Chernoff bound on E[X]and\nwork back to feandτsince by definition, X=R·fe.\nPr(fe≥2τ) = Pr(fe−τ≥τ)\n≤P[fe−qe≥τ], sinceeis non-crucial and qe< τ.\n= Pr/parenleftBig\nfe−E[fe]≥τ/parenrightBig\n= Pr/parenleftBig\nX−E[X]≥R·τ/parenrightBig\n, sinceX=fe·R.≤exp/parenleftBig\n−R·τ·log/parenleftbig\n1+(R·τ)/E[X]/parenrightbig\n2/parenrightBig\n,\nby Chernoff bound.\n≤exp/parenleftBig\n−R·τ·log(1+τ\nqe)\n2/parenrightBig\n,\nsinceE[X] =qe·R.\n≤exp/parenleftBig\n−50log(1/(ǫp2\nvpe))log(1+τ\nqe)/parenrightBig\n,\nsinceR·τ >100log(1 /(ǫp2\nvpe)).\n=1\nexp/parenleftBig\n50log(1/(ǫp2vpe))log(1+τ\nqe)/parenrightBig\n=1\n(1+τ\nqe)50log(1 /(ǫp2vpe))\n≤1\n(1+τ\nqe)(1+τ\nqe)49log(1 /(ǫp2vpe)),\nsinceǫ≤e−1and therefore log(1/(ǫp2\nvpe))≥1.\n≤1\n(1+τ\nqe)·249log(1 /(ǫp2vpe)),\nsinceτ > qefore∈Nand so,1+τ\nqe>2.\n=1\n(1+τ\nqe)·exp/parenleftbig\n49/log(2)·log(1/(ǫp2vpe))/parenrightbig\n≤1\n(1+τ\nqe)·exp/parenleftbig\n30log(1/(ǫp2vpe))/parenrightbig\n≤1\n(1+τ\nqe)·e10·exp/parenleftbig\n20log(1/(ǫp2vpe))/parenrightbig\n≤1\n20(1+τ\nqe)·exp/parenleftbig\n5log(1/(ǫp2vpe))/parenrightbig\n≤1\n20(1+τ\nqe)·log(1/ǫ)·exp/parenleftbig\n4log(1/(ǫp2vpe))/parenrightbig,\nsinceex≥xfor all real numbers xandpv,pe≤1.\n=ǫ4p8\nvp4\ne\n20(1+τ\nqe)·log(1/ǫ)\n≤ǫτ\n(1+τ\nqe), sinceτ=ǫ3p2\nvpe\n20log(1/ǫ).\n=ǫ·τ·qe\nτ+qe\n≤ǫ·τ·qe\nτ\n=ǫ·qe.\nNow, we can bound the size of ˜xe. First, note that by condi-\ntional expectation we know the following inequality\nE[min{fe,2τ}]≥Pr(fe≤2τ)·E[fe|fe≤2τ]. (3)\nAdditionally, we have\nE[fe] = Pr(fe≤2τ)E[fe|fe≤2τ]\n+Pr(fe>2τ)E[fe|fe>2τ].Subtracting inequality 3 from this equation yields\nE[fe]−E[min{fe,2τ}]≤Pr(fe>2τ)E[fe|fe>2τ].\nWe can then substitute the inequality Pr(fe≥2τ)≥ǫ·qe\nproven above to obtain\nE[fe]−E[min{fe,2τ}]≤(ǫ·qe)E[fe|fe>2τ]\n≤ǫ·qe\n=ǫ·E[fe]\n=⇒E[min{fe,2τ}]≥(1−ǫ)E[fe].\nNext, we will incorporate edge weights. Weights are not\nnecessary for this section since we are currently focused on\nunweighted graphs but will allow us to use this lemma later\nin the Weighted Approximations section. As such, we note\nthat\nE\n/summationdisplay\nEQ∩Nmin{fe,2τ}·we\n=/summationdisplay\nEQ∩NE[min{fe,2τ}]·we\n≥/summationdisplay\nEQ∩N(1−ǫ)E[fe]·we\n= (1−ǫ)/summationdisplay\nEQ∩Nqe·we\n= (1−ǫ)ϕ(N). (4)\nLastly for step 1, we can now bound the value of ˜xe. First,\nwe use1eto denote the indicator of the event that edge eis\nrealized. Note that any edge eis realized with probability pe\nonly if both of its vertices are realized with probability pv,\ni.e. edgeeis realized with probability p2\nvpe. Altogether, for\nstep 1 we have\nE\n/summationdisplay\ne∈EQ∩Nwe·˜xe\n\n≥E\n/summationdisplay\ne∈EQ∩Nwe·min{fe/(p2\nvpe),2τ/(p2\nvpe)}\n\n= 1/(p2\nvpe)E\n/summationdisplay\ne∈EQ∩Nwe·min{fe,2τ}\n\n= 1/(p2\nvpe)E\n/summationdisplay\ne∈EQ∩Nwe·min{fe,2τ}·1e\n.\nNote the switch from realized sub-graph QtoQby using\nthe indicator variable 1efor the edges of the sum. To finishup, we have\n= 1/(p2\nvpe)/summationdisplay\ne∈EQ∩NE[we·min{fe,2τ}·1e]\n= 1/(p2\nvpe)/summationdisplay\ne∈EQ∩Nwe·E[min{fe,2τ}]·E[1e]\n= 1/(p2\nvpe)/summationdisplay\ne∈EQ∩Nwe·E[min{fe,2τ}]·(p2\nvpe)\n=/summationdisplay\ne∈EQ∩Nwe·E[min{fe,2τ}]\n≥(1−ǫ)ϕ(N),from Inequality 4.\nProof of Lemma 15. First, note that the non-crucial edge\nprocedure only assigns non-zero values to edges in the re-\nalized sparse sub-graph Q. So, with this fact,\nE/bracketleftBigg/summationdisplay\ne∈Nwe·xe/bracketrightBigg\n=E\n/summationdisplay\ne∈EQ∩Nwe·xe\n\n=E\n/summationdisplay\ne∈EQ∩Nwe·se·˜xe\n.\nFrom Lemma 14, we know that for any edge e∈EQ∩N, the\nprobability that the scaling factor sefrom steps 2 and 3 of\nthe non-crucial edge procedure is less than 1−5ǫis at most\n2ǫgiven a vertex v∈e. Since there are two incident vertices\nfor edgee,se≥1−5ǫwith probability at least 1−4ǫ, and\nthus,\nE\n/summationdisplay\ne∈EQ∩Nwe·se·˜xe\n\n≥(1−4ǫ)(1−5ǫ)E\n/summationdisplay\ne∈EQ∩Nwe·˜xe\nBy Lemma 14.\n≥(1−9ǫ)E\n/summationdisplay\ne∈EQ∩Nwe·˜xe\n\n≥(1−9ǫ)(1−ǫ)ϕ(N) By Lemma 13.\n≥(1−10ǫ)ϕ(N).\nMissing Proofs of Crucial Edges and\nUnweighted Approximation\nProof of Lemma 17. From Lemma 11, the non-crucial edge\nprocedure creates a fractional matching with size at most\nǫ⌊|U|−1\n2⌋. Now, from the crucial edge procedure, since MC\nis an integral matching, it can only have at most ⌊|U|−1\n2⌋\nedges with each edge contributing at most 1−ǫto the frac-\ntional matching xafter step 2. In total, xwill have size at\nmost\nǫ⌊|U|−1\n2⌋+(1−ǫ)⌊|U|−1\n2⌋=⌊|U|−1\n2⌋Proof of Lemma 18. Since the crucial edge procedure builds\non the fractional matching of the non-crucial edge proce-\ndure, we have that/summationtext\ne∈Exe=/summationtext\ne∈Nxe+/summationtext\ne∈Cxe. Then,\nfrom Lemma 15, we have that/summationtext\ne∈Nxe≥(1−ǫ)ϕ(N) =\n(1−ǫ)q(N)sincewe= 1 for unweighted graphs. So, we\nwill be focused on showing the additional size from the cru-\ncial edges.\nGiven some crucial edge e, in order for xeto be assigned a\nnonzero value by the crucial edge, the edge must be included\ninQby Algorithm 1and then in the chosen matching µC\ngiven that eis in sub-graph Q. Sinceeis a crucial edge, by\ndefinition it is in Qwith probability greater than 1−ǫ. Given\nthateis inQ, someµCsuch that e∈µCwill be chosen\nby the crucial edge procedure in step 1 with probability at\nleastqesinceeappears in the maximum matching of Gwith\nprobability qeand has already been chosen by Q. The value\ngiven toxein step 2 will be (1−ǫ)min{1−qN\nu,1−qN\nv},\nand so,\nE[xe] = (1−ǫ)qe(1−ǫ)min{1−qN\nu,1−qN\nv}\n≥(1−2ǫ)qemin{1−qN\nu,1−qN\nv}. (5)\nTo simplify the inequality and remove the min function,\nthe crucial edges will be directed towards the endpoint with\nthe lower budget remaining prior to procedure 2. In other\nwords,e= (u,v)∈Cwill be directed to usuch that\nqN\nu> qN\nv, ties decided arbitarily. Let Ivdenote the set of\nincoming crucial edges to vertex vand letqI\nv:=/summationtext\ne∈Ivqe\nbe the matching probability of the edges directed towards v.\nNow, utilizing these definitions with inequality 5 yields\nE/bracketleftBigg/summationdisplay\ne∈Cxe/bracketrightBigg\n≥/summationdisplay\nv(1−2ǫ)(1−qN\nv)qI\nv\n= (1−2ǫ)/summationdisplay\nv(qI\nv−qN\nvqI\nv)\n= (1−2ǫ)/summationdisplay\nv(qI\nv−qN\nvqI\nv)\n= (1−2ǫ)q(C)−(1−2ǫ)/summationdisplay\nv∈VqN\nvqI\nv.(6)\nThen for the entire fractional matching, combining Inequal -\nity 6 with Lemma 15 we have\nE/bracketleftBigg/summationdisplay\ne∈Exe/bracketrightBigg\n=E/bracketleftBigg/summationdisplay\ne∈Nxe/bracketrightBigg\n+E/bracketleftBigg/summationdisplay\ne∈Cxe/bracketrightBigg\n≥(1−2ǫ)/parenleftBigg\nq(N)+q(C)−/summationdisplay\nv∈VqN\nvqI\nv/parenrightBigg\n.\nFinally, we will utilize this equation to bound the ap-\nproximation ratio. Observe that for unweighted graphs,\nE[µ(G)] =q(N) +q(C). Furthermore, since each crucial\nedge is directed, q(C) =/summationtext\nv∈VqI\nv, and similarly, 2q(N) =/summationtext\nv∈VqN\nvsince edge will be counted twice, once per inci-\ndent vertex. With these substitutions, the approximation r a-tio is as follows\nE[/summationtext\ne∈Exe]\nE[µ(G)]≥(1−2ǫ)/parenleftbig\nq(N)+q(C)−/summationtext\nv∈VqN\nvqI\nv/parenrightbig\nq(N)+q(C)\n≥(1−2ǫ)/parenleftBigg\n1−/summationtext\nv∈VqN\nvqI\nv/summationtext\nv∈VqIv+qN\nv\n2/parenrightBigg\n. (7)\nTo finish the proof off, we will need one algebraic\nlemma, which is written and proven as Lemma 5.4 in\n(Behnezhad et al. 2019b).\nLemma 27. Given sets of numbers a1,...,an,b1,...,bnsuch\nthat\n•∀i∈[1,n],ai≥0,bi≥0, andai+bi≤1\n•/summationtextn\ni=1ai+bi>0\nthen it is true that/summationtextn\ni=1aibi/summationtextn\ni=1ai+bi\n2≤6−4√\n2.\nUtilizing Lemma 27 and applying to Inequality 7 we have\nE[/summationtext\ne∈Exe]\nE[µ(G)]≥(1−2ǫ)(1−(6−4√\n2))\n≥(1−2ǫ)(4√\n2−5).\nIn the non-crucial and crucial edge procedures, we only as-\nsign fractional matching values to the edges of Q. Thus,\nE/bracketleftbig/summationtext\ne∈Exe/bracketrightbig\n=E/bracketleftBig/summationtext\ne∈EQxe/bracketrightBig\n, and we are finished.\nMissing Proofs of Weighted Approximation\nProof of Lemma 19. Given edge e∈C, letrebe the proba-\nbilityeis included in Qby Algorithm 1. Note that by con-\nstruction of Q,1−re= (1−qe)R, and since e∈C, we have\n1−re≤(1−τ)R. Furthermore, observe that R >log(1/ǫ)\nτ,\nand so,\n1−re≤((1−τ)(1/τ))log(1/ǫ)\n≤(1\ne)log(1/ǫ)=ǫ\n=⇒re≥1−ǫ. (8)\nNow, let1Q(e) := 1 ife∈EQ, 0 otherwise. With re≥1−ǫ,\nwe can finish the proof of the lemma using Inequality 8 in\nthe fifth line.\nE[µ(Q)]≥E[ϕ(EQ∩C)]\n=E/bracketleftBigg/summationdisplay\ne∈Cwe·qe·1Q(e)/bracketrightBigg\n=/summationdisplay\ne∈Cwe·qe·E[1Q(e)]\n=/summationdisplay\ne∈Cwe·qe·re\n≥(1−ǫ)/summationdisplay\ne∈Cwe·qe\n= (1−ǫ)ϕ(C).Lemma 28. Given fractional matching xfrom the weighted\ncrucial edge procedure. ∀U⊆Vwith|U| ≤1/ǫ,/summationtext\ne∈Exe≤⌊|U|/2⌋.\nProof. Proof is the same as proof of Lemma 17 by substi-\ntuting the weighted crucial edge procedure in for the crucia l\nedge procedure.\nProof of Lemma 21. As noted in the paragraphs preceding\nLemma 3.3, by definition of H, any heavy edge e= (u,v)∈\nHwill contribute to the fractional matching xoriginally\nfrom the non-crucial edge procedure with an amount at least\n(1−ǫ)we−(ϕN\nu+ϕN\nv). Again by definition of heavy edges,\nwe/(1+δ)≥(ϕN\nu+ϕN\nv), which implies that\n(1−ǫ)we−(ϕN\nu+ϕN\nv)≥(1−ǫ)we−1\n1+δwe\n= (δ\n1+δ−ǫ)we. (9)\nSimilarly, from the weighted crucial edge procedure, any\nsemi-heavy edge e= (u,v)∈H∗will contribute\n(1−ǫ)(1−qN\nu)we−(qN\nv−qN\nu)ϕN\nv\n≥(1−ǫ)(1−qN\nu)we−(1−qN\nu)ϕN\nv\n= (1−qN\nu)((1−ǫ)we−ϕN\nv).\nBy definition of semi-heavy edges, we≥2(1 +δ)ϕN\nvand\n1−qN\nu≥δ, and so,\n(1−qN\nu)((1−ǫ)we−ϕN\nv)\n≥(1−qN\nu)((1−ǫ)we−1\n2(1+δ)we)\n= (1−qN\nu)(1+2δ\n2(1+δ−ǫ)we\n≥δ(1+2δ\n2(1+δ−ǫ)we\n≥(δ+2δ2\n2(1+δ)−ǫ)we. (10)\nFrom inequalities 9and10, the total weight of the expected\nmatching will be at least\n(1−10ǫ)ϕ(N)+(δ\n1+δ−ǫ)ϕ(H)+(δ+2δ2\n2(1+δ)−ǫ)ϕ(H∗).\nSinceδ=.09, note thatδ\n1+δ≥0.048andδ+2δ2\n2(1+δ)≥0.048.\nThen, the fractional matching has an expected weight of at\nleast\n(1−10ǫ)ϕ(N)+(0.048−ǫ)(ϕ(H)+ϕ(H∗))\n≥(1−10ǫ)ϕ(N)+(0.048−ǫ)(0.09ϕ(C))\n= (1−10ǫ)(E[µ(G)]−ϕ(C))+(0.00432−ǫ)ϕ(C)\nsinceϕ(N)+ϕ(C) =E[µ(G].\n≥(1−10ǫ)E[µ(G]−ϕ(C)(1−0.0.00432)\n≥(1−10ǫ)E[µ(G]−0.5011·E[µ(G](1−0.00432)\nsinceϕ(C)≤.05011·E[µ(G].\n≥(0.50106−10ǫ)·E[µ(G].Thus, our fractional matching has the desired approxima-\ntion ratio. Note that fractional matching xsatisfies the re-\nquirements of Lemma 6 by Lemma 28. When converting\nthe fractional matching to an integral matching, by Lemma\n6we introduce another factor of (1−ǫ), and so, we will\nend up with an integral matching with approximation ratio\n(0.501−11ǫ).\nLemma 29. Lete= (u,v)∈C∗be directed towards v.\nThen,we≤2(1+δ)ϕN\nv.\nProof of Lemma 29. Let edge e= (u,v)be a type 1 edge\ndirected towards v; by definition we have ϕN\nv≥ϕN\nu. Ad-\nditionally, e∈C∗must not be a heavy edge, so we≤\n(1+δ)(ϕN\nu+ϕN\nv)≤2(1+δ)ϕN\nv.\nNow, ifeis a type 2 edge, then by definition we≤\n2(1 +δ)ϕN\nv. Finally, let e= (u,v)be a type 3 edge\ndirected towards v. Then, by definition of type 3 edges,\nϕN\nv> ϕN\nu, and similar to type 1 edges, e∈C∗must not\nbe a heavy edge. Therefore by definition of heavy edges,\nwe≤(1+δ)(ϕN\nu+ϕN\nv)≤2(1+δ)ϕN\nv.\nLemma 30. Lete= (u,v)∈C∗withqN\nv≥qN\nu. Ifeis\ndirected towards u, thenqN\nv≤qN\nu+δ.\nProof of Lemma 30. Given edge e= (u,v)∈C∗with\nqN\nv≥qN\nudirected towards u,emust be type 3 by defini-\ntion. So,\nwe>2(1+δ)ϕN\nv= 2(1+δ)qN\nv.\nAlso, since eis not semi-heavy, qN\nu>(1−δ). Combining\nthese two inequalities yields the result\nqN\nv≤1< qN\nu+δ.\nProof of Theorem 23. Given fractional matching xfrom the\nweighted crucial edge procedure on Q, we will bound the\nsize ofxto prove the desired approximation ratio. Addition-\nally, ifϕ(H) +ϕ(H∗)≥0.074ϕ(C), then we are done\nfrom21. Thus, we will work with the case when ϕ(C∗)>\n0.926ϕ(C).\nNow, let Ivbe the set of incoming crucial edges in C∗\ndirected towards vertex v, and similarly, ϕI\nv:=/summationtext\ne∈Ivϕe.\nAfter the non-crucial edge procedure, an edge may have a\nremaining budget of (1−ǫ)(1−max{qN\nu,qN\nv}). Ifeis di-\nrected towards v, then by Lemma 30the remaining budget is\nat least(1−ǫ)(1−δ−qN\nv). Also, recall that the probability\nthat any crucial edge eis included in Qby Algorithm 1is at\nleast(1−ǫ)by definition. So,\nE/bracketleftBigg/summationdisplay\ne∈C∗we·xe/bracketrightBigg\n≥/summationdisplay\nv∈V(1−ǫ)(1−ǫ)(1−δ−qN\nv)ϕI\nv\n≥(1−2ǫ)/summationdisplay\nv∈V(1−δ−qN\nv)ϕI\nv\n= (1−2ǫ)(1−δ)ϕ(C∗)−(1−2ǫ)/summationdisplay\nv∈VqN\nvϕI\nv (11)where the last line follows from the definition of ϕ(C∗).\nSo, for the entire matching using Lemma 15 and Inequality\n11 we have\nE/bracketleftBigg/summationdisplay\ne∈Exe/bracketrightBigg\n≥E/bracketleftBigg/summationdisplay\ne∈Nwe·xe/bracketrightBigg\n+E/bracketleftBigg/summationdisplay\ne∈C∗we·xe/bracketrightBigg\n≥(1−10ǫ)ϕ(N)\n+(1−2ǫ)/parenleftBigg\n(1−δ)ϕ(C∗)−/summationdisplay\nv∈VqN\nvϕI\nv/parenrightBigg\n.\nSinceϕ(N)≤E[µ(G)],\nE/bracketleftBigg/summationdisplay\ne∈Exe/bracketrightBigg\n−8ǫE[µ(G)]\n≥(1−2ǫ)/parenleftBigg\nϕ(N)+(1−δ)ϕ(C∗)−/summationdisplay\nv∈VqN\nvϕI\nv/parenrightBigg\n.\nAs in the unweighted case, we divide both sides by the\nexpected matching weight of non-crucial and crucial edges:\nE/bracketleftbig/summationtext\ne∈Exe/bracketrightbig\n−8ǫE[µ(G)]\nϕ(N)+ϕ(C∗)\n≥(1−2ǫ)/parenleftbig\nϕ(N)+(1−δ)ϕ(C∗)−/summationtext\nv∈VqN\nvϕI\nv/parenrightbig\nϕ(N)+ϕ(C∗)\n≥(1−2ǫ)/parenleftBigg\n1−/summationtext\nv∈Vδ·ϕI\nv+qN\nvϕI\nv\nϕ(N)+ϕ(C∗)/parenrightBigg\n≥(1−2ǫ)/parenleftBigg\n1−/summationtext\nv∈Vδ·ϕI\nv+qN\nvϕI\nv/summationtext\nv∈VϕIv+ϕNv\n2/parenrightBigg\n. (12)\nFrom Lemma 29, for edge e∈C∗directed to vertex v,\nwe≤2(1+δ)ϕN\nv. So,\nϕI\nv≤2(1+δ)qC\nvϕN\nv≤2(1+δ)(1−qN\nv)ϕN\nv.\nMoreover, the fraction\nδ·ϕI\nv+qN\nvϕI\nv\nϕIv+ϕNv\n2\nis increasing with ϕI\nv, and thus, by substituting in 2(1 +\nδ)(1−qN\nv)ϕN\nvas an upper bound for ϕN\nv, the fraction is\nat most\nδ·ϕI\nv+qN\nvϕI\nv\nϕIv+ϕNv\n2≤\n2(1+δ)(1−qN\nv)ϕN\nv(δ+qN\nv)\nϕNv(1\n2+2(1+δ)(1−qNv))\n=2(1+δ)(1−qN\nv)(δ+qN\nv)\n1\n2+2(1+δ)(1−qNv). (13)\nAs defined, we can substitute 0.09forδinto equation 13to\nobtain\n−2.18(qN\nv)2+1.9338qN\nv+0.1962\n−2.18qNv+2.68.This fraction can be found to be at most 0.43, however we\nskip the lengthy calculations for the sake of brevity. We wil l\nnext rely on the following algebraic lemma, whose proof is\ntrivial and excluded.\nLemma 31. For positive real values a, b, c, d, α, ifa\nb≤α\nandc\nd≤α, thena+c\nb+d≤α.\nThen, by Lemma 31and inequalities 12 and 13, we have\nthat\nE/bracketleftbig/summationtext\ne∈Exe/bracketrightbig\n−8ǫE[µ(G)]\nϕ(N)+ϕ(C∗)\n≥(1−2ǫ)/parenleftBigg\n1−/summationtext\nvδ·ϕI\nv+qN\nvϕI\nv/summationtext\nvϕIv+ϕNv\n2/parenrightBigg\n≥(1−2ǫ)(1−.43) = (1−2ǫ)(.57). (14)\nFinally, we can rearrange the terms of Inequality 14 and\nusing our assumption that ϕ(C∗)> .926ϕ(C), prove the\nfinal bound.\nE/bracketleftBigg/summationdisplay\ne∈Exe/bracketrightBigg\n≥(1−2ǫ)0.57(ϕ(N)+ϕ(C∗))+8ǫE[µ(G)]\n≥(1−2ǫ)(0.57·0.926)(ϕ(N)+ϕ(C))−8ǫE[µ(G)]\n≥(1−2ǫ)(0.528)(ϕ(N)+ϕ(C))−8ǫE[µ(G)]\n≥(.528−2ǫ)E[µ(G)]−8ǫE[µ(G)]\n= (.528−10ǫ)E[µ(G)]\nAs done previously, note that we can apply Lemma 6 since\nwe satisfy its requirements by Lemma 28. Thus,\nE[µ(Q)]≥(1−ǫ)E/bracketleftBigg/summationdisplay\ne∈Exe/bracketrightBigg\n≥(.528−11ǫ)E[µ(G)].\nMissing Proof of EDCS 2/3 Approximation\nProof of Lemma 26. We show the existence of two sub-\ngraphs˜Q⊆Q and˜G⊆G with the following properties.\n1.E[µ(˜G)]≥(1−ǫ)E[µ(G)], where the expectation is taken\nover the realization of graph.\n2.˜Qis an EDCS (˜G,(1+ǫ)pv·pe·β,(1−2ǫ)pv·pe·β)\nfor˜G.\nFirst we show how the existence of sub-graphs ˜Qand˜Gim-\nplies the lemma. By (2), we have that ˜Qis an EDCS for\n˜G. Also,(1+ǫ)pv·pe·β\n(1−2ǫ)pv·pe·β= 1 +O(ǫ), and(1 +ǫ)pv·pe·\nβ= Ω(ǫ2log(1/ǫ)). Therefore, by Theorem 25, we have\nµ(Q)≥µ(˜Q)≥(2/3−O(ǫ))µ(˜G). Combining this with\nthe property (1), gives us E[µ(Q)]≥(2/3−O(ǫ))E[µ(G)]\nwhich concludes the lemma.\nConsider a vertex v∈V, this vertex is in the realized\nsub-graphGwith the probability of pv. Therefore with the\nprobability of 1−pv, vertexvis not realized and we have\ndegQ(v) = 0 . Consider the case that vis realized. We then\nhaveE[degQ(v)] =pe·pv·degQ(v), since neighbors ofvare realized with the probability of pvand incident edges\nofvare realized with the probability of pe. In the follow-\ning lemma we show that only a small fraction of realized\nvertices can significantly deviate from this expectation.\nDefinition 32. LetV+⊆V be the set of realized vertices v\nsuch that degQ(v)> pe·pv·degQ(v) +ǫ·pv·pe·β/2.\nAlso, letV−⊆V be the set of realized vertices vsuch that\ndegQ(v)< pe·pv·degQ(v)−ǫ·pv·pe·β/2or there exists\nan edge(v,u)∈Qsuch that u∈V+.\nLemma 33. E[|V+|],E[|V−|]≤ǫ6·p6\nv·p6\ne·µ(G).\nProof. Consider a realized vertex v∈ V . We know that\ndegQ(v)≤β. Consider an edge (v,u)∈Q. This edge ap-\npears in the realized sub-graph with the probability of pv·pe\nsince vertex uis realized with the probability of pv, and edge\n(v,u)is realized with the probability of pe. Therefore we\nhaveE[degQ(v)] =pv·pe·degQ(v)≤pv·pe·β. By\nChernoff bound we have\nPr[v∈V+] = Pr[vis realized ]\n·Pr[degQ(v)> pe·pv·degQ(v)+ǫ·pv·pe·β/2]\n≤pv·e−O(ǫ2·pv·pe·β)\n≤e−O(log(1/(ǫ·pv·pe)))≤K−2·ǫ12·p12\nv·p12\ne,\nwhereKis a large constant and the last two inequalities\nfollow from the fact that β=Clog(1/(ǫ·pv·pe))\nǫ2pvpefor a large\nC. We can also set Clarge enough to get an arbitrary large\nconstantK > C . Therefore, each vertex is in V+with the\nprobability of K−2·ǫ12·p12\nv·p12\ne. On the other hand, Qhas\nat most2βµ(G)vertices with non-zero degree. The reason\nis that the graph Ghas a vertex cover of size 2µ(G)and\nall vertices in Qhave a degree of at most β. SinceQhas\nat most2βµ(G)vertices with non-zero degree, and each of\nthese vertices are in is in V+with the probability of K−2·\nǫ12·p12\nv·p12\ne, we have\nE[|V+|]≤2βµ(G)·K−2·ǫ12·p12\nv·p12\ne\n≤µ(G)·K−2·C·ǫ9·p9\nv·p9\ne\n≤µ(G)·K−1·ǫ9·p9\nv·p9\ne, SinceK > C .\nwhich shows that the number of vertices in |V+|is small.\nUsing a similar argument we can say that the number of ver-\ntices with the degree less than pe·pv·degQ(v)−ǫ·pv·pe·β/2\nis at most K−1·ǫ9·p9\nv·p9\ne. SinceV−is the set of low-degree\nvertices and neighbors of V+inQ, we have\nE[|V−|]≤µ(G)·K−1·ǫ9·p9\nv·p9\ne+β·E[|V+|]\n≤µ(G)·ǫ6·p6\nv·p6\ne,\nwhich proves the lemma.\nLemma 33 above shows that the sizes of V+andV−are\nvery small. We complete the proof of Lemma 26 by con-\nstructing sub-graphs ˜Qand˜Gas follows. Let ˜Ghave vertex\nsetVwhich are the set of realized vertices, and have the\nedge set equal toG, except we remove all edges incident to\nvertices inV+, and all edges (v,u)/∈Qthat are incidentto vertices inV−. Also let ˜Qbe the same asQexcept we\nremove all edges incident to V+. Now we show that these\nsub-graphs satisfy properties (1) and (2).\nFor property (1), note that ˜GandGare different only in\nthe vertices inV+andV−. Therefore,\nE[µ(˜G)]≥E[µ(G)]−E[|V+|]−E[|V−|]\n≥E[µ(G)]−ǫ5·p5\nv·p5\ne·µ(G) By Lemma 33.\n≥E[µ(G)]−ǫ3·p3\nv·p3\ne·E[µ(G)]\n≥(1−ǫ)E[µ(G)],,\nwhere the third inequality follows from the fact that\nE[µ(G)]≥p2\nv·pe·µ(G), since every edge in a maximum\nmatching is realized with the probability of p2\nv·pe.\nFor property (2), we have to show that ˜Qis an\nEDCS(˜G,(1+ǫ)pv·pe·β,(1−2ǫ)pv·pe·β)for˜G. To that\npurpose, we show that ˜Qsatisfies properties (1) and (2) of\nDefinition 24. Both ˜Gand˜Qdo not have any edges incident\ntoV+. Therefore, we can ignore these vertices. Therefore,\nfor all vertices vwe havedeg˜Q(v)≤pe·pv·degQ(v) +\nǫ·pv·pe·β/2, and for the vertices v /∈ V−we have\ndeg˜Q(v)≥pe·pv·degQ(v)−ǫ·pv·pe·β/2. Also˜G\\˜Q\nhas no edge incident to V−.\n1. For the property (1) of Definition 24: Consider an edge\n(v,u)∈˜Q, we then have\ndeg˜Q(v)+deg˜Q(u)\n≤pe·pv·degQ(v)\n+pe·pv·degQ(u)\n+ǫ·pv·pe·β\n≤(1+ǫ)pv·pe·β. SinceQis an EDCS of G.\n2. For the property (2) of Definition 24: Consider an edge\n(v,u)∈˜G\\˜Q, we then have\ndeg˜Q(v)+deg˜Q(u)\n≥pe·pv·degQ(v)\n+pe·pv·degQ(u)\n−ǫ·pv·pe·β\n≥(1−2ǫ)pv·pe·β, SinceQis an EDCS of G.\nwhich completes the proof of Lemma 26."    },    {        "title": "2206.02460v2.Probing_spin_dynamics_of_ultra_thin_van_der_Waals_magnets_via_photon_magnon_coupling.pdf",        "content": "Probing spin dynamics of ultra-thin van der Waals magnets via\nphoton-magnon coupling\nChristoph W. Zollitsch,1,a)Safe Khan,1Vu Thanh Trung Nam,2Ivan A. Verzhbitskiy,2Dimitrios Sagkovits,1, 3\nJames O’Sullivan,1Oscar W. Kennedy,1Mara Strungaru,4Elton J. G. Santos,4, 5John J. L. Morton,1, 6Goki\nEda,7, 2, 8and Hidekazu Kurebayashi1, 6, 9\n1)London Centre for Nanotechnology, University College London, 17-19 Gordon Street, London, WCH1 0AH,\nUK\n2)Department of Physics, Faculty of Science, National University of Singapore, 2 Science Drive 3, Singapore 117542,\nSingapore\n3)National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK\n4)Institute for Condensed Matter Physics and Complex Systems, School of Physics and Astronomy, The University of Edinburgh,\nEdinburgh EH9 3FD, UK\n5)Higgs Centre for Theoretical Physics, The University of Edinburgh, Edinburgh EH9 3FD,\nUK\n6)Department of Electronic & Electrical Engineering, UCL, London WC1E 7JE, United Kingdom\n7)Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546,\nSingapore\n8)Department of Chemistry, Faculty of Science, National University of Singapore, 3 Science Drive 3, Singapore 117543,\nSingapore\n9)WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1, Katahira, Sendai, 980- 8577,\nJapan\n(Dated: 1 May 2023)\nLayered van der Waals (vdW) magnets can maintain a\nmagnetic order even down to the single-layer regime and\nhold promise for integrated spintronic devices. While the\nmagnetic ground state of vdW magnets was extensively\nstudied, key parameters of spin dynamics, like the Gilbert\ndamping, crucial for designing ultra-fast spintronic de-\nvices, remains largely unexplored. Despite recent studies\nby optical excitation and detection, achieving spin wave\ncontrol with microwaves is highly desirable, as modern in-\ntegrated information technologies predominantly are op-\nerated with these. The intrinsically small numbers of\nspins, however, poses a major challenge to this. Here, we\npresent a hybrid approach to detect spin dynamics medi-\nated by photon-magnon coupling between high-Q super-\nconducting resonators and ultra-thin flakes of Cr 2Ge2Te6\n(CGT) as thin as 11 nm. We test and benchmark our tech-\nnique with 23 individual CGT flakes and extract an upper\nlimit for the Gilbert damping parameter. These results are\ncrucial in designing on-chip integrated circuits using vdW\nmagnets and offer prospects for probing spin dynamics of\nmonolayer vdW magnets.\nINTRODUCTION\nvan der Waals (vdW) materials1–3consist of individual\natomic layers bonded by vdW forces and can host different\ntypes of collective excitations such as plasmons, phonons and\nmagnons. Strong coupling between these excitation modes\nand electromagnetic waves (i.e. photonic modes) creates con-\nfined light-matter hybrid modes, termed polaritons. Polaritons\na)Electronic mail: c.zollitsch@ucl.ac.ukin vdW materials are an ideal model system to explore a va-\nriety of polaritonic states5,6, e.g. surface plasmon polaritons\nin graphene7,8and exciton polaritons in a monolayer MoS 2\nembedded inside a dielectric microcavity9. These states can\nbe further modified by electrostatic gating16, as well as by\nhetero-structuring with dissimilar vdW layers1.\nNumerous studies on magnon polaritons (MPs)11,12have\nbeen using macroscopic yttrium iron garnet (YIG) cou-\npled to either three-dimensional cavities13or to on-chip\nresonators14,15, with potential applications in ultra-fast infor-\nmation processing, non-reciprocity or microwave to optical\ntransduction. By reducing the number of excitations, MPs\nfind application in the quantum regime e.g., magnon number\ncounting via an electromagnetically coupled superconducting\nqubit16,17or as a building block for Bell state generation18.\nThe rapidly developing research around polaritons and\nspecifically MPs has so far, been little studied in magnetic\nvdW materials due to the relatively recent discoveries of long-\nrange magnetic order in vdW systems at the few monolayer\nregime9,20,21, in addition to its technically challenging real-\nization. Stable MP states are formed by strongly coupling the\nmagnetic field oscillation of a resonant photon to the collec-\ntive magnetization oscillation in a magnetic material. This\nstrong coupling is achieved when the collective coupling rate\ngeffis larger than the average of both system loss rates. In a\nsimplified picture, geffscales linearly with the strength of the\noscillating magnetic field of a resonator and the square root\nnumber of spins14. For studies involving bulk magnetic mate-\nrials and low quality and large microwave resonators, strong\ncoupling is achieved when geff=2pis in the MHz range, which\nis accomplished with relative ease due to the abundance of\nspins in bulk magnetic materials. A reduction of the bulk di-\nmensions down from mm to mm and nm scales, the typical\nlateral dimensions and thickness of vdW material monolay-arXiv:2206.02460v2  [cond-mat.mtrl-sci]  28 Apr 20232\ners, results in a decrease of the coupling strength by at least 6\norders of magnitude. Commonly used microwave resonators\nare not able to produce strong enough oscillating magnetic\nfields to compensate for such a reduction in absolute number\nof spins. Only by advanced resonator design and engineering\nthe regime of strongly coupled MPs in monolayer vdW mag-\nnetic materials can be accomplished, granting access to spin\ndynamic physics at a true 2d monolayer limit and research on\nMPs in nano-scale devices where the whole range of on-chip\ntuning and engineering tools, such as electric fields or device\ndesign, are available.\nMagnons or magnon polaritons have been observed in mag-\nnetic vdW materials, but it had been restricted to either to the\noptical frequency range22,23or a large thickness limit24,25, re-\nspectively. Here, we present our attempt of detecting spin\ndynamics in ultra-thin vdW magnetic materials and the cre-\nation of MPs by magnon-photon coupling in the microwave\nfrequency range, using superconducting resonators optimized\nfor increased magnon-photon coupling. By using microwave\nresonators with a small mode volume, we not only increased\nits oscillating magnetic field strength but also matched it more\nefficiently to the size of nanoscale vdW flakes. Our work\npresents a fundamental cornerstone for a general blueprint\nfor designing and developing magnon-photon hybrids for any\ntype of ultra-thin or monolayer vdW magnetic material, en-\nabling research on on-chip microwave applications for (quan-\ntum) information processing.\nRESULTS\nIn this article, we report on the observation of spin dynam-\nics and the creation of MPs at the onset of the high cooper-\nativity regime with the vdW ferromagnet CGT of nm scale\nthickness, demonstrating a pathway towards stable magnon-\nphoton polariton creation. We combine a precise transfer\nprocess of exfoliated CGT flakes and high sensitivity su-\nperconducting resonators, to access and study the dynami-\ncal response of coupled photon-magnon states in a small-\nvolume (nm-thick and \u0016m-sized) CGT flake (illustrated in\nFig. 1 (a)). High-quality-factor superconducting lumped el-\nement resonators are chosen to be the counterpart due to\ntheir extremely small mode volume ( \u00196000\u0016m3) and con-\nsequently strong oscillating magnetic fields ( B1\u001925nT, see\nSI for resonator quality-factors and B1-field distributions), re-\nsulting in high spin sensitivities4,26. At cryogenic temper-\natures, we perform low-power microwave spectroscopy on\nmultiple resonator-vdW-flake hybrids, covering a frequency\nrange from 12GHz to 18GHz for a variety of thickness. Sam-\nples consist of up to 12 resonators on a single chip, all capac-\nitively coupled to a common microwave transmission line for\nread-out (see SI for details). Multiple peaks of spin-wave res-\nonances are observed for each CGT flake measured. The spin-\nwave modes are closely spaced in frequency and show a large\noverlap. We employ a semi-optimized fitting model to pro-\nduce a good estimate for the collective coupling strength and\nmagnetic linewidth. By taking the resonance value of the most\nprominent peak of each spectrum, we find that all measuredpoints can be fitted very well by a single curve calculated by\nthe Kittel formula with bulk CGT parameters. Furthermore,\nwe extracted the linewidth for the thinnest CGT flake inves-\ntigated, 11nm or 15 monolayers (ML), the only device ex-\nhibiting well separated spin-wave modes. This allowed a fully\nquantitative analysis and we determined an upper limit of the\nGilbert damping parameter of 0 :02. This value is comparable\nto the damping reported for 3d transition metal ferromagnets,\nsuggesting that magnetic vdW flakes have the potential for the\nfabrication of functional spintronic devices.\nWe investigate the dynamics of nm-thick CGT flakes, us-\ning superconducting lumped element resonators made of NbN\n(see methods for fabrication details and SI and Ref. [28] for\nmore performance details). The advantages of a lumped ele-\nment design are the spatial separation of the oscillating mag-\nnetic field B1and electric field E1and the concentration of\nB1within a narrow wire section of the resonators, as indi-\ncated in Fig. 1 (a). Additionally, the B1field distribution is\nhomogeneous along the length of the narrow wire section (see\nfinite element simulations in SI). This magnetic-field concen-\ntration is our primary reason to use this type of resonator in\norder to reduce the photon mode volume as well as achieve\na considerable mode overlap between the resonator photon\nmode and CGT magnon mode, and consequently, a large cou-\npling strength. We therefore transfer CGT ���akes onto these\n5 μm\nB0CGTB1a\nbcE1\n0 4 8\nx (μm)y (nm)\n102030\n1240\nMCGT\nCrGeTe\nB1,extent  ≈ 2 μm\nFIG. 1. Magnon-photon coupling between thin CGT and a super-\nconducting resonator. a Schematic of a resonator shows the design\nin detail, indicating the areas of high E1-field (yellow) and B1-field\n(green) intensities, as well as the orientation of the externally applied\nfield B0. Finally, a schematic zoom in of the section loaded with a\nCGT flake is shown. The collective coupling between a microwave\nphoton and the magnetization of the CGT is illustrated, as well as the\napproximate extent of the microwave B1-field. bMicrograph image\nof a CGT flake transferred onto the narrow section of a resonator. c\nAFM image of the CGT flake together with a height profile along the\nblue solid line in the AFM image. The red solid line is a fit to the\nflake thickness. The results of this resonator are presented in Fig. 2.3\n12.8112.8212.83 ω/2π (GHz)\n560 580 600 620\nMagnetic Field (mT)640 66012.841.0 0.9 0.8|S21|20.7\n|S21|21.0\n0.9\n0.8a b\nc\n580 600\nMagnetic Field (mT)620234\nκeff/2π (MHz)0510 ωres/2π (MHz)\nd\n640\n+ 12820 MHz\n0.7550 mT\n598 mT\n614 mT\n670 mT\n12.81 12.82 12.83\nω/2π (GHz)12.84\nFIG. 2. Magnon-photon coupling observed in resonator microwave transmission. a jS21j2as a function static magnetic field B0and\nfrequency, with the microwave transmission encoded in the color. The results are obtained from the resonator shown in Fig 1 (b) and (c),\nfeaturing a loaded quality factor of QL=4600. bjS21j2as a function of frequency at fixed magnetic fields, indicated in aby dashed vertical\nlines. canddResonance frequency wresand effective loss rate keffas a function of magnetic field. Note the multiple resonance peaks,\nindicating multiple CGT FMRs. The dashed orange lines are results from the semi-optimized fit. dexemplary includes the individual peaks of\nwhich the orange dashed lines consists. The green bar in canddhighlights the main mode.\nnarrow sections (Fig. 1 (b)). Details of CGT flake transfers\nare described in the methods section. Optical imaging and\natomic force microscopy (AFM) measurements are used to\ncharacterise the size and thickness of the CGT flakes (see\nFig. 1 (c)). Measured thicknesses range from 153 \u000623nm\ndown to 11\u00061:8nm (15 ML), enabling a thickness dependent\nstudy of CGT flakes and their coupling to the resonators.\nWe measured the microwave transmission jS21j2as a func-\ntion of frequency and externally applied magnetic field B0for\neach resonator at a temperature of 1 :8K, using a microwave\npower of approximately \u000080dBm at the resonator chip. Fig-\nure 2 (a) shows the resulting 2D plot of jS21j2for a resonator\nloaded with a 17nm \u00060:8nm thick CGT flake (see Fig. 1 (b)\nand (c) for the respective micrograph and AFM images). A\nresonator peak can be clearly observed for each magnetic\nfield, with its resonance frequency wresdecreasing with in-\ncreasing magnetic field. The reduction of the frequency is\na result of a slow degradation of the superconductivity by\nB0, which in general exhibits a parabolic dependence29. For\n580mT\u0014B0\u0014630mT the resonator prominence is reduced,\nhighlighted byjS21j2as a function of frequency for four con-\nstant B0values in Fig. 2 (b). Within this field range, the mode\nresonance has been modified due to its hybridization with the\nmagnetic modes of the CGT flake. To further quantify the in-\nteraction, we fit each jS21j2profile by a Fano resonance line-\nshape (solid orange lines in Fig. 2 (b)) to account for an asym-\nmetric resonance peak due to additional microwave interfer-\nence in the circuitry30,31,\njS21j2=S0+A(qkeff=2+w\u0000wres)2\n(keff=2)2+ (w\u0000wres)2: (1)\nHere, S0is the microwave transmission baseline, Athe peak\namplitude, qdescribes the asymmetry of the lineshape and\nkeffrepresents the effective loss rate of the hybrid system (seeSI for resonator parameters before and after CGT transfer for\nall resonators). Figure 2 (c) shows wresof the hybrid system\nas a function of B0.wresexperiences a dispersive shift when\nthe photon mode and the magnon mode hybridize, indicating\nan onset of a strong interaction between the two individual\nsystems14,17,32–34. We observe multiple shifts in wres, suggest-\ning an interaction of several magnon modes with the resonator\nin our experiment.\nSignatures of the resonator–CGT-flake coupling are also\ncharacterised by keffof the hybrid system (Fig. 2 (d)). keffis\nenhanced from the value of the resonator loss rate k0due to\nan additional loss introduced by the magnon system charac-\nterized by the loss rate g14,32,35. Consistent with the B0de-\npendence of wres,keffshows a rich structure, having its main\npeak at 598mT, together with less prominent peaks distributed\naround it. Based on a formalism for coupled-harmonic-\noscillator systems in the high cooperativity regime32–34, we\nuse the following to analyse our experimental results with\nmultiple peaks:\nwres=wres;0+mB2\n0++n\nå\nk=\u0000ng2\neff;kDk\nD2\nk+g2; (2)\nkeff=k0++n\nå\nk=\u0000ng2\neff;kg\nD2\nk+g2: (3)\nwith the detuning factor for each resonance as Dk=\ngCGTmB\n¯h\u0000\nB0\u0000BFMR ;k\u0001\n. Here, wres;0is the resonator resonance\nfrequency at B0=0T and mrepresents the curvature of the\nresonance frequency decrease due to the applied magnetic\nfield. BFMR ;kis the CGT FMR field, gCGT the g-factor of\nCGT and geff;kgives the collective coupling strength between\nphoton and magnon mode. The summation is over all reso-\nnance modes kpresent on the low or high field (frequency)4\nside of the main resonance mode, where ngives the number\nof modes on one side. For simplicity, we assume a symmet-\nric distribution of modes about the main mode. The large\nnumber of multiple modes and their strong overlap prevent\na reliable application of a fully optimized fit to the data, due\nto the large number of free parameters required. In an ef-\nfort to gain a good estimate of the model parameters we ap-\nply the model functions Eq. (2) and (3) in a two-step semi-\noptimized fashion (see SI for details). With this approach, we\narrive at a model in good agreement with wresandkeff(see\norange dashed lines in Fig. 2 (c), (d), exemplary showing the\nindividual peaks of the orange dashed line in Fig. 2 (d) and\nthe SI for additional results and data). We can reproduce the\ndata using g=2p=94:03\u00065:95MHz and a collective cou-\npling strength of the main mode of 13 :25\u00061MHz. Together\nwith k0=2p=1:4\u00060:02MHz the system resides at the onset\nof the high cooperativity regime, classified by the cooperativ-\nityC=g2\neff=k0g=1:3>113,32. In this regime, magnon polari-\ntons are created and coherently exchange excitations between\nmagnons and resonator photons on a rate given by geff. The\ncreated MPs are, however, short lived and the excitations pre-\n100 200 500 700\nResonance Field BFMR (mT)300 400 600 0051015ωFMR/2π (GHz)500\nResonance Field BFMR (mT)600 7001518ωFMR/2π (GHz)\n12a\nb\n11 31 51 71 91 111 131151Flake Thickness (nm)\nFIG. 3. Summary of CGT-FMR conditions. a Extracted CGT res-\nonance fields and frequencies from the set of resonators loaded with\nCGT flakes of different thickness. Resonance values are taken from\nthe most prominent peaks in keff. The solid curve is calculated us-\ning the Kittel formalism presented in10, using same parameters, with\ngCGT =2:18,m0Ms=211:4mT and Ku=3:84\u0002104J=m3.bWider\nmagnetic field range of awhere the CGT flake thickness for the dif-\nferent symbols is indicated by the color gradient given in a.dominately dissipate in the magnonic system, as geff\u001cg.\nOur analysis suggests that the separation of the different\nFMR modes is of the same order of magnitude as the loss rate\n(see SI for additional data). We consider that these are from\nstanding spin wave resonances, commonly observed for thin\nmagnetic films12and with one reported observation in bulk of\nthe vdW material CrI 338. In thin-film magnets under a static\nmagnetic field applied in-plane, the magnetic-dipole interac-\ntion generates two prominent spin wave branches for an in-\nplane momentum, the backward volume spin wave (BVMSW)\nand magnetostatic surface spin wave (MSSW) modes39,40.\nThese spin wave modes have different dispersion relations,\nhaving higher (MSSWs) and lower (BVMSWs) resonance\nfrequencies with respect to that of the uniform FMR mode.\nWe calculate the distance of these standing spin-wave modes\nbased on magnetic parameters of bulk CGT as well as the lat-\neral dimensions of the flakes (see SI for more details). We\ncan find spin waves having a frequency separation within\n100MHz and 200MHz (3 :3mT to 6 :6mT in magnetic field\nunits), which are consistent with our experimental observa-\ntion in terms of its mode separation. However, the irregular\nshape of the CGT flakes renders exact calculations of spin\nwave mode frequencies very challenging. We also consid-\nered a possibility that each layer of CGT might have different\nmagnetic parameters (e.g. chemical inhomogeneity), and thus\nproducing different individual resonance modes. Our numer-\nical simulations based on atomistic spin dynamics14,15rule\nout this possibility, as resonance modes from individual lay-\ners average to a single mode as soon as a fraction of 10% of\ninter-layer exchange coupling is introduced (see SI for more\ndetails). Therefore, we speculate that the multiple mode na-\nture we observe in our experiments is likely originating from\nintrinsic properties of the CGT flakes.\nFigure 3 shows the extracted wFRM as a function of BFMR\nfor each resonator–CGT-flake hybrid. The experimental val-\nues are in excellent agreement with a curve calculated by the\nKittel equation with magnetic parameters for bulk CGT10,\nfrom which the data exhibits a standard deviation of less than\n5%. This agreement, achieved by independent characteri-\nzations of 23 CGT flakes measured by superconducting res-\nonators, is experimental evidence that the magnetic parame-\nters that determine the dispersion of wFRM (BFMR), i.e. the\nCGT g-factor gCGT, saturation magnetization Msand uniaxial\nanisotropy Ku, exhibit little thickness dependence in exfoli-\nated CGT flakes, and are not disturbed by the transfer onto\nthe resonator structure. We note, that this demonstrates that\nvdW magnetic materials are particularly attractive for device\napplications, as they are less prone for contamination from\nexfoliation.\nFinally, we present our analysis of kefffor a resonator with\na 11\u00061:8nm CGT flake in Fig.4. With the thickness of a\nsingle layer of CGT being 0 :7nm9, this flake consists of 15\nmonolayers and is the thinnest in our series. Figure 4 (a) and\n(b) show wresandkeffas a function of B0, respectively. While\nthe response of the CGT flake shows a prominent signature\ninkeff, the CGT FMR is considerably more subtle in wres.\nThis highlights the excellent sensitivity of the high-Q super-\nconducting resonators in our study. kefffeatures five well-5\nseparated peaks with the main peak at B0=547mT, which\nenables us to perform a single-peak fully optimized analy-\nsis for each, in contrast to our multi-step analysis for the re-\nmainder of the devices. We assume the additional peaks are\nBVMSW modes, as discussed in the previous section. How-\never, the splitting is about four times larger than compared to\nall other investigated devices, which would result in a signifi-\ncantly shorter wavelength. Thickness steps can lead to a wave-\nlength down-conversion13, however, due to the irregular shape\nandB1inhomogeneities it is difficult to exactly calculate the\nspin wave frequencies (see SI for further details). From the\nmain peak profile, we extract geff=2p=3:61\u00060:09MHz,\ng=2p=126:26\u00068:5MHz and k0=2p=0:92\u00060:05MHz. We\ncompare the experimental value of geffwith a numerically cal-\nculated geff;simu, using the dimensions of the CGT flake de-\ntermined by AFM measurements (see SI for details). The\ncalculation yields geff;simu=2p=8:94MHz, lying within the\nsame order of magnitude. The overestimation is likely due\nto in-perfect experimental conditions, like non-optimal place-\nment of the flake, uncertainties in the thickness and dimen-\nsion determination as well as excluding the additional modes\nin the calculation (see SI). With g\u001dgeffandC=0:11, the\nhybrid system is in the weak coupling regime13, but due to\nthe highly sensitive resonator with its small k0the response\nfrom the magnon system can still be detected. With the ex-\ntracted g=2pwe can give an upper limit of the Gilbert damp-\ning in CGT, by calculating aupper =g=wFMR. We find aupper as\n0:021\u00060:002, which is comparable to other transition metal\nmagnetic materials44, and is in very good agreement with a\npreviously reported effective Gilbert damping parameter de-\ntermined by laser induced magnetization dynamics45. Here,\nwe emphasise that the actual Gilbert damping value is lower\ndue to a finite, extrinsic inhomogeneous broadening contribu-\ntion.\nWe further use these results to benchmark the sensitivity\nof our measurement techniques. The detection limit is given\nby comparing the main peak height characterised by g2\neff=g\nand the median noise amplitude which is 18kHz in Fig. 4 (b)\nwhere g2\neff=2pg= 103 kHz. By assuming the same lateral di-\nmensions and scale the thickness down to a single monolayer,\nwhile keeping gconstant, we calculate the expected signal re-\nduction numerically by geff;simu;1ML=geff;simu;15ML to 0.26. We\nobtain (0:26geff)2=2pg=7kHz for the monolayer limit. Al-\nthough this suggests the noise amplitude is greater than the\nexpected peak amplitude, we can overturn this condition by\nimproving the coupling strength by optimising the resonator\ndesign, enhancing the exfoliation and flake transfer as well as\nby reducing the noise level by averaging a number of mul-\ntiple scans. Superconducting resonators with mode volumes\nof about 10 \u0016m3have been realised46, a reduction of 2 orders\nof magnitude compared to our current design. This would\ntranslate to an order of magnitude improvement in geff. Fur-\nthermore, this flake covers about 4% of the resonator. By\nassuming maximised coverage a 5 times enhancement of geff\ncan be achieved. Both approaches would make the detection\nof monolayer flakes possible.\nIn summary, we provide the first demonstration of photon-\nmagnon coupling between a superconducting resonator and\n520 540 600 6400.900.951.001.05\nκeff/2π (MHz)\nMagnetic Field (mT)560 580 620\nωres/2π (MHz)\n122801229012300a\nbFIG. 4. Magnon-photon coupling for the thinnest CGT flake. a\nResonance frequency wresandbeffective loss rate keffas a func-\ntion of magnetic field of a resonator loaded with the thinnest CGT,\nconsisting of 15 ML. The resonator’s loaded quality factor is 6938.\nThe solid orange lines are results a fit to Eq.(2) and (3), respectively.\nThe errorbars in brepresent the standard deviation from the Fano\nresonance lineshape fit to the resonator transmission.\nnm-thick vdW flakes of CGT, using a total of 23 devices\nwith different CGT flakes of thickness from 153nm down to\n11nm. By employing a coupled-harmonic-oscillator model,\nwe extract the coupling strength, magnetic resonance field\nand relaxation rates for both photon and magnon modes in\nour devices. From our semi-broadband experiments, we find\nthat the magnetic properties of exfoliated CGT flakes are ro-\nbust against the transfer process, with a standard deviation of\nless than 5% to expected resonance values from bulk param-\neters. Notably, this suggests that vdW magnetic materials can\nbe pre-screened at bulk to identify the most promising mate-\nrial for few layer device fabrication. The upper limit of the\nGilbert damping in the 15 ML thick CGT flake is determined\nto be 0 :021, which is comparable to commonly used ferro-\nmagnetic thin-films such as NiFe and CoFeB and thus mak-\ning CGT attractive for similar device applications. We high-\nlight that the damping parameter is key in precessional mag-\nnetisation switching47,48, auto-oscillations by dc currents49,50,\nand comprehensive spin-orbit transport in vdW magnetic sys-\ntems51. The presented techniques are readily transferable\nto other vdW magnetic systems to study spin dynamics in\natomically-thin crystalline materials. While creating stable\nmagnon polaritons is still an open challenge due to the large\nloss rate gof the CGT magnon system, this work offers an\nimportant approach towards its achievement. There are still\npotential improvements to the measurement sensitivity such\nas resonator mode volume reduction by introducing nm scale\nconstrictions52,53and use of exfoliation/transfer techniques to\nproduce larger flakes to enhance the mode overlap (hence cou-6\npling strength)54,55. With concerted efforts, the formation of\nmagnon polaritons in few layers vdW materials will become\nfeasible.\nMETHODS\nSuperconducting Resonators: The resonators were fab-\nricated by direct laser writing and a metal lift-off process.\nThe individual 5mm \u00025mm chips are scribed from an in-\ntrinsic, high resistivity ( r>5000Wcm) n-type silicon wafer\nof 250 \u0016m thickness. For a well defined lift-off, we use a\ndouble photoresist layer of LOR and SR1805. The resonator\nstructures are transferred into the resist by a Heidelberg Di-\nrect Writer system. After development, \u001850nm NbN are de-\nposited by magnetron sputtering in a SVS6000 chamber, at\na base pressure of 7 \u000210\u00007mbar, using a sputter power of\n200W in an 50:50 Ar/N atmosphere held at 5 \u000210\u00003mbar,\nwith both gas flows set to 50 SCCM28. Finally, the lift-off is\ndone in a 1165 solvent to release the resonator structures.\nCGT Crystal Growth: CGT crystals used in this study\nwere grown via chemical vapour transport. To this end, high-\npurity elemental precursors of Cr (chips, \u001599:995%), Ge\n(powder,\u001599:999%), and Te (shots, 99 :999%) were mixed\nin the molar weight ratio Cr:Ge:Te = 10:13.5:76.5, loaded into\na thick-wall quartz ampule and sealed under the vacuum of\n\u001810\u00005mbar. Then, the ampule was loaded into a two-zone\nfurnace, heated up and kept at 950\u000eC for 1 week to homog-\nenize the precursors. To ensure high-quality growth, the am-\npule was slowly cooled (0 :4\u000eC=h) maintaining a small tem-\nperature gradient between the opposite ends of the ampule.\nOnce the ampule reached 500\u000eC, the furnace was turned off\nallowing the ampule to cool down to room temperature nat-\nurally. The large ( \u00181cm) single-crystalline flakes were ex-\ntracted from the excess tellurium and stored in the inert envi-\nronment.\nCGT Flake Transfer: Devices for this study were made\nvia transfer of single-crystalline thin flakes on top of the super-\nconducting resonators. The flakes were first exfoliated from\nbulk crystals on the clean surface of a home-cured PDMS\n(polydimethylsiloxane, Sylgard 184) substrate. The thickness\nof the CGT flakes on PDMS was estimated through the con-\ntrast variation with transmission optical microscopy. Then,\nthe selected flake was transferred to a resonator. The trans-\nfer was performed in air at room temperature. To minimize\nthe air exposure, the entire process of exfoliation, inspection\nand transfer was reduced to 10-15 min per resonator. For\nthe flakes thicker than 50nm, the strong optical absorption of\nCGT prevented the accurate thickness estimation with optical\ncontrast. 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E.J.G.S. acknowledges computational resources\nthrough CIRRUS Tier-2 HPC Service (ec131 Cirrus Project)\nat EPCC (http://www.cirrus.ac.uk) funded by the University\nof Edinburgh and EPSRC (EP/P020267/1); ARCHER UK Na-\ntional Supercomputing Service (http://www.archer.ac.uk) via\nProject d429. E.J.G.S acknowledges the Spanish Ministry\nof Science’s grant program “Europa-Excelencia” under grant\nnumber EUR2020-112238, the EPSRC Early Career Fellow-\nship (EP/T021578/1), and the University of Edinburgh for\nfunding support. D.S. acknowledges EPSRC funding through\nthe Centre for Doctoral Training in Advanced Characteri-\nsation of Materials (EP/L015277/1) and European Union’s\nHorizon 2020 Research and Innovation program under grantagreement GrapheneCore3, number 881603 and the Depart-\nment for Business, Energy and Industrial Strategy through the\nNPL Quantum Program.\nAUTHOR CONTRIBUTION\nC.W.Z, S.K. and H.K. conceived the experimental project.\nResonator design and optimization was done by J.O’S.,\nO.W.K, C.W.Z and supervised by J.J.L.M. Resonator fabri-\ncation and characterization was done by C.W.Z. CGT crystals\nwere grown by I.A.V . and exfoliated and transferred by I.A.V .\nand N.V .T.T. and supervised by E.G.. D.S. measured AFM on\nthe CGT flakes on the resonators. C.W.Z. performed the ex-\nperiments and the data analysis with input from S.K. and H.K.\nAtomistic spin dynamics simulations were carried out by M.S.\nsupervised by E.J.G.S.. C.W.Z., M.S., I.A.V . and H.K. wrote\nthe manuscript with input from all authors.\nCOMPETING INTERESTS\nThe Authors declare no conflict of interests.\nSupplemental Material - Probing spin dynamics of ultra-thin van der Waals magnets via\nphoton-magnon coupling\nI. MICROWAVE SETUP AND MEASUREMENT\nVNA\nMW out MW in\n-20\ndB+32\ndB\nDUT\nB0Cryostat\nFIG. S1. Microwave delivery and detection setup. Schematic of the microwave delivery and detection circuit. The image shows the coplanar\nwaveguide transmission line. A resonator chip is placed on top of the transmission line for read out. On the right, a schematic layout of the\nresonators on a single chip is shown.2\nFigure S1 shows a schematic of the used microwave measurement setup. We are using a Keysight E5071C vector network\nanalyzer (VNA) to deliver and detect microwaves. The VNA is connected to a low temperature probe, fitted into a closed\ncycle helium cryostat and cooled to a base temperature of about 1 :8K. The microwave signal is transmitted into the cryostat\nand is attenuated by \u000020dB. The attenuator is positioned just before the sample box and provides a thermal anchoring for\nthe center conductor of the coaxial cable to minimize the thermal load onto the sample. The output line is equipped with a\nLow Noise Factory LNC6_20C cryogenic amplifier, operating between 6 \u000020GHz with an average amplification of +32dB.\nThe transmitted and amplified signal is finally detected by the VNA. Figure S1 also shows an image of the coplanar waveguide\ntransmission line PCB, loaded with a resonator ship, of which a schematic shows the resonator layout on a single chip. The\nresonators on the chip are capacitively coupled to the transmission line PCB. Upon resonance the transmission through the\nPCB is reduced, indicating the resonator resonance. The cryostat is equipped with a mechanical rotation stage and prior to the\nmeasurements the superconducting resonators are carefully aligned to the externally applied static magnetic field B0, such that\nthe field is in the plane of the superconductor and along the narrow section of the resonators.\nFigure S2 shows the raw uncalibrated microwave transmission, ranging from 10GHz to 18GHz. The transmission is domi-\nnated by imperfections in our microwave circuitry, masking the small signals from the superconducting resonators. Thus, we\nperformed a simple thru calibration of the microwave transmission to remove contributions from the setup, prior each magnetic\nfield dependent measurement. Here, we exploit the magnetic field tunability of our superconducting resonators. Before calibra-\ntion, we set the frequency range of the measurement. We change the applied magnetic field such that the resonator’s resonance\nfrequency is tuned out of the set frequency range. With a frequency window just showing the transmission of the setup we\nperform the thru calibration. After calibration we set the magnetic field back to its starting value, resulting in a background\ncorrected spectrum with just the resonator feature on it.\n10 11 12 13 14 15 16 17 18-1010\n0\nRaw Transmission |S21| (dB)\nFrequency (GHz)\nFIG. S2. Raw broadband microwave transmission signal. Logarithmic microwave transmission jS21jas a function of frequency between\n10GHz and 18GHz at a temperature of 1 :8K.\nII. RESONATOR CHARACTERIZATION\nIn this study, we fabricate twelve superconducting lumped element resonators on each of three resonator chips were fabricated\nusing the same design (see schematic Fig. 1 (a) in the main text). Prior to transfer of the CGT flakes, we characterized the res-\nonators at a temperature of 1 :8K and zero applied magnetic field, using microwave powers of about \u000080dBm at the resonators,\nwhich is well below the bifurcation limit starting above \u000060dBm. Due to finite fabrication tolerances the resonator parameters\nhave some variation, while some didn’t work at all. However, the targeted resonance frequencies are well reproducible and very\nsimilar for the 3 different chips. We compare the resonator parameters before and after transfer of the CGT flakes and collate the\nparameters in Tab. I. Note, the resonator parameters with the CGT flakes on were obtained with a static magnetic field applied\nin the plane of the superconductor, but far detuned from the CGT FMR. In addition, we add the respective thickness of the flake\non each resonator, acquired from AFM measurements. Here, we give the values of the thickest region of a given flake on a\nresonator, as the thickest region will dominate the FMR signal. Due to the arbitrary shape of exfoliated flakes, some exhibit\nregions of different thickness, as seen e.g. in Fig. S5 (h) and (i).3\nTABLE I. Resonator Parameters\nChip Number wres;before (MHz) QL;before wres;after(MHz) QL;after CGT Thickness (nm)\n1 12165 1978 12063 5733 16.2\u00061.3\n1 13303 7357 13177 4950 -\n1 13968 5575 13860 4679 49.4\u00063.5\n1 14184 6492 14048 5627 153.1\u000623.3\n1 16648 6606 16470 5021 23.5\u00062.5\n1 17431 3215 17237 6826 23.8\u00066.4\n1 17959 7595 17790 3963 26.2\u00064.1\n2 12285 360 12153 7135 49.1\u00069.1\n2 12669 3600 12548 6693 102.8\u00065.6\n2 12782 3448 12648 6557 105.9\u00063.9\n2 13393 4643 13244 4501 34.4\u00064.1\n2 13760 6858 13620 5488 95.9\u00065.9\n2 14395 9048 14201 4139 36.7\u00064.3\n2 16075 7283 - - -\n2 17048 6541 16811 4241 75.5\u00065.4\n3 12043 6114 11899 6044 59.7\u000632.8\n3 12456 2716 12314 6938 11.4\u00061.8\n3 12996 5828 12848 4600 17\u00060.8\n3 13422 6517 13272 5461 89.8\u00067.5\n3 13719 6800 13582 6608 -\n3 14238 9184 14064 5420 73.5\u00068.4\n3 15390 8680 15219 6030 30.5\u00064.2\n3 15821 2386 15604 4769 33.1\u00069.9\n3 16430 7518 16193 5780 30.1\u000638.1\n3 17308 6521 17054 5569 137.9\u00063.4\n3 18111 3542 17870 4643 50.2\u00066.9\nIII. RESONATOR AND COUPLING SIMULATION\nWe use finite element and numerical simulations to optimize our resonator design. Key requirements of our resonators are\na strong resilience to externally applied static magnetic fields and a small mode volume. To achieve a large field resilience we\nreduced the area of the resonator to minimize effects of the magnetic field on the superconducting film. Further, we designed the\nresonators such that they act as lumped element resonators. Here, the resonance frequency is given by the total capacitance and\ninductance of the structure, with wres=1=p\nLC, analogues to a parallel LC circuit. This allows us to locally separate oscillating\nelectric and magnetic fields and also to concentrate the magnetic fields in more confined regions, resulting in very small mode\nvolumes. To verify the lumped element nature of our resonators we performed finite element simulations, using CST Microwave\nStudio. Figure S3 shows the resulting magnitude of the E-field (left side) and H-field (right side) distribution along the resonator\nstructure for the resonator design producing the results shown in Fig. 2 in the main text. The E-field is concentrated along the\nparallel running wire sections, with its strength approaching zero along the narrow wire section. The opposite is the case for the\nH-field, where it is zero along the parallel wire sections and strongly concentrated along the narrow wire section. Note, that the\nH-field magnitude is homogeneous along the whole of the narrow wire section.\nThe CST Microwave Studio at hand allowed us a simulation with perfect electric conductors. This is sufficient to model\nthe general electric and magnetic energy distributions and resonance frequencies, however, not to simulate the corresponding\noscillating magnetic field distribution, created by a superconducting rectangular wire. To this end, we numerically solve the\nBiot-Savart law for a rectangular wire cross-sectionS1, assuming a superconducting current distribution Jx;zS2,\nB1;x;z=m0\n2pZw=2\n\u0000w=2Zd=2\n\u0000d=2J\u0002r\n(x\u0000x0)2+ (z\u0000z0)2dx0dz0; (S1)\nwith the vectors as J= (0;J(x;z);0)Tandr= (x\u0000x0;0;z\u0000z0)Tandm0being the magnetic constant. The integration is performed\nover the cross-section of the wire, of width wand thickness d. We define the wire cross-section in the x-z-plane, with win x-\ndirection and din z-direction. The length of the wire is along the y-direction. For a superconducting wire, the current is\nnot homogeneously distributed over the cross-section of the wire. Current is only flowing on the surface and is exponentially\ndecaying towards the center of the wire. The characteristic length scale is given by the London penetration depth lL. We use the4\nFIG. S3. Finite element simulations of resonator. CST Microwave Studio simulation of the distribution of E-fields and H-fields across the\nresonator structure. The color encoded fields represent the magnitude values.\nfollowing expression for the current distributionS2\nJ(x;z) =J1 \ncoshz0=lL\ncoshd=lL\"\nCcoshx0=l1\ncoshw=l1+1\u0000cosh x0=l2=coshw=l2p\n1\u0000(x0=w)2#\n+J2\nJ1coshx0=lL\ncoshw=lL!\n; (S2)\nwhere\nJ2\nJ1=1:008\ncoshd=lLs\nw=l?\n4\u0003l?=lL\u00000:08301lL=l?;\nC=\u0010\n0:506p\nw=2l?\u00110:75\n;\nl1=lLp\n2lL=l?;\nl2=0:774 l2\nL=l?+0:5152l?;\nl?=lL=2d:\nThe prefactors J1andJ2define the amplitude of the current density and hence the absolute value of the oscillating magnetic field\nB1. We define J1by normalizing the vacuum B1field to the energy density stored in the resonatorS3,S4\n1\n2¯hwres\n2=1\n2m0Z\nB2\n1dV=1\n2m0B2\n1Vm; (S3)\nwith Vmrepresenting the resonator mode volume. The additional factor of1=2on the left hand side of S3 takes into account that\nonly half of the total energy is stored in the magnetic fieldS5. As our resonator design is a quasi 1-dimensional structure we have\nto define boundaries for the mode volume in the x- and z-direction. A common assumption is to use the width of the conductor\nwire wS6. For simplicity, we approximate the x-z-area of the mode distribution with the area of an ellipse. For the last dimension\nwe use the length of the narrow wire section, supported by the CST Microwave Studio simulations (see Fig. S3). In total we find5\nthe mode volume to be Vm= ((p3:0\u0016m\u00022:025\u0016m)\u0000w\u0002d)\u0002300\u0016m=5696\u0016m3. Figure S4 shows the resulting distribution\nof the oscillating magnetic field for the cross-section of the rectangular wire of width w=2\u0016m and thickness d=50nm. The\nmagnitudejB1;x;zjis encoded in the color and the arrows indicate the B1;xandB1;zcomponents of the oscillating field.\n-2000200z (nm)400\n-400\nx (μm)0 -1 1 2 3 -2 -320\n15\n10\n5\n|B| (nT)\n30\n25\nFIG. S4. Cross-section of resonator magnetic field distribution. Calculated magnitude of the magnetic field distribution around the cross-\nsection of a rectangular superconducting wire. The wire cross-section lies in the center, indicated by the grey rectangular. The red arrows show\nthe direction of the magnetic field.\nWith the simulated B1field distribution we can calculate the position dependent single photon - single spin coupling strength\ng0(r)S3,S4for each magnetic moment per unit cell of CGT (ab-plane 0 :68nmS7,S8, along the c-axis 0 :7nmS9). Summation over\nall CGT unit cells Nwithin the mode volume of the resonator results in the collective coupling strength\ngeff=s\nN\nå\ni=1jg0(ri)j2=gCGTmB\n2¯hs\nN\nå\ni=1jB1(ri)j2=gCGTmB\n2¯hNys\nN\nå\ni=1h\n(B2\nx;i+B2\nz;i)i\n: (S4)\nHere, mBis the Bohr magneton, Nyis the number of unit cells along the y-direction and gCGTis the g-factor for CGT for which\na value of 2 :18S10is used. Note, we give the collective coupling strength for spin1=2and for linear polarized microwavesS3. For\nthe calculation of gefffor the resonator loaded with 15 monolayers of CGT we extracted its lateral dimension from the AFM\nmeasurements (see Fig. S4 (g)) to 2 \u0016m along the x-direction and 12 \u0016m along the y-direction. The flake is assumed to lie directly\non top of the superconducting wire without any gap in between. For these values the simulation yields geff=2p=8:94MHz,\nwhich is about a factor 2 :5 larger than the experimentally determined value of 3 :61MHz. The overestimation of the simulation\nmost likely results from non-ideal conditions in the experiment. The corresponding flake lies at the top end of the resonators\nnarrow wire section (see Fig. S5 (g)), where B1is concentrated. The finite element simulations show that in this area the field\nstrength is already declining, resulting in a reduced coupling strength. Further, AFM can overestimate the thickness of a flake\nslightly for when there is a gap between resonator surface and flakeS9. The calculation also not includes the multiple peaks\nobserved in the experiment, which - depending on their real nature - can distribute the magnon density over all resonant peaks.\nNevertheless, we can use the simulation to estimate the signal reduction by scaling down the thickness of the flake to a single\nmonolayer. Reducing the simulation to a single monolayer, while keeping the lateral dimensions, results in geff=2p=2:33MHz,\na reduction by a factor of 0 :26.\nIV. AFM MEASUREMENTS ON CGT FLAKES\nAfter the transfer of the CGT flakes onto the individual resonators and after measuring FMR, we characterized the thickness\nof the flakes by AFM. Figure S5 shows a selection of height profile maps from the three resonator chips, including a height\nprofile along the inductor wire of the resonator (blue line in the AFM profile images in Fig. S5). To extract the thickness we fit\nthe steps in the height profile (red or green lines in the height profiles in Fig. S5). Note, the height values are relative values with\nan arbitrary offset. Figure S5 (g) shows the thinnest flake of this study, where the processed FMR data is shown in Fig. 4 in the\nmain text.6\n10 μm\n5 μm0 5\nx (μm)y (nm)40506070\n5 μmy (nm)\n02040\n0 4\nx (μm)2 6\n5 μmy (nm)\n506070\n0 4\nx (μm)2 680\n8\n5 μm\ny (nm)\n02040\n0 10\nx (μm)5 1560\n2080\ny (nm)050100\n0 10\nx (μm)5\n5 μm\n5 μm\ny (nm)\n02040\n0 10\nx (μm)5 15\n5 μm 10 μmy (nm)101520\n0 6\nx (μm)325\ny (nm)\n03060\n0 20\nx (μm)1090\ny (nm)50100\n0 6\nx (μm)3150 120\n30(c) (b) (a)\n(f) (e) (d)\n(i) (h) (g)9\nFIG. S5. AFM measurements. AFM profile images with respective height profile (above) along the resonator inductor wire (blue and\npurple lines in profile images, with the arrow indicating scan direction). a-cfigures for resonator chip 1 (refer to Tab. I), having resonance\nfrequencies with CGT of 17237MHz, 17790MHz and 16470MHz, respectively. d-ffigures for resonator chip 2 (refer to Tab. I), having\nresonance frequencies with CGT of 13244MHz, 12063MHz and 13620MHz, respectively. g-iimages for resonator chip 3 (refer to Tab. I),\nhaving resonance frequencies with CGT of 12314MHz, 13272MHz and 17054 ;MHz, respectively. The red and green solid lines are fits to the\nheight profiles.\nV. ANALYSIS AND ADDITIONAL FMR DATA\nWe analyze our experimental data, using the model functions (2) and (3) from the main text in a two-step semi-optimized\nfashion. The main intention for this approach is to minimize the number of free parameters in our model functions. In a first7\ncoarse step, we match the collective coupling strength geff;kto fit the experimental data, assume a constant separation between\nthe individual magnon modes at BFMR ;kand the same magnon loss rate gfor all modes and determine the resonator loss rate\nk0from the resonator transmission far detuned from the FMR with the CGT flakes. This results in 3 free parameters for the\nfirst stage of our analysis, the magnon loss rate g,BFMR of the main mode and the constant separation between the BFMR ;k.\nAfter this first step we arrive at a best fit to the envelope of the experimental data, however with not matching amplitudes. In a\nconsecutive second step, we manually optimize the geff;kto arrive at a model in good agreement with wresandkeff(see dashed\nlines in Fig. S6).\nFig. S6 shows additional results from the corresponding FMR measurements performed on the in Fig. S5 showed resonators.\nAs described in the main text, the measurements were performed at a temperature of 1 :8K and recording the microwave trans-\nmissionjS21j2as a function of the static magnetic field. Analyzing the microwave transmission by fitting a Fano resonance\nlineshape to it we extract the effective loss rate of the resonator, interacting with the CGT keff. Figure S6 shows the resulting\nkeffas a function of the magnetic field. In general, the response of the CGT FMR is complex and varies for the different res-\nonators. The resonance lineshape is not well described by just a single Lorentzian and requires multiple peaks to produce a\ngood agreement. For some resonators, keffexhibits obvious peaks, residing on a broader spectrum (see Fig. S6 (c), (f) and (i)).\nTogether with the observation of well and clearly separated peaks for the resonator loaded with the thinnest CGT flake of 11nm,\nwe motivating the multiple peak analysis as presented in the main text. However, as the individual peaks are overlapping for the\nremainder of the resonators we only applied a qualitative analysis.\nκeff/2π (MHz)580 600\nMagnetic Field (mT)62036912\n525 550\nMagnetic Field (mT)5751.61.82.0\n600\n625\n675 700\nMagnetic Field (mT)725246\n7508\n10\n500 525\nMagnetic Field (mT)5501.251.301.35\n5751.40\n520 560\nMagnetic Field (mT)6000.900.951.00\n6401.05\n650 675\nMagnetic Field (mT)7004812\n72516\n675 700\nMagnetic Field (mT)7251.41.61.8\nκeff/2π (MHz)\n750 650 700\nMagnetic Field (mT)750369\n800\n12\n500 520\nMagnetic Field (mT)540123\n560\n(c) (b) (a)\n(f) (e) (d)\n(i) (h) (g)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nFIG. S6. Additional data on magnon-photon coupling of CGT-resonator devices. Results from FMR measurements with effective loss rate\nkeff=2pas a function of the static magnetic field. a-cresults for resonator chip 1 (refer to Tab. I), having resonance frequencies with CGT\nof 17237MHz, 17790MHz and 16470MHz, respectively. d-fresults for resonator chip 2 (refer to Tab. I), having resonance frequencies with\nCGT of 13244MHz, 12063MHz and 13620MHz, respectively. g-iresults for resonator chip 3 (refer to Tab. I), having resonance frequencies\nwith CGT of 12314MHz, 13272MHz and 11899MHz, respectively. The orange solid lines are semi-optimized fits, as described in the main\ntext. The errorbars in the figures represent the standard deviation from the Fano resonance lineshape fit to the respective resonator transmission.\nFigure S7 shows the extracted collective coupling strength geffas a function of the square root of the FMR active volume. We\ndefine the active volume as the overlap of the oscillating magnetic field B1and the CGT flake lying on the resonator. The B1\nfield distribution, discussed in Sec. III, is used to estimate the extend of the B1and is taken as 2 \u0016m. From AFM measurements\nand microscope images we extract the thickness and lateral dimensions of the flakes to calculate the final active volume. As\nthe collective coupling is proportional to the square root of the number of magnetic momentsS3, which are interacting with the\nresonator field, it follows that geffscales linearly with the square root of the active volume. This linear trend is highlighted by\nthe orange solid line in Fig. S7. The majority of the extracted data follows this linear trend very well, corroborating our analysis.\nOnly 3 data points deviate strongly from the rest of the data, which we attribute to significant inhomogeneities in the CGT-flakes,\nmaking the volume estimation inaccurate. These data points are highlighted in red in Fig. S7.8\n0 1 2 3 4 51030\n20Collective Coupling geff/2π (MHz)\nSquare Root of Active FMR Volume ( μm3/2)\nFIG. S7. Scaling of the collective coupling. Collective coupling strength geffas a function of the FMR active CGT-flake volume. The orange\nline highlights the linear trend of geffwith increasing volume. The red symbols are regarded as outliers, as these flakes show inhomogeneities,\nleading to inaccurate volume estimations. The star symbol represents data from the thinnest flake (see data in Fig. 4 in the main text) and the\npentagon symbol data from the 17nm flake (see data in Fig. 2 in the main text) The errorbars give confidence values for the extracted values.\nVI. MAGNETO-STATIC SPIN-WAVE DISPERSION IN THIN-FILM MAGNETS WITH PERPENDICULAR ANISOTROPY\nHere we describe the spin-wave mode frequency in a thin-film magnet with perpendicular anisotropy along the film normal.\nWe consider this at the magnetic-dipole limit where the wavelength is relatively large and the exchange interaction contribution\nto the spin-wave dispersion is neglected. Furthermore, standing spin-wave modes along the thickness direction are also ruled out\nsince these modes only appear at much higher frequencies than the main mode, where we consistently observe additional peaks\nat both higher and lower frequencies from the main mode. The mode (angular) frequency ( w) for wavevector k=0 when we\napply a magnetic field Balong one of the film plane directions can be given by Eq. 3d in Ref.S11as:\n\u0012w\ng\u00132\n=B\u0012\nB+m0Ms\u00002Ku\nMs\u0013\n: (S5)\nHere, g,MsandKuare the gyromagnetic ratio, saturation magnetization and the perpendicular anisotropy energy density, respec-\ntively. Note, that the total field within m0Ms\u00002Ku\nMsis negative for perpendicularly-magnetized films which we consider in this\nsection. Within the magnetic-dipole limit, the demagnetization term m0Msis modified for spin-waves with finite k, depending\non the relative orientation between the Msandkdirections. Here we follow the expression given in Serga et al.S12. For pure\nbackward volume magnetostatic modes where kkMs(illustrated in Fig. S8), the mode frequency becomes:\n\u0012wBVMSW\ng\u00132\n=B\u0012\nB+m0Ms\u00121\u0000e\u0000kt\nkt\u0013\n\u00002Ku\nMs\u0013\n; (S6)\nwhere tis the thickness of the magnet. Note, that this expression is only valid for the case where Msis colinear to B, meaning that\njBj>jm0Ms\u00002Ku\nMsj. To the limit of k!0, the term (1\u0000e\u0000kt)=ktis reduced to unity, consistent to Eq. (S5). When kis nonzero,\nwe can observe that wBVMSW becomes smaller than that for k= 0, exhibiting a negative group velocity for this spin-wave mode.\nAs the opposite extreme where k?Ms(illustrated in Fig. S8), the resonance frequency becomes larger than that for k= 0 and is\ncalled magneto-static surface spin-wave mode. The mode frequency expression for this mode is given by:\n\u0012wMSSW\ng\u00132\n=B\u0012\nB+m0Ms\u00002Ku\nMs\u0013\n+m2\n0M2\ns\u0010\n1\u0000e\u00002kt\u0011\n: (S7)\nHere, m2\n0M2\ns\u0000\n1\u0000e\u00002kt\u0001\nis the spin-wave correction term which goes to zero for k!0 (hence consistent to Eq. (S5)) and\nbecomes positive for k>0, meaning that wMSSW becomes larger as soon as spin-waves gain momentum along this direction.\nWe use these two expressions in an effort to explain the origin of the multiple peaks in our experiments. Figure S8 plots the\ncalculated wBVMSW =2pandwMSSW =2pas a function of wavevector k. The range of wavevector is chosen such that the resulting\nresonance frequencies are within the same order of magnitude as the observed mode splittings in the experiment ( µ100MHz).9\nWavevector (μm-1)6 4 8 10 12 2 012.712.812.9Resonance Frequency (GHz)13.0\n12.6\n12.5\n12.4Kittel\nBVMSW\nMSSWB0B0100 MHz\nFIG. S8. Spin-wave dispersion. Spin-wave resonance frequency for BVMSW (green solid line) and MSSW (yellow solid line) as a function\nof wavevector. The dashed blue line is the resonance frequency of the k=0 main mode. The parameters used are B0=598mT, gCGT =2:18,\nm0Ms=194:3mT and Ku=3:84\u0002104J=m3and a thickness of 17nm. The grey area highlights a 100MHz margin relative to the main mode,\nindicating the order of magnitude of the mode splitting observed in the experiment. The arrows on the right hand side illustrate the relative\nwavevector orientations of the BVMSW and MSSW spin-wave modes with respect to the static magnetic field.\nThe corresponding wavelength to a 100MHz resonance offset to the main mode are about 2 :2\u0016m and 620nm for wBVMSW\nandwMSSW , respectively. These values are within a reasonable scale for our different lateral CGT flake dimensions under\ninvestigation. This suggests that spin-wave modes are likely the origin of the multiple resonance peaks observed.\nThe thinnest CGT flake shows, however, a deviation from this behaviour. We only observe modes at lower frequencies, which\nwould indicate to BVMSW modes. Calculating the respective shortest wavelength results in 225nm, which is significantly\nshorter than for the other devices. We assume that the placement and irregular shape are likely to cause this difference. First,\nthis flake is placed at the very edge of the inductor wire, where the B1field strength is declining (see Fig. S3), reducing the FMR\nactive area. Thickness steps can lead to a wavelength down-conversionS13, however, with the overall irregular shape of the flake\nit is difficult to define a length scale for a standing spin wave mode.\nVII. ATOMISTIC SPIN DYNAMICS SIMULATIONS OF FMR\nTo study the ferromagnetic resonance in CGT we perform atomistic spin dynamics simulationsS14,S15. The magnetic Hamil-\ntonian employed in the simulations is given by:\nH=\u00001\n2å\ni;jSiJi jSj\u0000å\niDi(Si·e)2\u0000å\nimiSi·(B0+B1) (S8)\nwhere i,jrepresent the atoms index, Ji jrepresents the exchange interaction tensor, Dithe uniaxial anisotropy, which for\nCGT is orientated out of plane ( e= (0;0;1)) and B0the external static magnetic field applied in-plane during the ferromagnetic\nresonance simulations and B1=B1sin(2pnt)the oscillating field applied perpendicular with respect to B0. The CGT system\nhas been parameterized from first principle methodsS9, up to the third nearest neighbor intralayer and interlayer exchange. The\nexchange values have also been re-scaled by Gong et al.S9with a 0.72 factor to obtain the experimental TCand multiplied by\nS2to match the magnetic Hamiltonian. The magnetic moment or Cr is considered 3.26 mBS16and the uniaxial anisotropy has\na value of 0 :05 meV as extracted from first principle methodsS9. The parameters used in the simulations are given in Table II.\nFMR calculations have previously been employed for atomistic models, and can reproduced well the variation of linewidth with\ntemperature, for example, in recording media systemsS17. Hence, in the current simulations we use the same setup of frequency\nswept FMRS17and we obtain the spectra by performing a Fourier transform of the magnetisation component parallel to the\noscillating field. Since these calculations are done close to 0K, no averaging is require to reduce the thermal noise. To excite the\nFMR mode, we apply a DC field in-plane of 0.9 T on x-direction and an AC field perpendicular to the DC field, on y-direction.\nThe Fourier transform has been performed for the y-component of magnetisation for 5ns after an initial 1ns equilibration time.\nA thermal bath coupling has been chosen in agreement with the upper limit of the Gilbert damping observed in experiments.\nThe system size we performed FMR on is a 4-layer CGT system, with lateral size of 6 :91nm\u000211:97nm, periodic boundary\nconditions in xy and total of 1600 atoms. The small system size has been used to reduce the computational cost associated10\nQuantity Symbol quantity units\nTimestep ts 0.1 fs\nThermal bath coupling a 0.02\nGyromagnetic ratio ge 1.760859\u00021011rad s\u00001T\u00001\nMagnetic moment mB 3.26S16mB\nUniaxial anisotropy Di 0.05S9meV/link\nSimulation temperature T 0.001 K\nStatic magnetic field B0 0.9, 0.7 T\nOscillating magnetic field amplitude B0 0.001 T\nFMR frequency n varied GHz\nIntralayer exchange, NN J1 2.71S9meV/link\nIntralayer exchange, 2NN J2 - 0.058S9meV/link\nIntralayer exchange, 3NN J3 0.115S9meV/link\nInterlayer exchange, NN Jz\n1-0.036S9meV/link\nInterlayer exchange, 2NN Jz\n20.086S9meV/link\nInterlayer exchange, 3NN Jz\n30.27S9meV/link\nTABLE II. Simulation parameters for FMR on CGT system\n.\nwith FMR simulations. Experiments have showed modified g-factors due to photon-magnon coupling hence hereby we propose\na simple model where the properties of the individual layers have been modified to include different gyromagnetic ratio, as\nillustrated in Fig. S9 a.\nWe can define the resonance frequencies for each magnetic layer using the Kittel equation in the case of in-plane applied field\nwith perpendicular anisotropy B?u:\nw=gp\nB0(B0\u0000B?u) (S9)\nWe next investigate the FMR signal for a few cases assuming the CGT monolayers at low or strong interlayer exchange cou-\nplings J0\nz=0;0:1%;10%;100% Jz, where Jzcorresponds to the pristine interlayer exchange (Fig. S9 b-c). In the low interlayer\nexchange regime ( J0\nz=0;0:1%Jz), the CGT presents multiple peaks with each frequency corresponding to the layer dependent\ngyromagnetic ratio, g-n(g1) =16:81GHz, n(g2) =25:22GHz, n(g3) =33:62GHz. At J0\nz=0:1%J0\nz(Fig. S9 b) we can still\nobserve resonance peaks corresponding to each individual layer. However by increasing the exchange coupling to 10% J0\nzor\nhigher (Fig. S9 c) there is a single FMR peak indicating that the system behave coherently with all layers having the same FMR\nfrequency. The single FMR frequency corresponds to the average magnetic properties of the CGT layers. Small variations of\nthe resonance frequency as function of the inter-layer exchange coupling can be observed which these being correlated to the\ntransition of the system from the multi-peaks regime to a coherent excitation. By calculating the damping of the highest reso-\nnance peaks from a Lorenzian fit, we reobtain the damping corresponding to the input thermal bath coupling, 0 :02 with a relative\ntinny error\u00185%. Overall, the interlayer exchange coupling locks the dynamics of individual layers coherently together without\nallowing multiple frequencies at the FMR signalS18.\n[S1]A. E. Primenko, M. A. Osipov, and I. A. Rudnev, Technical Physics 62, 1346 (2017).\n[S2]L. H. Lee, T. P. Orlando, and W. G. Lyons, IEEE Transactions on Applied Superconductivity 4, 41 (1994).\n[S3]C. W. Zollitsch, K. Mueller, D. P. Franke, S. T. B. Goennenwein, M. S. Brandt, R. Gross, and H. Huebl, Applied Physics Letters 107, 142105 (2015).\n[S4]S. Weichselbaumer, P. Natzkin, C. W. Zollitsch, M. Weiler, R. Gross, and H. Huebl, Physical Review Applied 12, 024021 (2019).\n[S5]R. J. Schoelkopf and S. M. Girvin, Nature 451, 664 (2008).\n[S6]D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J.\nSchoelkopf, Nature 445, 515 (2007).\n[S7]Y . F. Li, W. Wang, W. Guo, C. Y . Gu, H. Y . Sun, L. He, J. Zhou, Z. B. Gu, Y . F. Nie, and X. Q. Pan, Physical Review B 98, 125127 (2018).\n[S8]Y . Sun, R. C. Xiao, G. T. Lin, R. R. Zhang, L. S. Ling, Z. W. Ma, X. Luo, W. J. Lu, Y . P. Sun, and Z. G. Sheng, Applied Physics Letters 112, 072409\n(2018).\n[S9]C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y . Xia, T. Cao, W. Bao, C. Wang, Y . Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature 546,\n265 (2017).\n[S10]S. Khan, C. W. Zollitsch, D. M. Arroo, H. Cheng, I. Verzhbitskiy, A. Sud, Y . P. Feng, G. Eda, and H. Kurebayashi, Physical Review B 100, 134437 (2019).\n[S11]M. Farle, Reports on Progress in Physics 61, 755 (1998).\n[S12]A. A. Serga, A. V . Chumak, and B. Hillebrands, Journal of Physics D: Applied Physics 43, 264002 (2010).\n[S13]J. Stigloher, T. Taniguchi, M. Madami, M. Decker, H. S. Körner, T. Moriyama, G. Gubbiotti, T. Ono, and C. H. Back, Applied Physics Express 11, 053002\n(2018).\n[S14]D. A. Wahab, M. Augustin, S. M. Valero, W. Kuang, S. Jenkins, E. Coronado, I. V . Grigorieva, I. J. Vera-Marun, E. Navarro-Moratalla, R. F. Evans, et al. ,\nAdvanced Materials 33, 2004138 (2021).\n[S15]A. Kartsev, M. Augustin, R. F. Evans, K. S. Novoselov, and E. J. G. Santos, npj Computational Materials 6, 1 (2020).\n[S16]I. A. Verzhbitskiy, H. Kurebayashi, H. Cheng, J. Zhou, S. Khan, Y . P. Feng, and G. Eda, Nature Electronics 3, 460 (2020).11\nFIG. S9. Atomistic simulations. a, Schematic of the crystal structure of CGT with atoms defined by different colours. b,FMR spectra of 4\nlayer CGT where the layers are low interayer exchange coupled (0 ;0:1%J0z, where J0zis the pristine CGT interlayer exchange). c,Similar as b,\nbut with the layers at a strong exchange coupling (10% ;100% J0z). The solid lines in b-crepresent a Lorenzian fit to the numerical data.\n[S17]M. Strungaru, S. Ruta, R. F. Evans, and R. W. Chantrell, Physical Review Applied 14, 014077 (2020).\n[S18]Data inputs/plots utilised for Supplementary Figure S7 (atomistic simulations) can be found at the following GitHub repository."    },    {        "title": "2206.04899v1.Spin_Pumping_into_Anisotropic_Dirac_Electrons.pdf",        "content": "Spin Pumping into Anisotropic Dirac Electrons\nTakumi Funato1;2, Takeo Kato3, Mamoru Matsuo2;4;5;6\n1Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n3Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan and\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: June 13, 2022)\nWe study spin pumping into an anisotropic Dirac electron system induced by microwave irra-\ndiation to an adjacent ferromagnetic insulator theoretically. We formulate the Gilbert damping\nenhancement due to the spin current \rowing into the Dirac electron system using second-order\nperturbation with respect to the interfacial exchange coupling. As an illustration, we consider the\nanisotropic Dirac system realized in bismuth to show that the Gilbert damping varies according to\nthe magnetization direction in the ferromagnetic insulator. Our results indicate that this setup can\nprovide helpful information on the anisotropy of the Dirac electron system.\nI. INTRODUCTION\nIn spintronics, spin currents are crucial in using elec-\ntrons' charge and spin. Spin pumping, the spin current\ngeneration of conduction electrons from nonequilibrium\nmagnetization dynamics at magnetic interfaces, is a pop-\nular method for generating and manipulating spin cur-\nrents. In previous experimental reports on spin pumping,\nthe enhancement of Gilbert damping in ferromagnetic\nresonance (FMR) was observed due to the loss of angu-\nlar momentum associated with the spin current injection\ninto the nonmagnetic layer adjacent to the ferromagnetic\nlayer1{9. Mizukami et al. measured the enhancement of\nthe Gilbert damping associated with the adjacent non-\nmagnetic metal. They reported that the strong spin-orbit\ncoupling in the nonmagnetic layer strictly a\u000bected the\nenhancement of the Gilbert damping3{5. Consequently,\nelectric detection by inverse spin Hall e\u000bect, in which the\ncharge current is converted from the spin current, led to\nspin pumping being used as an essential technique for\nstudying spin-related phenomena in nonmagnetic mate-\nrials10{24. Saitoh et al. measured electric voltage in a\nbilayer of Py and Pt under microwave application. They\nobserved that charge current converted because of inverse\nspin Hall e\u000bect from spin current injected by spin pump-\ning11.\nIn the \frst theoretical report on spin pumping, Berger\npredicted an increase in Gilbert damping due to the spin\ncurrent \rowing interface between the ferromagnetic and\nnonmagnetic layers25,26. Tserkovnyak et al. calculated\nthe spin current \rowing through the interface27{29based\non the scattering-matrix theory and the picture of adi-\nabatic spin pumping30{32. They introduced a complex\nspin-mixing conductance that characterizes spin trans-\nport at the interfaces based on spin conservation and no\nspin loss. The spin mixing conductance can represent\nthe spin pumping-associated phenomena and is quanti-\ntatively evaluated using the \frst principle calculation33.\nNevertheless, microscopic analysis is necessary to under-stand the detailed mechanism of spin transport at the in-\nterface34{44. It was clari\fed that spin pumping depends\non the anisotropy of the electron band structure and spin\ntexture. Spin pumping is expected to be one of the probes\nof the electron states41{44.\nBismuth has been extensively studied because of its at-\ntractive physical properties, such as large diamagnetism,\nlargeg-factor, high e\u000ecient Seebeck e\u000bect, Subrikov-de\nHaas e\u000bect, and de Haas-van Alphen e\u000bect45,46. The\nelectrons in the conduction and valence bands near the\nL-point in bismuth, which contribute mainly to the vari-\nous physical phenomena, are expressed as e\u000bective Dirac\nelectrons. Thus, electrons in bismuth are called Dirac\nelectrons45{47. The doping antimony to bismuth is known\nto close the gap and makes it a topological insulator48,49.\nBecause of its strong spin-orbit interaction, bismuth has\nattracted broad attention in spintronics as a high e\u000ecient\ncharge-to-spin conversion material50{55. The spin current\ngeneration at the interface between the bismuth oxide\nand metal has been studied since a signi\fcant Rashba\nMicrowaveDirac electron\nsystem\nInterfacial\nexchange\nFerromagnetic\ninsulator\nFIG. 1. Schematic illustration of a bilayer system composed\nof the Dirac electron system and ferromagnetic insulator. The\napplied microwave excited precession of the localized spin in\nthe ferromagnetic insulator and spin current is injected into\nthe Dirac electron system.arXiv:2206.04899v1  [cond-mat.mes-hall]  10 Jun 20222\nspin-orbit interaction appears at the interface56. The\nspin injection into bismuth was observed due to spin\npumping from yttrium iron garnet or permalloy57{59.\nNevertheless, microscopic analysis of spin pumping into\nbismuth has not been performed. The dependence of the\nspin pumping on the crystal and band structure of bis-\nmuth remains unclear.\nThis study aims at a microscopic analysis of spin in-\njection due to spin pumping into an anisotropic Dirac\nelectron system, such as bismuth, and investigates the\ndependence of spin pumping on the band structure. We\nconsider a bilayer system comprising an anisotropic Dirac\nelectron system and a ferromagnetic insulator where a\nmicrowave is applied (see Fig. 1). The e\u000bect of the inter-\nface is treated by proximity exchange coupling between\nthe Dirac electron spins and the localized spins of the\nferromagnetic insulator34{44. We calculate the Gilbert\ndamping enhancement due to spin pumping from the fer-\nromagnetic insulator into the Dirac electron system up to\nthe second perturbation of the interfacial exchange cou-\npling. For illustarion, we calculate the enhancement of\nthe Gilbert damping for an anisotropic Dirac system in\nbismuth.\nThis paper is organized as follows: Sec. II describes\nthe model. Sec. III shows the formulation of the Gilbert\ndamping enhancement and discuss the e\u000bect of the inter-\nfacial randomness on spin pumping. Sec. IV summarizes\nthe results and demonstration of the Gilbert damping\nenhancement in bismuth. Sec. V presents the conclu-\nsion. The Appendices show the details of the calcula-\ntion. Appendix A de\fnes the magnetic moment of elec-\ntrons in a Dirac electron system. Appendix B provides\nthe detailed formulation of the Gilbert damping modu-\nlation, and Appendix C presents the detailed derivation\nof Gilbert damping modulation.\nII. MODEL\nWe consider a bilayer system composed of an\nanisotropic Dirac electron system and a ferromagnetic\ninsulator under a static magnetic \feld. We evaluate a\nmicroscopic model whose Hamiltonian is given as\n^HT=^HD+^HFI+^Hex; (1)\nwhere ^HD,^HFI, and ^Hexrepresent an anisotropic Dirac\nelectron system, a ferromagnetic insulator, and an inter-\nfacial exchange interaction, respectively.\nA. Anisotropic Dirac system\nThe following Wol\u000b Hamiltonian models the\nanisotropic Dirac electron system46,47,50:\n^HD=X\nkcy\nk(\u0000~k\u0001v\u001a2+ \u0001\u001a3)ck; (2)where 2\u0001 (6= 0) is the band gap, cy\nk(ck) is the electrons'\nfour-component creation (annihilation) operator, and v\nis the velocity operator given by vi=P\n\u000bwi\u000b\u001b\u000bwith\nwi\u000bbeing the matrix element of the velocity operator.\n\u001b= (\u001bx;\u001by;\u001bz) are the Pauli matrices in the spin space\nand\u001a= (\u001a1;\u001a2;\u001a3) are the Pauli matrices specifying the\nconduction and valence bands.\nFor this anisotropic Dirac system, the Matsubara\nGreen function of the electrons is given by\ngk(i\u000fn) =i\u000fn+\u0016\u0000~~k\u0001\u001b\u001a2+ \u0001\u001a3\n(i\u000fn+\u0016)2\u0000\u000f2\nk; (3)\nwhere\u000fn= (2n+ 1)\u0019=\fis the fermionic Matsubara fre-\nquencies with nbeing integers, \u0016(>\u0001) is the chem-\nical potential in the conduction band ~kis de\fned by\n~k\u0001\u001b=~k\u000b\u001b\u000b=k\u0001v, and\u000fkis the eigenenergy given\nby\n\u000fk=p\n\u00012+ (~kiwi\u000b)2=q\n\u00012+~2~k2: (4)\nThe density of state of the Dirac electrons per unit cell\nper band and spin is givcen by\n\u0017(\u000f) =n\u00001\nDX\nk;\u0015\u000e(\u000f\u0000\u0015\u000fk); (5)\n=j\u000fj\n2\u00192~3s\n\u000f2\u0000\u00012\n\u00013det\u000bij\u0012(j\u000fj\u0000\u0001); (6)\nwherenDis the number of unit cells in the system and \u000bij\nis the inverse mass tensor near the bottom of the band,\nwhich characterize the band structure of the anisotropic\nDirac electron system:\n\u000bij=1\n~2@2\u000fk\n@ki@kj\f\f\f\f\nk=0=1\n\u0001X\n\u000bwi\u000bwj\u000b: (7)\nThe spin operator can be de\fned as\n^sq=X\nkcy\nk\u0000q=2sck+q=2; (8)\nsi=m\n\u0001Mi\u000b\u001a3\u001b\u000b;(i=x;y;z ); (9)\nwhereMi\u000bare the matrix elements of the spin magnetic\nmoment given as50,51\nMi\u000b=\u000f\u000b\f\r\u000fijkwi\fwj\r=2: (10)\nThe detailed derivation of the spin magnetic moment can\nbe found in Appendix A.\nB. Ferromagnetic insulator\nThe bulk ferromagnetic insulator under a static mag-\nnetic \feld is described by the quantum Heisenberg model\nas\n^HFI=\u00002JX\nhi;jiSi\u0001Sj\u0000g\u0016BhdcX\niSX\ni; (11)3\nFIG. 2. Relation between the original coordinates ( x;y;z ) and\nthe magnetization-\fxed coordinates ( X;Y;Z ). The direction\nof the ordered localized spin hSi0is \fxed to the X-axis.\u0012is\nthe polar angle and \u001eis the azimuthal angle.\nwhereJis an exchange interaction, gis g-factor of the\nelectrons,\u0016Bis the Bohr magnetization, and hi;jirepre-\nsents the pair of nearest neighbor sites. Here, we have in-\ntroduced a magnetization-\fxed coordinate ( X;Y;Z ), for\nwhich the direction of the ordered localized spin hSi0is\n\fxed to the X-axis. The localized spin operators for the\nmagnetization-\fxed coordinates are related to the ones\nfor the original coordinates ( x;y;z ) as\n0\n@Sx\nSy\nSz1\nA=R(\u0012;\u001e)0\n@SX\nSY\nSZ1\nA; (12)\nwhereR(\u0012;\u001e) =Rz(\u001e)Ry(\u0012) is the rotation matrix com-\nbining the polar angle \u0012rotation around the y-axisRy(\u0012)\nand the azimuthal angle \u001erotation around the z-axis\nRz(\u001e), given by\nR(\u0012;\u001e) =0\n@cos\u0012cos\u001e\u0000sin\u001esin\u0012cos\u001e\ncos\u0012sin\u001ecos\u001esin\u0012sin\u001e\n\u0000sin\u0012 0 cos \u00121\nA:(13)\nBy applying the spin-wave approximation, the spin op-\nerators are written as S\u0006\nk=SY\nk\u0006iSZ\nk=p\n2Sbk(by\nk) and\nSX\nk=S\u0000by\nkbkusing magnon creation/annihilation op-\nerators,by\nkandbk. Then, the Hamiltonian is rewritten\nas\n^HFI=X\nk~!kby\nkbk; (14)\nwhere ~!k=Dk2+~!0withD=zJSa2being the spin\nsti\u000bness and zbeing the number of the nearest neighbor\nsites, and ~!0=g\u0016Bhdcis the Zeeman energy.C. Interfacial exchange interaction\nThe proximity exchange coupling between the electron\nspin in the anisotropic Dirac system and the localized\nspin in the ferromagnetic insulator is modeled by\n^Hex=X\nq;k(Tq;k^s+\nqS\u0000\nk+ h.c.); (15)\nwhereTq;kis a matrix element for spin transfer through\nthe interface and ^ s\u0006\nq= ^sY\nq\u0006i^sZ\nqare the spin ladder\noperators of the Dirac electrons. According to the re-\nlation between the original coordinate ( x;y;z ) and the\nmagnetization-\fxed coordinate ( X;Y;Z ), the spin oper-\nators of the Dirac electrons are expressed as\n0\n@sX\nsY\nsZ1\nA=R\u00001(\u0012;\u001e)0\n@sx\nsy\nsz1\nA; (16)\nwhereR\u00001(\u0012;\u001e) =Ry(\u0012)Rz(\u0000\u001e) is given by\nR\u00001(\u0012;\u001e) =0\n@cos\u0012cos\u001ecos\u0012sin\u001e\u0000sin\u0012\n\u0000sin\u001e cos\u001e 0\nsin\u0012cos\u001esin\u0012sin\u001ecos\u00121\nA:(17)\nThe spin ladder operators are given by\ns+=m\n\u0001aiMi\u000b\u001b\u000b; s\u0000=m\n\u0001a\u0003\niMi\u000b\u001b\u000b; (18)\nwhereai(i=x;y;z ) are de\fned by\n0\n@ax\nay\naz1\nA=0\n@\u0000sin\u001e+isin\u0012cos\u001e\ncos\u001e+isin\u0012sin\u001e\nicos\u00121\nA: (19)\nIII. FORMULATION\nApplying a microwave to the ferromagnetic insulator\nincludes the localized spin's precession. The Gilbert\ndamping constant can be read from the retarded magnon\nGreen function de\fned by\nGR\nk(!) =\u0000i\n~Z1\n0dtei(!+i\u000e)th[S+\nk(t);S\u0000\nk]i; (20)\nwithS+\nk(t) =ei^HT=~S+\nke\u0000i^HT=~being the Heisenberg\nrepresentation of the localized spin, since one can prove\nthat the absorption rate of the microwave is proportional\nto ImGR\nk=0(!) (see also Appendix B). By considering the\nsecond-order perturbation with respect to the matrix el-\nement for the spin transfer Tq;k, the magnon Green func-\ntion is given by34{44\nGR\n0(!) =2S=~\n(!\u0000!0) +i(\u000b+\u000e\u000b)!: (21)\nHere, we introduced a term, i\u000b!, in the denominator\nto express the spin relaxation within a bulk FI, where4\n\u000bindicates the strength of the Gilbert damping. The\nenhancement of the damping, \u000e\u000b, is due to the adjacent\nDirac electron system, calculated by\n\u000e\u000b=2S\n~!X\nqjTq;0j2Im\u001fR\nq(!); (22)\nwhere\u001fR\nq(!) is the retarded component of the spin sus-\nceptibility (de\fned below). We assume that the FMR\npeak described by Im GR\nk=0(!) is su\u000eciently sharp, i.e.,\n\u000b+\u000e\u000b\u001c1. Then, the enhancement of the Gilbert damp-\ning can be regarded as almost constant around the peak\n(!'!0), allowing us to replace !in\u000e\u000bwith!0.\nThe retarded component of the spin susceptibility for\nthe Dirac electrons:\n\u001fR\nq(!) =i\n~Z1\n\u00001dtei(!+i\u000e)t\u0012(t)h[s+\nq(t);s\u0000\n\u0000q]i: (23)\nThe retarded component of the spin susceptibility is\nderived from the following Matsubara Green function\nthrough analytic continuation i!l!~!+i\u000e:\n\u001fq(i!l) =Z\f\n0d\u001cei!l\u001ch^s+\nq(\u001c)^s\u0000\n\u0000qi; (24)\nwhere!l= 2\u0019l=\f is the bosonic Matsubara frequency\nwithlbeing integers. According to Wick's theorem,\nthe Matsubara representation of the spin susceptibility\nis given by\n\u001fq(i!l)\n=\u0000\f\u00001X\nk;i\u000fntr[s+gk+q(i\u000fn+i!l)s\u0000gk(i\u000fn)];(25)\nwhereP\ni\u000fnindicates the sum with respect to the\nfermionic Matsubara frequency, \u000fn= (2\u0019+ 1)n=\f. The\nimaginary part of the spin susceptibility is given by\nIm\u001fR\nq(!) =\u0000\u0019F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015\n\u0002h\nf(\u00150\u000fk+q)\u0000f(\u0015\u000fk)i\n\u000e(~!\u0000\u00150\u000fk+q+\u0015\u000fk);(26)\nwheref(\u000f) = (e\f(\u000f\u0000\u0016)+ 1)\u00001is the Fermi distribution\nfunction,\u0015=\u0006is a band index (see Fig. 3), and F(\u0012;\u001e)\nis the dimensionless function which depends on the di-\nrection of the ordered localized spin, de\fned by\nF(\u0012;\u001e) =\u00122m\n\u0001\u00132X\n\u000baiMi\u000ba\u0003\njMj\u000b: (27)\nFor detailed derivation, see Appendix C.\nIn this paper, we model the interfacial spin transfer as a\ncombination of the clean and dirty processes. The former\ncorresponds to the momentum-conserved spin transfer\nand the latter to the momentum-nonconserved one41,44.\nBy averaging over the position of the localized spin at\nFIG. 3. Schematic illustration of the band structure of the\nanisotropic Dirac electron system. The red band represents\nthe conduction band with \u0015= +, and the blue band repre-\nsents the valence band with \u0015=\u0000. The chemical potential\nis in the conduction band.\nthe interface, we can derive the matrix elements of the\ninterfacial spin-transfer process as\njTq;0j2=T2\n1\u000eq;0+T2\n2; (28)\nwhereT1andT2are the averaged matrix elements con-\ntributing to the clean and dirty processes, respectively.\nThen, the enhancement of the Gilbert damping is given\nby\n\u000e\u000b=2S\n~!F(\u0012;\u001e)n\nT1Im ~\u001fR\nuni(!0) +T2Im ~\u001fR\nloc(!0)o\n;\n(29)\nwhere\u001fR\nuni(!) and\u001fR\nuni(!) are the local and uniform spin\nsusceptibilities de\fned by\n~\u001fR\nloc(!0) =F\u00001(\u0012;\u001e)X\nq\u001fR\nq(!0); (30)\n~\u001fR\nuni(!0) =F\u00001(\u0012;\u001e)\u001fR\n0(!0); (31)\nrespectively. From Eq. (26), their imaginary parts are\ncalculated as\nIm ~\u001fR\nloc(!0) =\u0000\u0019n2\nDZ\nd\u000f\u0017(\u000f)\u0017(\u000f+~!0)\n\u0002\u00141\n2+2\u00012+\u000f2\n6\u000f(\u000f+~!0)\u0015h\nf(\u000f+~!0)\u0000f(\u000f)i\n;\n(32)\nIm ~\u001fR\nuni(!0) =\u0000\u0019nD\u0017\u0000~!0\n2\u0001~2!2\n0\u00004\u00012\n3~2!2\n0\n\u0002h\nf(~!0\n2)\u0000f(\u0000~!0\n2)i\n: (33)\nThe enhancement of the Gilbert damping, \u000e\u000b, depends\non the direction of the ordered localized spin through the5\nFIG. 4. FMR frequency dependence of the (a) local\nand (b) uniform spin susceptibilities. The local spin sus-\nceptibility is normalized by \u0019n2\nD\u00172\n0and scaled by 106, and\nthe uniform spin susceptibility is normalized by \u0019nD\u00170with\n\u00170\u00111=2\u00192~3p\ndet\u000bij. Note that kBis the Boltzmann con-\nstant. The line with kBT=\u0001 = 0:001 is absent in (a) because\nthe local spin susceptibility approaches zero at low tempera-\nture.\ndimensionless function F(\u0012;\u001e) regardless of the interfa-\ncial condition.\nBy contrast, the FMR frequency dependence of \u000e\u000bre-\n\rects the interfacial condition; for a clean interface, it is\ndetermined mainly by Im \u001fR\nuni(!0), whereas for a dirty\ninterface, it is determined by Im \u001fR\nloc(!0). The FMR fre-\nquency dependence of the local and uniform spin sus-\nceptibilities, Im \u001fR\nloc(!0) and Im\u001fR\nuni(!0), are plotted in\nFigs. 4 (a) and (b), respectively. The local and uniform\nspin susceptibilities are normalized by \u0019n2\nD\u00172\n0and\u0019nD\u00170,\nrespectively, where \u00170\u00111=2\u00192~3p\ndet\u000bijis de\fned. In\nthe calculation, the ratio of the chemical potential to the\nenergy gap was set to \u0016=\u0001'4:61, which is the value in\nthe bismuth46. According to Fig. 4 (a), the local spin sus-\nceptibility increases linearly with the frequency !in the\nlow-frequency region. This !-linear behavior can be re-\nproduced analytically for low temperatures and ~!\u001c\u0016:\nIm ~\u001floc(!0)'~!0\u0019\n2n2\nD[\u0017(\u0016)]2\u0014\n1 +2\u00012+\u00162\n3\u00162\u0015\n:(34)Fig. 4 (b) indicates a strong suppression of the uniform\nspin susceptibility below a spin-excitation gap ( !0<2\u0016).\nThis feature can be checked by its analytic form at zero\ntemperature:\nIm ~\u001fR\nuni(!0) =\u0019nD\u0017\u0000~!0\n2\u0001~2!2\n0\u00004\u00012\n3~2!2\n0\u0012(~!0\u00002\u0016):\n(35)\nThus, the FMR frequency dependence of the enhance-\nment of the Gilbert damping depends on the interfacial\ncondition. This indicates that the measurement of the\nFMR frequency dependence may provide helpful infor-\nmation on the randomness of the junction.\nIV. RESULT\nWe consider bismuth, which is one of the anisotropic\nDirac electron systems45,46,52,60,61. The crystalline struc-\nture of pure bismuth is a rhombohedral lattice with the\nspace group of R\u00163msymmetry, see Figs. 5 (a) and (b).\nIt is reasonable to determine the Cartesian coordinate\nsystem in the rhombohedral structure using the trigonal\naxis withC3symmetry, the binary axis with C2symme-\ntry, and the bisectrix axis, which is perpendicular to the\ntrigonal and binary axes. Hereafter, we choose the x-axis\nas the binary axis, the y-axis as the bisectrix axis, and\nthez-axis as the trigonal axis. Note that the trigonal, bi-\nnary, and bisectrix axes are denoted as [0001], [1 \u0016210], and\n[10\u001610], respectively, where the Miller-Bravais indices are\nused. The bismuth's band structure around the Fermi\nsurface consists of three electron ellipsoids at L-points\nand one hole ellipsoid at the T-point. It is well known\nthat the electron ellipsoids are the dominant contribu-\ntion to the transport phenomena since electron's mass\nis much smaller than that of the hole, see Fig. 5 (c).\nTherefore, the present study considers only the electron\nsystems at the L-points. The electron ellipsoids are sig-\nni\fcantly elongated, with the ratio of the major to minor\naxes being approximately 15 : 1. Each of the three elec-\ntron ellipsoids can be converted to one another with 2 \u0019=3\nrotation around the trigonal axis. The electron ellipsoid\nalong the bisectrix axis is labeled as e1, and the other\ntwo-electron ellipsoids are labeled e2 ande3. The in-\nverse mass tensor for the e1 electron ellipsoids is given\nby\n\u000b$\ne1=0\nB@\u000b10 0\n0\u000b2\u000b4\n0\u000b4\u000b31\nCA: (36)\nThe inverse mass tensor of the electron ellipsoids e2 and\ne3 are obtained by rotating that of e1 by 2\u0019=3 rotation6\nas below:\n\u000b$\ne2;e3=1\n40\nBB@\u000b1+ 3\u000b2\u0006p\n3(\u000b1\u0000\u000b2)\u00062p\n3\u000b4\n\u0006p\n3(\u000b1\u0000\u000b2) 3\u000b1+\u000b2\u00002\u000b4\n\u00062p\n3\u000b4\u00002\u000b4 4\u000b31\nCCA:\n(37)\nLet us express the dimensionless function F(\u0012;\u001e) rep-\nresenting the localized spin direction dependence of the\ndamping enhancement on the inverse mass tensors.\nF(\u0012;\u001e) =\u00122m\n\u0001\u00132X\n\u000bh\n(sin2\u001e+ sin2\u0012cos2\u001e)M2\nx\u000b\n+(cos2\u001e+ sin2\u0012sin2\u001e)M2\ny\u000b\n+ cos2\u0012(M2\nz\u000b\u0000sin 2\u001eMx\u000bMy\u000b)\n+ sin 2\u0012Mz\u000b(Mx\u000bcos\u001e+My\u000bsin\u001e)i\n: (38)\nHere, we use the following calculations:\nX\n\u000bM2\nx\u000b=\u00012\n4(\u000byy\u000bzz\u0000\u000b2\nyz)total=\u00012\n4m2\u0016\u0014?;(39)\nX\n\u000bM2\ny\u000b=\u00012\n4(\u000bzz\u000bxx\u0000\u000b2\nzx)total=\u00012\n4m2\u0016\u0014?;(40)\nX\n\u000bM2\nz\u000b=\u00012\n4(\u000bxx\u000byy\u0000\u000b2\nxy)total=\u00012\n4m2\u0016\u0014k;(41)\nX\n\u000bMi\u000bMj\u000b=\u00012\n4(\u000bik\u000bjk\u0000\u000bij\u000bkk)total= 0;(42)\nwherei;j;k are cyclic. (\u0001\u0001\u0001)totalrepresents the summa-\ntion of the contributions of the three electron ellipsoids,\nand \u0016\u0014k, \u0016\u0014?(>0) are the total Gaussian curvature of the\nthree electron ellipsoids normalized by the electron mass\nm, given by\n\u0016\u0014k= 3m2\u000b1\u000b2; (43)\n\u0016\u0014?=3\n2m2[(\u000b1+\u000b2)\u000b3\u0000\u000b2\n4]: (44)\nHence, the dimensionless function Fis given by\nF(\u0012) = (1 + sin2\u0012)\u0016\u0014?+ cos2\u0012\u0016\u0014k: (45)\nThe results suggest that the variation of the damping\nenhancement depends only on the polar angle \u0012, which is\nthe angle between the direction of the ordered localized\nspinhSi0and the trigonal axis. It is also found that the \u0012\ndependence of the damping enhancement originates from\nthe anisotropy of the band structure. The dimensionless\nfunctionF(\u0012) is plotted in Fig. 6 by varying the ratio\nof the total Gaussian curvatures x= \u0016\u0014?=\u0016\u0014k, which cor-\nresponds to the anisotropy of the band structure. Fig-\nure 6 shows that the \u0012-dependence of the damping en-\nhancement decreases with smaller xand the angular de-\npendence vanishes in an isotropic Dirac electron system\nBinaryBisectrixTrigonal\ne�e�e�(c)\nBinary(x)(a)\nBisectrix(y)Trigonal(z)\nBinaryBisectrixTrigonal(b)FIG. 5. (a) The rhombohedral lattice structure of bismuth.\nThex-axis,y-axis, andz-axis are chosen as the binary axis\nwithC2symmetry, the bisectrix axis, and the trigonal axis\nwithC3symmetry, respectively. The yellow lines represents\nthe unit cell of the rhombohedral lattice. (b) The rhombohe-\ndral structure viewed from the trigonal axis. (c) Schematic\nillustration of the band structure at the Fermi surface. The\nthree electron ellispoids at L-points are dominant contribu-\ntion to the spin transport.\nx= 1. Bismuth is known to have a strongly anisotropic\nband structure. The magnitude of the matrix elements of\nthe inverse mass \u000b1-\u000b4was experimentally determined as\nm\u000b1= 806,m\u000b2= 7:95,m\u000b3= 349, and m\u000b4= 37:6.\nThe total Gaussian curvatures are evaluated as46\n\u0016\u0014k'1:92\u0002104; (46)\n\u0016\u0014?'4:24\u0002105: (47)\nThe ratio of the total Gaussian curvature is estimated\nasx'22:1. Therefore, the damping enhancement is\nexpected to depend strongly on the polar angle \u0012in a bi-\nlayer system composed of single-crystalline bismuth and\nferromagnetic insulator. Conversely, the \u0012-dependence of\nthe damping enhancement is considered to be suppressed\nfor polycrystalline bismuth.\nThe damping enhancement is independent of the az-\nimuthal angle \u001e. Therefore, it is invariant even on ro-\ntating the spin orientation around the trigonal axis. The\nreason is that the azimuthal angular dependence of the\ndamping enhancement cancels out when the contribu-\ntions of the three electron ellipsoids are summed over,\nalthough each contribution depends on the azimuthal an-\ngle. The azimuthal angular dependence of the damping\nenhancement is expected to remain when strain breaks\nthe in-plane symmetry. Additionally, suppose the spin\ncan be injected into each electron ellipsoid separately,\ne.g., by interfacial manipulation of the bismuth atoms.\nIn that case, the damping enhancement depends on the\nazimuthal angle of the spin orientation of the ferromag-\nnetic insulator39. This may be one of the probes of the\nelectron ellipsoidal selective transport phenomena.7\n- /2\n0 /2\ntheta1.01.52.0damping_modulation\nFIG. 6. The \u0012-dependence of the damping enhancement\nfor di\u000berent x. The ratio of the total Gaussian curvatures\nx= \u0016\u0014?=\u0016\u0014krepresents the anisotropy of the band structure.\nThe blue line with x= 22:1 corresponds to the damping en-\nhancement in single-crystalline bismuth, and the other lines\ncorrespond to that in the weakly anisotropic band structure.\nAs can be seen from the graph, the \u0012-dependence of the damp-\ning enhancement decreases as the more weakly anisotropic\nband structure, and the angular dependence turns out to van-\nish in an isotropic Dirac electron system with x= 1.\nIt is also noteworthy that the damping enhancement\nvaries according to the ordered localized spin direction\nwith both clean and dirty interfaces; that is independent\nof whether momentum is conserved in interfacial spin\ntransport. Conversely, it was reported that the spin ori-\nentation dependence of the damping enhancement due to\nthe Rashba and Dresselhaus spin-orbit interaction turned\nout to vanish by interfacial inhomogeneity42,43.\nV. CONCLUSION\nWe theoretically studied spin pumping from a ferro-\nmagnetic insulator to an anisotropic Dirac electron sys-\ntem. We calculated the enhancement of the Gilbert\ndamping in the second perturbation concerning the prox-\nimity interfacial exchange interaction by considering\nthe interfacial randomness. For illustration, we calcu-\nlated the enhancement of the Gilbert damping for an\nanisotropic Dirac system realized in bismuth. We showed\nthat the Gilbert damping varies according to the polar\nangle between the ordered spin hSi0and the trigonal axis\nof the Dirac electron system whereas it is invariant in its\nrotation around the trigonal axis. Our results indicate\nthat the spin pumping experiment can provide helpful in-\nformation on the anisotropic band structure of the Dirac\nelectron system.\nThe Gilbert damping is invariant in the rotation\naround the trigonal axis because the contributions of each\nelectron ellipsoid depend on the in-plane direction of theordered spinhSi0. Nevertheless, the total contribution\nbecomes independent of the rotation of the trigonal axis\nafter summing up the contributions from the three elec-\ntron ellipsoids that are related to each other by the C3\nsymmetry of the bismuth crystalline structure. If the spin\ncould be injected into each electron ellipsoid separately,\nit is expected that the in-plane direction of the ordered\nlocalized spin would in\ruence the damping enhancement.\nThis may be one of the electron ellipsoid selective spin in-\njection probes. The in-plane direction's dependence will\nalso appear when a static strain is applied. A detailed\ndiscussion of these e\u000bects is left as a future problem.\nACKNOWLEDGMENTS\nThe authors would like to thank A. Yamakage and Y.\nOminato for helpful and enlightening discussions. The\ncontinued support of Y. Nozaki is greatly appreciated.\nWe also thank H, Nakayama for the daily discussions.\nThis work was partially supported by JST CREST Grant\nNo. JPMJCR19J4, Japan. This work was supported by\nJSPS KAKENHI for Grants (Nos. 20H01863, 20K03831,\n21H04565, 21H01800, and 21K20356). MM was sup-\nported by the Priority Program of the Chinese Academy\nof Sciences, Grant No. XDB28000000.\nAppendix A: Magnetic moment of electrons in Dirac\nelectron system\nIn this section, we de\fne the spin operators in the\nDirac electron systems. The Wol\u000b Hamiltonian around\nthe L point is given by HD=\u001a3\u0001\u0000\u001a2\u0019\u0001v, where\nvi=P\n\u000bwi\u000b\u001b\u000bwithwi\u000bbeing the matrix component\nof the velocity vectors and \u0019=p+e\ncAis the momen-\ntum operator including the vector potential. It is rea-\nsonable to determine the magnetic moment of electrons\nin an e\u000bective Dirac system as the coe\u000ecient of the Zee-\nman term. The Wol\u000b Hamiltonian is diagonalized by the\nSchrie\u000ber-Wol\u000b transformation up to v=\u0001 as below:\nei\u0018HDe\u0000i\u0018'\u0014\n\u0001 +1\n2\u0001(\u0019\u0001v)2\u0015\n\u001a3; (A1)\nwhere\u0018=\u001a1\n2\u0001\u0019\u0001vis chosen to erase the o\u000b-diagonal\nmatrix for the particle-hole space. We can proceed cal-\nculation as follows:\n(\u0019\u0001v)2=\u0019i\u0019jwi\u000bwj\f(\u000e\u000b\f+i\u000f\u000b\f\r\u001b\r);\n= (\u0019iwi\u000b)2+i\n2\u000f\u000b\f\r\u001b\r[\u0019\u0002\u0019]i\u000fijkwj\u000bwk\f;\n= \u0001\u0012\n\u0019\u0001\u000b\u0001\u0019+~e\nc\u0001Mi\u000b\u001b\u000bBi\u0013\n; (A2)\nwhere we used ( \u0019\u0002\u0019) =e~\ncir\u0002AandMi\u000bis de\fned as\nMi\u000b=1\n2\u000f\u000b\f\r\u000fijkwj\fwk\r: (A3)8\nFinally, we obtain\nei\u0018HDe\u0000i\u0018'\u0014\n\u0001 +\u0019\u0001\u000b$\u0001\u0019\n2\u0015\n\u0000Bi\u0016s;i; (A4)\nwhere\u0016s;iis a magnetic moment of the Dirac electrons\nde\fned as\n\u0016s;i=\u0000~e\n2c\u0001Mi\u000b\u001a3\u001b\u000b=\u0000~e\n2c\u0001Mi\u000b\u0012\n\u001b\u000b0\n0\u0000\u001b\u000b\u0013\n:\n(A5)\nIn the main text, we de\fned the spin operator sas the\nmagnetic moment \u0016sdivided by the Bohr magnetization\n\u0016B=~e=2mc, i.e.,\nsi=\u0000\u0016s;i\n\u0016B=m\n\u0001Mi\u000b\u0012\n\u001b\u000b0\n0\u0000\u001b\u000b\u0013\n: (A6)\nFor an isotropic Dirac system, the matrix component is\ngiven bywi\u000b=v\u000ei\u000band Eq. (A6) reproduces the well-\nknown form of the spin operator\ns=g\u0003\n2\u0012\n\u001b0\n0\u0000\u001b\u0013\n; (A7)\nwhereg\u0003= 2m=m\u0003is the e\u000bective g-factor with m\u0003=\n\u0001=v2being e\u000bective mass.\nAppendix B: Linear Response Theory\nIn this section, we brie\ry explain how the microwave\nabsorption rate is written in terms of the uniform spincorrelation function. The Hamiltonian of an external\ncircular-polarized microwave is written as\n^Hrf=\u0000g\u0016Bhrf\n2X\ni(S\u0000\nie\u0000i!t+S+\niei!t)\n=\u0000g\u0016BhrfpnF\n2(S\u0000\n0e\u0000i!t+S+\n0ei!t); (B1)\nwherehrfis an amplitude of the magnetic \feld of the\nmicrowave, S\u0006\nkare the Fourier transformations de\fned\nas\nS\u0006\nk=1pnFX\niS\u0006\nie\u0000ik\u0001Ri; (B2)\nandRiis the position of the locazed spin i. Using the lin-\near response theory with respect to ^Hrf, the expectation\nvalue of the local spin is calculated as\nhS+\n0i!=GR\n0(!)\u0002g\u0016BhrfpnF\n2; (B3)\nwhereGR\nk(!) is the spin correlation function de\fned in\nEq. (20). Since the microwave absorption is determined\nby the dissipative part of the response function, it is\nproportional to Im GR\n0(!), that reproduces a Lorentzian-\ntype FMR lineshape. As explained in the main text, the\nchange of the linewidth of the microwave absorption, \u000e\u000b,\ngives information on spin excitation in the Dirac system\nvia the spin susceptibility as shown in Eq. (22).\nAppendix C: Spin susceptibility of Dirac electrons\nIn this section, we give detailed derivation of Eq. (26). The trace part in Eq. (25) is calculated as\ntr[s+gk+q(i\u000fn+i!l)s\u0000gk(i\u000fn)] =[(i\u000fn+i!l+\u0016)(i\u000fn+\u0016) + \u00012]tr[s+s\u0000]\u0000tr[s+~(~k+~q)\u0001\u001bs\u0000~~k\u0001\u001b]\n[(i\u000fn+i!l+\u0016)2\u0000\u000f2\nk+q][(i\u000fn+\u0016)2\u0000\u000f2\nk]; (C1)\nwhere ( ~k+~q)\u0001\u001b= (k+q)\u0001v. Using the following relations\ntr[s+s\u0000] =\u00122m\n\u0001\u00132X\n\u000baiMi\u000ba\u0003\njMj\u000b; (C2)\ntr[s+~(~k+~q)\u0001\u001bs\u0000~~k\u0001\u001b] =\u00122m\n\u0001\u00132X\n\u000b(2aiMi\u000b~~k\u000ba\u0003\njMj\f~~k\f\u0000~2~k2aiMi\u000ba\u0003\njMj\u000b); (C3)\nthe spin susceptibility is given by\n\u001fq(i!l) =\u00002F(\u0012;\u001e)X\nk\f\u00001X\ni\u000fn(i\u000fn+i!l+\u0016)(i\u000fn+\u0016) + \u00012+~2~k2=3\n[(i\u000fn+i!l+\u0016)2\u0000\u000f2\nk+q][(i\u000fn+\u0016)2\u0000\u000f2\nk]; (C4)\nwhere we dropped the terms proportional to ~k\u000b~k\f(\u000b6=\f) because they vanish after the summation with respect\nto the wavenumber k. Here, we introduced a dimensionless function, F(\u0012;\u001e) = (2m=\u0001)2P\n\u000baiMi\u000ba\u0003\njMj\u000b, which9\ndepends on the direction of the magnetization of the FI. Representing the Matsubara summation as the following\ncontour integral, we derive\n\u001fq(i!l) =\u00002F(\u0012;\u001e)X\nkIdz\n4\u0019itanh\u0012\f(z\u0000\u0016)\n2\u0013z(z+i!l) + \u00012+~2~k2=3\n[(z+i!l)2\u0000\u000f2\nk+q][z2\u0000\u000f2\nk]; (C5)\n= 2F(\u0012;\u001e)X\nkIdz\n2\u0019if(z)z(z+i!l) + \u00012+~2~k2=3\n[(z+i!l)2\u0000\u000f2\nk+q][z2\u0000\u000f2\nk]; (C6)\nWe note that tanh( \f(z\u0000\u0016)=2) has poles at z=i\u000fn+\u0016and is related to the Fermi distribution function f(z) as\ntanh[\f(z\u0000\u0016)=2] = 1\u00002f(z). Using the following identities\n1\nz2\u0000\u000f2\nk=1\n2\u000fkX\n\u0015=\u0006\u0015\nz\u0000\u0015\u000fk; (C7)\nz\nz2\u0000\u000f2\nk=1\n2X\n\u0015=\u00061\nz\u0000\u0015\u000fk; (C8)\nthe spin susceptibility is given by\n\u001fq(i!l) =F(\u0012;\u001e)X\nkIdz\n2\u0019if(z)X\n\u0015;\u00150=\u0006\"\n1\n2+(\u00012+~2~k2=3)\u0015\u00150\n2\u000fk\u000fk+q#\n1\nz\u0000\u0015\u000fk1\nz+i!l\u0000\u00150\u000fk+q; (C9)\n=F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015f(\u00150\u000fk+q)\u0000f(\u0015\u000fk)\ni!l\u0000\u00150\u000fk+q+\u0015\u000fk: (C10)\nBy the analytic continuation i!l=~!+i\u000e, we derive the retarded spin susceptibility as below:\n\u001fR\nq(!) =F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015f(\u00150\u000fk+q)\u0000f(\u0015\u000fk)\n~!+i\u000e\u0000\u00150\u000fk+q+\u0015\u000fk: (C11)\nThe imaginary part of the spin susceptibility is given by\nIm\u001fR\nq(!) =\u0000\u0019F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015h\nf(\u00150\u000fk+q)\u0000f(\u0015\u000fk)i\n\u000e(~!\u0000\u00150\u000fk+q+\u0015\u000fk): (C12)\nFrom this expression, Eqs. 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Rev.\nB84, 115137 (2011)."    },    {        "title": "2206.08820v2.Resolvent_estimates_for_the_one_dimensional_damped_wave_equation_with_unbounded_damping.pdf",        "content": "RESOLVENT ESTIMATES FOR THE ONE-DIMENSIONAL\nDAMPED WAVE EQUATION WITH UNBOUNDED DAMPING\nANTONIO ARNAL\nAbstract. We study the generator Gof the one-dimensional damped wave\nequation with unbounded damping. We show that the norm of the correspond-\ning resolvent operator, k(G\u0000\u0015)\u00001k, is approximately constant as j\u0015j!+1\non vertical strips of bounded width contained in the closure of the left-hand\nside complex semi-plane, C\u0000:=f\u00152C: Re\u0015\u00140g. Our proof rests on a\nprecise asymptotic analysis of the norm of the inverse of T(\u0015), the quadratic\noperator associated with G.\n1.Introduction\nThere is a well-developed theory for self-adjoint linear operators based on a num-\nber of key tools, notably the spectral theorem. This fundamental result underlies\nthe fact that the spectrum of a self-adjoint operator contains a signi\fcant amount\nof information which is of great value to understand its action. It is equally well-\nknown that there is no equivalent result for non-self-adjoint (NSA) linear operators.\nThis de\fciency is related to the spectral instability under small perturbations of-\nten exhibited by NSA operators. Such behaviour has prompted the development\nof new tools and techniques to study them, the pseudospectrum being one of the\nmost widely used (see e.g. [9, 11, 32]). We recall that, if His a closed linear oper-\nator acting in a Hilbert space Hand we take \">0, the\"-pseudospectrum of His\nde\fned as\n\u001b\"(H) :=\u001b(H)[f\u00152C:k(H\u0000\u0015)\u00001k>\"\u00001g:\nIt is immediate from its de\fnition that \u001b\"(H) is a family of nested open sets which\nincrease as \"!+1and approach \u001b(H) as\"!0. These sets can also be charac-\nterised as follows\n\u001b\"(H) =f\u00152C:\u00152\u001b(H+A) for somekAk<\"g\n(see [20, Thm. 13.2]), which makes apparent why one can expect the pseudospec-\ntrum to be more robust under linear perturbations than the spectrum.\nIn this context, it also becomes clear that the spectral analysis of any NSA\noperator must include at least some quantitative understanding of the behaviour\nof the norm of the resolvent operator k(H\u0000\u0015)\u00001kfor\u0015in the resolvent set \u001a(H).\nUsing general operator-theoretic arguments, it is possible to show that, if His\na closed operator whose numerical range, Num( H), satis\fes that each connected\nDate : October 19, 2022.\n2010 Mathematics Subject Classi\fcation. 35L05, 35P05, 34L40, 47A10.\nKey words and phrases. damped wave equation, unbounded damping, resolvent operator, re-\nsolvent bounds, pseudospectrum, Fourier transform.\nThe author should like to express his gratitude to his supervisor, P. Siegl, Graz University\nof Technology and Queen's University Belfast, for very valuable comments and suggestions in\nthe course of the work presented in this paper, and to B. Gerhat, Czech Technical University in\nPrague, for introducing the framework described in [19] to him.\n1arXiv:2206.08820v2  [math.SP]  18 Oct 20222 ANTONIO ARNAL\ncomponent of CnNum(H) has non-empty intersection with \u001a(H), then\nk(H\u0000\u0015)\u00001k\u00141\ndist(\u0015;Num(H)); \u00152\u001a(H);\n(see [15, Thm. III.2.3]). This estimate has the weakness that it provides very limited\ninformation about the behaviour of k(H\u0000\u0015)\u00001kwhen\u0015lies near the boundary of\nthe numerical range and none at all when it is inside. An aim of recent research in\nthis area has been to shed light on such behaviour, using both semi-classical (e.g.\n[8, 12, 29, 6]) and non-semi-classical (e.g. [26, 25, 3, 13, 5]) methods. One approach,\npioneered in [8] and subsequently developed non-semi-classically in [26, 3, 25, 13],\nrelies on the construction of pseudomodes (or approximate eigenfunctions) for the\noperator at hand (Schr odinger, damped wave equation, Dirac, biharmonic) inside\nthe numerical range thereby \fnding lower bounds on k(H\u0000\u0015)\u00001k. For Schr odinger\noperators with complex potentials, lower and upper bounds have recently been\nfound in [5] using di\u000berent (non-semi-classical) methods.\nThe aim of the work presented in this paper is to apply the new techniques\ndeveloped in [5] to the study of the resolvent of the NSA generator Gfor the one-\ndimensional damped wave equation (DWE) described by\n@2\ntu(t;x) + 2a(x)@tu(t;x) = (@2\nx\u0000q(x))u(t;x); t> 0; x2R; (1.1)\nwith non-negative damping aunbounded at in\fnity and non-negative potential q\nwhich may also be unbounded. There is a great deal of research literature covering\nthe case where ais a bounded function on a (possibly unbounded) domain \n in Rd,\nd\u00151, re\recting applications where the solution to the corresponding initial value\nproblem decays exponentially with time. On the other hand, recent research (see\n[30, 21, 18, 3]) has focused on the study of the equation when ais unbounded at\nin\fnity and on the impact of this feature on the spectral structure of the generator\nand/or the large-time behaviour of solutions. In [21], the existence and uniqueness\nof a weak solution to the initial value problem for (1.1), with some mild assumptions\non the initial data, were proven for continuous damping bounded below by a posi-\ntive constant in Rd,d\u00153. Furthermore, it was shown that both the solution and\nits energy decay polynomially with time. A similar result was presented in [30] for\ndampings of type a(x) =a0jxj\u000b, witha0;\u000b> 0, on exterior domains in Rd,d\u00152,\nalthough assuming more restrictive conditions on the initial data. In [24], the au-\nthors carry out an spectral analysis of the wave equation with distributional (Dirac\n\u000e) damping on a non-compact star graph that highlights the wild spectral behaviour\nassociated with its generator's non-self-adjointness; upper and lower bounds for the\nresolvent norm are also found (see [24, Thm. 2.3]). The perspective and methods\nused in [18] are closer to those applied in this paper, exploring as they do the im-\npact of the behaviour at in\fnity of aon the emergence of the essential spectrum\nofGand the stability of solutions. A similar spectral and stability analysis was\ncarried out in [17] for dampings of type a(x) =\u000b=x,\u000b > 0, on \n = (0 ;1)\u001aR.\nThe pseudospectrum of Gfor a wide class of unbounded dampings a(x); x2R;\nwas studied in [3] using a pseudomode construction f \u00152Dom(G) :\u00152\u0000\u001aCg\nof WKB type and estimates were obtained (as \u0015!1 ,\u00152\u0000) for the decay rate of\nk(G\u0000\u0015) \u0015kH\nk \u0015kH\n(withHdenoting the underlying Hilbert space where the operator Gacts) to yield\na lower bound on k(G\u0000\u0015)\u00001k.\nThe main \fnding in this paper is formulated in Theorem 3.5 and concerns the as-\nymptotic behaviour of k(G\u0000\u0015)\u00001kinC\u0000. For dampings aobeying Assumption 3.1,\nwhich encompasses smooth unbounded non-negative real functions with controlledRESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 3\nderivatives (e.g. a(x) =x2n; n2N, see Asm. 3.1 (ii)), we prove that k(G\u0000\u0015)\u00001kis\napproximately constant in any bounded-width vertical strip in C\u0000asjIm\u0015j!+1.\nOur result supports smooth non-negative potentials qwith controlled derivatives\n(Asm. 3.1 (iii)) as long as they are \"no stronger\" than aat in\fnity (Asm. 3.1 (iv)).\nIt encompasses cases where k(G\u0000\u0015)\u00001kcan be shown to diverge along any ray in\nthe second (or third) quadrant of the complex plane using the methods in [3] (see\nRemark 3.7). The key element in our proof of Theorem 3.5 is the derivation in\nTheorem 4.3 of an asymptotic estimate for the norm of the inverse of the quadratic\noperator\nT(\u0015) =\u0000@2\nx+q(x) + 2\u0015a(x) +\u00152; \u00152Cn(\u00001;0]:\nAlthough we shall defer a more rigorous de\fnition of this operator, along with an\nexplanation of how it relates to G, until Sub-section 2.4, we observe here that its\nstructure is that of a \u0015-dependent Schr odinger operator with the complex potential\nq(x) + 2\u0015a(x) +\u00152. Whereas the fact that qis \"no stronger\" than adiscourages\nus from (for example) attempting to recast the problem as a (relatively bounded)\nperturbation of a self-adjoint operator, it does on the other hand broadly \ft into\nthe framework used to prove [5, Thm. 4.2], where the asymptotic behaviour of\nk(H\u0000\u0015)\u00001kalong the real axis for a one-dimensional Schr odinger operator with\na complex potential Hwas determined. In order to adapt to T(\u0015) the strategy\nintroduced in that paper, we begin by transforming the problem to Fourier space\n(see (4.14)). The resulting pseudo-di\u000berential operator bT(\u0015) =bq+2\u0015ba+\u00182+\u00152has\nthe potential term \u00182+\u00152(for\u0015:=\u0000c+ib) with turning points \u0006\u0018b(where\u0018b:=b).\nWe subsequently carry out a separate analysis of kbT(\u0015)ukdepending on whether\nor not supp uis contained in certain neighbourhoods of \u0006\u0018bdesigned so that \u00182is\napproximately constant inside. More speci\fcally, the proof of Theorem 4.3 consists\nof the following steps (with \u0015=\u0000c+ib, wherec2K\u001aR+,Kbounded, and\nb2Rnf0g):\n(1) In Proposition 4.7, with \n0\nb;\u0006representing the neighbourhoods of \u0006\u0018bde\fned\nin (4.15), we use direct L2-norm estimates to \fnd that as b!+1\nb2.\u000einf(\nkbT(\u0015)uk\nkuk: 06=u2Dom(bT(\u0015));suppu\\(\n0\nb;+[\n0\nb;\u0000) =;)\n:\n(2) In Proposition 4.8, inside neighbourhoods \n b;\u0006of\u0006\u0018bde\fned in (4.22) and\nappropriately shifted, we Taylor-approximate bT(\u0015) with the (Fourier-space) pseudo-\ndi\u000berential version of the generalised Airy operator, A=\u0000@x+a(x), shifted by c\nto yield asb!+1\nk(A\u0000c)\u00001k\u000012b(1\u0000OK(b\u00001))\n\u0014inf(\nkbT(\u0015)uk\nkuk: 06=u2Dom(bT(\u0015));suppu\u001a\nb;\u0006)\n:\nThe norm resolvent convergence of (a localised realisation of) bT(\u0015) to the pseudo-\ndi\u000berential version of A\u0000cfollows from the second resolvent identity and several\ngraph norm estimates obtained by standard arguments.\n(3) In Proposition 4.9, we show that the estimate for kbT(\u0015)\u00001kobtained in\nStep (2) cannot be improved by \fnding functions ub2Dom(bT(\u0015)) such that as\nb!+1\nkbT(\u0015)ubk=k(A\u0000c)\u00001k\u000012b(1 +OK(b\u00001))kubk:\nThe proof relies on exploiting the localisation technique applied in Proposition 4.8\nand the fact that the operators involved have compact resolvent. Thus the norms\nof those resolvents can be obtained from the appropriate singular values and the4 ANTONIO ARNAL\ncorresponding eigenfunctions are used to determine the family ubwith the aid of\ncertain cut-o\u000b functions.\n(4) In our \fnal step, we combine the results from the previous ones with the aid\nof commutator estimates and a suitably constructed partition of unity.\nThe remainder of our paper is structured as follows. Section 2 describes our nota-\ntion and recalls some fundamental facts for the DWE and for the various tools (e.g.\ngeneralised Airy operators) used throughout. Section 3 formulates our assumptions,\nstates our main result for the generator Gand draws some consequences for the\nlong-time behaviour of the associated C0-semigroup that solves the corresponding\nCauchy problem. Section 4 is devoted to investigating a number of important prop-\nerties for the quadratic operator T(\u0015) associated with G, including a crucial result\nregarding the asymptotic behaviour of the norm of its inverse in bounded-width\nvertical strips inside C\u0000which is subsequently extended to general curves adjacent\nto the imaginary axis (see Sub-section 4.2). The proof of our main theorem can be\nfound in Section 5. Section 6 illustrates our results with a detailed analysis of an\nexample (with a(x) =x2andq(x) =\u0014x2,\u0014>0) where\u001b(G) is calculated and the\nstability of the C0-semigroup discussed.\n2.Notation and preliminaries\nWe write N0:=N[f0g,R+:= (0;+1),R\u0000:= (\u00001;0),C+:=f\u00152C: Re\u0015>\n0gandC\u0000:=f\u00152C: Re\u0015<0g. The characteristic function of a set Eis denoted\nby\u001fE. We shall use C1\nc(R) to represent the space of smooth functions of compact\nsupport and S(R) for the Schwartz space of smooth rapidly decreasing functions\n(with obvious adjustments for spaces in higher dimensions). The commutator of\ntwo operators A,Bis denoted by [ A;B] :=AB\u0000BA.\nIn the one-dimensional setting, we will refer to the \frst and second order dif-\nferential operators with @xand@2\nx, respectively, reserving the symbols rand \u0001\nfor statements in higher dimensions. When the relevant di\u000berentiation variable is\ntime, we shall use @tand@2\ntfor the \frst and second order derivatives, respectively.\nIfHdenotes a Hilbert space, we shall use h\u0001;\u0001iHandk\u0001kHto represent the inner\nproduct and norm on that space. The L2inner product shall be denoted by h\u0001;\u0001i2,\nor just byh\u0001;\u0001iif there is no ambiguity, and the L2norm byk\u0001k 2or just byk\u0001k.\nThe otherLpnorms will be represented by k\u0001kpwithL1denoting the space of\nessentially bounded functions endowed with the essential sup norm k\u0001k1.\nLet;6= \n\u001aRdbe open,k2Nandp2[1;+1]. We will denote the Sobolev\nspaces byWk;p(\n) andWk;p\n0(\n) (the latter representing as usual the closure of\nC1\nc(\n) inWk;p(\n), see e.g.[15, Sub-sec. V.3] for de\fnitions). We shall generally\nbe concerned with the particular cases where \n = R,k= 1 or 2 and p= 2.\nIfB1;B2are two Banach spaces, L(B1;B2) shall denote the (Banach) space of\nbounded linear operators from B1toB2. As it is customary, if Bis a Banach\nspace,L(B) means L(B;B). If the operator T2L(B), then rad( T) represents its\nspectral radius, i.e. rad( T) := supfjzj:z2\u001b(T)gwith\u001b(T) denoting the spectrum\nofT. Unless otherwise stated, for a closed, densely de\fned linear operator Ton\na Banach spaceB, we will use \u001be2(T) to denote the essential spectrum of Tas\ndetermined using singular sequences (see e.g. [15, Thm. IX.1.3]), a closed subset of\nC. As usual, \u001bp(T) will denote the set of eigenvalues of Tand\u001a(T) its resolvent\nset.\nIfHandH1are two linear operators acting in the Hilbert space H, we say that\nH1is an extension ofH, and write H1\u001bH, if Dom(H1)\u001bDom(H) andH1u=Hu\nfor allu2Dom(H). Note that our notation covers the case Dom( H1) = Dom(H),\ni.e.the extension does nothave to be proper.RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 5\nIfH1andH2represent two Hilbert spaces, we will denote by H1\bH2the product\nspace endowed with the inner product\nhu;viH1\bH 2:=hu1;v1iH1+hu2;v2iH2; u 1;v12H 1; u 2;v22H 2;\nwhich is also Hilbert, and k\u0001kH1\bH 2:=h\u0001;\u0001i1\n2\nH1\bH 2will represent the associated\nnorm.\nTo avoid introducing multiple constants whose exact value is inessential for our\npurposes, we write a.bto indicate that, given a;b\u00150, there exists a constant\nC > 0, independent of any relevant variable or parameter, such that a\u0014Cb. The\nrelationa&bis de\fned analogously whereas a\u0019bmeans that a.banda&b.\nWhen it becomes relevant to underlie the dependency of an implicit constant on\none or more parameters, p1;p2;:::, we will use the notation .p1;p2;:::,&p1;p2;:::or\n\u0019p1;p2;:::, as appropriate. We shall use Op1;p2;:::(big-O notation) with a similar\nmeaning.\nIn the rest of this section, we summarise the key properties of the damped wave\nequation and the main tools relied upon in the paper.\n2.1.Fourier transform and pseudo-di\u000berential operators. Foru2S(R),\nthe Fourier and inverse Fourier transforms read (with x;\u00182R)\nFu(\u0018) :=Z\nRe\u0000i\u0018xu(x)dx;F\u00001u(x) :=Z\nReix\u0018u(\u0018)d\u0018;d\u0001:=d\u0001p\n2\u0019;\nwe also use ^ u:=Fuand \u0014u:=F\u00001u, and retain the same notations to refer to the\ncorresponding isometric extensions to L2(R).\nWe recall that the Schwartz space, S(R), is endowed with the family of semi-\nnorms\njfjk;S:= max\n\u000b+\f\u0014ksup\nx2Rhxi\u000bj@\f\nxf(x)j; k2N0:\nGivenm2R, the symbol class Sm\n1;0(R\u0002R) is the vector space of smooth functions\np:R\u0002R!Csuch that for any \u000b; \f2N0there exists C\u000b;\f>0 satisfying\nj@\u000b\n\u0018@\f\nxp(\u0018;x)j\u0014C\u000b;\fhxim\u0000\f;(\u0018;x)2R\u0002R:\nThis space is endowed with a natural family of semi-norms de\fned by\njpj(m)\nk:= max\n\u000b;\f\u0014ksup\n\u0018;x2Rhxi\u0000m+\fj@\u000b\n\u0018@\f\nxp(\u0018;x)j; k2N0:\nWe associate a pseudo-di\u000berential operator with the symbol p2Sm\n1;0(R\u0002R) via\nOP(p)u(\u0018) :=Z\nRe\u0000i\u0018xp(\u0018;x)\u0014u(x)dx; \u00182R; u2S(R);\nand it can be shown that this is a bounded mapping on S(R) (see[1, Thm. 3.6]).\nThe following result will be used later on and we include it here for convenience.\nWe refer to [5, Lem. 4.4] for a proof.\nLemma 2.1. LetF2C1(R)andm> 0be such that\n8n2N0;9Cn>0;jF(n)(x)j\u0014Cnhxim\u0000n; x2R;\nand let\u001e2C1(R)\\L1(R)be such that supp\u001e0is bounded. For j2N0and\nu2S(R), we de\fne the operators (with P:=P(0)andQ:=Q(0))\nP(j)u:=FF(j)F\u00001u; Q(j)u:=\u001e(j)u:\nThen, for any N2N0, we have\n[P;Q]u=NX\nj=1ij\nj!Q(j)P(j)u+RN+1u; u2S(R); (2.1)6 ANTONIO ARNAL\nwhereRN+1is a pseudo-di\u000berential operator with symbol rN+12Sm\u0000N\u00001\n1;0 (R\u0002R)\nRN+1u(\u0018) :=Z\nRe\u0000i\u0018xrN+1(\u0018;x)\u0014u(x)dx: (2.2)\nMoreover, for every N2NwithN > m , there exist l=l(N)2NandKN>0,\nindependent of Fand\u001e, such that\nkRN+1uk\u0014KNmax\n0\u0014j\u0014ln\nk\u001e(N+1+j)k1o\nkuk: (2.3)\n2.2.Schr odinger operators with complex potentials. Let;6= \n\u001aRdbe\nopen. For a measurable function m: \n!C, we denote the maximal domain of\nthe multiplication operator determined by the function mas\nDom(m) =fu2L2(\n) :mu2L2(\n)g;\nthe Dirichlet Laplacian in L2(\n) is denoted by \u0000\u0001Dand\nDom(\u0001D) =fu2W1;2\n0(\n) : \u0001u2L2(\n)g:\nSuppose that the complex potential V: \n!C,V=Vu+Vb, satis\fes Re V\u00150,\nVu2C1\u0000\n\n\u0001\n,Vb2L1(\n) and, with \"crit= 2\u0000p\n2,\n9\"r2[0;\"crit);9Mr\u00150;jrVuj\u0014\"rjVuj3\n2+Mra.e. in \n:\nUnder these assumptions on Vone can \fnd the (Dirichlet) m-accretive realization\nH=\u0000\u0001D+V, with Dom( H) =fu2W1;2\n0(\n)\\Dom(jVj1\n2) : (\u0000\u0001+V)u2L2(\n)g,\nby appealing to a generalised Lax-Milgram theorem [2, Thm. 2.2]. It is also known\nthat the domain and the graph norm of Hseparate, i.e.Dom(H) = Dom(\u0001 D)\\\nDom(V) and\nkHuk2+kuk2&k\u0001Duk2+kVuk2+kuk2; u2Dom(H):\nFurthermore,\nC:=fu2Dom(H) : suppuis boundedg\nis a core of H. For details, see [2, 23, 28] and, for cases with minimal regularity of\nV, see [7, 22], [15, Chap. VI.2].\n2.3.Generalised Airy operators. In Section 4, we use operators in L2(R) of\ntype (with a2L1\nloc(R),a\u00150 a.e. and ess inf\njxj\u0015Na(x)!+1asN!+1)\nA=\u0000@x+a(x);Dom(A) =W1;2(R)\\Dom(a); (2.4)\nwhich we refer to as generalised Airy operators (on Fourier space). The adjoint\noperator is\nA\u0003=@x+a(x);Dom(A\u0003) =W1;2(R)\\Dom(a);\nand many properties of the usual complex Airy operators are preserved for Aand\nA\u0003. Namely they have compact resolvent, empty spectrum and\nkAuk2+kuk2&ku0k2+kauk2+kuk2; u2Dom(A);\nkA\u0003uk2+kuk2&ku0k2+kauk2+kuk2; u2Dom(A\u0003);(2.5)\nwhere the domain and graph norm separation require the additional assumptions\nthata2L1\nloc(R)\\C1(Rn[\u0000x0;x0]), with some x0>0, and that there exist\n\"2(0;1) andM > 0 such that\nja0(x)j\u0014\"(a(x))2+M;jxj>x0;\nsee [5, App. A] for details and [4] for resolvent norm estimates.RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 7\n2.4.The damped wave equation. The focus of our study shall be the linear\noperatorG(see below) associated with the one-dimensional DWE represented by\n(1.1). Following a standard procedure, we re-write the problem as a \frst order\nsystem of linear equations\n(@tu1(t;x) =u2(t;x);\n@tu2(t;x) = (@2\nx\u0000q(x))u1(t;x)\u00002a(x)u2(t;x);\nwhich leads naturally to the formal operator matrix\n\u00120I\n@2\nx\u0000q\u00002a\u0013\n:\nIn order to properly de\fne such a matrix as an unbounded (non-self-adjoint)\noperator, we follow Section 4 of [19] which specialises a new general framework\nfor the spectral analysis of operator matrices to the particular case of the DWE.\nAssuming that a;q2L1\nloc(R) witha;q\u00150 a.e., letH1:=W(R) represent the\ncompletion of C1\nc(R) with respect to the inner product\nhf;giW:=Z\nR@xf(x)@xg(x)dx+Z\nRq(x)f(x)g(x)dx\nand letH2:=L2(R). Furthermore, de\fne the Hilbert space\nDS:=W1;2(R)\\Dom(q1\n2)\\Dom(a1\n2);\nhf;giS:=Z\nR@xf(x)@xg(x)dx+Z\nRq(x)f(x)g(x)dx+Z\nRa(x)f(x)g(x)dx\n+Z\nRf(x)g(x)dx; f;g2DS;\nand letD\u0003\nSbe the space of bounded, conjugate-linear functionals on DS. It can be\nshown that the canonical embeddings DS\u001aH 2\u001aD\u0003\nSare continuous with dense\nrange, that C1\nc(R) is densely contained in DSand thatDScan also be continuously\nembedded inH1(see [19, Prop. 4.6]). Moreover, the operators\n02L(H1); I 2L(DS;H1);\n@2\nx\u0000q2L(H1;D\u0003\nS);\u00002a2L(DS;D\u0003\nS);\nwith@2\nx\u0000qandathe unique extensions of\n((@2\nx\u0000q)f;g)D\u0003\nS\u0002DS:=\u0000Z\nR\u0010\n@xf(x)@xg(x) +q(x)f(x)g(x)\u0011\ndx;\n(af;g)D\u0003\nS\u0002DS:=Z\nRa(x)f(x)g(x)dx;f;g2C1\nc(R);\nare well-de\fned (see [19, Prop. 4.9]). We are therefore in a position to introduce\nthe operator matrix\nbG:=\u00120I\n@2\nx\u0000q\u00002a\u0013\n2L(H1\bDS;H1\bD\u0003\nS); (2.6)\nits (second) Schur complement\nbS(\u0015) :=\u00002a\u0000\u0015+1\n\u0015(@2\nx\u0000q)jDS2L(DS;D\u0003\nS); \u00152Cnf0g;\nand the corresponding restrictions\nG:=bGjDom(G);\nDom(G) :=fu2H 1\bDS:bGu2H 1\bH 2g\n=fu:= (u1;u2)t2H 1\bDS: (@2\nx\u0000q)u1\u00002au22H 2g;(2.7)8 ANTONIO ARNAL\nand\nS(\u0015) :=bS(\u0015)jDom(S(\u0015));\nDom(S(\u0015)) :=fu2DS:bS(\u0015)u2H 2g\n=fu2DS: (@2\nx\u0000q\u00002\u0015a)u2H 2g:\nThe fundamental result derived from the general setting outlined above is that the\noperatorGis m-dissipative with dense domain in both H1\bDSandH1\bH 2and\nit therefore generates a C0-semigroup of contractions on H1\bH 2. Furthermore,\nfor all\u00152Cn(\u00001;0], it can also be shown that Dom( S(\u0015)) is dense inDSand\nthatGandS(\u0015) are spectrally equivalent in the following sense\n\u00152\u001b(G)() 02\u001b(S(\u0015)) (2.8)\n(see [19, Lem. 4.13, Thm. 4.2]). Moreover, if \u00152Cn(\u00001;0] and 02\u001a(S(\u0015)), the\noperator matrix (with I:=IH1!H 1)\nR\u0015:= \n\u00001\n\u0015I+1\n\u00152bS(\u0015)\u00001(@2\nx\u0000q)1\n\u0015S(\u0015)\u00001\n1\n\u0015bS(\u0015)\u00001(@2\nx\u0000q)S(\u0015)\u00001!\n2L(H1\bH 2) (2.9)\nis both a left and right inverse for G\u0000\u0015(see the proof of [19, Thm. 2.8] for details).\nLettingHqdenote the Friedrichs extension of \u0000@2\nx+qinitially de\fned on C1\nc(R),\ni.e.\nHq:=\u0000@2\nx+q;Dom(Hq) :=fu2W1;2(R)\\Dom(q1\n2) : (\u0000@2\nx+q)u2L2(R)g;\n(2.10)\nit has been proven in [18] under more restrictive assumptions on the damping\nand potential functions (which hold for the operators covered in this paper, see\nRemark 3.4) that the domain of the quadratic operator function in L2(R)\nT(\u0015) :=\u0000\u0015S(\u0015) =Hq+ 2\u0015a+\u00152; \u00152Cn(\u00001;0]; (2.11)\nseparates and does not depend on \u0015\nDom(T(\u0015)) = Dom(Hq)\\Dom(a)\u001aH 1\\DS; (2.12)\nmoreover, the subspace\nD:=fu2Dom(Hq) : suppuis compact in Rg\u001aDom(a) (2.13)\nis a core for T(\u0015) andT(\u0015)\u0003=T(\u0015), for\u00152Cn(\u00001;0] (see [18, Thm. 2.4]).\nIt also holds true that, if the damping asatis\fes [18, Asm. I] and is unbounded\n(see [18, Asm. II]), the set \u001b(G)\\Cn(\u00001;0] consists of at most a countable set\nof isolated eigenvalues of \fnite multiplicity which may only accumulate at ( \u00001;0]\n(see [18, Thm. 3.2]).\n3.Assumptions and statement of the main result\nWe begin by presenting the assumptions that aandqwill obey throughout the\nrest of the paper. We shall follow the notation introduced in Sub-section 2.4.\nAssumption 3.1. Leta;q2C1(R) such that a\u00150;q\u00150 and assume that the\nfollowing conditions are satis\fed for some x02R+:\n(i)ais unbounded:\nlim\njxj!+1a(x) = +1;\n(ii)ahas controlled derivatives:\n8n2N;9Cn>0;ja(n)(x)j\u0014Cn(1 +a(x))hxi\u0000n; x2R;\n(iii)qhas controlled derivatives:\n8n2N;9C0\nn>0;jq(n)(x)j\u0014C0\nn(1 +q(x))hxi\u0000n; x2R;RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 9\n(iv)qis eventually not bigger than a:\n9K > 0; q(x)\u0014Ka(x);jxj>x0:\nExample 3.2. Damping functions satisfying Assumption 3.1 (i)-(ii) include a(x) =\nx2n,n2N,a(x) =hxip,p>0, anda(x) = loghxip,p>0. The same functions are\nvalid potentials q(x) in addition to smooth, non-negative, bounded functions such\nasq(x) =k,k\u00150, andq(x) =hxip,p\u00140.\nRemark 3.3. It can be shown using Assumption 3.1 (ii) with n= 1 that there\nexistsma>0 such that\na(x).hxima; x2R;\n(e.g. see the comments following Example 3.1 in [26]). Furthermore, for any n2N\nja(n)(x)j.(1 +a(x))hxi\u0000n.hxima\u0000n; x2R;\nwhich shows that a2Sma\n1;0(R\u0002R). Similarly it follows from Assumption 3.1 (iii)\nthat there exists mq>0 such that q2Smq\n1;0(R\u0002R).\nRemark 3.4. Ifaandqsatisfy Assumption 3.1, then they automatically obey\nAssumptions I and II in [18] with \n = Randas= 0. Therefore the prop-\nerties ofGandT(\u0015) described in Sub-section 2.4 hold, in particular the do-\nmain separation Dom( T(\u0015)) = Dom( Hq)\\Dom(a). Furthermore, we also have\nDom(Hq) =W2;2(R)\\Dom(q) (refer to Sub-section 2.2). It therefore follows from\nAssumption 3.1 (iv) that Dom( T(\u0015)) =W2;2(R)\\Dom(a) and (2.13) simpli\fes to\nD=fu2W2;2(R) : suppuis compact in Rg:\nWe now state our main result regarding the asymptotic behaviour of the norm\nof the resolvent of Gin the left-hand side (with respect to the imaginary axis) of\nthe complex plane.\nTheorem 3.5. Letaandqsatisfy Assumption 3.1 and let Gbe the linear operator\n(2.6) -(2.7) acting inH:=H1\bH 2, withH1andH2as de\fned in Sub-section 2.4.\nLetK\u001aR+be a bounded subset and \u0015:=\u0000c+ib2Cwithc2Kandb2Rnf0g.\nThen asjbj!+1\nk(G\u0000\u0015)\u00001k\u0019K1: (3.1)\nRemark 3.6. The statement of Theorem 3.5 describes the asymptotic behaviour\nof the resolvent of Gas a function of the spectral parameter band it should be\nunderstood as follows: there exists b0(K)>0 such that for all jbj\u0015b0(K) and\nallc2K, then (3.1) holds. The same remark applies to other asymptotic results\ninvolving\u0015=\u0000c+ib(whether in relation to Gor the quadratic family T(\u0015))\nthroughout this paper.\nRemark 3.7. We note that the statement (3.1) is far from obvious. For example, it\nhas been shown (see [3, Ex. 3.9]) that for polynomial-like dampings and potentials,\ni.e. for functions a;q2Cn+1(R), withn>1, satisfying\n8x&1; a(x) =xp;jq(j)(x)j.xr\u0000j; p;r2R+;0\u0014j\u0014n; j2N0;\nthe norm of the resolvent of the corresponding generator Gdiverges to +1along\nanyray in the second (or third) quadrant\nk(G\u0000\u0015)\u00001k&b(n\u00001)(p+1)+2; b!+1;\nwhere\u0015=\u0000\u000b+i\f,\u000b=a(b),\f=k\u000b, andk2R+is arbitrary. A similar di-\nvergence (albeit with a di\u000berent rate: (log b)n\u00001bn+1) is observed for logarithmic\ndampings and potentials (see [3, Ex. 3.11]). By adding obvious restrictions, both\nsets of examples can be chosen so that they fall within the scope of Assumption 3.1\nand therefore Theorem 3.5 applies to them, meaning that k(G\u0000\u0015)\u00001kis (asymp-\ntotically) approximately constant on vertical lines.10 ANTONIO ARNAL\nWe close this section by drawing some consequences from the theorem that high-\nlight the dependency of the long-time behaviour of the corresponding semigroup on\nthe location of \u001b(G).\nLemma 3.8. Leta,q,GandHbe as in the statement of Theorem 3.5 and assume\nfurthermore that a6= 0almost everywhere. Then \u001bp(G)\\iR=;.\nProof. Assume \frstly that there exists 0 6=u:= (u1;u2)t2Dom(G)\u001aW (R)\bDS\n(see (2.7)) such that u2Ker(G). It follows\nkGukH= 0 =) k@xu2k2+kq1\n2u2k2+kHqu1+ 2au2k2= 0\nand hence we have k@xu2k= 0. Sinceu22DS\u001aW1;2(R)\u001aL2(R), we obtain that\nu2= 0 and therefore au2= 0. This shows that Hqu12L2(R) andHqu1= 0 and\nconsequently for any f2C1\nc(R)\n0 =hHqu1;fi=h@xu1;@xfi+hq1\n2u1;q1\n2fi=hu1;fiW:\nNoting that C1\nc(R) is dense inW(R) fork\u0001kW, we conclude that u1= 0 and hence\n0=2\u001bp(G).\nLet\u0015:=ib, withb2Rnf0g, and assume that there exists 0 6=u:= (u1;u2)t2\nDom(G) such that u2Ker(G\u0000\u0015). By Claim (ii) in the proof of [18, Thm. 3.2]\n(see Remark 3.4), we deduce that \u0015u1=u2andu22Ker(T(\u0015)). Then\nT(\u0015)u2= 0 =) hHqu2;u2i+ 2\u0015hau2;u2i+\u00152hu2;u2i= 0\n=) k@xu2k2+kq1\n2u2k2+ 2ibka1\n2u2k2\u0000b2ku2k2= 0\nand hence (note b6= 0) we haveka1\n2u2k= 0. Since a >0 a.e. by assumption, we\nconclude that u2= 0 and therefore u1= 0, which completes the proof. \u0003\nWe recall some de\fnitions and properties related to semigroups. The spectral\nbound of a linear operator Ais given by\ns(A) := supfRe\u0015:\u00152\u001b(A)g: (3.2)\nIf (St)t\u00150is aC0-semigroup acting on a Banach space, we de\fne its growth bound\nas\n!0:= inff!2R:9M!\u00151 s.t.kStk\u0014M!ewt;8t\u00150g: (3.3)\nThe following general relation holds between the growth bound of a C0-semigroup\n(St)t\u00150on a Banach space and the spectral bound of its generator A\n\u00001\u0014s(A)\u0014!0<+1\n(see [16, Cor. II.1.13]). Lastly the growth bound of a C0-semigroup on a Hilbert\nspace with generator Ais given by\n!0= inff!>s (A) : sup\ns2Rk(A\u0000(!+is))\u00001k<+1g (3.4)\n(see [16, Ex. V.1.13] or the proof of Theorem 2.3 in [17]).\nOur next result shows that, with additional conditions on a;q, the semigroup\ngenerated by the operator Gin Theorem 3.5 is uniformly exponentially stable and\ntherefore the solutions of the corresponding abstract Cauchy problem decay expo-\nnentially as t!+1.\nCorollary 3.9. Let the assumptions of Theorem 3.5 hold and assume furthermore\nthata6= 0almost everywhere and\n9K0>0; q(x)\u0015K0a(x);jxj>x0:\nThen we have\n!0=s(G)<0: (3.5)RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 11\nProof. Under the assumptions of the corollary and applying [18, Rmk. 3.3], there\nexists\u000bq>0 such that the spectral equivalence (2.8) holds for \u00152Cn(\u00001;\u0000\u000bq]\nand moreover \u001b(G)n(\u00001;\u0000\u000bq] consists of eigenvalues with \fnite multiplicity which\nmay only accumulate at points in ( \u00001;\u0000\u000bq]. This last observation combined with\nLemma 3.8 shows that s(G)<0.\nMoreover note that\nsup\ns2Rk(G\u0000(!+is))\u00001k<+1;8!>s (G): (3.6)\nIf! > 0, the above claim is a consequence of the fact that Gis m-dissipative,\nwhereas for s(G)<!\u00140, it follows from Theorem 3.5. Using (3.6) and (3.4), we\ndeduce that !0=s(G), as required. \u0003\n4.The associated quadratic operator\nIn this section, our aim is to formulate a number of properties of the operator\nfamilyT(\u0015) introduced in Sub-section 2.4 which will help us prove Theorem 3.5 in\nSection 5. We begin by studying the behaviour of kT(\u0015)\u00001kon the positive real\naxis.\nProposition 4.1. Letaandqsatisfy Assumption 3.1, Hqbe as in (2.10) and\n\u0016>0. De\fneT(\u0016)as in (2.11) -(2.12) , i.e.\nT(\u0016) :=Hq+ 2\u0016a+\u00162;Dom(T(\u0016)) :=W2;2(R)\\Dom(a):\nThen for every u2Dom(T(\u0016)), we have\nku00k2+kquk2+\u00162kauk2.kT(\u0016)uk2+\u00162kuk2; \u0016!+1: (4.1)\nFurthermore, the following inequalities hold\nkT(\u0016)\u00001k\u0014\u0016\u00002; \u0016> 0; (4.2)\nkHqT(\u0016)\u00001k+kT(\u0016)\u00001Hqk.1; \u0016!+1; (4.3)\nkH1\n2qT(\u0016)\u00001k+kT(\u0016)\u00001H1\n2qk.\u0016\u00001; \u0016!+1; (4.4)\nkH1\n2qT(\u0016)\u00001\n2k+kT(\u0016)\u00001\n2H1\n2qk.1; \u0016> 0: (4.5)\nProof. Letu2Dom(T(\u0016)) andH\u0016:=Hq+ 2\u0016awith Dom(H\u0016) := Dom(T(\u0016)).\nThenT(\u0016) andH\u0016are non-negative, self-adjoint operators and we have\nkT(\u0016)uk2=kH\u0016uk2+\u00164kuk2+ 2\u00162hH\u0016u;ui\u0015kH\u0016uk2: (4.6)\nFurthermore using q+ 2\u0016a\u00150\nkH\u0016uk2=ku00k2+k(q+ 2\u0016a)uk2+ 2 Rehu0;((q+ 2\u0016a)u)0i\n\u0015ku00k2+k(q+ 2\u0016a)uk2\u00002jhu0;(q0+ 2\u0016a0)uij:(4.7)\nApplying Assumptions 3.1 (ii), (iii), with n= 1, there exists C > 0 such that for\nany arbitrarily small \">0\njhu0;(q0+ 2\u0016a0)uij\u0014ku0kk(jq0j+ 2\u0016ja0j)uk\u0014Cku0k(k(1 + 2\u0016)uk+k(q+ 2\u0016a)uk)\n\u0014C\n2\u0000\nku0k2+ (1 + 2\u0016)2kuk2+\"\u00001ku0k2+\"k(q+ 2\u0016a)uk2\u0001\n\u0014C\n2\u0000\n\"ku00k2+\"k(q+ 2\u0016a)uk2\n+\u00162(C\"\u0016\u00002+ (2 +\u0016\u00001)2)kuk2\u0001\n;\nwith some (possibly large) constant C\">0. This shows that, for any small (but\n\fxed)\">0, we can \fnd constants C;C0\n\">0 (independent of \u0016) such that\n2jhu0;(q0+ 2\u0016a0)uij\u0014C\"\u0000\nku00k2+k(q+ 2\u0016a)uk2\u0001\n+C0\n\"\u00162kuk2; \u0016!+1:12 ANTONIO ARNAL\nHence by (4.7) we deduce\nkH\u0016uk2\u0015(1\u0000C\")\u0000\nku00k2+k(q+ 2\u0016a)uk2\u0001\n\u0000C0\n\"\u00162kuk2; \u0016!+1:\nSelecting an adequately small \"and substituting in (4.6), we \fnd\nku00k2+k(q+ 2\u0016a)uk2.kT(\u0016)uk2+\u00162kuk2; \u0016!+1: (4.8)\nIt is easy to see that\nk(q+ 2\u0016a)uk2\u0015kquk2+ 4\u00162kauk2;\nwhich, in combination with (4.8), yields (4.1).\nClearlyq+ 2\u0016a+\u00162\u0015\u00162>0 and hence \u001b(T(\u0016))\u001a[\u00162;1). It follows that\nT(\u0016) is invertible for all \u0016 > 0. Moreover by the Rellich's criterion (see [27,\nThm. XIII.65]) the set\nS=f 2L2(R) :Z\nRj (x)j2dx\u00141;Z\nR(q(x) + 2\u0016a(x))2j (x)j2dx\u00141;\nZ\nR\u00182j (\u0018)j2d\u0018\u00141g\nis compact and therefore, using the graph norm estimate (4.1), we conclude that\nT(\u0016) has compact resolvent. Since T(\u0016)\u00001is bounded and self-adjoint, we \fnd that\nkT(\u0016)\u00001k= rad(T(\u0016)\u00001)\u0014\u0016\u00002which proves (4.2).\nForu2Dom(T(\u0016))\u001aDom(Hq), appealing once again to (4.1),\nkHquk\u0014ku00k+kquk.kT(\u0016)uk+\u0016kuk; \u0016!+1; (4.9)\ntherefore, letting u:=T(\u0016)\u00001vwithv2L2(R) such thatkvk\u00141, we have by (4.2)\nkHqT(\u0016)\u00001vk.kvk+\u0016kT(\u0016)\u00001vk.1 +\u0016\u00001.1; \u0016!+1;\nwhich proves that kHqT(\u0016)\u00001k.1. Using the fact that ( HqT(\u0016)\u00001)\u0003is bounded\nand the property of adjoint ( AB)\u0003\u001bB\u0003A\u0003, ifABis densely de\fned, we deduce\nthatT(\u0016)\u00001Hqhas a bounded extension which completes the proof of (4.3).\nForu2Dom(T(\u0016))\u001aDom(Hq), we have as \u0016!+1\nkH1\n2quk2=hHqu;ui\u0014kHqukkuk=)\nkH1\n2quk\u0014kHquk1\n2kuk1\n2\u00141\n2\u0000\n\u0016\u00001kHquk+\u0016kuk\u0001\n.\u0016\u00001kT(\u0016)uk+\u0016kuk;\nusing (4.9) in the last step. Taking u:=T(\u0016)\u00001vas before and applying (4.2), we\nobtain\nkH1\n2qT(\u0016)\u00001vk.\u0016\u00001kvk+\u0016kT(\u0016)\u00001vk.\u0016\u00001; \u0016!+1;\nwhich shows that kH1\n2qT(\u0016)\u00001k.\u0016\u00001and, using adjoints as above, we deduce\n(4.4).\nFinally, for any \u0016>0, takingu2Dom(T(\u0016)1\n2), we have\nkT(\u0016)1\n2uk2=hT(\u0016)u;ui=hHqu;ui+h(2\u0016a+\u00162)u;ui\u0015hHqu;ui=kH1\n2quk2:\nLettingu:=T(\u0016)\u00001\n2vwithv2L2(R) such thatkvk\u00141, we deduce\nkH1\n2qT(\u0016)\u00001\n2vk\u0014kvk\u00141 =) kH1\n2qT(\u0016)\u00001\n2k\u00141;\nand, using adjoints, we obtain (4.5). \u0003\nProposition 4.2. Letaandqsatisfy Assumption 3.1 and let \u0015be as in the state-\nment of Theorem 3.5. If T(\u0015)is the family of operators (2.11) -(2.12) , then for any\nu2Dom(T(\u0015)) =W2;2(R)\\Dom(a), we have\nku00k2+kquk2+b2kauk2.kT(\u0015)uk2+b4kuk2;jbj!+1: (4.10)RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 13\nProof. For anyu2Dom(T(\u0015)), we have\nkT(\u0015)uk2=ku00k2+k(q+ 2\u0015a+\u00152)uk2+ 2 Reh\u0000u00;(q+ 2\u0015a+\u00152)ui\n=ku00k2+kquk2+ 4j\u0015j2kauk2+j\u0015j4kuk2+ 2 Rehqu;2\u0015aui\n+ 2 Rehqu;\u00152ui+ 2 Reh2\u0015au;\u00152ui+ 2 Rehu0;q0ui+ 2 Rehu0;qu0i\n+ 2 Rehu0;2\u0015a0ui+ 2 Rehu0;2\u0015au0i+ 2 Rehu0;\u00152u0i:\nThe term Rehu0;qu0i\u00150 can be dropped and, using integration by parts, we have\njRehu0;2\u0015au0ij=jhu0;2cau0ij\u0014jh 2ca0u;u0ij+jh2cau;u00ij:\nHence, taking C1;C0\n1>0 from Assumption 3.1 (ii), (iii) with n= 1, we get\nkT(\u0015)uk2\u0015ku00k2+kquk2+ 4j\u0015j2kauk2+j\u0015j4kuk2\u00002kqukk2cauk\n\u00002kqukkj\u0015j2uk\u00002k2j\u0015jaukkj\u0015j2uk\u00002ku0kkC0\n1(1 +q)uk\n\u00002ku0kk2j\u0015jC1(1 +a)uk\u00002ku0kk2cC1(1 +a)uk\n\u00002ku00kk2cauk\u00002j\u0015j2ku0k2:\nFixing a small \">0 to be chosen below and repeatedly applying estimates such as\n2kukkvk\u0014\"kuk2+\"\u00001kvk2, we obtain\nkT(\u0015)uk2\u0015ku00k2+kquk2+ 4j\u0015j2kauk2+j\u0015j4kuk2\u0000\"kquk2\u00004c2\"\u00001kauk2\n\u0000\"kquk2\u0000j\u0015j4\"\u00001kuk2\u00004j\u0015j2\"kauk2\u0000j\u0015j4\"\u00001kuk2\u0000ku0k2\n\u0000C02\n1kuk2\u0000\"\u00001ku0k2\u0000C02\n1\"kquk2\u0000ku0k2\u00004j\u0015j2C2\n1kuk2\n\u0000\"\u00001ku0k2\u00004j\u0015j2C2\n1\"kauk2\u0000ku0k2\u00004c2C2\n1kuk2\u0000ku0k2\n\u00004c2C2\n1kauk2\u0000\"ku00k2\u00004c2\"\u00001kauk2\u00002j\u0015j2ku0k2\n\u0015(1\u0000\")ku00k2\u00002j\u0015j2(1 + 2j\u0015j\u00002+\"\u00001j\u0015j\u00002)ku0k2\n+ (1\u00002\"\u0000C02\n1\")kquk2\n+ 4j\u0015j2(1\u00002c2j\u0015j\u00002\"\u00001\u0000\"\u0000C2\n1\"\u0000c2j\u0015j\u00002C2\n1)kauk2\n\u0000j\u0015j4(2\"\u00001\u00001 +C02\n1j\u0015j\u00004+ 4C2\n1j\u0015j\u00002+ 4C2\n1c2j\u0015j\u00004)kuk2:\nNote that for large enough jbj\n2j\u0015j2(1 + 2j\u0015j\u00002+\"\u00001j\u0015j\u00002)ku0k2\u00142j\u0015j2(1 + 2j\u0015j\u00002+\"\u00001j\u0015j\u00002)kukku00k\n\u0014\"ku00k2+j\u0015j4C\"kuk2\nfor someC\">0 independent of \u0015. Hence asjbj!+1\nkT(\u0015)uk2\u0015(1\u00002\")ku00k2+ (1\u00002\"\u0000C02\n1\")kquk2\n+ 4j\u0015j2(1\u00002c2j\u0015j\u00002\"\u00001\u0000\"\u0000C2\n1\"\u0000c2j\u0015j\u00002C2\n1)kauk2\n\u0000j\u0015j4(2\"\u00001+C\"\u00001 +C02\n1j\u0015j\u00004+ 4C2\n1j\u0015j\u00002+ 4C2\n1c2j\u0015j\u00004)kuk2:\nFurthermore, since c2K, a bounded subset of R, we havec.K1 andj\u0015j\u0019jbj\nasjbj!+1. Therefore, choosing an adequately small \", we obtain (with implicit\nconstant independent of \u0015, see also Remark 3.6)\nkT(\u0015)uk2&ku00k2+kquk2+b2kauk2\u0000b4kuk2;jbj!+1;\nwhich proves (4.10). \u0003\nOur main result in this section is an asymptotic estimate for kT(\u0015)\u00001kalong\nvertical strips inside C\u0000.14 ANTONIO ARNAL\nTheorem 4.3. Letaandqsatisfy Assumption 3.1 and let T(\u0015)be the family of\noperators (2.11) -(2.12) for\u0015:=\u0000c+ibas in the statement of Theorem 3.5. Then\nkT(\u0015)\u00001k=k(A\u0000c)\u00001k(2jbj)\u00001(1 +OK(jbj\u00001));jbj!+1; (4.11)\nwithAas in (2.4) .\nRemark 4.4. The conditions on ain Assumption 3.1 ensure that the generalised\nAiry operator Ain (4.11) satis\fes the properties listed in Sub-section 2.3 (see [5,\nProp. A.1, Prop. A.2] for details).\nRemark 4.5. Since\u001b(A) =;, it follows that there exists MK>0 such that\nk(A\u0000c)\u00001k\u0014MKfor allc2K.\nBefore proving Theorem 4.3, we present some immediate consequences.\nCorollary 4.6. Witha,q,\u0015andT(\u0015)as in Proposition 4.2 and Hqas in (2.10) ,\nthen forjbj!+1\nkHqT(\u0015)\u00001k+kT(\u0015)\u00001Hqk.Kjbj; (4.12)\nkH1\n2qT(\u0015)\u00001k+kT(\u0015)\u00001H1\n2qk.K1: (4.13)\nProof. Letu2Dom(T(\u0015))\u001aDom(Hq), then by (4.10)\nkHquk\u0014ku00k+kquk.kT(\u0015)uk+b2kuk;jbj!+1:\nTakingu:=T(\u0015)\u00001v, withv2L2(R),kvk\u00141, we have\nkHqT(\u0015)\u00001vk.kvk+b2kT(\u0015)\u00001vk.Kjbj;jbj!+1;\nwhere we have used kT(\u0015)\u00001k.Kjbj\u00001(see (4.11) and Remark 4.5). This proves\nthatkHqT(\u0015)\u00001k.Kjbj. A by now familiar use of adjoints and the fact that\nT(\u0015)\u0003=T(\u0015) (see our observations in Sub-section 2.4) yield (4.12).\nLetu2Dom(T(\u0015))\u001aDom(Hq), then applying once more (4.10) we derive\nkH1\n2quk2=hHqu;ui\u0014kHqukkuk\u0014(ku00k+kquk)kuk\n.(kT(\u0015)uk+b2kuk)kuk\nasjbj!+1. It follows\nkH1\n2quk.kT(\u0015)uk1\n2kuk1\n2+jbjkuk.kT(\u0015)uk+jbjkuk;jbj!+1;\nand therefore arguing as above and applying (4.11)\nkH1\n2qT(\u0015)\u00001vk.kvk+jbjkT(\u0015)\u00001vk.K1;jbj!+1;\nwhich, repeating previous arguments, proves (4.13). \u0003\n4.1.Proof of Theorem 4.3. The strategy of the proof follows the template laid\nout in [5, Sec. 4] to analyse the norm of the resolvent in the real axis for Schr odinger\noperators with complex potentials. We \frstly transform the problem to Fourier\nspace. We then study the resolvent norm of the transformed operator in four steps:\n\fnd an estimate away from the (asymptotic) zeroes of its potential function (i.e.\nthe non-pseudo-di\u000berential term), \fnd a local estimate near the zeroes, \fnd a lower\nbound for the norm and, \fnally, combine the previous results to prove the theorem.\nTo this end, let us introduce the operators in L2(R)\nbT(\u0015) :=FT(\u0015)F\u00001;Dom(bT(\u0015)) :=fu2L2(R) : \u0014u2Dom(T(\u0015))g;\nba:=FaF\u00001; Dom(ba) :=fu2L2(R) : \u0014u2Dom(a)g;\nbq:=FqF\u00001; Dom(bq) :=fu2L2(R) : \u0014u2Dom(q)g:(4.14)RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 15\nNotice thatbT(\u0015) =bq+2\u0015ba+\u00182+\u00152and, with\u0015as in the statement of Theorem 4.3,\nwe have\u00182+\u00152=\u00182\u0000b2(1 +O(jbj\u00001)) asjbj!+1.\nFurthermore, since T(\u0015) =T(\u0015)\u0003(refer to our remarks in Sub-section 2.4), it is\nenough to prove the theorem for b!+1and we will therefore assume b >0 in\nthe rest of the section. Let\n\n0\nb;\u0006:= (\u0006\u0018b\u0000\u000eb;\u0006\u0018b+\u000eb); \u0018b:=b; \u000eb:=\u000e\u0018b;0<\u000e<1\n4; (4.15)\nwhere the parameter \u000e=\u000e(K) will be speci\fed in Proposition 4.8.\n4.1.1. Step 1: estimate outside the neighbourhoods of \u0006\u0018b.\nProposition 4.7. Let\n0\nb;\u0006be de\fned by (4.15) , let the assumptions of Theorem 4.3\nhold and letbT(\u0015)be as in (4.14) . Then asb!+1\nb2.\u000einf(\nkbT(\u0015)uk\nkuk: 06=u2Dom(bT(\u0015));suppu\\(\n0\nb;+[\n0\nb;\u0000) =;)\n:\nProof. Let 06=u2Dom(bT(\u0015)) with supp u\\(\n0\nb;+[\n0\nb;\u0000) =;and consider\nkbT(\u0015)uk2=k(bq+ 2\u0015ba)uk2+k(\u00182+\u00152)uk2+ 2 Reh(bq+ 2\u0015ba)u;(\u00182+\u00152)ui\n\u0015k(bq+ 2\u0015ba)uk2+k(\u00182+\u00152)uk2\u00002jReh(bq+ 2\u0015ba)u;(\u00182+\u00152)uij:\n(4.16)\nNote that\nk(bq+ 2\u0015ba)uk2=kbquk2+ 4j\u0015j2kbauk2+ 2 Rehbqu;2\u0015baui\n=kbquk2+ 4j\u0015j2kbauk2\u00002hbqu;2cbaui\n\u0015kbquk2+ 4j\u0015j2kbauk2\u00002kbqukk2cbauk\n\u00151\n2kbquk2+ 4b2(j\u0015j2b\u00002\u00002c2b\u00002)kbauk2:\nHence we can \fnd C00\n1>0 such that\nkbqu+ 2\u0015bauk2\u0015C00\n1\u0000\nkbquk2+b2kbauk2\u0001\n; b!+1: (4.17)\nNext we estimate the third term in the right-hand side of (4.16). For any arbi-\ntrarily small \">0\n2jRehbqu;(\u00182+\u00152)uij\u00142kbqukk(\u00182+\u00152)uk\u0014\"\u00001kbquk2+\"k(\u00182+\u00152)uk2;\n2jReh2\u0015bau;(\u00182+\u00152)uij\u00144jRe\u0015hbau;\u00182uij+ 4cj\u0015j2hbau;ui\n\u00144cjRehbau;\u00182uij+ 4bjImhbau;\u00182uij+ 4cj\u0015j2hbau;ui\n\u00142kbaukk2c\u00182uk+ 4bjImha0\u0014u;\u0014u0ij+ 2j\u0015j2kbaukk2cuk\n\u0014\"b2kbauk2+\"\u00001b\u000024c2k\u00182uk2+ 4bjImha0\u0014u;\u0014u0ij\n+j\u0015j2(\"kbauk+\"\u000014c2kuk2):\nApplying Assumption 3.1 (ii) with n= 1, we obtain\n4bjImha0\u0014u;\u0014u0ij\u00144bC1k(1 +ba)ukk\u0018uk\u00142bC1(2kukk\u0018uk+ 2kbaukk\u0018uk)\n\u00142bC1\u0000\nkuk2+kbauk2+ 2k\u0018uk2\u0001\n\u00142bC1\u0000\nkuk2+kbauk2+ 2k\u00182ukkuk\u0001\n\u00142bC1\u0000\nkuk2+kbauk2+b\u00002k\u00182uk2+b2kuk2\u0001\n:\nHence there exists C00\n2>0 such that as b!+1\n2jReh2\u0015bau;(\u00182+\u00152)uij\u0014C00\n2(\"b2kbauk2+ (b\u00001+\"\u00001b\u00002c2)k\u00182uk2\n+b4(b\u00001+\"\u00001b\u00002c2)kuk2);16 ANTONIO ARNAL\nand consequently as b!+1\n2jReh(bq+ 2\u0015ba)u;(\u00182+\u00152)uij\u0014\"\u00001kbquk2+\"k(\u00182+\u00152)uk2\n+C00\n2(\"b2kbauk2+ (b\u00001+\"\u00001b\u00002c2)k\u00182uk2\n+b4(b\u00001+\"\u00001b\u00002c2)kuk2):(4.18)\nSubstituting (4.17) and (4.18) in (4.16), we have as b!+1\nkbT(\u0015)uk2\u0015(C00\n1\u0000\"\u00001)kbquk2+b2(C00\n1\u0000C00\n2\")kbauk2+ (1\u0000\")k(\u00182+\u00152)uk2\n\u0000C00\n2(b\u00001+\"\u00001b\u00002c2)(k\u00182uk2+b4kuk2)\nand therefore, choosing a small enough \", we can \fnd C00\n3;C00\n4>0 such that\nkbT(\u0015)uk2\u0015C00\n3(b2kbauk2+k(\u00182+\u00152)uk2)\n\u0000C00\n4(kbquk2+ (b\u00001+b\u00002c2)(k\u00182uk2+b4kuk2)); b!+1:(4.19)\nFinally, with \u00182+\u00152=\u00182\u0000b2+c2\u00002icb, we consider the term\nk(\u00182+\u00152)uk2=k(\u00182\u0000b2)uk2+c2(c2+ 4b2)kuk2+ 2c2h(\u00182\u0000b2)u;ui\n\u0015k(\u00182\u0000b2)uk2+ 4c2b2kuk2\u00002k(p\n2)\u00001(\u00182\u0000b2)ukkp\n2c2uk\n\u00151\n2k(\u00182\u0000b2)uk2+ 2c2b2(2\u0000c2b\u00002)kuk2\nand hence\nk(\u00182+\u00152)uk2\u00151\n2k(\u00182\u0000b2)uk2; b!+1:\nFurthermore, there exists C0\n\u000e>0, depending on \u000e, such that for any \u00182suppu\nj\u00182\u0000b2j=j\u0018+\u0018bjj\u0018\u0000\u0018bj\u0015\u000e2\nb=\u000e2b2;\nj\u0018j\u0014j\u0018\u0006\u0018bj+\u0018b\u0014(1 + 1=\u000e)j\u0018\u0006\u0018bj=) j\u00182\u0000b2j\u0015C0\n\u000e\u00182:\nConsequently there exists C00\n5;\u000e>0 such that for b!+1\nk(\u00182+\u00152)uk2\u00151\n4k(\u00182\u0000b2)uk2+1\n4k(\u00182\u0000b2)uk2\u0015C00\n5;\u000e(k\u00182uk2+b4kuk2):(4.20)\nReplacing (4.20) in (4.19), we deduce that there exists C00\n6;\u000e>0 such that\nkbT(\u0015)uk2\u0015C00\n6;\u000e(b2kbauk2+k\u00182uk2+b4kuk2)\n\u0000C00\n4(kbquk2+ (b\u00001+b\u00002c2)(k\u00182uk2+b4kuk2)); b!+1;\nand, noting that kbquk2.kbauk2+kuk2(see Assumption 3.1 (iv)) and b\u00002c2!0\nasb!+1forc2K, we conclude that there exists C0\n\u000e>0 such that\nkbT(\u0015)uk2\u0015C0\n\u000e(b2kbauk2+k\u00182uk2+b4kuk2); b!+1; (4.21)\nwhich proves the claim. \u0003\n4.1.2. Step 2: estimate near \u0006\u0018b.\nProposition 4.8. De\fne\n\nb;\u0006:= (\u0006\u0018b\u00002\u000eb;\u0006\u0018b+ 2\u000eb); (4.22)\nwith\u0018b,\u000ebas in (4.15) . Let the assumptions of Theorem 4.3 hold and let bT(\u0015)and\nAbe as in (4.14) and(2.4) , respectively. Then as b!+1\nk(A\u0000c)\u00001k\u000012b(1\u0000OK(b\u00001))\n\u0014inf(\nkbT(\u0015)uk\nkuk: 06=u2Dom(bT(\u0015));suppu\u001a\nb;\u0006)\n:(4.23)RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 17\nProof. We shall derive estimate (4.23) for usuch that supp u\u001a\nb;+. The procedure\nwhen suppu\u001a\nb;\u0000is similar (see our remarks at the end of the proof).\nWriting\u00182\u0000b2= 2\u0018b(\u0018\u0000\u0018b) + (\u0018\u0000\u0018b)2, we introduce\neVb(\u0018) :=c2\u00002ic\u0018b+ 2\u0018b(\u0018\u0000\u0018b) + (\u0018\u0000\u0018b)2\u001f\nb;+(\u0018); \u00182R:\nWithbqandbaas in (4.14), let us de\fne the following operator in L2(R)\neT(\u0015) =bq+ 2\u0015ba+eVb(\u0018);Dom(eT(\u0015)) =\b\nu2L2(R) : \u0014u2W1;2(R)\\Dom(a)\t\n:\nWe de\fne a translation on L2(R) by\n(Ubu)(\u0018) :=u(\u0018+\u0018b); \u00182R:\nThen, setting \n b:= (\u00002\u000eb;2\u000eb), we have\n1\n2\u0015UbeT(\u0015)U\u00001\nb=ba+1\n2\u0015bq+c2\u00002ic\u0018b\n2\u0015+\u0018b\n\u0015\u0018+1\n2\u0015\u00182\u001f\nb\n=ba\u0000i\u0018\u0000c+1\n2\u0015bq+c2\u00002ic\u0018b+ 2c\u0015\n2\u0015+\u0018b+i\u0015\n\u0015\u0018+1\n2\u0015\u00182\u001f\nb\n=ba\u0000i\u0018\u0000c+1\n2\u0015bq\u0000c2\n2\u0015\u0000ic\n\u0015\u0018+1\n2\u0015\u00182\u001f\nb\n=ba\u0000i\u0018\u0000c+1\n2\u0015bq+bRb(\u0018);\n(4.24)\nwith\nbRb(\u0018) :=\u0000c2\n2\u0015\u0000ic\n\u0015\u0018+1\n2\u0015\u00182\u001f\nb(\u0018); \u00182R: (4.25)\nFrom (4.15), we have \u000eb=\u000e\u0018b=\u000eband, sinceb\u0014j\u0015j, we \fnd\nk\u0018\u00001(bRb+c2\n2\u0015+ic\n\u0015\u0018)k1=k\u0018\u001f\nbk1\n2j\u0015j\u0014\u000e;\nk\u0018\u00002(bRb+c2\n2\u0015+ic\n\u0015\u0018)k1\u00141\n2j\u0015j\u00141\n2b:(4.26)\nUsing (4.24) and letting\nbS1:=FAF\u00001=ba\u0000i\u0018; Dom(bS1) =\b\nu2L2(R) : \u0014u2Dom(A)\t\n;(4.27)\nbSb:=1\n2\u0015UbeT(\u0015)U\u00001\nb+c=bS1+1\n2\u0015bq+bRb;Dom(bSb) = Dom(bS1);(4.28)\nour next aim is to show that the operator bSb\u0000cconverges tobS1\u0000cin the norm\nresolvent sense as b!+1.\nThe spectrum of A, and hence that of bS1, is empty (refer to Sub-section 2.3)\nand thereforek(bS1\u0000c)\u00001k.K1 for allc2K(see Remark 4.5). Moreover, using\nstandard arguments, the graph-norm inequalities (2.5) can be extended to show\nk(A\u0000c)uk2+hci2kuk2&ku0k2+kauk2+kuk2; u2Dom(A);\n(where the implicit constant is independent of c), which on Fourier space reads\nk(bS1\u0000c)uk2+hci2kuk2&k\u0018uk2+kbauk2+kuk2; u2Dom(bS1): (4.29)\nFrom (4.29), reasoning as in the proof of (4.3), we deduce\nk\u0018(bS1\u0000c)\u00001k+k(bS1\u0000c)\u00001\u0018k+kba(bS1\u0000c)\u00001k+k(bS1\u0000c)\u00001bak\n.1 +hcik(bS1\u0000c)\u00001k:(4.30)\nFurthermore, by Assumption 3.1 (iv), we have kbquk.kbauk+kukand hence\nkbq(bS1\u0000c)\u00001k+k(bS1\u0000c)\u00001bqk.1 +hcik(bS1\u0000c)\u00001k: (4.31)18 ANTONIO ARNAL\nLet us write\nbSb\u0000c=\u0012\nI+1\n2\u0015bq(bS1\u0000c)\u00001+bRb(bS1\u0000c)\u00001\u0013\n(bS1\u0000c): (4.32)\nNote that, by (4.25), (4.26), (4.30) and (4.31), we have\nk1\n2\u0015bq(bS1\u0000c)\u00001k.1 +hcik(bS1\u0000c)\u00001k\nb;\nkbRb(bS1\u0000c)\u00001k\u0014c2\n2j\u0015jk(bS1\u0000c)\u00001k+c\nj\u0015jk\u0018(bS1\u0000c)\u00001k\n+k\u0018\u00001(bRb+c2\n2\u0015+ic\n\u0015\u0018)k1k\u0018(bS1\u0000c)\u00001k\n.c2b\u00001k(bS1\u0000c)\u00001k+ (cb\u00001+\u000e)(1 +hcik(bS1\u0000c)\u00001k);\nand it therefore follows that there exists a large enough b0(K)>0 and a su\u000eciently\nsmall\u000e(K)>0 (independent of b) such that the operator I+ (2\u0015)\u00001bq(bS1\u0000c)\u00001+\nbRb(bS1\u0000c)\u00001is bounded (with kI+ (2\u0015)\u00001bq(bS1\u0000c)\u00001+bRb(bS1\u0000c)\u00001k\u00191) and\ninvertible for b\u0015b0. Hence using (4.29) and (4.32) we deduce\nk(bSb\u0000c)uk2+hci2kuk2&k\u0018uk2+kbauk2+kuk2; u2Dom(bSb); b!+1:(4.33)\nMoreover by (4.32) we \fnd that bSb\u0000cis invertible and\n(bSb\u0000c)\u00001= (bS1\u0000c)\u00001\u0012\nI+1\n2\u0015bq(bS1\u0000c)\u00001+bRb(bS1\u0000c)\u00001\u0013\u00001\n;(4.34)\nforb!+1. Therefore by (4.30) and (4.34)\nk(bSb\u0000c)\u00001k\u0019k (bS1\u0000c)\u00001k;\nk\u0018(bSb\u0000c)\u00001k+k(bSb\u0000c)\u00001\u0018k+kba(bSb\u0000c)\u00001k+k(bSb\u0000c)\u00001bak\n.1 +hcik(bS1\u0000c)\u00001k;(4.35)\nforb!+1.\nApplying the second resolvent identity, we have\nk(bSb\u0000c)\u00001\u0000(bS1\u0000c)\u00001k\u00141\n2j\u0015jk(bSb\u0000c)\u00001bq(bS1\u0000c)\u00001k\n+k(bSb\u0000c)\u00001(c2\n2\u0015+ic\n\u0015\u0018)(bS1\u0000c)\u00001k\n+k(bSb\u0000c)\u00001\u0018\u0018\u00002(bRb+c2\n2\u0015+ic\n\u0015\u0018)\u0018(bS1\u0000c)\u00001k\n.b\u00001k(bS1\u0000c)\u00001k(1 +hcik(bS1\u0000c)\u00001k)\n+c2b\u00001k(bS1\u0000c)\u00001k2\n+cb\u00001k(bS1\u0000c)\u00001k(1 +hcik(bS1\u0000c)\u00001k)\n+b\u00001(1 +hcik(bS1\u0000c)\u00001k)2\n.Kb\u00001k(bS1\u0000c)\u00001k;\n(4.36)\nasb!+1, where we have used (4.26), (4.30), (4.31), (4.35) and the fact that the\nresolvent ofbS1is bounded above and below on K. Thus\nk(bSb\u0000c)\u00001k=k(bS1\u0000c)\u00001k(1 +OK(b\u00001)); b!+1:RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 19\nSincebSb\u0000c= (2\u0015)\u00001UbeT(\u0015)U\u00001\nband moreoverkeT(\u0015)uk=kbT(\u0015)ukfor 06=u2\nDom(bT(\u0015)) such that supp u\u001a\nb;+, we arrive at\n2j\u0015jkuk= 2j\u0015jkeT(\u0015)\u00001eT(\u0015)uk\u0014k (bS1\u0000c)\u00001k(1 +OK(b\u00001))kbT(\u0015)uk; b!+1;\nas required.\nFor the case supp u\u001a\nb;\u0000, we repeat the above arguments but de\fning instead\neVb(\u0018) :=c2\u00002ic\u0018b\u00002\u0018b(\u0018+\u0018b) + (\u0018+\u0018b)2\u001f\nb;\u0000(\u0018), (Ubu)(\u0018) :=u(\u0018\u0000\u0018b),bRb(\u0018) :=\n\u0000(2\u0015)\u00001c2+i(\u0015)\u00001c\u0018+ (2\u0015)\u00001\u00182\u001f\nb(\u0018) andbS1=FA\u0003F\u00001=ba+i\u0018.\u0003\n4.1.3. Step 3: lower estimate.\nProposition 4.9. Let the assumptions of Theorem 4.3 hold and let bT(\u0015)andAbe\nas in (4.14) and(2.4) , respectively. Then there exist functions 06=ub2Dom(bT(\u0015))\nsuch that\nkbT(\u0015)ubk=k(A\u0000c)\u00001k\u000012b(1 +OK(b\u00001))kubk; b!+1:\nProof. We retain the notation introduced in the proof of Proposition 4.8; in par-\nticular,bS1andbSbare as in (4.27) and (4.28), respectively.\nWith a su\u000eciently large b0>0, theL2(R) operatorsbBb:= ((bS\u0003\nb\u0000c)(bSb\u0000c))\u00001,\nb2(b0;1], are compact, self-adjoint and non-negative. Let &2\nb>0 be their spectral\nradii andgb2Dom(bBb) be corresponding normalised eigenfunctions. Then gb2\nDom(bSb) and we have\nk(bSb\u0000c)gbk=&\u00001\nb=k(bSb\u0000c)\u00001k\u00001; b2(b0;+1]:\nMoreover from (4.36) we obtain\nj&b\u0000&1j=OK(&1b\u00001); b!+1: (4.37)\nConsider b2C1\nc((\u00002\u000eb;2\u000eb)), 0\u0014 b\u00141, b= 1 on (\u0000\u000eb;\u000eb) and such that\nk (j)\nbk1.(\u000eb)\u0000j; j2f1;2;:::;N + 1 +lg; (4.38)\nwithN:= maxfdmae;dmqeg+ 1 and su\u000eciently large l2N(see Remark 3.3 and\nthe statement of Lemma 2.1, in particular (2.3)). It is clear that  b!1 pointwise\ninRasb!+1.\nNext we justify that  bgb2Dom(ba) and therefore  bgb2Dom(bSb) (see (4.27)\nand (4.28)). Letting u2S(R), then bu2Dom(ba) and using the expansion (2.1)\nwe have\nba bu= bbau+ [ba; b]u= bbau+NX\nj=1ij\nj! (j)\nbba(j)u+RN+1u (4.39)\nand hence, applying Assumption 3.1 (ii), (4.38) and (2.3), there exists C > 0,\nindependent of b, such that\nkba buk\u0014kbauk+NX\nj=11\nj!k (j)\nbk1ka(j)\u0014uk+kRN+1uk\n\u0014kbauk+Cb\u00001(kbauk+kuk):\nButS(R) is a core for a, and hence for ba, and it therefore follows that\nkba bgbk\u0014kbagbk+Cb\u00001(kbagbk+kgbk):\nSincegb2Dom(ba), this shows that  bgb2Dom(ba).\nFurthermore\n(bSb\u0000c) bgb= (bSb\u0000c)gb+ ( b\u00001)(bSb\u0000c)gb+ [ba+ (2\u0015)\u00001bq; b]gb:20 ANTONIO ARNAL\nOur next goal is to estimate the second and third terms in the above equality.\nEmploying (4.37), (4.35) (and analogously for the adjoint bS\u0003\nb\u0000c) and expansions\nfor [ba; b] and [bq; b] such as (4.39), we obtain as b!+1\nk( b\u00001)(bSb\u0000c)gbk.k( b\u00001)\u0018\u00001k1k\u0018(bS\u0003\nb\u0000c)\u00001kk(bS\u0003\nb\u0000c)(bSb\u0000c)gbk\n.b\u00001(1 +hcik(bS1\u0000c)\u00001k)&\u00002\nb\n.Kb\u00001&\u00001\n1;\nk[ba+ (2\u0015)\u00001bq; b]gbk.b\u00001(kbagbk+kgbk) +b\u00002(kbqgbk+kgbk)\n.b\u00001(kbagbk+kgbk).b\u00001(k(bSb\u0000c)gbk+hcikgbk)\n.b\u00001(&\u00001\nb+hci).Kb\u00001&\u00001\n1;\nwhere in the two estimates before the last line we have also used Assumption 3.1 (iv)\nand (4.33). Hence k(bSb\u0000c) bgbk=&\u00001\nb+OK(&\u00001\n1b\u00001) asb!+1. Writing\n bgb=gb+ ( b\u00001)gb, we similarly obtain k bgbk= 1 +OK(&\u00001\n1b\u00001) asb!+1.\nThus applying (4.37), we arrive at\f\f\f\f\fk(bSb\u0000c) bgbk\nk bgbk\u00001\n&1\f\f\f\f\f=OK(&\u00001\n1b\u00001); b!+1:\nRecalling that bSb\u0000c= (2\u0015)\u00001UbeT(\u0015)U\u00001\nband letting ub:=U\u00001\nb bgb, then\nub2Dom(bT(\u0015)) with supp ub\u001a\nb;+. We therefore conclude\n\f\f\f\f\f(2j\u0015j)\u00001kbT(\u0015)ubk\nkubk\u00001\n&1\f\f\f\f\f=OK(&\u00001\n1b\u00001); b!+1\nand the claim follows. \u0003\n4.1.4. Step 4: combining the estimates. With \n0\nb;\u0006, \nb;\u0006and\u000ebas de\fned in (4.15),\n(4.22), let\u001eb;\u00062C1\nc(\nb;\u0006), 0\u0014\u001eb;\u0006\u00141, be such that\n\u001eb;\u0006(\u0018) = 1; \u00182\n0\nb;\u0006;k\u001e(j)\nb;\u0006k1.\u000e\u0000j\nb; j2f1;2;:::;N + 1 +lg; (4.40)\nwithN:= maxfdmae;dmqeg+ 1 and su\u000eciently large l2N(see Remark 3.3, the\nstatement of Lemma 2.1 and, in particular, the upper estimate (2.3)) and de\fne\n\u001eb;0(\u0018) := 1\u0000(\u001eb;+(\u0018) +\u001eb;\u0000(\u0018)); \u001eb;1(\u0018) :=\u001eb;+(\u0018);\n\u001eb;2(\u0018) :=\u001eb;\u0000(\u0018); \u00182R:(4.41)\nLemma 4.10. Let the assumptions of Theorem 4.3 hold, with baandbqas de\fned\nin(4.14) , and let\u001eb;k,k2f0;1;2g, be as de\fned in (4.41) . Then for all u2S(R),\nallk2f0;1;2g, we have\nk[bq+ 2\u0015ba;\u001eb;k]uk.\u000eb\u00001kbT(\u0015)uk+kuk; b!+1: (4.42)\nProof. Letu2S(R) andk2f0;1;2g, then by Lemma 2.1\n[ba;\u001eb;k]u=NX\nj=1ij\nj!\u001e(j)\nb;kba(j)u+Ra;N+1;ku: (4.43)\nNote that, since N\u00152, we have (see (2.3) and (4.40))\nkRa;N+1;kuk.b\u00003kuk; b!+1: (4.44)\nMoreover, using Assumption 3.1 (ii) with n=j, (4.40) and (4.10), we \fnd for\n2\u0014j\u0014Nandb!+1\nk\u001e(j)\nb;kba(j)uk\u0014k\u001e(j)\nb;kk1kba(j)uk.b\u00002k(1 +ba)uk\n.b\u00002(b\u00001kbT(\u0015)uk+bkuk):(4.45)RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 21\nIn order to estimate k\u001e0\nb;kba(1)uk, let us write \u001e0\nb;kba(1)u=ba(1)\u001e0\nb;ku\u0000[ba(1);\u001e0\nb;k]u.\nUsing Assumption 3.1 (ii) with n= 1 and (4.40) with j= 1, we deduce\nkba(1)\u001e0\nb;kuk.k(1 +ba)\u001e0\nb;kuk.b\u00001kuk+kba\u001e0\nb;kuk:\nFurthermore, noting that supp \u001e0\nb;ku\\(\n0\nb;+[\n0\nb;\u0000) =;and applying (4.21), we\nobtain asb!+1\nkba\u001e0\nb;kuk.\u000eb\u00001kbT(\u0015)\u001e0\nb;kuk.\u000eb\u00001(k\u001e0\nb;kbT(\u0015)uk+k[bT(\u0015);\u001e0\nb;k]uk)\n.\u000eb\u00001(b\u00001kbT(\u0015)uk+k[bq;\u001e0\nb;k]uk+bk[ba;\u001e0\nb;k]uk):\nApplying Lemma 2.1, Assumption 3.1 (ii) with n=j, (4.40) and (4.10), we have\nk[ba;\u001e0\nb;k]uk\u0014NX\nj=11\nj!k\u001e(j+1)\nb;kba(j)uk+kR0\na;N+1;kuk.b\u00002k(1 +ba)uk+b\u00004kuk\n.b\u00002(b\u00001kbT(\u0015)uk+bkuk); b!+1:\nMoreover, since 1 + q.1 +a(by Asumption 3.1 (iv)), we can similarly derive\nk[bq;\u001e0\nb;k]uk.b\u00002(b\u00001kbT(\u0015)uk+bkuk); b!+1:\nTherefore\nkba(1)\u001e0\nb;kuk.\u000eb\u00001(kuk+b\u00001kbT(\u0015)uk+b\u00002kbT(\u0015)uk+kuk)\n.\u000eb\u00001(b\u00001kbT(\u0015)uk+kuk); b!+1:(4.46)\nExpanding the term [ ba(1);\u001e0\nb;k]uas before, we have as b!+1\nk[ba(1);\u001e0\nb;k]uk\u0014NX\nj=11\nj!k\u001e(j+1)\nb;kba(j+1)uk+kR00\na;N+1;kuk\n.b\u00002k(1 +ba)uk+b\u00004kuk.b\u00002(b\u00001kbT(\u0015)uk+bkuk):(4.47)\nHence, combining (4.46) and (4.47), we obtain\nk\u001e0\nb;kba(1)uk.\u000eb\u00001(b\u00001kbT(\u0015)uk+kuk); b!+1: (4.48)\nSubstituting estimates (4.48), (4.45) and (4.44) in (4.43), we conclude\nk[ba;\u001eb;k]uk.\u000eb\u00001(b\u00001kbT(\u0015)uk+kuk); b!+1: (4.49)\nNote that repeating the above process for [ bq;\u001eb;k]u, and using 1+ q.1+afrom\nAssumption 3.1 (iv), we similarly \fnd\nk[bq;\u001eb;k]uk.\u000eb\u00001(b\u00001kbT(\u0015)uk+kuk); b!+1: (4.50)\nThe conclusion (4.42) follows from (4.49) and (4.50). \u0003\nLemma 4.11. Let the assumptions of Theorem 4.3 hold and let bT(\u0015)and\u001eb;k,\nk2f1;2g, be as de\fned in (4.14) and(4.41) , respectively. Then for all u2S(R),\nwe have as b!+1\n(kbT(\u0015)\u001eb;1uk2+kbT(\u0015)\u001eb;2uk2)1\n2=kbT(\u0015)(\u001eb;1+\u001eb;2)uk+O\u000e(b\u00001)(kbT(\u0015)uk+kuk):\n(4.51)\nProof. Letu2S(R) anduk:=\u001eb;kuwithk2f1;2g. Applying (2.1) to [ bq;\u001eb;k]\nand [ba;\u001eb;k], we have for any k2f1;2g\nbT(\u0015)uk=\u001eb;kbT(\u0015)u+ [bq+ 2\u0015ba;\u001eb;k]u\n=BN;k(\u0015)u+RN+1;k(\u0015)u;22 ANTONIO ARNAL\nwith\nBN;k(\u0015)u:=\u001eb;kbT(\u0015)u+NX\nj=1ij\nj!\u001e(j)\nb;k(bq(j)+ 2\u0015ba(j))u;\nRN+1;k(\u0015)u:=Rq;N+1;ku+ 2\u0015Ra;N+1;ku:\nThe remainders Rq;N+1;ku,Ra;N+1;kufor [bq;\u001eb;k], [ba;\u001eb;k], respectively, are de-\n\fned in (2.2). Noting that BN;1(\u0015)u\u001a\nb;+,BN;2(\u0015)u\u001a\nb;\u0000, and consequently\nBN;1(\u0015)u?BN;2(\u0015)uinL2, we deduce\nkbT(\u0015)(u1+u2)k2=kbT(\u0015)u1k2+kbT(\u0015)u2k2+ 2 RehBN;1(\u0015)u;RN+1;2(\u0015)ui\n+ 2 RehRN+1;1(\u0015)u;BN;2(\u0015)ui\n+ 2 RehRN+1;1(\u0015)u;RN+1;2(\u0015)ui:\nHence\nj(kbT(\u0015)u1k2+kbT(\u0015)u2k2)1\n2\u0000kbT(\u0015)(u1+u2)kj\n.kBN;1(\u0015)uk1\n2kRN+1;2(\u0015)uk1\n2+kRN+1;1(\u0015)uk1\n2kBN;2(\u0015)uk1\n2\n+kRN+1;1(\u0015)uk1\n2kRN+1;2(\u0015)uk1\n2:(4.52)\nSince (4.44) holds for Rq;N+1;kuandRa;N+1;ku, we \fnd for k2f1;2g\nkRN+1;k(\u0015)uk.b\u00002kuk; b!+1: (4.53)\nMoreover\nkBN;k(\u0015)u\u0000\u001eb;kbT(\u0015)uk\u0014k\u001e0\nb;k(bq(1)+ 2\u0015ba(1))uk+NX\nj=21\nj!k\u001e(j)\nb;k(bq(j)+ 2\u0015ba(j))uk:\nThe terms in the right-hand side of the above inequality have already been estimated\nin Lemma 4.10 (see (4.48), (4.45) and the comments regarding qat the end of the\nproof). Hence for k2f1;2g\njkBN;k(\u0015)uk\u0000kbT(\u0015)ukj.\u000eb\u00001kbT(\u0015)uk+kuk; b!+1: (4.54)\nApplying (4.53) and (4.54), we can estimate the \frst term in the right-hand side\nof (4.52) as b!+1\nkBN;1(\u0015)uk1\n2kRN+1;2(\u0015)uk1\n2.b\u00001kBN;1(\u0015)uk+bkRN+1;2(\u0015)uk\n.\u000eb\u00001(kbT(\u0015)uk+kuk):\nA similar estimate can be derived for kBN;2(\u0015)uk1\n2kRN+1;1(\u0015)uk1\n2which, combined\nwith (4.53), yields the desired result. \u0003\nProof of Theorem 4.3. Let 06=u2S(R)\u001aDom(bT(\u0015)) and let us write u=\nu0+u1+u2, whereuk:=\u001eb;kuwithk2f0;1;2gand\u001eb;kas de\fned in (4.41).\nThen\nbT(\u0015)uk=\u001eb;kbT(\u0015)u+ [bq+ 2\u0015ba;\u001eb;k]u; k2f0;1;2g;\nand therefore, noting that supp \u001eb;1\\supp\u001eb;2=;and applying Lemma 4.10, we\nobtain asb!+1\nkbT(\u0015)u0k\u0014(1 +O\u000e(b\u00001))kbT(\u0015)uk+O\u000e(1)kuk;\nkbT(\u0015)(u1+u2)k\u0014(1 +O\u000e(b\u00001))kbT(\u0015)uk+O\u000e(1)kuk:(4.55)RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 23\nFirstly, using the fact that u1?u2in combination with Proposition 4.8 and\nLemma 4.11, we \fnd as b!+1\nku1+u2k\u0014k (A\u0000c)\u00001k(2b)\u00001(1 +OK(b\u00001))(kbT(\u0015)u1k2+kbT(\u0015)u2k2)1\n2\n\u0014k(A\u0000c)\u00001k(2b)\u00001(1 +OK(b\u00001))(kbT(\u0015)(u1+u2)k\n+O\u000e(b\u00001)(kbT(\u0015)uk+kuk)):\nThus by (4.55) we have as b!+1\nku1+u2k\u0014k (A\u0000c)\u00001k(2b)\u00001(1 +OK(b\u00001))kbT(\u0015)uk+OK(b\u00001)kuk:(4.56)\nSecondly, since supp u0\\(\n0\nb;+[\n0\nb;\u0000) =;, then by Proposition 4.7\nb2ku0k.\u000ekbT(\u0015)u0k; b!+1;\nand applying (4.55) we have\nku0k.\u000eb\u00002(kbT(\u0015)uk+kuk); b!+1: (4.57)\nCombining (4.56) and (4.57), we \fnd that for b!+1\nkuk\u0014k (A\u0000c)\u00001k(2b)\u00001(1 +OK(b\u00001))kbT(\u0015)uk+OK(b\u00001)kuk\nand therefore\nkuk\u0014k (A\u0000c)\u00001k(2b)\u00001(1 +OK(b\u00001))kbT(\u0015)uk:\nSinceS(R) is a core for T(\u0015), and equivalently for bT(\u0015), we can extend the above\nestimate to any u2Dom(bT(\u0015)) relying on standard approximation arguments. The\nproof of the theorem follows by an appeal to Proposition 4.9 and the use of the\ninverse Fourier transform to take the result back to x-space. \u0003\n4.2.The norm of the resolvent along curves adjacent to the imaginary\naxis. As in the analysis for Schr odinger operators with complex potential carried\nout in [5, Sub-section 5.1], it is possible to extend the proof of Theorem 4.3 to more\ngeneral curves inside the left-hand side semi-plane C\u0000\n\u0015b:=\u0000c(b) +ib; (4.58)\nwhereb2Rnf0gandc:Rnf0g!R+satis\fes\ncbjbj\u00001=o(1);jbj!+1; (4.59)\n\bb:=hcbi2k(A\u0000cb)\u00001kjbj\u00001=o(1);jbj!+1; (4.60)\nwithAas de\fned in (2.4) and cb\u0011c(b). We are interested in two types of curves:\n(1)\u0015bwithcbsatisfying\ncb.1; b!+1; (4.61)\n(2)\u0015bwithcbsatisfying\nhcbik(A\u0000cb)\u00001k!+1; b!+1: (4.62)\nNote that, when (4.61) holds ( e.g.in the statement of Theorem 4.3), we have\nhcbi2k(A\u0000cb)\u00001k.1 and therefore conditions (4.59)-(4.60) are both automatically\nsatis\fed.\nWe also observe that, because of Assumption (4.59), we have \u00182+\u00152=\u00182\u0000\nb2(1 +o(1)) whenjbj!+1, as in the proof of Theorem 4.3.24 ANTONIO ARNAL\nProposition 4.12. Letaandqsatisfy Assumption 3.1 and let T(\u0015b)be the family\nof operators (2.11) -(2.12) for\u0015bde\fned by (4.58) . Assume furthermore that (4.59) -\n(4.60) hold withcbsatisfying either (4.61) or(4.62) . Then\nkT(\u0015b)\u00001k=k(A\u0000cb)\u00001k(2jbj)\u00001(1 +O(\bb));jbj!+1; (4.63)\nwithAas de\fned in (2.4) .\nSketch of proof. We shall closely follow the steps in Sub-section 4.1, keeping the\nnotation introduced there but omitting details whenever the arguments used earlier\nremain valid. As before, it is enough for us to consider the case b >0,b!+1.\nWe note that, for families \u0015bsatisfying (4.61) or (4.62), the choice of parameter \u000e\nin (4.15) is independent of b(see Step 2 below).\nStep 1\nAssumption (4.59) is enough to ensure that (4.21) continues to hold for \u0015band\nhence we have as b!+1\nb2.inf(\nkbT(\u0015b)uk\nkuk: 06=u2Dom(bT(\u0015b));suppu\\(\n0\nb;+[\n0\nb;\u0000) =;)\n:\nStep 2\nWe use the notation in Proposition 4.8, replacing \u0015with\u0015bandcwithcbwhere\nnecessary. From (4.24), (4.27) and (4.28), we have\nbSb\u0000cb=1\n2\u0015bUbeT(\u0015b)U\u00001\nb=bS1\u0000cb+1\n2\u0015bbq+bRb; (4.64)\nwithbRb(\u0018) as de\fned in (4.25). Our next aim is to prove that cb2\u001a(bSb) as\nb!+1. To do this, we argue as in Step 2 of [5, Prop. 5.1]. For any cb>0,\nthe operator bKb;1:=I\u0000cbbS\u00001\n1=bS\u00001\n1(bS1\u0000cb) = (bS1\u0000cb)bS\u00001\n1is bounded and\ninvertible and moreover by (4.30) (note also that either (4.61) or (4.62) holds by\nassumption) we have for b!+1\nkbK\u00001\nb;1k.hcbik(A\u0000cb)\u00001k: (4.65)\nRecalling from Proposition 4.8 that 0 2\u001a(bSb) for large enough band de\fning\nbKb:=I\u0000cbbS\u00001\nb=bS\u00001\nb(bSb\u0000cb) = (bSb\u0000cb)bS\u00001\nb, we \fnd\nbKb=bKb;1(I\u0000cbbK\u00001\nb;1(bS\u00001\nb\u0000bS\u00001\n1)):\nMoreover, by (4.36) with c= 0, (4.65) and (4.60), we have\nkcbbK\u00001\nb;1(bS\u00001\nb\u0000bS\u00001\n1)k.\bb=o(1); b!+1:\nIt follows that bKbis invertible andkbK\u00001\nbk\u0019kbK\u00001\nb;1kasb!+1. SincebSb\u0000cb=\nbKbbSb=bSbbKb, we conclude that cb2\u001a(bSb) forb!+1, as claimed. Moreover,\n(bSb\u0000cb)\u00001=bS\u00001\nbbK\u00001\nb=bK\u00001\nbbS\u00001\nband, applying (4.35) with c= 0 and (4.65), we\ndeduce asb!+1\nk\u0018(bSb\u0000cb)\u00001k+k\u0018(bS\u0003\nb\u0000cb)\u00001k.hcbik(A\u0000cb)\u00001k: (4.66)\nFurthermore, we have (see the argument in [5, Eq. (5.15)])\n((bSb\u0000cb)\u00001\u0000(bS1\u0000cb)\u00001)bKb=bK\u00001\nb;1(bS\u00001\nb\u0000bS\u00001\n1):\nHence\n(bSb\u0000cb)\u00001\u0000(bS1\u0000cb)\u00001=bK\u00001\nb;1(bS\u00001\nb\u0000bS\u00001\n1)bK\u00001\nb; b!+1;\nand therefore by (4.36) with c= 0 and (4.65), we have\nk(bSb\u0000cb)\u00001\u0000(bS1\u0000cb)\u00001k.k(A\u0000cb)\u00001k\bb; b!+1:RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 25\nIt follows that\nk(bSb\u0000cb)\u00001k=k(A\u0000cb)\u00001k(1 +O(\bb)); b!+1; (4.67)\nand hence from (4.64) and (4.67) as b!+1\n2j\u0015bjkeT(\u0015b)\u00001k=k(bSb\u0000cb)\u00001k=k(A\u0000cb)\u00001k(1 +O(\bb)):\nArguing as in the last stage of Proposition 4.8 and noting that j\u0015bj=bp\n1 +c2\nbb\u00002=\nb(1 +O(c2\nbb\u00002)) and furthermore, by (4.59) and (4.61)-(4.62), we have c2\nbb\u00002.\bb,\nwe deduce as b!+1\nk(A\u0000cb)\u00001k\u000012b(1\u0000O(\bb))\n\u0014inf8\n<\n:\r\r\rbT(\u0015b)u\r\r\r\nkuk: 06=u2Dom(bT(\u0015b));suppu\u001a\nb;\u00069\n=\n;:(4.68)\nStep 3\nWe follow the proof of Proposition 4.9, replacing bSb\u0000cwithbSb\u0000cb, to \fnd\ngb2Dom((bS\u0003\nb\u0000cb)(bSb\u0000cb)) such that\nk(bSb\u0000cb)gbk=&\u00001\nb=k(bSb\u0000cb)\u00001k\u00001; b!+1:\nMoreover, with &b;1:=k(A\u0000cb)\u00001k, we have (see (4.67))\n&b=&b;1(1 +O(\bb)); b!+1: (4.69)\nRecalling the cut-o\u000b functions  b, we write\n(bSb\u0000cb) bgb= (bSb\u0000cb)gb+ ( b\u00001)(bSb\u0000cb)gb+ [ba+ (2\u0015)\u00001bq; b]gb\nand we proceed to estimate as before the second and third terms in the right-hand\nside of the above equality, using also (4.66), (4.61)-(4.62) and (4.69), for b!+1\nk( b\u00001)(bSb\u0000cb)gbk.b\u00001hcbik(A\u0000cb)\u00001k&\u00002\nb.b\u00001hcbi&\u00001\nb;1.\bbhcbi\u00001&\u00002\nb;1\nk[ba+ (2\u0015)\u00001bq; b]gbk.b\u00001(k(bSb\u0000cb)gbk+hcbikgbk).b\u00001hcbi.\bbhcbi\u00001&\u00001\nb;1:\nHencek(bSb\u0000cb) bgbk=&\u00001\nb+O(\bbhcbi\u00001&\u00001\nb;1(1 +&\u00001\nb;1)) asb!+1. Writing\n bgb=gb+ ( b\u00001)gb, we similarly obtain k bgbk= 1 +O(\bbhcbi\u00001&\u00001\nb;1) as\nb!+1. Thus applying (4.69), we arrive at\n\f\f\f\f\fk(bSb\u0000cb) bgbk\nk bgbk\u00001\n&b;1\f\f\f\f\f=O(&\u00001\nb;1\bb); b!+1:\nRecalling that bSb\u0000cb= (2\u0015b)\u00001UbeT(\u0015b)U\u00001\nband letting ub:=U\u00001\nb bgb, then\nub2Dom(bT(\u0015b)) with supp ub\u001a\nb;+and we have\n\f\f\f\f\f(2j\u0015bj)\u00001kbT(\u0015b)ubk\nkubk\u00001\n&b;1\f\f\f\f\f=O(&\u00001\nb;1\bb); b!+1:\nHence\nkbT(\u0015b)ubk\nkubk=k(A\u0000cb)\u00001k\u000012b(1 +O(\bb)); b!+1: (4.70)\nStep 4\nIt is straightforward to verify that estimates (4.42) in Lemma 4.10 and (4.51) in\nLemma 4.11 continue to hold when we replace \u0015with\u0015b(with\u000eindependent of b).\nAs in the proof of Theorem 4.3, we have as b!+1\nkbT(\u0015b)u0k\u0014(1 +O(b\u00001))kbT(\u0015b)uk+O(1)kuk;\nkbT(\u0015b)(u1+u2)k\u0014(1 +O(b\u00001))kbT(\u0015b)uk+O(1)kuk:(4.71)26 ANTONIO ARNAL\nBy (4.68), (4.51) and (4.71), we obtain for b!+1\n2bku1+u2k\u0014k (A\u0000cb)\u00001k(1 +O(\bb))kbT(\u0015b)(u1+u2)k\n+O(k(A\u0000cb)\u00001kb\u00001)(kbT(\u0015b)uk+kuk)\n\u0014k(A\u0000cb)\u00001k(1 +O(\bb))kbT(\u0015b)uk+O(k(A\u0000cb)\u00001k)kuk:(4.72)\nBy (4.21) and (4.71), we have as b!+1\n2bku0k.b\u00001(kbT(\u0015b)uk+kuk): (4.73)\nCombining (4.72) and (4.73), we \fnd that as b!+1\n2bkuk\u00142b(ku0k+ku1+u2k)\n\u0014k(A\u0000cb)\u00001k(1 +O(\bb))kbT(\u0015b)uk+O(k(A\u0000cb)\u00001k)kuk\nand hence\n2b(1\u0000O(k(A\u0000cb)\u00001kb\u00001))kuk\u0014k (A\u0000cb)\u00001k(1 +O(\bb))kbT(\u0015b)uk:\nIt follows\nkuk\u0014k (A\u0000cb)\u00001k(2b)\u00001(1 +O(\bb))kbT(\u0015b)uk; b!+1:\nThis result combined with the lower bound (4.70) yields (4.63). \u0003\nAs an application of Proposition 4.12 and the resolvent norm estimate for gen-\neralised Airy operators found in [4, Thm. 4.2], we shall consider two examples of\ndamping functions that satisfy Assumption 3.1 and [4, Asm. 3.1] (note that the\nchoice ofqplays no role in the calculations provided that Assumptions 3.1 (iii)-(iv)\nare satis\fed).\n(i)a(x) = loghxip,p>0. We have (see [4, Ex. 4.3(i)])\nk(A\u0000cb)\u00001k=r\u0019\npexp \n2ps\nexp\u00122cb\np\u0013\n\u00001 +cb\n2p\u0000p\u0019!\n(1 +o(1)); b!+1;\n=)log log(k(A\u0000cb)\u00001k) =cb\np(1 +o(1)); b!+1:\nUsing (4.63) and substituting k(T(\u0015b))\u00001k=\"\u00001, with\" >0, we obtain the level\ncurves\ncb=plog log(2b\"\u00001)(1 +o(1)); b!+1:\nNote that, in terms of assumptions (4.59)-(4.60), we have cbb\u00001=o(1) and further-\nmore, since \b b\u0019(log logb)2b2p\u00001, we \fnd that the admissible curves are determined\nby 0<p< 1=2.\n(ii)a(x) =x2n,n2N. Applying [4, Ex. 4.3(ii)] with p= 2n, we have\nk(A\u0000cb)\u00001k=r\u0019\n2nc1\u00002n\n4n\nbexp\u00124n\n2n+ 1c2n+1\n2n\nb\u0013\n(1 +o(1)); b!+1:(4.74)\nLetting\u0010b:=c6n+1\n4n\nbk(A\u0000cb)\u00001k, we re-write (4.74) as\n4n\n2n+ 1c2n+1\n2n\nbexp\u00124n\n2n+ 1c2n+1\n2n\nb\u0013\n=4n\n2n+ 1r\n2n\n\u0019\u0010b(1 +o(1)); b!+1;RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 27\nwhich can be solved as highlighted in [5, Sub-sec. 5.1.1] via the Lambert function\n4n\n2n+ 1c2n+1\n2n\nb=W0 \n4n\n2n+ 1r\n2n\n\u0019\u0010b(1 +o(1))!\n=)\n4n\n2n+ 1c2n+1\n2n\nb= log(k(A\u0000cb)\u00001k)(1 +o(1)) =)\ncb=\u00122n+ 1\n4n\u0013 2n\n2n+1\u0000\nlog(k(A\u0000cb)\u00001k)\u00012n\n2n+1(1 +o(1));\nforb!+1. Applying (4.63) and substituting k(T(\u0015b))\u00001k=\"\u00001, with\">0, we\nobtain\ncb=\u00122n+ 1\n4n\u0013 2n\n2n+1\u0000\nlog(2b\"\u00001)\u00012n\n2n+1(1 +o(1)); b!+1: (4.75)\nClearly we have cbb\u00001=o(1) asb!+1. On the other hand, \b b\u0019c3p+1\n2p\nb\u0019\n(logb)3p+1\n2(p+1)!+1asb!+1for anyp>0. We shall therefore just put forward\nas a conjecture that (4.75) asymptotically describes the level curves in this case.\n5.The operator G\n5.1.Proof of Theorem 3.5.\nProof of Theorem 3.5. Our \frst goal is to \fnd a lower bound for k(G\u0000\u0015)\u00001k, with\n\u0015:=\u0000c+ibde\fned in the statement of the theorem, as jbj!+1.\nLet us take arbitrary u1;u22C1\nc(R) and letu:= (u1;u2)t. Thenu2Dom(G)\u001a\nHand\nkukH= (k@xu1k2+kq1\n2u1k2+ku2k2)1\n2;\nk(G\u0000\u0015)ukH= (k@x(\u0015u1\u0000u2)k2+kq1\n2(\u0015u1\u0000u2)k2+kHqu1+ (2a+\u0015)u2k2)1\n2:\nChoosingu2:=\u0015u1withu16= 0, we havekukH\u0015jbjku1kandk(G\u0000\u0015)ukH=\nkT(\u0015)u1k. Noticing that, by the spectral equivalence (2.8), ( G\u0000\u0015)\u00001exists if and\nonly ifT(\u0015)\u00001exists and that the existence of T(\u0015)\u00001is guaranteed by Theorem 4.3\nfor su\u000eciently large jbj, it follows that\nkukH\nk(G\u0000\u0015)ukH\u0015jbjku1k\nkT(\u0015)u1k=) k(G\u0000\u0015)\u00001k\u0015jbjkT(\u0015)\u00001k;\nwhere, for the last implication, we have used the fact that C1\nc(R) is densely con-\ntained inL2(R). An application of Theorem 4.3 shows that k(G\u0000\u0015)\u00001k&K1 as\njbj!+1.\nIn order to \fnd an upper bound for k(G\u0000\u0015)\u00001k, let 06=v2C1\nc(R)\u0002C1\nc(R)\u001aH\nand setu:= (G\u0000\u0015)\u00001v2Dom(G). Using (2.9) and (2.11), we have\nu=\u0012\u0000\u0015\u00001(I\u0000T(\u0015)\u00001Hq)\u0000T(\u0015)\u00001\nT(\u0015)\u00001Hq\u0000\u0015T(\u0015)\u00001\u0013\u0012v1\nv2\u0013\nand therefore\nu1=\u0000\u0015\u00001(I\u0000T(\u0015)\u00001Hq)v1\u0000T(\u0015)\u00001v2\nu2=T(\u0015)\u00001Hqv1\u0000\u0015T(\u0015)\u00001v2:(5.1)\nOur next task is to estimate kukHand to this end we shall \fnd upper bounds for\nkH1\n2qu1kandku2k, withu1andu2as in (5.1).\nConsidering \frstly u2and applying (4.13) and (4.11), we obtain as jbj!+1\nku2k.KkH1\n2qv1k+kv2k: (5.2)28 ANTONIO ARNAL\nTurning to H1\n2qu1we \fnd\nH1\n2qu1=\u0000\u0015\u00001(H1\n2qv1\u0000H1\n2qT(\u0015)\u00001Hqv1)\u0000H1\n2qT(\u0015)\u00001v2\n(note thatHqis a positive self-adjoint operator and Dom( T(\u0015))\u001aDom(Hq), hence\nthe above operations make sense). Applying (4.13), it follows as jbj!+1\nkH1\n2qu1k.Kjbj\u00001(kH1\n2qv1k+kH1\n2qT(\u0015)\u00001Hqv1k) +kv2k: (5.3)\nTo estimatekH1\n2qT(\u0015)\u00001H1\n2qk, we use the second resolvent identity with \u0016>0\nT(\u0015)\u00001=T(\u0016)\u00001+ (\u0016\u0000\u0015)T(\u0016)\u00001(2a+\u0016+\u0015)T(\u0015)\u00001\n=\u0016\n\u0015T(\u0016)\u00001+\u0016(\u0016\u0000\u0015)T(\u0016)\u00001T(\u0015)\u00001\u0000\u0016\u0000\u0015\n\u0015T(\u0016)\u00001HqT(\u0015)\u00001:\nHence\nH1\n2qT(\u0015)\u00001H1\n2q=\u0016\n\u0015H1\n2qT(\u0016)\u00001H1\n2q+\u0016(\u0016\u0000\u0015)H1\n2qT(\u0016)\u00001T(\u0015)\u00001H1\n2q\n\u0000\u0016\u0000\u0015\n\u0015H1\n2qT(\u0016)\u00001HqT(\u0015)\u00001H1\n2q:\nLettingz\u0015;\u0016:=\u0000\u0015=(\u0016\u0000\u0015) = (c2+c\u0016+b2\u0000i\u0016b)=((c+\u0016)2+b2), we deduce\n(H1\n2qT(\u0016)\u00001H1\n2q\u0000z\u0015;\u0016)H1\n2qT(\u0015)\u00001H1\n2q=\u0016\n\u0016\u0000\u0015H1\n2qT(\u0016)\u00001H1\n2q\n+\u0016\u0015H1\n2qT(\u0016)\u00001T(\u0015)\u00001H1\n2q:\nWe observe that H1\n2qT(\u0016)\u00001H1\n2qis self-adjoint, positive and it can be boundedly\nextended to L2(R) by (4.5). Furthermore, for b6= 0 we have z\u0015;\u0016=2Rand therefore\nthe operator H1\n2qT(\u0016)\u00001H1\n2q\u0000z\u0015;\u0016is invertible and\nk(H1\n2qT(\u0016)\u00001H1\n2q\u0000z\u0015;\u0016)\u00001k\u00141\njImz\u0015;\u0016j=(c+\u0016)2+b2\n\u0016jbj:\nHence\nH1\n2qT(\u0015)\u00001H1\n2q= (H1\n2qT(\u0016)\u00001H1\n2q\u0000z\u0015;\u0016)\u00001\u0012\u0016\n\u0016\u0000\u0015H1\n2qT(\u0016)\u00001H1\n2q\n+\u0016\u0015H1\n2qT(\u0016)\u00001T(\u0015)\u00001H1\n2q\u0013\n:\nChoosing\u0016=jbjand noticing\nk(H1\n2qT(jbj)\u00001H1\n2q\u0000z\u0015;jbj)\u00001k\u0014(c+jbj)2+b2\nb2.1;jbj!+1;\nwe obtain (applying (4.5), (4.4) and (4.13))\nkH1\n2qT(\u0015)\u00001H1\n2qk.jbj\n((c+jbj)2+b2)1\n2kH1\n2qT(jbj)\u00001\n2kkT(jbj)\u00001\n2H1\n2qk\n+jbj(c2+b2)1\n2kH1\n2qT(jbj)\u00001kkT(\u0015)\u00001H1\n2qk\n.K1 +b2jbj\u00001.Kjbj;jbj!+1:\nReturning with this estimate to (5.3), we obtain\nkH1\n2qu1k.Kjbj\u00001(kH1\n2qv1k+jbjkH1\n2qv1k) +kv2k\n.KkH1\n2qv1k+kv2k;jbj!+1:(5.4)RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 29\nCombining (5.2) and (5.4), we have\nkukH\nkvkH=(kH1\n2qu1k2+ku2k2)1\n2\nkvkH.K1;jbj!+1:\nSince 06=vis arbitrary and C1\nc(R)\u0002C1\nc(R) is dense inH, it follows thatk(G\u0000\n\u0015)\u00001k.K1 asjbj!+1, which concludes the proof. \u0003\n6.An example\nTo illustrate our results, we study the operator Gassociated with the damping\nand potential\na(x) =x2; q(x) =\u0014x2; \u0014> 0; x2R: (6.1)\nThese functions satisfy Assumption 3.1 and consequently we deduce that Gis m-\ndissipative and, in particular, \u001b(G)\u001aC\u0000. Our next result provides a description\nof\u001b(G) and the behaviour of the C0-semigroup of contractions generated by G.\nProposition 6.1. LetGdenote the linear operator de\fned in the statement of\nTheorem 3.5 with a(x)andq(x)determined by (6.1) . Then the following hold.\n(i)The spectrum of Gis\n\u001b(G) = (\u00001;\u0000\u0014=2]tf\u0015r\nn;\u0015i\nn;\u0015in:n2N0g\nwhere, for each n2N0, the numbers \u0015r\nn;\u0015i\nn;\u0015inare the solutions of the\nequation\n\u00154\u00002(2n+ 1)2\u0015\u0000(2n+ 1)2\u0014= 0\nsatisfying\u00152C\u0000n(\u00001;\u0000\u0014=2]. Moreover, as n!+1\n\u0015r\nn=\u0000\u0014\n2 \n1\u00002\u00008\n3\n3\u00142(2n+ 1)\u00004\n3+o(\u00142(2n+ 1)\u00004\n3)!\n; (6.2)\n\u0015i\nn= 21\n3(2n+ 1)2\n3 \n1\u00002\u00007\n3\n3\u0014(2n+ 1)\u00002\n3+o(\u0014(2n+ 1)\u00002\n3)!\nei(\u0019\u0000\u0012n);(6.3)\nwith\n\u0012n= arctan \np\n3 \n1 +2\u00001\n3\n3\u0014(2n+ 1)\u00002\n3+o(\u0014(2n+ 1)\u00002\n3)!!\n: (6.4)\n(ii)With de\fnitions (3.2) and(3.3) , we have\n!0=s(G)<0:\nRemark 6.2. It is clear from the above asymptotic expansions that \u0015r\nn!\u0000\u0014=2\nasn!+1(i.e. the sequence of real eigenvalues of Gaccumulates on\u0000\u0014=2, in\nline with Remark 3.3 in [18]). Furthermore, j\u0015i\nnj!+1and arg(\u0015i\nn)!2\u0019=3 as\nn!+1.\nProof of Proposition 6.1. (i) Sincea(x) =\u0014\u00001q(x) for every x2R, applying [18,\nThm. 3.2] and [18, Rmk. 3.3], we deduce that \u001be2(G)\u001a(\u00001;\u0000\u0014=2]. More-\nover,\u001be2(G) is closed and it is therefore su\u000ecient to show ( \u00001;\u0000\u0014=2)\u001a\u001be2(G)\nto conclude \u001be2(G) = (\u00001;\u0000\u0014=2]. In order to do this, selecting an arbitrary\n\u00152(\u00001;\u0000\u0014=2), we will construct a singular sequence (\b n)n2N\u001aDom(G) for\u0015\nadapting the proof of [18, Thm. 4.2].\nLetting\nA(x) :=\u0000(q(x) + 2\u0015a(x) +\u00152) = 2j\u0015+\u0014=2jx2\u0000\u00152; x2R;30 ANTONIO ARNAL\nwe note that A2C1(R) and\nlim\nx!+1A(x) = +1;lim\nx!+1jA0(x)j\nA(x)= 0: (6.5)\nOur main goal is to \fnd a sequence ( \u001en)n2N\u001aW2;2(R)\\Dom(x2) such that (the\nin\fmum of) supp \u001engoes to in\fnity in R+asn!+1and\nlim\nn!+1k@2\nx\u001en+A\u001enk\nk@x\u001enk= 0: (6.6)\nBy de\fning \b n:= (\u001en;\u0015\u001en)t, it follows from each of the above two properties of\n(\u001en)n2Nthat \bn=k\bnkHw!0 asn!+1and\nk(G\u0000\u0015)\bnkH\nk\bnkH\u0014k@2\nx\u001en+A\u001enk\nk@x\u001enk!0; n!+1;\nrespectively.\nLetting\u000b\u0015:=j\u0015j=p\n2j\u0015+\u0014=2jand applying (6.5), we have\nA(x)>0; x>\u000b \u0015;\n\u001an:= sup\nt>njA0(t)j\nA(t)!0; n>\u000b \u0015; n!+1:\nLet us de\fne \u001en(x) :='n(x) \u0015(x),x2R,n2N, where\n \u0015(x) := exp\u0012\niZx\n\u000b\u0015(A(t))1\n2dt\u0013\n; x\u0015\u000b\u0015;\n'n(x) :=\u001a1\n4n'(\u001a1\n2nx\u0000n); '2C1\nc((0;1));k'k= 1:\nFrom these de\fnitions, we immediately deduce: (i) \u001a\u00001\n2nn!+1, asn!+1, and\n(ii) supp'n\u001a(\u001a\u00001\n2nn;\u001a\u00001\n2n(n+ 1)),n2N. Furthermore\nk'nk2=\u001a1\n2nZ\nRj'(\u001a1\n2nx\u0000n)j2dx= 1; n2N;\nk'0\nnk2=\u001a3\n2nZ\nRj'0(\u001a1\n2nx\u0000n)j2dx=k'0k2\u001an=o(1); n!+1;\nk'00\nnk2=\u001a5\n2nZ\nRj'00(\u001a1\n2nx\u0000n)j2dx=k'00k2\u001a2\nn=o(1); n!+1;\nand forx\u0015\u000b\u0015\n 0\n\u0015(x) :=i(A(x))1\n2 \u0015(x);\n 00\n\u0015(x) :=i1\n2(A(x))\u00001\n2A0(x) \u0015(x)\u0000A(x) \u0015(x):\nStraightforward calculations show that as n!+1\nk@x\u001enk\u0015k'n 0\n\u0015k\u0000k'0\nnk&\u001a\u00001\n2nn=) k@x\u001enk\u00001=O(\u001a1\n2nn\u00001);\nand\nk@2\nx\u001en+A\u001enk\u0014k'00\nnk+ 2kA1\n2'0\nnk+1\n2kA\u00001\n2A0'nk.\u001an+n+ 1 + 1\n=) k@2\nx\u001en+A\u001enk=O(n):\nThereforek@2\nx\u001en+A\u001enk=k@x\u001enk=O(\u001a1\n2n) which shows that (6.6) holds. We con-\nclude that (\b n)n2Nis indeed a singular sequence and \u00152\u001be2(G) as claimed.RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 31\nTo determine the eigenvalues of G, we apply the spectral equivalence (2.8) (see\nalso [18, Thm. 3.2] and [18, Rmk. 3.3]) and seek to \fnd the set of \u00152C\u0000n\n(\u00001;\u0000\u0014=2] such that 02\u001bp(T(\u0015)), where\nT(\u0015) =\u0000@2\nx+ (\u0014+ 2\u0015)x2+\u00152;Dom(T(\u0015)) =W2;2(R)\\Dom(x2);\ni.e. we need to \fnd every \u00152C\u0000n(\u00001;\u0000\u0014=2] such that\n\u0000u00\n\u0015(x) + (2\u0015+\u0014)x2u\u0015(x) =\u0000\u00152u\u0015(x); x2R; (6.7)\nfor some 06=u\u00152Dom(T(\u0015)).\nTo this end, with \r2Candjarg(\r)j< \u0019, let us consider the \r-dependent\nSchr odinger operator family\nH\r:=\u0000@2\nx+\rx2;Dom(H\r) :=W2;2(R)\\Dom(x2):\nIt has been shown that H\ris a family of closed operators with compact resolvent\n(see [18, Lem. 2.3]). Furthermore, the spectrum of the rotated operator eH\r:=\n\u0000\r\u00001\n2@2\nx+\r1\n2x2is independent of \r(see [10, Lem. 5]). Since \u001b(eH1) =f2n+ 1 :\nn2N0g, with corresponding eigenfunctions ~ un(x) =Hn(x) exp(\u0000x2=2),x2R,\nn2N0, whereHnare the Hermite polynomials (see e.g. [20, Sec. 1.3]), it follows\nthat\u001b(H\r) =f(2n+ 1)\r1\n2:n2N0gwith eigenfunctions un(x) = ~un(\r1\n4x),x2R,\nn2N0. To verify that ( un)n2N0\u001aDom(H\r), consider u0(x) = exp(\u0000\r1\n2x2=2) and\nobserve thatjarg(\r)j< \u0019 andx2Rtogether imply that Re( \r1\n2)x2>0 (x6= 0).\nHence we conclude that u0, its derivatives and its product with any polynomial\nbelong toL2(R).\nTherefore, setting \r= 2\u0015+\u0014, the solutions of the eigenvalue problem (6.7) must\nsatisfy the family of equations\n\u00154= (2n+ 1)2(2\u0015+\u0014); n2N0: (6.8)\nEach of the above equations is a (reduced) quartic which can be solved in a standard\nway by Ferrari's method re-casting it as a product of two quadratics. To do this,\nwe re-write (6.8) as\n(\u00152+y)2=\u0012p\n2y\u0015+(2n+ 1)2\np2y\u00132\n;\nwherey6= 0 is a solution of\ny3+ (2n+ 1)2\u0014y\u00001\n2(2n+ 1)4= 0:\nOne can verify that this cubic has a (real) root given by Cardano's formula (with\n\u0014n:= (16=27)\u00143(2n+ 1)\u00002>0)\nyn= 2\u00002\n3(2n+ 1)4\n3\u0012\u0010\n(1 +\u0014n)1\n2+ 1\u00111\n3\u0000\u0010\n(1 +\u0014n)1\n2\u00001\u00111\n3\u0013\n; n2N0:(6.9)\nNoting that 0 < yn<2\u00001\n3(2n+ 1)4\n3, for every n2N0, the solutions of (6.8) are\n(withyn;\u0006:= (4(2n+ 1)2(2yn)\u00003\n2\u00061)1\n2>0)\n\u0015r\nn;\u0006:=\u00001\n2(2yn)1\n2(\u00001\u0007yn;\u0000);\n\u0015i\nn;\u0006:=\u00001\n2(2yn)1\n2(1\u0007iyn;+);\nfor everyn2N0. Next we examine each of these two sets of roots in turn.\nSinceGis m-dissipative, any eigenvalues must lie in the semi-plane f\u00152C:\nRe\u0015\u00140g. Hence, appealing once more to the spectral equivalence (2.8), we discard\n\u0015r\nn;+for everyn2N0as admissible solutions of the problem (6.7). Denoting32 ANTONIO ARNAL\n\u0015r\nn:=\u0015r\nn;\u0000, straightforward but somewhat lengthy calculations show that \u0015r\nn<0\nfor everyn2N0and we have asymptotically\n\u0015r\nn=\u0000\u0014\n2 \n1\u00002\u00008\n3\n3\u00142(2n+ 1)\u00004\n3+o(\u00142(2n+ 1)\u00004\n3)!\n; n!+1;\nas claimed in (6.2). Let us now consider whether any \u0015n2(\u00001;\u0000\u0014=2] can be a\nsolution of the eigenvalue problem. If \u0015n+\u0014=2 = 0, equation (6.7) becomes\n\u0000u00(x) +\u00152\nnu(x) = 0;\nwhose general solution un(x) =C1exp(j\u0015njx) +C2exp(\u0000j\u0015njx) does not belong to\nL2(R) unlessC1=C2= 0. For\u0015n+\u0014=2<0, applying the change of variable\ny=4p\n4j2\u0015n+\u0014jxenables us to re-write (6.7) as\nv00(y) +\u00121\n4y2\u0000bn\u0013\nv(y) = 0; (6.10)\nwithbn:=\u00152\nn=p\n4j2\u0015n+\u0014j>0. Note that equation (6.10) does not have L2\nsolutions. This is a standard result from the theory of Sturm-Liouville operators\nin the positive half-line case (see e.g. [31, Thm. 5.10] or [14, Thm. 3.5.6]). For\nthe whole line, the problem can be reduced to the positive half-line with Dirichlet\n(or Neumann) boundary condition at 0. From this analysis, we conclude that\nonly\u0015n2(\u0000\u0014=2;0) are admissible real solutions of (6.7). Hence (noting also\nRemark 6.2) we \fnd\n\u0000\u0014\n2<supf\u0015r\nn2\u001bp(G) :n2N0g<0: (6.11)\nOn the other hand, every imaginary solution \u0015i\nn;\u0006,n2N0, is admissible because\nRe\u0015i\nn;\u0006=\u0000(1=2)(2yn)1\n2<0 and Im\u0015i\nn;\u0006=\u0006(1=2)(2yn)1\n2yn;+6= 0. Moreover, it\nis straightforward to verify that ynis an increasing function of nand hence (with\n\u0015i\nn:=\u0015i\nn;+)\n\u00002\u00002\n3<supfRe\u0015i\nn:n2N0g=\u00001\n2(2y0)1\n2<0: (6.12)\nFurthermore we \fnd that for every n2N0\nj\u0015i\nnj= (2n+ 1)(2yn)\u00001\n4\u0012\n1 +1\n2(2n+ 1)\u00002(2yn)3\n2\u00131\n2\n;\narg(\u0015i\nn) =\u0019\u0000arctan\u00122(2n+ 1)yn;+\n(2yn)3\n4\u0013\n:\nAsymptotic expansions as n!+1yield\n2n+ 1\n(2yn)1\n4 \n1 +1\n2(2yn)3\n2\n(2n+ 1)2!1\n2\n= 21\n3(2n+ 1)2\n3 \n1\u00002\u00007\n3\n3\u0014(2n+ 1)\u00002\n3\n+o(\u0014(2n+ 1)\u00002\n3)!\n;\n2(2n+ 1)yn;+\n(2yn)3\n4=p\n3 \n1 +2\u00001\n3\n3\u0014(2n+ 1)\u00002\n3+o(\u0014(2n+ 1)\u00002\n3)!\n;\nwhich proves (6.3) and (6.4).\n(ii) The claim follows immediately from Corollary 3.9. \u0003\nTo illustrate this result, we plot the spectrum of Gcorresponding to q(x) = 10x2\nanda(x) =x2(see Fig. 6.1). In this case, \u001be2(G) = (\u00001;\u00005] and we calculateRESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 33\nsupf\u0015r\nn:n2N0g=\u00001:61326 and supfRe\u0015i\nn:n2N0g=\u00000:15809. This yields\nthe decay rate !0(G) =\u00000:15809 for the associated semigroup.\n-12 -10 -8 -6 -4 -2 0-20-1001020\nσe2\nσd\nFigure 6.1. The spectrum of Gforq(x) = 10x2anda(x) =x2.\n6.1.Further remarks. From (6.11), (3.2) and (3.5), it is clear that\n\u0014!0+=)!0!0\u0000:\nIt can also be veri\fed from (6.9) that y0!0 as\u0014!+1and therefore by (6.12),\n(3.2) and (3.5) we \fnd\n\u0014!+1=)!0!0\u0000:\nHence the exponential decay rates become progressively weaker in both limiting\ncases although for di\u000berent reasons from a spectral point of view: when \u0014ap-\nproaches zero, it is the (real) essential spectrum that approaches the imaginary\naxis whereas for very large \u0014the phenomenon is driven by the (non-real) eigenval-\nues.\n20 40 60 80 100κ\n-10-8-6-4-20λλ0rλ1rλ2rλ3rλ4r\n20 40 60 80 100κ\n-3.0-2.5-2.0-1.5-1.0-0.50.0Reλλ0iλ1iλ2iλ3iλ4i\n0 20 40 60 80 100κ0246810Imλλ0iλ1iλ2iλ3iλ4i\nFigure 6.2. First 5 real (top) and \frst 5 imaginary (bottom) eigenval-\nues ofGas a function of \u0014.\nWe also note that, when we remove the potential qaltogether (i.e. when \u0014= 0),\nwe recover the spectral con\fguration shown in Figure 1 in [18]. In terms of the34 ANTONIO ARNAL\nbehaviour of the solutions of the Cauchy problem, this of course implies that their\ntime-decay is no longer exponential.\nWe illustrate the dependency on \u0014of the \frst 5 real and the \frst 5 (upper-half\nplane) imaginary eigenvalues of Gin Fig. 6.2. It highlights the fact that \u001b(G) will\ntouch the imaginary axis when q(x) = 0 as the set of real eigenvalues vanishes\nand the essential spectrum \flls R\u0000. Note that the non-real point spectrum is not\nempty and it stays away from the imaginary axis (although as noted above it will\nget closer to it as \u0014becomes larger).\nWe conclude with the observation (without making speci\fc claims) that the\nabove analysis can also be applied to more general even monomials a(x) =x2m,\nq(x) =\u0014x2m,\u0014>0,m2N, similarly to the study in Proposition 6.1 in [18], with\nthe eigenvalues of the corresponding self-adjoint anharmonic oscillator replacing\n\u001b(eH1) above.\nReferences\n[1]Abels, H. Pseudodi\u000berential and Singular Integral Operators . De Gruyter, 2011.\n[2]Almog, Y., and Helffer, B. On the spectrum of non-selfadjoint Schr odinger operators with\ncompact resolvent. Comm. Partial Di\u000berential Equations 40 (2015), 1441{1466.\n[3]Arifoski, A., and Siegl, P. Pseudospectra of damped wave equation with unbounded damp-\ning. SIAM J. Math. Anal. 52 (2020), 1343{1362.\n[4]Arnal, A., and Siegl, P. Generalised Airy Operators. arXiv preprint arXiv:2208.14389\n(2022).\n[5]Arnal, A., and Siegl, P. Resolvent estimates for one-dimensional Schr odinger operators\nwith complex potentials. arXiv preprint arXiv:2203.15938 (2022).\n[6]Bordeaux Montrieux, W. Estimation de r\u0013 esolvante et construction de quasimode pr\u0012 es du\nbord du pseudospectre. arXiv:1301.3102, 2013.\n[7]Br\u0013ezis, H., and Kato, T. Remarks on the Schr odinger operator with singular complex\npotentials. J. Math. Pures Appl. 58 (1979), 137{151.\n[8]Davies, E. B. Semi-Classical States for Non-Self-Adjoint Schr odinger Operators. Comm.\nMath. Phys. 200 (1999), 35{41.\n[9]Davies, E. B. Pseudospectra of di\u000berential operators. J. Operator Theory 43 (2000), 243{262.\n[10]Davies, E. B. Wild spectral behaviour of anharmonic oscillators. Bull. Lond. Math. Soc. 32\n(2000), 432{438.\n[11]Davies, E. B. Non-Self-Adjoint Di\u000berential Operators. Bull. Lond. Math. Soc. 34 , 5 (2002),\n513{532.\n[12]Dencker, N., Sj ostrand, J., and Zworski, M. Pseudospectra of semiclassical (pseudo-)\ndi\u000berential operators. Commun. Pure Appl. Math. 57 (2004), 384{415.\n[13]Duc, T. N. Pseudomodes for biharmonic operators with complex potentials. arXiv preprint\narXiv:2201.03305 (2022).\n[14]Eastham, M. S. P., Kalf, H., et al. Schr odinger-type operators with continuous spectra ,\nvol. 65. Pitman Publishing, 1982.\n[15]Edmunds, D. E., and Evans, W. D. Spectral Theory and Di\u000berential Operators . Oxford\nUniversity Press, New York, 1987.\n[16]Engel, K.-J., and Nagel, R. One-parameter semigroups for linear evolution equations .\nSpringer-Verlag, New York, 2000.\n[17]Freitas, P., Hefti, N., and Siegl, P. Damped wave equation with singular damping. Proc.\nAmer. Math. Soc. 148 (2020), 4273{4284.\n[18]Freitas, P., Siegl, P., and Tretter, C. Damped wave equation with unbounded damping.\nJ. Di\u000berential Equations 264 (2018), 7023{7054.\n[19]Gerhat, B. Schur complement dominant operator matrices. arXiv preprint arXiv:2205.11653\n(2022).\n[20]Helffer, B. Spectral theory and its applications . Cambridge University Press, 2013.\n[21]Ikehata, R., and Takeda, H. Uniform energy decay for wave equations with unbounded\ndamping coe\u000ecients. Funkcialaj Ekvacioj (2018).\n[22]Kato, T. On some Schr odinger operators with a singular complex potential. Ann. Scuola\nNorm. Super. Pisa, Cl. Sci. IV 5 (1978), 105{114.\n[23]Krej \u0014ci\u0014r\u0013\u0010k, D., Raymond, N., Royer, J., and Siegl, P. Non-accretive Schr odinger operators\nand exponential decay of their eigenfunctions. Israel J. Math. 221 (2017), 779{802.\n[24]Krej \u0014ci\u0014r\u0013\u0010k, D., and Royer, J. Spectrum of the wave equation with Dirac damping on a\nnon-compact star graph. arXiv preprint arXiv:2204.12747 (2022).RESOLVENT ESTIMATES FOR THE DAMPED WAVE EQUATION 35\n[25]Krej \u0014ci\u0014r\u0013\u0010k, D., and Nguyen Duc, T. Pseudomodes for non-self-adjoint Dirac operators. J.\nFunct. Anal. 282 , 12 (2022), 109440.\n[26]Krej \u0014ci\u0014r\u0013\u0010k, D., and Siegl, P. Pseudomodes for Schr odinger operators with complex poten-\ntials. J. Funct. Anal. 276 (2019), 2856{2900.\n[27]Reed, M., and Simon, B. Methods of Modern Mathematical Physics, Vol. 4: Analysis of\nOperators . Academic Press, New York-London, 1978.\n[28]Semor \u0013adov \u0013a, I., and Siegl, P. Diverging eigenvalues in domain truncations of Schr odinger\noperators with complex potentials. SIAM J. Math. Anal. (to appear).\n[29]Sjostrand, J. Resolvent Estimates for Non-Selfadjoint Operators via Semigroups. In Around\nthe Research of Vladimir Maz'ya III . Springer New York, 2009, pp. 359{384.\n[30]Sobajima, M., and Wakasugi, Y. Di\u000busion phenomena for the wave equation with space-\ndependent damping term growing at in\fnity. Advances in Di\u000berential Equations 23 , 7/8\n(2018), 581{614.\n[31]Titchmarsh, E. C. Eigenfunction expansions associated with second-order di\u000berential equa-\ntions. Part I. Clarendon Press, Oxford, 1962.\n[32]Trefethen, L. N. Pseudospectra of linear operators. SIAM Rev. 39 (1997), 383{406.\nMathematical Sciences Research Centre, Queen's University Belfast, University\nRoad, Belfast BT7 1NN, UK\nEmail address :aarnalperez01@qub.ac.uk"    },    {        "title": "2206.09969v1.First_principles_calculation_of_the_parameters_used_by_atomistic_magnetic_simulations.pdf",        "content": "arXiv:2206.09969v1  [cond-mat.mtrl-sci]  20 Jun 2022APS/123-QED\nFirst-principles calculation of the parameters used by ato mistic magnetic simulations\nSergiy Mankovsky and Hubert Ebert\nDepartment of Chemistry/Phys. Chemistry, LMU Munich,\nButenandtstrasse 11, D-81377 Munich, Germany\n(Dated: June 22, 2022)\nWhile the ground state of magnetic materials is in general we ll described on the basis of spin den-\nsity functional theory (SDFT), the theoretical descriptio n of finite-temperature and non-equilibrium\nproperties require an extension beyond the standard SDFT. T ime-dependent SDFT (TD-SDFT),\nwhich give for example access to dynamical properties are co mputationally very demanding and can\ncurrently be hardly applied to complex solids. Here we focus on the alternative approach based on\nthe combination of a parameterized phenomenological spin H amiltonian and SDFT-based electronic\nstructure calculations, giving access to the dynamical and finite-temperature properties for example\nvia spin-dynamics simulations using the Landau-Lifshitz- Gilbert (LLG) equation or Monte Carlo\nsimulations. We present an overview on the various methods t o calculate the parameters of the\nvarious phenomenological Hamiltonians with an emphasis on the KKR Green function method as\none of the most flexible band structure methods giving access to practically all relevant parameters.\nConcerning these, it is crucial to account for the spin-orbi t coupling (SOC) by performing rela-\ntivistic SDFT-based calculations as it plays a key role for m agnetic anisotropy and chiral exchange\ninteractions represented by the DMI parameters in the spin H amiltonian. This concerns also the\nGilbert damping parameters characterizing magnetization dissipation in the LLG equation, chiral\nmultispin interaction parameters of the extended Heisenbe rg Hamiltonian, as well as spin-lattice\ninteraction parameters describing the interplay of spin an d lattice dynamics processes, for which an\nefficient computational scheme has been developed recently b y the present authors.\nPACS numbers: 71.15.-m,71.55.Ak, 75.30.Ds\nI. INTRODUCTION\nDensity functional theory (DFT) is a ’formally exact\napproachto the static electronic many-body problem’ for\ntheelectrongasintheequilibrium, whichwasadoptedfor\na huge number of investigations during the last decades\nto describe the ground state of solids, both magnetic and\nnon-magnetic,aswellasvariousgroundstateproperties1.\nHowever, dealing with real systems, the properties\nin an out-of-equilibrium situation are of great interest.\nAn example for this is the presence of external pertur-\nbation varying in time, which could be accounted for\nby performing time-dependent first-principles electronic\nstructure calculations. The time-dependent extension\nof density functional theory (TD-DFT)2is used suc-\ncessfully to study various dynamical processes in atoms\nand molecules, in particular, giving access to the time\nevolution of the electronic structure in a system af-\nfected by a femtosecond laser pulse. However, TD-DFT\ncan be hardly applied to complex solids because of the\nlack of universal parameter-free approximations for the\nexchange-correlation kernel. Because of this, an ap-\nproach based on the combination of simulation methods\nfor spin- and lattice dynamics, using model spin and lat-\ntice Hamiltonians is more popular for the moment. A\ngreat progress with this approach has been achieved dur-\ning last decade due to the availability of parameters for\nthe model Hamiltonians calculated on a first principles\nlevel, that is a central issue of the present contribution.\nAs it was pointed out in Ref. 1, this approach has the ad-\nvantage, that the spin-related many-body effects in thiscase are much simpler to be taken into account when\ncompared to the ab-initio approach. Thus, the isotropic\nexchangecouplingparameters JijfortheclassicalHeisen-\nberg Hamiltonian worked out Liechtenstein et al.3,4have\nbeen successfully used by many authors to predict the\nground state magnetic structure of material and to in-\nvestigateitsfinite-temperatureproperties. Dependingon\nthe materials, the isotropic Jijcan exhibit only spatial\nanisotropy. Extension of the Heisenberg Hamiltonian ac-\ncounting for anisotropy in spin subspace is often done by\nadding the so-called Dzyaloshinskii-Moriya interactions\n(DMI) and the magnetic anisotropy term,\nHH,rel=−/summationdisplay\ni,jJij(ˆei·ˆej)−/summationdisplay\ni,j/vectorDij(ˆei׈ej)+/summationdisplay\niˆeiKiiˆei.\n(1)\nwith ˆei(j)the orientation of the spin magnetic moment at\nsitei(j). Alternatively, one may describe exchange inter-\nactions in the more general tensorial form, Jij, leading\nto:\nHH,rel=−/summationdisplay\ni,jˆeiJijˆej+/summationdisplay\niˆeiKiiˆei,(2)\nIn the second case the DMI is represented as the an-\ntisymmetric part of the exchange tensor, i.e. Dα\nij=\n1\n2(Jβγ\nij−Jγβ\nij)ǫαβγ. It should be stressed, that calcula-\ntions of the spin-anisotropic exchange interaction param-\neters as well as of the magnetic anisotropy parameters\nrequire a relativistic treatment of the electronic struc-\nture in contrast to the case of the isotropic exchange pa-\nrameters which can be calculated on a non-relativistic2\nlevel. Various schemes to map the dependence of the\nelectronicenergyonthemagneticconfigurationweresug-\ngested in the literature to calculate the parameters of the\nspin Hamiltonians5–8, depending of its form given in Eqs.\n(1) or (2).\nDespite of its simplicity, the spin Hamiltonian gives\naccess to a reasonable description of the temperature\ndependence of magnetic properties of materials when\ncombined with Monte Carlo (MC) simulations9, or non-\nequilibrium spin dynamics simulations based on the phe-\nnomenological Landau-Lifshitz-Gilbert equations10,11\n1\nγd/vectorM\ndτ=−/vectorM×/vectorHeff+/vectorM×/bracketleftBigg˜G(/vectorM)\nγ2M2sd/vectorM\ndτ/bracketrightBigg\n.(3)\nHere/vectorHeffis the effective magnetic field defined as /vectorHeff=\n−1\nM∂F\n∂ˆm, whereFis the free energy of the system and\nˆm=/vectorM\n/vectorMswithMsthesaturationmagnetizationtreatedat\nfirst-principles level, and γis the gyromagnetic ratio and\n˜Gis the Gilbert damping parameter. Alternatively, the\neffective magnetic field can be representedin terms ofthe\nspin Hamiltonian in Eq. (2), i.e. /vectorHeff=−1\nM∂/angbracketleftHH,rel/angbracketrightT\n∂ˆm,\nwith/an}b∇acketle{t.../an}b∇acket∇i}htTdenoting the thermal averagefor the extended\nHeisenberg Hamiltonian HH,rel.\nThe first-principles calculation of the parameters for\nthe Heisenberg Hamiltonian as well as for the LLG equa-\ntion for spin dynamics have been reported in the litera-\nture by various groups who applied different approaches\nbased on ab-initio methods. Here we will focus on calcu-\nlations based on the Green function multiple-scattering\nformalism being a rather powerful tool to supply all pa-\nrameters for the extended Heisenberg Hamiltonian as\nwell as for the LLG equation.\nA. Magnetic anisotropy\nLet’s first consider the magnetic anisotropy term in\nspin Hamiltonian, characterized by parameters (written\nintensorialforminEqs.(1)and(2))deducedfromtheto-\ntalenergydependentontheorientationofthemagnetiza-\ntion ˆm. The latter is traditionallysplit into the magneto-\ncrystalline anisotropy(MCA) energy, EMCA(ˆm), induced\nby spin-orbit coupling (SOC) and the shape anisotropy\nenergy,Eshape(ˆm), caused by magnetic dipole interac-\ntions,\nEA(ˆm) =EMCA(ˆm)+Eshape(ˆm). (4)\nAlthough a quantum-mechanical description of the mag-\nneticshapeanisotropydeservesseparatediscussion12this\ncontribution can be reasonably well estimated based on\nclassical magnetic dipole-dipole interactions. Therefore,\nwe will focus on the MCA contribution which is fully\ndetermined by the electronic structure of the considered\nsystem. In the literature the focus is in general on the\nMCA energy of the ground state, which can be estimated\nstraightforwardlyfromthe totalenergycalculatedfordif-\nferent orientations of the magnetization followed by amapping onto a model spin Hamiltonian, given e.g. by\nan expansion in terms of spherical harmonics Ylm(ˆm)13\nEMCA(ˆm) =/summationdisplay\nlevenm=l/summationdisplay\nm=−lκm\nlYlm(ˆm).(5)\nAlternative approach to calculate the MCA parameters\nis based on magnetic torque calculations, using the defi-\nnition\nTˆm(θˆu) =−∂E(ˆm)\n∂θˆu, (6)\navoiding the time-consuming total energy calculations.\nThis scheme is based on the so-called magnetic force the-\noremthatallowstorepresenttheMCAenergyintermsof\na correspondingelectronic single-particleenergies change\nunder rotation of magnetization, as follows14:\n∆ESOC(ˆm,ˆm′) =−/integraldisplayEˆm\nF\ndE/bracketleftBig\nNˆm(E)−Nˆm′(E)/bracketrightBig\n−1\n2nˆm′(Eˆm′\nF)(Eˆm\nF−Eˆm′\nF)2\n+O(Eˆm\nF−Eˆm′\nF)3(7)\nwithNˆm(E) =/integraltextEdE′nˆm(E′) the integrated DOS for\nthe magnetization along the direction ˆ m, andnˆm(E) the\ndensityofstates(DOS) representedin termsofthe Green\nfunction as follows\nnˆm(E) =−1\nπIm TrGˆm(E). (8)\nThis expressioncan be used in a very efficient way within\nthe framework of the multiple-scattering formalism. In\nthis case the Green function is given in terms of the scat-\ntering path operator τ(E)nn′connecting the sites nand\nn′as follows\nG0(/vector r,/vector r′,E) =/summationdisplay\nΛΛ′Zn\nΛ(/vector r,E)τnn′\nΛΛ′(E)Zn′×\nΛ′(/vector r′,E)\n−/summationdisplay\nΛ/bracketleftBig\nZn\nΛ(/vector r,E)Jn×\nΛ(/vector r′,E)Θ(r′−r)\n+Jn\nΛ(/vector r,E)Zn×\nΛ(/vector r′,E)Θ(r−r′)/bracketrightBig\nδnn′,(9)\nwhere the combined index Λ = ( κ,µ) represents the rela-\ntivistic spin-orbit and magnetic quantum numbers κand\nµ, respectively15;Zn\nΛ(/vector r,E) andJn\nΛ(/vector r,E) are the regular\nand irregular solutions of the single-site Dirac equation\n(27)16–18. The scattering path operator is given by the\nexpression\nτ(E) = [m(E)−G0(E)]−1(10)\nwithm(E) =t−1(E) andG0(E) the inverse single-site\nscattering and structure constant matrices, respectively.\nThe double underline used here indicates matrices with\nrespect to site and angular momentum indices17.3\nUsing the Lloyd’s formula that gives the integrated\nDOS in terms of the scattering path operator, Eq. (7)\ncan be transformed to the form\n∆ESOC(ˆm,ˆm′) =−1\nπIm Tr/integraldisplayEF\ndE\n×/parenleftbig\nlnτ(ˆm,E)−lnτ(ˆm′,E)/parenrightbig\n(11)\nwith the scattering path operator evaluated for the mag-\nnetization along ˆ mand ˆm′, respectively.\nWith this, the magnetic torque T(θ) can be expressed\nby means of multiple scattering theory leading for the\ntorque component with respect to a rotation of the mag-\nnetization around an axis ˆ u, to the expression19\nTˆm(θˆu) =−1\nπℑ/integraldisplayEF\ndE∂\n∂θˆu/bracketleftbig\nlndet/parenleftbig\nt(ˆm)−1−G0/parenrightbig/bracketrightbig\n.\n(12)\nMapping the resulting torque onto a corresponding pa-\nrameterized expression as for example Eq. (5), one ob-\ntains the corresponding parameters of the spin Hamilto-\nnian.\nHowever,oneshouldnotethatthemagneticanisotropy\nof materials changes when the temperature increases.\nThis occurs first of all due to the increasing amplitude\nof thermally induced spin fluctuations responsible for a\nmodification of the electronic structure. A correspond-\ning expression for magnetic torque st finite temperature\nwas worked out by Staunton et al.19, on the basis of the\nrelativistic generalization of the disordered local moment\n(RDLM) theory20. To perform the necessary thermal av-\neraging over different orientational configurations of the\nlocal magnetic moments it uses a technique similar to\nthe one used to calculate the configurational average in\nthe case of random metallic alloys, so-called Coherent\nPotential Approximation (CPA) alloy theory21,22. Ac-\ncordingly, the free energy difference for two different ori-\nentations of the magnetization is given by\n∆F(ˆm,ˆm′) =−/integraldisplay\ndEfFD(E,ˆm) (13)\n/bracketleftbigg\n/an}b∇acketle{tNˆm/an}b∇acket∇i}ht(E)−/an}b∇acketle{tNˆm′/an}b∇acket∇i}ht(E)/bracketrightbigg\n.(14)\nBy using in this expression the configurational aver-\naged integrated density of states20,23given by Lloyd’s\nformula, the corresponding expression for the magnetic\ntorque at temperature T\nTˆm,T(θˆu) =−∂\n∂θˆu/parenleftbigg/summationdisplay\ni/integraldisplay\nPˆm\ni(ˆei)/an}b∇acketle{tΩˆm/an}b∇acket∇i}htˆeidˆei/parenrightbigg\n.(15)\ncan be written explicitly as:\nTˆm,T(θˆu) =−1\nπIm/integraldisplayEF\ndEfFD(E,ˆm)\n/parenleftbigg/summationdisplay\ni/integraldisplay∂Pˆm\ni(ˆei)\n∂θˆuln detMˆm\ni(ˆei,E)dˆei/parenrightbigg\n.(16)where\nMˆm\ni(ˆei,E) = 1+([ti(ˆei)]−1−tˆm\ni,c(ˆei)]−1)τˆm\nii,c,(17)\nand\nτˆm\nii,c= ([tˆm\ni,c(ˆei)]−1−G0)−1. (18)\nwhere the index cindicates quantities related to the CPA\nmedium.\nFig. 1 (top) shows as an example the results for\nthe temperature-dependent magnetization ( M(T)) cal-\nculated within the RDLM calculations for L10-ordered\nFePt24. Fig. 1 (bottom) gives the corresponding param-\neterK(T) for a uni-axial magneto-crystalline anisotropy,\nwhich is obviously in good agreement with experiment.\n200 400 600800\nTemperature T (K)00.20.40.60.8M(T)\n0.2 0.4 0.60.8\n(M(T))2-2-1.5-1-0.5∆ESOC (meV)\nFIG. 1. RDLM calculations on FePt. Top: the magneti-\nzationM(T) versus Tfor the magnetization along the easy\n[001] axis (filled squares). The full line shows the mean field\napproximation to a classical Heisenberg model for compar-\nison. Bottom: the magnetic anisotropy energy ∆ ESOCas\na function of the square of the magnetization M(T). The\nfilled circles show the RDLM-based results, the full line giv e\nK(T)∼[M(T)/M(0)]2, and the dashed line is based on the\nsingle-ion model function. All data taken from24.\nB. Inter-atomic bilinear exchange interaction\nparameters\nMost first-principles calculations on the bilinear ex-\nchange coupling parameters reported in the literature,4\nFIG. 2. Adiabatic spin-wave dispersion relations along hig h-\nsymmetry lines of the Brillouin zone for Ni. Broken line:\nfrozen-magnon-torque method, full line: transverse susce pti-\nbility method31. All data are taken from Ref. 31.\nare based on the magnetic force theorem (MFT) by\nevaluating the energy change due to a perturbation on\nthe spin subsystem with respect to a suitable reference\nconfiguration25. Many results are based on calculations\nof the spin-spiral energy ǫ(/vector q), giving access to the ex-\nchange parameters in the momentum space, J/vector q7,26–28,\nfollowed by a Fourier transformation to the real space\nrepresentation Jij. Alternatively, therealspaceexchange\nparameters are calculated directly by evaluating the en-\nergy change due to the tilting of spin moments of inter-\nacting atoms. The corresponding non-relativistic expres-\nsion (so-called Liechtenstein or LKAG formula) has been\nimplemented based on the KKR as well as LMTO Green\nfunction (GF)3,4,25,29band structure methods. It should\nbe noted that the magnetic force theorem provides a rea-\nsonable accuracy for the exchange coupling parameters\nin the case of infinitesimal rotations of the spins close to\nsome equilibrium state, that can be justified only in the\nlong wavelength and strong-coupling limits30. Accord-\ningly, calculations of the exchange coupling parameters\nbeyond the magnetic force theorem, represented in terms\nof the inverse transverse susceptibility, were discussed\nin the literature by various authors25,30–33. Grotheer et\nal., for example, have demonstrated31a deviation of the\nspin-wave dispersion curves away from Γ point in the\nBZ, calculated for fcc Ni using the exchange parameters\nJ/vector q∼χ−1\n/vector q, from the MFT-based results for J/vector q. On the\notherhand, the resultsareclosetoeachotherin the long-\nwavelength limit (see Fig. 2). The calculations beyond\nthestandardDFTaredonebymakinguseoftheso-called\nconstrained-field DFT. The latter theory was also used\nby Bruno33who suggested the ’renormalization’ of the\nexchange coupling parameters expressed in terms of non-\nrelativistic transverse magnetic susceptibility, according\ntoJ=1\n2Mχ−1M=1\n2M(˜χ−1−Ixc)M, with the various\nquantities defined as follows\n˜χ−1\nij=2\nπ/integraldisplayEF\ndE/integraldisplay\nΩid3r/integraldisplay\nΩjd3r′(19)\n×Im[G↑(/vector r,/vector r′,E)G↓(/vector r′,/vector r,E)],(20)Mi=/integraldisplay\nΩid3rm(/vector r), (21)\nand\n˜Ixc\nij=δij∆i\n2Mi, (22)\nwith ∆ i=4\nMi/summationtext\nj˜Jij, where\n˜Jij=1\nπIm/integraldisplayEF\ndE/integraldisplay\nΩid3r/integraldisplay\nΩjd3r′(23)\n×[Bxc(/vector r)G↑(/vector r,/vector r′,E)Bxc(/vector r′)G↓(/vector r′,/vector r,E)].(24)\nThis approach results in a Curie temperature of 634 K\nfor fcc Ni (vs. 350 K based on the MFT) which is in good\nagreement with the experimental value of (621 −631 K).\nAs was pointed out by Solovyev30, such a corrections can\nbe significant only for a certain class of materials, while,\nfor instance, the calculations of spin-wave energies31and\nTC33for bcc Fe demonstrate that these corrections are\nquite small. As most results in the literature were ob-\ntained using the exchange parameters based on the mag-\nnetic force theorem, we restrict below to this approxima-\ntion.\nSimilar to the case of the MCA discussed above, ap-\nplication of the magnetic force theorem gives the energy\nchange due to tilting of two spin moments represented in\nterms of the integrated DOS4. Within the multiple scat-\ntering formalism, this energy can be transformed using\nthe Lloyd’s formula leading to the expression\n∆E=−1\nπIm Tr/integraldisplayEF\ndE/parenleftbig\nlnτ(E)−lnτ0(E)/parenrightbig\n(25)\nwithτ(0)(E) andτ(E) the scattering path operators for\nnon-distorted and distorted systems, respectively.\nAs reported in Ref. 4, the expression for Jijrepresent-\ning the exchange interaction between the spin moments\non sitesiandj, is given by the expression\nJij=−1\n4πImTrL/integraldisplayEF\ndE∆iτ↑\nij∆jτ↓\nji,(26)\nwith ∆i(j)= ([t↑]−1\ni(j)−[t↓]−1\ni(j)), wheret↑\ni(j)andt↓\ni(j)are\nthespin-upandspin-downsingle-sitescatteringmatrices,\nrespectively, while τ↑\nijandτ↓\njiare the spin-up and spin-\ndown, respectively, scattering path operators. As rela-\ntivistic effects are not taken into account, the exchange\ninteractions are isotropic with respect to the orientation\nof the magnetization as well as with respect to the di-\nrection of the spin tilting. On the other hand, spin-orbit\ncoupling gives rise to an anisotropy for exchange inter-\nactions requiring a representation in the form of the ex-\nchange tensor Jijwith its antisymmetric part giving ac-\ncess to the Dzyaloshinskii-Moriya (DM) interaction /vectorDij.\nUdvardi et al.5and later Ebert and Mankovsky6sug-\ngested an extension of the classical Heisenberg Hamilto-\nnianbyaccountingforrelativisticeffects forthe exchange\ncoupling (see also Ref. 25). These calculations are based5\nonafullyrelativistictreatmentoftheelectronicstructure\nobtained by use of of the Dirac Hamiltonian\nHD=−ic/vector α·/vector∇+1\n2c2(β−1)\n+¯V(/vector r)+β/vector σ·/vectorB(/vector r)+e/vector α·/vectorA(/vector r).(27)\nHere,αiandβare the standard Dirac matrices15while\n¯V(/vector r) and/vectorB(/vector r) are the spin independent and spin depen-\ndent parts of the electronic potential.\nConsidering a ferromagnetic (FM) state as a reference\nstate with the magnetization along the zdirection, a tilt-\ning of the magnetic moments on sites iandjleads to a\nmodification ofthe scattering path operatorimplying the\nrelation\nlnτ−lnτ0=−ln/parenleftbig\n1+τ[∆mi+∆mj+...]/parenrightbig\n,(28)\nwithmi=t−1\ni. This allows to write down the expression\nfor the energy change due to a spin tilting on sites iand\njas follows\nEij=−1\nπImTr/integraldisplayEF\ndE∆miτij∆mjτji(29)\nWithin the approach of Udvardi et al.5, the depen-\ndence of the single-site inverse scattering matrix mion\nthe orientation of magnetic moment ˆ eiis accounted for\nby performing a corresponding rotation operation us-\ning the rotation matrix R(θ,φ), i.e., one has mi(θ,φ) =\nR(θ,φ)m0\niR+(θ,φ). The change of the scattering matrix\nmiunder spin rotation, ∆ mi, linearized with respect to\nthe rotation angles, is given by the expression\n∆mi=R(θi,φi)m0\niR+(θi,φi)−m0\ni\n=mθ\niδθi+mφ\niδφi (30)\nwith\nmθ\ni=∂\n∂θmi=∂R\n∂θmiR++Rmi∂R+\n∂θ,\nmφ\ni=∂\n∂φmi=∂R\n∂φmiR++Rmi∂R+\n∂φ.(31)\nTo calculate the derivatives of the rotation matrix, the\ndefinition\nˆR(αˆn,ˆn) =eiαˆn(ˆn·ˆ/vectorJ)(32)\nfor the corresponding operator is used, withˆ/vectorJthe total\nangular momentum operator. ˆR(αˆn,ˆn) describes a rota-\ntion of the magnetic moment ˆ mby the angle αˆnabout\nthe direction ˆ n⊥ˆm, that gives in particular R(θ,ˆn) for\nˆn= ˆyandR(φ,ˆn) for ˆn= ˆz.\nThis leads to the second derivatives of the total energy\nwith respect to the titling angles αi={θi,φi}andβj=\n{θj,φj}\n∂2E\n∂αi∂βj=−1\nπImTr/integraldisplayEF\ndEmα\niτijmβ\njτji(33)As is discussed by Udvardi et al.5, these derivatives give\naccess to all elements Jµν\nijof the exchange tensor, where\nµ(ν) ={x,y,z}. Note, however, that only the tensor el-\nements with µ(ν) ={x,y}can be calculated using the\nmagnetization direction along the ˆ zaxis, giving access to\nthezcomponent Dz\nijof the DMI. In order to obtain all\nother tensor elements, an auxiliary rotation of the mag-\nnetization towards the ˆ xand ˆydirections of the global\nframe of reference is required. For example, the com-\nponentDx\nijif the DMI vector can be evaluated via the\ntensor elements\nJzy\nij=∂2E\n∂θi∂φjandJyz\nij=∂2F\n∂φi∂θj(34)\nforθ=π\n2andφ= 0.\nAn alternative expression within the KKR multiple\nscattering formalism has been worked out by Ebert and\nMankovsky6, by using the alternative convention for the\nelectronicGreenfunction(GF) assuggestedbyDederichs\nand coworkers34. According to this convention, the off-\nsite part of the GF is given by the expression:\nG(/vector ri,/vector rj,E) =/summationdisplay\nΛΛ′Ri\nΛ(/vector ri,E)Gij\nΛΛ′(E)Rj×\nΛ′(/vector rj,E),(35)\nwhereGij\nΛΛ′(E) is the so-called structural Green’s func-\ntion,Ri\nΛisaregularsolutiontothesingle-siteDiracequa-\ntionlabeledbythecombinedquantumnumbersΛ15. The\nenergy change ∆ Eijdue to a spin tilting on sites iandj\n, given by Eq. (29), transformed to the above mentioned\nconvention is expressed as follows\n∆Eij=−1\nπImTr/integraldisplay\ndE∆tiGij∆tjGji,(36)\nwhere the change of the single-site t-matrix ∆ tican be\nrepresented in terms of the perturbation ∆ Vi(/vector r) at site\niusing the expression\n∆ti\nΛ′Λ=/integraldisplay\nd3rRi×\nΛ′(r)∆V(r)Ri\nΛ(r) = ∆V(R)i\nΛ′Λ,(37)\nwherethe perturbation causedby the rotationof the spin\nmagnetic moment ˆ eiis represented by a change of the\nspin-dependent potential in Eq. (27) (in contrast to the\napproach used in Ref. 5)\n∆V(r) =Vˆn(r)−Vˆn0(r) =β/vector σ(ˆn−ˆn0)B(r).(38)\nUsing again the frozen potential approximation implies\nthat the spatial part of the potential Vˆn(r) does not\nchange upon rotation of spin orientation.\nComing back to the convention for the GF used by\nGy¨ orffy and coworkers35according to Eq. (9) the expres-\nsion for the elements of the exchange tensor represented\nin terms of the scattering path operator τij\nΛ′Λ(E) has the\nform\nJαiαj\nij=−1\nπImTr/integraldisplay\ndETαiτijTαjτji,(39)6\nwhere\nTαi\nΛΛ′=/integraldisplay\nd3rZ×\nΛ(/vector r)βσαB(r)ZΛ′(/vector r).(40)\nWhen compared to the approach of Udvardi et al.5,\nthe expression in Eq. (39) is given explicitly in Cartesian\ncoordinates. However, auxiliary rotations of the magne-\ntization are still required to calculate all tensor elements,\nand as a consequence, all components of the DMI vec-\ntor. This can be avoided using the approach reported\nrecently36for DMI calculations.\nIn this case, using the grand-canonical potential in the\noperator form\nK=H−µN, (41)\nwithµthe chemical potential, the variation of single-\nparticle energy density ∆ E(/vector r) caused by a perturbation\nis written in terms of the electronic Green function for\nT= 0 K as follows\n∆E(/vector r) =−1\nπImTr/integraldisplayµ\ndE(E−µ)∆G(/vector r,/vector r,E).(42)\nAssuming the perturbation ∆ Vresponsible for the\nchange of the Green function ∆ G=G−G0(the in-\ndex 0 indicates here the collinear ferromagnetic reference\nstate) to be small, ∆ Gcan be expanded up to any order\nw.r.t. the perturbation\n∆G(E) =G0∆VG0\n+G0∆VG0∆VG0\n+G0∆VG0∆VG0∆VG0\n+G0∆VG0∆VG0∆VG0∆VG0+...,(43)\nleading to a corresponding expansion for the energy\nchange with respect to the perturbation as follows\n∆E= ∆E(1)+∆E(2)+∆E(3)+∆E(4)+...,(44)\nHere and below we drop the energy argument for the\nGreen function G(E) for the sake of convenience. This\nexpression is completely general as it gives the energy\nchange as a response to any type of perturbation. When\n∆Vis associated with tiltings of the spin magnetic mo-\nments, it can be expressed within the frozen potential\napproximation and in line with Eq. (38) as follows\n∆V(/vector r) =/summationdisplay\niβ/parenleftbig\n/vector σ·ˆsi−σz/parenrightbig\nBxc(/vector r).(45)\nWith this, the energy expansion in Eq (44) gives access\nto the bilinear DMI as well as to higher order multispin\ninteractions37. To demonstrate the use of this approach,\nwe start with the xandycomponents of the DMI vector,\nwhich can be obtained by setting the perturbation ∆ Vin\nthe form of a spin-spiral described by the configuration\nof the magnetic moments\nˆmi=/parenleftBig\nsin(/vector q·/vectorRi),0,cos(/vector q·/vectorRi)/parenrightBig\n,(46)with the wave vector /vector q= (0,q,0). As it follows from\nthe spin Hamiltonian, the slope of the spin wave energy\ndispersion at the Γ point is determined by the DMI as\nfollows\nlim\nq→0∂E(1)\nDM\n∂qy= lim\nq→0∂\n∂qy/summationdisplay\nijDy\nijsin(/vector q·(/vectorRj−/vectorRi))\n=/summationdisplay\nijDy\nij(/vectorRj−/vectorRi)y. (47)\nIdentifying this with the corresponding derivative of the\nenergy ∆ E(1)in Eq. 44\n∂∆E(1)\n∂qα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0=∂E(1)\nDM\n∂qα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0, (48)\nand equating the corresponding terms for each atomic\npair (i,j), one obtains the following expression for the y\ncomponent of the DMI vector:\nDy\nij=/parenleftbigg\n−1\n2π/parenrightbigg\nImTr/integraldisplayµ\ndE(E−µ)\n×/bracketleftbigg\nOj(E)τji(E)Ti,x(E)τij(E)\n−Oi(E)τij(E)Tj,x(E)τji(E)/bracketrightbigg\n,(49)\nIn a completely analogous way one can derive the x-\ncomponent of the DMI vector, Dx\nij. The overlap inte-\ngralsOj\nΛΛ′and matrix elements Ti,α\nΛΛ′of the operator\nTi,α=βσαBi\nxc(/vector r) (which are connected with the compo-\nnents of the torque operator β[/vector σ׈m]Bi\nxc(/vector r)) are defined\nas follows:6\nOj\nΛΛ′=/integraldisplay\nΩjd3rZj×\nΛ(/vector r,E)Zj\nΛ′(/vector r,E) (50)\nTi,α\nΛΛ′=/integraldisplay\nΩid3rZi×\nΛ(/vector r,E)/bracketleftBig\nβσαBi\nxc(/vector r)/bracketrightBig\nZi\nΛ′(/vector r,E).(51)\nAs is shown in Ref. 37, the Dz\nijcomponent of the DMI,\nas well isotropic exchange parameter Jijcan also be ob-\ntained on the basis of Eqs. (43) and (44) using the second\norder term w.r.t. the perturbation, for a spin spiral with\nthe form\nˆsi= (sinθcos(/vector q·/vectorR),sinθsin(/vector q·/vectorR),cosθ).(52)\nIn this case case, the DMI component Dz\nijand the\nisotropic exchange interaction are obtained by taking the\nfirst- and second-orderderivatives of the energy ∆ E(2)(/vector q)\n(see Eq. (44)), respectively, with respect to /vector q:\n∂\n∂/vector q∆EH(/vector q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0=−sin2θN/summationdisplay\ni/negationslash=jDz\nijˆq·(/vectorRi−/vectorRj) (53)\nand\n∂2\n∂/vector q2∆EH(/vector q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0= sin2θ/summationdisplay\ni,jJij(ˆq·(/vectorRi−/vectorRj))2(54)7\nwith ˆq=/vector q/|/vector q|the unit vector giving the direction of the\nwave vector /vector q. Identifying these expressions again with\nthe corresponding derivatives of ∆ E(2)(/vector q), one obtains\nthe following relations for Dz\nij\nDz\nij=1\n2(Jxy\nij−Jyx\nij) (55)\nand forJij\nJij=1\n2(Jxx\nij+Jyy\nij), (56)\nwhere the tensor elements Jαβare given by Eqs. (39)\nand (40).\nSimilar to the magnetic anisotropy, the exchange cou-\npling parameters depend on temperature, that should be\ntaken into account within the finite temperature spin dy-\nnamic simulations. An approach that gives access to\ncalculations of exchange coupling parameters for finite\ntemperature has been reported in Ref. 37. It accounts\nfor the electronic structure modification due to temper-\nature induced lattice vibrations by using the alloy anal-\nogy model in the adiabatic approximation. This implies\ncalculations of the thermal average /an}b∇acketle{t.../an}b∇acket∇i}htTas the configu-\nrational average over a set of appropriately chosen set of\natomic displacements, using the CPA alloy theory38–40.\nTo make use of this scheme to account for lattice vi-\nbrations, a discrete set of Nvvectors ∆/vectorRq\nv(T) is intro-\nduced for each atom, with the temperature dependent\namplitude, which characterize a rigid displacement of\nthe atomic potential in the spirit of the rigid muffin-\ntin approximation41,42. The corresponding single-site t-\nmatrix in the common global frame of the solid is given\nby the transformation:\ntq\nv=U(∆/vectorRv)tq,locU(∆/vectorRv)−1, (57)\nwith the so-called U-transformation matrix U(/vector s) given in\nits non-relativistic form by:41,42\nULL′(/vector s) = 4π/summationdisplay\nL′′il+l′′−l′CLL′L′′jl′′(|/vector s|k)YL′′(ˆs).(58)\nHereL= (l,m) represents the non-relativistic angu-\nlar momentum quantum numbers, jl(x) is a spheri-\ncal Bessel function, YL(ˆr) a real spherical harmonics,\nCLL′L′′a corresponding Gaunt number and k=√\nEis\nthe electronic wave vector. The relativistic version of\nthe U-matrix is obtained by a standard Clebsch-Gordan\ntransformation.15\nEvery displacement characterized by a displacement\nvectors ∆/vectorRv(T) can be treated as a pseudo-component\nof a pseudo alloy. Thus, the thermal averaging can be\nperformed as the site diagonal configurational average\nforasubstitutional alloy,bysolvingthe multi-component\nCPA equations within the global frame of reference40.\nThe same idea can be used also to take into account\nthermalspinfluctuations. Asetofrepresentativeorienta-\ntion vectors ˆ ef(withf= 1,...,Nf) for the local magneticmoment is introduced. Using the rigid spin approxima-\ntion, the single-site t-matrix in the global frame, corre-\nsponding to a given orientation vector, is determined by:\ntq\nf=R(ˆef)tq,locR(ˆef)−1, (59)\nwheretq,locis the single-site t-matrix in the local frame.\nHere the transformation from the local to the global\nframe of reference is expressed by the rotation matrices\nR(ˆef) that are determined by the vectors ˆ efor corre-\nsponding Euler angles.15Again, every orientation can be\ntreated as a pseudo-component of a pseudo alloy, that\nallows to use the alloy analogy model to calculate the\nthermal average over all types of spin fluctuations40.\nThe alloy analogy for thermal vibrations applied to\nthe temperature dependent exchange coupling parame-\nters leads to\n¯Jαiαj\nij=−1\n2πℑ/integraldisplay\ndETrace/an}b∇acketle{t∆Vαiτij∆Vαjτji/an}b∇acket∇i}htc,(60)\nwhere/an}b∇acketle{t.../an}b∇acket∇i}htcrepresents the configurational average with\nrespect to the set of displacements. In case of the ex-\nchange coupling parameters one has to distinguish be-\ntween the averaging over thermal lattice vibrations and\nspin fluctuations. In the first case the configurational av-\nerage is approximated as follows /an}b∇acketle{t∆Viτij∆Vjτji/an}b∇acket∇i}htvib≈\n/an}b∇acketle{t∆Viτij/an}b∇acket∇i}htvib/an}b∇acketle{t∆Vjτji/an}b∇acket∇i}htvib, assuming a negligible impact of\nthe so-called vertex corrections43. This averaging ac-\ncounts for the impact of thermally induced phonons on\nthe exchange coupling parameters for every temperature\nbefore their use in MC or spin dynamics simulations that\ndealsubsequentlywith thethermalaveraginginspinsub-\nspace. The impact of spin fluctuations can be incorpo-\nrated as well within the electronic structure calculations.\nFor a non-polarized paramagnetic reference state, this\ncan be done, e.g., by using the so-called disorder local\nmoment (DLM) scheme formulated in general within the\nnon-relativistic (or scalar-relativistic) framework. Mag-\nnetic disorder in this case can be modeled by creating a\npseudo alloy with an occupation of the atomic sites by\ntwo types of atoms with opposite spin moments oriented\nupwards,M↑and downwards M↓, respectively, i.e. con-\nsidering the alloy M↑\n0.5M↓\n0.5. In the relativistic case the\ncorresponding RDLM scheme has to describe the mag-\nnetic disorder by a discrete set of Nforientation vectors,\nand as a consequence, the average /an}b∇acketle{tτij/an}b∇acket∇i}htspinhas to be\ncalculated taking into account all these orientations. A\ncomparison of the results obtained for the isotropic ex-\nchange coupling constants Jijfor bcc Fe using the DLM\nand RDLM schemes is shown in Fig. 3, demonstrating\nclose agreement, with the small differences to be ascribed\nto the different account of relativistic effects, i.e. in par-\nticular the spin-orbit coupling.8\n11.5 22.5 3\nRij/a051015202530Jij (meV)SR-DLM\nRDLMFe (bcc), T = 1500 K\nFIG. 3. Isotropic exchange coupling parameters calculated\nfor the disordered magnetic state of bcc Fe within the scalar -\nrelativistic approach, using the DLM scheme (circles, SR-\nDLM) and within the fully-relativistic approach, using the\nRDLM scheme19,24(squares, RDLM).\nC. Multi-spin expansion of spin Hamiltonian:\nGeneral remarks\nDespite the obvious success of the classical Heisenberg\nmodel for many applications, higher-order multi-spin ex-\npansionHmsof the spin Hamiltonian H, given by the\nexpression\nHms=−1\n3!/summationdisplay\ni,j,kJijkˆsi·(ˆsj׈sk),\n−2\np!/summationdisplay\ni,j,k,lJs\nijkl(ˆsi·ˆsj)(ˆsk·ˆsl)\n−2\np!/summationdisplay\ni,j,k,l/vectorDijkl·(ˆsi׈sj)(ˆsk·ˆsl)+...,\n=H3+H4,s+H4,a+... (61)\ncan be of great importance to describe more subtle prop-\nerties of magnetic materials44–56.\nThis concerns first of all systems with a non-collinear\nground state characterized by finite spin tilting angles,\nthat makes multispin contributions to the energy non-\nnegligible. Inparticular,manyreportspublishedrecently\ndiscuss the impact of the multispin interactions on the\nstabilization of exotic topologically non-trivial magnetic\ntextures, e.g. skyrmions, hopfions, etc.57–59\nCorresponding calculations of the multi-spin exchange\nparameters have been reported by different groups. The\napproach based on the Connolly-Williams scheme has\nbeen used to calculate the four-spin non-chiral (two-site\nand three-site) and chiral interactions for Cr trimers52\nand for a deposited Fe atomic chain60, respectively, for\nthe biquadratic, three-site four spin and four-site four\nspin interaction parameters58,61. The authors discuss\nthe role of these type of interactions for the stabilization\nof different types of non-collinear magnetic structures as\nskyrmions and antiskyrmions.\nA more flexible mapping scheme using perturbation\ntheory within the KKR Green function formalism wasonly reported recently by Brinker et al.62,63, and by the\npresent authors37. Here we discuss the latter approach,\ni.e. the energy expansion w.r.t. ∆ Vin Eq. (44). One\nhas to point out that a spin tilting in a real system has a\nfinite amplitude and therefore the higher order terms in\nthis expansion might become non-negligible and in gen-\neral should be taken into account. Their role obviously\ndepends on the specific materialandshould increasewith\ntemperature that leads to an increasing amplitude of the\nspin fluctuations. As these higher-order terms are di-\nrectly connected to the multispin terms in the extended\nHeisenberg Hamiltonian, one has to expect also a non-\nnegligibleroleofthe multispin interactionsforsomemag-\nnetic properties.\nExtending the spin Hamiltonian to go beyond the clas-\nsical Heisenberg model, we discuss first the four-spin ex-\nchange interaction terms Jijkland/vectorDijkl. They can be\ncalculated using the fourth-order term of the Green func-\ntion expansion ∆ E(4)given by:\n∆E(4)=−1\nπImTr/integraldisplayEF\ndE\n×(E−EF)∆VG∆VG∆VG∆VG\n=−1\nπImTr/integraldisplayEF\ndE∆VG∆VG∆VG∆VG.\n(62)\nwhere the sum rule for the Green functiondG\ndE=−GG\nfollowed by integration by parts was used to get a more\ncompact expression. Using the multiple-scattering repre-\nsentation for the Green function, this leads to:\n∆E(4)=/summationdisplay\ni,j,k,l−1\nπImTr/integraldisplayEF\ndE\n×∆Viiτij∆Vjjτjk∆Vkkτkl∆Vllτli.(63)\nwith the matrix elements ∆ Vii=/an}b∇acketle{tZi|∆V|Zi/an}b∇acket∇i}ht. Using the\nferromagnetic state with /vectorM||ˆzas a reference state, and\ncreating the perturbation ∆ Vin the form of a spin-spiral\naccording to Eq. (52), one obtains the corresponding /vector q-\ndependent energy change ∆ E(4)(/vector q), written here explic-9\nitly as an example\n∆E(4)=−1\nπ/summationdisplay\ni,j,k,lImTr/integraldisplayEF\ndEsin4θ\n×/bracketleftbigg\nIxxxx\nijklcos(/vector q·/vectorRi)cos(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Ixxyy\nijklcos(/vector q·/vectorRi)cos(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin/vector q·/vectorRl)\n+Iyyxx\nijklsin(/vector q·/vectorRi)sin(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Iyyyy\nijklsin(/vector q·/vectorRi)sin(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin(/vector q·/vectorRl)\n+Ixyxx\nijklcos(/vector q·/vectorRi)sin(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Iyxyy\nijklsin(/vector q·/vectorRi)cos(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin/vector q·/vectorRl)\n+Iyxxx\nijklsin(/vector q·/vectorRi)cos(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Ixyyy\nijklcos(/vector q·/vectorRi)sin(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin(/vector q·/vectorRl)+.../bracketrightbigg\n(64)\nwhere\nIαβγδ\nijkl=Ti,α(E)τij(E)Tj,β(E)τjk(E)\n×Tk,γ(E)τkl(E)Tl,δ(E)τli(E).(65)\nAs is shownin Ref. 37, the four-spinisotropicexchange\ninteraction Jijklandz-component of the DMI-like in-\nteraction Dz\nijklcan be obtained calculating the energy\nderivatives∂4\n∂q4∆E(4)and∂3\n∂q3∆E(4)in the limit of q= 0,\nand then identified with the corresponding derivatives of\nthe termsH4,sandH4,ain Eq. (61). These interaction\nterms are given by the expressions\nJs\nijkl=1\n4/bracketleftbigg\nJxxxx\nijkl+Jxxyy\nijkl+Jyyxx\nijkl+Jyyyy\nijkl/bracketrightbigg\n(66)\nand\nDz\nijkl=1\n4/bracketleftbigg\nJxyxx\nijkl+Jxyyy\nijkl−Jyxxx\nijkl−Jyxyy\nijkl)/bracketrightbigg\n,(67)\nwhere the following definition is used:\nJαβγδ\nijkl=1\n2πImTr/integraldisplayEF\ndETα\niτijTβ\njτjkTγ\nkτklTδ\nlτli(68)\nThese expressionobviously give also access to a special\ncases, i.e. the four-spin three-site interactions with l=j,\nand the four spin two-site, socalled biquadratic exchange\ninteractions with k=iandl=j.\nThe scalar biquadratic exchange interaction parame-\ntersJs\nijijcalculated on the basis of Eq. (66) for the three\n3dbulk ferromagnetic systems bcc Fe, hcp Co and fcc Ni\nhave been reported in Ref. 37. The results are plotted in\nFig. 4 as a function of the distance Rij+Rjk+Rkl+Rli.\nFor comparison, the insets give the corresponding bilin-\near isotropic exchange interactions for these materials.\nOne can see rather strong first-neighbor interactions for\nbcc Fe, demonstrating the non-negligible characterof the\n✵ \u0000 ✁\n✶✶ \u0000 ✁\n✷✷\n\u0000 ✁\n✸✸\n\u0000 ✁❘✐ ✂\n✴ ✄\n✵\n✵ \u0000 ✁\n✶\n✶ \u0000 ✁\n✷\n✷\n\u0000 ✁\n✸\n✸\n\u0000 ✁\n✹\n✹\n\u0000 ✁\n✁❏\n☎✆☎✆s\n✥✝✞✟✠✡\n① ①✡\n② ②\n☛☛☞ ✌\n✍✍☞ ✌\n✎✎☞ ✌✏✑ ✒\n✓✔\n✕\n✌\n☛✕\n☛✌✖\n✗✘\n✙✚✛✜✢❜ ✣✣ ✤✦\n(a)✶✶\u0000 ✁\n✷✷\u0000 ✁\n✸✸\u0000 ✁❘✐ ✂\n✴ ✄\n✲ ☎ \u0000 ☎✷\n☎\n☎\u0000 ☎ ✷☎\u0000 ☎✵\n☎\u0000 ☎\n✆\n☎\u0000 ☎\n✝\n☎ \u0000\n✶❏\n✞✟✞✟s\n✥✠✡☛☞\n✌\n① ①✌\n② ②\n✍✍✎ ✏\n✑✑✎ ✏\n✒✒✎ ✏✓✔ ✕\n✖✗\n✘\n✏\n✍✘\n✍✏✙\n✚✛\n✜✢✣✤✦❤ ✧★ ✩ ✪\n(b)✵ \u0000 ✁\n✶✶ \u0000 ✁\n✷✷\n\u0000 ✁\n✸❘✐ ✂\n✴ ✄\n✲\n✵ \u0000 ✵✷\n✲\n✵ \u0000 ✵✶ ✁\n✲\n✵ \u0000 ✵✶\n✲\n✵ \u0000 ✵✵ ✁\n✵❏\n☎✆☎✆s\n✥✝✞✟✠✡\n① ①✡\n② ②\n☛☛☞ ✌\n✍✍☞ ✌\n✎✏✑ ✒\n✓✔\n✕\n☛\n✍\n✎✖\n✗✘\n✙✚✛✜✢❢ ✣✣ ✤✦\n(c)\nFIG. 4. Scalar biquadratic exchange interactions Js\nijijin bcc\nFe (a), hcp Co (b) and fcc Ni (Ni). For comparison, the insets\nshow the bilinear exchange interaction parameters calcula ted\nfor the FM state with the magnetization along the ˆ z-axis. All\ndata are taken from Ref. 37.\nbiquadratic interactions. This is of course a material-\nspecific property, and one notes as decrease for the bi-\nquadratic exchange parameters when going to Co and Ni\nas shown in Fig. 4 (b) and (c), respectively.\nIn order to calculate the xandycomponents of\nthe four-spin and as a special case the three-site-DMI\n(TDMI) and biquadratic-DMI (BDMI) type interactions,\nthe scheme suggested in Ref. 37 for the calculation of the\nDMI parameters36,64can be used, which exploited the\nDMI-governed behavior of the spin-wave dispersion hav-\ning a finite slope at the Γ point of the Brillouin zone.10\nNote, however, that a more general form of perturbation\nisrequiredin thiscasedescribedbya2Dspin modulation\nfield according to the expression\nˆsi=/parenleftbig\nsin(/vector q1·/vectorRi) cos(/vector q2·/vectorRi),sin(/vector q2·/vectorRi),\ncos(/vector q1·/vectorRi)cos(/vector q2·/vectorRi)/parenrightbig\n, (69)\nwhere the wave vectors /vector q1and/vector q2are orthogonal to each\nother, as for example /vector q1=q1ˆyand/vector q2=q2ˆx.\nTaking the second-order derivative with respect to the\nwave-vector /vector q2and the first-order derivative with respect\nto the wave-vectors /vector q1and/vector q2, and considering the limit\nq1(2)→0, one obtains\n∂3\n∂q3\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq2=0H4,a=/summationdisplay\ni,j,k,lDx\nijkl(ˆq2·/vectorRij)(ˆq2·/vectorRlk)2,\nand\n∂\n∂q1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq1=0∂2\n∂q2\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq2=0H4,a=/summationdisplay\ni,j,k,lDy\nijkl(ˆq1·/vectorRij)(ˆq2·/vectorRlk)2,\nwhere/vectorRij=/vectorRj−/vectorRiand/vectorRlk=/vectorRk−/vectorRl.\nThe microscopic expressions for the xandycompo-\nnents of/vectorDijkldescribing the four-spin interactions is de-\nrived on the basis of the third-order term in Eq. (43)\n∆E(3)=−1\nπImTr/integraldisplayEF\ndE(E−EF)\n×G0∆VG0∆VG0∆VG0. (70)\nThe final expression for Dα\nijklis achieved by taking the\nsecond-order derivative with respect to the wave-vector\n/vector q2and the first-orderderivative with respect to the wave-\nvectors/vector q1(2), considering the limit q1(2)→0, i.e. equat-\ning within the ab-initio and model expressions the cor-\nresponding terms proportional to ( /vectorRi−/vectorRj)y(/vectorRk−/vectorRl)2\nx\nand (/vectorRi−/vectorRj)x(/vectorRk−/vectorRl)2\nx(we keep a similar form in both\ncasesforthe sakeofconvenience)givestheelements Dy,x\nijkl\nandDy,y\nijkl, as well as Dx,x\nijklandDx,y\nijkl, respectively, of the\nfour-spin chiral interaction as follows\nDα,β\nijkj=ǫαγ1\n8πImTr/integraldisplayEF\ndE(E−EF)\n/bracketleftBig\nOiτijTj,γτjkTk,βτklTl,βτli\n−Ti,γτijOjτjkTk,βτklTl,βτli/bracketrightBig\n+/bracketleftBig\nOiτijTj,βτjkTk,βτklTl,γτli\n−Ti,γτijTj,βτjkTk,βτklOlτli/bracketrightBig\n(71)\nwithα,β=x,y, andǫαγthe elements of the transverse\nLevi-Civita tensor ǫ=/bracketleftbigg\n0 1\n−1 0/bracketrightbigg\n. The TDMI and BDMI\nparameterscan be obtained as the special cases l=jand\nl=j,k=i, respectively, from Eq. (71).The expression in Eq. (71) gives access to the xandy\ncomponents of the DMI-like three-spin interactions\nDα\nijkj=Dα,x\nijkj+Dα,y\nijkj. (72)\nFinally, three-spin chiral exchange interaction (TCI)\nrepresented by first term in the extended spin Hamilto-\nnianhas been discussedin Ref. 37. As it followsfrom this\nexpression, the contribution due to this type of interac-\ntion is non-zero only in case of a non-co-planar and non-\ncollinear magnetic structure characterized by the scalar\nspatial type product ˆ si·(ˆsj׈sk) involving the spin mo-\nments on three different lattice sites.\nIn order to work out the expression for the Jijkinter-\naction, one has to use a multi-Q spin modulation65–67\nwhich ensure a non-zero scalar spin chirality for every\nthree atoms. The energy contribution due to the TCI,\nis non-zero only if Jijk/ne}ationslash=Jikj, etc. Otherwise, the\ntermsijkandikjcancel each other due to the relation\nˆsi·(ˆsj׈sk) =−ˆsi·(ˆsk׈sj).\nAccordingly, the expression for the TCI is derived us-\ning the 2Q non-collinear spin texture described by Eq.\n(69), which is characterized by two wave vectors oriented\nalong two mutually perpendicular directions, as for ex-\nample/vector q1= (0,qy,0) and/vector q2= (qx,0,0). Applying such a\nspin modulation in Eq. (69) for the term H3associated\nwith the three-spin interaction in the spin Hamiltonian\nin Eq. (61), the second-order derivative of the energy\nE(3)(/vector q1,/vector q2) with respect to the wave vectors q1andq2is\ngiven in the limit q1→0,q2→0 by the expression\n∂2\n∂/vector q1∂/vector q2H(3)\n=−/summationdisplay\ni/negationslash=j/negationslash=kJijk/parenleftbig\nˆz·[(/vectorRi−/vectorRj)×(/vectorRk−/vectorRj)]/parenrightbig\n.(73)\nThe microscopic energy term of the electron system,\ngiving access to the chiral three-spin interaction in the\nspin Hamiltonian is described by the second-order term\n∆E(2)=−1\nπImTr/integraldisplayEF\ndE(E−EF)\nG0∆VG0∆VG0 (74)\nof the free energy expansion. Taking the first-order\nderivative with respect to q1andq2in the limit q1→\n0,q2→0, and equating the terms proportional to/parenleftbig\nˆz·[(/vectorRi−/vectorRj)×(/vectorRk−/vectorRj)]/parenrightbig\nwith the correspondingterms\ninthespinHamiltonian,oneobtainsthefollowingexpres-\nsion for the three-spin interaction parameter\nJijk=1\n8πImTr/integraldisplayEF\ndE(E−EF)\n/bracketleftBig\nTi,xτijTj,yτjkOkτki−Ti,yτijTj,xτjkOkτki\n−Ti,xτijOjτjkTk,yτki+Ti,yτijOjτjkTk,xτki\n+OiτijTi,xτjkTk,yτki−OjτijTi,yτjkTk,xτki/bracketrightBig\n,(75)11\ngiving access to the three-spin chiral interaction deter-\nmined asJ∆=Jijk−Jikj. Its interpretation was dis-\ncussed in Ref. 68, where its dependence on the SOC as\nwell as on the topological orbital susceptibility χTO\n∆=\nχTO\nijk−χTO\nikjwas demonstrated. In fact that the expres-\nsion forχTO\nijkworked out in Ref. 68 has a rather similar\nform asJijk, as that can be seen from the expression\nχTO\nijk=−1\n4πImTr/integraldisplayEF\ndE\n×/bracketleftBig\nTi,xτijTj,yτjklk\nzτki−Ti,yτijTj,xτjklk\nzτki\n−Ti,xτijlj\nzτjkTk,yτki+Ti,yτijlj\nzτjkTk,xτki\n+li\nzτijTj,xτjkTk,yτki−li\nzτijTj,yτjkTk,xτki/bracketrightBig\n.\n(76)\nFor everytrimerofatoms, both quantities, χTO\nijkandJijk,\nare non-zero only in the case of non-zero scalar spin chi-\nrality ˆsi·(ˆsj׈sk) and depend on the orientation of the\ntrimermagneticmomentwith respecttothetrimerplain.\nThis is shown in Fig. 668representing ∆ Jand ∆χTOas\na function of the angle between the magnetization and\nnormal ˆnto the surface, which are calculated for the two\nsmallest trimers, ∆ 1and ∆ 2, centered at the Ir atom and\nthe hole site in the Ir surface layer for 1ML Fe/Ir(111),\nrespectively (Fig. 5).\nFIG. 5. Geometry of the smallest three-atom clusters in the\nmonolayer of 3 d-atoms on M(111) surface ( M= Au, Ir): M-\ncentered triangle ∆ 1and hole-centered triangle ∆ 2.\nThe role of the SOC for the three-site 4-spin DMI-like\ninteraction, Dz\nijik, and the three-spin chiral interaction,\nJ∆is shown in Fig. 7. These quantities are calculated\nfor 1ML Fe on Au (111), for the two smallest triangles\n∆1and ∆ 2centered at an Au atom or a hole site, re-\nspectively (see Fig. 5). Here, setting the SOC scaling\nfactorξSOC= 0 implies a suppression of the SOC, while\nξSOC= 1 corresponds to the fully relativistic case. Fig.\n7 (a) shows the three-site 4-spin DMI-like interaction pa-\nrameter, Dz\nijik(ξSOC) when the SOC scaling parameter\nξSOCapplied to all components in the system, shown by\nfull symbols, and with the SOC scaling applied only to\nthe Au substrate. One can see a dominating role of the\nSOC of substrate atoms for Dz\nijik. Also in Fig. 7 (b), a\nnearly linear variation can be seen for J∆(ξSOC) when\nthe SOC scaling parameter ξSOCis applied to all com-\nponents in the system (full symbols). Similar to Dz\nijik,\n✵ ✷✵ ✹✵✻✵\n✽✵❣ ✥ \u0000 ✁✂ ✄\n✵\n✵ ☎ ✆\n✵ ☎ ✷\n✵ ☎ ✝\n✵ ☎ ✹\n✵ ☎ ✞\n✵ ☎ ✻✲✟\n❉ ❚\n✠\n✡☛☞✌✍\n❏✎✶\n✥ ❣ ✄ ✥ ✮ ✏ ✑✒ ✄❏✎\n✓\n✥ ❣ ✄ ✥ ✔ ✑✕ ✖✄❏✎✶\n✥ ✭ ✄ ✗ ✑✘✥ ❣ ✄❏✎\n✓\n✥ ✭ ✄ ✗ ✑✘✥ ❣ ✄\n(a)✵ ✷✵ ✹✵✻✵\n✽✵❣ ✥ \u0000✁✂ ✄\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(b)\nFIG. 6. (a) Three-spinchiral exchange interaction paramet ers\n−J∆(γ) (’minus’ is used to stress the relation between J∆and\nχTO\n∆), and (bc) topological orbital susceptibility (TOS, for\nSOC = 0), calculated for Fe on Ir (111), as a function of the\nangle between the magnetization and normal ˆ nto the surface,\nfor the smallest triangles ∆ 1and ∆ 2. The dashed lines rep-\nresentJ∆(0) cos(γ) (a) and χTO\n∆(0) cos(γ) (b), respectively.\nAll data are taken from Ref. 68.\nthis shows that the SOC is an ultimate prerequisite for a\nnon-vanishing TCI J∆. When scaling the SOC only for\nAu (open symbols), Fig. 7 (b) show only weak changes\nfortheTCIparameters J∆(ξSOC), demonstratingaminor\nimpact of the SOC of the substrate on these interactions,\nin contrast to the DMI-like interaction shown in Fig. 7\n(a). One can see also that Dz\nijikis about two orders of\nmagnitude smaller than J∆for this particular system.\nThe origin of the TCI parameters have been discussed\nin the literature suggesting a different interpretation of\nthe correspondingterms derivedalsowithin the multiple-\nscattering theory Green function formalism62,69,70. How-\never, the expression worked out in Ref. 69 has obviously\nnot been applied for calculations so far. As pointed out\nin Ref. 68, the different interpretation of this type of in-\nteractions can be explained by their different origin. In\nparticular, one has to stress that the parameters in Refs.\n68 and 69 were derived in a different order of pertur-\nbation theory. On the other hand, the approach used\nfor calculations of the multispin exchange parameters re-\nported in Ref. 62, 69, and 71 is very similar to the one\nused in Refs. 37 and 68. The corresponding expressions\nhave been worked out within the framework of multiple-\nscattering Green function formalism using the magnetic\nforce theorem. In particular, the Lloyd formula has been\nused to express the energy change due to the perturba-\ntion ∆Vleading to the expression\n∆E=−1\nπIm Tr/integraldisplayEF\ndE/summationdisplay\np1\npTr/bracketleftbig\nG(E)∆V/bracketrightbigp.(77)\nUsing the off-site part of the GF in Eq. (35), as defined12\n0 0.2 0.4 0.6 0.8 1\nξSOC-0.004-0.00200.0020.0040.0060.008Dijik  (meV)\nDx(∆1), SOC\nDx(∆1), SOC(Au)Dx(∆2), SOC(Au) Dx(∆2), SOC\n(a)\n0 0.2 0.4 0.6 0.8 1\nξSOC-0.4-0.3-0.2-0.10J∆ (meV)∆1, SOC\n∆2, SOC\n∆1, SOC(Au)\n∆2, SOC(Au)\n(b)\nFIG. 7. (a) Three-site 4-spin DMI-like interaction, Dz\nijkjand\n(c) three-spin chiral exchange interaction (TCI) paramete rs\nJ∆calculated for Fe on Au (111) on the basis of Eq. (75) as\na function of SOC scaling parameter ξSOCfor the smallest\ntriangles ∆ 1and ∆ 2. In figure (b), full symbols represent\nthe results obtained when scaling the SOC for all elements in\nthe system, while open symbols show the results when scaling\nonly the SOC for Au. All data are taken from Ref. 68.\nby Dederichs et al.34, Eq. (77) is transformed to the form\n∆E=−1\nπIm Tr/integraldisplayEF\ndE/summationdisplay\np1\npTr/bracketleftBig\nGstr(E)∆t(E)/bracketrightBigp\n.(78)\nBy splitting the structural Green function Gstr\nijinto a\nspin-dependent ( /vectorBstr\nij) and a spin-independent ( Astr\nij)\nparts according to\nGstr\nij=Astr\nijσ0+/vectorBstr\nij·/vector σ (79)\nand expressing the change of the single-site scattering\nmatrix\n∆ti(E) = (t↑\ni(E)−t↓)δˆsi×/vector σ, (80)\nby means of the rigid spin approximation, the different\nterms in Eq. (78) corresponding to different numbers p\ngive access to corresponding multispin terms, chiral and\nnon-chiral, in the extended spin Hamiltonian. In particu-\nlar, the isotropic six-spin interactions, that are responsi-\nbleforthenon-collinearmagneticstructureofB20-MnGe\naccording to Grytsiuk et al69, is given by the expression\nκ6−spin\nijklmn=1\n3πIm Tr/integraldisplayEF\ndE\n×Aijtσ\njAjktσ\nkAkltσ\nlAlmtσ\nmAmntσ\nnAnitσ\ni.(81)A rather different point of view concerning the multi-\nspin extension of the spin Hamiltonian was adopted by\nStreib et al.72,73, who suggested to distinguish so-called\nlocal and global Hamiltonians. According to that classi-\nfication, a global Hamiltonian implies to include in prin-\nciple all possible spin configurations for the energy map-\nping in orderto calculate exchangeparametersthat char-\nacterize in turn the energy of any spin configuration. On\nthe other hand, a local Hamiltonian is ’designed to de-\nscribe the energetics of spin configurations in the vicinity\nof the ground state or, more generally, in the vicinity of a\npredefined spin configuration’72. This implies that taking\nthe ground state as a reference state, it has to be deter-\nmined first before the calculating the exchange parame-\nters which are in principle applicable only for small spin\ntiltings around the reference state and can be used e.g.\nto investigate spin fluctuations around the ground state\nspin configuration. In Ref. 72, the authors used a con-\nstrainingfieldtostabilizethenon-collinearmagneticcon-\nfiguration. This leads to the effective two-spin exchange\ninteractions corresponding to a non-collinear magnetic\nspin configuration72,73. According to the authors, ’lo-\ncal spin Hamiltonians do not require any spin interac-\ntions beyond the bilinear order (for Heisenberg exchange\nas well as Dzyaloshinskii-Moriya interactions)’ . On the\nother hand, they point out the limitations for these ex-\nchange interactions in the case of non-collinear system in\nthe regime when the standard Heisenberg model is not\nvalid73, and multi-spin interactions get more important.\nII. GILBERT DAMPING\nAnother parameter entering the Landau-Lifshitz-\nGilbert (LLG) equation in Eq. (3) is the Gilbert damping\nparameter ˜Gcharacterizingenergydissipation associated\nwith the magnetization dynamics.\nTheoretical investigations on the Gilbert damping pa-\nrameter have been performed by various groups and ac-\ncordinglythepropertiesofGDisdiscussedindetailinthe\nliterature. Many of these investigations are performed\nassuming a certain dissipation mechanism, like Kamber-\nsky’sbreathingFermisurface(BFS)74,75, ormoregeneral\ntorque-correlationmodels (TCM)76,77to be evaluated on\nthe basis of electronic structure calculations. The earlier\nworks in the field relied on the relaxation time param-\neter that represents scattering processes responsible for\nthe energy dissipation. Only few computational schemes\nfor Gilbert damping parameter account explicitly for dis-\norderin the systems, which is responsible forthe spin-flip\nscatteringprocess. This issuewasaddressedin particular\nby Brataas etal.78who described the Gilbert damping\nmechanism by means of scattering theory. This develop-\nmentsuppliedtheformalbasisforthefirstparameter-free\ninvestigations on disordered alloys38,39,79.\nA formalism for the calculation of the Gilbert damping\nparameter based on linear response theory has been re-\nported in Ref. 39 and implemented using fully relativistic13\nmultiple scattering or Korringa-Kohn-Rostoker (KKR)\nformalism. Considering the FM state as a reference state\nof the system, the energy dissipation can be expressed in\nterms of the GD parameter by:\n˙Emag=/vectorHeff·d/vectorM\ndτ=1\nγ2˙/vector m[˜G(/vector m)˙/vector m].(82)\nOn the other hand, the energy dissipation in the elec-\ntronic system is determined by the underlying Hamilto-\nnianˆH(τ) as follows ˙Edis=/angbracketleftBig\ndˆH\ndτ/angbracketrightBig\n. Assuming a small\ndeviation of the magnetic moment from the equilibrium\n/vector u(τ), the normalized magnetization /vector m(τ) can be written\nin a linearized form /vector m(τ) =/vector m0+/vector u(τ), that in turn leads\nto the linearized time dependent electronic Hamiltonian\nˆH(τ)\nˆH=ˆH0(/vector m0)+/summationdisplay\nµ/vector uµ∂\n∂/vector uµˆH(/vector m0).(83)\nAs shown in Ref. 38, the energy dissipation within the\nlinear response formalism is given by:\n˙Edis=−π/planckover2pi1/summationdisplay\nij/summationdisplay\nµν˙uµ˙uν/angbracketleftBigg\nψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg\nψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ej).(84)\nIdentifying it with the corresponding phenomenological\nquantity in Eq. (82), ˙Emag=˙Edisone obtains for the GD\nparameterαa Kubo-Greenwood-like expression:\nαµν=−/planckover2pi1γ\nπMsTrace/angbracketleftBigg\n∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBigg\nc,\n(85)\nwhereα=˜G/(γMs), and/an}b∇acketle{t.../an}b∇acket∇i}htcindicates a configura-\ntional average required in the presence of chemical or\nthermally induced disorder responsible for the dissipa-\ntion processes. Within the multiple scattering formalism\nwith the representation of the Green function given by\nEq. (9), Eq. (85) leads to\nαµµ=g\nπµtot/summationdisplay\nnTrace/angbracketleftbig\nT0µ˜τ0nTnµ˜τn0/angbracketrightbig\nc(86)\nwith the g-factor 2(1 + µorb/µspin) in terms of the spin\nand orbital moments, µspinandµorb, respectively, the\ntotal magnetic moment µtot=µspin+µorb, and ˜τ0n\nΛΛ′=\n1\n2i(τ0n\nΛΛ′−τ0n\nΛ′Λ) and with the energy argument EFomit-\nted. The matrix elements Tnµare identical to those oc-\ncurring in the context of exchange coupling6and can be\nexpressed in terms of the spin-dependent part Bof the\nelectronic potential with matrix elements:\nTnµ\nΛ′Λ=/integraldisplay\nd3rZn×\nΛ′(/vector r) [βσµBxc(/vector r)]Zn\nΛ(/vector r).(87)\nAs is discussed in Ref. 39, fora system havingchemical\ndisorder, the configurational average is performed using00.10.2 0.3 0.4 0.5\nconcentration xV02040α × 103without vertex corrections\nwith vertex correctionsFe1-xVx\n(a)\nFIG. 8. The Gilbert damping parameter for (a) bcc Fe 1−xVx\n(T= 0 K) as a function of V concentration. Full (open)\nsymbols give results with (without) the vertex corrections .\nAll data are taken from Ref. 39.\nthe scattering path operators evaluated on the basis of\nthe coherent potential approximation (CPA) alloy the-\nory. In the case of thermally induced disorder, the al-\nloy analogy model is used, which was discussed already\nabove. When evaluating Eq. (86), the so-called vertex\ncorrections have to be included43that accounts for the\ndifference between the averages /an}b∇acketle{tTµImG+TνImG+/an}b∇acket∇i}htcand\n/an}b∇acketle{tTµImG+/an}b∇acket∇i}htc/an}b∇acketle{tTνImG+/an}b∇acket∇i}htc. Within the Boltzmann formal-\nism these corrections account for scattering-in processes.\nThe crucial role of these corrections is demonstrated39\nin Fig. 8 representing the Gilbert damping parameter\nfor an Fe 1−xVxdisordered alloy as a function of the con-\ncentrationx, calculated with and without vertex correc-\ntions. As one can see, neglect of the vertex corrections\nmay lead to the nonphysical result α <0. This wrong\nbehavior does not occur when the vertex corrections are\nincluded, that obviously account for energy transfer pro-\ncesses connected with scattering-in processes.\nThe impact of thermal vibrations onto the Gilbert\ndamping can be taken into account within the alloy-\nanalogy model (see above) by averaging over a discrete\nset of thermal atom displacements for a given temper-\natureT. Fig. 9 represents the temperature dependent\nbehavior of the Gilbert damping parameter αfor bcc Fe\nwith 1% and 5% of impurities of Os and Pt38,39. One can\nseeastrongimpactofimpuritiesonGD.Inthecaseof1%\nof Pt in Fig. 9 (a), αdecreases in the low-temperature\nregime much steeper upon increasing the temperature,\nindicating that the breathing Fermi surface mechanism\ndominates. When the concentration of the impurities in-\ncreases up to 5% (Fig. 9 (a)), the spin-flip scattering\nmechanism takes the leading role for the magnetization\ndissipation practically for the whole region of tempera-\ntures under consideration. The different behavior of GD\nforFe with OsandPt isaresult ofthe different densityof\nstates (DOS) of the impurities at the Fermi energy (see\nRef. 39 for a discussion).\nThe role of the electron-phonon scattering for the ul-\ntrafast laser-induced demagnetization was investigated14\n0100200 300 400 500\ntemperature (K)12345α × 103Fe0.99Me0.01Pt\nOs\n(a)\n0100200 300 400 500\ntemperature (K)22.533.54α × 103Fe0.95Me0.05\nPtOs\n(b)\nFIG. 9. Gilbert damping parameter for bcc Fe 1−xMxwith\nM= Pt (circles) and M= Os (squares) impurities as a func-\ntion of temperature for 1% (a) and 5% (b) of the impurities.\nAll data are taken from Ref. 39.\nby Carva et al.80based on the Elliott-Yafet theory of\nspin relaxation in metals, that puts the focus on spin-\nflip(SF) transitionsupon theelectron-phononscattering.\nAs the evaluation of the spin-dependent electron-phonon\nmatrix elements entering the expression for the rate of\nthe spin-flip transition is a demanding problem, various\napproximations are used for this. In particular, Carva et\nal.80,81use the so-called Elliott approximation to evalu-\nate a SF probability Pb\nS=τ\nτsfwith the spin lifetime τsf\nand a spin-diagonal lifetime τ:\nPb\nS=τ\nτsf= 4/an}b∇acketle{tb2/an}b∇acket∇i}ht (88)\nwith the Fermi-surface averaged spin mixing of Bloch\nwave eigenstates\n/an}b∇acketle{tb2/an}b∇acket∇i}ht=/summationdisplay\nσ,n/integraldisplay\nd3k|bσ\n/vectorkn|δ(Eσ\n/vectorkn−EF).(89)\nIn the case of a non-collinear magnetic structure, the\ndescription of the Gilbert damping can be extended byadding higher-order non-local contributions. The role of\nnon-local damping contributions has been investigated\nby calculating the precession damping α(/vector q) for magnons\nin FM metals, characterized by a wave vector /vector q. Follow-\ning the same idea, Thonig et al.82used a torque-torque\ncorrelationmodel based on atight binding approach,and\ncalculated the Gilbert damping for the itinerant-electron\nferromagnets Fe, Co and Ni, both in the reciprocal, α(/vector q),\nand realαijspace representations. The important role\nof non-local contributions to the GD for spin dynam-\nics has been demonstrated using atomistic magnetization\ndynamics simulations.\nAformalismforcalculatingthe non-localcontributions\nto the GD has been recently worked out within the KKR\nGreen function formalism83. Using linear response the-\nory for weakly-noncollinear magnetic systems it gives ac-\ncess to the GD parameters represented as a function of\na wave vector /vector q. Using the definition for the spin sus-\nceptibility tensor χαβ(/vector q,ω), the Fourier transformation\nof the real-space Gilbert damping can be represented by\nthe expression84,85\nααβ(/vector q) =γ\nM0Vlim\nω→0∂ℑ[χ−1]αβ(/vector q,ω)\n∂ω.(90)\nHereγ=gµBis the gyromagneticratio, M0=µtotµB/V\nis the equilibrium magnetization and Vis the volume of\nthe system. As is shown in Ref. 83, this expression can\nbe transformed to the form which allows an expansion of\nGD in powers of wave vector /vector q:\nα(/vector q) =α+/summationdisplay\nµαµqµ+1\n2/summationdisplay\nµναµνqµqν+....(91)\nwith the following expansion coefficients:\nα0±±\nαα=g\nπµtot1\nΩBZTr/integraldisplay\nd3k/angbracketleftbigg\nTβτ(/vectork,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc\nαµ±±\nαα=g\nπµtot1\nΩBZTr/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβτ(/vectork,E±\nF)/angbracketrightbigg\nc\nαµν±±\nαα=−g\n2πµtot1\nΩBZ\n×Tr/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβ∂τ(/vectork,E±\nF)\n∂kν/angbracketrightbigg\nc.(92)\nFor the prototype multilayer system\n(Cu/Fe 1−xCox/Pt)nthe calculated zero-order (uni-\nform) GD parameter αxxand the corresponding\nfirst-order (chiral) αx\nxxcorrection term for /vector q/ba∇dblˆxare\nplotted in Fig. 10 top and bottom, respectively, as a\nfunction of the Co concentration x. Both terms, αxx\nandαx\nxx, increase approaching the pure limits w.r.t. the\nFe1−xCoxalloy subsystem. As is pointed out in Ref. 83,\nthis increase is associated with the dominating so-called\nbreathing Fermi-surface damping mechanism due to the\nmodification of the Fermi surface (FS) induced by the\nSOC, which follows the magnetization direction that\nslowly varies with time. As αis caused for a ferromagnet15\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. 10. The Gilbert damping parameters αxx(top) and\nαx\nxx(bottom) calculated for the model multilayer system\n(Cu/Fe 1−xCox/Pt)nusing first and second expressions in Eq.\n(92), respectively. All data are taken from Ref. 83.\nexclusively by the SOC one can expect that it vanishes\nfor vanishing SOC. This was indeed demonstrated\nbefore39. The same holds also for αxthat is caused by\nSOC as well.\nAlternatively, a real-space extension for classical\nGilbert dampingtensorwasproposedrecentlybyBrinker\net al.86, by introducing two-site Gilbert damping tensor\nGijentering the site-resolved LLG equation\n1\nγd/vectorMi\ndτ=−γ/vectorMi×/parenleftbigg\n/vectorHi,eff+/summationdisplay\nj/bracketleftBigg\nGij(/vectorM)·d/vectorMi\ndτ/bracketrightBigg/parenrightbigg\n,(93)\nwhich is related to the inverse dynamical susceptibility\nχijvia the expression\nd\ndωIm[χ]αβ\nij=δij/parenleftbigg1\nγMiǫαβγ/parenrightbigg\n+/parenleftbigg\nRiGijRT\nj/parenrightbigg\nαβ,(94)\nwhereRiandRjarethe rotationmatricesto gofromthe\nglobal to the local frames of reference for atoms iandj,\nrespectively, assuming a non-collinear magnetic ground\nstate in the system. Thus, an expression for the GD\ncan be directly obtained using the adiabatic approxima-\ntion for the slow spin-dynamics processes. This justifies\nthe approximation ([ χ]−1(ω))′\nω≈([χ0]−1(ω))′\nω, with the\nun-enhanced dynamical susceptibility given in terms ofelectronic Green function Gij\nχαβ\nij(ω+iη) =−1\nπTr/integraldisplayEF\ndE\n/bracketleftbigg\nσαGij(E+ω+iη)σβImGij(E)\n+σαGij(E)σβImGij(E−ω−iη)/bracketrightbigg\n,(95)\nwith the Green function G(E±iη) = (E− H ±iη)−1\ncorresponding to the Hamiltonian H.\nMoreover, this approach allows a multisite expansion\noftheGDaccountingforhigher-ordernon-localcontribu-\ntions for non-collinearstructures86. For this purpose, the\nHamiltonian His split into the on-site contribution H′\nand the intersite hopping term tij, which is spin depen-\ndent in the general case. The GF can then be expanded\nin a perturbative way using the Dyson equation\nGij=G0\niδij+G0\nitijG0\nj+G0\nitikG0\nktkjG0\nj+....(96)\nAs a result, the authors generalize the LLG equation\nby splitting the Gilbert damping tensor in terms pro-\nportional to scalar, anisotropic, vector-chiral and scalar-\nchiral products of the magnetic moments, i.e. terms like\nˆei·ˆej, (ˆnij·ˆei)(ˆnij·ˆej), ˆnij·(ˆei׈ej), etc.\nIt should be stressed that the Gilbert damping param-\neter accounts for the energy transfer connected with the\nmagnetization dynamics but gives no information on the\nangular momentum transfer that plays an important role\ne.g. for ultrafast demagnetization processes. The formal\nbasis to account simultaneously for the spin and lattice\ndegrees of freedom was considered recently by Aßmann\nand Nowak87and Hellsvik et al.88. Hellsvik et al.88,89re-\nportonanapproachsolvingsimultaneouslytheequations\nfor spin and lattice dynamics, accounting for spin-lattice\ninteractions in the Hamiltonian, calculated on a first-\nprinciples level. These interactions appear as a correc-\ntion to the exchange coupling parameters due to atomic\ndisplacements. As a result, this leads to the three-body\nspin-lattice coupling parameters Γαβµ\nijk=∂Jαβ\nij\n∂uµ\nkand four-\nbody parameters Λαβµν\nijkl=∂Jαβ\nij\n∂uµ\nk∂uν\nlrepresented by rank 3\nand rank 4 tensors, respectively, entering the spin-lattice\nHamiltonian\nHsl=−1\n2/summationdisplay\ni,j,k,αβ,µΓαβµ\nijkeα\nieβ\njuµ\nk\n−1\n4/summationdisplay\ni,j,k,l,αβ,µ,νΛαβµν\nijkleα\nieβ\njuµ\nkuν\nl.(97)\nThe parameters Γαβµ\nijkin Ref. 88 are calculated using a\nfinite difference method, using the exchange coupling pa-\nrametersJijfor the system without displacements ( J0\nij)\nand with a displaced atom k(J∆\nij(/vector uk)), used to estimate\nthe coefficient Γαβµ\nijk≈(J∆\nij(/vector uk)−J0\nij)\nuµ.16\nAlternatively, to describe the coupling of spin and spa-\ntial degrees of freedom the present authors (see Ref. 90)\nadopt an atomistic approach and start with the expan-\nsion of a phenomenological spin-lattice Hamiltonian\nHsl=−/summationdisplay\ni,j,α,β/summationdisplay\nk,µJαβ,µ\nij,keα\nieβ\njuµ\nk\n−/summationdisplay\ni,j/summationdisplay\nk,lJαβ,µν\nij,kleα\nieβ\njuµ\nkuν\nl,(98)\nthat can be seen as a lattice extension of a Heisenberg\nmodel. Accordingly, the spin and lattice degrees of free-\ndom are represented by the orientation vectors ˆ eiof the\nmagnetic moments /vector miand displacement vectors /vector uifor\neach atomic site i. The spin-lattice Hamiltonian in Eq.\n(98) is restricted to three and four-site terms. As rel-\nativistic effects are taken into account, the SLC is de-\nscribed in tensorial form with Jαβ,µ\nij,kandJαβ,µν\nij,klrepre-\nsented by rank 3 and rank 4 tensors, similar to those\ndiscussed by Hellsvik et al.88.\nThesamestrategyasforthe exchangecouplingparam-\netersJij4orJαβ\nij5,6, is used to map the free energy land-\nscapeF({ˆei},{/vector ui}) accounting for its dependence on the\nspin configuration {ˆei}as well as atomic displacements\n{/vector ui}, making use of the magnetic force theorem and the\nLloyd formulato evaluate integrated DOS ∆ N(E). With\nthis, the free energy change due to any perturbation in\nthe system is given by Eq. (25).\nUsing as a reference the ferromagnetically ordered\nstate of the system with a non-distorted lattice, and the\nperturbed state characterized by finite spin tiltings δˆei\nand finite atomic displacements /vector uiat sitei, one can\nwrite the corresponding changes of the inverse t-matrix\nas ∆s\nµmi=mi(δˆeµ\ni)−m0\niand ∆u\nνmi=mi(uν\ni)−m0\ni.\nThis allows to replace the integrand in Eq. (11) by\nlnτ−lnτ0=−ln/parenleftBig\n1+τ[∆s\nµmi+∆u\nνmj+...]/parenrightBig\n,(99)\nwhere all site-dependent changes in the spin configura-\ntion{ˆei}and atomic positions {/vector ui}are accounted for in\na one-to-one manner by the various terms on the right\nhand side. Due to the use of the magnetic force theorem\nthese blocks may be written in terms of the spin tiltings\nδˆeµ\niand atomic displacements of the atoms uν\nitogether\nwith the corresponding auxiliary matrices Tµ\niandUν\ni,\nrespectively, as\n∆s\nµmi=δˆeµ\niTµ\ni, (100)\n∆u\nνmi=uν\niUν\ni. (101)\nInserting these expressionsinto Eq. (99) and the result in\nturnintoEq.(25)allowsustocalculatetheparametersof\nthe spin-lattice Hamiltonian as the derivatives of the free\nenergy with respect to tilting angles and displacements.\nThis way one gets for example for the three-site term:\nJαβ,µ\nij,k=∂3F\n∂eα\ni∂eβ\nj∂uµ\nk=1\n2πIm Tr/integraldisplayEF\ndE\n×/bracketleftBig\nTα\niτijTβ\njτjkUµ\nkτki+Tα\niτikUµ\nkτkjTβ\njτji/bracketrightBig\n(102)\nFIG. 11. The absolute values of site-off-diagonal and site-\ndiagonal SLC parameters: DMI |/vectorDx\nij,j|and isotropic SLC\nJiso,x\nij,j(top), anti-symmetric diagonal components Jdia−a,x\nij,j\nandJdia−a,x\nii,k(middle), and symmetric off-diagonal compo-\nnentsJoff−s,x\nij,jandJoff−s,x\nii,k(bottom) for bcc Fe, as a function\nof the interatomic distance rij\nand for the four-site term:\nJαβ,µν\nij,kl=∂4F\n∂eα\ni∂eβ\nj∂uµ\nk∂uν\nl=1\n4πIm Tr/integraldisplayEF\ndE\n×/bracketleftBigg\nUµ\nkτkiTα\niτijTβ\njτjlUν\nlτlk\n+Tα\niτikUµ\nkτkjTβ\njτjlUν\nlτli\n+Uµ\nkτkiTα\niτilUν\nlτljTβ\njτjk\n+Tα\niτikUµ\nkτklUν\nlτljTβ\njτji/bracketrightBigg\n.(103)\nFig. 11 shows corresponding results for the SLC pa-\nrameters of bcc Fe, plotted as a function of the distance\nrijfori=kwhich implies that a displacement along the\nxdirection is applied for one of the interacting atoms.\nThe absolute values of the DMI-like SLC parameters\n(DSLC) |/vectorD|µ=x\nij,k(note that Dz,µ\nij,k=1\n2(Jxy,µ\nij,k− Jyx,µ\nij,k) )\nshow a rather slow decay with the distance rij. The\nisotropic SLC parameters Jiso,µ=x\nij,j, which have only a\nweak dependence on the SOC, are about one order\nof magnitude larger than the DSLC. All other SOC-\ndriven parameters shown in Fig. 11, characterizing the\ndisplacement-induced contributions to MCA, are much\nsmaller than the DSLC.17\nIII. SUMMARY\nTo summarize, we have considered a multi-level atom-\nistic approach commonly used to simulate finite temper-\natureand dynamical magneticpropertiesof solids, avoid-\ning in particular time-consuming TD-SDFT calculations.\nTheapproachisbasedonaphenomenologicalparameter-\nized spin Hamiltonian which allows to separate the spin\nand orbital degrees of freedom and that way to avoid the\ndemanding treatment of complex spin-dependent many-\nbody effects. As these parameters are fully determined\nby the electronic structure of a system, they can be de-\nduced from the information provided by relativistic band\nstructure calculations based on SDFT. 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Ebert, arXiv:2203.16144v1 (2022),\n10.48550/ARXIV.2203.16144."    },    {        "title": "2208.02136v2.On_ergodic_invariant_measures_for_the_stochastic_Landau_Lifschitz_Gilbert_equation_in_1D.pdf",        "content": "arXiv:2208.02136v2  [math.PR]  28 Dec 2023On ergodic invariant measures for the stochastic\nLandau-Lifschitz-Gilbert equation in 1D\nEmanuela Gussetti\nBielefeld University, Germany\nDecember 29, 2023\nAbstract\nWe establish existence of an ergodic invariant measure on H1(D,R3)∩L2(D,S2)for the stochastic\nLandau-Lifschitz-Gilbert equation on a bounded one dimens ional interval D. The conclusion is achieved by\nemploying the classical Krylov-Bogoliubov theorem. In con trast to other equations, verifying the hypothesis\nof the Krylov-Bogoliubov theorem is not a standard procedur e. We employ rough paths theory to show that\nthe semigroup associated to the equation has the Feller prop erty inH1(D,R3)∩L2(D,S2). It does not\nseem possible to achieve the same conclusion by the classica l Stratonovich calculus. On the other hand, we\nemploy the classical Stratonovich calculus to prove the tig htness hypothesis. The Krein-Milman theorem\nimplies existence of an ergodic invariant measure. In case o f spatially constant noise, we show that there\nexists a unique Gibbs invariant measure and we establish the qualitative behaviour of the unique stationary\nsolution. In absence of the anisotropic energy and for a spat ially constant noise, we are able to provide a\npathwise long time behaviour result: in particular, every s olution synchronises with a spherical Brownian\nmotion and it is recurrent for large times.\n1 Introduction\nThe Landau-Lifschitz-Gilbert (LLG) equation is a phenomen ological model that describes the time evolution\nof the magnetization of a ferromagnetic material for small t emperatures below the Curie temperature, namely\nthe temperature above which the material loses its permanen t magnetic properties. The LLG equation is given\nby\ndu=λ1u×H(u)−λ2u×(u×H(u)), (1.1)\nwhereλ1∈Randλ2>0. Before introducing H(φ), we need to define the energy functional. Given a bounded\nintervalD⊂R, we define the energy functional for all φ∈L2(D;S2)∩H1(D;R3)and by\nE(φ) :=1\n2/integraldisplay\nD/bracketleftbig\n|∂xφ(x)|2+g(φ(x))·φ(x)/bracketrightbig\ndx, (1.2)\nwhereg:R3→R3is a linear function. The gradient part in (1.2) represents t he exchange energy, which is a\nquantum mechanical effect responsible of the ferromagneti c phenomenon. The bilinear term in (1.2) takes into\naccount the anisotropic energy, which describes the fact th at some properties of the material occur differently\nin different directions. An example of the anisotropic ener gy for the current system is for β2,β3∈R\ng(φ)≡g((φ1,φ2,φ3)) = (0,β2φ2,β3φ3),\nwhere we could read that two directions are getting a push in c omparison to the first one, where no linear push\nis applied (of course other choices of the “privileged” axis are possible). The operator H(φ)is minus the L2\ngradient of the energy functional, namely\nH(φ) =−∇E(φ) =∂2\nxφ−g′(φ),\nMathematics Subject Classification (2022) — 60G10, 60H15, 60L50, 60L90.\nKeywords and phrases — Landau-Lifschitz-Gilbert equation, harmonic map heat flow , invariant measures, ergodic measures, spherical\nBrownian motion, stationary solutions, Gibbs measure, rou gh PDEs, long time behaviour.\n1for allφ∈L2(D;S2)∩H1(D;R3). Landau and Lifschitz [41] provided for the first time a pheno menological\nderivation of the equation in 1935 and later, in 1954, Gilber t [26] considered and other damping parameter. The\ntwo equations proposed by Landau and Lifschitz and Gilbert h ave been proved to be mathematically equivalent\nby Brown [10]. See [40] for an overview on the physics.\nMathematically, the deterministic equation (1.1) has been studied extensively. The existence and non\nuniqueness of weak solution on a two and three dimensional do main has been established by Alouges and\nSoyeur [2]. For the existence and uniqueness of strong solut ions on a one dimensional domain, we refer to\nY . Zhu, B. Guo, S. Tan [48]. See also De Laire [20] for a review o n recent developments in the deterministic\nframework as well as for the connection with other equations . In the deterministic setting, we mention existing\nresults on steady states (i.e. stationary in time solutions ): Carbou, Labbe [16] for the one dimensional LLG\nand Van den Berg, J. Williams [15, 47] for the two and three dim ensional case. We stress here that, if λ1= 0,\nλ2= 1 and we do not consider the anisotropic energy, (1.1) coincid es with the harmonic map flow on the\nsphereS2⊂R3given by\n∂tu=∂2\nxu+u|∂xu|2,\nvia the equality −ut×(ut×∂2\nxut) =∂2\nxut+ut|∂xut|2inH−1forut(x)∈S2(see [12, Lemma 2.4]). Thus,\nthe results of this paper apply also to the harmonic map flow.\nThere are also several studies on the stochastic version of ( 1.1). Indeed the introduction of a noise in (1.1)\nmodels the fact that ferromagnetic materials are susceptib le to temperature changes. As observed by Brown\n[9, 10], a way to do so is to include a Brownian fluctuation Winto the gradient of the energy: in this way\nthe random changes of temperature are taken into account. In troducing a thermal fluctuation leads to a (not\nanymore phenomenological) description of the magnetizati on as a solution on [0,T]×Dto the equation\ndu= [λ1u×∂2\nxu−λ2u×(u×∂2\nxu)+λ1u×g′(u)−λ2u×(u×g′(u))]dt+u×◦dW , (1.3)\nfor an initial condition u0=u0, with null Neumann boundary conditions and a saturation con dition on the\nmaterial (which gives the mathematical spherical constrai nt)\n∂xut(x) = 0∀(t,x)∈[0,T]×∂D,|ut(x)|= 1∀(t,x)∈[0,T]×D.\nIn this work the noise Wis chosen to be white in time and more regular in space and the i ntegral is understood\nin the sense of Stratonovich. More precisely, throughout th is introduction, we assume the following conditions\non the noise.\nAssumption 1.1. We consider noises either of the form Wt(x) :=h1(x)Bt, whereh1∈H1(D;R3),Bis a\nreal valued Brownian motion, or of the form Wt(x) =h2(x)¯Bt, whereh2∈H1(D;R)and¯Bis aR3-valued\nBrownian motion.\nIt is possible to consider linear combinations of the two noi ses in what follows, but we consider only one\nnoise type at the time for clarity of exposition. The bounds i n the theorems would change accordingly.\nThe addition of the anisotropic energy as a linear operator i nto the equation does not constitute a mathemat-\nical difficulty when studying existence and uniqueness of th e equation via our method (also in the stochastic\ncase). On the other hand, as proved also in Brze´ zniak, Goldy s and Jegaraj [12], the anisotropic energy is deter-\nminant to capture some physical behaviours of the magnetiza tion, like the magnetization reversal phenomenon.\nThe anisotropic energy gives an accurate description of the stray energy for one dimensional magnets. We\ninclude the anisotropic energy in the study. Throughout thi s introduction, we assume the following condition\non the anisotropic energy.\nAssumption 1.2. Assume that the anisotropy is of the form g′(x) :=Ax+b, whereAis a real valued 3×3\nmatrix and b∈R3.\nConcerning existence of weak solutions to the stochastic LL G in more dimensions, we mention the works\n[11, 30, 28, 27] (but the list is not exhaustive). Existence o f a martingale solution uto (1.3) on a one dimensional\ndomain is established by Brze´ zniak, Goldys, Jegaraj [12] a nd Brze´ zniak, Manna, Mukherjee [14]. The authors\nalso prove pathwise uniqueness. Gussetti, Hocquet [29] pro ve existence and uniqueness of a pathwise energy\nsolutionuinL∞(H1)∩L2(H2)∩C([0,T],L2)by means of rough paths theory.\n2The existence of a unique solution to (1.3) implies the exist ence of a Markov semigroup associated to u,\nthat we denote by (Pt)t. From a physical perspective, studying the semigroup (Pt)tcorresponds to looking at\nthe average behaviour of the trajectories at every time t >0. In other words, we observe where the trajectories\nare concentrated on the sphere at time tor at the average configuration.\nA typical question in this framework concerns the existence of a configuration on the sphere which does\nnot vary in time, that we can refer to as stationary configurat ion. This translates into searching for the so-called\ninvariant measures, which correspond to equilibrium state s of the system (see e.g. [42]). The existence of an\ninvariant measure µassociated to (Pt)tonH1(S2) :=L2(D;S2)∩H1(D;R3)and the rigorous identification\nwith the Gibbs measure\nexp(−E(y))dy/integraltext\nH1(S2)exp(−E(v))dv(1.4)\nfor some of those invariant measures, are open problems. Not e that the formulation in (1.4) is only formal.\nIndeed the only existing translation invariant Borel measu res on an infinite dimensional separable Banach space\nare either the null measure or the measure that assigns to eve ry open set +∞. As a consequence, there is no\nnatural extension (in this sense) of a “Lebesgue”-like meas ure to an infinite dimensional separable Banach\nspace. There are nevertheless different ways of constructi ng measures on an infinite dimensional space. A\nformal derivation of the fact that the invariant measure sho uld be a Gibbs measure of the form (1.4) is established\nby Garc´ ıa-Palacios, L´ azaro [25].\nIn the framework of ferromagnetism, it is worth mentioning t he Landau-Lifschitz-Bloch (LLB) equation,\nwhich is an alternative model to the LLG equation that holds b elow, close and above the Curie temperature.\nThe LLB equation interpolates between the LLG at low tempera tures and the Bloch equation, which holds\nfor very high temperatures. In the LLB model, the magnetizat ion is no more constrained to the sphere. The\nLLB equation has been introduced by Garanin [23, 24] and its s tochastic version has been studied by Jiang,\nJu, Wang [38] and Brze´ zniak, Goldys and K.-N. Le [13]. The fir st named authors establish global existence of\nmartingale weak solutions to the LLB equations. The second n amed authors establish existence and uniqueness\nof probabilistically strong martingale solutions to the eq uation and the existence of an invariant measure in one\nand two dimensions.\nDespite the numerous studies on the stochastic LLG equation and related equations in ferromagnetism,\nthe existence of invariant measures for the LLG has not been a ddressed for the full equation (1.3). The only\nresult approaching an approximated version of the problem i s from Neklyudov, Prohl [43]. The authors show\nrigorously the existence of a Gibbs invariant measure of a fin ite system of spins behaving like a simplified\nLLG equation: the construction occurs by means of a Lyapunov function. The authors also make crucial\nconsiderations on the role of the noise and its implications on invariant measures. This fact is explored in\nwhat follows. We refer to the monograph [6] and the reference s therein for further details on the numerical\napproaches and the open problems concerning the LLG equatio n. We mention the study\nThe main results The aim of this paper is to prove existence of an ergodic invar iant measures for the stochas-\ntic LLG equation on a one dimensional domain. We prove existe nce of an ergodic invariant measure in presence\nof both exchange energy and anisotropic energy (in the form o f a bilinear operator). In some special cases, we\ncan identify the invariant measure with the Gibbs distribut ion. We state the main results and the procedures.\nExistence of an ergodic invariant measure and stationary so lutions.\nTheorem. The semigroup (Pt)tonH1(S2)associated to (1.3) admits an invariant measure.\nWe prove existence of an invariant measure (refer to Theorem 4.14) by means of the classical Krylov-\nBogoliubov theorem. The rough path formulation is useful wh en dealing with pathwise properties: here the\ndeterministic calculus can be employed and one does not have to take in account integrability and measurability\nissues with respect to the probability measure. On the other hand classical stochastic calculus is of use when\ndealing with probabilistic properties, like the martingal e property of the Itˆ o integral.\nWe use rough paths theory to show that the semigroup (Pt)thas the Feller property: the proof of this fact\nis a consequence of the continuity of the Itˆ o-Lyons map with respect to the initial condition and the noise, as\nalready proved in [29]. We include the proof and statements a lso in this paper for completeness of exposition.\n3The Feller property is achieved pathwise in H1(S2)and no condition on the intensity of the noise is required.\nRough paths theory is determinant to show the pathwise bound\nsup\nt∈[0,T]/ba∇dblut−vt/ba∇dbl2\nH1/lessorsimilarexp(T),W/bracketleftbig\n/ba∇dblu0/ba∇dbl4\nH1+/ba∇dblv0/ba∇dbl4\nH1/bracketrightbig\n/ba∇dblu0−v0/ba∇dbl2\nH1, (1.5)\nwhereu,vare solutions to (1.1) started at u0,v0∈H1(S2). The continuity with respect to the initial condition\ninH1(S2)implies the Feller property in Theorem 4.3. In other words, t he semigroup maps continuous bounded\nfunctions from H1(S2)toRto continuous bounded functions from H1(S2)toR:\nTheorem. The semigroup (Pt)thas the Feller property, i.e. Pt:Cb(H1(S2))→Cb(H1(S2))for allt >0.\nIt is fundamental to stress that classical Stratonovich cal culus does not allow to achieve easily (1.5): indeed\none needs to employ the Itˆ o’s isometry to estimate the stoch astic integral and therefore it is needed to take the\nexpectation of the inequality. But the drift of the equation is multiplied by the initial conditions to the power\nfour. As a consequence, by applying expectation, one is not a ble to close the estimate by means of the classical\nGronwall’s Lemma. We expand on this point in Remark 4.5.\nThe proof of tightness is achieved by the classical Itˆ o-Str atonovich calculus and, in presence of anisotropic\nenergy, by means of Poincar´ e’s inequality; we need to use th at the Itˆ o integral is a martingale. Note that the\nStratonovich solution of (1.3) introduced in [14] and [12] c oincides with the pathwise solution in [29], a part\nfrom a set of null measure, it is possible to pass from one form ulation to the other, under the hypothesis that\nu0∈ L2(Ω;H1(S2)). We formulate in a compact form the technical estimates nece ssary to conclude tightness\n(contained in Lemma 4.9, Lemma 4.12).\nLemma. There exists C >0independent on time, so that for every t >0it holds\nsup\nr∈[0,t]E/bracketleftbig\n/ba∇dbl∂xur/ba∇dbl2\nL2/bracketrightbig\n+/integraldisplayt\n0E/bracketleftig\n/ba∇dblur×∂2\nxur/ba∇dbl2\nL2+/ba∇dbl∂2\nxur/ba∇dbl1/2\nL2/bracketrightig\ndr/lessorsimilar∂xu0,∂xh,g,λ1,λ2C(1+t).\nNote that it is possible to bound the norm /ba∇dbl∂2\nxu/ba∇dblL2with respect to the expectation only with the power 1/2.\nThis bound requires to use the geometry of the equation.\nLetµbe an invariant measure and consider an initial condition di stributed like µ. Then there exists a path-\nwise stationary solution wwith initial condition w0. If the initial condition u0used in the Krylov-Bogolioubov\ntheorem has bounded second moment, then each initial condit ion distributed like µhas bounded second mo-\nment. It is also possible to conclude that supp µ⊂H2(S2). These results are contained in Theorem 5.1\nand Theorem 5.2. The condition E[/ba∇dblw0/ba∇dbl2\nH1(S2)]<+∞allows to show that the set of invariant measures is\ncompact. Since the set of invariant measures is not empty and convex, from the Krein-Milman theorem the\nfollowing result (contained in Theorem 5.6) follows:\nTheorem. The semigroup (Pt)tonH1(S2)associated to (1.3) admits an ergodic invariant measure.\nGibbs measures for spatially constant noise and small aniso tropic energy. Up to this point, the previous\nresults do not require any restrictive assumption on the noi se nor on the drift of equation (1.1). The shape of\nthe noise does not play any role in the previous results, but i t is important in what follows. We also employ\nthe results in [43] for some considerations on invariant mea sures of the LLG. We can interpret the Stratonovich\nintegral in two basic ways\n/integraldisplayt\n0ur×h1◦dBr,/integraldisplayt\n0h2ur×◦d¯Br,\nwhereh1∈H1(D;R3),Bis a real valued Brownian motion, h2∈H1(D;R)and¯Bis aR3-valued Brownian\nmotion. One could also consider linear combinations of the n oises above. We can define precisely the shape of\nstationary solutions, by assuming ∂xh1,∂xh2= 0. Consider the equations\n(A)δvs,t=D(v)s,t+/integraldisplayt\nsvr×h1◦dBr, (B)δ¯vs,t=D(¯v)s,t+/integraldisplayt\nsh2¯vr×◦d¯Br,\nwhereD(˜v)s,t=/integraltextt\ns[λ2˜vr×(˜vr×A˜vr)−λ1˜vr×A˜vr]drfor˜vr∈S2is the contribution coming from the\nanisotropic energy. We need a smallness assumption on the an isotropy.\n4Assumption 1.3. Assume that the anisotropy is of the form g′(x) :=Ax+b, whereAis a real valued 3×3\nmatrix and b∈R3. Assume further that ¯G:= 2supi,j|Ai,j|2+|b|< λ2/2Cp(2λ2+|λ1|), whereCpis the\nPoincar ´e’s constant associated to the domain D.\nRefer to Theorem 6.5 and Theorem 6.7 for the next result.\nTheorem. Assume Assumption 1.3 and ∂xh1= 0,h1/\\e}atio\\slash= 0 ( resp.∂xh2= 0,h2/\\e}atio\\slash= 0) . Then every stationary\nsolution to (1.3) is constant in space and it is a solution to (A) (resp. (B)).\nWe follow the discussion by Neklyudov, Prohl [43] on the choi ce of the noise for (1.3) and its implications.\nThe deterministic LLG equation admits steady states (i.e. s tationary in time solutions), which are in general\nnot unique. Denote by ¯ua steady state to the deterministic equation and set h1= ¯u. Then both ¯uand−¯uare\nsolutions to equation (A) and, in general, we can not expect t o have a unique stationary solution to (1.3). For\nother examples, see Proposition 1.17 in [6] (see also [43]).\nIf we consider instead equation (B)and we assume h2/\\e}atio\\slash= 0, then the stationary in time solutions to the\ndeterministic LLG are not solutions to equation (B). We also observe that equation (B)describes the so called\n“spherical Brownian motion” with an anisotropic drift. In p articular, equation (B)has a unique solution for\nh2/\\e}atio\\slash= 0 and admits a unique invariant measure. The proof of the prope rties of(B)is contained in Neklyudov,\nProhl [43] and applies to this setting. We state the main resu lt, connecting the solution to equation (B)with the\nLLG equation. Refer to Theorem 6.7.\nTheorem. Assume Assumption 1.3 and h2/\\e}atio\\slash= 0,∂xh2= 0. Then there exists a unique ergodic invariant\nmeasureµfor(1.3) . The unique invariant measure µcan be identified with a Gibbs measure on (S2,BS2), i.e.\nµ(dy) =exp(−λ2\nh2¯E(y))dy\n/integraltext\nS2exp(−λ2\nh2¯E(v))dv, (1.6)\nwhere¯E(v) =/integraltext\nDg′(v)·vdx=|D|g′(v)·v, forv∈S2.\nNote that the Gibbs measure above contains the anisotropic t erms, but no derivatives. If there is no\nanisotropy, then the Gibbs measure reduces to the uniform me asure on the sphere. Observe also that the mea-\nsureµin (1.6) on (S2,BS2)is consistent with (1.4). Indeed consider the restriction o fL2(D;S2)∩H1(D;R3)\nto the space of constant maps φv:D→S2so thatφv(x) :=vfor allx∈Dand for all v∈S2. The space\nof constant maps {φv}v∈S2can be identified with the sphere S2. Thus the energy (1.2) evaluated in each φv\ncoincides with\nE(φv) =/integraldisplay\nDg(φv)·φvdx=|D|g(φv)·φv=|D|g(v)·v=¯E(v),∀v∈S2.\nThis shows the identification of the integration domain H1(S2)withS2in this particular case, hence the corre-\nspondence of the formal Gibbs measure in (1.4) with (1.6).\nWe can compare the stationary solutions to the LLG with the wo rks of M. R¨ ockner, B. Wu, R. Zhu, X. Zhu\n[44], [45] on stochastic heat equation taking values on a Rie mannian manifold. The authors employ the theory\nof Dirichlet forms to prove existence of martingale solutio ns to the stochastic heat equation taking values on a\nRiemannian manifold. More specifically, they prove that the Wiener (Brownian bridge) measure is an invariant\nmeasure on the Riemannian path loop space. In a simplified cas e, we are able to prove that the Brownian motion\non the sphere is a stationary solutions to (1.3) and thus that the invariant measure associated to the LLG is the\none of a Brownian motion on the sphere. This fact is in accorda nce to the results obtained in [44], [45].\nLong time behaviour for ∂xh2= 0,h2/\\e}atio\\slash= 0 andg≡0.We establish also a time behaviour result under\nthe condition ∂xh2= 0: every solution converges to its spatial average, as a conse quence of the Poincar´ e-\nWirtinger theorem. In absence of anisotropic energy and pro vided∂xh2= 0, we observe that every solution\nto (1.3), independently on the initial condition, is synchr onizing with the stationary solution up to a constant\n(independent on space and time). The result is contained in P roposition 6.12 and modifies the proof for the\nstochastic mean curvature flow from Dabrock, Hofmanov´ a, Ro gers [17].\n5Proposition. Assumeh2/\\e}atio\\slash= 0,∂xh2= 0and no anisotropic energy, i.e. g≡0. For large times every solution\nuto(1.3) converges to a Brownian motion wwith values on the sphere, where wis the unique solution to\nδws,t=h2/integraldisplayt\nswr×◦d¯Br.\nThe convergence occurs up to a constant, i.e. for T→+∞fort >0\n[uT+t−wT+t]2−[uT−wT]2= 0\ninCb([0,+∞);L1)P-a.s.\nWe refer to Appendix 7.4 for a rigorous definition of the spher ical Brownian motion and for further refer-\nences. Since the spherical Brownian motion is recurrent, th en every solution to (1.3) satisfying the hypothesis\nof Proposition 6.12 undergoes magnetization reversal for l arge times.\nCorollary. Assumeh2/\\e}atio\\slash= 0,∂xh2= 0and no anisotropic energy, i.e. g≡0. Every solution to (1.3) undergoes\nmagnetization reversal for large times.\nWe conclude with several questions left open, namely the ide ntification of an ergodic invariant measure\nwith the Gibbs measure if ∂xh2is not null and the number of invariant measures in that case. It would also be\ninteresting to study the long time behaviour for (1.3) if ∂xh2/\\e}atio\\slash= 0and in presence of anisotropic energy.\nOrganisation of the paper: In Section 2 we introduce some basic notations and the rough d river. In Section 3,\nwe introduce the definition of solution to (1.3) and we recall some known results. In Section 4, we recall the\nbasics concerning invariant measures and the Krylov-Bogol iubov theorem: in Section 4.1 we show that the\nFeller property holds and in Section 4.2 we prove the tightne ss of the sequence of measures in the Krylov-\nBogoliubov theorem. In Section 5, we employ the Krein-Milma n theorem to achieve the existence of an ergodic\nmeasure. We identify the invariant measure with the unique G ibbs measure in absence of anisotropy and if\n∂xh2= 0,h2/\\e}atio\\slash= 0 in Section 6. In presence of anisotropy and provided ∂xh2= 0,h2/\\e}atio\\slash= 0, we determine the\ninvariant measure to be the Gibbs measure in Section 6. In Sec tion 6.1, we address the long time behaviour\nproblem by showing the convergence of the solutions to the LL G to the stationary solution, under the condition\n∂xh2= 0,h2/\\e}atio\\slash= 0and in absence of anisotropic energy.\nAcknowledgements: The author is warmly grateful to Professor M. Hofmanov´ a for her many advices and\nconstant help to run this project and for pointing out the pro cedure for the long time behaviour in [17]. The\nauthor is also thankful to Professor B. Goldys many interest ing discussions in the last phase of this work and\nfor pointing out Remark 6.10, the sensitivity of the equatio n to the choice of the noise and for pointing out\nseveral references [44, 45]. The author thankfully acknowl edges the financial support by the German Science\nFoundation DFG via the Collaborative Research Center SFB12 83, Project B7. The author is also thankful to\nthe organizers of the GDR-TRAG Young researchers meeting in Paris in December 2021. The author wishes to\nthank the anonymous reviewer for the useful remarks.\n2 Notations and setting\n2.1 Frequently used notations.\nFora,b∈R3, we denote by a·bthe inner product in R3, and by| · |the norm inherited from it (we will not\ndistinguish between the different dimensions, it will be cl ear from the context). We recall the definition of cross\nproducta×b:= (a2b3−a3b2,a3b1−a1b3,a1b2−a2b1), fora≡(a1,a2,a3),b≡(b1,b2,b3)∈R3. We denote\nbyS2:={a∈R3:|a|R3= 1}the unit sphere in R3. The space of bounded linear operators from a Banach\nspaceEto itself is denoted by L(E)and by13×3the identity operator in L(R3). The space La(R3)is the\nsubspace of L(R3)of antisymmetric matrices, i.e. Γ∈L(R3)such that Γv·w=−v·(Γw)∀v,w∈R3.\nGivena,b∈R3, we denote by a⊗bthe tensor product in R3, i.e.a⊗b= (aibj)i,j=1,2,3.\n6Paths and controls LetJ⊂[0,T]be a subinterval and J2:=J×J. For convenience of notations, we\nwill write ‘ ∀s≤t∈J’ instead of ‘ ∀(s,t)∈J2such that s≤t.’ For a one-index map hdefined on J,\nwe use the notation δhs,t:=ht−hsfors≤t∈J. For a two-index map Hdefined on J2, we define\nδHs,u,t:=Hs,t−Hs,u−Hu,tfors≤u≤t∈J. We call increment any two-index map which is given by\nδhs,tfor someh=ht. Observe that increments are exactly those two-index eleme ntsHfor which δHs,u,t≡0.\nWe say that a continuous map ω:{(s,t)∈J:s≤t} →[0,+∞)is acontrol on Jifωis continuous,\nω(t,t) = 0 for anyt∈Jand if it is super-additive , namely for all s≤u≤t\nω(s,u)+ω(u,t)≤ω(s,t).\nGiven a control ωonJ:= [s,t], we also denote ω(J) :=ω(s,t).\nLet(E,/ba∇dbl·/ba∇dblE)be a Banach space and p >0, we denote by Vp\n2(J;E)the set of two-index maps H:{(s,t)∈\nJ2:s≤t} →Ethat are continuous in both the components such that Ht,t= 0for allt∈Jand there exists a\ncontrolωonJsuch that\n/ba∇dblHs,t/ba∇dblE≤ω(s,t)1/p(2.1)\nfor alls≤t∈J. The space Vp\n2(J;E)is equivalently defined as the space of two-index maps of finit e\np-variation, namely Hbelongs to Vp\n2(J;E)if and only if\n/ba∇dblH/ba∇dblVp\n2(J;E):= sup\nP/parenleftig/summationdisplay\n[s,t]∈π/ba∇dblHs,t/ba∇dblp\nE/parenrightig1\np<+∞,\nwhere the supremum is taken over the set of partitions P={[s0,s1],...[sn−1,sn]}ofJ. Moreover, the semi-\nnorm/ba∇dbl·/ba∇dblVp\n2(J;E)coincides with the infimum of ω(J)pover the set of controls ωsuch that the condition (2.1)\nholds (see [31] or [22, Paragraph 8.1.1]). Analogously we ca n defineVp(J;E)as the space of all continuous\npathsg:J→Esuch that δh∈ Vp\n2(J;E), equipped with the norm /ba∇dblh/ba∇dblVp(J;E)= supt∈J/ba∇dblht/ba∇dblE+/ba∇dblδh/ba∇dblVp\n2(J;E).\nWe will sometimes need to work with local versions of the prev ious spaces: we define Vp\n2,loc(J;E)as the\nspace of maps H:{(s,t)∈J2:s≤t} →Esuch that there exists a finite covering (Jk)k∈KofJ,K⊂N, so\nthatH∈ Vp\n2(Jk;E)for allk∈K. We define the linear space\nV1−\n2,loc(J;E) :=/uniondisplay\n0<p<1Vp\n2,loc(J;E).\nFixT >0. We denote by Vp(E) =Vp([0,T];E),. Analogously, we use the short hand Vp\n2(E) =Vp\n2([0,T];E).\nWe denote the space of continuous functions defined on [0,T]with values in EbyC([0,T];E). The space\nCk\nb([0,+∞);R)indicates the space of bounded and continuously differenti able functions with bounded deriva-\ntives, defined on [0,+∞)with real values, for k∈N. The notation ˙hdenotes the time derivative of h∈\nCk\nb([0,+∞);R).\nSobolev spaces LetD⊂Rbe an open bounded interval of R. Denote by Nthe space of natural numbers\nandN0:=N∪{0}. Forn∈N, we consider the usual Lebesgue spaces Lp:=Lp(D;Rn), forp∈[1,+∞]\nendowed with the norm /ba∇dbl·/ba∇dblLpand the classical Sobolev spaces Wk,q:=Wk,q(D;Rn)for integer q∈[1,+∞]\nandk∈Nendowed with the norm /ba∇dbl · /ba∇dblWk,q. We also denote by Hk:=Wk,2(D;Rn). We need to consider\nalso functions taking values in S2⊂R3: we therefore introduce the notation\nHk(S2) :=Hk(D;R3)∩{g:D→R3s.t.|g(x)|= 1 a.e.x∈D}, (2.2)\nfork∈N0. Finally, we will denote by Lp(Wk,q) :=Lp([0,T];Wk,q(D;Rn)). We indicate with Ck\n0(D)the\nspace of real valued functions with compact support on D,k-times continuously differentiable and such that\nevery derivative is compactly supported on D. Let(Ω,F,P)be a probability space. We denote by Lp(Ω;E)\nthe usual Lebesgue space with respect to the probability mea sureP.\n2.2 Construction of the rough driver.\nWe briefly introduce the noise, as constructed in [29]. We are interested in interpreting the Stratonovich integral\n/integraldisplayt\nshur×◦dWr, (2.3)\n7by means of rough path theory. We do not employ the classical d efinition of rough path, but we use the notion\nof rough driver (see [4], [19, 31, 32, 35, 33]), that we introd uce in Definition 2.1.\nDefinition 2.1 (Rough driver) .Letn∈Nandp∈[2,3). A pair\nB:= (B,B)∈ Vp\n2/parenleftbig\n[0,T];L2(D;L(Rn))/parenrightbig\n×Vp/2\n2/parenleftbig\n[0,T];L2(D;L(Rn))/parenrightbig\n(2.4)\nis said to be a n-dimensional rough driver provided δBs,u,t= 0 and the relation is fulfilled as the sense of\ncomposition of linear maps in Rn:\nδBs,u,t(x) =Bu,t(x)Bs,u(x),for alls < u < t ∈[0,T],Lebesgue-a.e. x∈D. (2.5)\nWe refer to (2.5) as Chen’s relation. We introduce other prop erties of a rough driver.\n(a)Bisgeometric if it can be obtained as the limit with respect to the (p,p/2)-variation topology, of a\nsequence of smooth rough drivers Bǫ= (Bǫ,Bǫ),whereǫ >0, explicitly defined for s≤t∈[0,T]as\nBǫ\ns,t:= Γǫ\nt−Γǫ\ns,Bǫ\ns,t:=/integraldisplayt\nsdΓǫ\nr(Γǫ\nr−Γǫ\ns), (2.6)\nfor some smooth path Γǫ: [0,T]→L2(D;L(Rn)).\n(b)Bisanti-symmetric if it is geometric and such that the approximating sequence (Γǫ\nt(x))in (2.6) can be\ntaken with values in the space La(Rn)⊂L(Rn)(this implies in particular that the first component\nBs,t(x)takes values in La(Rn)).\nGiven a rough driver B= (B,B), we callBfirst iterated integral andBsecond iterated integral .\nRemark 2.2. An equivalent definition of geometric rough driver is that Sy m(Bs,t) = (Bs,t+BT\ns,t)/2 =\nBs,tBs,t/2. This corresponds to the usual rough paths framework ([21]) in the following way: the main differ-\nence is that the first iterated integral of a rough path is norm ally a vector, thus the matrix product Bs,tBs,tis\nreplaced with a cross product (see e.g. [21]). The geometric ity in (a) property is thus directly inherited from\nthe rough path setting and adapted to the fact that the first it erated integral of the rough driver is a matrix.\nWe construct a rough driver so that the first iterated integra l in (2.3) takes value on the sphere S2: this is\ndone by using the structure of the cross product. Let (Ω,F,(Ft)t,P)be a filtered probability space. Consider\na Brownian motion w≡(w1,w2,w3) : Ω×[0,T]→R3on(Ω,F,(Ft)t,P)and a map F:R3→La(R3)\ndefined by F(ξ) :=·×ξ. LetΩStr⊂Ωbe the set of full measure such that wi,,j\ns,t(ω) =/integraltextt\nswi\ns,r◦dwj\nr(ω)exists\nfor allω∈ΩStrand for all i,j= 1,2,3, i.e. the set of ω∈Ωsuch that the Stratonovich integral of wagainst\nitself exists.\nFor all fixed ω∈ΩStr, we define the first iterated integral of the rough driver W(ω) = (W(ω),W(ω))as\nWs,t(ω) :=F(δws,t(ω)) =\n0δw3\ns,t(ω)−δw2\ns,t(ω)\n−δw3\ns,t(ω) 0 δw1\ns,t(ω)\nδw2\ns,t(ω)−δw1\ns,t(ω) 0\n, (2.7)\nand the second iterated integral as\nWs,t(ω) =\n−w3,3\ns,t(ω)−w2,2\ns,t(ω) w1,2\ns,t(ω) w1,3\ns,t(ω)\nw2,1\ns,t(ω) −w3,3\ns,t(ω)−w1,1\ns,t(ω) w2,3\ns,t(ω)\nw3,1\ns,t(ω) w3,2\ns,t(ω) −w2,2\ns,t(ω)−w1,1\ns,t(ω)\n. (2.8)\nThe couple W≡(W,W)is a three dimensional random anti-symmetric geometric rou gh driver, i.e. for all\nω∈Ωstrthe couple W(ω)is a three dimensional anti-symmetric geometric rough driv er. A way to add spatial\ndependency to the previous example is to let, for some h∈Hk(D;R),\nWh≡/parenleftbig\nh(x)W,h(x)2W/parenrightbig\n. (2.9)\n8We denote by RDp\na(Hk), fork∈N, the space of 3-dimensional anti-symmetric rough drivers such that for\nh∈Hk(D;R)\nWh∈ Vp\n2/parenleftig\n0,T;Hk(D;La(R3))/parenrightig\n×Vp/2\n2/parenleftig\n0,T;Hk(D;L(R3))/parenrightig\n,\nwhose coordinates belong to the k-th order Sobolev space Hk.We introduce also the following controls for\ns≤tand0≤γ≤k\nωW,Hγ(s,t) :=/ba∇dblW/ba∇dblp\nVp\n2(s,t;Hγ(D;La(R3))), ωW,Hγ(s,t) :=/ba∇dblW/ba∇dblp/2\nVp/2\n2(s,t;Hγ(D;L(R3))),\nωW,Hγ(s,t) :=/ba∇dblW/ba∇dblp\nVp\n2(s,t;Hγ(D;La(R3)))+/ba∇dblW/ba∇dblp/2\nVp/2\n2(s,t;Hγ(D;L(R3))).\nWe could consider rough drivers that are more general than th e construction in (2.9), but we restrict our inves-\ntigation to rough drivers constructed as above by a three dim ensional Brownian motion: indeed we rely on the\nmartingale properties of the stochastic Itˆ o integral.\n2.3 Other constructions of the stochastic integral.\nWe mention an other possible construction of the noise in (1. 1). Leth∈Hk(D;R3)and the Stratonovich lift\nto a rough path of a real valued Brownian motion W= (W,W). We can interpret the stochastic integral in the\nclassical rough path setting as\n/integraldisplayt\nsur×h◦dWr=Ws,tus×h+Ws,tus×h+(u×h)♮\ns,t.\nThe same integral can be interpreted in the Stratonovich sen se. One can consider also linear combination\nof different Brownian motions, by using the formalism above . For what concerns the proof of existence of\ninvariant measures, the choice of the noise has no consequen ces. On the other side, the choice of the noise\naffects the number of invariant measures. We return to this p oint in Section 6.\n3 The stochastic Landau-Lifschitz-Gilbert equation in one dimension\nWe introduce the notion of solution of stochastic LLG that we employ in what follows.\nDefinition 3.1. Consider a Brownian motion wwith values in R3defined on a probability space (Ω,F,P). We\nconsider the filtered probability space (Ω,F,(Ft)t,P)endowed with the complete natural filtration induced by\nw. Letω/ma√sto→Wh(ω)≡(hW(ω),h2W(ω))be a pathwise random rough driver belonging to RDp\na(Hk)for\nsomep∈[2,3), constructed from was in (2.9). Let D⊂Ran open bounded interval. We say that a stochastic\nprocessu: Ω×[0,T]→L2(D;R3)is apathwise solution of (1.3) if it fulfils\n(i)ut(ω,x)∈S2for a.e.(ω,t,x)∈Ω×[0,T]×D;\n(ii)u∈L∞(0,T;H1)∩L2(0,T;H2)P-a.s. ;\n(iii) there exists q <1and a random variable u♮∈ L0(Ω;Vq\n2,loc(0,T;L2))such that\nδus,t+/integraldisplayt\ns/bracketleftbig\nλ2ur×(ur×∂2\nxur)−λ1ur×∂2\nxur/bracketrightbig\ndr\n+/integraldisplayt\ns/bracketleftbig\nλ2ur×(ur×g′(ur))−λ1ur×g′(ur)/bracketrightbig\ndr=hWs,tus+h2Ws,tus+u♮\ns,t,(3.1)\nas an equality in L2(D;R3), for every s≤t∈[0,T]andP-a.s.\n(iv) (initial condition) u0=u0, whereu0∈H1(S2)P-a.s.\n(v) (boundary conditions) ∂xut(x) = 0 for all(t,x)∈∂D×[0,T]P-a.s.\n9Observe that, if u∈L∞(0,T;H1)∩L2(0,T;H2), then the equality −ut×(ut×∂2\nxut) =∂2\nxut+ut|∂xut|2\ninH−1forut(x)∈S2(see [12, Lemma 2.4]) holds in L2for a.e.t∈[0,T]: indeed, since u∈L∞(0,T;L∞)\nand∂2\nxu∈L2(0,T;L2), thenu×(u×∂2\nxu)∈L2(0,T;L2). Moreover, u|∂xu|2∈L2(0,T;L2)from the\nspherical bound and from the interpolation inequality in Le mma 7.6.\nIn [29], it is proved that there exists a unique solution to (1 .3) in the sense of Definition 3.1, provided g≡0.\nAgain under the assumption g≡0, the solution is also continuous with respect to the noise an d to the initial\ndata inH1(S2)with respect to the norm L∞(0,T;H1)∩L2(0,T;H2)∩ Vp(W2,+∞)(see Theorem 5.5 in\n[29]). The continuity with respect to the initial data sugge sts that the semigroup associated to the solution uhas\nthe Feller property and that we can try to employ the Krylov-B ogoliubov Theorem to show the existence of an\ninvariant measure. We remark that the solution in Definition 3.1 coincides with the solution of (1.3) where the\nnoise is interpreted with respect to the classical Stratono vich calculus (as studied in [14, 12]): we can therefore\npass from one formulation to the other. In [29], the solution to (1.3) is studied on the one-dimensional torus T\nand in absence of anisotropic energy.\nWe are interested in dealing with non null anisotropic energ y, i.e.g/\\e}atio\\slash= 0. The addition of anisotropic energy\ndoes not imply any significant change in all the results in [29 ]: indeed the solution u∈L∞(D;R3)and the\nanisotropy is trivially in L∞(L2).\nThe change of domain from the torus to a domain with boundary r equires a slightly modified product\nformula, as we discuss in Section 4.1. The proofs present min or changes, for instance when the divergence the-\norem is involved, but also in this case the null Neumann bound ary conditions makes the change not noticeable.\nWe refer to the proof of Lemma 4.4, where we apply these change s to the continuity with respect to the initial\ndatum in H1(S2). The changes regarding the proof of existence and the proof o f uniqueness are similar as in\nthe the proof of Lemma 4.4.\n4 Existence of an invariant measure\nWe introduce some definitions and basic results regarding in variant measures associated to Markov processes.\nLetEbe a Polish space endowed with the Borel σ-algebraBE. LetBb(E)be the space of bounded Borel\nmeasurable functions defined on Ewith values in Rand letCb(E)be the space of continuous bounded functions\nfromEwith values in R. Let(Pt)tbe a Markov semigroup defined on Bb(E). If(Pt)tmapsCb(E)intoCb(E),\nwe say that (Pt)thas the Feller property . Denote by P(E)be the space of probability measures on (E,BE).\nThe semigroup (Pt)tinduces a dual semigroup (P∗\nt)tonP(E), defined for all ν∈ P(E)andφ∈Cb(E)by\n/integraldisplay\nEφd(P∗\ntν) =/integraldisplay\nEPtφdν.\nWe say that a measure µ∈ P(E)isinvariant for the semigroup (Pt)tifP∗\ntµ=µfor allt >0. The Krylov-\nBogoliubov theorem furnishes a way to prove constructively the existence of an invariant measure (see e.g.\n[8]).\nTheorem 4.1. LetEbe a Polish space and let (Pt)tbe a Markov semigroup with the Feller property on Cb(E).\nConsider a random variable u0with values in Ewith lawµand denote by µu0\nt:=P∗\ntµ.\nAssume that there exists a divergent monotone increasing se quence of times (tn)nso that the sequence of\nprobability measures (µtn)n⊂ P(E)defined for all A∈ BEby\nµtn(A) :=1\ntn/integraldisplaytn\n0µu0\ns(A)ds\nis tight. Then there exists at least one invariant measure fo r(Pt)t.\nLetu0be a random variable with values in H1(S2)and independent of the Brownian motion W. Consider\nu=π(u0,W)to be the unique solution to (1.3), then the stochastic proce ss(uu0\nt)tis Markov process with\nrespect to the filtration generated by Wandu0. In our context, we are interested in studying the existence\nof an invariant measure on H1(S2)for the Markov semigroup of linear operators (Pt)tdefined for all φ∈\nBb(H1(S2))by\nPtφ(x) :=E[φ(ux\nt)], (4.1)\n10whereux\ntis the solution to (1.3) at time twith initial condition x∈H1(S2). As stated in the Krylov-Bogoliubov\ntheorem, we need to show that the semigroup (Pt)thas the Feller property and the tightness property of (µtn)n\ninH1(S2), where we can rewrite µtnas\nµtn(·) =1\ntn/integraldisplaytn\n0P(ux\ns∈ ·)ds. (4.2)\nIn Section 4.1, we prove that (Pt)thas the Feller property by means of a rough path approach. The inequality\nwhich leads to the Feller property of the semigroup is alread y contained in [29] for the solution to (1.3) defined\non the one dimensional torus and in absence of anisotropic en ergy. We recall the proof for the reader’s conve-\nnience by adding the anisotropic energy (which does not cons titute a mathematical problem) and the boundary\ncondition (which requires a slightly modified product formu la, in Proposition 7.2). In Section 4.2, we prove\nthat the sequence (µtn)nis tight in H1(S2): this fact follows from some geometric properties of the equ ation,\na proper a posteriori estimate of solution to the equation an d the Markov property.\nRemark 4.2. Note that the semigroup (Pt)texists. Indeed the unique solution uon[0,T]is adapted to (Ft)t(it\nis a continuous function of the rough path, as a consequence o f the Wong-Zakai result in [29]). The solution is\nalso continuous in time with values in L2. Sinceφ∈Bb(H1(S2))isF-measurable, also the composition φ(u)\nisF-measurable. We also know that φis bounded, therefore Ptφ(x)is well defined for every time t∈[0,T].\nFor every initial condition x∈H1(S2)there exists a unique solution u=π(u0,W)on every fixed time\ninterval[0,T]. As a consequence of uniqueness, for every s≤t∈[0,T], it holds that for all φ∈Cb(H1(S2))\nE[ux\ns+t|Fs] =E[φ(ux\ns+t)] =Ptφ(ux\ns),P−a.s.,\nwhich shows that (Pt)tis a Markov semigroup on H1(S2)with respect to (Ft)t. We do not specify the time\ninterval, which always coincides with the interval of exist ence of the solution to (1.3).\n4.1 Feller property in the H1(S2)-norm via rough paths.\nIn order achieve the Feller property of the semigroup (Pt)t, we need a small extension of the continuity with\nrespect to the initial datum contained in [29, Theorem 5.5, e quation (5.13)], which exploits the rough path\nformulation of the equation: for fixed T >0we know that the pathwise solution depends continuously on\nthe initial datum, namely given two initial conditions u0,v0∈H1(S2)and the corresponding solutions u=\nπ(u0,W)andv=π(v0,W)to (1.3), the following inequality holds\nsup\nt∈[0,T]/ba∇dblut−vt/ba∇dblH1≤Cexp/parenleftig\n[/ba∇dblu0/ba∇dbl2\nH1+/ba∇dblv0/ba∇dbl2\nH1]t+ω1/p\nW/parenrightig\n/ba∇dblu0−v0/ba∇dblH1, (4.3)\nP−a.s. and where the constant C >0is independent on time and on the initial condition. We note t hat the\nconstant depends exponentially on time and exponentially o n the initial conditions. More specifically, if we\nconsider two initial conditions u0,v0∈BR(0)⊂H1(S2)the ball in H1(S2)of radius R >0centred in 0,\nthen the constant in (4.3) depends on the initial condition o nly as a positive power of R.\nThe pathwise convergence in (4.3) implies the Feller proper ty, as stated in Theorem 4.3.\nTheorem 4.3. The semigroup (Pt)thas the Feller property, i.e. Pt:Cb(H1(S2))→Cb(H1(S2))for allt >0.\nProof. Assume indeed that (xn)nconverges to xinH1(S2)(note that for each converging sequence (xn)n,\nthere exists a radius R >0so that(xn)n⊂BR(0)). For every φ∈Cb(H1(S2)), it follows from the dominated\nconvergence theorem and the continuity in (4.3) that\nlim\nn→+∞(Ptφ)(xn) = lim\nn→+∞E/bracketleftbig\nφ(uxn\nt)/bracketrightbig\n=E/bracketleftbigg\nlim\nn→+∞φ(uxn\nt)/bracketrightbigg\n=E[φ(ux\nt)] = (Ptφ)(x), (4.4)\nwhich means that Ptφ:H1(S2)→H1(S2)is continuous for every fixed t >0andφ∈Cb(H1(S2)). Thus\nPtφ∈Cb(H1(S2))(where the boundedness follows from the boundedness of φ).\nFor the reader’s convenience, we briefly proof of (4.3) in the following Lemma 4.4: we mainly highlight\nthe parts of the proof where the anisotropic term and the boun dary conditions play a role. For a more detailed\nproof see [29, Theorem 5.5]\n11Lemma 4.4. The solution map u= Φ(u0)to(1.3) is locally Lipschitz continuous as a function of u0∈H1(S2)\nfor every fixed T >0. Namely, fix R >0and letu0,v0∈BR(0) :={x∈H1(S2) :/ba∇dblx/ba∇dblH1(S2)< R}. Denote\nthe respective solutions to (1.3) byu= Φ(u0)andv= Φ(v0), then\nsup\nt∈[0,T]/ba∇dblut−vt/ba∇dbl2\nH1/lessorsimilarg,|D|exp/parenleftig\nRkTk+Rkω1/p\nW(0,T)/parenrightig\n/ba∇dblu0−v0/ba∇dbl2\nH1,\nfor allu0,v0∈BR(0)and for some power k≥4.\nProof. In this proof, the parameters λ1,λ2do not play any role, therefore we simplify the computations by\nsettingλ1=λ2= 1. As a consequence of the uniqueness proof (which we modify by employing Proposition\n7.2), we know that udepends continuously on the initial data in L2. More specifically, given two initial\nconditions u0,v0∈H1(S2), it follows from [29, Theorem 4.1, equation (4.8)] that\nsup\nt∈[0,T]/ba∇dblut−vt/ba∇dbl2\nL2/lessorsimilarg,|D|exp(Tω1/p\nW(0,T)+T/ba∇dblu0/ba∇dbl4\nH1(S2)+/ba∇dblv0/ba∇dbl4\nH1(S2))/ba∇dblu0−v0/ba∇dbl2\nL2, (4.5)\nwhich is actually local Lipschitz continuity of the solutio n with respect to the initial datum in L2(remark that\nthe initial datum needs to belong to H1(S2), in order to achieve this estimate). We notice that the Lipsc hitz\nconstant depends on the time exponentially: this implies in particular that we can not use this kind of estimate\nto study the long time behaviour of the equation.\nSince the initial datum lies in H1(S2), we need to prove also continuity with respect to the initial datum of\nthe gradient of the solution. Set z:=u−v. In order to achieve the conclusion, we need to study the syst em of\nequations\nδ(z⊗2)s,t= 2Ds,t(0,0)+(I+I+˜I)(0,0)s,t+(z⊗2)♮\ns,t (4.6)\nδ(z⊗∂xz)s,t=Ds,t(1,0)+Ds,t(0,1)+(I+I+˜I)(0,1)s,t+(z⊗∂xz)♮\ns,t (4.7)\nδ(∂xz⊗z)s,t=Ds,t(1,0)+Ds,t(0,1)+(I+I+˜I)(1,0)s,t+(∂xz⊗z)♮\ns,t (4.8)\nδ(∂xz⊗∂xz)s,t= 2Ds,t(1,1) +(I+I+˜I)(1,1)s,t+(∂xz⊗∂xz)♮\ns,t, (4.9)\nwhere we used the product formula in Proposition 7.2 and the f ollowing notations\nD(l,m) :=/integraldisplayt\ns∂l\nx[(u×∂2\nxu−∂2\nxu+u|∂xu|2)−(v×∂2\nxv−∂2\nxv+v|∂xv|2)]⊗∂m\nxzdr\n+/integraldisplayt\ns∂l\nx[u×g′(u)−v×g′(v)]⊗∂m\nxzdr\n−/integraldisplayt\ns∂l\nx[u×(u×g′(u))−v×(v×g′(v))]⊗∂m\nxzdr,\nIs,t(l,m) :=∂l\nx(Ws,tzs)⊗∂m\nxzs+∂l\nxzs⊗∂m\nx(Ws,tzs),\nIs,t(l,m) :=∂l\nx(Ws,tzs)⊗∂m\nxzs+∂l\nxzs⊗∂m\nx(Ws,tzs),\n˜Is,t(l,m) :=∂l\nx(Ws,tzs)⊗∂m\nx(Ws,tzs),\nform,l∈ {0,1}. We notice that if we test the system above by 1= (1l=m), it is possible to simplify the\nequations.\nWe need to consider the above system of equation since, while estimating the remainder term (∂xz⊗∂xz)♮,\nthe remainders (∂xz⊗z)♮,(z⊗∂xz)♮,(z⊗z)♮appear. As a difference from the classical Itˆ o-Stratonovi ch\ncalculus, again in the estimation of the remainder, the mixe d terms(∂xzi∂xzj)♮appear, for i/\\e}atio\\slash=j: this more\ncomplicated setting is the price to pay to get the pathwise co nvergence.\nWe recall the structure of the noise (W,W). In our context the first iterated integral Wis an anti-symmetric\nmatrix, namely WT=−W. As a consequence Ws,ta·a= 0for alla∈R3. The second iterated integral can\nbe decomposed as\nWs,t=1\n2Ws,tWs,t+Ls,t,\n12whereLis an anti-symmetric matrix. Thus for all a∈R3it follows that\nWs,ta·a+a·Ws,ta= 0,\nWs,ta·a+a·Ws,ta+Ws,ta·Ws,ta=−Ws,ta·Ws,ta+Ws,ta·Ws,ta= 0.\nThese symmetries lead to the conclusion that, if we test the s ystem by 1, we can reduce the noises of the\nequations as follows\n/a\\}b∇acketle{t(I+I+˜I)s,t(0,0),1/a\\}b∇acket∇i}ht= 0,/a\\}b∇acketle{t(I+I+˜I)s,t(1,0),1/a\\}b∇acket∇i}ht= 0,/a\\}b∇acketle{t(I+I+˜I)s,t(0,1),1/a\\}b∇acket∇i}ht= 0,\n/a\\}b∇acketle{tIs,t(1,1),1/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t∂xWs,tzs⊗∂xzs+∂xzs⊗∂xWs,tzs,1/a\\}b∇acket∇i}ht,\n/a\\}b∇acketle{tIs,t(1,1)+˜Is,t(1,1),1/a\\}b∇acket∇i}ht=/a\\}b∇acketle{t∂xWs,tzs⊗∂xzs+∂xzs⊗∂xWs,tzs+∂xWs,tzs⊗∂xWs,tzs,1/a\\}b∇acket∇i}ht\n+/a\\}b∇acketle{tWs,t∂xzs⊗∂xWs,tzs+∂xWs,tzs⊗Ws,t∂xzs,1/a\\}b∇acket∇i}ht.\nIn particular, one has that in (4.6) the equation is determin istic and also the remainder term cancels (one already\nobserves this fact in the uniqueness proof). In (4.7) and (4. 8), the noise is also vanishing as well as the drifts\n(from integrations by parts): thus also the mixed equations forl=mare deterministic (which is consistent\nwith the fact that the solution lies on the sphere and thus its derivative lies in the tangent plane). Finally, the\nnoises in (4.9) simplify and the equation tested by 1inL2, has the form\nδ/a\\}b∇acketle{t∂xz⊗∂xz,1/a\\}b∇acket∇i}hts,t=/a\\}b∇acketle{t(D+I+I+˜I)s,t(1,1),1/a\\}b∇acket∇i}ht+/a\\}b∇acketle{t(∂xz)♮,2\ns,t,1/a\\}b∇acket∇i}ht. (4.10)\nBy estimating equation (4.10), we achieve the conclusion th at the gradient of the equation is continuous with\nrespect to the initial data in H1(S2): the main parts to bound are the drift /a\\}b∇acketle{tD,1/a\\}b∇acket∇i}ht, the noise terms /a\\}b∇acketle{t(I+I+\n˜I)s,t(1,1),1/a\\}b∇acket∇i}htand the remainder /a\\}b∇acketle{t(∂xz)♮,2\ns,t,1/a\\}b∇acket∇i}ht. We briefly recall the estimate of the drift: we remember from the\nproof of existence of the solution that u∈L∞(H1)∩L2(H2)∩C(L2)pathwise. We exploit this boundedness,\nwhich holds for every fixed time interval [0,T], forT >0(note that the bounds on the solution depend\nexponentially on time, on the initial condition u0∈H1(S2)and on the dimension of the domain). We write the\ndrift tested by 1,\n/a\\}b∇acketle{tDs,t(1,1),1/a\\}b∇acket∇i}ht=/integraldisplayt\ns/integraldisplay\nD∂x[∂2\nxzr+zr|∂xur|2+vr(|∂xur|2−|∂xvr|2)+zr×∂2\nxur+vr×∂2\nxzr]·∂xzrdxdr\n+/integraldisplayt\ns/integraldisplay\nD∂x[zr×g′(ur)+vr×(g′(ur−vr))]·∂xzrdxdr\n+/integraldisplayt\ns/integraldisplay\nD∂x[ur×(ur×g′(ur))−vr×(vr×g′(vr))]·∂xzrdxdr,\nwhich from integration by parts and by using the null Neumann boundary conditions, lead to\n/a\\}b∇acketle{tDs,t(1,1),1/a\\}b∇acket∇i}ht=−/integraldisplayt\ns/ba∇dbl∂2\nxzr/ba∇dbl2\nL2dr+/integraldisplayt\ns/integraldisplay\nD∂x[zr×g′(ur)+vr×g′(ur−vr)]·∂xzrdxdr\n+/integraldisplayt\ns/integraldisplay\nD∂x[ur×(ur×g′(ur))−vr×(vr×g′(vr))]·∂xzrdxdr\n−/integraldisplayt\ns/integraldisplay\nD[zr|∂xur|2·∂2\nxzr+vr(|∂xur|2−|∂xvr|2)·∂2\nxzr+zr×∂2\nxur·∂2\nxzr]dxdr.\nThe estimate of the different terms of the drift follows from the one dimensional Agmon’s inequality /ba∇dblz/ba∇dblL∞/lessorsimilar|D|\n/ba∇dblz/ba∇dbl1/2\nL2/ba∇dblz/ba∇dbl1/2\nH1,\n/integraldisplayt\ns/integraldisplay\nD[z|∂xu|2·∂2\nxz+v(|∂xu|2−|∂xv|2)·∂2\nxz+z×∂2\nxu·∂2\nxz]dxdr\n≤3\n4/integraldisplayt\ns/ba∇dbl∂2\nxz/ba∇dbl2\nL2dr+/ba∇dblz/ba∇dblL∞([s,t],L2)/ba∇dblz/ba∇dblL∞([s,t],H1)/integraldisplayt\ns/ba∇dbl∂xu/ba∇dbl4\nL4dr\n+/ba∇dbl∂xz/ba∇dbl2\nL∞([s,t],L2)/integraldisplayt\ns[/ba∇dbl∂xu/ba∇dblL2/ba∇dbl∂xu/ba∇dblH1+/ba∇dbl∂xv/ba∇dblL2/ba∇dbl∂xv/ba∇dblH1]dr\n+/ba∇dblz/ba∇dblL∞([s,t];L2)/ba∇dblz/ba∇dblL∞([s,t];H1)/integraldisplayt\ns/ba∇dbl∂2\nxu/ba∇dbl2\nL2dr.\n13Hence we conclude from Young’s inequality and the uniquenes s inequality (4.5) that\n/integraldisplayt\ns/integraldisplay\nD[z|∂xu|2·∂2\nxz+v(|∂xu|2−|∂xv|2)·∂2\nxz+z×∂2\nxu·∂2\nxz]dxdr\n≤3\n4/integraldisplayt\ns/ba∇dbl∂2\nxz/ba∇dbl2\nL2dr+CT/bracketleftbig\n/ba∇dblu0/ba∇dbl4\nH1+/ba∇dblv0/ba∇dbl4\nH1/bracketrightbig\n(/ba∇dblz/ba∇dbl2\nL∞(s,t;H1)+/ba∇dblz0/ba∇dbl2\nL2),\nwhereCT>0is a constant depending exponentially on time, on the dimens ion of the domain and on some\npowers of the H1(S2)norm of the initial conditions u0,v0∈H1(S2)such that\n/bracketleftig\n(1+T)(/ba∇dblu/ba∇dbl2\nL∞(H1)∩L2(H2)+/ba∇dblv/ba∇dbl2\nL∞(H1)∩L2(H2))/bracketrightig\n< CT/bracketleftbig\n/ba∇dblu0/ba∇dbl4\nH1+/ba∇dblv0/ba∇dbl4\nH1/bracketrightbig\n.\nHere we used that /ba∇dbl∂xu/ba∇dbl4\nL4/lessorsimilar/ba∇dbl∂xu/ba∇dblH1/ba∇dbl∂xu/ba∇dbl3\nL2. With analogous computations, the first anisotropic part of\nthe drift can be bounded by\n/integraldisplayt\ns/integraldisplay\nD∂x[ur×(ur×g′(ur))−vr×(vr×g′(vr))]·∂xzrdxdr/lessorsimilar/ba∇dblg′/ba∇dblL∞/integraldisplayt\ns/ba∇dbl∂xzr/ba∇dbl2\nL2dr,\nas well as the other anisotropic part. We now turn to the estim ation of the noise /a\\}b∇acketle{t(I+I+˜I)s,t(1,1),1/a\\}b∇acket∇i}ht, which\nfrom Young’s inequality and (4.5) leads to\n/a\\}b∇acketle{t(I+I+˜I)s,t(1,1),1/a\\}b∇acket∇i}ht ≤ω1/p\nW(s,t)\n2/bracketleftig\n/ba∇dblz/ba∇dbl2\nL∞([s,t],L2)+/ba∇dblz/ba∇dbl2\nL∞([s,t],H1)/bracketrightig\n/lessorsimilarω1/p\nW(s,t)\n2/bracketleftig\n/ba∇dblz0/ba∇dbl2\nL2+/ba∇dblz/ba∇dbl2\nL∞([s,t],H1)/bracketrightig\n.\nWe now pass to the estimate of the remainder /a\\}b∇acketle{t(∂xz)♮,2\ns,t,1/a\\}b∇acket∇i}ht: this term depends on the mixed equations (4.7),\n(4.8) and on the first level equation (4.6). In particular, fo r technical reasons, we estimate /ba∇dbl(∂xz)♮,2\ns,t/ba∇dblH−1:\nthis allows to conclude that /a\\}b∇acketle{t(∂xz)♮,2\ns,t,1/a\\}b∇acket∇i}ht ≤ /ba∇dbl(∂xz)♮,2\ns,t/ba∇dblH−1/ba∇dbl1/ba∇dblH1. We employ the sewing Lemma 7.1, after\napplying the operator δ(f)s,u,t:=fs,t−fs,u−fu,tto each component (∂xz)♮,2, we obtain\n/ba∇dbl(∂xz)♮,2\ns,t/ba∇dblH−1/lessorsimilarω1/p\nW(s,t)/bracketleftbig\n/ba∇dblz/ba∇dbl2\nL∞(s,t;H1)+[ωD(1,0);H−1+ωD(0,1);H−1+ωD(0,0);L2](s,t)/bracketrightbig\n,\nwhere the H−1norm inD(1,0)is bounded by D(0,1). By noticing that from the uniqueness proof we can\nboundωD(0,0);L2/lessorsimilarT/ba∇dblz0/ba∇dbl2\nL2and from similar computations as for D(1,1), we can conclude that\nωD(0,1);L2(s,t)/lessorsimilarǫ\n2/integraldisplayt\ns/ba∇dbl∂2\nxzr/ba∇dbl2\nL2dr+CT/bracketleftbig\n/ba∇dblu0/ba∇dbl4\nH1+/ba∇dblv0/ba∇dbl4\nH1/bracketrightbig\nω1/p\nW(s,t)[/ba∇dblz/ba∇dbl2\nL∞(s,t;H1)+/ba∇dblz0/ba∇dbl2\nL2].\nIn conclusion, by choosing ǫ >0suitably small, we can rewrite the equation as\nδ/bracketleftbig\n/ba∇dbl∂xz/ba∇dbl2\nL2/bracketrightbig\ns,t+1\n8/integraldisplayt\ns/ba∇dbl∂2\nxzr/ba∇dbl2\nL2dr/lessorsimilar/ba∇dbl∂xz0/ba∇dbl2\nL2\n+CT/bracketleftig\n1+ω1/p\nW/bracketrightig/bracketleftbig\n/ba∇dblu0/ba∇dbl4\nH1+/ba∇dblv0/ba∇dbl4\nH1/bracketrightbig/bracketleftig\n/ba∇dblz/ba∇dbl2\nL∞(s,t;H1)+/ba∇dblz0/ba∇dbl2\nL2/bracketrightig\n,(4.11)\nwhich from the rough Gronwall’s Lemma 7.3 leads to the conclu sion.\nRemark 4.5. On Feller’s property with classical Stratonovi ch calculus. For every fixed T >0it is possible,\nby means of the rough path theory, to prove the local Lipschit z continuity of the solution map with respect to\nthe initial condition in H1(S2)P-a.s. We show that it does not seem possible to conclude the sa me by means\nof the classical Itˆ o-Stratonovich calculus. We keep the no tations of the proof of Lemma 4.4. Assume that u,v\nare martingale solutions to (1.3) with initial conditions u0,v0∈ L4(Ω;H1(S2)). We apply the Itˆ o’s formula to\nthe equation for the difference ∂xz=∂x(u−v)\n/ba∇dbl∂xzt/ba∇dbl2\nL2−/ba∇dbl∂xz0/ba∇dbl2\nL2=/a\\}b∇acketle{tD0,t(1,1)[u]−D0,t(1,1)[v],1/a\\}b∇acket∇i}ht+/integraldisplay\nD/integraldisplayt\n0∂xhzr×dWrdx\n+2/integraldisplayt\n0/integraldisplay\nD∂xh2[z·z−z·∂xz]dxdr+2/integraldisplay\nD/integraldisplayt\n0∂xh∂xz·z×dWrdx.\n14If we first look at the drift estimate, we can conclude with the same estimates as in Lemma 4.4, more specifically\n/a\\}b∇acketle{tD0,t(1,1)[u]−D0,t(1,1)[v],1/a\\}b∇acket∇i}ht ≤ −1\n4/integraldisplayt\n0/ba∇dbl∂2\nxz/ba∇dbl2\nL2dr+CT/bracketleftbig\n/ba∇dblu0/ba∇dbl4\nH1+/ba∇dblv0/ba∇dbl4\nH1/bracketrightbig/parenleftig\n/ba∇dblz/ba∇dbl2\nL∞(H1)+/ba∇dblz0/ba∇dbl2\nL2/parenrightig\n.\nSince we look for the L∞(H1(S2))-norm ofz, we take the supremum in time into the equation and we estimat e\nthe noises by the Burkholder-Davis-Gundy inequality. To do so, we take the expectation of the energy and,\nfrom the estimate of the drifts, we conclude that we can not ap ply Gronwall’s Lemma to achieve the required\nbound. This is due to the elements/bracketleftbig\n/ba∇dblu0/ba∇dbl4\nH1+/ba∇dblv0/ba∇dbl4\nH1/bracketrightbig\n, which do not allow to pass to get the correct powers\nfor the Grownall’s Lemma.\n4.2 Tightness of (µT)T>0inH1(S2)via Stratonovich calculus.\nIn Lemma 4.6 we observe an orthogonality property for the sol utionuto (1.3), which leads to an equality useful\nin the following.\nLemma 4.6. Letu∈L∞(H1)such that |ut(x)|R3= 1 for a.e.(t,x)∈[0,T]×D, then\nut(x)·∂xut(x) = 0a.e.(t,x)∈[0,T]×D.\nIn particular, it follows that |ut(x)×∂xut(x)|R3=|∂xut(x)|R3for a.e.(t,x)∈[0,T]×D.\nProof. Recall that |ut(x)|= 1 for a.e.(t,x)∈[0,T]×D. Then from the product rule for Sobolev functions\n(sinceut∈H1) it follows that for a.e. t∈[0,T]\n∂x|ut|2= 2∂xut·ut.\nWe apply the product rule in the first equality and we observe t hat\n2/integraldisplay\nDφu·∂xudx=/integraldisplay\nDφ∂x|u|2dx=−/integraldisplay\nD∂xφ|u|2dx= 0,\nfor every φ∈C1\n0(D)(the equality on the right hand side is 0since|u|= 1 for a.e.(t,x)∈[0,T]×Dand\nφis compactly supported on D): thus from the fundamental lemma of calculus of variations it follows that\nut(x)·∂xut(x) = 0 for a.e.(t,x)���[0,T]×D. As a consequence, uis orthogonal to ∂xuand|ut(x)×\n∂xut(x)|=|ut(x)||∂xut(x)||sin(π/2)|=|∂xut(x)|for a.e.(t,x)∈[0,T]×D.\nWe derive first a linear bound in time for the gradient norm of t he solution, in absence of anisotropic energy.\nLemma 4.7. Letube the unique solution to (1.3) in the sense of Definition 3.1 with g≡0. For every t >0,\nthe bound holds\nsup\nr∈[0,t]E/bracketleftbig\n/ba∇dbl∂xur/ba∇dbl2\nL2/bracketrightbig\n+2λ2/integraldisplayt\n0E/bracketleftbig\n/ba∇dblur×∂2\nxur/ba∇dbl2\nL2/bracketrightbig\ndr≤E/bracketleftbig\n/ba∇dbl∂xu0/ba∇dbl2\nL2/bracketrightbig\n+t/ba∇dbl∂xh/ba∇dbl2\nL2. (4.12)\nProof. The rough integral coincides with the Stratonovich stochas tic integral apart from a set of null measure,\ntherefore we can switch to the classical Itˆ o-Stratonovich calculus. We write the equation for the derivative in\nStratonovich form\nδ∂xus,t=/integraldisplayt\ns∂x[λ1ur×∂2\nxur−λ2ur×(ur×∂2\nxur)]dr+/integraldisplayt\nsh∂xur×◦dWr+/integraldisplayt\ns∂xhur×◦dWr.\nFirst we convert the equation from Stratonovich integratio n to Itˆ o integration. We would like to determine\nc(x)≡(c1(x),c2(x),c3(x))defined for all i= 1,2,3by\nci(x) =3/summationdisplay\nk=13/summationdisplay\nj=1∂γi,j\n∂xj(x)γj,k,\n15wherex≡(x1,x2,x3)and the map γ(x)≡(γi,j(x))i,j=1,2,3is given by\nγ(x) =x×·=\n0−x3x2\nx30−x1\n−x2x10\n\nwherex≡(x1,x2,x3)∈R3. Thus we can rephrase the integrals as\n/integraldisplayt\nsγ(ur)◦dWr=1\n2/integraldisplayt\nsc(ur)dr+/integraldisplayt\nsγ(ur)dWr.\nBy using the above formula, we conclude that c(x) = [γ2,3−γ3,2,γ3,1−γ1,3,γ1,2−γ2,1](x), which leads to\nc(X) =−2X. Hence\n/integraldisplayt\nsh∂xur×◦dWr+/integraldisplayt\ns∂xhur×◦dWr=/integraldisplayt\nsh∂xur×dWr−/integraldisplayt\nsh2∂xurdr\n+/integraldisplayt\ns∂xhur×dWr−/integraldisplayt\ns∂xh2urdr.\nFrom Itˆ o’s formula applied to f(X)≡f(X1,X2,X3) =X·X, where we use that ∇Xf(X) = 2Xand\n∇2\nXf(X) = 2I∈R3⊗R3. This leads to\n/ba∇dbl∂xut/ba∇dbl2\nL2+2λ2/integraldisplayt\n0/ba∇dblu×∂2\nxu/ba∇dbl2\nL2dr=/ba∇dbl∂xu0/ba∇dbl2\nL2−2/integraldisplayt\n0/integraldisplay\nDh2∂xu·∂xudxdr−2/integraldisplayt\n0/integraldisplay\nD∂xh2u·∂xudxdr\n+2/integraldisplayt\n0/integraldisplay\nDh2∂xu·∂xudxdr+2/integraldisplayt\n0/integraldisplay\nD∂xh2u·udxdr\n+2/integraldisplay\nD/integraldisplayt\n0h∂xu·∂xu×dWrdx+2/integraldisplay\nD/integraldisplayt\n0∂xh∂xu·u×dWrdx.\nBy taking expectation into the above equation, the energy in equality takes the form\nE/bracketleftbig\n/ba∇dbl∂xut/ba∇dbl2\nL2/bracketrightbig\n+2λ2/integraldisplayt\n0E/bracketleftbig\n/ba∇dblu×∂2\nxu/ba∇dbl2\nL2/bracketrightbig\ndr=E/bracketleftbig\n/ba∇dbl∂xu0/ba∇dbl2\nL2/bracketrightbig\n+E/bracketleftbigg\n2/integraldisplayt\n0/integraldisplay\nD∂xh2u·udxdr/bracketrightbigg\n(4.13)\n+E/bracketleftbigg\n2/integraldisplay\nD/integraldisplayt\n0h∂xu·∂xu×dWrdx+2/integraldisplay\nD/integraldisplayt\n0∂xh∂xu·u×dWrdx/bracketrightbigg\n. (4.14)\nThe integral in (4.13) is deterministic, since |ut(x)|2\nR3= 1for a.e.(x,t)∈D×[0,T]andP-a.s. and therefore\nwe rewrite it as\nE/bracketleftbigg\n2/integraldisplayt\n0/integraldisplay\nD∂xh2ur·urdxdr/bracketrightbigg\n= 2t/ba∇dbl∂xh/ba∇dbl2\nL2. (4.15)\nThe stochastic integrals in (4.14) are an Itˆ o integrals and thus it have null expectation, which concludes the\nproof.\nWe introduce now Lemma 4.8, which we need in Lemma 4.9.\nLemma 4.8. LetCp>0be the Poincar ´e’s constant associated to Dandube the unique solution to (1.3) , then\nthe inequality holds\nC−1\np/ba∇dblu×∂xu/ba∇dblL2≤ /ba∇dblu×∂2\nxu/ba∇dblL2.\nProof. As a consequence of Lemma 4.6, it follows that /ba∇dbl∂xu/ba∇dbl2\nL2=/ba∇dbl∂xu×u/ba∇dbl2\nL2for a.e.t >0andP−a.s. We\nalso observe that, from a×a= 0for alla∈R3, it holds in L2that\n∂x(u×∂xu) =∂xu×∂xu+u×∂2\nxu=u×∂2\nxu.\nSince we are dealing with a one dimensional domain, from Morr ey’s inequality H1(S2)is continuously em-\nbedded into C(D;R): as a consequence both utand∂xutare continuous in the space variable for a.e. t≥0\n16andP-a.s. Thus we can infer that we can extend the derivative ∂xucontinuously to 0on the boundary. Because\nof the continuity of ∂xuon the boundary and of uon the boundary, also u×∂xuis null on the boundary: thus\nu×∂xu∈H1\n0. We are therefore in the conditions to apply Poincar´ e’s ine quality: there exists Cp>0such that\nC−1\np/ba∇dblu×∂xu/ba∇dblL2≤ /ba∇dbl∂x(u×∂xu)/ba∇dblL2=/ba∇dblu×∂2\nxu/ba∇dblL2.\nwhich leads to the conclusion.\nWe add the anisotropic energy and observe how inequality (4. 12) changes.\nLemma 4.9. For every t >0and for a positive constant C(λ1,λ2)>0, it holds\nsup\nr∈[0,t]E/bracketleftbig\n/ba∇dbl∂xur/ba∇dbl2\nL2/bracketrightbig\n+3λ2\n2/integraldisplayt\n0E/bracketleftbig\n/ba∇dblur×∂2\nxur/ba∇dbl2\nL2/bracketrightbig\ndr\n≤E/bracketleftbig\n/ba∇dbl∂xu0/ba∇dbl2\nL2/bracketrightbig\n+t/bracketleftigg\n/ba∇dbl∂xh/ba∇dbl2\nL2+[sup\ni,j|Ai,j|2+|b|2]C(λ1,λ2)/bracketrightigg\n.(4.16)\nProof. Recall the shape of the anisotropic energy: for A∈ L(R3)andb∈R3asg′(x) =Ax+b, for all\nx∈R3. We turn to the drift elements appearing in (1.3), which from the orthogonality in (4.17) leads to\n2λ1/integraldisplayT\n0/integraldisplay\nD∂x(ur×g′(ur))·∂xurdxdr=−2λ1/integraldisplayT\n0/integraldisplay\nD(ur×g′(ur))·∂2\nxurdxdr\n=−2λ1/integraldisplayT\n0/integraldisplay\nD(ur×g′(ur))·(ur×∂2\nxur)dxdr.\nIn the case λ1= 0, the other term does not appear. From the weighted Young’s in equality, we can bound this\nterm forǫ >0\n2λ1/integraldisplayT\n0/integraldisplay\nD∂x(ur×g′(ur))·∂xurdxdr≤2λ2\n1ǫ\n4/integraldisplayT\n0/ba∇dblur×∂2\nxur/ba∇dbl2\nL2dr+2·4\n3ǫ/integraldisplayT\n0/ba∇dblur×g′(ur)/ba∇dbl2\nL2dr,\nwhere we use ǫ=λ2/λ2\n1. With analogous considerations and ǫ= 1/λ2, we obtain the bound\n2λ2/integraldisplayT\n0/integraldisplay\nD∂x(u×(u×g′(u)))·∂xurdxdr≤2λ2\n4/integraldisplayT\n0/ba∇dblu×∂2\nxu/ba∇dbl2\nL2dr+2·4λ2\n3/integraldisplayT\n0/ba∇dblu×g′(u)/ba∇dbl2\nL2dr.\nRemark 4.10. On a different choice of the noise. Assume now that equation (1.3) is driven by the noise\n/integraldisplayt\n0ur×h◦dWr,\nwhereh∈H1(D;R3)andWis a real valued Brownian motion. The Itˆ o formula leads, in t his case, to a\ndifferent outcome. Nevertheless the final estimate coincid es, up to a non relevant constant, to (6.5). The only\ndifference is that one needs to pass through the Poincar´ e’s inequality to absorb to the left hand side\n/integraldisplayt\n0/integraldisplay\nD|h||∂xur||∂xh||ur|dxdr.\nThis fact holds true also in absence of anisotropic energy.\nLemma 4.11. It holds for a.e. t >0andP-a.s. that\n/ba∇dbl∂2\nxu/ba∇dbl2\nL2=/ba∇dbl∂xu/ba∇dbl4\nL4+/ba∇dblu×∂2\nxu/ba∇dbl2\nL2.\n17Proof. We recall from Lemma 2.4 in [12]. It holds in L2that\n−u×(u×∂2\nxu) =∂2\nxu+u|∂xu|2. (4.17)\nSince∂2\nxu∈L2, we can test (4.17)\n−/integraldisplay\nT1u×(u×∂2\nxu)·∂2\nxudx=/integraldisplay\nT1(∂2\nxu+u|∂xu|2)·∂2\nxudx,\nBy recalling that a×(a×b)·b=−|a×b|2for alla,b∈R3and by using that |u|2= 1, we conclude that\n/ba∇dbl∂2\nxu/ba∇dbl2\nL2=/ba∇dblu×∂2\nxu/ba∇dbl2\nL2+/ba∇dbl∂xu/ba∇dbl4\nL4,\nwhere we used the equality |∂xu|2=−∂2\nxu·ufor a.e.t >0,x∈DandP-a.s.\nLemma 4.12. There exists a constant C≡C(|D|,λ1,λ2,¯G,/ba∇dblu0/ba∇dblH1)>0independent on the time, such that\nfor allt >0\n/integraldisplayt\n0E/bracketleftig\n/ba∇dbl∂2\nxur/ba∇dbl1/2\nL2/bracketrightig\ndr≤CE/bracketleftbig\n/ba∇dbl∂xu0/ba∇dbl2\nL2/bracketrightbig\n+Ct. (4.18)\nProof. From Lemma 4.11, it holds for a.e. t >0fixed and P-a.s. that\n/ba∇dbl∂2\nxut/ba∇dbl2\nL2=/ba∇dbl∂xut/ba∇dbl4\nL4+/ba∇dblu×∂2\nxut/ba∇dbl2\nL2.\nBy employing the previous relation, by taking the power 1/4it follows that\n/ba∇dbl∂2\nxu/ba∇dbl1/2\nL2= (/ba∇dbl∂2\nxu/ba∇dbl2\nL2)1/4= (/ba∇dbl∂xu/ba∇dbl4\nL4+/ba∇dblu×∂2\nxu/ba∇dbl2\nL2)1/4≤(/ba∇dbl∂xu/ba∇dblL4+/ba∇dblu×∂2\nxu/ba∇dbl1/2\nL2). (4.19)\nThus we need to estimate the norms on the right hand side. Reca ll the one dimensional interpolation inequality\n/ba∇dblz/ba∇dblL4≤C(D)/ba∇dblz/ba∇dbl3/4\nL2/ba∇dblz/ba∇dbl1/4\nH1,\nwhereC(D)>0is a constant depending only on the dimension of the domain. W e integrate in time in (4.19)\nand we estimate first the L4-norm: from H¨ older’s inequality and the one dimensional in terpolation inequality\n/integraldisplayt\n0/ba∇dbl∂xur/ba∇dblL4dr≤C(D)/integraldisplayt\n0/ba∇dbl∂xur/ba∇dbl3/4\nL2/ba∇dbl∂2\nxur/ba∇dbl1/4\nL2dr.\nBy taking expectation and integrating in time in (4.19) and f rom the weighted Young’s inequality with ǫ >0\n/integraldisplayt\n0E[/ba∇dbl∂2\nxur/ba∇dbl1/2\nL2]dr≤ǫC(D)\n2/integraldisplayt\n0E[/ba∇dbl∂xur/ba∇dbl3/2\nL2]dr+C(D)\n2ǫ/integraldisplayt\n0E[/ba∇dbl∂2\nxur/ba∇dbl1/2\nL2]dr+/integraldisplayt\n0E[/ba∇dblur×∂2\nxur/ba∇dbl1/2\nL2]dr.\nBy choosing ǫ=C(D), we can absorb the Laplacian to the left hand side and we obtai n\n1\n2/integraldisplayt\n0E[/ba∇dbl∂2\nxur/ba∇dbl1/2\nL2]dr≤C(D)2\n2/integraldisplayt\n0E[/ba∇dbl∂xur/ba∇dbl3/2\nL2]dr+/integraldisplayt\n0E[/ba∇dblur×∂2\nxur/ba∇dbl1/2\nL2]dr.\nFrom Young’s inequality and Lemma 4.7 or Lemma 4.9 (which Lem ma to use, depends on the presence or not\nof anisotropic energy: this affects the constant Cin the statement), it follows that\n/integraldisplayt\n0E[/ba∇dbl∂xur/ba∇dbl3/2\nL2]dr/lessorsimilar/integraldisplayt\n0E[/ba∇dbl∂xur/ba∇dbl2\nL2]dr+Ct/lessorsimilarCE[/ba∇dbl∂xu0/ba∇dbl2\nL2]+Ct,\n/integraldisplayt\n0E[/ba∇dblur×∂2\nxur/ba∇dbl1/2\nL2]dr/lessorsimilar/integraldisplayt\n0E[/ba∇dblur×∂2\nxur/ba∇dbl2\nL2]dr+Ct/lessorsimilarCE[/ba∇dbl∂xu0/ba∇dbl2\nL2]+Ct,\nwhich concludes the proof.\n18Lemma 4.13. The sequence (µT)Tis tight in H1(S2).\nProof. The space H2(S2)is compactly embedded in H1(S2), therefore the ball BR:={x∈H2(S2) :\n/ba∇dblx/ba∇dblH2(S2)≤R}is compact in H1(S2), for some R >0. We use this compact set to prove the tightness\nof(µT)T>0, whereµTis defined in (4.2). By evaluating each µTinBC\nR:=H2(S2)\\BR,\nµT(BC\nR) =µT(/ba∇dblx/ba∇dblH2> R) =1\nT/integraldisplayT\n0P(/ba∇dblux\nt/ba∇dblH2(S2)> R)dt=1\nT/integraldisplayT\n0P(/ba∇dblux\nt/ba∇dbl1/2\nH2(S2)>√\nR)dt, (4.20)\nwhere we used that φ(w) =√wis monotone increasing. From Markov’s inequality applied t o (4.20) with the\npositive non-decreasing function φ(w)and from the estimate in Lemma 4.12,\nµT(BC\nR)≤1\nT√\nR/integraldisplayT\n0E/bracketleftig\n/ba∇dblux\nt/ba∇dbl1/2\nH2/bracketrightig\ndt≤C(1+T)√\nRT≤2C√\nR,\nwhere we used that T >1. By taking the limit for R→+∞, we conclude that µT(BC\nR)converges to 0and\nthus that(µT)T>0is tight in H1(S2).\n4.3 Existence of an invariant measure.\nIn Theorem 4.14 we conclude that there exists an invariant me asure for the semigroup (Pt)t.\nTheorem 4.14. There exists at least an invariant measure for the semigroup (Pt)tassociated to (1.3) on\nH1(S2).\nProof. As a consequence of Lemma 4.4, the semigroup (Pt)thas the Feller property in H1(S2). From\nLemma 4.13, (µT)Tis tight in H1(S2): from the Krylov-Bogoliubov Theorem 4.1, there exists at le ast an\ninvariant measure µonH1(S2)associated to the semigroup (Pt)t.\nRemark 4.15. We prove existence of a stationary solution as limit in the we ak-star topology of a subsequence\nin (4.2). This implies that the set of invariant measures con structed as weak-star limits of subsequences of the\nform (4.2) is not empty. A posteriori, we observe that every i nvariant measure can be built with the Krylov-\nBogoliubov procedure, since (4.2) reduces to the constant s equence for an invariant measure. Thus the set of\ninvariant measures for (Pt)tcoincides with the set con measures constructed by means of ( 4.2).\n5 Ergodic measures and stationary solutions\n5.1 Existence of stationary solutions\nWe look at stationary pathwise solutions to the equation. Gi ven an invariant measure µ∈ I, there exists\na random variable w0distributed like µon a probability space (˜Ω,˜F,˜P). From Skorohod’s representation\ntheorem, there exists a filtered probability space (Ω,F,(Ft)t,P)where the Brownian motion W, the initial\ncondition u0and the initial condition w0are adapted. With abuse of notation, we will not distinguish between\nthe different probability spaces. We discuss the regularit y of stationary solutions.\nTheorem 5.1. Letw0be an initial condition distributed like an invariant measu reµof the semigroup (Pt)ton\nH1(S2). Then there exists a pathwise stationary solution wsuch that w(ω)∈L∞(H1)∩L2(H2)∩C([0,T];L2)\nP−a.s.\nTheorem 5.1 states that for each invariant measure µthe equation admits a pathwise stationary solution. We\nstill do not have informations on the integrability with res pect to the probability space. If we use the classical\nStratonovich calculus, we need to require u0∈ L4(Ω;H1(S2)). We are not able to prove that there exists an\ninitial condition w0distributed like an invariant measure µsuch that w0∈ L4(Ω;H1(S2)), which means that\nwe do not know whether there exists a stationary solution tha t can be interpreted as a martingale solution. We\ncan nevertheless prove that w0∈ L2(Ω;H1(S2))and interpret stationary solution as pathwise solutions in the\nsense of Definition 3.1.\n19Theorem 5.2. Assumeu0∈ L2(Ω;H1(S2))and letw0be distributed like an invariant measure µto(Pt)t.\nEvery stationary solution wstarted in w0has bounded second moment in H1(S2). The constant K >0is\ncommon to every stationary solution, i.e. it holds\nE[/ba∇dblw0/ba∇dbl2\nH1(S2)] =E[/ba∇dblw/ba∇dbl2\nH1(S2)]≤(/ba∇dbl∂xh/ba∇dbl2\nL2+¯G2C(λ1,λ2)) =:K,\nfor every stationary solution w. There exists a constant C >0common to each initial condition so that\nE[/ba∇dblw0/ba∇dbl1/2\nH2(S2)]< C. Moreover, each invariant measure is supported on H2(S2), i.e.H2(S2)⊆suppµ.\nProof. In order to consider µ∈ I as distribution for an initial condition w0to (1.3), we look at its regularity.\nTo this aim, we need w0∈ L2(Ω;H1(S2)). We look therefore at the second moment of the H1(S2)ofw0,\nE[/ba∇dblw0/ba∇dbl2\nH1(S2)] =/integraldisplay\nΩ/ba∇dblw0(ω)/ba∇dbl2\nH1(S2)dP(ω) =/integraldisplay\nH1(S2)/ba∇dblv/ba∇dbl2\nH1(S2)dµ(v),\nwhich needs to be finite. Let (µtnk)nkthe subsequence converging weakly to µ, then from Fatou’s Lemma\n/integraldisplay\nH1(S2)/ba∇dblv/ba∇dbl2\nH1(S2)dµ(v)≤liminf\nR→+∞lim\nk→+∞/integraldisplay\nH1(S2)/ba∇dblv/ba∇dbl2\nH1(S2)∧Rdµtnk(v), (5.1)\nwhere∧denotes the minimum of the two quantities (in particular the integrand is a continuous and bounded\nfunction from H1(S2)with real values). By the definition of µtnk, it follows that\n/integraldisplay\nH1(S2)/ba∇dblv/ba∇dbl2\nH1(S2)∧Rdµtnk(v) =1\ntnk/integraldisplaytnk\n0/integraldisplay\nH1(S2)/ba∇dblv/ba∇dbl2\nH1(S2)∧Rd(P◦(uu0\nr)−1)(v)dr. (5.2)\nWe recognise on the right hand side of (5.2) the definition of e xpectation of /ba∇dbluu0\nr/ba∇dblH1(S2)∧Rwith respect to P,\nwhich leads from the monotone convergence theorem and Lemma 4.7 to\nE[/ba∇dblw0/ba∇dbl2\nH1(S2)]≤1\ntnk/integraldisplaytnk\n0E[/ba∇dbluu0\nr/ba∇dbl2\nH1(S2)]dr/lessorsimilarE[/ba∇dblu0/ba∇dbl2\nH1(S2)]+tnk/ba∇dbl∂xh/ba∇dbl2\nL2\ntnk,\nwhich is bounded. We have concluded that we can choose as init ial condition to (1.3) any random variable w0\ndistributed like an invariant measure µ∈ I, sincew0∈ L2(Ω;H1(S2)). This implies existence of stationary\nsolutions to (1.3). An analogous procedure and Lemma 4.18 le ads to\nE[/ba∇dblw0/ba∇dbl1/2\nH2(S2)] =/integraldisplay\nH2(S2)/ba∇dblv/ba∇dbl1/2dµ(v)< C,\nwhich allows us to conclude that H2(S2)⊆suppµ. Ifw0∈ L2(Ω;H1(S2)), the associated stationary solution\nwsatisfies the energy inequality (4.16). From the stationari ty ofw, we deduce that\nE[/ba∇dbl∇w0/ba∇dbl2\nL2]≤(/ba∇dbl∂xh/ba∇dbl2\nL2+¯G2C(λ1,λ2)).\nThis implies that every stationary solution is bounded by th e same constant K >0.\nRemark 5.3. Also assuming u0∈ L4(Ω;H1(S2)), it is not possible to achieve w0∈ L4(Ω,H1(S2)). This is\nreflected from the estimate in Lemma 4.12. As already mention ed, it does not seem possible to sample initial\nconditions from the invariant measure for martingale solut ions. Namely, we can not employ initial conditions\ndistributed like invariant measures w0as initial conditions for martingale solutions. We can neve rtheless inter-\npret the stationary solutions as pathwise solutions, were n o integrability requirement for the initial condition is\nneeded.\n205.2 Existence of ergodic measures\nDenote by Ithe set of all invariant measures for (Pt)ton(H1(S2),BH1(S2)). We address the problem of\nexistence of ergodic invariant measures in I. Denote by L2(H1(S2),µ)the space of maps φ:H1(S2)→Rso\nthat the expectation with respect to µof the second moment of φis bounded. Recall that a measure µis ergodic\nprovided for all φ∈L2(H1(S2),µ)\nlim\nT→+∞1\nT/integraldisplayT\n0Ptφdt=/integraldisplay\nH1(S2)φ(v)dµ(v).\nWe prove existence of an ergodic invariant measure in the set Iby means of two well known results in Propo-\nsition 5.4 and Theorem 5.5.\nProposition 5.4. (e.g. Proposition 3.2.7 in [18] ) An invariant measure for th e semigroup (Pt)tis ergodic if\nand only if it is an extremal point of the set I.\nRecall some basic definitions. Consider a convex subset Kof a Hausdorff topological vector space. The\nextreme points of the set Kare elements of the set which do not lie in any open line segmen t joining two points\nofK. The convex hull of Kis the smallest convex set containing K. We now state the Krein-Milman theorem.\nTheorem 5.5. (Krein-Milman Theorem, [39]) Any compact convex subset Kof a Hausdorff locally convex\ntopological vector space is equal to the closed convex hull o f its extreme points.\nWe approach the main result of this section. We assume again t o work on a probability space (Ω,F,(Ft)t,P)\nwhere the Brownian motion W, the initial condition u0and the initial condition w0are adapted (which can be\nconstructed via Skorohod representation theorem).\nTheorem 5.6. Assumeu0∈ L2(Ω;H1(S2)). Then, there exists at least an ergodic invariant measure fo r the\nsemigroup (Pt)tassociated to (1.3) .\nProof. From Theorem 4.14, I /\\e}atio\\slash=∅. We aim to apply the Krein-Milman Theorem 5.5 and prove that t he set of\nextremal points of Iis not empty. This implies existence at least an ergodic inva riant measure from Proposition\n5.4 and concludes the proof.\nIn our framework, the Hausdorff locally convex topological vector space is P(H1(S2))equipped with the\ntopology of the weak convergence and Icorresponds to K.\nIndeed, the set Iis convex in P(H1(S2)). Indeed for all α∈[0,1]and for all µ,ν∈ I the sum of the\nmeasures αµ+(1−α)νis also probability measure on H1(S2). The probability measure αµ+ (1−α)νis\nalso an element of I, indeed from the linearity of (P∗\nt)tand the invariance of µ,νit follows that\n/integraldisplay\nEφd(P∗\nt(αµ+(1−α)ν)) =α/integraldisplay\nEφd(P∗\ntµ)+(1−α)/integraldisplay\nEφd(P∗\ntν) =α/integraldisplay\nEφdµ+(1−α)/integraldisplay\nEφdν,\nwhich shows that αµ+(1−α)ν∈ I.\nWe show now that the set of invariant measures Iis compact. As a consequence of the Feller’s property,\nIis closed: therefore we just need to prove that Iis precompact. This reduces to show that Iis tight in\nP(H1(S2)). Letµ∈ I and consider for R >0the measurable set {/ba∇dblx/ba∇dblH2> R} ⊂ B(H1(S2)). It follows\nfrom the invariance of µand from the a priori bounds for the Krylov-Bogoliubov theor em (applied to the\nstationary solutions). Consider an initial condition w0adapted on the joint probability space given by w0and\nthe Brownian motion W, so thatw0is distributed like µ(via Skorohod). From Theorem 5.1, there exists a\npathwise stationary solution. We consider\nµ({/ba∇dblx/ba∇dblH2> R}) =/integraldisplay\nH1(S2)1{/bardblx/bardblH2>R}(v)dµ(v) =/integraldisplay\nΩ1{/bardblw0(ω)/bardblH2>R}(v)dP(ω) =P({/ba∇dblw0/ba∇dblH2> R}).\nFrom Theorem 5.2, w0∈ L2(Ω;H1(S2))and there exists a common constant C >0such thatE[/ba∇dblw0/ba∇dbl2\nH1]< C,\nfor any given initial condition distributed like an invaria nt measure. By employing the stationarity of the\nsolution, Lemma 4.12 and the uniform bound in Theorem 5.2\nP({/ba∇dblw0/ba∇dblH2> R}) =1\nT/integraldisplayT\n0P(/ba∇dblw0/ba∇dblH2> R)dt=1\nT/integraldisplayT\n0P(/ba∇dbluw0\nt/ba∇dbl1/2\nH2> R1/2)dt\n≤1\nTR1/2/integraldisplayT\n0E[/ba∇dbluw0\nt/ba∇dbl1/2\nH2]dt≤C(/ba∇dbl∂xh/ba∇dbl2\nL∞+E[/ba∇dblw0/ba∇dbl2\nH1])\nR1/2,\n21which shows tightness of the set I. We are therefore in the situation of Krein-Milman theorem a nd it follows\nthatIhas extremal points. In conclusion, the semigroup (Pt)tadmits at least an ergodic measure.\nRemark 5.7. The choice of the definition of pathwise solution is determin ant: indeed the tightness of the set I\nrelies on the boundedness of E[/ba∇dbluw0\nt/ba∇dbl1/2\nH2]from Theorem 5.2.\n6 On uniqueness and non uniqueness of stationary solutions a nd invariant\nmeasures.\nThroughout this section, we assume the noise to has either sh ape (6.1) or shape (6.2), where\n/integraldisplayt\n0ur×h1◦dBr, (6.1)/integraldisplayt\n0h2ur×◦d¯Br, (6.2)\nh1∈H1(D;R3),Bis a real valued Brownian motion, h2∈H1(D;R)and¯Bis aR3-valued Brownian\nmotion. We make further assumptions on h1,h2to determine explicitly stationary solutions to (1.1), nam ely\nh1/\\e}atio\\slash= 0, ∂xh1= 0, (6.3) h2/\\e}atio\\slash= 0, ∂xh2= 0. (6.4)\nNote that if h1= 0 (resp,h2= 0), it is known that static solutions to (1.1) exist and are not unique. The\nassociated invariant measures are Dirac measures centred i n those static solutions. Under (6.3), the choice of\nthe noise (6.1) leads to non uniqueness of stationary soluti ons for some particular values of h1. Instead, by\nemploying noise (6.2) under (6.4) there exists a unique stat ionary solution. Define for all the maps v: [0,T]→\nR3the drift\nD(v)s,t:=/integraldisplayt\ns/bracketleftbig\nλ1vr×g′(vr)−λ2vr×(vr×g′(vr))/bracketrightbig\ndr.\nWe state the following additional condition on the anisotro py, which is a smallness condition on the coefficients\nof the matrix and the additive vector b.\nAssumption 6.1. Assume that the anisotropy is of the form g′(x) :=Ax+b, whereAis a real valued 3×3\nmatrix and b∈R3. Assume further that ¯G:= 2supi,j|Ai,j|2+|b|< λ2/2Cp(2λ2+|λ1|), whereCpis the\nPoincar ´e’s constant associated to the domain D.\nUnder Assumption 6.1, we can achieve an improved a priori bou nd, which we state. Without loss of\ngenerality, we prove the result only for the noise (6.1) (see the discussion in Remark 4.10).\nProposition 6.2. Consider a solution to the stochastic LLG equation driven by the noise (6.1) . Under Assump-\ntion 6.1, every solution to the stochastic LLG equation sati sfies\nsup\nr∈[0,t]E/bracketleftbig\n/ba∇dbl∂xur/ba∇dbl2\nL2/bracketrightbig\n+C(λ1,λ2)/integraldisplayt\n0E/bracketleftbig\n/ba∇dblur×∂xur/ba∇dbl2\nL2/bracketrightbig\ndr+λ2/integraldisplayt\n0E/bracketleftbig\n/ba∇dblur×∂2\nxur/ba∇dbl2\nL2/bracketrightbig\ndr\n≤E/bracketleftbig\n/ba∇dbl∂xu0/ba∇dbl2\nL2/bracketrightbig\n+t/ba∇dbl∂xh1/ba∇dbl2\nL2.(6.5)\nProof. We look again at inequality (4.16). We ask whether it is possi ble to obtain an estimate where the linear\ngrowth depends only on the derivative of the spatial compone nt of the noise. We look again at the drift\n2λ1/integraldisplayT\n0/integraldisplay\nD∂x(ur×(Aur+b))·∂xurdxdr= 2λ1/integraldisplayT\n0/integraldisplay\nDur×A∂xur·∂xurdxdr\n≤2|λ1|sup\ni,j|Ai,j|2/integraldisplayT\n0/ba∇dbl∂xur/ba∇dbl2\nL2dr.\n22Now, by recalling that a×(b×c) =b(c·a)−c(a·b)for alla,b,c∈R3, we observe that\n∂x(u×(u×g′(u)))·∂xu=u×(∂xu×g′(u))·∂xu+u×(u×A∂xu)·∂xu\n= [∂xu(u·g′(u))−g′(u)(∂xu·u)]·∂xu+u×(u×A∂xu)·∂xu\n=|∂xu|2(u·g′(u))−A∂xu·∂xu,\nwhere in the last equality we used, from the same vectorial re lation, that\n[u×(u×A∂xu)]·∂xu= [u(A∂xu·u)−A∂xu]·∂xu=−A∂xu·∂xu.\nThis implies the estimate\n2λ2/integraldisplayT\n0/integraldisplay\nD∂x(ur×(ur×g′(ur)))·∂xurdxdr≤4λ2¯G/integraldisplayT\n0/ba∇dbl∂xur/ba∇dbl2\nL2dr.\nBy following the steps of Lemma 4.7 and by applying Lemma 4.8, we obtain the inequality\nsup\nr∈[0,T]E/bracketleftbig\n/ba∇dbl∂xur/ba∇dbl2\nL2/bracketrightbig\n+λ2C−1\np/integraldisplayT\n0E/bracketleftbig\n/ba∇dblur×∂xur/ba∇dbl2\nL2/bracketrightbig\ndr+λ2/integraldisplayt\n0E/bracketleftbig\n/ba∇dblur×∂2\nxur/ba∇dbl2\nL2/bracketrightbig\ndr\n≤E/bracketleftbig\n/ba∇dbl∂xu0/ba∇dbl2\nL2/bracketrightbig\n+t/ba∇dbl∂xh1/ba∇dbl2\nL2+[4λ2¯G+2|λ1sup\ni,j|Ai,j|2|]/integraldisplayT\n0/ba∇dbl∂xur/ba∇dbl2\nL2dr.(6.6)\nFrom Lemma 4.6, the equality /ba∇dbl∂xur/ba∇dbl2\nL2=/ba∇dblur×∂xur/ba∇dbl2\nL2holds. By assuming ¯G < λ 2/2Cp(2λ2+|λ1|),\ninequality (6.5) follows.\nRemark 6.3. The difference between (4.16) and (6.5) is that the second in equality the right hand side depends\non time only through ∂xh1. Hence, the linear growth in (4.16) is the only one needed to p rove existence of an\ninvariant measure: there is no restriction on the anisotrop y. The second estimate in (6.5) is needed for the long\ntime behaviour.\nRemark 6.4. We observe that, if ∂xh1= 0, then the estimate (4.12) is uniform in time. By looking at th e proof\nof estimate (4.12), we realize that the estimate is actually pathwise (due to the orthogonality of the noise) and\nreduces to\nλ2/integraldisplayt\n0/ba∇dblur×∂2\nxur/ba∇dbl2\nL2dr≤ /ba∇dbl∂xu0/ba∇dbl2\nL2, (6.7)\nnamely we have a uniform in time estimate, which leads to\n/integraldisplay+∞\n0/ba∇dblur×∂2\nxur/ba∇dbl2\nL2dr <+∞,/integraldisplay+∞\n0/ba∇dbl∂xur/ba∇dbl2\nL2dr <+∞.\nWe state the main result of this section: we describe explici tly the shape of the stationary solutions under\nthe above conditions on the noise and on the smallness condit ion on the anisotropy.\nTheorem 6.5. Assume that (1.3) is driven by the noise (6.1) , under condition (6.3) , and assume Assumption\n6.1 on the anisotropy. Then every stationary solution to (1.3) is a solution to\nδws,t=D(w)s,t+/integraldisplayt\nswr×h1◦dBr. (6.8)\nFor every fixed h1, there exist more solutions to (6.8) and thus more invariant measures µto(1.3) .\nRemark 6.6. One can check that, for instance, that +h1and−h1are solutions to (6.8), provided |h1|= 1(this\nis without lost of generality, since it is enough to normaliz eh1). The invariant measures to the system are Dirac\ndeltas centred in h1,−h1. There are more invariant measures, as proved in [43] (see al so Proposition 1.17 in\n[6]). This situation can be avoided by taking a sum of differe nt independent Brownian motion and associated\nvectorsh2(more than two).\n23Proof. We aim to show that every stationary solution to (1.3), under the current hypothesis on the noise, is a\nsolution to (6.8). Let µbe an invariant measure and consider w0∈ L2(Ω;H1(S2))as an initial condition to\n(1.3). Denote by wthe solution to (1.3) started in w0. Sincewis stationary, then for all t >0\nE[/ba∇dblwt/ba∇dbl2\nH1] =E[/ba∇dblw0/ba∇dbl2\nH1].\nThe stationary solution whas to satisfy the inequality in Proposition 6.2. Combining these two facts and the\ncondition on the noise ∂xh1= 0, it follows for every t >0that\nC(λ1,λ2)/integraldisplayt\n0E/bracketleftbig\n/ba∇dblwr×∂xwr/ba∇dbl2\nL2/bracketrightbig\ndr+λ2/integraldisplayt\n0E/bracketleftbig\n/ba∇dblwr×∂2\nxwr/ba∇dbl2\nL2/bracketrightbig\ndr= 0.\nIn particular, for a.e. (x,t)∈D×[0,+∞)andP-a.s. ti holds wt(x)×∂xwt(x) = 0 .\nSincewand∂xware orthogonal and wt(x)∈S2, it follows that ∂xw≡0P-a.s. Analogously it follows\nwt(x)×∂2\nxw(x) = 0 for a.e.(x,t)∈D×[0,+∞)andP-a.s. From the orthogonality relation relation, for a.e.\n(x,t)∈D×[0,+∞)andP-a.s.\n∂2\nxwt(x) =wt(x)|∂xwt(x)|2+wt(x)×∂2\nxwt(x),\nit follows that ∂2\nxwt(x) = 0 for a.e.(x,t)∈D×[0,+∞)andP-a.s.. As a consequence of the continuity of\nthe derivative (because of the continuous embedding of H1in the space of continuous functions), we need to\nconclude that we have to take ∂xw0= 0on¯D. In particular, wis a solution to (6.8). As proved in [43], (6.8)\nadmits more more solutions for different values of h1, thus more stationary solutions to (1.3). In particular, fo r\neach fixed h1there are more Dirac delta measures centred in the stationar y measures.\nIn contrast to Theorem 6.5, when considering noise (6.2) the re exists only a stationary measure and a unique\ninvariant measure.\nTheorem 6.7. Assume that (1.3) is driven by the noise (6.2) , under condition (6.4) . Let Assumption 6.1 hold\non the anisotropy. Then every stationary solution to (1.3) is a solution to\nδvs,t=D(v)s,t+/integraldisplayt\nsh2vr×◦dBr. (6.9)\nThere exists a unique solution vto(6.9) . Hence (1.3) admits as unique stationary solution vand a unique\ninvariant measure µ. Moreover, ¯µis a Gibbs measure on (S2,BS2)\n¯µ[dv] =exp(−λ2\nh2¯E(v))dv\n/integraltext\nS2exp(−λ2\nh2¯E(z))dz, (6.10)\nwhere¯E(v) =/integraltext\nDg′(v)·vdx=|D|g′(v)·v, forv∈S2(the integral on the sphere is with respect to the\nRiemmanian volume measure).\nProof. By following the lines of Theorem 6.5 with the estimate in Lem ma 4.12, we can show that every sta-\ntionary solution to (1.3) endowed with the noise (6.4), unde r condition (6.4), fulfils equation (6.9). For every\nfixed constant h2/\\e}atio\\slash= 0, there exists a unique solution vto (6.9) (see Section 1.2.2 in [6]). Thus (1.3) admits vas\nunique stationary solution. Furthermore, (6.9) has a uniqu e invariant measure of the form (6.10), as proved by\nNeklyudov, Prohl [43]. The authors prove that the infinitesi mal generator of the semigroup associated to (6.9)\nhas the form\nA=h2\n2\n2∆S2−/bracketleftbig\nλ1vr×g′(vr)−λ2vr×(vr×g′(vr))/bracketrightbig\n·∇,\nand that the unique invariant measure is a Gibbs measure (see Section 1.2, Theorem 1.7 in [6]).\nRemark 6.8. In absence of anisotropic energy, (6.9) reduces to the so cal led “spherical Brownian motion”\n(see the Appendix for more details on this equation). A funda mental behaviour of the Brownian motion on the\nsphere is the existence of a unique probabilistic invariant measure of the form µ[dv] = dv/|S2|, namely the\nuniform distribution (as proved by Van den Berg, Lewis [46]) . The uniform distribution is trivially a Gibbs\nmeasure.\n24Remark 6.9. (On the stochastic LLG in more dimensions) Observe that, in absence of anisotropic energy and\nprovided the equation (1.3) is driven by noise (6.2) under th e conditions (6.4), the solution to equation (6.9)\nis also a probabilistically and analytically strong soluti on to (1.3) in every space dimensions. The associated\nmeasure is, also in this case, a Gibbs measure as above.\nRemark 6.10. In Theorem 6.7 the invariant measure does not depend on λ1. In particular, it is possible to\nobtain the same results if λ1= 0. This goes in the direction of looking only at the simplified m odel with\nλ1= 0, since this term does not seem to contribute significantly.\nRemark 6.11. A second interesting behaviour, already observed by Brze´ z niak, Goldys and Jegaraj [12] in a\nparticular case, is the phenomenon of magnetization revers al: the Brownian motion with values on the sphere\nis a recurrent Markov process (since it is valued on a compact manifold), hence also the solution to uto (1.3)\nin the large time exhibits a recurrent behaviour. The Browni an motion vis recurrent and from the continuity of\nthe solution with respect to the initial condition in L2, namely\nsup\n0≤t≤T/ba∇dblut−vt/ba∇dbl2\nL2/lessorsimilar/ba∇dblu0−v0/ba∇dbl2\nL2.\nThis implies that if the initial conditions are close, the so lutions will be close. In particular the paths of the two\nsolutions are also close.\n6.1 Long time behaviour if ∂xh2= 0(resp∂xh1= 0).\nWe address the so called “long time behaviour” problem for (1 .3): do the global solutions solution uto (1.3)\nstarted in u0∈H1(S2)converge to the unique stationary solution vstarted in an initial condition v0distributed\nlike the invariant measure for long times? Provided ∂xh2= 0(resp∂xh1= 0) andg≡0, we can say that\nthe solutions to (1.3) started in u0converges to its average /a\\}b∇acketle{tu/a\\}b∇acket∇i}ht“at time+∞”. This space average is itself a\nsolution to the equation satisfied by the stationary solutio ns: hence every solution, for big times, converges to a\nstationary solution.\n6.2 Some heuristics on the long time behaviour also in presen ce of anisotropic energy.\nIn this section, we derive an intuition on the limit behaviou r of the trajectories of the stochastic LLG. In this\nSection 6.2, we assume that the anisotropic energy is non zer o, i.e.g/\\e}atio\\slash= 0and satisfied Assumption 6.1. From\nRemark 6.4, the integral\n/integraldisplay+∞\n0/ba∇dblur×∂xur/ba∇dbl2\nL2dr <+∞\nis bounded; this implies that there exists a monotonic incre asing subsequence (tk)k⊂[0,+∞)such that P-a.s.\nlim\nk→+∞/ba∇dblutk×∂xutk/ba∇dbl2\nL2= lim\nk→+∞/ba∇dbl∂xutk/ba∇dbl2\nL2= 0, (6.11)\nwhere the last equality follows from Lemma 4.6. From the path wise energy inequality (6.7), we observe that\nsup\nt≥T/ba∇dbl∂xut/ba∇dbl2\nL2/lessorsimilar/ba∇dbl∂xuT/ba∇dbl2\nL2/lessorsimilar/ba∇dbl∂xutk/ba∇dbl2\nL2,\nP-a.s. fortk< T. From Poincar´ e-Wirtinger inequality (see Theorem 7.5), e ach solution fulfilling (6.7) satisfies\nfor big times\nsup\nt≥T/ba∇dblut−/a\\}b∇acketle{tut/a\\}b∇acket∇i}ht/ba∇dbl2\nL2≤Cpsup\nt≥T/ba∇dbl∂xut/ba∇dbl2\nL2/lessorsimilar/ba∇dbl∂xutk/ba∇dbl2\nL2, (6.12)\nP-a.s fortk< T and where /a\\}b∇acketle{tut/a\\}b∇acket∇i}htis the spatial average. Thus every solution uconverges for big times to its\nmean value /a\\}b∇acketle{tu∞/a\\}b∇acket∇i}ht(if it exists).\n25Intuitively, since ufor big times converges to its limit spatial average, the lim it spatial average itself U:=\n/a\\}b∇acketle{tu∞/a\\}b∇acket∇i}ht= limt→+∞/a\\}b∇acketle{tut/a\\}b∇acket∇i}htneeds to be a solution to the stochastic LLG. The terms in the d rift where a derivative\nappear need to vanish and Uis the unique solution to\nδUs,t=/integraldisplayt\ns[Ur×g′(Ur)−Ur×(Ur×g′(Ur))]dr+/integraldisplayt\nsh2Ur×◦dWr. (6.13)\nIf the last part can be made rigorous, it proves that in the lon g time every trajectory to the stochastic LLG\nconverges to a stationary solution (indeed recall that (6.1 3) coincides with the equation fulfilled by the stationary\nsolutions in Theorem 6.7).\n6.3 A rigorous proof in case of null anisotropic energy.\nWe assume now that there is no anisotropic energy in the syste m, i.e.g≡0. The following Proposition 6.12\nshows rigorously that the limit behaviour of each solution uis indeed a SDE (no spatial dependence in the\nlimit), where the derivative terms vanish. In particular, w e conclude that udiffers from Bup to a constant\nvector for large times.\nProposition 6.12. (Long time behaviour) Under the condition h2/\\e}atio\\slash= 0,∂xh2= 0 (resp.h2/\\e}atio\\slash= 0,∂xh2= 0)\nand forg≡0, there exists a random variable αsuch that\nlim\nT→+∞sup\nt≥T/ba∇dbl(ut−Bt)·(ut−Bt)−α/ba∇dbl2\nL1= 0.\nIn particular, for large times the solution uconverges P-a.s. to a Brownian motion Bwith values on the sphere\nup to a constant, i.e. for T→+∞fort >0\n|uT+t−BT+t|2−|uT−BT|2= 0\ninCb([0,+∞);L1).\nProof. We follow the strategy of Theorem 3.6 in [17]. The proof below works with small adaptations for the\nnoise (6.1) under the assumptions (6.3). We want to prove the convergence of utoBfor large times, therefore\nwe look at the equation of the difference u−B, which reads in the Stratonovich formulation (respectivel y in\nthe Itˆ o formulation)\nδ(u−B)s,t=/integraldisplayt\nsb(ur)dr+/integraldisplayt\nsh2/bracketleftbig\nur−Br/bracketrightbig\n×◦dWr\n=/integraldisplayt\nsb(ur)dr+/integraldisplayt\nsh2\n2/bracketleftbig\nur−Br/bracketrightbig\ndr+/integraldisplayt\nsh2/bracketleftbig\nur−Br/bracketrightbig\n×dWr.\nConsider the squared equation (from the Itˆ o’s formula on th e second formulation of the equation as in Lemma\n4.12, where ∂xh2is assumed to be 0)\nδ/integraldisplay\nD(u−B)·(u−B)s,tdx=/integraldisplayt\ns/integraldisplay\nDb(ur)·(ur−Br)dxdr.\nWe introduce now a random variable α, which is independent on time and space\nα:=1\n|D|/integraldisplay\nD(u0−B0)·(u0−B0)dx+/integraldisplay+∞\n01\n|D|/integraldisplay\nDb(ur)·(ur−Br)dxdr. (6.14)\nWe prove that |α|<+∞. Recall the equivalent formulation of the drift b(ur) =ur×∂xur+∂2\nxu2+ur|∂xur|2.\nBy integrating by parts, the fact that ∂xBr= 0for allx∈D, from Stokes theorem and from the null Neumann\nboundary conditions,\n/integraldisplay\nD[ur×∂2\nxur+∂2\nxur]·Brdx=/integraldisplay\nDur×∂2\nxur·Brdx+/integraldisplay\nD∂2\nxur·Brdx\n=−/integraldisplay\nDur×∂xur·∂xBrdx+/integraldisplay\n∂D∂u(y)\n∂n(y)·B×u(y)dσ(y)\n−/integraldisplay\nD∂xur·∂xBrdx+/integraldisplay\n∂D∂u(y)\n∂n(y)·Bdσ(y) = 0.\n26Hence, we are left with\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n|D|/integraldisplay+∞\n0/integraldisplay\nDur|∂xur|2·Brdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n|D|/integraldisplay+∞\n0/ba∇dbl∂xur/ba∇dbl2\nL2dr≤Cp\n|D|/integraldisplay+∞\n0/ba∇dblur×∂2\nxur/ba∇dbl2\nL2dr≤Cp\n|D|/ba∇dbl∂xu0/ba∇dbl2\nL2.\nIn conclusion, the random variable αfulfilsP-a.s.\n|α| ≤4+Cp\n|D|/ba∇dbl∂xu0/ba∇dbl2\nL2.\nNote that, with the same computations, it holds that E[|α|]<+∞. The expression (ut−Bt)·(ut−Bt)−α\nis meant to look at the difference ut−Btfor times bigger that t: we write\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nD[(ut−Bt)·(ut−Bt)−α]dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/integraldisplay+∞\nt/integraldisplay\nDb(ur)·(ur−Br)dxdr/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nNotice that it holds that\nlim\nT→+∞E/bracketleftigg\nsup\nt≥T/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nD[(ut−Bt)·(ut−Bt)−α]dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightigg\n= 0. (6.15)\nFrom the triangular inequality, we obtain\n/ba∇dbl(ut−Bt)·(ut−Bt)−α/ba∇dblL1≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble(ut−Bt)·(ut−Bt)−α−1\n|D|/integraldisplay\nD[(ut−Bt)·(ut−Bt)−α]dx/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nD[(ut−Bt)·(ut−Bt)−α]dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nBy applying the Poincar´ e-Wirtinger inequality, we conclu de that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble(ut−Bt)·(ut−Bt)−α−1\n|D|/integraldisplay\nD[(ut−Bt)·(ut−Bt)−α]dx/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1≤Cp/ba∇dbl∂x[(ut−Bt)·(ut−Bt)]/ba∇dblL1\n≤CpC/ba∇dbl∂xut/ba∇dblL2,\nwhereCp>0is the Poincar´ e-Wirtinger constant and we used that Bandαare constant in space in the last\ninequality. Thus, by taking the supremum for big times, it fo llows that\nsup\nt≥T/ba∇dbl(ut−Bt)·(ut−Bt)−α/ba∇dblL1/lessorsimilarsup\nt≥T/ba∇dbl∂xut/ba∇dblL1+sup\nt≥T/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nD[(ut−Bt)·(ut−Bt)−α]dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nBy passing to the limit for T→+∞in the left and in the right hand side of the inequality, the av erage on the\nright hand side tends to 0as a consequence of (6.15). From the computations in Lemma 4. 7, the bound holds\n/integraldisplay+∞\n0/ba∇dbl∂xur/ba∇dbl2\nL2dr <+∞,\nhence there exists a divergent sequence (tk)ksuch that\nlim\nk→+∞/ba∇dbl∂xutk/ba∇dbl2\nL2= 0. (6.16)\nFrom the inequalities in Lemma 4.7, it follows that\nsup\nt≥T/ba∇dbl∂xut/ba∇dblL2≤ /ba∇dbl∂xuT/ba∇dblL2≤ /ba∇dbl∂xutk/ba∇dblL2,\nfortk< T. Hence we conclude that\nlim\nT→+∞sup\nt≥T/ba∇dbl(ut−Bt)·(ut−Bt)−α/ba∇dblL1= 0.\n27Observe also that for all t≥0\n/ba∇dbluT+t−BT+t/ba∇dbl2\nL2=/integraldisplay\nD[uT+t−BT+t]·[uT+t−BT+t]dx\n=/integraldisplay\nD[uT−BT]·[uT−BT]dx+/integraldisplayT+t\nT/integraldisplay\nDb(ur)·[ur−Br]dxdr.\nThe conclusion is achieved by observing that\nsup\nt≥0/ba∇dbl|uT+t−BT+t|2−|uT−BT|2/ba∇dblL1= sup\nt≥T/ba∇dbl(ut−Bt)·(ut−Bt)−α/ba∇dblL1,\nwhich converges to 0forT→+∞(we used that the domain Dis bounded). Observe that the convergence\noccurs also in expectation, by following the same lines of th e above proof. This concludes the proof.\nRemark 6.13. (Every solution is synchronised with Brownian motion for bi g times in case of the noise (6.2)).\nIn Remark 6.11, we observe that the solution to the stationar y solution and the solutions to (1.3) under the\nhypothesis of Proposition 6.12 are close. As a consequence o f Proposition 6.12, the stationary solution and\nthe other solutions are synchronised for big times. This is d ue to the fact that the constant cappearing in\nProposition 6.12 is independent on time and space. One could think that both the solution to (1.3) and the\nspherical Brownian motion Bare valued on the sphere. Hence their distance is trivially a lways bounded by a\nconstant. But since the constant is time and space independe nt, then the motions are synchronised.\nSince the spherical Brownian motion is recurrent, then for b ig times every solution started in a different\ninitial condition is recurrent on the sphere for large times .\nCorollary 6.14. Consider the stochastic LLG driven by the noise (6.2) , under condition (6.4) , andg≡0. For\nlarge times, every solution to (1.3) is recurrent.\n7 Appendix\n7.1 Useful results\nWe list some useful result from rough path theory. The follow ing classical result (see e.g. [21]) enables us to\nestimate the remainder term.\nLemma 7.1 (Sewing lemma) .Fix an interval J, a Banach space Eand a parameter ζ >1. Consider a map\nG:I3→Esuch that G∈ {δH;H:J2→E}and for every s < u < t ∈J,\n|Gsut|/lessorequalslantω(s,t)ζ,\nfor some regular control ωonJ. Then there exists a unique element g∈ V1/ζ\n2(J;E)such that δg=Gand for\neverys < t∈J,\n|gst|/lessorequalslantC(ζ)ω(s,t)ζ, (7.1)\nfor some universal constant Cζ.\nWe introduce a product formula, which is the equivalent of th e Stratonovich product rule in this framework:\nwe employ Proposition 4.1 in [32] and the modification introd uced in [29].\nProposition 7.2 (Product formula) .Fix an integer n≥1and leta= (ai)n\ni=1: [0,T]→L2(D;Rn)(resp.\nb= (bi)n\ni=1: [0,T]→L2(D;Rn)) be a bounded path, given as a weak solution of the system\nδas,t=/integraldisplayt\nsfdt+As,tas+As,tas+a♮\ns,t,/parenleftbigg\nresp.δbs,t=/integraldisplayt\nsgdt+Bs,tbs+Bs,tbs+b♮\ns,t/parenrightbigg\n,\non[0,T]×D,for somef∈L2(L2)(resp.g∈L2(L2)). We assume that both\nA=/parenleftig\nAi,j\ns,t(x),Ai,j\ns,t(x)/parenrightig\n1≤i,j≤n;\ns≤t∈[0,T];x∈DB=/parenleftig\nBi,j\ns,t(x),Bi,j\ns,t(x)/parenrightig\n1≤i,j≤n;\ns≤t∈[0,T];x∈D\naren-dimensional geometric rough drivers of finite (p,p/2)-variation with p∈[2,3)and with coefficients in\nH1(D). Then the following holds:\n28(i) The two parameter mapping ΓA,B≡(ΓA,B,LA,B)defined for s≤t∈[0,T]as\nΓA,B\ns,t:=As,t⊗1+1⊗Bs,t,LA,B\ns,t:=As,t⊗1+As,t⊗Bs,t+1⊗Bs,t,(7.2)\nwhere1≡1n×n∈ L(Rn)is the identity, is a n2-dimensional rough driver (in the sense of Definition\n2.1),\n(ii) The product v⊗2\nt(x) = (ai\nt(x)bj\nt(x))1≤i,j≤nis bounded as a path in L1(D;Rn×n).Moreover, it is a weak\nsolution, in L1,of the system\nd(a⊗b) = (a⊗g+f⊗b)dt+dΓA,B[a⊗b]. (7.3)\nThe following Lemma is a rough path equivalent of the Gronwal l’s lemma (see [19]).\nLemma 7.3 (Rough Gronwall’s lemma) .LetE: [0,T]→R+be a path such that there exist constants κ,ℓ >0,\na super-additive map ϕand a control ωsuch that:\nδEs,t≤/parenleftbigg\nsup\ns≤r≤tEr/parenrightbigg\nω(s,t)κ+ϕ(s,t), (7.4)\nfor every s≤t∈[0,T]under the smallness condition ω(s,t)≤ℓ.\nThen, there exists a constant τκ,ℓ>0such that\nsup\n0≤t≤TEt≤exp/parenleftbiggω(0,T)\nτκ,ℓ/parenrightbigg/bracketleftigg\nE0+ sup\n0≤t≤T|ϕ(0,t)|/bracketrightigg\n. (7.5)\n7.2 Useful inequalities.\n7.2.1 On Poincar ´e and Poincar ´e-Wirtinger inequalities.\nWe recall the classical Poincar´ e inequality\nTheorem 7.4. (Poincar ´e inequality) Let D⊂Rnbe an open bounded subset. Then for every p∈[1,∞)there\nexists a constant C≡C(p,D,n)>0, depending only on p,D,n , such that\n/ba∇dblv/ba∇dblLp(D)≤C/ba∇dbl∇v/ba∇dblLp(D),∀v∈W1,p\n0(D).\nThe application of the Poincar´ e’s inequality is restricte d to functions which are null on the boundary in the\nsense of the trace. The Poincar´ e-Wirtinger inequality is a n extension of the Poincar´ e’s inequality to the whole\nW1,p(D)(see e.g. [3, Corollary 5.4.1]), which we recall.\nTheorem 7.5. LetD⊂Rnbe an open connected bounded subset with C1boundary. Then there exists a\nconstantC≡C(p,D,n)>0, depending only on p,D,n , such that for all v∈W1,p(D)\n/vextenddouble/vextenddouble/vextenddouble/vextenddoublev−1\n|D|/integraldisplay\nDv(y)dy/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nLp(D)≤C/ba∇dbl∇v/ba∇dblLp(Ω).\n7.2.2 Interpolation inequalities on a one dimensional boun ded domain.\nRecall that from the Gagliardo-Nirenberg-Sobolev inequal ity on a bounded one dimensional domain, the fol-\nlowing inequality holds.\nLemma 7.6. LetD⊂Rbe a bounded connected open domain with C1boundary. Then for all v∈W1,2(D)\nthere exists a constant C >0depending on the domain such that it holds\n/ba∇dblv/ba∇dblL4(D)≤C/ba∇dblv/ba∇dbl1\n4\nH1(D)/ba∇dblv/ba∇dbl3\n4\nL2(D).\n29Proof. The assertion follows from e.g. [1, Theorem 5.8] with n= 1,q= 4,m= 1,p= 2.\nWe recall also the Agmon’s interpolation inequality in one d imension.\nLemma 7.7. LetD⊂Rbe an open bounded domain with C1boundary, then there exists a constant C >0\nsuch that for all v∈H1(D),\n/ba∇dblv/ba∇dblL∞(D)≤C/ba∇dblv/ba∇dbl1/2\nH1(D)/ba∇dblv/ba∇dbl1/2\nL2(D).\n7.3 Global in time existence and uniqueness of a solution if ∂xh1= 0 (or∂xh2= 0) and in\nabsence of anisotropic energy.\nIn the particular case of constant space component of the noi se and in absence of anisotropic energy, it is\npossible to show global in time existence and uniqueness of a solution to (1.3). In particular, as a consequence\nof the orthogonality of the noise, the estimates hold pathwi se and the regularity of the moments of the initial\ncondition passes directly to the L∞(H1(S2))norm of the solution. Assume that u0∈H1(S2), then the\nfollowing inequalities hold at the level of the approximati ons and for the solution P-a.s.\nsup\n0≤t≤T/ba∇dbl∂xut/ba∇dbl2\nL2+/integraldisplayT\n0/ba∇dblur×∂2\nxur/ba∇dbl2\nL2dr≤ /ba∇dbl∂xu0/ba∇dbl2\nL2, (7.6)\n/integraldisplayT\n0/ba∇dbl∂xur/ba∇dbl4\nL4dr/lessorsimilarsup\n0≤t≤T/ba∇dbl∂xut/ba∇dbl3\nL2/integraldisplayT\n0/ba∇dblur×∂2\nxur/ba∇dblL2dr≤ /ba∇dbl∂xu0/ba∇dbl4\nL2, (7.7)\nforT <1, where we employed Jensen’s inequality in (7.7) to obtain th at\n/parenleftbigg/integraldisplayT\n0/ba∇dblur×∂2\nxur/ba∇dblL2dr/parenrightbigg2\n≤1\nT/parenleftbigg/integraldisplayT\n0/ba∇dblur×∂2\nxur/ba∇dbl2\nL2dr/parenrightbigg\n≤/integraldisplayT\n0/ba∇dblur×∂2\nxur/ba∇dbl2\nL2dr≤ /ba∇dbl∂xu0/ba∇dbl2\nL2.\nWe aim to get existence and uniqueness on the whole time inter val[0,+∞): since the proof of uniqueness on\neach compact interval relies on the Gronwall’s Lemma, we can not use the same proof. We employ a classical\nargument, where we paste intervals [0,T], for fixed T <1, and cover the whole [0+∞). On the first interval\n[0,T], there exists a unique strong solution to (1.3) starting in u0and ending in uTwhich satisfies the energies\n(7.6) and (7.7). Now we consider uTto be the new initial condition on the interval [T,2T], then there exists a\nunique solution uon[T,2T]starting at uTand ending in u2Tsuch that\nsup\nT≤t≤2T/ba∇dbl∂xut/ba∇dbl2\nL2+/integraldisplay2T\nT/ba∇dblur×∂2\nxur/ba∇dbl2\nL2dr≤ /ba∇dbl∂xuT/ba∇dbl2\nL2≤ /ba∇dbl∂xu0/ba∇dbl2\nL2,\nwhere the last inequality follows as a consequence of (7.6). Similarly, by employing (7.7), we establish\n/integraldisplay2T\nT/ba∇dbl∂xur/ba∇dbl4\nL4dr/lessorsimilarsup\nT≤t≤2T/ba∇dbl∂xut/ba∇dbl3\nL2/integraldisplay2T\nT/ba∇dblur×∂2\nxur/ba∇dblL2dr≤ /ba∇dbl∂xuT/ba∇dbl4\nL2≤ /ba∇dbl∂xu0/ba∇dbl4\nL2.\nThis procedure can be iterated on each interval [(n−1)T,nT], forn∈Nand leads to existence and uniqueness\nof a pathwise global solution to (1.3). Under the assumption that the initial condition u0∈ L4(Ω;H1(S2)), we\nobserve that the global in time solution uto (1.3) belongs to the spaces\nL2(Ω;L∞([0,+∞);H1)∩L2([0,+∞);H2))∩L4(Ω;L∞(H1))\n(the last bound comes from the pathwise estimate in (7.6)). N ote that if u0∈ Lk(Ω;H1(S2)), it also holds that\nLk(Ω;L∞(H1)), for allk≥4.\n307.4 Spherical Brownian motion: existence, uniqueness and u nique invariant measure.\nThe content of this section is not new in the literature: we re fer for instance to Chapter V of [37], to [46] or to\n[36]. We sketch an alternative proof of existence and unique ness of the Brownian motion with values on the\nsphere using the same techniques of this paper, whereas we fo llow [46] to show that the process is a diffusion.\nWe describe the evolution of a Brownian motion on a sphere by\nBt=Bs+/integraldisplayt\nsBr×◦dWr, (7.8)\nwith initial condition B0∈S2. The existence and uniqueness of the solution Bto (7.8) follows from the\nclassical rough path theory, indeed it is a linear equation: uniqueness holds pathwise and, from the continuity\nof the Itˆ o-Lyons map, the solution is also adapted (this tri vially implies a large deviations result and a sup-\nport theorem). It also follows that Blies for every t >0on the sphere S2, provided the initial condition lies\non the sphere. The equation is linear and therefore continuo us with respect to the initial condition in the eu-\nclidean norm. In order to conclude existence of an invariant measure, we can employ again Krylov-Bogoliubov,\nprovided we prove tightness of (µTn)ndefined by\nµTn(BC\nR) =1\nTn/integraldisplayTn\n0P(|Br|2\nR3> R)dr.\nSince the solution Blies on a sphere, we observe that µT(BC\nR) = 1 forR∈[0,1]andµT(BC\nR) = 0 orR >1.\nThis implies that limR→+∞µT(BC\nR) = 0 , thus(µT)Tis tight. Thus Badmits at least one invariant measure\nµBand (7.8) admits a stationary solution.\nUnique invariant measure and recurrence With similar steps as in Lemma 4.7, we observe that the Itˆ o\nformulation of (7.8) has the form\nBt=Bs−/integraldisplayt\nsBrdr+/integraldisplayt\nsBr×dWr\nand we are in the context of [46]. The authors in [46] prove tha t the generator of the semigroup of (7.8) is\na spherical Laplacian (Laplace-Beltrami operator on the sp hereS2): thus the solution process Bto (7.8) is a\nBrownian motion with values on the sphere. We recall briefly t he proof. For a map in C2(M;R)we can apply\nItˆ o’s formula and obtain\nf(Bt) =f(Bs)−/integraldisplayt\ns∇Xf(Br)·Brdr+1\n2/integraldisplayt\nstr[γ(Br)T∇2f(Br)γ(Br)]dr+/integraldisplayt\ns∇Xf(Br)γ(Br)dWr,\nwhereγ(x) =x×·for allx∈R3. In particular, we observe that\ntr(γ(B)T∇2f(B)γ(B)) = ∆f−3/summationdisplay\ni=1Bi(B·[∂i,1f,∂i,2f,∂i,3f]).\nFrom a geometrical point of view, DVf:=∇f(B)−B(B·∇f)and∆Vf:= tr(D2\nVf)is the Laplace-Beltrami\noperator, which defines a diffusion operator on the sphere. I n other terms, the process Bis a diffusion with\ngenerator ∆S2. SinceBis a Brownian motion on S2, it follows from Chapter V , Theorem 4.6 (i) - (iii) in [37]\nthat there exists a unique invariant probability measure µof the form\nµ[dv] =exp(−F(v))dv/integraltext\nS2exp(−F(z))dz,\nsuch that dF= 0: hence in this case F(x)≡C∈[0,+∞). 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Existence and uniqueness of smoot h solution for system of ferromagnetic chain.\nScience in China Series A-Mathematics ,34: 257 , 1991.\n34"    },    {        "title": "2209.00558v1.Growth_parameters_of_Bi0_1Y2_9Fe5O12_thin_films_for_high_frequency_applications.pdf",        "content": "1 \n Growth parameters of Bi 0.1Y2.9Fe5O12 thin films for high frequency \napplications  \n \nGanesh  Gurjar1,4, Vinay Sharma2, S. Patnaik1,*, Bijoy K. Kuanr3 \n1School of Physical S ciences, Jawaharlal Neh ru University, New Delhi, INDIA 110067  \n2Department of Physics, Morgan State University, Baltimore, MD, USA 21251  \n3Special C entre for Nanosciences, Jawaharlal Nehru University, New Delhi , INDIA 110067  \n4Shaheed Rajguru College of Applied Sciences for Women, University of Delhi, INDIA 110096  \n \n \nAbst ract \n \nThe growth and characterization of Bismuth  (Bi) substituted YIG ( Bi-YIG, Bi0.1Y2.9Fe5O12) thin \nfilms are reported.  Pulsed laser deposited (PLD) films with thicknesses ranging from 20 to 150 nm \nwere grown o n Gadolinium Gallium Garnet substrates . Two substrate orientations of (100) and \n(111)  were considered . The enhanced distribution of Bi3+ ions at dodecahedral site along (111) is \nobserved to lead to  an increment in lattice constant from 12.379 Å in (1 00) to 12.415 Å in (1 11) \norient ed films.  Atomic force microscopy images show ed decreasing roughness with increasing \nfilm thickness.  Compared to  (100) grown films, (111) orient ed films  showed an increase in \nferromagnetic resonance  linewid th and consequent increase in Gilbert dampin g. The lowest  \nGilbert damping values are found  to be  (1.06±0. 12) × 10-4 for (100)  and (2.30±0. 36) × 10-4 for (111)  \noriented  films  with thickness  of ≈150 nm . The observed value s of extrinsic linewidth,  effective \nmagnetization , and anisotropic field  are related  to thickness  of the films  and substrate orientation. \nIn addition, the in-plane angular variation establishe d four-fold symmetry for the (100) deposited \nfilms unlike the case of (111) deposited films.  This study prescribes growth condition s for PLD \ngrow n single-crystalline Bi -YIG films towards desired high frequency  and magneto -optic al device \napplications.  \n \nKeyword s: Bi-Yttrium iron oxide; Thin film; Lattice mismatch; Pulsed Laser Deposition; \nFerromagnetic resonance; Gilbert damping; Inhomogeneous br oadening . \nCorresponding authors:  spatnaik@mail.jnu.ac.in  2 \n 1.1 Introduction  \n  \n One of the most important magnetic materials for studying high frequency magnetization \ndynamics is the Yttrium Iron Garnet (YIG, Y 3Fe5O12). Thin film  form  of YIG have attracted  a \nhuge attention in the field of spintronic devices due to its  large spin -wave propagation length , high \nCurie temperature T c ≈ 560 K  [1], lowest Gilbert damping and strong magneto -crystalline \nanisotropy  [2-7]. Due to these merits of YIG, it finds  several ap plications such as in magneto -\noptical (MO) devices, spin-caloritronics  [8,9] , and microwave resonators  and filters  [10-14].  \n The crystal structure of YIG is body centered cubic  under Ia3̅d space group . In Wyckoff \nnotation, t he yttrium (Y) ions are located at the dodecahedral 24c sites, whereas the Fe ions are \nlocated at two  distinct sites ; octahedral 16a and tetrahedral 24d .  The  oxygen ions are located in \nthe 96h sites  [7]. The ferrimagnetism of YIG is induced via a super -exchange interaction at the ‘d’ \nand ‘a’ site between the non-equivalent Fe3+ ions. It has already been observed that substituting \nBi/Ce for Y in YIG improves magneto -optical responsiv ity [13,15 -21]. In addition, Bi substitution \nin YIG (Bi -YIG) is known to generate growth -induced anisotropy, therefore, perpendicular \nmagnetic anisotropy (PMA) can be achieved in Bi doped YIG, which is beneficial in applications \nlike magnetic memory and logic devices  [7,22,23] . Due to its u sage in magnon -spintronics and \nrelated  disciplines such as caloritronics, the study of fundamental characteristics of Bi -YIG \nmaterials is of major current interest due to their high uniaxial anisotropy and F araday rotation  \n[17, 24-27]. Variations in the concentration of Bi3+ in YIG, as well as substrate orientation and \nfilm thickness, can improve strain tuned structural properties and magneto -optic characteristics . \nAs a result, selecting the appropriate substrate orientation and film thickness is important for \nidentifying the  growth of Bi-YIG thin films.  3 \n  The structural and magnetic characteristics of Bi -YIG [Bi 0.1Y2.9Fe5O12] thin film have been \nstudied in the current study. Gadolinium Gallium Garnet (GGG) substrates with orientations of \n(100) and (111)  were used to grow thin films . The Bi-YIG films of  four different thickness  (≈20 \nnm, 50 nm, 100 nm and 150 nm ) were deposited in -situ by pulsed laser deposit ion (PLD) method  \n[19,2 8] over single -crystalline GGG  substrates . Along with structural characterization  of PLD  \ngrown films , magnetic properties were ascertained by using  vibrating sample magnetometer \n(VSM) in conjunction with ferromagnetic resonance (FMR) techniques.  FMR is a highly effective \ntool for studying magnetization dynamics. The FMR response not only provides information about \nthe magnetization dynamic s of the material such as Gilbert damping and anisotropic field, but also \nabout the static magnetic properties such as saturation magnetization and anisotropy field.  \n \n1.2 Experiment  \nPolycrystalline YIG an d Bi -YIG targets were synthesized via the solid -state reaction \nmethod. Briefly, yttrium oxide (Y 2O3) and iron oxide (Fe 2O3) powders from Sigma -Aldrich were \ngrounded for ≈14 hours before calcination at 1100 ℃. The calcined powders were pressed into \npellets of one inch and sintered at 1300 ℃.  Using these polycrystalline YIG and Bi -YIG targets, \nthin films of four thicknesses ( ≈20 nm, 50 nm, 100 nm, and 150 nm) were synthesized in -situ on \n(100) - and (111) -oriented GGG substrates using the PLD method.  The samples are labelled in the \ntext as 20 nm (100), 20 nm (111) , 50 nm (100) , 50 nm (111),  100 nm (100) , 100 nm (111) , 150 nm \n(100),  and 150 nm (111) . Before deposition,  GGG substrates were cleaned in an ultrasonic bath \nwith acetone and isopropanol for 30  minute s.  The deposition chamber was cleaned and evacuated  \nto 5.3×10-7 mbar. For PLD growth, a 248 nm KrF excimer laser  (Laser fluence (2.3 J cm-2) with \n10 Hz pulse rate was used to ablate the material from the target . Oxygen pressure, target -to-4 \n substrate distance, and substrate temperature were maintained at 0.15 mbar, 5.0 cm, and 825 oC, \nrespectively.  Growth  rate of deposited films were  6 nm/min . The as -grown films were annealed \nin-situ for 2 hours at 825 oC in the presence of oxygen (0.15 mbar).  The structural characterization \nof thin films were ascertained  using X -ray diffraction (XRD)  with Cu-Kα radiation  (1.5406  Å). We \nhave performed the XRD me asurement at room temperature in -2 geometry and incidence angle \nare 20 degrees.   The film's surface morphology and thickness were estimated using atomic force \nmicroscopy  (AFM)  (WITec GmbH , Germany ). The magnetic properties were studied using a  \nvibrating sample magnetomet ry (VSM) in  Cryogenic 14 Tesla Physical Property Measurement \nSystem (PPMS).  FMR measurements were done on a coplanar waveguide (CPW) in a flip -chip \narrangement with a dc magnetic field applied perpendicular to the high -frequency  magnetic field \n(hRF). A Keysight Vector Network Analyzer was used  for this purpose. The CPW was rotated in \nthe film plane from 0º to 360º  for in -plane () measurement s and from 0º to 18 0º for out of plane \n(θ) measurement.  \nIn this study, the thickness of Bi -YIG was determined by employing methods such as laser \nlithography and AFM. We have calibrated the thickness of thin films with PLD laser shots. \nPhotoresist by spin coating is applied to a silicon substrate, and then straight-line patterns were \ndrawn on the photoresist coated substrates using laser photolithography. The PLD technique was \nused to deposit thin films of the required material onto a pattern -drawn substrate. It is then \nnecessary to wet etch the PLD grown thin fi lm in order to remove the photoresist coating. Then, \nAFM tip is scanned over the line pattern region in order to estimate the thickness of the grown \nsamples from the AFM profile image.  \n \n 5 \n 1.3 Results and Discussion  \n \n1.3.1  Structural properties  \n \nFigure  1 (a)-(d) show the XRD pattern of  (100)- and (111)-oriented Bi-YIG grown thin \nfilms  with thickness ≈20-150 nm  (Insets depict the zoomed image of XRD patterns) . XRD data \nindicate single -crystalline growth of Bi -YIG thin films . Figures 1 (e) and 1 (f) show the l attice \nconstant and lattice mismatch  (with respect to substrate) determined from XRD data, respectively . \nThe cubic lattice constant 𝒂 is calculated using the formula , \n𝒂=𝜆√ℎ2+𝑘2+𝑙2\n2sin𝜃                                                        (1) \n where the wavelength of Cu -Kα radiation  is represented by 𝜆, diffraction angle  by 𝜃, and the Miller \nindices of the corresponding XRD peak  by [h, k, l] . Further, the l attice  mismatch  parameter (𝛥𝑎\n𝑎) \nis calculated  using  the equation , \n   𝛥𝑎\n𝑎=(𝑎𝑓𝑖𝑙𝑚 − 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 )\n𝑎𝑓𝑖𝑙𝑚 100               (2) \nHere lattice constant of film and substrate are represented by  𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 , respectively . \nThe reported lattice constant values are consistent with prior findings  [15,17,21].  Lattice constant \nslightly increases with the increase in thickness of the film in the case of (111) as compared to \n(100) . Since the distribution of Bi3+ in the dodecahedral sit e is dependent on the substrate \norientation  [7,23,2 9], the (111) oriented films show an increase in the lattice constant . In Bi -YIG \nfilms, this slight increase in the lattice constant (in the 111 direction) leads to a \ncompar atively  larger  lattice mismatch  as seen in Fig. 1 (f). For 50 nm (111) Bi -YIG film, we \nachieved a lattice mismatch  of ~0.47 , which is close to what has been reported earlier  [30,31]. 6 \n Smaller value of lattice mismatch can reduc e the damping constant of the film  [31]. We want to  \nunderline the importance of lattice plane dependen t growth in conjunction with film thickness in \nindicating structural and magnetic property changes.  \n \n1.3.2  Surface morphology  \n  \nFigure 2 (a) -(h) shows room temperature AFM images with root mean square (RMS) \nroughness. Roughness is essential from an application standpoint because the roughness directly \nimpacts the inhomogeneous linewidth broadening which leads to increase in the Gil bert damping. \nWe have observed RMS roughness  around  0.5 nm  or less  for all grown Bi -YIG films which are \ncomparable to previous reported YIG films  [32,33]. We have observed that  RMS  roughness  \ndecreases with increase  in thickness of the film. With (100) and (111) orientations, there is no \ndiscernible difference in roughness. Furthermore, roughness would be more affected by changes \nin growth factors and by substrate orientation  [7,33,34]. \n \n1.3.3  Static magnetization study  \n \n The room temperature ( ≈296 K ) VSM magnetization measurements were carried out with \napplied  magnetic field parallel to the film plane  (in-plane) . The paramagnetic contribution s from \nthe GGG substrate were carefully subtracted. F igure 3 (a)-(h) show s the magnetization plot s of Bi-\nYIG thin films of thickness ≈20-150 nm . Inset of Fig. 3 (i) shows the measured saturation \nmagnetization ( µ0MS) data of as-grown (100) and (111) -oriented Bi -YIG films which are \nconsistent with the previous reports  [6,17,22,3 5,36]. Figure 3 ( i) shows plot of µ0Ms × t Vs. t, \nwhere ‘t’ is film thickness . This is done t o determine thickness of dead layer via linear 7 \n extrapolati on plot to the x -axis. The obtaine d magnetic dead -layer for (100) and (111) -oriented \nGGG substrates are 2.88 nm and 5.41 nm , which are  comparable to previous reports  [37-39]. The \nsaturation magnetization of Bi -YIG films increases as the thickness of the films increases . The \nincrease in saturation magnetization with increase in thickness can be understood by the following \nways .  Firstly, ferromagnetic thin films are generally deposited with a thin magnetically dead layer \nover the interface with the substrate. This magnetic dead layer effect is larger in thinner films  that \nleads to the decrease in net magnetization with the decrease in thickness  [40,41]. Figure 3 (i) shows \nthe effect of magnetic dead laye r region near to the substrate.  Secondly, t hicker films exhibit the \nbulk effect of YIG which, in turn, results in increas ed magnetization.  \n \n1.3.4 Ferromagnetic r esonance study  \n \n Figure 4 (a) -(d) shows  the FMR absorption spectra  of (100) and (111)  -oriented films that  \nare labeled with open circle ( Ο) and open triangle ( Δ) respectively . FMR experiment s were carried \nout at room temperature. In -plane dc magnetic field was a pplied  parallel to film surface . To find \nthe effective magnetization and Gilbert damping, the  FMR linewidth (∆H) and resonance magnetic \nfield (H r) are calculated using a Lorentzian fit of the FMR absorption spectra measured at 𝑓 = 1 \nGHz to 12 GHz. Effective magnetization field ( 0𝑀𝑒𝑓𝑓) were obtained from the fitting of Kittel's \nin-plane equation (Eq. 3) [42].  \n𝑓=𝛾\n2𝜋0√(𝐻𝑟)(𝐻𝑟+𝑀𝑒𝑓𝑓)     (3), \nHere, 0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖), anisotropy field 𝐻𝑎𝑛𝑖=2𝐾1\n0𝑀𝑠, and   𝛾 being the gyromagnetic \nratio. Further, the dependence of FMR linewidth on microwave frequency shows a linear variation \n(Eq. 4) [42]  from which the Gilbert damping parameter (α) and FMR linewidth broad ening ( 𝛥𝐻 0) \nwere obtained:  \nµ0𝛥𝐻=µ0𝛥𝐻 0+4𝜋𝛼\n𝛾𝑓                                     (4) 8 \n where, 𝛥𝐻 0 is the inhomogeneous broadening linewidth and α is the Gilbert damping. Figures 4 \n(e) and 4 (f) show Kittel and linewidth fitted graphs, respectively . Figure 5 (a) -(d) shows the \nderived parameters acquired from the FMR  study. The estimated Gilbert damping is con sistent \nwith data reported for sp in-wave propagation  [3,22] . The value of α decreases as the thickness of \nthe film increases ( Fig. 5 (c)). Howe ver, in the instance of Bi -YIG with (111) orientation, there is \na substantial increase. This might be attributed qualitatively to the presence of Bi3+ ions, which \ncause strong spin-orbit coupling  [43-45] as well as electron scattering inside the lattice when the \nlattice mismatch (or strain) increases  [46]. Our earlier study  [7] revealed a clear  distribution  of \nBi3+ ions along (11 1) planes, as well as slightly  larger lattice mismatch  in Bi -YIG (111). These \nresults  explain the larger values of Gilbe rt damping,  0𝑀𝑒𝑓𝑓, and ΔH 0 values in Bi -YIG (111)  (Fig. \n5). The change in 0𝑀𝑒𝑓𝑓 is due to u niaxial in -plane magne tic anisotropy and it  is observed from \nmagnetization measurements  using  0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖) [36,47,48]. The enhanced \nanisotropy field in the lower thic kness of Bi -YIG ( Fig. 5 (d) ) signifies the effect of dead magnetic \nlayer at the interface.  The lattice mismatch between films and GGG substrates induces uniaxial in -\nplane magnetic anisotropy  [36,47]. ΔH 0 has a magnitude that is similar to previously published \nvalues for the same substrate orientation  [7,47]. In conclusion, Bi -YIG with (100) orientation \nproduces  the lowest Gilbert damping facto r and inhomogeneous broadening linewidth . These are \nthe required  optimal parameters for  spintronics  based devices.  \n Figure 6 (a) shows the variation of resonance field with polar angle ( ) for the grown 20 \nnm-150 nm films , H is the angle measured between applied magnetic field and surface of film \n(shown in inset of Fig. 4 (a)).  The FMR linewidth (ΔH) were extracted fr om fitting of FMR spectra \nwith L orentzian absorption functions.  From Fig. 6 (a), we observe change in H r value for 50 nm \nBi-YIG film as  0.22 T and 0.27 T for (100) and  (111)  orientation respectively. Similarly, 0.21 T \nand 0.31 T  change is observed in (100) and (111) orientation respectively  for 100 nm Bi -YIG film .  \nWe see that H r increases slightly in case o f (111) oriented film by changing  the di rection of H from \n0º to 90º  with regard  to sample surface (inset of Fig. 4 (a)).  The change in H r decreases with \nincrease in film thickness in cas e of (100) while it is reversed  in case of (111 ). Figure 6 (b) shows 9 \n the variation of FMR linewidth with polar angle  for 150 nm Bi -YIG film . Maximum FMR  \nlinewidth is observed at 90º  and it is slightly more as compared with (100) orientation. The \nenhanced variation of FMR linewidth in (111) oriented samples is generated due to the higher \ncontribution of two -magnon scatte ring in perpendicular geometry  [49]. This can be understood \ndue to the higher anisotropy field in (111) oriented samples ( Fig. 5 (d)). \n Figure 6 (c) & (e) shows the azimuthal angle ( ) variation of H r. Frequency of 5 GHz  is \nused in the measurement . From  variation data  (by changing  the direction of H  from 0 º to 360  \nwith regard  to sample  surface  (inset of Fig. 4 (a)). We can see clearly  in-plane anisotropy of  four-\nfold in Bi-YIG (100)  (Fig. 6 (c)) unlike in  Bi-YIG (111)  (Fig. 6 (e)). According to crystalline \nsurface symmetry there would be six -fold in -plane anisotropy in case of (111) orientation but we \nhave not observe d it, based  on previous reports, it can be superseded by a mis cut-induced uniaxial \nanisotropy  [33,50]. This reinforces our grown films' single -crystalline nature . The observed change \nin H r (H=0 to 45) is 6 .6 mT in 50 nm (100 ), 0.17 mT for 50 nm (111) , 6.2 mT in 100 nm (100) , \n0.17 mT for 100 nm (111) ) and 5.1 mT in 150 nm (100 ). As a result, during in -plane rotation, the  \nhigher FMR field change observed  along the (100) orientation.  The  dependent FMR field data \nshown in figure 6 (c) were fitted using the following Kittel relation [50]  \n𝑓=𝛾\n2𝜋0√([𝐻𝑟cos(𝐻−𝑀)+𝐻𝑐cos4(𝑀−𝐶)+𝐻𝑢cos2(𝑀−𝑢)])×\n(𝐻𝑟cos(𝐻−𝑀)+𝑀𝑒𝑓𝑓+1\n4𝐻𝑐(3+cos4(𝑀−𝐶))+𝐻𝑢𝑐𝑜𝑠2(𝑀−𝑢))        (5) \nWith respect to the [100] direction of the GGG substrate, in -plane directions of the magnetic field, \nmagnetization, uniaxial, and cubic anisotropies are given by H, M, u and c, respectively. \n𝐻𝑢=2𝐾𝑢\nµ0𝑀𝑠  and 𝐻𝑐=2𝐾𝑐\nµ0𝑀𝑠 correspond to the uniaxial and cubic anisotropy fields, respectively, \nwith 𝐾𝑢 and 𝐾𝑐 being the uniaxial and cubic magnetic anisotropy constants, respectively.  10 \n Figure 6 (d) shows t he obtained uniaxial anisotropy field, cubic anisotropy field and saturation \nmagnetization field for (100) orientation. The obtained saturation magnetization field follows the \nsame pattern as we have obtained from the VSM measurements. The cubic anisotropy  field \nincreases and then saturates with the thickness of the film. A large drop in the uniaxial anisotropy \nfield is observed with the thickness of the grown films. We have not got the in -plane angular \nvariation data for the 20 nm thick Bi -YIG sample and m ay be due to the low thickness of the Bi -\nYIG, it is not detected by our FMR setup.  \n \n1.4 Conclusion  \n In conclusion, we compare  the properties of high-quality Bi -YIG thin films of four distinct \nthicknesses (20 nm, 50 nm, 100 nm, and 150 nm) grown on GGG substrates with orientations of \n(100) and (111). Pulsed laser deposition was used to synthesize the se films. AFM and XRD \ncharacterizations reveal th at the deposited thin films have  smooth surfaces and are phase pure. \nAccording to FMR data, t he Gilbert  damping value  decreases with increase in film  thickness . This \nis explained i n the context of a dead m agnetic layer . The (100) orientation has a lower va lue of \nGilbert damping, indicating that it is the preferable substrate for doped YIG thin films  for high \nfrequency application . Bi-YIG on (111) orientation , on the other hand, exhibits anisotropic \ndominance, which is necessary for magneto -optic devices. Th e spin -orbit coupled Bi3+ ions are \nresponsible  for the enhanced Gilbert damping in (111). We have also correlated  ∆H 0, anisotropic \nfield, and effective magnetization  to the variations in film thickness and substrate ori entation . In \n(100) oriented films, there is unambiguous observation of four-fold in -plane anisotropy. In  \nparticular, Bi-YIG grown on (111) GGG substrates  yields best result for optim al magnetization \ndynamics. This is linked to an enhanced  magnetic anisotropy.  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Ishibashi, L. Beran, M. Zahradnik, R. Antos, M. \nKučera, M. Veis, Optical and magneto -optical properties of Bi substituted yttrium iron \ngarnets prepared by metal organic decomposition, Opt. Mater. Express. 6 (2016) 1986 –\n1997.  https://doi.org/10.1364/ OME.6.001986  \n[45] J. Yang, Y. Xu, F. Zhang, M. Guillot, The effects of the spin --orbit coupling strength of \nthe two configurations of rare -earth ions on the magneto -optical properties in garnets, J. \nPhys. Condens. Matter. 18 (2006) 9287.  https://doi.org/10 .1088/0953 -8984/18/40/013  \n[46] M.A.W. Schoen, D. Thonig, M.L. Schneider, T.J. Silva, H.T. Nembach, O. Eriksson, O. \nKaris, J.M. Shaw, Ultra -low magnetic damping of a metallic ferromagnet, Nat. Phys. 12 \n(2016) 839 –842. https://doi.org/10.1038/nphys3770  \n[47] N. S. Sokolov, V. V. Fedorov, A. M. Korovin, S. M. Suturin, D. A. Baranov, S. V Gastev, \nB. B. Krichevtsov, K. Yu. Maksimova, A. I. Grunin, V. E. Bursian, L. V. Lutsev, and M. \nTabuchi , Thin yttrium iron garnet films g rown by pulsed laser deposition: Crystal structure, 19 \n static, and dynamic magnetic properties, J. Appl. Phys. 119 (2016) 23903. \nhttps://doi.org/10.1063/1.4939678.  \n[48] P. Hansen, K. Witter, Growth -induced uniaxial anisotropy of bismuth -substituted iron -\ngarne t films, J. Appl. Phys. 58 (1985) 454 –459. https://doi.org/10.1063/1.335645  \n[49] M.J. Hurben, C.E. Patton, Theory of two magnon scattering microwave relaxation and \nferromagnetic resonance linewidth in magnetic thin films, J. Appl. Phys. 83 (1998) 4344 –\n4365 . https://doi.org/10.1063/1.367194  \n[50] S. Akansel, A. Kumar, N. Behera, S. Husain, R. Brucas, S. Chaudhary, and P. Svedlindh, \nThickness -dependent enhancement of damping in Co 2FeAl/β -Ta thin films , Physical \nReview B 97.13 (2018): 134421. https://doi.org/10 .1103/PhysRevB.97.134421  \n \n \n \n \n \n \n  20 \n List of f igure caption s \n \nFigure 1:  (a)-(d) X -ray diffraction (XRD)  patterns of 20 nm -150 nm Bi -substituted YIG films in \n(100) and (111) orientations. Insets in (a) -(d) depict the zoomed image of XRD patterns. Variation \nof lattice constant (e) and (f) lattice mismatch with thickness  are shown . \n \nFigure 2:  (a)-(h) A tomic force microscopy  images of 20 nm -150 nm Bi -YIG film in (100) and \n(111) orientations  are shown . \n \nFigure 3:  (a)-(h) Static magnetization graph of 2 0 nm -150 nm Bi-substituted YIG  (Bi-YIG) films \nin (100) and (111) orientations. ( i) Graph to determine the magnetic dead -layer thickness of Bi -\nYIG films on (100) and (111) -oriented GGG substrates is depicted (inset shows the variation of  \nsaturation magnetiz ation value with the film thickness).  \n \nFigure 4:  (a)-(d) Ferromagnetic resonance ( FMR ) absorption spectra of 20 nm -150 nm Bi-\nsubstituted YIG  films with (100) and (111) orientations. Inset in (a) shows the geometry of an \napplied field angle measured from the sample surface.  (e) shows frequency -dependent FMR \nmagnetic field data fitted with Kittel Eq. 3 . (f) shows frequency -dependent FMR linewidth  data \nfitted with Eq. 4 . \n \nFigure 5:  Variation s of (a) extrinsic linewidth, (b) effective magnetization, (c) Gilbert damping, \nand (d) magnetic anisotropy with thickness for (100) and (111) oriented Bi-substituted YIG  films  \nare depicted .  21 \n Figure 6:  (a) Angular variation of Ferromagnetic resonance (FMR) magnetic field for 20 nm -150 \nnm Bi -substituted YIG (Bi -YIG) film with (100) and (111) orientations is shown. (b) Angular \nvariation of FMR linewidth of 150 nm thick Bi -YIG film with (100) and (111) orientation is \nshown. Variations of FMR magnetic field as a  function of azimuthal angle ( ) for (c) 50 nm, 100 \nnm and 150 nm Bi -YIG film with (100) orientation is depicted (d) obtained uniaxial anisotropy \nfield, cubic anisotropy field and saturation magnetization field for (100) orientation. (e)  \ndependent FMR fi eld data for 50 nm and 100 nm Bi -YIG film with (111) orientation is depicted.  \n \n \n  22 \n Figure 1  \n \n \n \n \n \n \n23 \n  \nFigure 2  \n \n \n24 \n  \nFigure 3  \n  \n25 \n Figure  4 \n \n26 \n  \nFigure 5  \n \n \n \n \n \n \n \n \n \n \n27 \n  \nFigure 6  \n \n \n"    },    {        "title": "2209.02914v2.Convergence_analysis_of_an_implicit_finite_difference_method_for_the_inertial_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "CONVERGENCE ANALYSIS OF AN IMPLICIT FINITE\nDIFFERENCE METHOD FOR THE INERTIAL\nLANDAU-LIFSHITZ-GILBERT EQUATION\nJINGRUN CHEN, PANCHI LI, AND CHENG WANG\nAbstract. The Landau-Lifshitz-Gilbert (LLG) equation is a widely used model\nfor fast magnetization dynamics in ferromagnetic materials. Recently, the iner-\ntial LLG equation, which contains an inertial term, has been proposed to cap-\nture the ultra-fast magnetization dynamics at the sub-picosecond timescale.\nMathematically, this generalized model contains the \frst temporal derivative\nand a newly introduced second temporal derivative of magnetization. Conse-\nquently, it produces extra di\u000eculties in numerical analysis due to the mixed\nhyperbolic-parabolic type of this equation with degeneracy. In this work, we\npropose an implicit \fnite di\u000berence scheme based on the central di\u000berence in\nboth time and space. A \fxed point iteration method is applied to solve the im-\nplicit nonlinear system. With the help of a second order accurate constructed\nsolution, we provide a convergence analysis in H1for this numerical scheme, in\nthe`1(0;T;H1\nh) norm. It is shown that the proposed method is second order\naccurate in both time and space, with unconditional stability and a natural\npreservation of the magnetization length. In the hyperbolic regime, signi\fcant\ndamping wave behaviors of magnetization at a shorter timescale are observed\nthrough numerical simulations.\n1.Introduction\nThe Landau-Lifshitz-Gilbert (LLG) equation [15, 19] describes the dissipative\nmagnetization dynamics in ferromagnetic materials, which is highly nonlinear and\nhas a non-convex constraint. Physically, it is widely used to interpret the experi-\nmental observations. However, recent experiments [5, 16, 17] con\frm that its valid-\nity is limited to timescales from picosecond to larger timescales for which the angular\nmomentum reaches equilibrium in a force \feld. At shorter timescales, e.g. \u0018100 fs,\nthe ultra-fast magnetization dynamics has been observed [17]. To account for this,\nthe inertial Landau-Lifshitz-Gilbert (iLLG) equation is proposed [6, 10, 12]. As a\nresult, the magnetization converges to its equilibrium along a locus with damping\nnutation simulated in [21], when the inertial e\u000bect is activated by a non-equilibrium\ninitialization or an external magnetic \feld.\nFor a ferromagnet over \n 2Rd;d= 1;2;3, the observable states are depicted by\nthe distribution of the magnetization in \n. The magnetization denoted by m(x;t) is\na vector \feld taking values in the unit sphere S2ofR3, which indicates that jmj= 1\nin a point-wise sense. In micromagnetics, the evolution of mis governed by the\nLLG equation. In addition to experiment and theory, micromagnetics simulations\nDate : September 13, 2022.\n2010 Mathematics Subject Classi\fcation. Primary 35K61, 65M06, 65M12.\nKey words and phrases. Convergence analysis, inertial Landau-Lifshitz-Gilbert equation, im-\nplicit central di\u000berence scheme, second order accuracy.\n1arXiv:2209.02914v2  [math.NA]  12 Sep 20222 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nhave become increasingly important over the past several decades. Therefore, nu-\nmerous numerical approaches have been proposed for the LLG equation and its\nequivalent form, the Landau-Lifshitz (LL) equation; see [9, 18] for reviews and ref-\nerences therein. In terms of time marching, the simplest explicit methods, such\nas the forward Euler method and Runge-Kutta methods, were favored in the early\ndays, while small time step size must be used due to the stability restriction [22].\nOf course, implicit methods avoid the stability constraint and these methods pro-\nduce the approximate solution in H1(\n) [1, 2]. However, in order to guarantee the\nconvergence of the schemes, a step-size condition k=O(h2) must be satis\fed in\nboth the theoretical analysis and numerical simulations. To obtain the weak solu-\ntion in the \fnite element framework, an intermediate variable vwith the de\fnition\nv=@tmrepresenting the increment rate at current time is introduced, and to solve\nvin the tangent space of mwhere it satis\fes v\u0001m= 0 in a point-wise sense, then\nthe con\fguration at the next time step can be obtained. Directly, the strong solu-\ntion can be obtained through solving the implicit mid-point scheme [4] and the im-\nplicit backward Euler scheme [13] using \fxed-point iteration methods. By contrast,\nthe semi-implicit methods have achieved a desired balance between stability and\ne\u000eciency for the micromagnetics simulations. The Gauss-Seidel projection meth-\nods [11, 20, 27], the linearized backward Euler scheme [8, 14], the Crank-Nicolson\nprojection scheme [3], and the second order semi-implicit backward di\u000berentiation\nformula projection scheme [7, 28] have been developed in recent years. In prac-\ntice, all these semi-implicit methods inherit the unconditional stability of implicit\nschemes, and achieve the considerable improvement in e\u000eciency.\nThe LLG equation is a nonlinear parabolic system which consists of the gyro-\nmagnetic term and the damping term. It is a classical kinetic equation that only\ncontains the velocity; no acceleration is included in the equation. When relaxing\nthe system from a non-equilibrium state or applying a perturbation, it is natural\nthat an acceleration term will be present, resulting in the inertial term in the iLLG\nequation. More speci\fcally, the time evolution of m(x;t) is described by @tmand\nm\u0002@tmwith the addition of an inertial term m\u0002@ttm. Thus, the iLLG equa-\ntion is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy. To\nnumerically study the hyperbolic behaviors of the magnetization, the \frst-order\naccuracy tangent plane scheme (TPS) and the second-order accuracy angular mo-\nmentum method (AMM) are proposed in [23]. The \fxed-point iteration method is\nused for the implicit marching. These two methods aim to \fnd the weak solution.\nFurthermore, a second-order accurate semi-implicit method is presented in [21], and\n@ttmand@tmare approximated by the central di\u000berence.\nIn this work, we provide the convergence analysis of the implicit mid-point\nscheme on three time layers for the iLLG equation. Subject to the condition\nk\u0014Ch2, it produces a unique second-order approximation in H1(\nT). Owing to\nthe application of the mid-point scheme, it naturally preserves the magnetization\nlength. Moreover, we propose a \fxed-point iteration method to solve the nonlinear\nscheme, which converges to a unique solution under the condition of k\u0014Ch2.\nNumerical simulations are reported to con\frm the theoretic analysis and study the\ninertial dynamics at shorter timescales.\nThe rest of this paper is organized as follows. The iLLG equation and the\nnumerical method are introduced in Section 2. The detailed convergence analysis\nis provided in Section 3. In addition, a \fxed-point iteration method for solvingCONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 3\nthe implicit scheme is proposed in Section 4, and the convergence is established\nupon the condition k\u0014Ch2. Numerical tests, including the accuracy test and\nobservation of the inertial e\u000bect, are presented in Section 5. Concluding remarks\nare made in Section 6.\n2.The physical model and the numerical method\nThe intrinsic magnetization of a ferromagnetic body m=m(x;t) : \nT:=\n\n\u0002(0;T)!S2is modeled by the conventional LLG equation:\n@tm=\u0000m\u0002\u0001m+\u000bm\u0002@tm; (x;t)2\nT; (2.1a)\nm(x;0) =m(0); x2\n; (2.1b)\n@\u0017m(x;t) = 0; (x;t)2@\n\u0002[0;T]; (2.1c)\nwhere\u0017represents the unit outward normal vector on @\n, and\u000b\u001c1 is the\ndamping parameter. If the relaxation starts from a non-equilibrium state or a\nsudden perturbation is applied, the acceleration should be considered in the kinetic\nequation, which is the inertial e\u000bect observed in various experiments at the sub-\npicosecond timescale. In turn, its dynamics is described by the iLLG equation\n@tm=\u0000m\u0002(\u0001m+He) +\u000bm\u0002(@tm+\u001c@ttm); (x;t)2\nT; (2.2a)\nm(x;0) =m(0); x2\n; (2.2b)\n@tm(x;0) = 0; x2\n; (2.2c)\n@\u0017m(x;t) = 0; (x;t)2@\n\u0002[0;T]; (2.2d)\nwhere\u001cis the phenomenological inertia parameter, and Heis a perturbation of\nan applied magnetic \feld. To ease the discussion, the external \feld is neglected in\nthe subsequent analysis and is only considered in micromagnetics simulations. An\nadditional initial condition @tm(x;0) = 0 is added, which implies that the velocity\nis 0 att= 0 and it is a necessary condition for the well-posedness. Then the energy\nis de\fned as\n(2.3)E[m] =1\n2Z\n\n\u0010\njrmj2\u00002m\u0001He+\u000b\u001cj@tmj2\u0011\ndx:\nFor constant external magnetic \felds, it satis\fes the energy dissipation law\n(2.4)d\ndtE[m] =\u0000\u000bZ\n\nj@tmj2dx\u00140:\nTherefore, under the condition of (2.2c), for almost all T02(0;T), we have\n(2.5)1\n2Z\n\n\u0010\njrm(x;T0)j2+\u000b\u001cj@tm(x;T0)j2\u0011\ndx\u00141\n2Z\n\n\u0000\njrm(x;0)j2\u0001\ndx:\nBefore the formal algorithm is presented, here the spatial di\u000berence mesh and\nthe temporal discretization have to be stated. The uniform mesh for \n is con-\nstructed with mesh-size hand a time step-size k > 0 is set. Let Lbe the set of\nnodesfxl= (xi;yj;zk)gin 3-D space with the indices i= 0;1;\u0001\u0001\u0001;nx;nx + 1;j=\n0;1;\u0001\u0001\u0001;ny;ny +1 andk= 0;1;\u0001\u0001\u0001;nz;nz +1, and the ghost points on the bound-\nary of \n are denoted by ix= 0;nx+ 1,jy= 0;ny+ 1 andkz= 0;nz+ 1. We\nuse the half grid points with mi;j;k=m((i\u00001\n2)hx;(j\u00001\n2)hy;(k\u00001\n2)hz). Here\nhx= 1=nx,hy= 1=ny,hz= 1=nzandh=hx=hy=hzholds for uniform4 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nspatial meshes. Due to the homogeneous Neumann boundary condition (2.2d), the\nfollowing extrapolation formula is derived:\n(2.6)mix+1;j;k=mix;j;k;mi;jy+1;k=mi;jy;k;mi;j;k z+1=mi;j;k z;\nfor any 1\u0014i\u0014nx;1\u0014j\u0014ny;1\u0014k\u0014nz. Meanwhile, the temporal derivatives\nare discretized by the central di\u000berence, with the details stated in the following\nde\fnition.\nDe\fnition 2.1. For\u001en+1=\u001e(x;tn+1)and n+1= (tn+1), de\fne\nd+\nt\u001en=\u001en+1\u0000\u001en\nk; d\u0000\nt\u001en=\u001en\u0000\u001en\u00001\nk;\nand\nD+\nt n= n+1\u0000 n\nk; D\u0000\nt n= n\u0000 n\u00001\nk:\nConsequently, we denote\ndt\u001en+1=1\n2(d+\nt\u001en+d\u0000\nt\u001en); Dt n+1=1\n2(D+\nt n+D\u0000\nt n):\nIn particular, the second time derivative is approximated by the central di\u000berence\nform\n(2.7) dtt\u001e=\u001en+1\u00002\u001en+\u001en\u00001\nk2:\nThen for the initial condition (2.2c), there holds\n(2.8) m(xl;0) =m(xl;k);8l2L;\nwhereL=f(i;j;k )ji= 1;\u0001\u0001\u0001;nx;j= 1;\u0001\u0001\u0001;ny;k= 1;\u0001\u0001\u0001;nz:g. Denotemn\nh(n\u0015\n0) as the numerical solution. Given grid functions fh;gh2`2(\nh;R3), we list\nde\fnitions of the discrete inner product and norms used in this paper.\nDe\fnition 2.2. The discrete inner product h\u0001;\u0001iin`2(\nh;R3)is de\fned by\n(2.9) hfh;ghi=hdX\nl2Lfh(xl)\u0001gh(xl):\nThe discrete `2norm andH1\nhnorm ofmhare\n(2.10) kfhk2\n2=hdX\nl2Lfh(xl)\u0001fh(xl);\nand\n(2.11) kfhk2\nH1\nh=kfhk2\n2+krhfhk2\n2\nwithrhrepresenting the central di\u000berence stencil of the gradient operator.\nBesides, the norm k\u0001k1in`1(\nh;R3) is de\fned by\n(2.12) kfhk1= max\nl2Lkfh(xl)k1:\nTherefore, the approximation scheme of the iLLG equation is presented below.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 5\nAlgorithm 2.1. Givenm0\nh;m1\nh2W1;2(\nh;S2). Letmn\u00001\nh;mn\nh2W1;2(\nh;S2),\nwe computemn+1\nhby\n(2.13) dtmn+1\nh\u0000\u000b\u0016mn\nh\u0002\u0000\ndtmn+1\nh+\u001cdttmn\nh\u0001\n=\u0000\u0016mn\nh\u0002\u0001h\u0016mn\nh;\nwhere \u0016mn\nh=1\n2(mn+1\nh+mn\u00001\nh), and \u0001hrepresents the standard seven-point stencil\nof the Laplacian operator.\nThe corresponding fully discrete version of the above (2.13) reads as\nmn+1\nh\u0000mn\u00001\nh\n2k\u0000\u000bmn+1\nh+mn\u00001\nh\n2\u0002\u0012mn+1\nh\u0000mn\u00001\nh\n2k+\u001cmn+1\nh\u00002mn\nh+mn\u00001\nh\nk2\u0013\n=\u0000mn+1\nh+mn\u00001\nh\n2\u0002\u0001h\u0012mn+1\nh+mn\u00001\nh\n2\u0013\n: (2.14)\nWithin three time steps, there have not been many direct discretization methods\nto get the second-order temporal accuracy. Due to the mid-point approximation\nfeature, this implicit scheme is excellent in maintaining certain properties of the\noriginal system.\nLemma 2.1. Given\f\fm0\nh(xl)\f\f= 1, then the sequence fmn\nh(xl)gn\u00150produced by\n(2.13) satis\fes\n(i)jmn\nh(xl)j= 1;8l2L;\n(ii)1\n2Dtkrhmn+1\nhk2\n2+\u000bkdtmn+1\nhk2\n2+1\n2\u000b\u001cD\u0000\ntkd+\ntmn\nhk2\n2= 0.\nProof. On account of the initial condition (2.2c), we see that m0(xl) =m1(xl)\nholding for all l2L. Taking the vector inner product with (2.13) by ( mn+1\nh(xl) +\nmn\u00001\nh(xl)), it obvious that we can get\njmn+1\nhj=jmn\nhj=\u0001\u0001\u0001=jm1\nhj=jm0\nhj= 1;\nin the point-wise sense. This con\frms (i). In order to verify (ii), we take inner\nproduct with (2.13) by \u0000\u0001h\u0016mn\nhand get\n1\n2Dtkrhmn+1\nhk2\n2\u0000\u000bh\u0016mn\nh\u0002dtmn+1\nh;\u0000\u0001h\u0016mn\nhi\u0000\u000b\u001chmn\nh\u0002dttmn\nh;\u0000\u0001h\u0016mn\nhi= 0:\nSubsequently, taking inner products with dtmn+1\nhanddttmn+1\nhseparately leads to\nthe following equalities:\nkdtmn+1\nhk2\n2\u0000\u000b\u001chmn\nh\u0002dttmn\nh;dtmn+1\nhi=\u0000h\u0016mn\nh\u0002dtmn+1\nh;\u0000\u0001h\u0016mn\nhi;\nand\n1\n2D\u0000\ntkd+\ntmn\nhk2\n2+\u000bhmn\nh\u0002dttmn\nh;dtmn+1\nhi=\u0000hmn\nh\u0002dttmn\nh;\u0000\u0001h\u0016mn\nhi:\nA combination of the above three identities yields (ii). \u0003\nIn lemma 2.1, taking k!0 gives\n(2.15)d\ndt\u00121\n2krhmn+1\nhk2\n2+\u000b\u001c\n2k@tmn\nhk2\n2\u0013\n=\u0000\u000bk@tmn+1\nhk2\n2;\nwhich is consistent with the continuous energy law (2.4). Accordingly, in the ab-\nsence of the external magnetic \feld, the discretized version energy dissipation law\nwould be maintained with a modi\fcation\n(2.16)E(mn+1\nh;mn\nh) =\u000b\u001c\n2\r\r\rmn+1\nh\u0000mn\nh\nk\r\r\r2\n2+1\n4(krhmn+1\nhk2\n2+krhmn\nhk2\n2):6 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nTheorem 2.1. Givenmn\u00001\nh;mn\nh;mn+1\nh2W1;2(\nh;S2), we have a discrete energy\ndissipation law, for the modi\fed energy (2.16) :\n(2.17) E(mn+1\nh;mn\nh)\u0014E(mn\nh;mn\u00001\nh):\nProof. Denote a discrete function\n\u0016n:=\u000b\u0010mn+1\nh\u0000mn\u00001\nh\n2k+\u001cmn+1\nh\u00002mn\nh+mn\u00001\nh\nk2\u0011\n\u00001\n2\u0001h(mn+1\nh+mn\u00001\nh):\nTaking a discrete inner product with (2.13) by \u0016ngives\n\u000b\n4k2hmn+1\nh\u0000mn\u00001\nh;mn+1\nh\u0000mn\u00001\nhi+\u000b\u001c\n2k3hmn+1\nh\u0000mn\u00001\nh;mn+1\nh\u00002mn\nh+mn\u00001\nhi(2.18)\n+\u000b\n4kD\nmn+1\nh\u0000mn\u00001\nh;\u0000\u0001h(mn+1\nh+mn\u00001\nh)E\n=D\n\u0000mn+1\nh+mn\u00001\nh\n2\u0002\u0016n;\u0016nE\n= 0:\nMeanwhile, the following estimates are available:\nhmn+1\nh\u0000mn\u00001\nh;mn+1\nh\u0000mn\u00001\nhi=kmn+1\nh\u0000mn\u00001\nhk2\n2\u00150; (2.19)\nhmn+1\nh\u0000mn\u00001\nh;mn+1\nh\u00002mn\nh+mn\u00001\nhi\n=D\n(mn+1\nh\u0000mn\nh) + (mn\nh\u0000mn\u00001\nh);(mn+1\nh\u0000mn\nh)\u0000(mn\nh\u0000mn\u00001\nh)E\n;\n=kmn+1\nh\u0000mn\nhk2\n2\u0000kmn\nh\u0000mn\u00001\nhk2\n2;(2.20)\nD\nmn+1\nh\u0000mn\u00001\nh;\u0000\u0001h(mn+1\nh+mn\u00001\nh)E\n=D\nrh(mn+1\nh\u0000mn\u00001\nh);rh(mn+1\nh+mn\u00001\nh)E\n=krhmn+1\nhk2\n2\u0000krhmn\u00001\nhk2\n2\n=(krhmn+1\nhk2\n2+krhmn\nhk2\n2)\u0000(krhmn\nhk2\n2+krhmn\u00001\nhk2\n2):(2.21)\nGoing back to (2.18), we arrive at\n\u000b\u001c\n2k\u0010\r\r\rmn+1\nh\u0000mn\nh\nk\r\r\r2\n2\u0000\r\r\rmn\nh\u0000mn\u00001\nh\nk\r\r\r2\n2\u0011\n(2.22)\n+1\n4k\u0010\n(krhmn+1\nhk2\n2+krhmn\nhk2\n2)\u0000(krhmn\nhk2\n2+krhmn\u00001\nhk2\n2)\u0011\n=\u0000\u000b\r\r\rmn+1\nh\u0000mn\u00001\nh\n2k\r\r\r2\n2\u00140;\nwhich is exactly the energy dissipation estimate (2.17). This \fnishes the proof of\nTheorem 2.1. \u0003\nMeanwhile, it is noticed that, given the initial pro\fle of matt= 0, namelym0,\nan accurate approximation to m1andm2has to be made. In more details, an\nO(k2+h2) accuracy is required for both m1,m2andm1\u0000m0\nk,m2\u0000m1\nk, which is\nneeded in the convergence analysis.\nThe initial pro\fle m0could be taken as m0=m(\u0001;0). This in turn gives a\ntrivial zero initial error for m0. Form1andm2, a careful Taylor expansion revealsCONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 7\nthat\nm1=m0+k@tm0+k2\n2@ttm0+O(k3)\n=m0+k2\n2@ttm0+O(k3); (2.23)\nm2=m0+ 2k@tm0+ 2k2@ttm0+O(k3)\n=m0+ 2k2@ttm0+O(k3); (2.24)\nin which the initial data (2.2c), @tm(\u0001;0)\u00110, has been applied in the derivation.\nTherefore, an accurate approximation to m1andm2relies on a precise value of\n@ttmatt= 0. An evaluation of the original PDE (2.2a) implies that\nm0\u0002(@ttm0) =1\n\u000b\u001cm0\u0002(\u0001m0+H0\ne); (2.25)\nin which the trivial initial data (2.2c) has been applied again. Meanwhile, motivated\nby the point-wise temporal di\u000berentiation identity\n(2.26) m\u0001@ttm=\u0000(@tm)2+1\n2@tt(jmj2) =\u0000(@tm)2;\nand the fact that jmj\u00111, we see that its evaluation at t= 0 yields\n(2.27) m0\u0001@ttm0=\u0000(@tm0)2= 0:\nSubsequently, a combination of (2.26) and (2.27) uniquely determines @ttm0:\n(2.28) @ttm0=\u00001\n\u000b\u001cm0\u0002(m0\u0002(\u0001m0+H0\ne));\nand a substitution of this value into (2.23), (2.24) leads to an O(k3) approximation\ntom1andm2.\nMoreover, with spatial approximation introduced, an O(k2+h2) accuracy is\nobtained for both m1,m2andm1\u0000m0\nk,m2\u0000m1\nk. This \fnishes the initialization\nprocess.\n3.Convergence analysis\nThe theoretical result concerning the convergence analysis is stated below.\nTheorem 3.1. Assume that the exact solution of (2.2) has the regularity me2\nC3([0;T]; [C0(\u0016\n)]3)\\C2([0;T]; [C2(\u0016\n)]3)\\L1([0;T]; [C4(\u0016\n)]3). Denote a nodal\ninterpolation operator Phsuch thatPhmh2C1(\n), and the numerical solution mn\nh\n(n\u00150) obtained from (2.13) with the initial error satisfying kep\nhk2+krhep\nhk2=\nO(k2+h2), whereep\nh=Phme(\u0001;tp)\u0000mp\nh,p= 0;1;2, andkeq+1\nh\u0000eq\nh\nkk2=O(k2+h2),\nq= 0;1. Then the following convergence result holds for 2\u0014n\u0014\u0004T\nk\u0005\nash;k!0+:\nkPhme(\u0001;tn)\u0000mn\nhk2+krh(Phme(\u0001;tn)\u0000mn\nh)k2\u0014C(k2+h2); (3.1)\nin which the constant C>0is independent of kandh.\nBefore the rigorous proof is given, the following estimates are declared, which\nwill be utilized in the convergence analysis. In the sequel, for simplicity of notation,\nwe will use a uniform constant Cto denote all the controllable constants throughout\nthis part.8 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nLemma 3.1 (Discrete gradient acting on cross product) .[7]For grid functions fh\nandghover the uniform numerical grid, we have\nkrh(fh\u0002gh)k2\u0014C\u0010\nkfhk2\u0001krhghk1+kghk1\u0001krhfhk2\u0011\n: (3.2)\nLemma 3.2 (Point-wise product involved with second order temporal stencil) .For\ngrid functions fhandghover the time domain, we have\nfn+1\nh\u00002fn\nh+fn\u00001\nh\nk2\u0001gn\nh=\u0000fn\nh\u0000fn\u00001\nh\nk\u0001gn\nh\u0000gn\u00001\nh\nk\n+1\nk\u0010fn+1\nh\u0000fn\nh\nk\u0001gn\nh\u0000fn\nh\u0000fn\u00001\nh\nk\u0001gn\u00001\nh\u0011\n: (3.3)\nNow we proceed into the convergence estimate. First, we construct an approxi-\nmate solution m:\n(3.4) m=me+h2m(1);\nin which the auxiliary \feld m(1)satis\fes the following Poisson equation\n\u0001m(1)=^Cwith ^C=1\nj\njZ\n@\n@3\n\u0017meds; (3.5)\n@zm(1)jz=0=\u00001\n24@3\nzmejz=0; @zm(1)jz=1=1\n24@3\nzmejz=1;\nwith boundary conditions along xandydirections de\fned in a similar way.\nThe purpose of such a construction will be illustrated later. Then we extend the\napproximate pro\fle mto the numerical \\ghost\" points, according to the extrapo-\nlation formula:\n(3.6) mi;j;0=mi;j;1;mi;j;nz +1=mi;j;nz;\nand the extrapolation for other boundaries can be formulated in the same man-\nner. Subsequently, we prove that such an extrapolation yields a higher order\nO(h5) approximation, instead of the standard O(h3) accuracy. Also see the re-\nlated works [24, 25, 26] in the existing literature.\nPerforming a careful Taylor expansion for the exact solution around the boundary\nsectionz= 0, combined with the mesh point values: z0=\u00001\n2h,z1=1\n2h, we get\nme(xi;yj;z0) =me(xi;yj;z1)\u0000h@zme(xi;yj;0)\u0000h3\n24@3\nzme(xi;yj;0) +O(h5)\n=me(xi;yj;z1)\u0000h3\n24@3\nzme(xi;yj;0) +O(h5); (3.7)\nin which the homogenous boundary condition has been applied in the second step.\nA similar Taylor expansion for the constructed pro\fle m(1)reveals that\nm(1)(xi;yj;z0) =m(1)(xi;yj;z1)\u0000h@zm(1)(xi;yj;0) +O(h3)\n=m(1)(xi;yj;z1) +h\n24@3\nzme(xi;yj;0) +O(h3); (3.8)\nwith the boundary condition in (3.5) applied. In turn, a substitution of (3.7)-(3.8)\ninto (3.4) indicates that\n(3.9) m(xi;yj;z0) =m(xi;yj;z1) +O(h5):\nIn other words, the extrapolation formula (3.6) is indeed O(h5) accurate.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 9\nAs a result of the boundary extrapolation estimate (3.9), we see that the discrete\nLaplacian of myields the second-order accuracy at all the mesh points (including\nboundary points):\n(3.10)\n\u0001hmi;j;k= \u0001me(xi;yj;zk)+O(h2);80\u0014i\u0014nx+1;0\u0014j\u0014ny+1;0\u0014k\u0014nz+1:\nMoreover, a detailed calculation of Taylor expansion, in both time and space, leads\nto the following truncation error estimate:\nmn+1\nh\u0000mn\u00001\nh\n2k=mn+1\nh+mn\u00001\nh\n2\u0002\u0010\n\u000bmn+1\nh\u0000mn\u00001\nh\n2k+\u000b\u001cmn+1\nh\u00002mn\nh+mn\u00001\nh\nk2\n\u0000\u0001h\u0010mn+1\nh+mn\u00001\nh\n2\u0011\u0011\n+\u001cn; (3.11)\nwherek\u001cnk2\u0014C(k2+h2). In addition, a higher order Taylor expansion in space\nand time reveals the following estimate for the discrete gradient of the truncation\nerror, in both time and space:\n(3.12) krh\u001cnk2;k\u001cn\u0000\u001cn\u00001\nkk2\u0014C(k2+h2):\nIn fact, such a discrete k\u0001kH1\nhbound for the truncation comes from the regularity as-\nsumption for the exact solution, me2C3([0;T]; [C0(\u0016\n)]3)\\C2([0;T]; [C2(\u0016\n)]3)\\\nL1([0;T]; [C4(\u0016\n)]3), as stated in Theorem 3.1, as well as the fact that m(1)2\nC1([0;T]; [C1(\u0016\n)]3)\\L1([0;T]; [C2(\u0016\n)]3), as indicated by the Poisson equation (3.5).\nWe introduce the numerical error function en\nh=mn\nh\u0000mn\nh, instead of a di-\nrect comparison between the numerical solution and the exact solution. The error\nfunction between the numerical solution and the constructed solution mhwill be\nanalyzed, due to its higher order consistency estimate (3.9) around the boundary.\nTherefore, a subtraction of (2.14) from the consistency estimate (3.11) leads to the\nerror function evolution system:\nen+1\nh\u0000en\u00001\nh\n2k=mn+1\nh+mn\u00001\nh\n2\u0002~\u0016n\nh+en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh+\u001cn;(3.13)\n\u0016n\nh:=\u000b\u0010mn+1\nh\u0000mn\u00001\nh\n2k+\u001cmn+1\nh\u00002mn\nh+mn\u00001\nh\nk2\u0011\n\u0000\u0001h\u0010mn+1\nh+mn\u00001\nh\n2\u0011\n;(3.14)\n~\u0016n\nh:=\u000b\u0010en+1\nh\u0000en\u00001\nh\n2k+\u001cen+1\nh\u00002en\nh+en\u00001\nh\nk2\u0011\n\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\n:(3.15)\nBefore proceeding into the formal estimate, we establish a W1\nhbound for\u0016n\nh,\nwhich is based on the constructed approximate solution m(by (3.14)). Because of\nthe regularity for me, the following bound is available:\nk\u0016`\nhk1;krh\u0016`\nhk1;k\u0016n\nh\u0000\u0016n\u00001\nh\nkk1\u0014C; ` =n;n\u00001: (3.16)\nIn addition, the following preliminary estimate will be useful in the convergence\nanalysis.10 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nLemma 3.3 (A preliminary error estimate) .We have\nke`\nhk2\n2\u00142ke0\nhk2\n2+ 2Tk`\u00001X\nj=0kej+1\nh\u0000ej\nh\nkk2\n2;8`\u0001k\u0014T: (3.17)\nProof. We begin with the expansion:\ne`\nh=e0\nh+k`\u00001X\nj=0ej+1\nh\u0000ej\nh\nk;8`\u0001k\u0014T: (3.18)\nIn turn, a careful application of the Cauchy inequality reveals that\nke`\nhk2\n2\u00142\u0010\nke0\nhk2\n2+k2k`\u00001X\nj=0ej+1\nh\u0000ej\nh\nkk2\n2\u0011\n; (3.19)\nk2k`\u00001X\nj=0ej+1\nh\u0000ej\nh\nkk2\n2\u0014k2\u0001`\u0001`\u00001X\nj=0kej+1\nh\u0000ej\nh\nkk2\n2\u0014Tk`\u00001X\nj=0kej+1\nh\u0000ej\nh\nkk2\n2; (3.20)\nin which the fact that `\u0001k\u0014Thas been applied. Therefore, a combination of\n(3.19) and (3.20) yields the desired estimate (3.17). This completes the proof of\nLemma 3.3. \u0003\nTaking a discrete inner product with the numerical error equation (3.13) by ~\u0016n\nh\ngives\n1\n2khen+1\nh\u0000en\u00001\nh;~\u0016n\nhi=hmn+1\nh+mn\u00001\nh\n2\u0002~\u0016n\nh;~\u0016n\nhi\n+hen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;~\u0016n\nhi+h\u001cn;~\u0016n\nhi: (3.21)\nThe analysis on the left hand side of (3.21) is similar to the ones in (2.19)-(2.21):\n1\n2khen+1\nh\u0000en\u00001\nh;~\u0016n\nhi=\u000b\u001c\n2k3hen+1\nh\u0000en\u00001\nh;en+1\nh\u00002en\nh+en\u00001\nhi\n+\u000b\n4k2hen+1\nh\u0000en\u00001\nh;en+1\nh\u0000en\u00001\nhi\n+1\n4kD\nrh(en+1\nh\u0000en\u00001\nh);rh(en+1\nh+en\u00001\nh)E\n; (3.22)\nhen+1\nh\u0000en\u00001\nh;en+1\nh\u0000en\u00001\nhi=ken+1\u0000en\u00001\nhk2\n2\u00150; (3.23)\nhen+1\nh\u0000en\u00001\nh;en+1\nh\u00002en\nh+en\u00001\nhi\n=ken+1\nh\u0000en\nhk2\n2\u0000ken\nh\u0000en\u00001\nhk2\n2; (3.24)\nD\nen+1\nh\u0000en\u00001\nh;\u0000\u0001h(en+1\nh+en\u00001\nh)E\n=D\nrh(en+1\nh\u0000en\u00001\nh);rh(en+1\nh+en\u00001\nh)E\n=krhen+1\nhk2\n2\u0000krhen\u00001\nhk2\n2\n=(krhen+1\nhk2\n2+krhen\nhk2\n2)\u0000(krhen\nhk2\n2+krhen\u00001\nhk2\n2): (3.25)\nThis in turn leads to the following identity:\n1\n2khen+1\nh\u0000en\u00001\nh;~\u0016n\nhi=1\nk(En+1e;h\u0000Ene;h) +\u000b\n4k2ken+1\nh\u0000en\u00001\nhk2\n2; (3.26)\nEn+1e;h=\u000b\u001c\n2ken+1\nh\u0000en\nh\nkk2\n2+1\n4(krhen+1\nhk2\n2+krhen\nhk2\n2): (3.27)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 11\nThe \frst term on the right hand side of (3.21) vanishes, due to the fact that\nmn+1\nh+mn\u00001\nh\n2\u0002~\u0016n\nhis orthogonal to ~\u0016n\nh, at a point-wise level:\n(3.28) hmn+1\nh+mn\u00001\nh\n2\u0002~\u0016n\nh;~\u0016n\nhi= 0:\nThe second term on the right hand side of (3.21) contains three parts:\nhen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;~\u0016n\nhi=I1+I2+I3; (3.29)\nI1=\u000bhen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;en+1\nh\u0000en\u00001\nh\n2ki; (3.30)\nI2=\u000b\u001chen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;en+1\nh\u00002en\nh+en\u00001\nh\nk2i; (3.31)\nI3=hen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\ni: (3.32)\nThe \frst inner product, I1, could be bounded in a straightforward way, with the\nhelp of discrete H older inequality:\nI1=\u000bhen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;en+1\nh\u0000en\u00001\nh\n2ki\n\u0014\u000b\n4ken+1\nh+en\u00001\nhk2\u0001k\u0016n\nhk1\u0001ken+1\nh\u0000en\u00001\nh\nkk2\n\u0014Cken+1\nh+en\u00001\nhk2\u0001ken+1\nh\u0000en\u00001\nh\nkk2\n\u0014C(ken+1\nhk2\n2+ken\u00001\nhk2\n2+ken+1\nh\u0000en\u00001\nh\nkk2\n2): (3.33)\nFor the second inner product, I2, we denotegn\nh:=en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh. An application\nof point-wise identity (3.3) (in lemma 3.2) reveals that\nI2=\u000b\u001chgn\nh;en+1\nh\u00002en\nh+en\u00001\nh\nk2i\n=\u0000\u000b\u001chen\nh\u0000en\u00001\nh\nk;gn\nh\u0000gn\u00001\nh\nki\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;gn\nhi\u0000hen\nh\u0000en\u00001\nh\nk;gn\u00001\nhi\u0011\n: (3.34)\nMeanwhile, the following expansion is observed:\ngn\nh\u0000gn\u00001\nh\nk=1\n4(en+1\nh\u0000en\nh\nk+en\u00001\nh\u0000en\u00002\nh\nk)\u0002(\u0016n\nh+\u0016n\u00001\nh)\n+en+1\nh+en\nh+en\u00001\nh+en\u00002\nh\n4\u0002\u0016n\nh\u0000\u0016n\u00001\nh\nk: (3.35)12 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nThis in turn indicates the associated estimate:\nkgn\nh\u0000gn\u00001\nh\nkk2\u00141\n4(ken+1\nh\u0000en\nh\nkk2+ken\u00001\nh\u0000en\u00002\nh\nkk2)\u0001(k\u0016n\nhk1+k\u0016n\u00001\nhk1)\n+ken+1\nhk2+ken\nhk2+ken\u00001\nhk2+ken\u00002\nhk2\n4\u0001k\u0016n\nh\u0000\u0016n\u00001\nh\nkk1\n\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n+ken+1\nhk2+ken\nhk2+ken\u00001\nhk2+ken\u00002\nhk2\u0011\n; (3.36)\nin which the bound (3.16) has been applied. Going back to (3.34), we see that\n\u0000\u000b\u001chen\nh\u0000en\u00001\nh\nk;gn\nh\u0000gn\u00001\nh\nki\u0014\u000b\u001cken\nh\u0000en\u00001\nh\nkk2\u0001kgn\nh\u0000gn\u00001\nh\nkk2\n\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2+ken\u00001\nh\u0000en\u00002\nh\nkk2+ken+1\nhk2\n+ken\nhk2+ken\u00001\nhk2+ken\u00002\nhk2\u0011\nken\nh\u0000en\u00001\nh\nkk2\n\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2\n2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n2+ken+1\nhk2\n2\n+ken\nhk2\n2+ken\u00001\nhk2\n2+ken\u00002\nhk2\n2+ken\nh\u0000en\u00001\nh\nkk2\n2\u0011\n; (3.37)\nI2\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2\n2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n2+ken+1\nhk2\n2\n+ken\nhk2\n2+ken\u00001\nhk2\n2+ken\u00002\nhk2\n2+ken\nh\u0000en\u00001\nh\nkk2\n2\u0011\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;gn\nhi\u0000hen\nh\u0000en\u00001\nh\nk;gn\u00001\nhi\u0011\n: (3.38)\nFor the third inner product part, I3, an application of summation by parts formula\ngives\nI3=hen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\ni\n=hrh\u0010en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh\u0011\n;rh\u0010en+1\nh+en\u00001\nh\n2\u0011\ni: (3.39)\nMeanwhile, we make use of the preliminary inequality (3.2) (in lemma 3.1) and get\nkrh\u0010en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh\u0011\nk2\n\u0014C\u0010\nken+1\nh+en\u00001\nh\n2k2\u0001krh\u0016n\nhk1+k\u0016n\nhk1\u0001krh(en+1\nh+en\u00001\nh\n2)k2\u0011\n\u0014C\u0010\nken+1\nh+en\u00001\nh\n2k2+krh(en+1\nh+en\u00001\nh\n2)k2\u0011\n\u0014C\u0010\nken+1\nhk2+ken\u00001\nhk2+krhen+1\nhk2+krhen\u00001\nhk2\u0011\n: (3.40)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 13\nAgain, the bound (3.16) has been applied in the derivation. Therefore, the following\nestimate is available for I3:\nI3\u0014krh\u0010en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh\u0011\nk2\u0001krh\u0010en+1\nh+en\u00001\nh\n2\u0011\nk2\n\u0014C\u0010\nken+1\nhk2+ken\u00001\nhk2+krhen+1\nhk2+krhen\u00001\nhk2\u0011\n\u0001\u0010\nkrhen+1\nhk2+krhen\u00001\nhk2\u0011\n\u0014C\u0010\nken+1\nhk2\n2+ken\u00001\nhk2\n2+krhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n: (3.41)\nThe estimate of I3can also be obtained by a direct application of discrete H older\ninequality:\nI3=h\u0010en+1\nh+en\u00001\nh\n2\u0002rh\u0016n\nh\u0011\n;rh\u0010en+1\nh+en\u00001\nh\n2\u0011\ni\n\u00141\n4ken+1\nh+en\u00001\nhk2\u0001krh\u0016n\nhk1\u0001krh(en+1\nh+en\u00001\nh)k2\n\u0014C\u0010\nken+1\nhk2\n2+ken\u00001\nhk2\n2+krhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n: (3.42)\nA substitution of (3.33), (3.38) and (3.42) into (3.29) yields the following bound:\nhen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;~\u0016n\nhi=I1+I2+I3\n\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2\n2+ken\nh\u0000en\u00001\nh\nkk2\n2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n2\n+ken+1\nhk2\n2+ken\nhk2\n2+ken\u00001\nhk2\n2+ken\u00002\nhk2\n2+krhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;gn\nhi\u0000hen\nh\u0000en\u00001\nh\nk;gn\u00001\nhi\u0011\n: (3.43)\nThe third term on the right hand side of (3.21) could be analyzed in a similar\nfashion:\nh\u001cn;~\u0016n\nhi=I4+I5+I6; (3.44)\nI4=\u000bh\u001cn;en+1\nh\u0000en\u00001\nh\n2ki; I 5=\u000b\u001ch\u001cn;en+1\nh\u00002en\nh+en\u00001\nh\nk2i; (3.45)\nI6=h\u001cn;\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\ni; (3.46)\nI4=\u000bh\u001cn;en+1\nh\u0000en\u00001\nh\n2ki\u0014\u000b\n2k\u001cnk2\u0001ken+1\nh\u0000en\u00001\nh\nkk2\n\u0014\u000b\n4(k\u001cnk2\n2+ken+1\nh\u0000en\u00001\nh\nkk2\n2); (3.47)\nI5=\u000b\u001ch\u001cn;en+1\nh\u00002en\nh+en\u00001\nh\nk2i\n=\u0000\u000b\u001chen\nh\u0000en\u00001\nh\nk;\u001cn\u0000\u001cn\u00001\nki\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;\u001cni\u0000hen\nh\u0000en\u00001\nh\nk;\u001cn\u00001i\u0011\n; (3.48)14 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\n\u0000hen\nh\u0000en\u00001\nh\nk;\u001cn\u0000\u001cn\u00001\nki\u0014ken\nh\u0000en\u00001\nh\nkk2\u0001k\u001cn\u0000\u001cn\u00001\nkk2\n\u0014C(k2+h2)ken\nh\u0000en\u00001\nh\nkk2\u0014C(k4+h4) +1\n2ken\nh\u0000en\u00001\nh\nkk2\n2; (3.49)\nI5\u0014C(k4+h4) +\u000b\u001c\n2ken\nh\u0000en\u00001\nh\nkk2\n2\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;\u001cni\u0000hen\nh\u0000en\u00001\nh\nk;\u001cn\u00001i\u0011\n; (3.50)\nI6=h\u001cn;\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\ni=hrh\u001cn;rh\u0010en+1\nh+en\u00001\nh\n2\u0011\ni\n\u0014krh\u001cnk2\u0001krh\u0010en+1\nh+en\u00001\nh\n2\u0011\nk2\u0014C(k2+h2)krh\u0010en+1\nh+en\u00001\nh\n2\u0011\nk2\n\u0014C(k4+h4) +1\n2\u0010\nkrhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n: (3.51)\nNotice that the truncation error estimate (3.12) has been repeatedly applied in the\nderivation. Going back to (3.44), we obtain\nh\u001cn;~\u0016n\nhi=I4+I5+I6\n\u0014C(k4+h4) +\u000b\n2ken+1\nh\u0000en\nh\nkk2\n2+\u000b(\u001c+ 1)\n2ken\nh\u0000en\u00001\nh\nkk2\n2\n+1\n2\u0010\nkrhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;\u001cni\u0000hen\nh\u0000en\u00001\nh\nk;\u001cn\u00001i\u0011\n: (3.52)\nFinally, a substitution of (3.26)-(3.27), (3.28), (3.43) and (3.52) into (3.21) leads\nto the following inequality:\n1\nk(En+1e;h\u0000Ene;h) +\u000b\n4k2ken+1\nh\u0000en\u00001\nhk2\n2\n\u0014C(k4+h4) +C\u0010\nken+1\nh\u0000en\nh\nkk2\n2+ken\nh\u0000en\u00001\nh\nkk2\n2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n2\n+ken+1\nhk2\n2+ken\nhk2\n2+ken\u00001\nhk2\n2+ken\u00002\nhk2\n2+krhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;gn\nh+\u001cni\u0000hen\nh\u0000en\u00001\nh\nk;gn\u00001\nh+\u001cn\u00001i\u0011\n: (3.53)\nSubsequently, a summation in time yields\nEn+1e;h\u0014E2e;h+CT(k4+h4) +Ck\u0010nX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2+n+1X\nj=0(kej\nhk2\n2+krhej\nhk2\n2)\u0011\n+\u000b\u001c\u0010\nhen+1\nh\u0000en\nh\nk;gn\nh+\u001cni\u0000he2\nh\u0000e1\nh\nk;g1\nh+\u001c1i\u0011\n: (3.54)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 15\nFor the term \u000b\u001chen+1\nh\u0000en\nh\nk;gn\nh+\u001cni, the following estimate could be derived\n\u000b\u001chen+1\nh\u0000en\nh\nk;gn\nh+\u001cni\u0014\u000b\u001c\n4ken+1\nh\u0000en\nh\nkk2\n2+ 2\u000b\u001c(kgn\nhk2\n2+k\u001cnk2\n2); (3.55)\nkgn\nhk2=ken+1\nh+en\u00001\nh\n2\u0002\u0016n\nhk2\u0014ken+1\nh+en\u00001\nh\n2k2\u0001k\u0016n\nhk1\n\u0014Cken+1\nh+en\u00001\nh\n2k2\u0014C(ken+1\nhk2+ken\u00001\nhk2); (3.56)\nin which the bound (3.16) has been used again. Then we get\n\u000b\u001chen+1\nh\u0000en\nh\nk;gn\nh+\u001cni\u0014\u000b\u001c\n4ken+1\nh\u0000en\nh\nkk2\n2+ 2\u000b\u001ck\u001cnk2\n2\n+C(ken+1\nhk2\n2+ken\u00001\nhk2\n2)\n\u00141\n2En+1e;h+ 2\u000b\u001ck\u001cnk2\n2+C(ken+1\nhk2\n2+ken\u00001\nhk2\n2); (3.57)\nin which the expansion identity, En+1e;h=\u000b\u001c\n2ken+1\nh\u0000en\nh\nkk2\n2+1\n4(krhen+1\nhk2\n2+krhen\nhk2\n2)\n(given by (3.27)), has been applied. Its substitution into (3.54) gives\nEn+1e;h\u00142E2e;h+CT(k4+h4) +Ck\u0010nX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2+n+1X\nj=0(kej\nhk2\n2+krhej\nhk2\n2)\u0011\n+C(ken+1\nhk2\n2+ken\u00001\nhk2\n2) + 4\u000b\u001ck\u001cnk2\n2\u00002\u000b\u001che2\nh\u0000e1\nh\nk;g1\nh+\u001c1i: (3.58)\nMoreover, an application of the preliminary error estimate (3.17) (in Lemma 3.3)\nleads to\nEn+1e;h\u00142E2e;h+CT(k4+h4) +C(T2+ 1)knX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2+CTke0\nhk2\n2\n+Ckn+1X\nj=0krhej\nhk2\n2+ 4\u000b\u001ck\u001cnk2\n2\u00002\u000b\u001che2\nh\u0000e1\nh\nk;g1\nh+\u001c1i; (3.59)\nin which we have made use of the following fact:\nkn+1X\nj=0kej\nhk2\n2\u0014k\u0001(n+ 1)\u0010\n2ke0\nhk2\n2+ 2TknX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2\u0011\n\u00142Tke0\nhk2\n2+ 2T2knX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2: (3.60)16 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nIn addition, for the initial error quantities, the following estimates are available:\nE2e;h=\u000b\u001c\n2ke2\nh\u0000e1\nh\nkk2\n2+1\n4(krhe2\nhk2\n2+krhe1\nhk2\n2)\u0014C(k4+h4); (3.61)\nke0\nhk2\n2\u0014C(k4+h4); (3.62)\n4\u000b\u001ck\u001cnk2\n2\u0014C(k4+h4); (3.63)\nkg1\nhk2=ke2\nh+e0\nh\n2\u0002\u00161\nhk2\u0014ke2\nh+e0\nh\n2k2\u0001k\u00161\nhk1\u0014C(k2+h2); (3.64)\n\u00002\u000b\u001che2\nh\u0000e1\nh\nk;g1\nh+\u001c1i\u00142\u000b\u001cke2\nh\u0000e1\nh\nkk2\u0001(kg1\nhk2+k\u001c1k2)\n\u0014C(k4+h4); (3.65)\nwhich comes from the assumption in Theorem 3.1. Then we arrive at\nEn+1e;h\u0014C(T2+ 1)knX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2+Ckn+1X\nj=0krhej\nhk2\n2+C(T+ 1)(k4+h4)\n\u0014C(T+ 1)(k4+h4) +C(T2+ 1)knX\nj=0Ej+1e;h; (3.66)\nin which the fact that Ej+1e;h=\u000b\u001c\n2kej+1\nh\u0000ej\nh\nkk2\n2+1\n4(krhej+1\nhk2\n2+krhej\nhk2\n2), has been\nused. In turn, an application of discrete Gronwall inequality results in the desired\nconvergence estimate:\nEn+1e;h\u0014CTeCT(k4+h4);for all (n+ 1) :n+ 1\u0014\u0016T\nk\u0017\n; (3.67)\nken+1\nh\u0000en\nh\nkk2+krhen+1\nhk2\u0014C(k2+h2): (3.68)\nAgain, an application of the preliminary error estimate (3.17) (in Lemma 3.3) im-\nplies that\nken+1\nhk2\n2\u00142ke0\nhk2\n2+ 2TknX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2\u0014C(k4+h4);\nso thatken+1\nhk2\u0014C(k2+h2): (3.69)\nA combination of (3.68) and (3.69) \fnishes the proof of Theorem 3.1.\n4.A numerical solver for the nonlinear system\nIt is clear that Algorithm 2.1 is a nonlinear scheme. The following \fxed-point\niteration is employed to solve it.\nAlgorithm 4.1. Setmn+1;0\nh= 2mn\nh\u0000mn\u00001\nhandp= 0.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 17\n(i)Computemn+1;p+1\nhsuch that\n(4.1)mn+1;p+1\nh\u0000mn\u00001\nh\n2k=\u0000mn+1;p+1\nh+mn\u00001\nh\n2\u0002\u0001h \nmn+1;p\nh+mn\u00001\nh\n2!\n+\u000bmn+1;p+1\nh+mn\u00001\nh\n2\u0002 \nmn+1;p+1\nh\u0000mn\u00001\nh\n2k+\u001cmn+1;p+1\nh\u00002mn\nh+mn\u00001\nh\nk2!\n:\n(ii)Ifkmn+1;p+1\nh\u0000mn+1;p\nhk2\u0014\u000f, then stop and set mn+1\nh=mn+1;p+1\nh.\n(iii) Setp p+ 1and go to (i).\nDenote the operator\n(4.2)Lp=I\u0000\u000bmn\u00001\nh\u0002\u00002\u000b\u001c\nkmn\nh\u0002\u0000k\n2\u0001h(mn+1;p\nh+mn\u00001\nh)\u0002;\nand make the \fxed-point iteration solve the following equation\n(4.3)Lpmn+1;p+1\nh=mn\u00001\nh+2\u000b\u001c\nkmn\nh\u0002mn\u00001\nh\u0000k\n2mn\u00001\nh\u0002\u0001h(mn+1;p\nh+mn\u00001\nh);\nin its inner iteration. Under the condition k\u0014Ch2withCa constant, the following\nlemma con\frms the convergence of Algorithm 4.1. For any l2Land owing to the\nproperty ofjmh(xl)j= 1, it is clear that 0 <kmhk1\u00141. For the discretized `2\nnorm of ^m2C3([0;T]; [C0(\u0016\n)]3)\\C2([0;T]; [C2(\u0016\n)]3)\\L1([0;T]; [C4(\u0016\n)]3), we\nhave\n(4.4) krh^mk2\u00142h\u00001k^mk2:\nThen\n(4.5)k\u0001h^mk2\n2=\u0000hrh^m;rh\u0001h^mi\n\u0014krh^mk2krh\u0001h^mk2\u00142h\u00001krh^mk2k\u0001h^mk2;\nwhich in turn implies the following inverse inequality:\n(4.6) k\u0001h^mk2\u00144h\u00002k^mk2:\nLemma 4.1. Letjmn\u00001\nhj=jmn\nhj= 1, there exists a constant c0such thatkmn\u00001\nhk1,\nkmn\nhk1\u0014c0. The solution mn+1;p\nhcalculated by (4.1) satis\fesjmn+1;p\nhj=jmn\u00001\nhj\nforp= 1;2;\u0001\u0001\u0001, which means that we can still \fnd the constant c0\u00141satisfying\nkmn+1;p\nhk1\u0014c0. Then, for all p\u00151, there exists a unique solution mn+1;p\nhin\n(4.1) and the following inequality is valid:\n(4.7)kmn+1;p+1\nh\u0000mn+1;p\nhk2\u00144c0kh\u00002kmn+1;p\u0000mn+1;p\u00001k2:\nProof. For anymh2S2, the following identity is clear:\nhmh;Lpmhi= 1;\nfor allp\u00151. Thus the operator Lpis positive de\fnite for all p\u00151, which\nprovides the unique solvability of (4.1). Taking the discrete inner product with\n(4.1) bymn+1;p+1\nh+mn\u00001\nh, we havejmn+1;p+1\nhj= 1 in a point-wise sense, which\nmeans that the length of the magnetization is preserved at each step in the inner18 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\niteration. Thus, we can \fnd a constant c0\u00141 to control the `1norm ofmn\u00001\nh,\nmn\nhandmn+1;p\nhforp= 1;2;\u0001\u0001\u0001simultaneously.\nSubtraction of two subsequent equations in the \fxed-point iteration yields\n1\n2k(mn+1;p+1\nh\u0000mn+1;p\nh) =\u00001\n4(mn+1;p+1\nh\u0000mn+1;p\nh)\u0002\u0001hmn+1;p\nh\n\u00001\n4mn+1;p\nh\u0002\u0001h(mn+1;p\nh\u0000mn+1;p\u00001\nh)\n\u00001\n4mn\nh\u0002\u0001h(mn+1;p\u0000mn+1;p\u00001)\n\u00001\n4(mn+1;p+1\nh\u0000mn+1;p)\u0002\u0001hmn\u00001\nh\n+\u000b\n2kmn\u00001\nh\u0002(mn+1;p+1\nh\u0000mn+1;p\nh) +\u000b\u001c\n2k2mn\nh\u0002(mn+1;p+1\nh\u0000mn+1;p\nh):\nTaking the inner product with ( mn+1;p+1\nh\u0000mn+1;p\nh) by the above equation produces\nkmn+1;p+1\nh\u0000mn+1;p\nhk2\u0014k\n2kmn+1;p\nhk1k\u0001h(mn+1;p\nh\u0000mn+1;p\u00001\nh)k2\n+k\n2kmn\nhk1k\u0001h(mn+1;p\nh\u0000mn+1;p\u00001\nh)k2\n\u0014c0kk\u0001h(mn+1;p\u0000mn+1;p\u00001)k2:\nIn turn, the convergence result becomes\n(4.8)kmn+1;p+1\nh\u0000mn+1;p\nhk2\u00144c0kh\u00002kmn+1;p\nh\u0000mn+1;p\u00001\nhk2;\nwhich completes the proof of Lemma 4.1. \u0003\n5.Numerical experiments\n5.1.Accuracy tests. Consider the 1-D iLLG equation\n@tm=\u0000m\u0002@xxm+\u000bm\u0002(@tm+\u001c@ttm) +f:\nThe exact solution is chosen to be me= (cos(\u0016x) sin(t2);sin(\u0016x) sin(t2);cos(t2))T\nwith \u0016x=x2(1\u0000x)2, and the forcing term is given by f=@tme+me\u0002@xxme\u0000\n\u000bme\u0002(@tme+\u001c@ttme). Fixing the tolerance \u000f= 1.0e-07 for the \fxed-point\niteration, we record the discrete `2and`1errors between the exact solution and\nnumerical solution with a sequence of temporal step-size and spatial mesh-size. The\nparameters in the above 1-D equation are set as: \u000b= 0:1,\u001c= 10:0, and the \fnal\ntimeT= 0:01. The temporal step-sizes and spatial mesh-sizes are listed in the\nTable 1 and Table 2.\nTable 1. The discrete `2and`1errors in terms of the temporal\nstep-size. The spatial mesh-size is \fxed as h= 0:001 over \n =\n(0;1) and the \fnal time is T= 0:01.\nkkmh\u0000mek2kmh\u0000mek1\nT/40 1.2500e-11 1.2584e-11\nT/60 5.6887e-12 5.6024e-12\nT/80 3.2008e-12 3.1525e-12\nT/100 2.0487e-12 2.0174e-12\norder 1.97 2.00CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 19\nTable 2. The discrete `2and`1error in terms of the spatial\nmesh-size. The parameters are set as: the temporal step-size k=\n2.0e-06, \n = (0 ;1) and the \fnal time T= 0:5.\nhkmh\u0000mek2kmh\u0000mek1\n1/20 1.9742e-05 2.5320e-05\n1/40 4.9846e-06 6.3459e-06\n1/60 2.2340e-06 2.8201e-06\n1/80 1.2720e-06 1.5853e-06\norder 1.98 2.00\nIn addition, the 3-D iLLG equation is also considered:\n@tm=\u0000m\u0002\u0001m+\u000bm\u0002(@tm+\u001c@ttm) +f:\nThe exact solution is chosen to be me= (cos(\u0016x\u0016y\u0016z) sin(t2);sin(\u0016x\u0016y\u0016z) sin(t2));cos(t2))T\nwith \u0016y=y2(1\u0000y)2and \u0016z=z2(1\u0000z)2, and the forcing term f=@tme+me\u0002\n\u0001me\u0000\u000bme\u0002(@tme+\u001c@ttme). Similarly, we record the discrete `2and`1er-\nrors between exact and numerical solutions with a sequence of temporal step-sizes\nand spatial mesh-sizes. The corresponding parameters are set as: \u000b= 0:01 and\n\u001c= 1000:0. Besides, the \fnal time of this simulation is given by T= 0:01, with\nthe temporal step-size and spatial mesh-size listed in Table 3 and Table 4.\nTable 3. The discrete `2and`1errors in terms of the temporal\nstep-size. The spatial mesh-size is \fxed as h= 0:001 and \fnal time\nisT= 0:01.\nkkmh\u0000mek2kmh\u0000mek1\nT/100 1.2678e-05 1.2765e-05\nT/120 8.8067e-06 8.8830e-06\nT/140 6.4725e-06 6.5419e-06\nT/160 4.9576e-06 5.0224e-06\norder 2.00 1.98\nTable 4. The discrete `2and`1errors in terms of spatial mesh-\nsize. The temporal step-size is \fxed as k= 2.0e-06.\nhkmh\u0000mek2kmh\u0000mek1\n1/8 1.4392e-07 3.4940e-07\n1/10 9.6832e-08 2.2864e-07\n1/12 6.9825e-08 1.6079e-07\n1/14 5.2828e-08 1.1895e-07\norder 1.79 1.92\n5.2.Micromagnetics tests. The inertial e\u000bect can be observed during the relax-\nation of a system with a non-equilibrium initialization. To visualize this, we conduct\nmicromagnetics simulations for both the LLG equation and the iLLG equation.20 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nIn the following simulations, a 3-D domain \n = [0 ;1]\u0002[0;1]\u0002[0;0:4] is uniformly\ndiscretized into 10 \u000210\u00024 cells, with uniform initialization m0= (p\n2=2;p\n2=2;0)T.\nFor comparison, the LLG equation is discretized by the mid-point scheme proposed\nin [4] with the \fxed-point iteration solver proposed in this work. The damping pa-\nrameter is\u000b= 0:5 and the \feld is \fxed as He= (10;0;0)T, which indicates that the\nsystem shall converge to m= (1;0;0)T. Here the relaxation of the magnetization\nbehavior controlled by the LLG equation is visualized in Figure 1.\nTime0 1 2 3 4 5/angbracketleftm/angbracketright\n-0.500.51\n/angbracketleftm1/angbracketright\n/angbracketleftm2/angbracketright\n/angbracketleftm3/angbracketright\nFigure 1. The relaxation of the spatially averaged magnetization\ncontrolled by the LLG equation. The \fnal time is T= 5:0 with\nk= 0:001, and the damping parameter is \u000b= 0:5.\nAs for the counterpart of the LLG equation, wtih a given reference \feld He, the\ndiscrete energy of the iLLG equation becomes\n(5.1) E[mn+1;mn] =1\n4(krhmn+1\nhk2\n2+krhmn\nhk2\n2)+\n\u000b\u001c\n2\r\r\rmn+1\nh\u0000mn\nh\nk\r\r\r2\n2\u00001\n2hmn+1\nh+mn\nh;Hei:\nSetting the inertial parameter \u001c= 1:0, the spatially averaged magnetization is\nrecorded to depict the inertial e\u000bect in Figure 2(a). Meanwhile, the energy decay is\nalso numerically veri\fed by Figure 2(b). The inertial e\u000bect is observed at shorter\ntimescales for magnetization dynamics during the relaxation of the system with a\nnon-equilibrium initialization.\nFurthermore, the inertial e\u000bect can also be activated by an external perturbation\napplied to an equilibrium state. Here we set the damping parameter \u000b= 0:02 and\n\u001c= 0:5, then the time step-size must be reduced to 0.001 with T= 3:0. For the\nequilibrium state m0= (1;0;0)T, the perturbation 4 :0\u0002sin(2\u0019ft) is applied along\nydirection over the time interval [0 ;0:05], withf= 20. The relaxation of the\niLLG equation, revealed by the evolution of the spatially averaged magnetization,\nis visualized in Figure 3.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 21\nTime0 5 10 15 20/angbracketleftm/angbracketright\n-0.500.51\n/angbracketleftm1/angbracketright\n/angbracketleftm2/angbracketright\n/angbracketleftm3/angbracketright\n(a)Spatially averaged magnetization\nTime0 2 4 6 8 10 12 14 16 18 20E[mn+1,mn]\n-4-3.8-3.6-3.4-3.2-3-2.8\n(b)Energy decay\nFigure 2. The spatially averaged magnetization (A) and the en-\nergy evolution (B) in the iLLG equation. Parameters setting:\nT= 20,k= 0:02,\u001c= 1:0 and\u000b= 0:5.\n6.Conclusion\nIn this work, an implicit mid-point scheme, with three time steps, is proposed to\nsolve the inertial Landau-Lifshitz-Gilbert equation. This algorithm preserves the\nproperties of magnetization dynamics, such as the energy decay and the constant\nlength of magnetization and is proved to be second-order accurate in both space\nand time. In the convergence analysis, we \frst construct a second-order approxi-\nmationmof the exact solution. It is found that mproducesO(h5) accuracy at\nthe mesh points around the boundary sections, which simpli\fes the estimation at\nboundary points. Then, by analyzing the error function between the numerical22 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nTime0 0.5 1 1.5 2 2.5 3/angbracketleftm/angbracketright\n-0.200.20.40.60.8/angbracketleftm1/angbracketright\n/angbracketleftm2/angbracketright\n/angbracketleftm3/angbracketright\nFigure 3. The response of the spatially averaged magnetization\nfor the magnetic perturbation in the presence of the inertial e\u000bect.\nFor the equilibrium initialization m0= (1;0;0)T, a perturbation\n4:0\u0002sin(2\u0019ft) is applied along ydirection during time interval\n0\u00180:05, withf= 20. The basic simulation parameters are:\n\u000b= 0:02,\u001c= 0:5,T= 3:0 andk= 0:001.\nsolution and the constructed solution mh, we derive the convergence result in the\nH1(\nT) norm. Furthermore, a \fxed-point iteration method is proposed to solve\nthis implicit nonlinear scheme under the time-step restriction k\u0014Ch2. Numerical\nresults con\frm the theoretical analysis and clearly show the unique inertial e\u000bect\nin micromagnetics simulations.\nAcknowledgments\nThis work is supported in part by the grant NSFC 11971021 (J. Chen), the\nprogram of China Scholarships Council No. 202106920036 (P. Li), the grant NSF\nDMS-2012269 (C. Wang).\nReferences\n1. F. Alouges and P. 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Phys. 404(2020), 109104.24 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nSchool of Mathematical Sciences and Suzhou Institute for Advanced Research, Uni-\nversity of Science and Technology of China, China\nSuzhou Institute for Advanced Research, University of Science and Technology of\nChina, Suzhou, Jiangsu 215123, China\nEmail address :jingrunchen@ustc.edu.cn\nSchool of Mathematical Sciences, Soochow University, Suzhou, 215006, China\nEmail address :lipanchi1994@163.com\nMathematics Department, University of Massachusetts, North Dartmouth, MA 02747,\nUSA\nEmail address :cwang1@umassd.edu"    },    {        "title": "2209.07908v1.Pseudo_PT_symmetric_Dirac_equation___effect_of_a_new_mean_spin_angular_momentum_operator_on_Gilbert_damping.pdf",        "content": "arXiv:2209.07908v1  [quant-ph]  16 Sep 2022Pseudo- PTsymmetric Dirac equation : effect of a new mean\nspin angular momentum operator on Gilbert damping\nY. Bouguerra, S. Mehani, K. Bechane and M. Maamache\nLaboratoire de Physique Quantique et Syst` emes Dynamiques ,\nFacult´ e des Sciences, Universit´ e Ferhat Abbas S´ etif 1, S ´ etif 19000, Algeria\nP. -A. Hervieux\nInstitut de Physique et Chimie des Mat´ eriaux de Strasbourg ,\nCNRS and Universit´ e de Strasbourg BP 43, F-67034 Strasbour g, France\n(Dated: September 19, 2022)\nAbstract\nThe pseudo- PTsymmetric Dirac equation is proposed and analyzed by using a non-unitary\nFoldy-Wouthuysen transformations. A new spin operator PTsymmetric expectation value (called\nthe mean spin operator) for an electron interacting with a ti me-dependent electromagnetic field is\nobtained. Weshowthatspinmagnetization -whichisthequan tityusuallymeasuredexperimentally\n- is not described by the standard spin operator but by this ne w mean spin operator to properly\ndescribe magnetization dynamics in ferromagnetic materia ls and the corresponding equation of\nmotion is compatible with the phenomenological model of the Landau-Lifshitz-Gilbert equation\n(LLG).\nPACS numbers: Pseudo PT Symmetry; Non-Hermitian Dirac equation ; Foldy-Wouthuysen transformation;\nLandau-Lifshitz-Gilbert equation\n1In the field of micromagnetism ,which provides the physical framework for understanding\nand simulating ferromagnetic materials ,there is a fundamental unsolved problem which is\nthe microscopic origin of the intrinsic Gilbert damping. However, this d amping mechanism\nhas been introduced phenomenologically by T. L. Gilbert in 1955 for de scribing the spatial\nand temporal evolution of the magnetization (known as the LLG equ ation), a vector field\nwhich determines the properties of ferromagnetic materials on the sub-micron length scale\n[1]. Let us stress that this equation leads to the conservation of th e magnetization modulus.\nThis phenomenological model has since been validated by numerous e xperimental data and\nconstitutes the foundation of micromagnetism [2]. Moreover, magn etic damping plays a\ncrucial role in the operation of magnetic devices .The scattering theory can be used to\ncompute the Gilbert damping tensor [3].\nSpinisaquantumconcept [4]thatarisesnaturallyfromtheDiracth eoryandisassociated\nwith the operator ˆΣD≡1\n2iα∧α=\nσ0\n0σ\nwhereα=\n0σ\nσ0\nandσare the usual\n2×2 Pauli matrices [5]. For a classical version of the spin (see Supplementary\nMaterials). Usually, spin magnetization M(r,t) (the quantity which is experimentally\nmeasured) is defined as the expectation value of the spin angular mo mentum given by\nµB/angbracketleftBig\nΨD|ˆΣD|ΨD/angbracketrightBig\nwithµB≡e/planckover2pi1\n2mthe Bohr’s magneton ( e <0) and where ΨDisasolution\nof the Dirac equation. Indeed, in most magnetic materials the orbita l moment is quenched\nand therefore magnetism is only due to the spins [6].\nWhile for a free electron the spin angular momentum in the Heisenberg picture is not a\nconstant of motion/parenleftBigg\ndˆΣD\ndt/ne}ationslash= 0/parenrightBigg\n[5], there exists another spin operatorˆΣD\n,considered to\nbe a constant of motion ,(called the mean spin operator [7])\ndˆΣD\ndt= 0\n. In the presence\nof an electromagnetic field, which is relevant for exploring the micros copic origin of the\nLandau-Lifshitz-Gilbert (LLG) equation, a satisfactory result ha s not yet given.\nKnowingthatthespinorsintheDiractheoryconsistoffourcompon ents, itisimportantto\ncheck whether the Dirac equation yields physically reasonable result s in the non-relativistic\nexpansion case and to show that the Dirac equation reproduces th e two-component Pauli\nequation. We transform the Hamiltonian in such a way that all operat ors of the type αthat\ncouple the large to the small components will be removed. This can be achieved by a Foldy\nWouthuysen transformation [7–9] which is a non-relativistic expans ion of the Hamiltonian\n2into series of the particle ′s Compton wave lengths λC≡h\nmc.\nHickey and Moodera [10] have proposed that the spin-orbit interac tion, which arises from\nthe non-relativistic expansion of the Dirac equation, may be respon sible for the intrinsic\nferromagnetic line width. In their work, the term containing the curl of the electric field\nwhen coupled to Maxwell’s equations lead to a time-varying magnetic ind uction,and the\ntheoretical methods employed involve previously developed formalis ms in which an effective\nnon-Hermitian and time-dependent Hamiltonian is used. However, th e non-Hermiticity of\nthe Hamiltonian imposes new rules which are modified with respect to th ose of standard\nquantum mechanics. This fact was not explicitly taken into account b y the authors of [10]\nand therefore, their derivation of the intrinsic damping process is u nfortunately incorrect.\nMoreover, there is another fundamental issue which emerges fro m this work [10] concerning\nhow to properly perform the coupling between the classical Maxwell equations and the\nquantum evolution resulting from the non-relativistic limit of the Dirac equation. In what\nfollows, we show how to overcome this difficulty by using the well-known correspondence\nprinciple. Intheref. [11], themain goalwastodemonstratethatt hereisawaytoderivethe\nLLG equation coming from a non Hermitian quantum mechanics and to s park a discussion\nabout the connection between quantum and classical spin dynamics . Unfortunately, the\nquantum Heisenberg equation for a non-Hermitian Hamiltonian opera tor describing the\ndamping process is not compatible with the time-evolution operators for non-Hermitian\nHamiltonian operator.\nFromthe relativistic Diracequation, performing a Foldy-Wouthuyse n transformationand\nusing the Heisenberg equation of spin motion, Mondal et al [12–14] derive general relativistic\nexpressions for the Gilbert damping, but the term involving the cros s-product between the\nmagnetisationandthetime-derivativeofthemagneticfieldispurelyim aginary,andtherefore\nappears not to correspond to damping.\nIn the seminal work made by Dirac on relativistic quantum mechanics, the corresponding\nHamiltonian would be Hermitian. We stress that this property is very u seful to have in a\nphysical system, however we argue that the same features can b e achieved when starting\nfromnon-HermitianHamiltonian systems. These features canalso b eobtainedfromtheories\nbased on non-Hermitian Hamiltonians that have been considered in diff erent contexts. And\nwe distinguish three separate regimes: i) The PT-symmetric regime where the eigenval-\nues are real, ii) The spontaneously broken PTregime where the eigenvalues are complex\n3conjugate pairs and iii) The regime with complex, unrelated, eigenvalu es in which the PT\n-broken regime.\nOur objective here is to derive the LLG equation based on a non-Her mitian Dirac Hamil-\ntonian when compared to the most common standard approaches [ 12–14].\nTherefore, the Dirac equation in its fundamental representation is not unique to either\nHermitian quantum mechanics orquantum field theory. By relaxing the assumption of Her-\nmiticity andadoptinginstead the principles of P0T0-symmetry quantum mechanics outlined\nin the following paragraph, we will not make any modifications to the Dir ac equation. By\nP0T0-symmetry we mean reflection in space, with a simultaneous reversa l of time. The\nfundamental representation of the Dirac equation emerges comp letelyintact,identical in\nevery aspect to the Dirac equation derived from Hermitian theory. Before constructing the\nanalogous 4-d representation using the principles of P0T0quantum mechanics, let us briefly\nrecall the notion of P0T0-symmetry.\nThe Hermiticity of quantum Hamiltonians depends on the choice of the inner product\nof the states in the physical Hilbert space. This point was first point ed out by Bender et\nal [16, 17]. They showed that a wide class of Hamiltonians that respec tP0T0-symmetry\ncan exhibit entirely real spectra. Since then P0T0-symmetry has been a subject of intense\ninterest in the field of quantum mechanics.\nWhile any evidence of P0T0-symmetry has remained out of reach due to the hermitian\nnatureofthequantummechanics theory,opticshave providedafe rtilegroundforobservation\nof this property- P0T0-symmetry-since this field mainly relies on the presence of gain and\nloss.Note that even though HandP0T0commute, they do not continuously have identical\neigenvectors, as a result of the anti-linearity of the P0T0operator. If HandP0T0don’t have\nthe same eigenvectors, we say that the P0T0−symmetry is broken. The parity operator P0\neffects the momentum operator pand the position operator ras (P0:r→ −r,p→\n−p). This parity transformation has the following effect on the various vector potentials\nP0A(r,t)(P0)−1=−A(r,t) andP0Φ(r,t)(P0)−1= Φ(r,t), expressing thus their scalar\nand vector nature.\nThe anti-linear time reversal operator T0has the effect of changing the sign of the mo-\nmentum operator p,the pure imaginary complex quantity iand the time t(T0:r→r,\np→ −p, i→ −i, t→ −t). Since A(r,t) is generated by currents, which reverse s\nsignswhen the sense of time is reversed, it holds that T0A(r,t)(T0)−1=−A(r,t)\n4andT0Φ(r,t)(T0)−1= Φ(r,t).The two reflection operators commute with each other:\nP0T0=T0P0.\nTherefore, it is natural to introduce a modified Hilbert space, which is now endowed with\nP0T0-inner product, for the P0T0-symmetric nonself-adjoint theories. In such a Hilbert\nspace, the time evolution becomes unitary as the Hamiltonian is self- P0T0-adjoint and\nthe eigenfunctions form a complete set of orthonormal functions . But the norms of the\neigenfunctions have alternate signs even in the new Hilbert space en dowed with the P0T0-\ninnerproducts. Infact, anytheoryhavinganunbroken P0T0-symmetryitexistsasymmetry\nof the Hamiltonian associated with the fact that there are equal nu mbers of positive-norm\nand negative-norm states [17]:\n/an}b∇acketle{tψm,ψn/an}b∇acket∇i}htP0T0=/integraldisplay\ndx/bracketleftbig\nP0T0ψn(x)/bracketrightbig\nψm(x) =/integraldisplay\ndxψ∗\nn(−x)ψm(x)\n= (−1)nδmn (1)\nThe situation here is analogous to the problem that Dirac encounter ed in formulating the\nspinor wave equation in relativistic quantum theory [18] .\nThis again raises an obstacle in probabilistic interpretation in spite of t he systembeingin\nanunbroken P0T0phase. Afterwards, a new symmetry C, inherent to all P0T0-symmetric\nnon-Hermitian Hamiltonians, has been introduced [17]. Ccommutes with both HandP0T0\nand fixes the problem of negative norms of the eigenfunctions when the inner products are\ntaken with respect to CP0T0-adjoint.\nDoes aP0T0-symmetric Hamiltonian Hspecify a physical quantum theory in which the\nnorms of states are positive and time evolution is unitary? The answe r is that if Hhas an\nunbroken P0T0symmetry, then it has another symmetry represented by a linear o perator\nC.Thereforewecanconstruct atime-independent inner productwit hapositive-definite norm\nin terms of C.\nAnother possibility to explain the reality of the spectrum is making use of the\npseudo/quasi-Hermiticity transformations which do not alter the e igenvalue spectra. It was\nshown by Mostafazadeh [19] that P0T0-symmetric Hamiltonians are only aspecific class\nof the general families of thepseudo-Hermitian operators. A Hamiltonian is said to be\nη-pseudo-Hermitian if:\nH†=ηHη−1, (2)\n5whereηisametric operator . Theeigenvaluesofpseudo-HermitianHamiltoniansareeither\nreal or appear in complex conjugate pairs whilethe eigenfunctions satisfy bi-orthonormality\nrelations in the conventional Hilbert space. Due to this reason, suc h Hamiltonians do not\npossessacomplete set of orthogonal eigenfunctions in the conventional Hilb ert space and\nhence the probabilistic interpretation and unitarity of time evolution have not been satisfied\nby these pseudo-Hermitian Hamiltonians.\nHowever, like the case of P0T0-symmetric non-Hermitian systems, thepresence of the\nadditional operator ηin the pseudo-Hermitian theories allows usto define a new inner\nproduct in the fashion\n/an}b∇acketle{tφ|ψ/an}b∇acket∇i}htη=/an}b∇acketle{tηφ|ψ/an}b∇acket∇i}ht=/integraldisplay\n(ηφ(x))ψ(x)dx=/an}b∇acketle{tφ|ηψ/an}b∇acket∇i}ht. (3)\nLateranovel concept ofthepseudoparity-time (pseudo -P0T0)symmetry wasintroduced\nin [15] to connect the non-Hermitian Hamiltonian Hto its Hermitian conjugate H†\nH†=/parenleftbig\nP0T0/parenrightbig\nH/parenleftbig\nP0T0/parenrightbig−1(4)\nwhere in the expression of the inner product (3), the metric ηis replaced by P0T0. We now\nturn our attention to the main topic of interest, the spatial reflec tion and the time-reversal\ninvariance of the Dirac equation. The complete spatial reflection (p arity) transformation\nfor spinors and the complete time-inversion operator are denoted asP=γ0P0andT=\n−iα1α3T0=iγ1γ3T0such that\nPγ0P−1=γ0,PγiP−1=−γi\nTγ0T−1=γ0,TαT−1=−α (5)\nwhere the Hermitian matrices β=γ0andαsatisfy the ‘Dirac algebra’ {αi,αj}= 2δij;\n{αi,β}= 0.\nInwhat follows, we will denote by\nˆHD(t) =i/parenleftbig\ncα.(ˆp+ieA(r,t))−eΦ(r,t)+mc2β/parenrightbig\n(6)\nthe non-Hermitian DiracHamiltonian for a single electron in thepresence of a classical time-\ndependent external electromagnetic field defined by ( A(r,t),Φ(r,t)). The associated non-\nHermitian Dirac equation for a single electron in thepresence of a classical time-dependent\n6external electromagnetic field reads\ni/planckover2pi1∂ΨD(r,t)\n∂t=ˆHD(t)ΨD(r,t)\n=i/parenleftBig\nˆT+ˆV+mc2β/parenrightBig\nΨD(r,t), (7)\nwhere ΨD(r,t) = col(u(r,t),v(r,t)),is bispinors verifying the pseudo-othogonality relation\n/angbracketleftbig\nΨD|ΨD/angbracketrightbig\nPT=I,ˆV≡ −eΦ andˆT≡cα.ˆπ=cα.(ˆp+ieA(r,t)) is the kinetic energy which\nproduces a coupling between small and large components of the Dirac wavefun ction ΨD,\nα,βandγ5(see Eq.(15)) are the Dirac matrices [20]. The PTsymmetry condition\n(PT)ˆHD(t)(PT)−1=ˆH†D(t), (8)\nconnects the non-Hermitian Dirac Hamiltonian ˆHD(t) to its Hermitian conjugate\nˆH†D(t).This observation leads us to introduce a novel concept of the pseu do-parity-time\n(pseudo-PT) symmetry, where ( PT) is interpreted as a metric. Thus as the case of pseudo-\nhermiticity, the bispinor ΨD(r,t) = col(u(r,t),v(r,t)),verifiesthe pseudo-othogonality\nrelation/angbracketleftbig\nΨD|ΨD/angbracketrightbig\nPT=I. Note that, there is another situation which differs from that\ndescribed above ,also called pseudo- PTsymmetry which means that the system can have\na real eigenvalues whether or not the original system is PT- symmetric [21, 22].\nAs we are dealing with the non-relativistic expansion of the Dirac equa tion, the following\ndecomposition ΨD(r,t) = Π(r,t)×χD(us,t) [23] can be used, where χD(us,t) is the time-\ndependent bi-spinor representing the spin state oriented in the dir ection defined by usand\nΠ(r,t) thescalarpart of the wave function.\nWe argue that, in the non-relativistic limit, the operatorˆΣD\nis the one which must be\ninterpreted as the spin operator in the Pauli theory and used to de fine the magnetization as\nM(r,t)≡µB/angbracketleftbigg\nχD|ˆΣD\n|χD/angbracketrightbigg\nPT=µB/angbracketleftbigg\nχD|ˆU−1ˆΣFWˆU|χD/angbracketrightbigg\nPT=µB/an}b∇acketle{tχFW|ˆΣFW|χFW/an}b∇acket∇i}htPTwhere\nχFWis the spin part of the Dirac bi-spinor wave function ΨDin a non-relativistic expansion,\nobtained by using the Foldy-Wouthuysen (FW) transformation [7] andˆUis the associated\noperator (see the definition in the following). It worth mentioning th at,according to the\nabove definition ,expanding the spin operator (to a consistent order in h/mc) and using the\nDirac representation of the wave function is equivalent to expand ingthe wave function (also\nto a consistent order in h/mcusing the FW transformation) and keep ingthe original spin\noperator in the Dirac representation. In this work we have chosen to expand the mean value\noperator. The classical magnetization, M(r,t), is obtained by using the correspondence\n7principle. Indeed, we will show in what follows that the equation of mot ion of the mean\nspin operator for an electron interacting with a time-dependent ele ctromagnetic field leads\nto the LLG equation of motion revealing thus its microscopic origin.\nIna seminal work, FoldyandWouthuysen (FW)solved theproblem of finding a canonical\ntransformation thatallows to obtain a two-component theory in the low-energy limit (Pauli\napproximation) ,in the case of the Dirac equation coupled to an electromagnetic field [7 ].\nUnfortunately, contrarytothefree-electroncase, thesolutio ncannotbeexpressed inaclosed\nform. However, FW showed how to obtain successive approximation s of this transformation\nas a power series expansion in powers of the Compton wave length of the particle λC≡\nh\nmc. This procedure, generally restricted to the second-order in 1 /m, is presented in many\ntextbookson relativistic quantum mechanics [8, 20, 24, 25] and has been exte nded to fifth\norder in powers of 1 /m[26].\nIn what follows, the symbol [ ˆC,ˆD] ({ˆC,ˆD}) denotes the commutator (anticommutator)\nof the operators ˆCandˆD. We shall also use the following notations: ˆX≡X(r,t) and\nˆY≡Y(r,t).\nIn the Hermitian Dirac representation\nˆHhD(t) =/parenleftbig\ncα.(ˆp−eA(r,t))+eΦ(r,t)+mc2β/parenrightbig\n, (9)\nthe Heisenberg equation of motion for the spin operator reads as f ollows\ndˆΣD\ndt=i\n/planckover2pi1/bracketleftBig\nˆHhD,ˆΣD/bracketrightBig\n=−2c\n/planckover2pi1[α∧(ˆp−eA(r,t)]. (10)\nIt is well established that the expectation value onto/vextendsingle/vextendsingleΨD/angbracketrightbig\nof the above equation does\nnot lead to the LLG equation for the magnetization. However, as it w illbeshown in the\nfollowing, the latter can be obtained by using the non unitary FW tran sformation and the\nnew definitionof themagnetizationasanexpectation valueof means pinoperator. Thefirst-\nand second-order terms of the FW expansion in powers of 1 /mcorrespond, respectively, to\nthe precessional motion of the magnetization around an effective m agnetic field and its\ndamping.\nSince the Hamiltonian ˆHD(t) has a similar structure to the one in the Dirac case, by\nanalogy with the latter, we use for ˆU(t) the form eˆS(t)whereˆSis a non self-adjoint\n8operator. Therefore, the transformation ΨFW(r,t) =eˆS(t)ΨD(r,t)≡ˆU(t)ΨD(r,t) leads to a\nnew Hamiltonian\nˆHFW(t) =eˆS(t)/parenleftbigg\nˆHD(t)−i/planckover2pi1∂\n∂t/parenrightbigg\ne−ˆS(t), (11)\nwhereˆSis a non self-adjoint operator. More generally, any operator inthe FW\nrepresentation ,that is not explicitly time dependent, ˆOFWwill be transformed in the Dirac\nrepresentation as ˆOD(t) =ˆU−1(t)ˆOFWˆU(t).\nThe most natural extension oftheEhrenfest equation tonon-He rmitian pseudo- PTsym-\nmetric systems is by replacing a Hermitian ˆHhDwitha non-Hermitian one. The structure\nof the Ehrenfest equation does not change, having assumed that part of the action of T\n(i.e.T0) is to send t→ −t, i.e.;we show that it anticommute swith the operator ∂/∂t,\nconsequently, the operator PTcommuteswithi∂/∂t.Which immediately leads us to\ndeduce the Ehrenfest equation of motion for the diagonal matrix e lement of an operator\nˆOD(t)\nd\ndt/angbracketleftBig\nΨD|ˆOD|ΨD/angbracketrightBig\nPT=/angbracketleftBigg\nΨD|i\n/planckover2pi1/bracketleftBig\nˆHD,ˆOD/bracketrightBig\n+∂ˆOD\n∂t|ΨD/angbracketrightBigg\nPT. (12)\nIt’s straightforward to show that the equation of motion (12) lead s to\nd\ndt/angbracketleftBig\nΨFW|ˆOFW|ΨFW/angbracketrightBig\nPT=/angbracketleftbigg\nΨFW|i\n/planckover2pi1/bracketleftBig\nˆHFW,ˆOFW/bracketrightBig\n|ΨFW/angbracketrightbigg\nPT. (13)\nBy expanding ˆS=ˆS1m−1+ˆS2m−2+ˆS3m−3+...withˆS1=β\n2c(α.ˆπ),ˆS2=−i/planckover2pi1e\n4c3(α.E),\nˆS3=−β\n8c6/parenleftBig\n4c3\n3(α.ˆπ)(α.ˆπ)(α.ˆπ)+iec/planckover2pi12(α.∂tE)/parenrightBig\nandE=−∇Φ−∂A\n∂t[26], the mean spin\noperator is computed using the inverse FW transformation of the s pin operator asˆΣD\n=\ne−(ˆS1/m+ˆS2/m2+ˆS3/m3)ˆΣDe(ˆS1/m+ˆS2/m2+ˆS3/m3)and may be expanded in power series of (1 /m)\nleading toˆΣD\n=ˆΣD\n0+ˆΣD\n1m−1+ˆΣD\n2m−2+ˆΣD\n3m−3+...\nˆΣD\n0=ˆΣD,\nˆΣD\n1=−iβ\nc(α׈π),\nˆΣD\n2=1\n8c2/parenleftBig\n−4ie/planckover2pi1B+2e/planckover2pi1/parenleftBig\nˆΣD×B/parenrightBig\n−4/parenleftBig\nˆπ×/parenleftBig\nˆΣD׈π/parenrightBig/parenrightBig/parenrightBig\n−e/planckover2pi1\n2c3(α×E),\nˆΣD\n3=β\n48c3/bracketleftBig\n(α.ˆπ),/bracketleftBig\n(α.ˆπ),/bracketleftBig\n(α.ˆπ),ˆΣD/bracketrightBig/bracketrightBig/bracketrightBig\n+β\n6c3/bracketleftBig\n(α.ˆπ)(α.ˆπ)(α.ˆπ),ˆΣD/bracketrightBig\n−e/planckover2pi12β\n4c5(α×(∂tE)), (14)\n9whereB=∇ ×A.The free case which is investigated in [7] (may be obtained in closed\nform in this case) is recovered from the above formula by substitut ingE=B= 0 ,iβ→β\nandiˆπ→ˆpleading to\nˆΣD\n=ˆΣD−iβ(α∧ˆp)\nEp−/parenleftBig\nˆp∧/parenleftBig\nˆΣD∧ˆp/parenrightBig/parenrightBig\nEp(Ep+mc)\nwhereEp=/radicalbig\nm2c2+p2.\nThe pseudo- PTequation of motion (12) for the mean spin operator is\nd\ndt/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT\n=/angbracketleftbig\nΨD/vextendsingle/vextendsingleeβ\nm/parenleftBig\nˆΣD∧B/parenrightBig\n−e/planckover2pi1\n4m2c2/parenleftBig\nˆΣD∧∂tB/parenrightBig\n−ie\n2m2c2/parenleftBig\nˆΣD∧(E∧ˆπ)/parenrightBig\n+1\n4/planckover2pi1m2c/parenleftBig/bracketleftBig\n(α.π)ˆΣD(α.π),(α.π)/bracketrightBig\n−/bracketleftbig\n(α.π)(α.π)(α.π),ΣD/bracketrightbig/parenrightBig\n+ϑ(m−3)/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT.(15)\n(In order) to check this result, we have applied to the above equat ion the direct FW trans-\nformation. It leads to\nd\ndt/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT=d\ndt/angbracketleftbig\nΨFW/vextendsingle/vextendsinglee(ˆS1/m+ˆS2/m2+ˆS3/m3)ˆΣD\ne−(ˆS1/m+ˆS2/m2+ˆS3/m3)/vextendsingle/vextendsingleΨFW/angbracketrightbig\nPT\n≡d\ndt/angbracketleftbig\nΨFW/vextendsingle/vextendsingleˆΣ/vextendsingle/vextendsingleΨFW/angbracketrightbig\nPT=/angbracketleftbig\nΨFW/vextendsingle/vextendsinglei\n/planckover2pi1/bracketleftbigg\nˆHFW,ˆΣFW/bracketrightbigg/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT(16)\nwhereˆΣFW≡ˆΣand the expression of ˆHFWis given by\nˆHFW=iβmc2−ieˆΦ+iβˆπ2\n2m+iβˆπ4\n8m3c2+βe/planckover2pi12\n16m3c4{ˆπ,∂tE}\n−βe/planckover2pi1\n2mˆΣ./bracketleftbigg\nB−β/planckover2pi1\n4mc2/parenleftbigg\n(∇∧E)+2i\n/planckover2pi1E∧ˆπ/parenrightbigg\n+i/planckover2pi1\n8m2c4(∂tE∧ˆπ)+ ˆπ∧∂tE)/bracketrightbigg\n−βe/planckover2pi1\n8m3c2/braceleftBig\nˆπ2,ˆΣ.B/bracerightBig\n−iβ/parenleftbigge/planckover2pi1\n2m/parenrightbigg2B2\n2mc2+ie/planckover2pi12\n8m2c2∇.E+ϑ(m−4). (17)\nNow, the equation of motion for the mean spin operator (15) can be written in a simpler\nway being obtained as the pseudo- PTexpectation value a spin Dirac states |χD/an}b∇acket∇i}htassociated\ntoˆHD, one gets\nd\ndt/an}b∇acketle{tχD|ˆΣD\n|χD/an}b∇acket∇i}htPT\n=eβ\nm/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧B−e/planckover2pi1\n4m2c2/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧∂tB\n−e\n2mc2/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧/parenleftbigg\nE∧/an}b∇acketle{tχD|i\nmˆπ|χD/an}b∇acket∇i}htPT/parenrightbigg\n+ϑ(m−3). (18)\n10The above expression has been obtained by using/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT=/an}b∇acketle{tχD|ˆΣD\n|χD/an}b∇acket∇i}htPT. It\nis of interest to mention that the non-diagonal terms which appear at thesecondline of\n(15) are due to the Zitterbewegung phenomenon [5]. They cancel o ut when they are pseudo\naveraged out in a Dirac states.\nLet us note that, it is more convenient to use the FW representatio n to find the evolution\nof spin, (indeed) from Eq.(7) whered\ndt/angbracketleftbig\nΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig\nPT≡d\ndt/angbracketleftbig\nΨFW/vextendsingle/vextendsingleˆΣFW/vextendsingle/vextendsingleΨFW/angbracketrightbig\nPT, we get\nd\ndt/an}b∇acketle{tχFW|ˆΣ|χFW/an}b∇acket∇i}htPT=eβ\nm/an}b∇acketle{tχFW|ˆΣ∧/parenleftbigg\nB+1\n2c2/bracketleftbigg\nE∧iβ\nmˆπ/bracketrightbigg/parenrightbigg\n−e/planckover2pi1\n4m2c2/bracketleftBig\nˆΣ∧(∇∧E)/bracketrightBig\n|χFW/an}b∇acket∇i}htPT\n(19)\nConcerning the damping process, as previously explained, the only t erm of importance\nin the expression (15) is −e/planckover2pi1\n4m2c2/parenleftBig\nˆΣD∧∂tB/parenrightBig\nwhich has been obtained from the commutator\n[(α×E),(α.ˆπ)] coming from/bracketleftBig\nˆHD,ˆΣD/bracketrightBig\nand the partial derivative with respect to time\nof the mean spin angular momentum operator at second order in 1 /m[27] given in Eq.\n(14). In addition, classical Maxwell equations have been also employ ed. Similarly to the\nBreit Hamiltonian ,which is obtained from the classical Darwin Lagrangian (which also\noriginates from Maxwell equations) by using the correspondence p rinciple (CP) [25] ,we\nhere resort to the same procedure (in its inverse form, from quan tum to classical) for the\nMaxwell equations. According to this principle, the quantum counte rpartsˆf, ˆgof classical\nobservables f,gsatisfy/angbracketleftBig/bracketleftBig\nˆf,ˆg/bracketrightBig/angbracketrightBig\n=i/planckover2pi1{f,g}p,qwhere/angbracketleftBig/bracketleftBig\nˆf,ˆg/bracketrightBig/angbracketrightBig\nis the expectation value of\nthe commutator and the symbol {}p,qdenotes the Poisson bracket [28–31]. Let’s take for\ninstance the Maxwell-Faraday equation, we have\n∇∧E(r,t) =−∂B(r,t)\n∂t(20)\nwhich can be rewritten as\nǫijk{pi,Ej}p,qek=−∂B(r,t)\n∂t, (21)\nand using the CP one gets\nǫijk[ˆpi,Ej]\ni/planckover2pi1ek=−∂B(r,t)\n∂t≡ −∂tB. (22)\nConsequently, by using our definition of the magnetization M (r,t)≡\nµB/angbracketleftbigg\nχD|ˆΣD\n|χD/angbracketrightbigg\nPT≡µB/angbracketleftBig\nχFW|ˆΣ|χFW/angbracketrightBig\nPTand the CP, the equation of motion (18)\n11may be rewritten for the electron part as\ndM(r,t)\ndt=e\nmM(r,t)∧B(r,t)−e\n4m2c2M(r,t)∧∂tB(r,t)\n+e\n2mc2M(r,t)∧(E(r,t)∧v)+ϑ(m−3). (23)\nThe above equation constitutes the main result of this work.\nMoreover, if the electron is embedded in a magnetically polarizable med ium,defined\nby its magnetic polarizability χm,then∂M\n∂tgenerates a time-dependent magnetic induction\naccording to the relation ∂tB(r,t) =1\nχm∂M\n∂tand the equation (23) can be rewritten as\ndM(r,t)\ndt=−γM(r,t)∧Beff(r,t)−αG\nM/parenleftbigg\nM(r,t)∧∂M(r,t)\n∂t/parenrightbigg\n(24)\nwithγ=−e\nm>0 the gyromagnetic ratio for an isolated electron, Beff≡B−1\n2c2v∧E\nandαG≡eM\n4m2c2χm. The first term describes the precessional motion of the magnetiz ation\nvector around the direction of the effective magnetic field and the s econd term represents\nits damping ,characterized by the Gilbert’s constant αG.\nLet us stress that the first term in the right hand side of equation ( 24) can be retrieved\nfrom the non-relativistic expansion of the Bargmann-Michel-Telegd i’s equation [24, 32, 33]\nwhichrepresents the relativistic equation of motion of a classical magnetic dipole momen t\n[34]. However, the damping term cannot be obtained from this classic al description due to\nits quantum origin.\nIn summary, the mean spin angular momentum operator introduced for the first time\nby Foldy and Wouthuysen for the case of a free electron has been e xtended to the non\n-Hermitian or precisely to a pseudo PT-symmetric case of an electron interacting with\na time-dependent electromagnetic field. The expectation equation of the motion of the\nlatter leads to the Landau-Lifshitz-Gilbert equation revealing thus its microscopic origin.\nWe therefore argue that the expectation value of the pseudo-me an spin operator with the\nnew definition of PT-inner product must be used instead of the usual one to properly\ndescribe the dynamics of the spin magnetization.\n[1] T. L. Gilbert, IEEE Transactions on magnetics 40, 3443 (2 004). In this paper published in\nClassics in Magnetics, the key results of the Gilbert’s thes is are reproduced.\n12[2] H. Kronm¨uller and M. F¨ahnle, ”Micromagnetism and the M icrostructure of Ferromagnetic\nSolids”, Cambridge (2003).\n[3] Arne Brataas, Yaroslav Tserkovnyak, and Gerrit E. W. Bau er, Phys. Rev. Lett. 101, 037207\n(2008).\n[4] Jean-Marc L´ evy-Leblond, Commun. math. Phys. 6, 286-311 (1967).\n[5] J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wes ley (1967).\n[6] S. Blundell, Magnetism in Condensed Matter, Oxford Univ ersity Press, (2001).\n[7] L. Foldy and S. Wouthuysen, Phys. Rev. 78, 29 (1950).\n[8] W. Greiner, Relativistic quantum mechanics. - Wave equa tions, Springer, (2000).\n[9] J. D. Bjorken and S.D. Drell, Relativistic Quantum Mecha nics. McGraw-Hill Book Company,\nNew York (1964).\n[10] M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102, 1376 01 (2009).\n[11] R. Wieser, Phys. Rev. Lett. 110, 147201 (2013).\n[12] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppeneer, P hys. Rev. B 96, 024425 (2017).\n[13] R. Mondal, Marco Berritta, Peter M. Oppeneer, J. Phys.: Condens. Matter 30, 165801 (2018).\n[14] R. Mondal, Peter M. Oppeneer, J. Phys.: Condens. Matter 32, 455802 (2020).\n[15] Naima Mana and Mustapha Maamache, International Journ al of Modern Physics A, 35, No.1,\n2075001 (2020)\n[16] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5234 ( 1998).\n[17] C. M. Bender, Dorje C. Brody, and Hugh F. Jones, Phys. Rev . Lett.89, 270401 (2002).\n[18] P. A. M. Dirac, Proc. R. Soc. Lond. A 180,1 (1942).\n[19] A. Mostafazadeh, J. Math. Phys. 43, 205 (2002).\n[20] P. Strange, Relativistic Quantum Mechanics, Cambridg e University Press, (2005).\n[21] X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar an d C. Lee, Phys. Rev. Lett. 110,\n243902 (2013).\n[22] M. Maamache, S. Lamri and O. Cherbal, Annals Phys. 378, 1 50 (2017).\n[23] J. S. Roman, L. Roso and L. Plaja, J. Phys. B 37, 435 (2004).\n[24] C. Itzykson, J. -B. Zuber, Quantum Field Theory, McGraw -Hill (1985).\n[25] M. Reiher and A. Wolf, Relativistic Quantum Chemistry, Wiley-VCH (2009).\n[26] Y. Hinschberger and P.-A. Hervieux, Physics Letters A 3 76, 813 (2012).\n13[27]ˆBof/parenleftBig\nˆΣD׈B/parenrightBig\nin Eq. (14) also originates from a commutator [ˆ πi,ˆπj].\n[28] R. L. Liboff, Foundations of Physics 17, 981 (1987).\n[29] L´ evy-Leblond, J.M. The pedagogical role and epistemological significance of gro up theory in\nquantum mechanics . Riv. Nuovo Cim. 4, 99–143 (1974)\n[30] Hove, L´ eon Van. “ Sur le probl` eme des relations entre les transformations un itaires de la\nm´ ecanique quantique et les transformations canoniques de la m´ ecanique classique .” (1951).\n[31] D. Sen, S. K. Das, A. N. Basu and S. Sengupta, Current Scie nce, Vol. 80, No.4,536-541\n(2001).\n[32] V. Bargmann, L. Michel, and V. L. Telegdi, Phys. Rev. Let t. 2, 435 (1959).\n[33] J. D. Jackson, Classical Electrodynamics, John Wiley ( 1998).\n[34] In order to be compatible with our description which doe s not include QED effects, the Land´ e\nfactor must be g= 2.\nSupplementary Materials\nIn terms of the conjugate variable ( q,p) the classical spin− →Sis desribed by [1, 2]\n\n\nSx=/radicalbig\nS2−p2cosq\nSy=/radicalbig\nS2−p2sinq\nSz=p(25)\nthe Poisson brackets {Si,Sj}=εijkSk(i,j,karex,yorz) are analogous to the same rela-\ntionships one has with spin components and commutators in quantum mechanics.\nSuppose we have the following Hamiltonian\nH=− →B.− →S (26)\nwhich is formally identical to the Hamiltonian for a spin 1 /2 system in a uniform magnetic\nfield. We can calculate the evolution of the vector components using the standard Hamil-\ntonian techniques and The motion of spin− →Son the sphere (phase space) with (conserved)\nradiusS=/vextendsingle/vextendsingle/vextendsingle− →S/vextendsingle/vextendsingle/vextendsinglegenerated by (26), can be obtained by regarding H(26) as classical hamil-\ntonian . It may be confirmed that Hamilton’s equation reproduce exa ctly what spin does in\na magnetic field i.e,− →·\nS=− →B∧− →S.\nThe two-level spin system can be written as a classical model if we em ploy the anticom-\nmuting Grassmann variables [3–6]− →ζwhich are transformed to the spin operator after the\n14quantization∧− →ζ=∧− →S/√\n2 . Unlike the classical spin defined in the equation ((25)) which\ndoes not tranformed into a spin operator after the quantization∧− →S/ne}ationslash=− →S.\n[1] M. V. Berry, in ”Fundamental Aspects of Quantum” (Edited by V. Gorini and A. Frigerio),\nPlenum, Nato ASI series vol. 144, 267-278 (1986)).\n[2] M. Maamache, exact solution and geometic Angle for the cl assical spin system, Phys. Scr. 54,\n21 (1996).\n[3] R. Casalbuoni, On the quantization of systems with antic ommuting variables, Nuovo Cimento\nA 33, 115 (1976).\n[4] F.A. Berezin and M.S. Marinov, Particle Spin Dynamics as the Grassmann Variant of Classical\nMechanics, Ann. Phys. (N.Y) 104, 336 (1977).\n[5] E. Gozzi and W. D. Thacker, Classical adiabatic holonomy in a Grassmannian system, Phys.\nRev. D 35, 2388 (1987).\n[6] M. Maamache and O. Cherbal, Evolution of Grassmannian in variant-angle coherent states and\nnonadiabatic Hannay’s angle, Eur. Phys. J. D 6, 145 (1999).\n15"    },    {        "title": "2209.14179v1.Unidirectional_magnetic_coupling.pdf",        "content": "Unidirectional magnetic coupling\nH. Y. Yuan,1R. Lavrijsen,2and R. A. Duine1, 2\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584 CC Utrecht, The Netherlands\n2Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: September 29, 2022)\nWe show that interlayer Dzyaloshinskii-Moriya interaction in combination with non-local Gilbert\ndamping gives rise to unidirectional magnetic coupling. That is, the coupling between two magnetic\nlayers | say the left and right layer | is such that dynamics of the left layer leads to dynamics of\nthe right layer, but not vice versa. We discuss the implications of this result for the magnetic sus-\nceptibility of a magnetic bilayer, electrically-actuated spin-current transmission, and unidirectional\nspin-wave packet generation and propagation. Our results may enable a route towards spin-current\nand spin-wave diodes and further pave the way to design spintronic devices via reservoir engineering.\nIntroduction. | Non-reciprocal transmission of elec-\ntrical signals lies at the heart of modern communication\ntechnologies. While semi-conductor diodes, as an exam-\nple of an electronic component that underpins such non-\nreciprocity, have been a mature technology for several\ndecades, new solutions are being actively pursued [1, 2].\nSuch research is spurred on by the emergence of quan-\ntum technologies that need to be read out electrically\nbut should not receive unwanted back-action from their\nelectronic environment.\nComplementary to these developments, spintronics has\nsought to control electronic spin currents and, more\nrecently, spin currents carried by spin waves | i.e.,\nmagnons | in magnetic insulators [3]. Devices that im-\nplement non-reciprocal spin-wave spin currents have been\nproposed [4{7]. Most of these proposals rely on dipolar\ninteractions [8{11] or Dzyaloshinskii-Moriya interactions\n(DMI) [12{16]. Other proposals involve the coupling of\nthe spin waves to additional excitations such that the spin\nwaves are endowed with non-reciprocity. Examples are\nthe coupling of the spin waves to magnetoelastic, optical,\nand microwave excitations [17{22].\nMost of these proposals have in common that they con-\nsider spin-wave dispersions that are asymmetric in wave\nvector. For example, due to the DMI spin waves at one\nparticular frequency have di\u000berent wave numbers and ve-\nlocities for the two di\u000berent directions. There are there-\nfore spin waves travelling in both directions. This may\nbe detrimental for some applications. For example, one\nwould like to shield quantum-magnonic technologies from\nspin-current noise [23], and completely quench the spin-\ncurrent transmission in one of the two directions along a\nwire.\nHere, we propose a set-up that realizes unidirectional\nmagnetic coupling between two magnetic layers or be-\ntween two magnetic moments. The ingredients are DMI\nand dissipative coupling between the two layers or mo-\nments. The dissipative coupling takes the form of a non-\nlocal Gilbert damping and may arise, for example, from\nthe combined action of spin pumping and spin transfer.Then, one magnet emits spin current when it precesses,\nwhich is absorbed by the other. The resulting dissipa-\ntive coupling turns out to, for certain parameters, pre-\ncisely cancel the DMI in one direction. As a result, an\nexcitation of one of the magnets leads to magnetization\ndynamics of the other, but not vice versa. This yields\nspin-wave propagation that is truly uni-directional: for\nspeci\fc direction and magnitude of the external \feld, all\nspin waves travel in one direction only.\nMinimal model. | Let us start with the minimal set-\nup that demonstrates the unidirectional coupling. We\n\frst consider two identical homogeneous magnetic lay-\ners that are coupled only by an interlayer DMI with\nDzyaloshinskii vector Dand by interlayer spin pumping\n(see Fig. 1). The magnetization direction in the layers\nis denoted by mi, wherei2f1;2glabels the two lay-\ners. We also include an external \feld H. The magnetic\nenergy is given by\nE[m1;m2] =D\u0001(m1\u0002m2)\u0000\u00160MsH\u0001(m1+m2);(1)\nwhereMsis the saturation magnetization of both layers\nand\u00160is the vacuum susceptibility. The magnetization\ndynamics of layer 1 is determined by the Landau-Lifshitz-\nGilbert (LLG) equation\n@m1\n@t=\r\nMsm1\u0002\u000eE\n\u000em1+\u000bnlm1\u0002@m2\n@t; (2)\nwhere\ris the gyromagnetic ratio and \u000bnlcharacterizes\nthe strength of the non-local damping that in this set-up\nresults from the combination of spin pumping and spin\ntransfer torques, as described in the introduction. The\nequation of motion for the magnetization dynamics of the\nsecond layer is found by interchanging the labels 1 and\n2 in the above equation. Working out the e\u000bective \feldsarXiv:2209.14179v1  [cond-mat.mes-hall]  28 Sep 20222\nFM\nFM\nm2\nm1xy\nz, H, D\nFIG. 1. Schematic of two magnetic moments coupled by an\ninterlayer DMI and by interlayer spin pumping. The dynam-\nics of m1induces the motion of m2, but not vice versa for\nappropriate parameters.\n\u000eE=\u000emiyields\n@m1\n@t=\r\nMsm1\u0002(m2\u0002D\u0000\u00160MsH) +\u000bnlm1\u0002@m2\n@t;\n(3a)\n@m2\n@t=\r\nMsm2\u0002(D\u0002m1\u0000\u00160MsH) +\u000bnlm2\u0002@m1\n@t;\n(3b)\nwhere the sign di\u000berence in e\u000bective-\feld contribution\nfrom the DMI stems from the asymmetric nature of the\nDMI. We show now that depending on the magnitude and\ndirection of the e\u000bective \feld, this sign di\u000berence leads\nfor one of the layers to cancellation of the torques due\nto interlayer DMI and non-local damping. As the can-\ncellation does not occur for the other layer, and because\nthe DMI and non-local damping are the mechanisms that\ncouple the layers in the model under consideration, this\nleads to uni-directional magnetic coupling.\nTaking the external \feld to be much larger than the\ninterlayer DMI, i.e., \u00160jHj\u001djDj=Ms, and taking \u000bnl\u001c\n1, we may replace @mi=@tby\u0000\r\u00160mi\u0002Hon the right-\nhand side of Eqs. (3) because the external \feld then is\nthe dominant contribution to the precession frequency.\nFor the \feld H=D=\u000bnl\u00160Ms, one then \fnds that\n@m1\n@t=\u0000\r\n\u000bnlMsm1\u0002D; (4a)\n@m2\n@t=2\r\nMsm2\u0002(D\u0002m1)\u0000\r\n\u000bnlMsm2\u0002D:(4b)\nHence, the coupling between the two magnetic layers is\nunidirectional at the \feld H=D=\u000bnl\u00160Ms: the magne-\ntization dynamics of layer 1 leads to dynamics of layer\n2 as evidenced by Eq. (4b), but not vice versa as im-\nplied by Eq. (4a). This one-way coupling is reversed by\nchanging the direction of the \feld to \u0000Hor the sign of\nthe non-local coupling \u000bnl.\nMagnetic susceptibility. | Let us now take into ac-\ncount the Gilbert damping within the layers, exchange,\nand anisotropies and discuss the in\ruence of the unidi-\nrectional coupling on the magnetic susceptibility. Theenergy now reads\nE[m1;m2] =\u0000Jm1\u0001m2+D\u0001(m1\u0002m2)\n\u0000\u00160MsH\u0001(m1+m2)\u0000K\n2\u0000\nm2\n1;z+m2\n2;z\u0001\n;(5)\nwith the constant Kcharacterizing the strength of the\nanisotropy and Jthe exchange. We shall focus on the\nferromagnetic coupling ( J >0) without loss of generality.\nThe LLG equation now becomes\n@m1\n@t=\r\nMsm1\u0002@E\n@m1+\u000bm1\u0002@m1\n@t\n+\u000bnlm1\u0002@m2\n@t; (6)\nwith\u000bthe Gilbert damping constant of each layer, and\nwhere the equation for the second layer is obtained from\nthe above by interchanging the labels 1 and 2. We\ntake the external \feld in the same direction as the\nDzyaloshinskii vector and D=D^z,H=H^z, while\n\u00160MsH;K\u001dD, so that the magnetic layers are aligned\nin the ^z-direction. Linearizing the LLG equation around\nthis direction we write mi= (mi;x;mi;y;1)Tand keep\nterms linear in mi;xandmi;y. Writing\u001ei=mi;x\u0000imi;y,\nwe \fnd, after Fourier transforming to frequency space,\nthat\n\u001f\u00001(!)\u0012\u001e1(!)\n\u001e2(!)\u0013\n= 0: (7)\nTo avoid lengthy formulas, we give explicit results below\nfor the case that J= 0, while plotting the results for\nJ6= 0 in Fig. 2. The susceptibility tensor \u001fij, or magnon\nGreen's function, is given by\n\u001f(!) =1\n((1 +i\u000b)!\u0000!0)2\u0000(\rD=Ms)2\u0000\u000b2\nnl!2)\n\u0002\u0012(1 +i\u000b)!\u0000!0i(\rD=Ms\u0000\u000bnl!)\n\u0000i(\rD=Ms+\u000bnl!) (1 +i\u000b)!\u0000!0\u0013\n;(8)\nwith!0=\r(\u00160H+K=Ms) the ferromagnetic-resonance\n(FMR) frequency of an individual layer. The poles of\nthe susceptibility determine the FMR frequencies of the\ncoupled layers and are, for the typical case that \u000b;\u000b nl\u001c\n1, given by\n!\u0006=!r;\u0006\u0000i\u000b!r;\u0006; (9)\nwith resonance frequency\n!r;\u0006=\r(\u00160H+K=Ms\u0006D=Ms): (10)\nWhen\r\u00160H= (1\u0007\u000bnl)D=(\u000bnlMs)\u0000K=Ms\u0019\nD=(\u000bnlMs)\u0000K=Mswe have for J= 0 that\u001f12(!r;\u0006) = 0\nwhile\u001f21(!r;\u0006)6= 0, signalling the non-reciprocal cou-\npling. That is, the excitation of layer 1 by FMR leads\nto response of magnetic layer 2, while layer 1 does not\nrespond to the excitation of layer 2. For opposite direc-\ntion of \feld the coupling reverses: the excitation of layer3\n|χ21(J=0)|\n|χ21(J=0.5D)|\n|χ21(J=15D)|\n|χ12(J=0)|\n|χ12(J=0.5D)|\n|χ12(J=15D)|\n0.96 0.98 1.00 1.02 1.040100200300400\nω/ωH\nFIG. 2. Magnetic susceptibilities of two magnetic layers as a\nfunction of frequency at di\u000berent exchange couplings. !H\u0011\n\r(\u00160H+K=M s). The resonance frequencies are located at\nthe peak positions. The parameters are D=! H= 0:001;\u000bnl=\n0:001;\u000b= 0:002.\n2 by FMR leads in that case to response of magnetic\nlayer 1, while layer 2 does not respond to the excitation\nof layer 1. As is observed from Fig. 2, for \fnite but\nsmallJ\u001cD, the coupling is not purely unidirectional\nanymore but there is still a large non-reciprocity. For\nJ\u001dD, this non-reciprocity is washed out.\nElectrically-actuated spin-current transmission. | In\npractice, it may be challenging to excite the individual\nlayers independently with magnetic \felds, which would\nbe required to probe the susceptibility that is determined\nabove. The two layers may be more easily probed inde-\npendently by spin-current injection/extraction from ad-\njacent contacts. Therefore, we consider the situation that\nthe two coupled magnetic layers are sandwiched between\nheavy-metal contacts (see Fig. 3(a)). In this set-up, spin\ncurrent may be transmitted between the two contacts\nthrough the magnetic layers.\nFollowing the Green's function formalism developed by\nZheng et al. [24], the spin-current from the left (right)\nlead to its adjacent magnetic layer is determined by the\ntransmission function of the hybrid system T12(T21)\ngiven by\nTij(!) = Trh\n\u0000i(!)G(+)(!)\u0000j(!)G(\u0000)(!)i\n: (11)\nHere,G(+)(!) is the retarded Green's function for\nmagnons in contact with the metallic leads that is de-\ntermined by Dyson's equation\u0002\nG(+)\u0003\u00001(!) =\u001f\u00001(!)\u0000\n\u0006(+)\n1(!)\u0000\u0006(+)\n2(!), where the retarded self energy\n~\u0006(+)\ni(!) accounts for the contact with the metallic lead\ni. These self energies are given by\n~\u0006(+)\n1(!) =\u0000i~\u000b0\n1\u0012\n!0\n0 0\u0013\n; (12)and\n~\u0006(+)\n2(!) =\u0000i~\u000b0\n2\u00120 0\n0!\u0013\n: (13)\nThe rates for spin-current transmission from the heavy\nmetal adjacent to the magnet iinto it, are given by\n\u0000i(!) =\u00002Imh\n\u0006(+)\ni(!)i\n=~. The couplings \u000b0\ni=\n\rRe[g\"#\ni]=4\u0019Msdiare proportional to the real part of the\nspin-mixing conductance per area g\"#\nibetween the heavy\nmetal and the magnetic layer i, and further depend on\nthe thickness diof the magnetic layers. Finally, the ad-\nvanced Green's function is G(\u0000)(!) =\u0002\nG(+)\u0003y.\nIn the analytical results below, we again restrict our-\nselves to the case that J= 0 for brevity, leaving the\ncaseJ6= 0 to plots. Using the above ingredients,\nEq. (11) is evaluated. Taking identical contacts so that\n\u000b0\n1=\u000b0\n2\u0011\u000b0, we \fnd that\nT12=4(\u000b0)2!2(\rD=Ms+\u000bnl!)2\njC(!)j2; (14)\nwhile\nT21=4(\u000b0)2!2(\rD=Ms\u0000\u000bnl!)2\njC(!)j2; (15)\nwith\nC(!) = [!H\u0000(1 +i(\u000b\u0000\u000bnl+\u000b0))!]\u0001\n[!H\u0000(1 +i(\u000b+\u000bnl+\u000b0))!]\u0000(\rD=Ms)2:(16)\nFrom the expression for C(!) it is clear that, since\n\u000b;\u000b nl;\u000b0\u001c1, the transmission predominantly occurs\nfor frequencies equal to the resonance frequencies !r;\u0006\nfrom Eq. (9). Similar to the discussion of the suscepti-\nbilities, we have for \felds \r\u00160H=D=\u000b nl\u0000K=Msthat\nthe transmission T12(!=D=\u000b nl)6= 0, while T21(!=\nD=\u000b nl) = 0. As a result, the spin-current transmis-\nsion is unidirectional at these \felds. For the linear spin-\nconductances Gij, given byGij=R\n~!(\u0000N0(~!))Tij(!),\nwe also have that G126= 0, while G21= 0. Here,\nN(~!) = [e~!=kBT\u00001]\u00001is the Bose-Einstein distri-\nbution function at thermal energy kBT. For the oppo-\nsite direction of external \feld we have G12= 0, while\nG216= 0. Like in the case of the susceptibility discussed\nin the previous section, a \fnite but small exchange cou-\npling makes the spin current transport no longer purely\nunidirectional, while maintaining a large non-reciprocity\n(see Fig. 3(b)).\nSpin-wave propagation. | Besides the unidirectional\ncoupling of two magnetic layers, the above results may\nbe generalized to a magnetic multilayer, or, equivalently,\nan array of coupled magnetic moments that are labeled\nby the index isuch that the magnetization direction of\nthei-th layer is mi. This extension allows us to engi-\nneer unidirectional spin-wave propagation as we shall see4\nm1\nm2\nLead Lead FM FM(a)\n(b)\nT21(J=0)\nT21(J=0.5D)\nT21(J=15D)\nT12(J=0)\nT12(J=0.5D)\nT12(J=15D)\n0.96 0.98 1.00 1.02 1.040.0000.0020.0040.0060.008\nω/ωH\nFIG. 3. (a) Schematic of the system that the two coupled\nmagnetic layers are sandwiched between two heavy-metal con-\ntacts. (b) Transmission of the hybrid system as a function of\nfrequency.\nbelow. We consider the magnetic energy\nE[m] =X\nk[D\u0001(mk\u0002mk+1)\u0000\u00160MsH\u0001mk];(17)\nand \fnd | within the same approximations as for our\ntoy model above | for the magnetization dynamics that\n@mk\n@t=2\r\nMsmk\u0002(D\u0002mk\u00001)\u0000\r\n\u000bnlMsmk\u0002D;(18)\nfor the \feld H=D=\u000bnl\u00160Ms. This shows that for these\n\felds the magnetic excitations travel to the right | cor-\nresponding to increasing index k| only. The direction\nof this one-way propagation is reversed by changing the\nmagnetic \feld to \u0000Hor by changing the sign of the non-\nlocal damping.\nTo study how spin waves propagate in an array of cou-\npled magnetic moments described by the Hamiltonian in\nEq. (17). We start from the ground state mk= (0;0;1)T\nand perturb the left-most spin ( k= 0) to excite spin\nwaves. Since the dynamics of this spin is not in\ruenced\nby the other spins for the \feld H=D=\u000b nl\u00160Ms, its\nsmall-amplitude oscillation can be immediately solved\nas\u001e0(t) =\u001e0(t= 0) exp(\u0000i!0t\u0000\u000b!0t) with\u001ek=\nmk;x\u0000imk;yas used previously. The dynamics of the\nspins to the right of this left-most spin is derived by solv-\ning the LLG equation (18) iteratively, which yields\n\u001ek(t) =\u001e0(t= 0)e\u0000i!0te\u0000\u000b!0t\nk!(\u00002\u000bnl!0t)k;(19)wherek= 0;1;2;:::N\u00001.\nTo guarantee the stability of the magnetization dynam-\nics, the dissipation matrix of the N-spin system should\nbe negative-de\fnite, which imposes a constraint on the\nrelative strength of Gilbert damping and non-local damp-\ning, i.e.,\u000b > 2\u000bnlcos\u0019\nN+1. For an in\fnitely-long chain\nN!1 , we have\u000b>2\u000bnl. Physically, this means that\nthe local dissipation of a spin has to be strong enough to\ndissipate the spin current pumped by its two neighbors.\nFor a spin chain with \fnite number of spins, \u000b= 2j\u000bnljis\nalways su\u000ecient to guarantee the stability of the system.\nTaking this strength of dissipation simpli\fes Eq. (19) to\n\u001ek(t) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)e\u0000t=\u001c\nk!(\u0000t=\u001c)k; (20)\nwhere\u001c\u00001=\u000b!0is the inverse lifetime of the FMR\nmode. This spatial-temporal pro\fle of spins is the same\nas a Poisson distribution with both mean and variance\nequal to\u001b=t=\u001cexcept for a phase modulation, and it\ncan be further approximated as a Gaussian wavepacket\non the time scale t\u001d\u001c, i.e.\n\u001e(x) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)\np\n2\u0019\u001be\u0000(x\u0000\u001b)2\n2\u001b: (21)\nSuch similarity suggests that any local excitation of the\nleft-most spin will generate a Gaussian wavepacket prop-\nagating along the spin chain. The group velocity of the\nmoving wavepacket is v=a=\u001c, whereais the distance\nbetween the two neighboring magnetic moments. The\nwidth of the wavepacket spreads with time as ap\nt=\u001c,\nwhich resembles the behavior of a di\u000busive particle. Af-\nter su\u000eciently long time, the wavepacket will collapse.\nOn the other hand, the excitation is localized and can-\nnot propagate when the right-most spin ( k=N\u00001) is\nexcited, because its left neighbor, being in the ground\nstate, has zero in\ruence on its evolution. These results\ndemonstrate the unidirectional properties of spin-wave\ntransport in our magnetic array.\nDiscussion, conclusion, and outlook. | We have\nshown that the ingredients for unidirectional coupling be-\ntween magnetic layers or moments are that they are cou-\npled only by DMI and non-local Gilbert damping. While\nin practice it may be hard to eliminate other couplings,\nthe DMI and non-local coupling need to be su\u000eciently\nlarger than the other couplings to observe unidirectional\ncoupling.\nThere are several systems that may realize the unidi-\nrectional coupling we propose. A \frst example is that of\ntwo magnetic layers that are coupled by a metallic spacer.\nSuch a spacer would accommodate non-local coupling via\nspin pumping and spin transfer. For a spacer that is\nmuch thinner than the spin relaxation length, we \fnd,\nfollowing Refs. [25{27], that \u000bnl=\r~Re[~g\"#]=4\u0019dMs,\nwith ~g\"#the spin-mixing conductance of the interface\nbetween the magnetic layers and the spacer, dthe thick-\nness of the magnetic layers. For simplicity, we took the5\nmagnetic layers to have equal properties. The two mag-\nnetic layers may be coupled by the recently-discovered\ninterlayer DMI [28, 29], tuning to a point (as a function\nof thickness of the spacer) where the ordinary RKKY ex-\nchange coupling is small. We estimate \u000bnl= 4:5\u000210\u00003\nford= 20 nm, Re[~g\"#] = 4:56\u00021014\n\u00001m\u00002and\nMs= 1:92\u0002105A=m (YIGjPt). The required mag-\nnetic \feld for unidirectional magnetic coupling is then\naround 4.5 T for D= 1 mT. Another possible platform\nfor realizing the unidirectional coupling is the system of\nFe atoms on top of a Pt substrate that was demonstrated\nrecently [30]. Here, the relative strength of the DMI and\nexchange is tuned by the interatomic distance between\nthe Fe atoms. Though not demonstrated in this experi-\nment, the Pt will mediate non-local coupling between the\natoms as well. Hence, this system may demonstrate the\nunidirectional coupling that we proposed.\nThe non-local damping is expected to be generically\npresent in any magnetic material and does not require\nspecial tuning, though it may be hard to determine its\nstrength experimentally. Hence, an attractive implemen-\ntation of the unidirectional coupling would be a magnetic\nmaterial with spins that are coupled only via DMI, with-\nout exchange interactions. While such a material has\nto the best of our knowledge not been discovered yet,\nit is realized transiently in experiments with ultrafast\nlaser pulses [31]. Moreover, it has been predicted that\nhigh-frequency laser \felds may be used to manipulate\nDMI and exchange, even to the point that the former is\nnonzero while the latter is zero [32, 33].\nPossible applications of our results are spin-wave and\nspin-current diodes and magnetic sensors, where a weak\n\feld signal can be ampli\fed and transported through\nthe unidirectional coupling to the remote site to be read\nout without unwanted back-action. Finally, we remark\nthat the unidirectional magnetic coupling that we pro-\npose here may be thought of as reservoir engineering, cf.\nRef. [34]. In our proposal, the reservoir is made up by the\ndegrees of freedom that give rise to the non-local damp-\ning, usually the electrons. We hope that this perspective\nmay pave the way for further reservoir-engineered mag-\nnetic systems\nAcknowledgements. | It is a great pleasure to\nthank Mathias Kl aui and Thomas Kools for discus-\nsions. 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X 5,021025 (2015)."    },    {        "title": "2210.00366v1.Nonlinear_features_of_the_superconductor__ferromagnet__superconductor___varphi_0__Josephson_junction_in_ferromagnetic_resonance_region.pdf",        "content": "Nonlinear features of the superconductor{ferromagnet{superconductor '0Josephson\njunction in ferromagnetic resonance region\nAliasghar Janalizadeh1, Ilhom R. Rahmonov2;3;4, Sara A.\nAbdelmoneim5, Yury M. Shukrinov2;3;4, and Mohammad R. Kolahchi1\n1Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45137-66731, Zanjan, Iran\n2BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n3Dubna State University, Dubna, 141980, Russia\n4Moscow Institute of Physics and Technology, Dolgoprudny, 141700, Moscow Region, Russia\n5Physics department, Meno\fya University, Faculty of Science, 32511, Shebin Elkom,Egypt\n(Dated: October 4, 2022)\nWe demonstrate the manifestations of the nonlinear features in magnetic dynamics and IV-\ncharacteristics of the '0Josephson junction in the ferromagnetic resonance region. We show that\nat small values of system parameters, namely, damping, spin-orbit interaction, and Josephson to\nmagnetic energy ratio, the magnetic dynamics is reduced to the dynamics of the scalar Du\u000eng os-\ncillator, driven by the Josephson oscillations. The role of increasing superconducting current in the\nresonance region is clari\fed. Shifting of the ferromagnetic resonant frequency and the reversal of\nits damping dependence due to nonlinearity are demonstrated by the full Landau-Lifshitz-Gilbert-\nJosephson system of equations, and in its di\u000berent approximations. Finally, we demonstrate the\nnegative di\u000berential resistance in the IV{characteristics, and its correlation with the foldover e\u000bect.\nI. I. INTRODUCTION\nThe coupling of superconducting phase di\u000berence with\nmagnetic moment of ferromagnet in the '0junction leads\nto a number of unique features important for supercon-\nducting spintronics, and modern information technology\n[1{5]. It allows to control the magnetization preces-\nsion by superconducting current and a\u000bects the current{\nvoltage (IV) characteristics by magnetic dynamics in the\nferromagnet, in particular, to create a DC component in\nthe superconducting current [6{8]. A remarkable mani-\nfestation of such coupling is the possibility to stimulate\na magnetization reversal in the ferromagnetic layer by\napplying current pulse through the '0-junction [3, 9{13].\nThere are two features of our Josephson junction that\ncome into play in our study. One is the broken inver-\nsion symmetry in the weak link of the Josephson junc-\ntion, when the link is magnetic, which introduces an ex-\ntra phase in the current|-phase relation, preventing it\nfrom being antisymmetric. Such Josephson junctions are\nnamed'0junctions [1], and examples exist such as MnSi\nand FeGe. Second is the nonlinear property of the system\nthat makes for an anomalous resonance behavior [14].\nWe couple such a Josephson junction to the model\nthat describes the magnetodynamics in thin \flms or\nheterostructure, to form the Landau-Lifshitz-Gilbert-\nJosephson model (LLGJ)[14{16]. It is shown that for\na particular set of parameters, the coupled equations\nreduce to the dynamics of a Du\u000eng oscillator [14].\nThe cubic nonlinearity in this oscillator has applications\nin describing several e\u000bects in other models too [17].\nOne being the resonance e\u000bects in the antiferromagnetic\nbimeron in response to an alternating current, which has\napplications in the detection of weak signals [15, 18, 19].\nThe Gilbert damping term is added phenomenologi-\ncally to the Landau|-Lifshitz model, to reproduce the\ndamping of the precessing magnetic moment. Gilbertdamping is important in modeling other resonance fea-\ntures too, as its temperature dependence a\u000bects them\n[20, 21], and in return in the superconducting correla-\ntions that a\u000bect it [22]. The magnetization precession\nin the ultra thin Co20Fe60B20layer stimulated by mi-\ncrowave voltage under a large angle, needs modeling by\nDu\u000eng oscillator too. This gets help from the so called\nfoldover features, again due to nonlinearity [16, 23, 24].\nThe consequences of the nonlinear nature of the cou-\npled set of LLGJ system of equations in the weak cou-\npling regime was demonstrated recently in Ref. [14]. We\nshowed in this regime, where the Josephson energy is\nsmall compared to the magnetic energy, the '0Joseph-\nson junction is equivalently described by a scalar non-\nlinear Du\u000eng equation. An anomalous dependence of\nthe ferromagnetic resonant frequency (FMR) with the\nincrease of the Gilbert damping was found. We showed\nthat the damped precession of the magnetic moment is\ndynamically driven by the Josephson supercurrent, and\nthe resonance behavior is given by the Du\u000eng spring.\nThe obtained results were based on the numerical simu-\nlations. The role of dc superconducting current, and the\nstate with negative di\u000berential resistance (NDR) in IV-\ncharacteristic were not clari\fed. Also, the e\u000bects of the\nJosephson to magnetic energy ratio and the spin-orbit\ncoupling (SOC) were not investigated at that time.\nIn the present paper, we study the nonlinear aspects\nof the magnetic dynamics and IV-characteristics of the\n'0Josephson junction in the ferromagnetic resonance re-\ngion. We compare description of the anomalous damp-\ning dependence (ADD) exhibited by full LLGJ system\nof equations with approximated equations and demon-\nstrate the Du\u000eng oscillator features in the small param-\neter regime. E\u000bects of the Josephson to magnetic energy\nratio, and the spin-orbit coupling on the ADD, referred\nto earlier as the \u000b-e\u000bect [14] are demonstrated. By de-\nriving the formula which couples the dc superconduct-arXiv:2210.00366v1  [cond-mat.supr-con]  1 Oct 20222\ning current and maximal amplitude of magnetization we\ndiscuss the correlation of superconducting current and\nthe negative di\u000berential resistance in the resonance re-\ngion. Finally, we discuss the experimentally important\nfeatures by emphasizing the details of the magnetization\ndynamics and the IV-characteristics of the '0junction.\nWe have shown that in the limit of small system pa-\nrameters; that is, the Josephson to magnetic energy ra-\ntioG, the damping \u000b, and the spin-orbit coupling r, the\ndynamics is given by the Du\u000eng spring [14]. We focus\non the shift in resonance and the e\u000bects of nonlinear in-\nteractions. We give semi-analytic models to explain our\nresults in various limits.\nThe paper is organized as follows. In Section II we\noutline the theoretical model and discuss the methods\nof calculations. The ferromagnetic resonance and ef-\nfects of system parameters on the anomalous damping\ndependence are considered in Subsection A of Section\nIII. In Subsection B we present analytical description of\nthe dynamics and IV-characteristics of the '0junction\nat small system parameters. Manifestation of the nega-\ntive di\u000berential resistance in IV-characteristics through\nthe foldover e\u000bect is discussed. We compare the de-\nscription of the anomalous damping dependence by full\nLLGJ system of equation with approximated equation,\nand show how the Du\u000eng oscillator captures the non-\nlinearities in the small parameter regime in Subsection\nC. We present results on the critical damping and de-\nrive the formula which couples the dc superconducting\ncurrent and maximal amplitude of magnetization in the\nferromagnetic layer. Finally, in Section IV we concludes\nthe paper.\nII. II. MODELS AND METHOD\nThe following section is closely related to our work\nin [13]. The '0junction [6, 12, 25] that we study is shown\nin Fig.1. The current-phase relation in varphi 0junction\nhas the form Is=Icsin ('\u0000'0), where'0=rMy=M0,\nMydenotes the component of magnetic moment in ^ ydi-\nrection,M0is the modulus of the magnetization. The\nphysics of'0Josephson juncton is determined by system\nof equations which consists of Landau-Lifshits-Gilbert\n(LLG), resistively capacitively shunted junction (RCSJ)\nmodel expression with current-phase relation ( Is) de-\nscribed above, and Josephson relation between phase dif-\nference and voltage.\nThe dynamics of the magnetic moment Mis described\nby the LLG equation [26]\ndM\ndt=\u0000\rM\u0002Heff+\u000b\nM0\u0012\nM\u0002dM\ndt\u0013\n; (1)\nwhere Mis the magnetization vector, \ris the gyromag-\nnetic relation, Heffis the e\u000bective magnetic \feld, \u000bis\nGilbert damping parameter, M0=jMj.\nFigure 1. Schematic view of SFS '0Josephson junction. The\nexternal current applied along x direction, ferromagnetic easy\naxis is along z direction.\nIn order to \fnd the expression for the e\u000bective mag-\nnetic \feld we have used the model developed in Ref.[6],\nwhere it is assumed that the gradient of the spin-orbit\npotential is along the easy axis of magnetization taken to\nbe along ^z. In this case the total energy of the system\ncan be written as\nEtot=\u0000\b0\n2\u0019'I+Es(';' 0) +EM('0); (2)\nwhere'is the phase di\u000berence between the supercon-\nductors across the junction, Iis the external current,\nEs(';' 0) =EJ[1\u0000cos ('\u0000'0)], andEJ= \b 0Ic=2\u0019\nis the Josephson energy. Here \b 0is the \rux quantum,\nIcis the critical current, r=l\u001dso=\u001dFl= 4hL=~\u001dF,L\nis the length of Flayer,his the exchange \feld of the\nFlayer,EM=\u0000KVM2\nz=(2M2\n0), the parameter \u001dso=\u001dF\ncharacterizes a relative strength of spin-orbit interaction,\nKis the anisotropic constant, and Vis the volume of the\nferromagnetic ( F) layer.\nThe e\u000bective \feld for LLG equation is determined by\nHe\u000b=\u00001\nV@Etot\n@M\n=\nF\n\r\u0014\nGrsin\u0012\n'\u0000rMy\nM0\u0013\nby+Mz\nM0bz\u0015\n(3)\nwhere \n F=\rK=M 0is frequency of ferromagnetic reso-\nnance andG=EJ=(KV) determines the ratio of Joseph-\nson energy to magnetic one.\nIn order to describe the full dynamics '0junction the\nLLG equations should be supplemented by the equation\nfor phase di\u000berence ', i.e. equation of RCSJ model for\nbias current and Josephson relation for voltage. Accord-\ning to the extended RCSJ model, which takes into ac-\ncount derivative of '0phase shift, the current \rowing\nthrough the system in underdamped case is determined\nby\nI=~C\n2ed2'\ndt2+~\n2eR\u0014d'\ndt\u0000r\nM0dMy\ndt\u0015\n(4)\n+Icsin\u0012\n'\u0000r\nM0My\u0013\n:\nwhereIis the bias current, CandRare the capacitance\nand resistance of Josephson junction respectively. The3\nJosephson relation for voltage is given by :\n~\n2ed'\ndt=V: (5)\nWe note that in the framework of RCSJ{model the\ndisplacement current is proportional to the \frst deriva-\ntive of voltage (or second derivative of phase di\u000berence).\nFrom the other hand, the magnetization dynamics plays\nrole of the external force and \frst order derivative of '0\nis a source of external current for JJ. This was demon-\nstrated in Ref.[25, 27] where the authors included the \frst\nderivative of '0as the source of the electromotive force.\nVoltage is determined by the phase di\u000berence, and does\nnot depend on '0. From this point of view, in the frame-\nwork of RCSJ model the external current source cannot\nmodify the expression for displacement current. That's\nwhy we do not include the second derivative of varphi 0\nin our model.\nUsing (1), (3), (4) and (5) we can write the system of\nequations, in normalised variables, which describes the\ndynamics of '0junction\n_mx=!F\n1 +\u000b2f\u0000mymz+Grm zsin('\u0000rmy)\n\u0000\u000b[mxm2\nz+Grm xmysin('\u0000rmy)]g;\n_my=!F\n1 +\u000b2fmxmz\n\u0000\u000b[mym2\nz\u0000Gr(m2\nz+m2\nx) sin('\u0000rmy)]g;\n_mz=!F\n1 +\u000b2f\u0000Grm xsin('\u0000rmy)\n\u0000\u000b[Grm ymzsin('\u0000rmy)\u0000mz(m2\nx+m2\ny)]g;\n_V=1\n\fc[I\u0000V+r_my\u0000sin('\u0000rmy)];\n_'=V(6)\nwheremx;y;z =Mx;y;z=M0and satisfy the constraintP\ni=x;y;zm2\ni(t) = 1,\fc= 2eIcCR2=~is McCumber pa-\nrameter. In order to use the same time scale in the\nLLG and RCSJ equations in this system of equations\nwe have normalized time to the !\u00001\nc, where!c=2eIcR\n~,\nand!F= \n F=!cis the normalized frequency of ferro-\nmagnetic resonance \n F=\rK=M 0. Bias current is nor-\nmalized to the critical current Icand voltage V{ to the\nVc=IcR. The system of equations (6), is solved numer-\nically using the fourth-order Runge-Kutta method(see\nRef.[14]).\nIII. III. RESULTS AND DISCUSSION\nA. A. E\u000bect of system parameters on the\nanomalous damping dependence\nADD of the FMR frequency with increasing \u000bwas dis-\ncussed in Ref. [14]. It was found that the resonance\ncurves demonstrate features of Du\u000eng oscillator, re-\n\recting the nonlinear nature of Landau-Lifshitz-Gilbert-\n 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14\n 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8mymax\nValpha=0.01\nalpha=0.02\nalpha=0.03\nalpha=0.04\nalpha=0.05\nalpha=0.06\nalpha=0.07\nalpha=0.08\nalpha=0.09\nalpha=0.1\n 0 0.1\n 0.5Figure 2. Maximal amplitude of magnetization\nmy\u0000component at each values of bias current and voltage\nalong IV-characteristics of the '0junction in the ferromag-\nnetic resonance region for various \u000b. Inset enlarges the main\nmaximum. Parameters: \fc= 25,G=0.05,r=0.05, !F= 0:5.\nJosephson (LLGJ) system of equations. There is a criti-\ncal damping value at which anomalous dependence comes\ninto play. This critical value depends on the system pa-\nrameters. Here we present the details of such transforma-\ntion from usual to anomalous dependence with variation\nin spin-orbit coupling and ratio of Josephson to magnetic\nenergy.\nTo investigate the e\u000bect of damping, we calculate\nthe maximal amplitude of magnetization component my\ntaken at each value of the bias current based on the\nLLGJ system of equations (6). In Fig.2 we show the\nvoltage dependence of maximal amplitude mmax\nyin the\nferromagnetic resonance region at di\u000berent damping pa-\nrameter and small values of Josephson to magnetic en-\nergy ratio G=0.05 and spin-orbit coupling r= 0:05. We\nfound that the ferromagnetic resonance curves demon-\nstrate the di\u000berent forms. An increase in damping shows\na nonuniform change in the resonant frequency: it is ap-\nproaching the !Finstead of moving away with increase\nin\u000b. We stress that this happens at small Gandr. We\nconsider that such behavior can be explained by the non-\nlinear nature of the LLGJ system of equations. There is\na manifestation of subharmonics of the FMR in Fig.2 at\n!= 1=2;1=3;1=4.\nWe usually expect the resonance peak to move away\nfrom resonance as the \u000bincreases. Figure 2 shows that\nthis normal e\u000bect is accompanied with an anomalous be-\nhaviour as can be seen in the inset to this \fgure, where\nthe resonance peak approaches !Fas\u000bincreases [14].\nThe manifestation of FMR in IV-characteristics of the\n'0junction at three values of damping parameter is\ndemonstrated in Fig. 3. The strong deviation of the\nIV-curve is observing at \u000b= 0:01, which is characteristic4\nFigure 3. Part of the IV characteristic of the '0junction\natG= 0:05;r= 0:05 and di\u000berent values of Gilbert damp-\ning. The numbers show \u000bvalue. Inset shows the total IV-\ncharacteristic and arrow indicates the resonance region\nvalue for many magnetic materials. This fact indicates\nthat ADD can be observed experimentally by measuring\nIV-characteristics in wide interval of the damping param-\neter.\nInteresting features of ADD appear by a variation of\nspin-orbit coupling. As it was demonstrated in Ref.[28],\nan increase in SOC leads to the essential change in IV-\ncharacteristics and magnetization precession in the fer-\nromagnetic resonance region. The nonlinearity is going\nstronger and the state with negative di\u000berential resis-\ntance appears at large SOC.\nFigure 4(a) demonstrates results of numerical simu-\nlations ofmmax\nydependence on \u000bat di\u000berent values of\nSOC parameter r. It shows two speci\fc features of ADD.\nFirst, with an increase in r, the critical value of Vpeakis\ndecreasing (the curve moves away from !F). The sec-\nond important feature is an increasing of \u000bcritwhich is\ndemonstrated in this \fgure by arrows.\nAnother model parameter which a\u000bects the phe-\nnomenon discussed in the present paper is the ratio G\nof Josephson to magnetic energies. Figure 4(b) demon-\nstrates the results of numerical simulations of mmax\nyde-\npendence on \u000bat di\u000berent values of G.\nSimilar to the e\u000bect of r, increasing Galso causes the\nvalue of\u000bcritto increase. By changing the volume of the\nferromagnetic layer, the ferromagnetic energy and con-\nsequently the value of G can be changed [6]. For small\nG, i.e. a situation where the magnetic energy is much\nlarger than the Josephson energy, the magnetic layer re-\nceives less energy, and its amplitude decreases in the y\ndirection, and also the maximum value of the oscillation\nfrequency is closer to the magnetic frequency, !F.\nVpeakα\n0.46 0.47 0.48 0.49 0.500.050.10.150.2\nαcrit\n1 23\nVpeakα\n0.46 0.47 0.48 0.49 0.500.050.10.150.2\nαcrit\n1 23Figure 4. (a) Demonstration of ADD at di\u000berent values of\nSOC parameter ratG= 0:05. Numbers indicate: 1 -\nr= 0:05; 2 -r= 0:1; 3 -r= 0:5; Arrows show critical \u000b\nvalue, corresponded to the reversal in the \u000bdependence (b)\nDemonstration of ADD at di\u000berent values of the Josephson\nto magnetic energy ratio Gatr= 0:05. Numbers indicate: 1\n-G= 0:01; 2 -G= 0:1; 3 -G= 1.\nB. B. Dynamics and IV-characteristics of the '0\njunction at small system parameters\nAs it was discussed in Refs.[6, 29, 30], in case of\nG;r;\u000b<< 1 andmz\u00191, \frst three equations of the sys-\ntem (6) can be simpli\fed. Taking into account '=!Jt\nand neglecting quadratic terms of mxandmy, we get\n(\n_mx=!F[\u0000my+Grsin(!Jt)\u0000\u000bmx]\n_my=!F[mx\u0000\u000bmy];(7)\nThis system of equations can be written as the second\norder di\u000berential equation with respect to my\nmy+ 2\u000b!F_my+!2\nFmy=!2\nFGrsin!Jt: (8)5\nCorresponding solution for myhas the form\nmy(t) =!+\u0000!\u0000\nrsin(!Jt)\u0000\r++\r\u0000\nrcos(!Jt);(9)\nwhere\n!\u0006=Gr2!F\n2!J\u0006!F\n\n\u0006; (10)\nand\n\r\u0006=Gr2!F\n2\u000b!J\n\n\u0006: (11)\nwith \n \u0006= (!J\u0006!F)2+ (\u000b!J)2(see Ref.[6] and corre-\nsponded Erratum[31]).\nWhen the Josephson frequency !Jis approaching the\nferromagnetic one !F,mydemonstrates the damped fer-\nromagnetic resonance. Di\u000berential resistance in the res-\nonance region is decreasing and it is manifested in the\nIV{characteristic as a resonance branch [7].\nTaking into account rmy<<1, we rewrite expression\nfor superconducting current as\nIs(t) = sin(!Jt\u0000rmy(t))\n= sin(!Jt)\u0000rmycos(!Jt) (12)\nUsing solution (9) we can obtain\nIs(t) = sin!Jt\u0000!+\u0000!\u0000\n2sin 2!Jt\n+\r++\r\u0000\n2cos 2!Jt+I0(\u000b) (13)\nwhere\nI0=\r++\r\u0000\n2: (14)\nThis superconducting current explains the appearance\nof the resonance branch in the IV{characteristic. The\ngenerated current I0can be expressed through the am-\nplitude ofmyand SOI parameter r\nI0=r\n2mmax\ny(!J); (15)\nwithmmax\ny(!J) being the frequency response of my.\nAt small model parameters \u000b<<Gr<< 1 of SFS'0\nJosephson junction, the states with a negative di\u000berential\nresistance appear in the IV-characteristics in the FMR\nregion. Due to the nonlinearity, the resonance peak is\nasymmetric. Increasing of nonlinearity leads to the bista-\nbility (foldover e\u000bect). A natural questions appears if the\nstates with a negative di\u000berential resistance are the origin\nof the foldover and ADD. In order to clarify this question,\nwe show in Fig.5 a part of IV{characteristics of '0junc-\ntion together with IV-characteristic of SIS junction in\nthe ferromagnetic resonance region and numerically cal-\nculated superconducting current through this junction.\nThe total IV{characteristics are demonstrated in the in-\nset to this \fgure.\nFigure 5. IV-characteristic of '0and SIS junctions and calcu-\nlated average superconducting current through the '0junc-\ntion.\nWe see the correlation of the foldover e\u000bect in super-\nconducting current (blue) with the NDR part of IV-curve.\nThe peak in the superconducting current and minimum\nof IV-curve are at the same voltage value. So, both ef-\nfects re\rect the nonlinear features of the ferromagnetic\nresonance in '0junction. But in contrast to the foldover\nand ADD e\u000bects, which start manifesting themselves at\nrelatively small deviations from linear case, the nonlin-\nearity in case of the NDR plays a more essential role.\nWe note that in the resonance region for considered\nlimit of model parameters the myamplitude are coupled\nto the value of superconducting current (see Eq.(15)). We\nstress the importance of the performed analysis demon-\nstrating the analytical coupling of time independent su-\nperconducting current and magnetization, re\recting the\nDu\u000eng oscillator features of the '0junction.\nAs it is well known, the states with negative di\u000beren-\ntial resistance appear in IV-characteristics of Josephson\nstructures in di\u000berent physical situations. In particular,\nthe nonlinear superconducting structures being driven\nfar from equilibrium exhibit NDR states [32]. The NDR\nstates plays an essential role in the applications related\nto the THz radiation emission [33]. A detailed expla-\nnation for di\u000berent types of negative di\u000berential resis-\ntance (NDR) in Josephson junction (i.e., N-shaped and\nS-shaped) is introduced in[34]. The authors emphasize\nthat the nonlinear behavior of the Josephson junction\nplays a key role in the NDR feature. In our case, the\nNDR states appear as a result of system's nonlinearity\nat small values of '0junction parameters, such as SOC,\nratio of Josephson to magnetic energy and Gilbert damp-\ning. We demonstrate these e\u000bects here by by presented\nresults of detailed investigation of the NDR state at dif-\nferent system parameters and discuss possibility of their\ncontrol near the ferromagnetic resonance.\nFigure 6 shows the e\u000bect of the spin-orbit coupling on\nthe IV-characteristic at G= 0:05 and\u000b= 0:01. We\nsee the NDR feature which is getting more pronounced\nwith increase in r. A further increase in rleads to the\njump down in voltage and then practically linear growth6\nIV\n0.49 0.495 0.5 0.505 0.510.460.470.480.490.09\n0.15\nIV\n0 0.5 100.20.40.60.811.2r =0.15\nFigure 6. Enlarged parts of the IV-curves in the resonance\nregion at di\u000berent values of SOC parameter rat\u000b= 0:01\nandG= 0:05. The numbers indicate the increasing of rfrom\n0.09 to 0.15 with an increment 0.01. Inset demonstrates total\nIV-characteristic ar r= 0:15\nof IV-characteristic.\nAn interesting question is concerning the e\u000bect of\nGilbert damping. Results of IV-characteristics simula-\ntions in the resonance region at a certain range of the\ndamping parameter \u000batG= 0:05 andr= 0:13 are\nshown in Fig.7(a). In this case, the most pronounced\ncharacteristic appears at \u000b= 0:01. At chosen set of pa-\nrametersG= 0:05 andr= 0:13, the range of \u000bwith\npronounced NDR features is 0 :016\u000b<0:014.\nThe maximal amplitude mmax\nyas a function of volt-\nage is shown in the Fig.7(b). Based on the results, pre-\nsented in Fig.7(a) and Fig.7(b), we came to the important\nconclusion that the foldover e\u000bect (bistability) and NDR\nstate have strong correlations and have the same origin\nrelated to the nonlinearity at small system parameters.\nBut anomalous damping dependence does not show a\none-to-one correlation with either negative di\u000berential re-\nsistance or foldover e\u000bect. The resonance peak positions\nofmmax\nyin bias current Ipeak, and in voltage Vpeakas the\nfunctions of \u000bare demonstrated in Fig.7(c). According\nto our results, we can divide \u000binterval into two regions\n(see Fig.7(c)). Region I introduces the values of \u000bwhere\nthe NDR feature is present, while in the Region II it dis-\nappears. In the Region II the foldover e\u000bect (bistability)\ndisappears as well, but ADD is realized.\nC. C. Du\u000eng oscillator features of '0junction and\ncritical damping\nThe system of equations (6) is nonlinear and very com-\nplex, so in order to provide analytical study of dynamics\nof the'0junction we need to derive an approximated\nequation for some limited values of model parameters.\nIn Ref. [14] was shown that the resonance curves demon-\nstrate features of Du\u000eng oscillator, re\recting the nonlin-\near nature of Landau-Lifshitz-Gilbert-Josephson (LLGJ)\nIV\n0.495 0.5 0.505 0.510.4650.470.4750.480.4850.49\n0.01(a)\n0.018\nIV\n0 0.5 100.20.40.60.811.2\nα= 0.01\nαIpeak, Vpeak\n0.01 0.012 0.014 0.016 0.018 0.020.470.480.490.5Ipeak\nVpeak(c)\nΙΙΙFigure 7. (a) Enlarged parts of the IV-curves at di\u000berent\nvalues of\u000b; (b) Voltage dependence of mmax\nyat di\u000berent \u000b;\n(c)\u000b-dependence of the resonance curve maximum in current\n(Ipeak) and voltage ( Vpeak). The numbers indicate the value\nof\u000bfrom 0.01 to 0.018 in (a) and from 0.01 to 0.02 in (b)\nby an increment 0.001. Results are obtained at r= 0:13, and\nG= 0:05.\nsystem of equations. In this section, we present analyt-\nical approach to describe the nonlinear dynamics of '0-\njunction and compare analytical results followed from ap-\nproximated Du\u000eng equation with numerical simulations\nof total system of equations (6). We show that in the\nlimit of\u000b << G;r << 1, we arrive to the Du\u000eng os-\ncillator. We start by \frst three equations of the (6) for7\nmagnetization components\n_mx\n!F=\u0000mymz+Grm zsin('\u0000rmy)\u0000\u000bmxm2\nz\n_my\n!F=mxmz\u0000\u000bmym2\nz (16)\n_mz\n!F=\u0000Grm xsin('\u0000rmy) +\u000bmz(m2\nx+m2\ny)\nSimplifying this system of equations by the same pro-\ncedure as it was done in Ref.[14] we can write equation\nformyas\nmy+2\u0018\u000b_my+\u00182(1+\u000b2)my\u0000\u00182(1+\u000b2\u0000\u000b4)m3\ny=\u00182\rsin':\n(17)\nFinally, by neglecting the \u000b2and\u000b4terms which are\nmuch smaller than 1, we come to well known Du\u000eng\nequation,\nmy+ 2!F\u000b_my+!2\nFmy\u0000!2\nFm3\ny=!2\nFGrsin': (18)\nIn the small parameter range, this Du\u000eng equation\ncan describe the dynamics of my. We will have the full\ndynamics, once we consider the coupling with the Joseph-\nson equation,\n'+1\n\fc[ _'\u0000r_my+ sin('\u0000rmy)] =1\n\fcI: (19)\nThe system of equations, (18) and (19) can replace the\nLLGJ equations in the limit of G;r<< 1 andG;r>>\u000b .\nTaking into account '=!Jtwe can right analytically\nobtained frequency response for equation (18)\n(mmax\ny)2=\u0000\nGr\u00012\n\u0002\n!2\u00001 +3\n4(mmaxy)2\u00032+\u0000\n2\u000b!\u00012\n(20)\nwhere!=!J=!F. From Eq. (20) we get\n(mmax\ny)6+8\n3(!2\u00001)(mmax\ny)4\n+\u00124\n3\u00132\u0014\n(!2\u00001)2+\u0000\n2\u000b!\u00012\u0015\n(mmax\ny)2\n\u0000\u00124\n3Gr\u00132\n= 0: (21)\nThis equation allows to determine analytically fre-\nquency dependence of the mmax\nyamplitude. To \fnd it\nwe solve the equation (21) by the Newton method. Re-\nsults of analytical calculations (blue dots) corresponded\nto (21) and numerical one (red doted line) corresponded\nto the full system of equation (6) are demonstrated in\nFig.8.\nFigure 8. Numerically (curve 1) and analytically (curve 2)\ncalculated amplitude dependence of my.\nFigure 9. Numerically calculated superconducting current for\nSFS junction (plot 1) and analytical I0(plot 2) and super-\nconducting current for SIS junction (plot 3).\nWe can see that they are close to each other which\nproves the correctness of the chosen approximation.\nBoth curves demonstrate an asymmetric resonance peak,\nwhich is common for Du\u000eng oscillator. When a role of\nthe cubic term is getting larger, we observe a bistability\nof the resonance curve, which is usually called a foldover\ne\u000bect. Note that the foldover e\u000bect can be also achieved\nby the damping decreasing; i.e., by the decreasing of dis-\nsipative term in (18), we can increase the in\ruence of the\ncubic term in this equation.\nThe comparison of analytically and numerically cal-\nculated superconducting current as a function of the\nJosephson frequency is demonstrated in Fig. 9. We note\nthat in our normalization V=!J. We can see the man-\nifestation of the asymmetric resonance peak in the fre-\nquency dependence of superconducting current. So, the\napproximated system of equations 7 re\rects one of the\nmain feature of Du\u000eng oscillator.\nFigure (10) compares anomalous damping dependence\nof the resonance peak of mmax\ny(V) calculated numeri-\ncally according to the full LLGJ system of equations (6)\nwith calculated numerically according to the generalized\nDu\u000eng model (equations (17, 19)). We see that in the\ndamping parameter interval [0.001 { 0.2] the coincidence8\nFigure 10. The \u000b-dependence of the resonance maximum of\nmmax\ny(V) in the damping parameter interval [0.001 { 0.12].\nGreen squares show results calculated numerically according\nto the full system of equations (6), blue circles show results\ncalculated numerically according to the generalized Du\u000eng\nand Josephson equations (17,19). The dashed line connects\nthe symbols to guide eyes. Solid line show analytical \u000b-\ndependence calculated according to the Eq. (22). All calcu-\nlation have been done at \fc= 25, G=0.05, r=0.05, !F= 0:5.\nof the dependences is enough good.\nUsing equation (18) with '=!Jt, we can \fnd (see\nSupplementary materials ??) a relation between posi-\ntion of the resonance peak in mmax\ny(V) dependence and\ndamping\n!peak=s\n1\u00003\u000b2\n2+1\n2r\n(1\u0000\u000b2)2\u000012(Gr\n4\u000b)2(22)\nwhere!peak=!J;peak\n!Fdetermines the position of the res-\nonance peak.\nEquation (22) allows to \fnd the formula for critical\ndamping\u000bcritwhich is an important parameter deter-\nmining the reversal point in damping dependence of the\nresonance peak of mmax\ny(V) .\nTaking into account equation (22) we can write equa-\ntion with respect of Gr=(4\u000b) (See supplementary mate-\nrials??).\n9\u0012Gr\n4\u000bcrit\u00134\n+ 3\u000b2\ncrit(10\u000b2\ncrit\u00001)\u0012Gr\n4\u000bcrit\u00132\n(23)\n\u00002\u000b4\ncrit(\u000b2\ncrit\u00001)2= 0\nUsing approximation 10 \u000b2\ncrit<<1 and\u000b2\ncrit<<1 it\ngives (see Supplementary Materials)\n\u000bcrit\u00191\n2sr\n3\n2Gr (24)\nFigure 11. Numerical calculations according to Eq. (6)\n(squares), analytical according to Eq. (23)(solid line) and\napproximated analytical according to Eq. (24) (dashed line).\nTable 1: A comparison between the numerical and an-\nalytical values of \u000bcrit:at di\u000berent values of Gandr.\nG r Gr\u000bcrit:;numerics \u000bcrit:;analytics\n0.01 0.05 0.0005 0.0100 0.0123\n0.05 0.05 0.0025 0.0300 0.0276\n0.05 0.10 0.0050 0.0400 0.0391\n0.05 0.30 0.0150 0.0700 0.0677\n0.05 0.50 0.0250 0.0900 0.0874\n0.10 0.05 0.0050 0.0391 0.0391\n0.60 0.05 0.0300 0.0950 0.0958\n0.70 0.05 0.0350 0.1000 0.1035\n1.00 0.05 0.0500 0.1200 0.1237\nFigure 11 presents comparison of numerical and ana-\nlytical results \u000bcritversusGr.\nAs we see, it shows a good agreement of numerical\nand analytical results of calculations at small product of\nJosephson to magnetic energy ratio and spin-orbit inter-\naction.\nIV. IV. CONCLUSIONS\nThe understanding of the nonlinear features of\nmagnetization dynamics in superconductor-ferromagnet-\nsuperconductor Josephson junction and their manifesta-\ntion in the IV-characteristics has implications for super-\nconductor spintronics, and modern information technol-\nogy. In'0junctions the nonlinear features can a\u000bect the\ncontrol of magnetization precession by superconducting\ncurrent and external electromagnetic radiation [28].\nHere, using numerical and analytic approaches, we\nhave demonstrated that at small system parameters,9\nnamely, the damping, spin-orbit interaction and Joseph-\nson to magnetic energy ratio in '0junction, magnetic dy-\nnamics is reduced to the dynamics of the scalar Du\u000eng\noscillator, driven by the Josephson oscillations. We have\nclari\fed the role of increasing superconducting current\nin the resonance region leading to the foldover e\u000bect in\nthe ferromagnet magnetization. We have demonstrated\nthe parameter dependence of the anomalous ferromag-\nnetic resonant shifting with anomalous damping depen-\ndence due to nonlinearity of the full LLGJ equation and\nin its di\u000berent approximations. We have derived the an-\nalytical expression for critical damping value. Also, we\ndemonstrated appearance of negative di\u000berential resis-\ntance in the IV-characteristics and the correlation with\noccurrence of the foldover e\u000bect in the magnetization of\nferromagnet.\nWe have stressed that the manifestation of negative\ndi\u000berential resistance is related to the nonlinear features\nof the system[34, 35]. It was demonstrated that in the\nsmall model parameters case the equation for magnetic\nsubsystem takes form of Du\u000eng equation where nonlin-\nearity manifest itself as the cubic term. We have shown\nthat the appearance of negative di\u000berential resistance in\nthe I-V curve is related to the appearance of foldover inthemmax\ny-Vcurve.\nWe believe that the experimentally measured IV-\ncharacteristics of '0junction with manifestations dis-\ncussed in detail in the present paper, would allow close\ninvestigations of its nonlinear features important for su-\nperconductor electronics and spintronics.\nV. SUPPLEMENTARY\nIn supplementary material are presented the details of\ncalculations for Eq.22 and Eq.24.\nVI. FUNDING\nNumerical simulations were funded by Project No. 18-\n71-10095 of the Russian Science Foundation. The pre-\nsented results concerning the calculations of DC super-\nconducting current in the section V are supported by the\nRussian Science Foundation in the framework of project\n22-42-04408. A.J. and M.R.K. are grateful to IASBS for\n\fnancial support.\n[1] Buzdin, A. Physical Review Letters 2008 ,101 (10),\n107005.\n[2] Linder, J., Robinson, J. W. 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P.,\nYuan, W., Ma, Y., Yun, Y., Xie, X., Moodera, J. S.,\nHan, W. Physical Review B 2018 ,97(22), 224414.\n[22] Silaev, M. Physical Review B 2020 ,102(14), 144521.\n[23] Nayfeh, A., Mook, D. Nonlinear Oscillations. John Wil-\nley & Sons , 1979.\n[24] Chen, W., De Loubens, G., Beaujour, J.-M., Sun, J.,\nKent, A. Applied Physics Letters 2009 ,95(17), 172513.\n[25] Guarcello, C., Bergeret, F. Physical Review Applied\n2020 ,13(3), 034012.\n[26] Landau, L., Lifshitz, E. Phys. Rev 1955 ,100, 1243.\n[27] Rabinovich, D. S., Bobkova, I. V., Bobkov, A. M.,\nSilaev, M. A. Phys. Rev. Lett. 2019 ,123, 207001. doi:\n10.1103/PhysRevLett.123.207001 .\n[28] Abdelmoneim, S., Shukrinov, Y. M., Kulikov, K., ElSam-\nman, H., Nashaat, M. Phys. Rev. B 2022 ,106, 014505.\n[29] Shukrinov, Y. M., Rahmonov, I. Physics of Particles and\nNuclei 2020 ,51(4), 816{822.\n[30] Shukrinov, Y. M., Rahmonov, I., Botha, A. Low Temper-\nature Physics 2020 ,46(9), 932{938.\n[31] Konschelle, F., Buzdin, A. Physical Review Letters 2019 ,\n123(16), 169901.10\n[32] Pedersen, N. F., Filatrella, G., Pierro, V., S\u001crensen, M. P.\nPhysica C: Superconductivity and its Applications 2014 ,\n503, 178{182.\n[33] Kadowaki, K., Yamaguchi, H., Kawamata, K., Ya-\nmamoto, T., Minami, H., Kakeya, I., Welp, U.,\nOzyuzer, L., Koshelev, A., Kurter, C., Gray, K.,\nKwok, W.-K. Physica C: Superconductivity and its ap-\nplications 2008 ,468(7-10), 634{639.[34] Filatrella, G., Pierro, V., Pedersen, N. F., Sorensen, M. P.\nIEEE Transactions on Applied Superconductivity 2014 ,\n24(6), 1{7.\n[35] Nagel, J., Speer, D., Gaber, T., Sterck, A., Eichhorn, R.,\nReimann, P., Ilin, K., Siegel, M., Koelle, D., Kleiner, R.\nPhysical Review Letters 2008 ,100, 217001."    },    {        "title": "2210.08429v1.Magnetic_damping_anisotropy_in_the_two_dimensional_van_der_Waals_material_Fe__3_GeTe__2__from_first_principles.pdf",        "content": "Magnetic damping anisotropy in the two-dimensional van der Waals material\nFe3GeTe 2from \frst principles\nPengtao Yang, Ruixi Liu, Zhe Yuan, and Yi Liu\u0003\nThe Center for Advanced Quantum Studies and Department of Physics,\nBeijing Normal University, 100875 Beijing, China\n(Dated: October 18, 2022)\nMagnetization relaxation in the two-dimensional itinerant ferromagnetic van der Waals ma-\nterial, Fe 3GeTe 2, below the Curie temperature is fundamentally important for applications to\nlow-dimensional spintronics devices. We use \frst-principles scattering theory to calculate the\ntemperature-dependent Gilbert damping for bulk and single-layer Fe 3GeTe 2. The calculated damp-\ning frequency of bulk Fe 3GeTe 2increases monotonically with temperature because of the dominance\nof resistivitylike behavior. By contrast, a very weak temperature dependence is found for the damp-\ning frequency of a single layer, which is attributed to strong surface scattering in this highly con\fned\ngeometry. A systematic study of the damping anisotropy reveals that orientational anisotropy is\npresent in both bulk and single-layer Fe 3GeTe 2. Rotational anisotropy is signi\fcant at low tem-\nperatures for both the bulk and a single layer and is gradually diminished by temperature-induced\ndisorder. The rotational anisotropy can be signi\fcantly enhanced by up to 430% in gated single-layer\nFe3GeTe 2.\nI. INTRODUCTION\nNewly emerged intrinsic two-dimensional (2D) ferro-\nmagnetic (FM) van der Waals (vdW) materials1{6have\nbecome the subject of intense research. Weak vdW\nbonding facilitates the extraction of thin layers down to\natomic thicknesses, whereas strong magnetocrystalline\nanisotropy protects long-range magnetic order. These\nmaterials provide an exciting arena to perform funda-\nmental investigations on 2D magnetism and promis-\ning applications of low-dimensional spintronics devices.\nAmong these materials, Fe 3GeTe 2(FGT) is especially\nattractive for its itinerant ferromagnetism and metal-\nlicity, such that both spin and charge degrees of free-\ndom can be exploited for designing functional devices.\nBulk FGT has a relatively high Curie temperature ( TC)\nof approximately 220-230 K.7{11Atomically thin lay-\ners of FGT have lower TCs, which, however, have been\nraised to room temperature (by ionic gating4) and be-\nyond (by patterning12). As a FM metal at reasonably\nhigh temperature, FGT opens up vast opportunities for\napplications.13{23\nThe dynamical properties of FGT critically a\u000bect the\napplicability and performance of these proposed low-\ndimensional spintronics devices. The most salient of\nthese properties is the dynamical dissipation of mag-\nnetization. It is usually described using a phenomeno-\nlogical parameter called Gilbert damping, which char-\nacterizes the e\u000eciency of the instantaneous magneti-\nzation to align eventually with the e\u000bective magnetic\n\feld during its precessional motion. Although this pa-\nrameter has been extensively studied in conventional\nFM materials, such as 3 dtransition metals and alloys,\ntwo key issues with the Gilbert parameter of FGT re-\nmain to be addressed: the temperature dependence and\nanisotropy (one naturally expects anisotropic damping\nin FGT because of its layered structure and the strong\nmagnetocrystalline anisotropy). Temperature-dependentGilbert damping was \frst observed in Fe24and later more\nsystematically in Fe, Co and Ni.25{27A nonmonotonic\ntemperature dependence has been found, for which a so-\ncalled \\conductivitylike\" component decreases with in-\ncreasing temperature, usually at low temperatures, and a\n\\resistivitylike\" component increases with temperature,\nusually at high temperatures. This nonmonotonic be-\nhavior has been successfully described by the torque-\ncorrelation model28and reproduced by \frst-principles\ncomputations.29{32Anisotropic damping was \frst theo-\nretically predicted in FM metals33and in noncollinear\nmagnetic textures.34With di\u000berent orientation of the\nequilibrium magnetization with respect to the crystal-\nlographic axes, the damping parameter can be quanti-\ntatively di\u000berent in general. This is referred to as the\norientational anisotropy. Even for the same equilibrium\nmagnetization orientation in a single crystalline lattice,\nthe magnetization may precess instantaneously along\ndi\u000berent directions resulting in the so-called rotational\nanisotropy.33The orientational anisotropy of damping\nhas been observed in recent experiments on single-crystal\nFM alloys,35{37but the underlying physical mechanism\nremains unclear.\nThe dimensionless Gilbert damping parameter \u000bcan\nbe expressed in terms of a frequency \u0015via\u0015=\u000b\rM ,38\nwhereM=jMjis the magnetization magnitude and \ris\nthe gyromagnetic ratio. Despite of the di\u000berent dimen-\nsions, these two parameters are equivalent39and both\npresent in literature for experimental24{27,35{37and the-\noretical studies.28,29,31,33,34,40{42\nIn this study, we systematically investigate\ntemperature-dependent Gilbert damping in single-\nlayer (SL) and bulk FGT using \frst-principles scattering\ntheory. Considering that the magnetization perpen-\ndicular to the 2D atomic planes is favored by the\nstrong magnetocrystalline anisotropy, we calculate the\ndamping as a function of temperature below TCand\n\fnd nearly temperature-independent damping in thearXiv:2210.08429v1  [cond-mat.mes-hall]  16 Oct 20222\n(a)\nFeⅠ\nFeⅡ\nGe\nTe(b)\nFIG. 1. (a) Side and (b) top view of the lattice structure\nfor bulk Fe 3GeTe 2. The black dashed frame delineates the\nin-plane unit cell.\nSL and damping dominated by resistivitylike behavior\nin the bulk. Varying the equilibrium direction of the\nFGT magnetization produces a twofold symmetry in\ndamping. When the magnetization is aligned inside\nthe 2D planes, a remarkable rotational anisotropy in\nthe Gilbert damping is present for in- and out-of-plane\nrotating magnetization.\nThis paper is organized as follows. The crystalline\nstructure of SL and bulk FGT is brie\ry introduced in Sec.\nII, followed by a description of our theoretical methods\nand computational details. The calculated temperature-\ndependent damping in SL and bulk FGT is presented in\nSec. III. The two types of damping anisotropy, i.e., orien-\ntational and rotational anisotropy, are analyzed in Sect.\nIV. Conclusions are drawn in Sec. V.\nII. GEOMETRIC STRUCTURE OF FGT AND\nCOMPUTATIONAL METHODS\nThe lattice structure of FGT is shown in Fig. 1. Two\ndi\u000berent types of Fe atoms occupy inequivalent Wycko\u000b\nsites and are denoted as FeI and FeII. Five atomic layers\nstack along the caxis to form an SL of FGT: Ge and\nFeII constitute the central atomic layer perpendicular to\nthecaxis, and two FeI layers and two Te layers are lo-\ncated symmetrically above and beneath the central layer,\nrespectively. Single layers with ABAB :::stacking form\nthe bulk FGT, where Layer A is translated in plane with\nrespect to Layer B, such that the Ge atoms in Layer A\nlie on top of the Te and FeII atoms in Layer B.\nThe electronic structure of bulk and SL FGT has\nbeen determined using the linear augmented plane wave\nmethod43within the local density approximation (LDA).\nDi\u000berent types of exchange-correlation functionals have\nbeen investigated in the literature, among which LDA\nwas found to yield satisfactory structural and magnetic\nproperties for FGT.44We employ experimentally ob-\ntained lattice constants7for bulk FGT calculations and\nobtain magnetic moments of 1.78 \u0016Band 1.13\u0016Bfor\nthe two types of Fe, respectively. The initial structure\nof a single layer is taken from the bulk lattice and fully\nrelaxed, resulting in an in-plane constant a= 3:92\u0017A.\nA vacuum spacing of 11.76 \u0017A is chosen to exclude theinterlayer interaction under periodic boundary condi-\ntions. The magnetic moments for the Fe atoms in SL\nFGT are obtained as 1.72 \u0016Band 1.01\u0016B. All the\ncalculated magnetic moments are in good agreement\nwith experimental7,8,45,46and calculated values44,47,48re-\nported in the literature.\nThe Gilbert damping calculation is performed using\nthe scattering theory of magnetization dissipation pro-\nposed by Brataas et al.49Within this theory, a single do-\nmain FM metal is sandwiched between two nonmagnetic\n(NM) metal leads. The Gilbert damping that charac-\nterizes the energy dissipation during magnetization dy-\nnamics can be expressed in terms of a scattering ma-\ntrix and its derivative with respect to the magnetiza-\ntion direction. We thus construct a two-terminal trans-\nport structure as Au jFGTjAu, where the Au lattice is\nslightly deformed to match that of FGT: we use 3 \u00021 and\n4\u00021 unit cells (UCs) of Au (001) to match the UCs of\nSL and bulk FGT, respectively. To investigate the ef-\nfect of temperature on Gilbert damping, we use a frozen\nthermal lattice and spin disorder31,40,50to mimic lattice\nvibration and spin \ructuation at \fnite temperatures in\nFGT. The measured Debye temperature \u0002 D= 232 K\nand temperature-dependent magnetization for the bulk8\nand SL3are employed to model the lattice and spin disor-\nder. In the scattering calculations, lateral supercells are\nemployed to satisfy periodic boundary conditions perpen-\ndicular to the transport direction. The electronic poten-\ntials required for the transport calculation are calculated\nself-consistently using a minimal basis of tight-binding\nlinear mu\u000en-tin orbitals (TB-LMTOs), and the result-\ning band structures for SL and bulk FGT e\u000bectively re-\nproduce those obtained using the linear augmented plane\nwave method. Then, the scattering matrices consisting\nof re\rection and transmission probability amplitudes for\nthe Bloch wave functions incident from the NM leads are\ndetermined by the so-called \\wave function matching\"\nmethod, which is also implemented using TB-LMTOs.40\nOther computational details can be found in our previous\npublications.31,40{42In this work, we focus on the damp-\ning with collective magnetization dynamics in the long-\nwave limit corresponding to the reported values in ex-\nperiment via ferromagnetic resonance and time-resolved\nmagneto-optical Kerr e\u000bect. The damping with a \f-\nnite wavelength can be determined in our framework of\nscattering calculation42or using the torque-correlation\nmodel,51but the wavelength dependence of damping is\nbeyond the scope of the current study.\nIII. TEMPERATURE-DEPENDENT DAMPING\nThe strong magnetocrystalline anisotropy of FGT re-\nsults in the equilibrium magnetization being naturally\nperpendicular to the atomic layers. Slightly excited mag-\nnetization deviates from the plane normal (denoted as ^ z)\nand relaxes back by dissipating energy and angular mo-\nmentum, as schematized in the inset of Fig. 2(a). The3\n𝛂∥𝛂∥𝑴(𝑡)𝑥𝑦𝑧\n5101520F|| (10-3)\n0 0.2 0.4 0.6 0.8 1T/TC468Q|| (108 Hz)Lattice disorder only(b)(a)\nBulkSingle layer\nBulkSingle layer\nFIG. 2. The calculated dimensionless Gilbert damping pa-\nrameter\u000bk(a) and corresponding damping frequency \u0015k(b)\nfor single-layer and bulk Fe 3GeTe 2as a function of tempera-\nture. The relaxation of the instantaneous magnetization M(t)\nresults in a change in the in-plane magnetization component,\nwhich is parallel to the atomic planes, as schematized in the\ninset of (a). The empty symbols in (b) denote the damping\nfrequencies that are calculated considering only thermal lat-\ntice disorder. The green line indicates the linear temperature\ndependence.\nGilbert damping parameter \u000bkdescribes the e\u000eciency\nof such a dissipative process. The calculated \u000bkof SL\nand bulk FGT is plotted in Fig. 2(a) as a function of\ntemperature. The damping for both increases monoton-\nically with the temperature. This behavior resembles\nthe so-called \\resistivitylike\" damping observed in many\nsingle-crystal FM metals.24{26However, the damping \u000bk\nfor the bulk tends to diverge as the temperature ap-\nproachesTC. This divergence originates from vanishing\nmagnetization, as has been found in three-dimensional\nFM alloys.42Therefore, as temperatures approaching TC,\nit is more appropriate to use the damping frequency pa-\nrameter\u0015=\u000b\rM .\nThe calculated damping frequencies are shown in\nFig. 2(b). The damping of a SL FGT, \u0015S\nk, is larger\nthan the damping of the bulk, \u0015B\nk, especially at low\ntemperatures. This di\u000berence can be attributed to the\nstrong surface e\u000bect of highly con\fned SL FGT. The\nlowered symmetry at the surface signi\fcantly enhances\nspin-orbit coupling (SOC),52which enables the dissipa-\ntion of angular momentum from electronic spins to the\norbital degree of freedom and then into the lattice reser-voir. In addition, as the thickness of a single layer is\nconsiderably smaller than the electronic mean free path,\nconduction electrons in FGT are strongly scattered by\nthe surface. Therefore, the two necessary ingredients for\nGilbert damping, namely, SOC and electronic scattering,\nare both enhanced in the SL compared with the bulk, re-\nsulting in a larger damping for the SL.\nThe calculated damping frequency \u0015S\nkremains nearly\nconstant with increasing temperature, except for a mi-\nnor increase at T > 0:6TC. To gain further insight into\nthe temperature e\u000bect, we perform the damping calcu-\nlation considering only lattice disorder, where the calcu-\nlated\u0015S\nlatare plotted as red empty circles in Fig. 2(b).\nLattice-disorder-induced damping in the SL FGT, \u0015S\nlat,\nexhibits a very weak temperature dependence, indicating\nthat increasing lattice vibration does not in\ruence the\ndamping frequency. The di\u000berence between \u0015S\nlatand\u0015S\nk\nincreases slightly only near TC, which can be attributed\nto the strong spin \ructuation. The overall weak tem-\nperature dependence in the damping for a single layer\nindicates that a non-thermal disorder scattering mecha-\nnism is dominant: the strong surface scattering in such\na thin layer (only a few \u0017A) combined with the enhanced\nSOC at the surfaces is the main channel for the magnetic\ndamping in the SL FGT instead of spin \ructuation and\nlattice vibration. Gilbert damping with a similarly weak\ntemperature dependence has also been found in a permal-\nloy,40,53where chemical disorder scattering overwhelms\nthermally induced disorder.\nThe temperature dependence of the bulk damping fre-\nquency is signi\fcantly di\u000berent from that of the SL. The\ncalculated bulk damping, \u0015B\nk, (shown by the black solid\ndiamonds in Fig. 2(b)) increases linearly with the temper-\nature. This typical resistivitylike behavior suggests that\nthe interband transition in bulk FGT is the dominant\ndamping mechanism.54We also calculate the damping\nfrequency\u0015B\nlatconsidering only lattice disorder, as shown\nas the black empty diamonds in Fig. 2(b). Comparing the\nresults corresponding to the solid and empty diamonds\nleads us to conclude that both lattice and spin disorder\nsubstantially contribute to damping in bulk FGT. As the\ntemperature approaches TC, the bulk damping is compa-\nrable with that in the single layer.\nIV. ANISOTROPIC DAMPING\nThe damping torque exerted on the magnetization\nin the Landau-Lifshitz-Gilbert equation has the general\nform of M(t)\u0002[~\u000b\u0001_M(t)], where the Gilbert damping\nparameter ~\u000bor the corresponding frequency is a tensor.\nThis tensor and its elements depend on both the instan-\ntaneous M(t) and its time derivative _M(t), where the\nanisotropy has been extensively analyzed using theoret-\nical models55and \frst-principles calculations.33,34Fol-\nlowing the de\fnition given by Gilmore et al. ,33we call\nthe anisotropic damping that depends on the equilibrium\norientation of Meqthe orientational anisotropy and that4\n𝑴𝐞𝐪𝑥𝑦𝑧𝜃\n-U/2 -U/40U/4U/2V81216F|| (10-3)Single layerBulk\nFIG. 3. The calculated Gilbert damping parameter \u000bkfor SL\n(red circles) and bulk FGT (black diamonds) as a function\nof the angle between the equilibrium magnetization Meqand\nthe atomic layer normal (^ z) of Fe 3GeTe 2. The lines are \ftted\nusingC0+C2cos 2\u0012.\ndepending on _M(t) the rotational anisotropy. Consider-\ning the layered structure of vdW materials, the lowered\nsymmetry should result in remarkable anisotropy for the\nmagnetization relaxation. Both the orientational and ro-\ntational anisotropy in bulk and SL FGT have been sys-\ntematically analyzed in this section. Notably, the damp-\ning tensor is reduced to a scalar for the con\fguration\nshown in Fig. 2.\nUnder a large in-plane magnetic \feld, the perpendicu-\nlar magnetization of FGT can be tilted toward the exter-\nnal \feld direction, which is de\fned as the y-axis without\nloss of generality. Thus, the angle between the equilib-\nrium magnetization Meqand the plane normal ^ zis re-\nferred to as \u0012, as shown in the inset of Fig. 3. At \u0012= 0,\nas studied in Sec. III, \u000bxx=\u000byy=\u000bk. For\u00126= 0,\n\u000bxx=\u000bkstill holds, whereas the other diagonal element\n\u000byydepends on speci\fc values of \u0012. Here, we focus on \u000bk\nto study the orientational anisotropy of damping. The\ncalculated in-plane damping \u000bkis plotted as a function\nof\u0012in Fig. 3 for a SL at 77 K and bulk FGT at 100\nK. The temperature is chosen in this way to obtain the\nsame relative magnetization for the two systems, namely,\nM=M s= 88%, according to the experimentally measured\nmagnetization as a function of temperature.3,8The same\ntwofold symmetry is found for the damping parameters\nof both SL and bulk FGT, which can be e\u000bectively \ftted\nusing a cos 2 \u0012term. As the magnetization rotates away\nfrom the easy axis, \u000bkincreases and reaches a maximum\nwhen the magnetization aligns inside the FGT layer. The\nchanges, [\u000b(\u0012=\u0006\u0019=2)\u0000\u000b(\u0012= 0)]=\u000b(\u0012= 0), are 62% for\nthe SL and 39% for the bulk. A similar dependence of\nthe damping on the magnetization orientation has been\nrecently observed in single-crystal CoFe alloys.35,36The\npredicted anisotropic damping of FGT shown in Fig. 3\nshould analogously be experimentally observable.\nThe rotational anisotropy of damping33in FGT is most\nsigni\fcant when the equilibrium magnetization lies in-\nside the atomic plane of FGT (along the hard axis), i.e.,\n0 0.20.40.6 0.81T/TC120150180QC/Q|| (%)\n0 0.2 0.4 0.6 0.8 1T/TC10152025Q (108 Hz)Bulk\nSingle layerBulkQCQ||Single layer\nBulkBulk\n𝛂∥𝑴(𝐭)\n𝑥𝑦𝑧(a)𝜶\"(b)FIG. 4. (a) Schematic of damping with the equilibrium mag-\nnetization Meqlying inside the atomic plane. Then, the in-\nstantaneous magnetization M(t) dissipates both the in- and\nout-of-plane spin angular momentum. The two types of dis-\nsipation are denoted as \u000bk(\u0015k) and\u000b?(\u0015?). (b) The calcu-\nlated Gilbert damping frequency \u0015k(?)as a function of tem-\nperature for single-layer and bulk Fe 3GeTe 2. The inset shows\nthe ratio of the two frequencies \u0015?=\u0015k.\n\u0012=\u0006\u0019=2. As schematized in Fig. 4(a), the magne-\ntization M(t) loses its in- or out-of-plane components\ndepending on the instantaneous precessional direction\n_M(t). In this case, one has \u000bxx=\u000bkand\u000bzz=\u000b?,\nwhereas the o\u000b-diagonal elements of the damping ten-\nsor are guaranteed to remain zero by symmetry.40The\ncalculated\u0015kand\u0015?for SL and bulk FGT are shown\nas a function of temperature in Fig. 4(b). For SL FGT,\n\u000bk(as shown by the circles with horizontal hatching) is\nnearly independent of temperature, which is the same as\nforMeqalong the easy axis. This result suggests that de-\nspite the sizable orientational anisotropy in the damping\nof SL FGT, the temperature has very little in\ruence on\nthe speci\fc values of the damping frequency. The calcu-\nlated\u0015?for the SL (shown by the red circles with vertical\nhatching) is considerably larger than \u0015kat low tempera-\ntures but decreases with increasing temperature. \u0015?be-\ncomes comparable with \u0015knear the Curie temperature,\nindicating that the rotational anisotropy is signi\fcantly\ndiminished by temperature.\nThe calculated \u0015kfor bulk FGT with Meqalong the\nhard axis (shown by the black diamonds with horizontal\nhatching) is temperature-independent, in sharp contrast\nto the linear temperature dependence of \u0015kwith Meq\nalong the easy axis shown in Fig. 2(b). This result sug-\ngests that the damping is already saturated in this case5\n-0.4 -0.2 0 0.2 0.4\nE-EF (eV)100200300400500λ⊥/λ|| (%)50 K\n77 K\n100 KSingle layer\nFIG. 5. The calculated rotational damping anisotropy for\nsingle-layer Fe 3GeTe 2as a function of the Fermi energy at\ndi\u000berent temperatures.\nat a su\u000eciently large scattering rate, where saturated\ndamping has also been found in FM Ni.25The calculated\n\u0015?of bulk FGT is also larger than \u0015kat low tempera-\ntures and slightly decreases with increasing temperature.\nWe summarize the results for the rotationally anisotropic\ndamping frequency by plotting the ratio between \u0015?and\n\u0015kin the inset of Fig. 4(b). The ratio for both SL and\nbulk FGT decreases with increasing temperature and\napproaches unity near TC. This behavior is consistent\nwith the results calculated using the torque-correlation\nmodel,33where rotationally anisotropic damping disap-\npears gradually as the scattering rate increases. In highly\ndisordered systems, the damping is more isotropic, as in-\ntuitively expected.\nWe emphasize that the calculated \u0015?values are dis-\ntinct from those reported in previous studies in the\nliterature,55that is,\u0015?was found to vanish in single-\ncrystal monoatomic FM layers based on the breathing\nFermi surface model.56{58Interband scattering is ne-\nglected in the breathing Fermi surface model. However,\nthe resistivitylike behavior of our calculated \u0015kfor bulk\nFGT shows that interband scattering plays an important\nrole in this vdW FM material.\nOne of the unique advantages of 2D vdW materials\nis the tunability of the electronic structure via electri-\ncal gating.4,59To simulate such a scenario, we slightly\nadjust the Fermi level EFof SL FGT without changing\nthe band structure for simplicity. The calculated rota-\ntional anisotropy in the damping \u0015?=\u0015kof SL FGT is\nshown as a function of the Fermi energy in Fig. 5. At all\nthe temperatures considered, the anisotropy ratio \u0015?=\u0015k\nincreases dramatically as EFis lowered by 0.3 eV, es-\npecially at low temperatures, and only exhibits minor\nchanges when EFis increased. At 50 K, the ratio \u0015?=\u0015k\nbecomes as high as 430%, which is almost three times\nlarger than that obtained without gating. This result\nsuggests that a small quantity of holes doped into SLFGT at low temperatures remarkably enhances the rota-\ntional damping anisotropy.\nV. CONCLUSIONS\nWe have systematically studied Gilbert damping in a\n2D vdW FM material Fe 3GeTe 2by using \frst-principles\nscattering calculations where the temperature-induced\nlattice vibration and spin \ructuation are modeled by\nfrozen thermal lattice and spin disorder. When the mag-\nnetization is perpendicular to the 2D atomic plane, the\ndamping frequency of bulk FGT increases linearly with\nthe temperature, whereas that of SL FGT exhibits a\nweak temperature dependence. The di\u000berence can be\nattributed to surface scattering (which is absent in the\nbulk) dominating scattering due to temperature-induced\ndisorder in SLs, which have a thickness smaller than the\nelectronic mean free path. The anisotropy of Gilbert\ndamping in this 2D vdW material has also been thor-\noughly investigated. The orientational anisotropy, which\ndepends on the direction of the equilibrium magnetiza-\ntion with respect to the atomic planes, exhibits twofold\nrotational symmetry in both the bulk and SL. When\nthe equilibrium magnetization is parallel to the atomic\nplane, the damping is signi\fcantly enhanced compared to\nthat with the magnetization perpendicular to the atomic\nplane. The rotational anisotropic damping depending on\nthe direction of motion of the instantaneous magnetiza-\ntion is remarkable with the equilibrium magnetization ly-\ning inside the atomic plane. With an out-of-plane compo-\nnent in the timederivative of the precessional magnetiza-\ntion, the damping frequency ( \u0015?) is much larger than the\none where only in-plane magnetization is varying ( \u0015k).\nThe ratio\u0015?=\u0015kis larger than unity for both the bulk\nand a single layer and decreases with increasing temper-\nature. In SL FGT, \u0015?=\u0015kcan be enhanced up to 430%\nby slight holedoping at 50 K.\nAntiferromagnetic order has recently been discovered\nin 2D vdW materials (as reviewed in Ref. 60 and the ref-\nerences therein) and some intriguing properties are found\nin their damping behaviors.61,62Owing to the more com-\nplex magnetic order, more than a single parameter is\nnecessary in describing the damping in antiferromagnetic\ndynamics.63,64It would be very interesting to study the\nmagnetization relaxation in these 2D materials with more\ncomplex magnetic order.\nACKNOWLEDGMENTS\nThe authors are grateful to Professor Xiangang Wan\nat Nanjing University for his support and helpful dis-\ncussions. Financial support for this study was provided\nby the National Natural Science Foundation of China\n(Grants No. 11734004 and No. 12174028).6\n\u0003yiliu@bnu.edu.cn\n1Cheng Gong, Lin Li, Zhenglu Li, Huiwen Ji, Alex Stern,\nYang Xia, Ting Cao, Wei Bao, Chenzhe Wang, Yuan\nWang, Z. Q. Qiu, R. J. 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Wang,\n\\Proper dissipative torques in antiferromagnetic dynam-\nics,\" EPL (Europhysics Letters) 126, 67006 (2019)."    },    {        "title": "2211.01714v5.Skyrmion_Jellyfish_in_Driven_Chiral_Magnets.pdf",        "content": "Skyrmion Jellyfish in Driven Chiral Magnets\nNina del Ser1and Vivek Lohani1\n1Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany\n(Dated: July 20, 2023)\nChiral magnets can host topological particles known as skyrmions, which carry an exactly quan-\ntised topological charge Q=−1. In the presence of an oscillating magnetic field B1(t), a single\nskyrmion embedded in a ferromagnetic background will start to move with constant velocity vtrans.\nThe mechanism behind this motion is similar to the one used by a jellyfish when it swims through\nwater. We show that the skyrmion’s motion is a universal phenomenon, arising in any magnetic\nsystem with translational modes. By projecting the equation of motion onto the skyrmion’s transla-\ntional modes and going to quadratic order in B1(t), we obtain an analytical expression for vtransas a\nfunction of the system’s linear response. The linear response and consequently vtrans are influenced\nby the skyrmion’s internal modes and scattering states, as well as by the ferromagnetic background’s\nKittel mode. The direction and speed of vtrans can be controlled by changing the polarisation, fre-\nquency and phase of the driving field B1(t). For systems with small Gilbert damping parameter α,\nwe identify two distinct physical mechanisms used by the skyrmion to move. At low driving frequen-\ncies, the skyrmion’s motion is driven by friction, and vtrans∼α, whereas at higher frequencies above\nthe ferromagnetic gap, the skyrmion moves by magnon emission, and vtrans becomes independent\nofα.\nI. INTRODUCTION\nMedusas, commonly known as jellyfish, are remarkable\ncreatures. At over 500 million years of age, which is one\nhundred times older than Homo sapiens, they are the old-\nest multi-organ animal on Earth. Jellyfish usually con-\nsist of a bell-like structure with tentacles attached to it.\nWhile the tentacles serve to stun and catch prey, the jel-\nlyfish actually relies on its bell to swim. By periodically\nrelaxing and contracting the bell, the jellyfish generates\nmini-vortices in the surrounding water which help propel\nit forward. Over the course of this paper we will show\nthat a magnetic skyrmion, a kind of microscopic particle\nwhich occurs naturally in certain classes of magnets, can\nalso move through its environment using a mechanism\nvery similar to that of the jellyfish.\nChiral magnets are predominantly ferromagnetic ma-\nterials where the inversion symmetry of the crystal lattice\nis broken. The energy term responsible for this is known\nas the Dzyaloshinskii-Moriya Interaction (DMI) and orig-\ninates from weak spin-orbit interactions [1]. Physically\nthe DMI favours neighbouring spins to be perpendicular\nto each other, thus encouraging twisting in the magnetic\ntexture. Chiral magnets can host different textures in-\ncluding helical and conical phases [2, 3], where the spins\nwind around a pitch vector q. The reciprocal lattices for\nthese phases carry just one finite momentum ±qmode,\nwhich also means they are topologically trivial textures.\nIn a small phase pocket near a critical temperature Tc,\nand in the presence of a stabilising external magnetic\nfield, chiral magnets can also host a skyrmion phase, con-\nsisting of a hexagonal lattice of topologically quantised\nmagnetic whirls called skyrmions [4]. Single skyrmions\ncan also be created by irradiating ferromagnetic samples\nwith spin-polarised currents using STM tips [5, 6]. Due\nto its non trivial spatial structure, a single skyrmion has\ninfinitely many kmodes in the Fourier domain, unlikethe helical and conical textures, and additionally carries\na finite quantised topological charge Q=−1.\nGenerally, a skyrmion will start to move in the presence\nof external forces if enough symmetries are broken in the\nsystem. One option is to break the translational symme-\ntry, for instance by subjecting the skyrmion to magnetic\n[7–9] or temperature [10, 11] field gradients, electric or\nspin currents [12, 13], by driving it with an oscillating\nmagnetic field near a wall [14, 15], or even by firing at\nit with magnons [16, 17]. Another possibility is to break\nFIG. 1. A two-dimensional N´ eel skyrmion driven by a tilted\noscillating field B1(t) =B1(cos(Ω t),sin(Ω t) cos(Ω t)) starts to\nmove in the xy-plane with constant velocity vtrans∝B2\n1. The\nskyrmion moves through the ferromagnetic bulk as a result of\nperiodic deformations arising at order O(B1) from the exci-\ntation of its own internal breathing mode, the ferromagnetic\nbackground’s Kittel mode, and scattering states above the fer-\nromagnetic gap. Here the driving frequency Ω = Ω Kit., which\nresonantly excites the m=±1 momentum sectors, as a result\nof which the rotational symmetry of the skyrmion gets visi-\nbly broken. A jellyfish swims through water using a similar\nmechanism of cyclic asymmetric contractions and expansions\nof its bell.\nthe combined time translation and rotational (around the\nezaxis) symmetries. There are several numerical studiesarXiv:2211.01714v5  [cond-mat.mes-hall]  19 Jul 20232\nexploiting this second mechanism, including [18], where\na skyrmion was driven with a homogeneous out-of-plane\noscillating field in the presence of an in-plane static mag-\nnetic field Bxex, as well as [14], where there was no Bxex,\nbut the driving field was tilted. It is also possible to break\nthe rotational symmetry of the skyrmion by coupling its\nelectric dipole moment to a homogeneous oscillating elec-\ntric field, see [19]. In this work, we will investigate the\nsame setup as [14], but our results of the skyrmion veloc-\nities differ and the full analytical treatment we will pro-\nvide is novel. We will explain precisely how the skyrmion\nmoves as a result of periodic asymmetric deformations in\nits shape, see Fig. 1, and why the details of this mecha-\nnism share similarities with a jellyfish swimming through\nwater.\nWhen a weakly oscillating spatially homogeneous mag-\nnetic field B1(t) is applied to a magnetic system, to linear\norder in perturbation theory magnons oscillating at fre-\nquency ±Ω, where Ω is the driving frequency, will be\nexcited. Going beyond linear response, at order O(B2\n1)\nwe would na¨ ıvely expect frequency responses at 0 and 2Ω.\nUpon closer inspection, the O(B2\n1) response in fact also\nadmits a mode which grows linearly in time. In a previ-\nous work [20], we studied this in the case of driven helical\nand conical phases, and showed that this linear in tgrow-\ning mode corresponds precisely to the translational mode\nof the helix in the ez-direction. Due to the screw sym-\nmetry, the translational motion can equivalently be inter-\npreted as the helix rotating on its axis at constant angular\nvelocity Ω screw, mimicking an Archimedean screw. In the\ncase of the Archimedean screw, the translational and ro-\ntational modes of the magnetic texture coincide, however\nthis is not generally the case. For instance, in a skyrmion\nlattice the ezaxis rotational mode is completely separate\nfrom any of its translational modes. Driving a skyrmion\nlattice with an oscillating B1(t)ezwill in fact also ac-\ntivate this rotational mode. Recently, this phenomenon\nwas observed in experiments with femtosecond lasers [21],\nwith large rotation speeds of 2 ×107–108deg s−1achieved\nnear the breathing resonance of the skyrmion lattice. In\nthe present work we will be interested in what happens\nwhen we drive a single skyrmion. We will show that in\nthis case, the two translational modes of the skyrmion in\nthexy-plane get activated and the skyrmion acquires a\nconstant velocity vtrans= (vx\ntrans, vy\ntrans,0)T.\nIn what follows, we will first show that an oscillating\nspatially homogeneous magnetic field B1(t) universally\nactivates the translational mode(s) of chiral magnets.\nThis manifests as an additional O(B2\n1) force in the Thiele\nequation, which we derive as a function of the linear or-\nder response of the magnet. We then calculate this lin-\near response specifically for a driven single skyrmion and\nuse it to evaluate its second order translational velocity\nvtrans. We show that the skyrmion has two different ways\nof achieving maximal vtrans, related to either activating\nits own internal breathing resonance, or the background\nferromagnetic Kittel resonance. This is analogous to a\njellyfish swimming by itself versus relying on backgroundocean currents to help it swim, except of course that in\nthe case of the skyrmion all the power comes from the\ndriving magnetic field. Just like for the jellyfish, friction\nvia the phenomenological Gilbert damping term plays a\nfundamental role in our theory for the skyrmion. Para-\ndoxically, we even find that in some driving regimes (be-\nlow the ferromagnetic gap), the skyrmion actually moves\nfaster in systems with more damping. In the final part of\nthe paper we discuss what happens to vtrans in various\nexperimentally accessible driving frequency and damp-\ning limits, and obtain an excellent match between our\nanalytical predictions and micromagnetic simulations.\nII. MODEL\nWe consider a chiral magnet whose free energy is given\nby\nF=Z\nd3r\u0014\n−J\n2ˆM· ∇2ˆM+DˆM·(∇ × ˆM)−M·Bext\u0015\n,\n(1)\nwhere ˆM=M/M0, encodes the unit local magnetisa-\ntion and we are using spins of fixed length M0=|M|.\nFor simplicity we neglect dipole-dipole interactions. We\nconsider an external magnetic field consisting of a static\ncomponent B0and a dynamic driving component B1(t),\nBext=B0+ϵB1(t), B0= (0,0, B0)T,\nB1(t) = (Bx\n⊥cos(Ω t), By\n⊥sin(Ω t), Bzcos(Ω t+δ))T.(2)\nWe will concentrate on the weak driving limit B1/B0≪\n1, using ϵas a book-keeping parameter in perturbation\ntheory in the later sections. In the absence of driving,\nB1= 0, and close to the critical temperature Tc, the\nground state of Eq. (1) is a hexagonal skyrmion lattice. If\nthe stabilising B0field is large enough, a single skyrmion\nalso becomes a stable excitation of the ferromagnetic\nphase [17]. This is the texture we will be interested in\nhere. Due to its rotational symmetry, a single skyrmion\nis most naturally parametrised using polar coordinates,\nˆM(0)=\nsin (θ0(r)) cos( χ+h)\nsin (θ0(r)) sin( χ+h)\ncos (θ0(r))\n, (3)\nwhere r=p\nx2+y2is the radial coordinate and χ=\narctan ( y/x) is the polar angle. To model the ferromag-\nnetic background, where ˆM(0)=ez, we set the boundary\ncondition lim r≫r0θ0(r) = 0, where r0is the radius of the\nskyrmion. In Eq. (3), his short for helicity, a parameter\ndetermined by the type of DMI used in the model. In\nEq. (1), we used the bulk form of DMI, in which case\nthe free energy is minimised when h=π/2, giving rise\nto Bloch skyrmions. We could instead have used interfa-\ncial DMI, given by D\u0010\nˆMz∇ ·ˆM−(ˆM· ∇)ˆMz\u0011\n. In that\ncase, h= 0 minimises F, giving rise to N´ eel skyrmions.3\nEq. (3) is translationally invariant in the ez-direction, so\nthat in 3D bulk materials the texture forms skyrmion\ntubes.\nTo determine the radial dependence of the angle θ0(r),\nwe substitute the skyrmion ansatz Eq. (3) into Eq. (1)\nand solve the Euler-Lagrange equation\nδF\nδθ0=1\nrd\ndr\u0012\nrδF\nδθ′\n0\u0013\n, (4)\nwhere θ′\n0=dθ0\ndrand the extra r-factor on the right\nside comes from using polar coordinates. For nota-\ntional clarity, we now switch to dimensionless radial\nunits ˜ r= (D/J)rand reduced magnetic field units\nb0=\u0000\nM0J/D2\u0001\nB0. Eq. (4) then evaluates to\nθ′′\n0+1\nrθ′\n0−sin(θ0)\nr2\u0000\nb0r2−2rsin(θ0) + cos( θ0)\u0001\n= 0,(5)\nwith boundary conditions θ0(r= 0) = πandθ0(r=\n∞) = 0, where for simplicity we dropped the tilde on ˜ r.\nEq. (5) is identical for both Bloch and N´ eel skyrmions, so\nconveniently only needs to be solved once. The asymp-\ntotic limits r→0 and r→ ∞ admit the leading order\nsolutions\nθ0(r→0)≈π+θ′\n0(0)r,\nθ0(r≫r0)≈Ae−√b0r\n√r. (6)\nIn the intermediate regime 0 < r < r 0,θ0(r) has no\nknown analytical solution, so we solve for it numer-\nically using a shooting method implemented on julia\n[17, 22, 23]. Using the initial condition θ0(0) = π, we\nvary θ′\n0(0) until θ0(r≫r0) decays to zero. For large\nenough r, we match this numerical solution to the ana-\nlytical asymptotic Eq. (6), giving us a numerical value for\nA. The resulting profile θ0(r) depends only on the static\nexternal field: the larger b0is, the faster θ0(r) decays to\nzero, and therefore the smaller the skyrmion. For simplic-\nity, we set b0= 1 in the plots and numerical simulations\nshown in later parts of the text, safe in the knowledge\nthat changing the value of b0will shift the resonances in\nthe frequency spectrum but not introduce any other new\nfeatures.\nIII. ACTIVATING THE TRANSLATIONAL\nMODE(S)\nWe now turn on the driving field B1(t). The magnetic\ntexture evolves in time according to the Landau-Lifshitz-\nGilbert (LLG) equation,\n˙M=γM×Beff−γ\n|γ|αˆM×˙M, (7)\nwhere Beff=−δF[M]\nδMis the effective magnetic field, cal-\nculated by taking the functional derivative of the freeenergy, Eq. (1). The term proportional to αis a phe-\nnomenological damping term and γ=qg/(2m) is the gy-\nromagnetic ratio. In our convention, γe=−|e|g/(2me)\nis negative for an electron with charge −|e|, mass meand\ng-factor g, which is also the case in most magnetic sys-\ntems. Nevertheless we have included the prefactor γ/|γ|\nto ensure that Eq. (7) and all ensuing expressions are also\nvalid for systems with positive γ. Note that Eq. (7) is\nthe general equation of motion for anymagnetic texture,\nnot just the skyrmion. Consequently, the phenomenon\nwe are deriving is universal.\nAsB1(t) is assumed to be weak, we can expand the\nmagnetisation perturbatively,\nˆM(r, t) =ˆM(0)(r) +ϵM(1)(r, t) +ϵ2M(2)(r, t) +O(ϵ3),\n(8)\nwhere ϵis the same book-keeping parameter we intro-\nduced in Eq. (2) to keep track of the powers of B1. Linear\nresponse dictates that M(1)(r, t) should exactly mimic\nthe frequency dependence of the drive. For the type of\nmonochromatic driving field we are using in Eq. (2), this\nimplies that M(1)(r, t) must also be purely oscillatory,\nwith frequency components ±Ω. Thus any net transla-\ntional motion, which is linear in t, can only enter at O(ϵ2)\nor above. A more accurate ansatz describing a magnetic\ntexture which is being translated in time at a constant\nO(ϵ2) velocity vtrans is given by\nˆM(r, t) =ˆM(0)\u0000\nr−ϵ2vtranst\u0001\n+ϵM(1)\nosc.(r, t)\n+ϵ2\u0010\nM(2)\nosc.(r, t) +M(2)\nstat.(r)\u0011\n+O(ϵ3),(9)\nwhere M(1)\nosc.∼e±iΩt,ˆM(2)\nosc.∼e±2iΩt, while ˆM(2)\nstat.∼\ne0iΩtis time-independent. Comparing Eq. (9) and\nEq. (8), we see that Eq. (8) is only a valid approximation\nat short times, as the O(ϵ2) terms grow linearly in tand\nwill eventually get larger than the O(ϵ0, ϵ1) terms. We\ncan get rid of this problem by differentiating everything\nonce with respect to time. Taking the time derivatives of\nEq. (8) and Eq. (9) and collecting the O(ϵ1) and O(ϵ2)\nterms separately, we have\n˙M(1)(r, t) =˙M(1)\nosc.(r, t), (10)\n˙M(2)(r, t) =−(vtrans·∇)ˆM(0)(r) +˙M(2)\nosc.(r, t).(11)\nSubstituting Eq. (11) into Eq. (7), operating on it withR\nd3r∇iˆM·(ˆM×) and time averaging over one period of\noscillation, T= 2π/Ω, see App. B.1 for technical details,\nwe obtain a Thiele-inspired equation for vtrans,\n−sgn(γ)G×vtrans+αDvtrans=Ftrans[ˆM(0),M(1)],\n(12)\nwhere the gyrocoupling vector Gand the dissipation ma-\ntrixDare defined in the usual way as functions of the\nstatic texture ˆM(0),\nGα=1\n2ϵαβγZ\nd3rˆM(0)·\u0010\n∇βˆM(0)×∇γˆM(0)\u0011\n,\nDαβ=Z\nd3r∇αˆM(0)· ∇βˆM(0),(13)4\nand the second order force is given by\nFi\ntrans. =D\n−sgn(γ)Z\nd3rˆM(0)·\u0010\n˙M(1)×∇iM(1)\u0011\n+αZ\nd3r˙M(1)· ∇iM(1)E\nt, (14)\nwhere ⟨. . .⟩tdenotes time averaging over one period of\noscillation T= 2π/Ω. Note that Eq. (12) is a completely\ngeneral Thiele-inspired equation, valid for anymagnetic\ntexture driven by a homogeneous oscillating magnetic\nfield. Conveniently, the effective force Ftrans only de-\npends on the static texture ˆM(0)and the linear response\nM(1)— this means we don’t need to know anything\nabout the oscillatory and static responses M(2)\nosc.,M(2)\nstat.\nto calculate vtrans. Remarkably, there is also no trace\nofBeff, which disappeared from the equation when we\nintegrated over space due to the translational symmetry\nof the free energy, see App. B.1. There is an alternative\n(longer) way to derive Eq. (12), which involves writing\ndown a continuity equation in terms of the stress energy\ntensor of the system, see App. C. This approach is useful\nfor investigating the momentum and current densities of\nthe magnons excited by the driving and will be exploited\nin Sec. V.\nThe presence of the ∇iM(1)terms in Eq. (14) implies\nthat we need some spatial modulation in M(1)if we want\nto activate the translational modes of the texture. Imme-\ndiately, we can conclude that vtrans = 0 for a ferromag-\nnet, as driving a ferromagnet with a homogeneous B1(t)\nwill generate a spatially homogeneous M(1)(t). The next\nsimplest magnetic textures to consider are the helical and\nconical phases of chiral magnets. If we set up our axes\nsuch that the helical pitch vector q=qez, the helical\nand conical phases are translationally invariant in the xy-\nplane, but spatially modulated in the ez-direction. Driv-\ning a helical or conical phase generates magnons travel-\nling in the ±ez-direction at linear order, M(1)∼ei(qz±ωt)\n[20]. Substituting this into Eq. (14), we immediately no-\ntice that the ˆM(0)·(˙M(1)×∇zM(1)) term vanishes, as\n˙M(1)∥ ∇ zM(1). This leaves only the two dissipation\nterms in Eq. (12), and the common factor αcancels to\ngive a velocity\nvtrans=ezRλ\n0dzD\n˙M(1)· ∇zM(1)E\ntRλ\n0dz|∇zˆM(0)|2, (15)\nwhere λ= 2π/qis the helix wavelength. The transla-\ntional velocity vtrans can equivalently be interpreted as\na rotational angular velocity Ωscrew =qvtransezdue to\nthe screw symmetry of the helical and conical phases.\nThe resulting magnetisation dynamics is reminiscent of\na rotating Archimedean screw, see also [20] for a video\nand additional details. Notice that in Eq. (15), we used\nthe discrete lattice symmetry of the helical and conical\nphases to integrate only over one helix winding λ, rather\nthan the whole volume of the magnet. This trick can\ngenerically be used in other repeating textures, such asthe skyrmion lattice, but will not work in systems lacking\nlattice symmetry, such as a single skyrmion.\nDespite the multiple qualities of Eq. (14), using it to\ncalculate vtrans doesn’t always work in practice. This is\ndue to problems of convergence which arise in the inte-\ngration step during the calculation of Ftrans. If there is\nno underlying lattice symmetry and the linear response\nM(1)decays very slowly over space, the integration do-\nmain needs to be huge, making a numerical implementa-\ntion impractical. Luckily, there is an alternative way of\ncalculating the force which bypasses this difficulty. By\nprojecting ∇iˆM(0)·(ˆM(0)×onto Eq. (7), see App. B.2\nfor details, we again obtain the Thiele equation Eq. (12),\nbut the force is calculated in a different way,\n˜Fi\ntrans=|γ|DZ\nd3rh\u0010\n∇iˆM(0)·M(1)\u0011\u0010\nˆM(0)·B(1)\neff\u0011\n+1\n2\u0010\nM(1)·M(1)\u0011\n∇i\u0010\nˆM(0)·B(0)\neff\u0011iE\nt.(16)\nEq. (16) is not valid in general for any magnetic texture\nas it requires some surface terms vanishing at infinity,\nsee App. B.2. However, it is always valid if ∇iM(0),\nwhich parametrises the texture’s translational modes,\nis bounded. This is the case for example for a single\nskyrmion, where as we leave the skyrmion and enter the\nspatially homogeneous ferromagnetic bulk, ∇iM(0)→0.\nIf this condition is fulfilled, then the integrands con-\ntained in Eq. (16) are also always bounded, as a bounded\n∇iM(0)also implies a bounded ∇iB(0)\neff. Hence, irrespec-\ntive of the behaviour of M(1), the integrands in Eq. (16)\nare always bounded by ∇iM(0). This makes a practi-\ncal numerical implementation of the integral achievable.\nThe price to pay when using this method compared to\nEq. (14) is that one also needs to calculate the B(1)\neffterms\n— overall, the integrand in Eq. (16) is therefore a much\nmore complicated object. Also, investigating what hap-\npens to the force in various limiting values of α,Ω is very\nchallenging using Eq. (16), but easy to do using Eq. (14).\nWhen using the effective Thiele equation Eq. (12), it is\nimportant to be aware of the requirements for its validity.\nThe main assumptions used in the derivation of this equa-\ntion are i), that the system has already reached its steady\nstate (meaning all transients generated in the response\nat the start have by now decayed) and ii), that we are\nworking on time scales much larger than the other char-\nacteristic time scales of the system, including the driving\nfrequency Ω. Effectively, we are “integrating out” the\nshort-timescale fluctuations in favour of calculating just\ntheO(ω0) drift of the magnetic texture. We have also not\nincluded any effects caused by inertia terms [24, 25], ex-\npected to arise as a consequence of the periodic deforma-\ntions of the driven texture. This is because in the steady-\nstate limit we are considering, vtrans =˙R= const.,\nso that any contribution from a mass term would van-\nish,m¨R= 0. Under these conditions, we will use both\nEq. (14) and Eq. (16) to calculate and understand vtrans\nfor a single skyrmion in Sec. V. To do this, we first need5\nthe linear response M(1), which we calculate in the next\nsection.\nIV. LINEAR RESPONSE FOR THE DRIVEN\nSKYRMION\nThe goal of this section is to solve Eq. (7) to order\nO(ϵ) to obtain the linear response M(1). Instead of doing\nthis directly, we will first rewrite our problem in the lan-\nguage of bosonic fields, inspired by (but not identical to)\nthe Holstein-Primakoff approach for quantum spins. Our\napproach will also include the effects of the phenomeno-\nlogical damping α, which is not as evident to describe\nusing quantum Hamiltonians. We begin by parametris-\ning the magnetisation as\nM=M0 \ne3(1−a∗a) +r\n1−a∗a\n2(e−a+e+a∗)!\n,\n(17)\nwhere a(r, t), a∗(r, t) are complex time- and space- de-\npendent fields. Note that, contrary to the quantum case,\nthe ordering of a∗andais unimportant here, as they\nare complex numbers rather than operators. In the co-\nordinate system we use, e3is parallel to the zeroth order\nmagnetisation ˆM(0)given in Eq. (3), while e±span the\nplane perpendicular to ˆM(0),\ne3=ˆM(0),e∓=1√\n2\ncos(θ0) cos( ϕ)±isin(ϕ)\ncos(θ0) sin(ϕ)∓icos(ϕ)\n−sin(θ0)\n,\n(18)\nwith ϕ=χ+h. Using Eq. (18), it can be checked that the\nimportant property |M|=√\nM·M=M0is preserved by\nthe parametrisation introduced in Eq. (17). By imposing\nthe Poisson bracket {a(r), a∗(r′)}=δ(r−r′), we further\nensure that\n{ˆMi(r),ˆMj(r′)}=iϵijkˆMkδ(r−r′) (19)\ntoallorders of a, a∗. Eq. (19) implies that i{F,ˆM(r)}=\nM×Bext, letting us rewrite Eq. (7) as\nsgn(γ)˙ˆM=i{F,ˆM} −αˆM×˙ˆM, (20)\nwhere we used dimensionless time units defined by\n˜t=D2|γ|/(JM0)t, with corresponding dimensionless fre-\nquency ω=JM0/(D2|γ|)Ω, rescaled the free energy\n˜F= (D/J2)F, and then immediately dropped the tildes\non˜t,˜Fto make the formulas neater. Eq. (20) resem-\nbles Hamilton’s equation of motion, familiar to us from\nclassical mechanics, but additionally includes the phe-\nnomenological damping α. Projecting Eq. (20) onto e±,\nwe obtain non-linear equations of motion for aanda∗,\nrespectively, which are listed in Eq. (D1). We can expandaanda∗perturbatively in powers of ϵ,\na=ϵa(1)+ϵ2a(2)+O(ϵ3),\na∗=ϵa∗(1)+ϵ2a∗(2)+O(ϵ3).(21)\nNotice that a, a∗is to lowest order O(ϵ1) and notO(ϵ0),\nunlike the expansion for ˆMin Eq. (8). This is because\nby definition, aanda∗vanish when the driving field is\nturned off, B1(t) = 0. Within our perturbative scheme,\nit is sufficient to solve Eq. (D1) to order O(ϵ1), which\nmeans we only need to retain those terms which are at\nmost linear in a, a∗. This leads to the following linearised\nequations\n(sgn( γ) +iα) ˙a=in\nF(1)\ndrive+F(2)\nno drive, ao\n,\n(sgn( γ)−iα) ˙a∗=in\nF(1)\ndrive+F(2)\nno drive, a∗o\n,(22)\nwhere F(1)\ndriveandF(2)\nno drive, refer to the O(a1) and O(a2)\ncontributions in the free energy given in Eq. (E2)\nand (E4), respectively. In our notation, F(n)\ndriverefers\nto terms which are explicitly dependent on the exter-\nnal driving field, while F(n)\nno driverefers to all the other\nfree energy terms. After the Poisson bracket operation\nin Eq. (22), there will be O(a1) terms and O(a0) terms\ncoming from {F(2)\nno drive, a}and{F(1)\ndrive, a}, respectively.\nTheO(a0) terms coming from {F(1)\ndrive, a}provide a driv-\ning force f, given in Eq. (F1), which carries only three\nazimuthal angle Fourier components eimχ:m=−1,0\nand 1. We use the Fourier convention\na(r, χ) =X\nmeimχam(r),\na∗(r, χ) =X\nmeimχa∗\n−m(r),\nf(r, χ) =X\nmeimχfm(r),(23)\nand accompanying Poisson bracket,\n{am(r), a∗\nm′(r′)}=1\n2πrδmm′δ(r−r′) (24)\n(note the factor of 1 /r, a consequence of using polar co-\nordinates!), to write Eq. (22) as a single matrix equation\ni(sgn( γ) +iασz) \n˙a(1)\nm\n˙a∗(1)\n−m!\n=σzHm \na(1)\nm\na∗(1)\n−m!\n+\u0012\nfm\n−f∗\n−m\u0013\n,\n(25)\nwhere Hmis a 2×2 matrix defined in Eq. (F2) and\nf−1=−1\n4√\n2(cos(θ0) + 1)\u0010\nbRe−i(h−ωt)+bLe−i(h+ωt)\u0011\nf0=1\n2√\n2bzsin(θ0)\u0010\nei(ωt+δ)+e−i(ωt+δ)\u0011\n(26)\nf1=−1\n4√\n2(cos(θ0)−1)\u0010\nbRei(h−ωt)+bLei(h+ωt)\u0011\n.6\nAs we are only interested in solutions which oscillate in\ntime at frequency ±ω, we need only solve Eq. (25) for\nthe values of mwhere fmis non-zero. This means we\nonly need to solve the three cases m=−1,0 and +1.\nInserting the ansatz\n \na(1)\nm(t)\na∗(1)\n−m(t)!\n= \na(1)\nm,+ω\na∗(1)\n−m,−ω!\neiωt+ \na(1)\nm,−ω\na∗(1)\n−m,ω!\ne−iωt,\nfm=fm,+ωeiωt+fm,−ωe−iωt,(27)\nwhere a(1)\nm,±ω,a∗(1)\n−m,∓ω,fm,±ωare complex fields which\nonly depend on r, into Eq. (25), and collecting the coeffi-\ncients of the e±iωtterms, we obtain the time-independent\nmatrix equation\n∓ω(sgn( γ) +iασz) \na(1)\nm,±ω\na∗(1)\n−m,∓ω!\n=σzHm \na(1)\nm,±ω\na∗(1)\n−m,∓ω!\n+\u0012fm,±ω\n−f∗\n−m,∓ω\u0013\n.(28)\nWe will solve Eq. (28) by expanding a(1)\nm,±ω,a(1)\n−m,∓ωinto\nthe eigenbasis of σzHm. Sec. IV.1 explains how to ob-\ntain this eigenbasis including the effects of the Gilbert\ndamping α.\nIV.1. Damped eigenbasis of σzHm\nUsing the notation |m, k, i ⟩andEm,k,i to designate the\neigenvectors and eigenvalues respectively, the eigenvalue\nequation reads\nEm,k,i(sgn( γ) +iασz)|m, k, i ⟩=σzHm|m, k, i ⟩,(29)\nwhere kparametrises the energy and i={1,2}labels\nwhich of the two eigenvectors of σzHmwe are referring\nto. Using the property σxHmσx=H−m, it can be shown\nthat if |m, k, i ⟩is an eigenvector of σzHmwith eigenvalue\nEm,k,i, then σx|m, k, i ⟩∗is also an eigenvector of σzH−m\nwith eigenvalue −E∗\nm,k,i, see App. 1 for the derivation.\nThis means that |−m, k, 2⟩=σx|m, k, 1⟩∗andE−m,k,2=\n−E∗\nm,k,1, and we can write everything in terms of only\nthei= 1 eigenvector and eigenvalue, suppressing the\ni-index in the expressions that follow.\nDue to the presence of damping αin Eq. (29),\n|m, k⟩andEm,kwill in general be complex. Physically,\nRe(Em,k) corresponds to the frequency of the spin wave\nwhile Im( Em,k) quantifies its damping. Assuming that\nα≪1, we can expand everything perturbatively in pow-\ners of α,\n|m, k⟩=\f\f\fm, k(0)E\n+iαsgn(γ)\f\f\fm, k(1)E\n+O(α2),\nEm,k= sgn( γ)ϵ(0)\nm,k−iαϵ(1)\nm,k+O(α2).(30)\nStopping at linear order in αin Eq. (30) is sufficient to\ntake into account the leading order effects of damping,\n0 2 4 6 8 10 12 14−0.4−0.200.20.4\nr1\n2√\n2sin(θ0)\n−1\n4√\n2(cos(θ0) + 1)\n−1\n4√\n2(cos(θ0)−1)\n0 2 4 6 8 10 12 1400.20.40.60.81\nr/vextendsingle/vextendsingle/vextendsingle0,br.(0)/angbracketrightBig\nu/vextendsingle/vextendsingle/vextendsingle0,br.(0)/angbracketrightBig\nl/vextendsingle/vextendsingle1,trans.(0)/angbracketrightbig\nu/vextendsingle/vextendsingle1,trans.(0)/angbracketrightbig\nl\n0 2 4 6 8 10 12 14−0.500.511.52\nr/vextendsingle/vextendsingle−1,1.5(0)/angbracketrightbig\nu/vextendsingle/vextendsingle−1,1.5(0)/angbracketrightbig\nl/vextendsingle/vextendsingle0,1.5(0)/angbracketrightbig\nu/vextendsingle/vextendsingle0,1.5(0)/angbracketrightbig\nl/vextendsingle/vextendsingle1,1.5(0)/angbracketrightbig\nu/vextendsingle/vextendsingle1,1.5(0)/angbracketrightbig\nl\n0 1 2 3 4 5 6−0.200.20.40.6\nkk2˜c(0)\n−1,k,±ω\nk2˜c(2)\n−1,k,±ω\nc(0)\n0,k,±ω\nc(1)\n0,k,±ω\n˜c(0)\n1,k,±ω\n˜c(1)\n1,k,±ω(a) (b)\n(c) (d)FIG. 2. a) r-dependent factors of f0,f±1. While the factors\ninf0andf1are bounded and decay as r≫r0∼6, the factor\ninf−1tends to a finite constant value −1/(2√\n2) as r→ ∞ .\nb) Bound modes of the skyrmion. Only the m= 0 breathing\nmode and m=±1 translational modes are excited by spa-\ntially homogeneous driving. “ u” and “ l” stand for “upper”\nand “lower” component of the eigenvector, respectively. c)\nScattering states of the skyrmion. Here we show one example\nof a scattering state with m= 0,±1 and kset to 1 .5. A con-\ntinuum of such states with k >0 is excited by homogeneous\ndriving. d) Scattering state coefficients c(0,1)\n0,k,±ω, ˜c(0,1)\n±1,k,±ωfor\nbz, bR, bL= 1 and δ, h= 0, plotted as a function of k.\nthus we will only discuss how to obtain the O(α0) and\nO(α1) eigenvectors and eigenvalues. The problem of cal-\nculating\f\fm, k(0)\u000b\n,ϵ(0)\nm,khas been solved before, see [17]\nfor a detailed account. Very briefly, it involves using the\nshooting method with some initial conditions at r= 0,\nand imposing that the lower component of |m, k⟩decays\nto zero for r≫r0. The eigenspectrum consists of dis-\ncrete bound modes with ϵ(0)\nm< b0and a continuum of\nscattering states with ϵ(0)\nm> b0. For m= 0, there is a\nsingle bound mode known as the breathing mode, which\nwe label\f\f\f0,br.(0)E\n, with energy ϵ(0)\nbr.= 0.839 for b0= 1.\nForm= 1, there is also a a single bound mode, which\nhappens to be one of the skyrmion’s two translational\nmodes (the other one being in the m=−1 sector), with\nenergy ϵ(0)\ntrans= 0. This translational mode is known an-\nalytically,\n\f\f\fm= 1,trans.(0)E\n=1\n2√\n2 \nθ′\n0−sin(θ0)\nr\nθ′\n0+sin(θ0)\nr!\n, (31)\nand can be used to obtain the other translational\nmode via\f\fm=−1,trans.(0)\u000b\n=σx\f\fm= 1,trans.(0)\u000b\n.\nThe upper and lower components of\f\f\f0,br.(0)E\nand\f\fm= 1,trans.(0)\u000b\nare plotted as a function of rin\nFig. 2(b) — note how they are all confined to the\nskyrmion radius r0≃6. For the scattering states7\n\f\fm, k(0)\u000b\n, the energy is independent of m, and exists\non a continuum parametrised by ϵm,k=b0+k2. We\nshow some examples of the scattering states from the\nm= 0,±1 sectors in Fig. 2(c). Note that, unlike the\nbound modes, the upper components are notconfined\nto the skyrmion radius but instead penetrate far into\nthe ferromagnetic bulk. Far away from the skyrmion\nand after a few oscillations, the behaviour of the upper\ncomponents can be described analytically by Bessel\nfunctions,\nlim\nr≫r0,\n2π/k\f\f\fm, k(0)E\nu=Am,kJm+1(kr)−Bm,kYm+1(kr),\n(32)\nwhere Am,k= cos( δm,k),Bm,k= sin( δm,k), and δm,kare phase shifts required to match the near and far-field\nsolutions. The near-field ( r < r 0) behaviour and δm,k\nare obtained numerically during the shooting procedure.\nTogether, the bound and scattering states constitute a\ncomplete orthogonal eigenbasis for each m-sector, with\nthe eigenvectors satisfying the inner products listed in\nApp. G2. We can use this orthogonal eigenbasis to do\nperturbation theory in orders of α, see App. G3. The\nresulting first order corrections to the eigenstates and\neigenenergies,\f\fm, k(1)\u000b\nandϵ(1)\nm,k, are given in Eq. (G2)\nand (G3). Armed with the orthogonal eigenbasis of\nσzHm, it becomes — at least at first glance — much eas-\nier to solve Eq. (28). The idea is to expand the am,±ω,\na∗\n−m,∓ωfields in the bound and scattering states of the\nrelevant m-sectors,\n \na(1)\nm,±ω\na∗(1)\n−m,∓ω!\n=cm,bd.,±ω\n∓ω−Em,bd.|m,bd.⟩ −c∗\n−m,bd.,∓ω\n∓ω+E∗\n−m,bd.σx|−m,bd.⟩∗\n+Z∞\n0k dk\u0012cm,k,±ω\n∓ω−Ek|m, k⟩ −c∗\n−m,k,∓ω\n∓ω+E∗\nkσx|−m, k⟩∗\u0013\n,(33)\nwhere the coefficients cm,bd.,±ω,cm,k,±ωare complex\nnumbers which still need to be identified. Just like the\neigenbasis in Eq. (30), we can expand the c’s perturba-\ntively in α,\nc±m,bd.,±ω= sgn( γ)c(0)\n±m,bd.,±ω−iαc(1)\n±m,bd.,±ω+O(α2)\ncm,k,ω = sgn( γ)c(0)\n±m,k,±ω−iαc(1)\n±m,k,±ω+O(α2).\n(34)\nSubstituting Eq. (33) and (34) into Eq. (28), project-\ningD\nm,bd.(0)\f\f\for\nm, k(0)\f\fand using the orthogonality\nproperties given in App. 2, we find, at order O(α0),\nc(0)\nm,bd.,±ω=D\nm,bd.(0)\f\f\fσz\f\f\f\f\u0012fm,±ω\n−f∗\n−m,∓ω\u0013\u001d\n,\nc(0)\nm,k,±ω=D\nm, k(0)\f\f\fσz\f\f\f\f\u0012fm,±ω\n−f∗\n−m,∓ω\u0013\u001d\n.(35)\nFollowing the same procedure at order O(α1), we calcu-\nlatec(1)\nm,bd.,±ωandc(1)\nm,k,±ω, listing the resulting expres-\nsions in Eq. (G4). Let us take a closer look at Eq. (34)\nand (G4) to check that these coefficients are well-defined.\nForc(0,1)\nm,bd.,±ω, the presence of the bound state ⟨m,bd.|\nensures that the integrands are bounded, consequently\nthec(0,1)\nm,bd.,±ωcoefficients are always well-defined. For the\nscattering state coefficients c(0)\nm,k,±ωthis is no longer true,\nas\nm, k(0)\f\fonly decays with amplitude 1 /√r, so that the\nintegrand has overall r-dependence√rfm,±ω(r). We thus\nneed lim r≫r0fm,±ωto decay faster than 1 /√rfor the in-\ntegral to be well-defined. In Fig. 2(a), we plot fm,±ωas a\nfunction of rform= 0,±1, using the definitions given inEq. (26). While f0, f1both decay to zero as r≫r0,f−1\ntends to a finite constant value. Physically, this comes\nabout because f−1excites the ferromagnetic background\nrather than the skyrmion core. One way to check this\nis to consider what happens if we repeat our calculation\nfor a system without a skyrmion. In this case we would\nhave a simple ferromagnetic state with ˆM(0)∥ezevery-\nwhere, and consequently θ0(r) = 0, from which we can\nconfirm that the only driving component seen at linear\norder by the ferromagnet is f−1. The technical conclu-\nsion from this discussion is that while we can use Eq. (34)\nand (G4) successfully to determine c(0,1)\n0,±ωin Eq. (33), we\nneed a different approach for calculating c(0,1)\n±1,±ω. As we\nwill show in Sec. IV.2, we can solve the problem caused\nby the f−1component by separately defining and con-\nstructing a |m=−1, k= 0⟩mode.\nIV.2. |m=−1, k= 0⟩mode\nAs we are driving our system with a spatially homo-\ngeneous oscillating magnetic field, we excite the k= 0\nmagnon modes of the ferromagnetic bulk. Given that the\nscattering modes oscillate with wavelength λ= 2π/k, the\nk= 0 magnons have infinite wavelength. In other words,\nforr≫r0the|m, k = 0⟩wave-functions must reach\na constant r-independent value. To find out whether\nthis constant value is finite or zero, we can consider the\nscattering states of a ferromagnet. These are given by\n|m, k⟩u=Jm+1(kr),|m, k⟩l= 0. Using Jn(0) = δn0, we\nconclude that |m,0⟩u,lis only finite when m=−1. Thus,\nthe only relevant k= 0 mode in a system driven by a8\n0 2 4 6 8 1005101520\nrδM/ 10−2bz= 0.01,bR=bL= 0,m= 0\nω= 0.76\nω= 0.84\nω= 0.95\nω= 1.00\nω= 1.05\nω= 1.10\n0 10 20 30 4005101520\nrbR= 0.01,bL=bz= 0,m=±1\n0 10 20 30 4000.10.20.30.4\nrbL= 0.01,bR=bz= 0,m=±1\n(a) (b) (c)\nFIG. 3. δMas a function of rfor different driving frequencies ωand polarisations bz, bR, bL, with α= 0.03,b0= 1 and\nγ <0. a) Out-of-plane driving bz= 0.01, bR=bL= 0. δMis largest at the breathing mode resonance ωbr.= 0.839 and is\nmostly confined to the region of the skyrmion, r≲r0. b) In-plane right-polarised driving bR= 0.01, bz=bL= 0. The Kittel\n(background ferromagnet) mode at ωKittel = 1 is resonantly excited. c) In-plane left-polarised driving bL= 0.01, bz=bR= 0.\nThe Kittel mode is notresonantly excited, consequently δMis two orders of magnitude smaller than for right-polarised driving.\nIn a material with γ >0 the situation would be reversed, with the Kittel mode being resonantly excited by left-polarised driving.\nIn the in-plane-polarised driving case, δMtends to a constant value in the bulk rather than decaying to zero, due to the k= 0\nmode. For all three driving polarisations, when ω > b 0= 1 we resonantly excite the relevant scattering mode |m, k⟩with\nk=√ω−b0, but the time-averaging washes these out in panels (a) and (c). Meanwhile, interference between the aconst\n−1,+ωand\nascatt\n−1,+ωgives visible oscillations in panel (b). Finite damping ensures that these oscillations decay with length scale l∼vg/(αω).\nspatially homogeneous magnetic field is |m=−1, k= 0⟩.\nNote that the arguments made in this paragraph are per-\nfectly general and apply not only to a single skyrmion,\nbut also to any other localised magnetic defect embedded\nin a ferromagnet.\nWhile the need for a |m=−1, k= 0⟩skyrmion scat-\ntering mode is conceptually clear, trouble arises when we\nattempt to implement it numerically using the shooting\nmethod. The reason is the following: for the scattering\nmodes, as long as kis not extremely small, the oscil-\nlations of |m=−1, k⟩uconveniently penetrate into the\nbulk of the ferromagnet without significant numerical er-\nrors. Indeed, in this case, the energy term in the eigen-\nvalue equation starts to dominate the potentials (evalu-\nated at machine precision) at a finite distance from the\nskyrmion core. However, as k→0, we need to go further\nand further away from the skyrmion core for the energy\nterm to completely overshadow the potentials and ex-\nhibit the oscillatory nature of the wavefunction in the\nbulk. Unfortunately, this very fact renders the shoot-\ning numerics invalid at tiny values of k, as small inac-\ncuracies in the numerical value of the potentials cannot\nbe eliminated up to very large distances and they end\nup fundamentally changing the nature of the numerical\nsolution. The manifestly unstable nature of the shoot-\ning numerics in the small kregime can actually be at-\ntributed to the fact that the |m=−1, k= 0⟩mode with\nϵ=b0is a boundary mode across which the long-distance\nasymptotics of the (upper) wavefunction changes from\nexponential (for 0 < ϵ < b 0) to decaying oscillatory (for\nϵ > b 0). In fact, precisely at k= 0, the eigenvalue equa-\ntion Eq. (29) admits two allowed asymptotic solutions,\nlimr≫r0|m=−1, k= 0⟩u=A+Blog(r). Physical in-\ntuition then dictates that B= 0, but numerical inaccu-\nracies in the code (which are unavoidable) always resultin a finite B. An alternative approach to the shooting\nmethod is therefore needed. Here we will propose a so-\nlution where we write the |m=−1, k= 0,±ω⟩mode as\na superposition of f±1,±ωand all the other modes from\nthem=−1 sector,\n|−1,0,±ω⟩= (sgn( γ) +iασz)−1\u0012f−1,±ω\n−f∗\n1,∓ω\u0013\n+Z∞\nk>0k dk\u0000\n˜c−1,k,±ω|−1, k⟩ −˜c∗\n1,k,∓ωσx|1, k⟩∗\u0001\n+ ˜c−1,trans. ,±ω|−1,trans.⟩, (36)\nwhere f−1,±ωtakes care of the upper component tending\nto a constant value as r≫r0. The role of all the other\nm=−1 modes is to ensure that the eigenvalue equation,\nE0(sgn( γ) +iασz)|−1,0,±ω⟩=σzH−1|−1,0,±ω⟩,\n(37)\nis fulfilled. Perturbatively expanding the ˜ c(0,1)\nm,k,±ω’s anal-\nogously to the c(0,1)\nm,k,±ω’s in Eq. (34), we find\n˜c(0)\n−1,k,±ω=−1\nk2D\n−1, k(0)\f\f\f(H−1−b0σz)\f\f\f\f\u0012f−1,±ω\n−f∗\n1,∓ω\u0013\u001d\n,\n(38)\n˜c∗(0)\n1,k,∓ω=1\n2b0+k2D\n1, k(0)\f\f\fσx(H−1−b0σz)\f\f\f\f\u0012f−1,±ω\n−f∗\n1,∓ω\u0013\u001d\nand ˜ c(0,1)\n−1,trans. ,±ω= 0, with the ˜ c(1)\n±1,k,±ω’s given in\nEq. (G6). To check that the integrands in all the ˜ c(0,1)’s\nare now bounded, let us look at the r≫r0limit of the\nH−1−b0σzand ( b0−H−1)σzoperators found inside these\nexpressions. Using lim r≫r0H−1=1b0+2\nr2(1−σz) we9\nhave\nlim\nr≫r0(H−1−b0σz) = (1−σz)\u0012\nb0+2\nr2\u0013\n,\nlim\nr≫r0(b0−H−1)σz=−2\nr2(1−σz).\nThe presence of the ( 1−σz) factor in both of these terms\nguarantees that the f−1,±ωcomponent, which was caus-ing the problematic unbounded behaviour in the inte-\ngrands of Eq. (35) and (G4), gets pushed out of the in-\ntegrand at large r. This makes the integrands overall\nbounded, as they vanish for r≳r0. Consequently, the\n˜c(0,1)\n±1,±ω’s coefficients are completely well-defined.\nWe can use the newly defined |−1,0,±ω⟩mode from\nEq. (36) to write an ansatz for the a1,±ω,a∗\n−1,±ωfields,\n \na(1)\n1,±ω\na∗(1)\n−1,∓ω!\n=−1\n∓ω+E∗\n0σx|−1,0,∓ω⟩∗−Z∞\nk>0k dk\u0012˜c1,k,±ω\n∓ω−Ek|1, k⟩ −˜c∗\n−1,k,∓ω\n∓ω+E∗\nkσx|−1, k⟩∗\u0013\n. (39)\nUsing the eigenvalue equation Eq. (29), it is straightfor-\nward to verify that Eq. (39) solves Eq. (28). The tech-\nnique which we used here of separating the k= 0 mode\nfrom all the finite kmodes is inspired by the well-known\ntrick of separating the BEC mode from all the finite en-\nergy excitations in a bosonic gas [26]. Here, however,\nthe defining feature of the k= 0 mode is not that it\ncosts zero energy (it costs finite energy b0) — rather, its\n“specialness” is observed in the first order dynamics of\nthe driven system. Inspecting Eq. (39), we can see that\nwhen driven on-resonance, only the k= 0 mode will be\nresonantly excited and grow like 1 /αin the limit of small\ndamping. On the other hand, any resonance excited in\nthe finite k-modes subsequently gets smeared out by the\nintegration over k.\nIn Fig. 2(d) we plot c(0,1)\n0,k,±ω, ˜c(0,1)\n±1,k,±ωas a function of k.\nAll the coefficients decay to zero for k≳6, which provides\na natural upper k-cutoff for the numerical implementa-\ntion of expressions Eq. (33) and (39). Note the presence\nof the 1 /k2factor in ˜ c(0,1)\n−1,k,∓ω, which causes a singularity\nask→0. While this would be problematic if we really\nwanted to obtain the k= 0 mode in isolation, when com-\nbined with the finite kscattering modes in Eq. (39) the\n1/k2singularities vanish, because for each ˜ c(0,1)\n−1,k,∓ωthere\nis a pre-factor 1 /(±ω+E∗\n0)−1/(±ω+E∗\nk)∼k2. As\nwe have full numerical knowledge of the c0,k,±ω, ˜c±1,k,±ω\ncoefficients, we can calculate the linear response fields\na(1)\n0,±ω,a(1)\n±1,±ωfor any rusing Eq. (33) and (39).\nWe now consider the far-field limit r≫r0, where only\nthe constant component of the k= 0 mode and the finite\nkscattering mode contributions survive. We denote these\nasa(1),const\nm,±ωanda(1),scatt\nm,±ω, respectively. a(1),const\n−1,±ωcan be\ndirectly read off from Eq. (36) and (39),\na(1),const\n−1,±ω=−e−ihbR/L\n2√\n21\n±ω+ sgn( γ)b0−ib0α,(40)\nwhile a(1),const\n1,±ω,a(1),const\n0,±ω = 0 as there is no finite k= 0\nmode in those m-sectors. As the scattering modes adopt\nthe free Bessel form in Eq. (32), a(1),scatt\nm,±ωcan be furthersimplified using a contour integral, see App. I for tech-\nnical details. We then obtain, for γ <0 and to leading\norder in α,\na(1),scatt\n0,+ω=−rπ\n2k0rc(0)\n0,k0,+ωe−iδ0,k0ei(π\n4−k0r)e−αωr/v g,\n(41)\na(1),scatt\n±1,+ω=rπ\n2k0r˜c(0)\n±1,k0,+ωei(±π\n2−δ±1,k0)ei(π\n4−k0r)e−αωr/v g,\nanda(1),scatt\n0,−ω =a(1),scatt\n±1,−ω= 0. Here k0=√ω−b0is\nthe momentum of the resonantly selected spin-wave and\nvg= 2√ω−b0is its group velocity. The physical inter-\npretation is that when driven with ω > b 0, the skyrmion\nacts like a resonant antenna, sending out magnons with\nmomentum k0which get damped over a length scale\nl∼vg/(αω).\nVideos of the time-dependent linear response of the\nskyrmion to out-of-plane as well as in-plane right-\npolarised driving are provided as supplementary mate-\nrials, see App. A. Another way to visualise the linear\nresponse of the system is to consider the time- and polar\nangle- averaged local deviation from the static skyrmion\ntexture M(0), defined as ⟨|δˆM|2⟩=⟨|ˆM(t)−ˆM(0)|2⟩χ,t=\n2⟨a∗a⟩χ,t, using Eq. (17). To leading order O(ϵ2), this is\ngiven by\n⟨|δˆM|2⟩=X\nm=0,±1\nω′=±ω2a∗(1)\nm,ω′(r)a(1)\nm,ω′(r). (42)\nIn Fig. 3 we plot δM=q\n⟨|δˆM|2⟩as a function of rfor\nvarious driving frequencies and polarisations with γ <0\nandα= 0.03. For out-of-plane driving, bL=bR=\n0,δMis mostly confined to the skyrmion radius and\nlargest when we drive at the breathing mode frequency,\nω=ωbr.= 0.839, see Fig. 3(a). For in-plane driving,\nbz= 0, the resonant frequency is the Kittel frequency,\nωKit.=b0= 1, but it is only resonantly excited by right-\npolarised driving, as can be seen by comparing the scales\nin Fig. 3(b) and (c). In-plane driving excites the ferro-\nmagnetic bulk, hence δMtends to a constant value in the10\nferromagnetic bulk rather than decaying to zero, as was\nthe case with out-of-plane driving. Interference between\nthea(1),const\n−1,+ωanda(1),scatt\n−1,+ωresults in visible oscillations of\nwavelength 2 π/k0in Fig. 3(b) when ω > ω Kit..\nIn this section we developed all the machinery neces-\nsary to obtain the full linear response of the skyrmion to\nexternal driving by a homogeneous magnetic field B1(t).\nNext, we will use this to calculate the second order trans-\nlational velocity vtrans in Sec. V.\nV. SWIMMING SKYRMION\nIn the linear response framework of Sec. IV, the\nskyrmion was able to undergo periodic contractions, di-\nlations and rotational-symmetry breaking deformations,\nbut its (time-averaged) centre remained firmly stuck in\none place. At order O(ϵ2), however, the skyrmion will\nstart to “swim” with constant velocity vtrans, see also\nthe video provided in App. A. This motion happens as\naconsequence of the linear order cyclic asymmetric de-\nformations, a kind of mechanism also used by jellyfish\nwhen they swim through water. In this section we will\ncalculate vtrans and understand how it is influenced by\nthe driving field’s polarisation and frequency, as well as\nthe amount of damping in the system.\nWe wish to solve Eq. (12) for vtrans. On the LHS of\nthis equation we have the familiar gyrocoupling and dis-\nsipation terms of the Thiele equation, which for a single\nskyrmion are given by\n˜G=−4πez,\n˜Dxx=˜Dyy=πZ∞\n0dr r\u0012\nθ′2\n0+1\nr2sin2(θ0)\u0013\n,(43)\nwhere we used the skyrmion’s translational symmetry in\ntheez-direction to define ˜G=G/L z,˜D=D/Lz,Lz\nbeing the sample length in the ez-direction. Due to the\nskyrmion’s rotational symmetry, ˜Dxxand ˜Dyyare the\nonly non-zero entries of the dissipation matrix ˜Dij. As\na result, the vz\ntrans component vanishes completely and\nEq. (12) reduces to a 2 ×2 matrix equation,\n\u0012\nα˜Dxx−4πsgn(γ)\n4πsgn(γ)α˜Dyy\u0013\u0012\nvx\ntrans\nvy\ntrans\u0013\n=\u0012\nFx\ntrans/Lz\nFy\ntrans/Lz\u0013\n,\n(44)\nwhich we can easily invert to evaluate vtrans=vx\ntransex+\nvy\ntransey. As discussed in Sec. III, there are two different\nways of evaluating the force on the RHS of Eq. (44),\ngiven in Eq. (14) and (16). In both methods, the force\ndensities are functions of ˆM(0),M(1), which carry no\nz-dependence. This means we can replace1\nLzR\nd3r→R\nd2rin Eq. (14) and (16).\nLet us first develop some intuition for Ftrans using\nthe first method, Eq. (14). It turns out that the first\nterm in Eq. (14) actually vanishes in the case of a drivenskyrmion,\nZ\nd2r⟨ˆM(0)·(˙M(1)×∇iM(1))⟩t= 0. (45)\nThis result relies on αbeing finite so that all the linear re-\nsponse, with the exception of the spatially constant bulk\ncontribution coming from the k= 0 mode, decays to\nzero as r→ ∞ , see App. J for details. For this reason,\nEq. (45) is in fact a general result, valid for any driven\nmagnetic system with some localised topological charge\nembedded in a ferromagnetic sea, as long as there is finite\ndamping. Physically, Eq. (45) has two interesting conse-\nquences. The first involves the emergent electric field,\nwhich is given by\nEi=ℏ\n2|e|ˆM·(∇iˆM×˙ˆM) (46)\nfor any general moving magnetic texture M[27]. To\norder O(ϵ2), Eq. (46) has two contributions,ℏ\n2|e|ˆM(0)·\n(∇iM(1)×˙M(1)) and\nℏ\n2|e|ˆM(0)·(∇iM(0)×˙M(2)) =ℏ\n2|e|(qtop×vtrans)i,\nwhere qtopis the topological charge density defined in\nEq. (C8). When time-averaged and integrated over all\nspace, the first contribution vanishes and we conclude\nthat the O(ϵ2) time-averaged, spatially integrated elec-\ntric field is just given by\nZ\nd2r⟨E(2)⟩t=ℏ\n2|e|G×vtrans. (47)\nThis means that an electron far away will only feel\nan electric field from the net time-averaged motion of\nthe skyrmion at velocity vtrans. Eq. (47) is exactly\nthe electric field predicted by Faraday’s law of induc-\ntion ˙B=−∇ × E, if we model the moving skyrmion\nas a point flux with an emergent magnetic field, B=\n−h\n|e|δ(r−vtranst)ez.\nThe second consequence of Eq. (45) is that the time-\naveraged total rate of change of the magnon momentum,R\nd2r⟨˙Pm\ni⟩, defined in Eq. (C9), also vanishes. The phys-\nical interpretation of this is that all the force generated\nby the time-averaged rate of change of momentum of the\nemitted magnons eventually gets absorbed by the bulk\ndue to the finite damping α.\nGetting back to our Thiele equation, we only have the\nsecond force term to worry about in Eq. (44),\nFi\ntrans=αZ\nd2r⟨˙M(1)· ∇iM(1)⟩t\n=αZ\nd2rD\n˙a(1)∇ia∗(1)+ ˙a∗(1)∇ia(1)(48)\n−icos(θ0)∇i(χ)(˙a(1)a∗(1)−˙a∗(1)a(1))E\nt,11\nwhere we rescaled Fi\ntrans/Lz→Fi\ntransto avoid extra clut-\nter, and then substituted the expansion Eq. (17) to write\nthe integrand in terms of the a(1),a∗(1)fields. Due to\nthe spatial derivative ∇iand integration over the po-\nlar angle χ, only terms carrying a net angular momen-\ntum m=±1 in the integrand of Eq. (48) will survive.\nFor example, terms such as a(1)\n0,+ωa∗(1)\n1,+ω,a(1)\n0,+ωa∗(1)\n−1,+ωsur-\nvive but a(1)\n0,+ωa∗(1)\n0,+ω,a(1)\n−1,+ωa∗(1)\n−1,−ωare killed by the in-\ntegration over χ. All surviving terms in the integrand of\nEq. (48) are therefore products of one a(1)\n0field and one\na(1)\n±1field — we thus require a tilted driving field b1(t),\nwith bz>0 and at least one of bR, bL>0, to get a finite\nFtrans.\nNext, we consider what happens to Ftrans as function\nof the driving frequency ωand damping α. If we drive\nbelow the gap, ω < ω Kit., none of the scattering modes\nare resonantly excited, so a(1)\n0,±ω, and consequently the\nintegrand of Ftrans, are bounded to the skyrmion radius\nr0. We choose to discuss in particular the case of a N´ eel\nskyrmion, h= 0, driven with a general tilted field b1(t)\nas defined in Eq. (2), with bz, bR, bL>0 and phase shift\nδ= 0. A different choice of δorhwould have an effect on\nthe orientation, but not the magnitude of Ftrans. With\nthis particular choice, the forces off and on resonance in\nthe limit of small damping are given by\nlim\nα→0Ftrans(ω̸=ωbr., ωKit.)∼αey,\nlim\nα→0Ftrans(ω=ωbr., ωKit.)∼ex,(49)\nrespectively, see App. K for the derivation of this result.\nIn the resonant driving cases, ω=ωbr., ωKit., the pre-\nfactor αgets cancelled by a 1 /iαsingularity entering via\neither the a(1)\n0,±ωora(1)\n−1,±ωfields, depending on whether\nwe are at the breathing or Kittel resonance. Meanwhile,\non the other side of the Thiele equation only the gyro-\ncoupling term survives in the limit of small α, resulting\nin velocities\nlim\nα→0vtrans(ω̸=ωbr., ωKit.)∼αex,\nlim\nα→0vtrans(ω=ωbr., ωKit.)∼ey,(50)\noff and on resonance. We conclude that when driven\nnon-resonantly below the gap, the skyrmion curiously\n“swims” faster in more damped environments! As damp-\ning needs to be non-zero for the skyrmion to move in\nthis driving frequency range, we call it the friction-driven\nregime. In other parts of the natural world, snails, cater-\npillars and other molluscs are also known to rely on sim-\nilar friction-driven mechanisms for their motion.\nIf we instead drive above the gap, ω > ω Kit.,\nthe skyrmion turns into a magnon antenna, emitting\nmagnons of radial momentum k0=√ω−ωKit.and an-\ngular momentum m= 0,±1 in a non-uniform way. This\nmeans that the integrand of Ftrans is no longer bounded\ntor0, but remains finite up to the magnon’s decay length,\nl=vg/(αω). Thus, for small αmost of the integrand isactually outside the skyrmion radius r0. In the region\nr≫r0,2π/k0we can substitute the far-field expressions\na(1),const\n0,±1anda(1),scatt\n0,±1, given in Eq. (40) and (41), to eval-\nuate the integrand. The slowest decaying terms in the\nintegrand come from products such as a(1),scatt\n0,+ωa∗(1),scatt\n1,+ω ,\nand have radial dependence ∼e−2αωr/v g/r. From dimen-\nsional analysis, we know that the radial integral of this\nquantity scales as\nZ∞\nr0r dr e−2αωr/v g\nr∼1\nα.\nTheαpre-factor in Eq. (48) will cancel the 1 /αsingu-\nlarity coming from the integral and we conclude that\nlim\nα→0Ftrans(ω > ω Kit.)∼const. (51)\nIn the limit of small damping, the force is therefore\nindependent ofαwhen we drive above the ferromag-\nnetic gap. On the other side of the Thiele equation\nthe gyrocoupling term still dominates for small α, so\nlimα→0vtrans(ω > ω Kit.)∼const.\nCollecting all these results in one expression for the\nspeed of the skyrmion as a function of the driving fre-\nquency ω, we have\nlim\nα→0vtrans∼(\nα, ω < ω Kit., ω̸=ωbr.,\nconst., ω=ωbr.orω≥ωKit..(52)\nThe fact that the skyrmion’s speed is independent of\nαwhen we drive it above the ferromagnetic gap sug-\ngests that it cannot be the same friction-driven mech-\nanism that is causing the skyrmion’s motion in this fre-\nquency range. We can gain a better understanding of the\nmechanism by looking at the O(ϵ2) contributions of the\ntime-averaged magnon rate-of-change-of-momentum and\ncurrent densities, ⟨˙Pm\nν⟩tand⟨J(2)\nν⟩t, which we defined\nin Eq. (C5) and (C9). These quantities enter the total\nforce density Ftotin Eq. (C10), but disappear from the\nequation once we integrate Eq. (C10) over all space. In\nfact we can also integrate Eq. (C10) over a disk of finite\nradius, rather than all space. We expect that at some\nfinite radius rbulk(ω) the time-averaged magnon rate-of-\nchange-of-momentum and current densities will reach the\nferromagnetic bulk values\n⟨˙Pm\nν⟩bulk\nt= 0,\n⟨J(2)\nν⟩bulk\nt= const. eν.\nOnce they have reached these bulk values, ⟨˙Pm\nν⟩tand\n∇ · ⟨J(2)\nν⟩will once again vanish after the integration\nstep, so it is in fact sufficient to integrate Eq. (C10) in\nthe region 0 < r < rbulk(ω).rbulk(ω) depends strongly\non the driving frequency ω. In Fig. 5 we plot the\ndriven skyrmion spin texture together with ⟨J(2)\nν⟩tand\n⟨˙Pm\nν⟩tforν=x, y, at three different driving frequencies,12\nFIG. 4. A N´ eel skyrmion driven (a) below the gap, ω= 0.84, (b) at the gap, ω= 1, and (c) above the gap, ω= 3. All\nother parameters are fixed, with bz=bR= 0.03,bL= 0, α= 0.01 and δ=h= 0. In each panel, the top layer shows the\nskyrmion spin texture. The next two layers show the time-averaged O(ϵ2) magnon rate-of-change-of-momentum and current\ndensities ⟨˙Pm\nν⟩tand⟨J(2)\nν⟩t, defined in Eq. (C5) and (C9), for ν=x, y, respectively. Purple and yellow indicate the minimum\nand maximum values of ⟨˙Pm\nν⟩t, respectively, and white corresponds to ⟨˙Pm\nν⟩t= 0. The arrows indicate the size and direction\nof the time-averaged local magnon current ⟨J(2)\nν⟩t, which for r≳rbulk(ω) tends to ⟨J(2)\nν⟩bulk\nt=−1\n2b0⟨M(1)·M(1)⟩teν. In panel\n(a),⟨˙Pm\nν⟩tand⟨J(2)\nν⟩treach their bulk values already at r≳r0. This is not the case when ω≥ωKit., panels (b) and (c). In\npanel (b), we are resonantly driving the k= 0 mode. This resonance manifests as a very obvious decrease in the time-averaged\nz-component of the ferromagnetic spins, ⟨ˆMz⟩t, and is signalled by the lighter green colour of the bulk spins in panel (b). In\npanel (c), we excite a scattering state magnon of frequency k0=√ω−ωKit., whose decay length l∼vg/(αω)≫r0. We can\nconclude that in the vicinity of the skyrmion r≳r0, the magnon density and current make no contribution to the total force\ndensity Ftot\nν(defined in Eq. (C10)) when ω < ω Kit., but do contribute significantly when ω > ω Kit..\nω= 0.84,1 and 3. In Fig. 5(a), where ω < ω Kit., we\nsee that ⟨˙Pm\nν⟩tand⟨Jν⟩talready reach their bulk values\natr∼r0, as no finite kmagnons are excited below the\ngap. On the other hand, in Fig. 5(c), where ω > ω Kit.,\nmagnons of finite k0=√ω−ωKit.get excited. As a con-\nsequence, ⟨˙Pm\nν⟩tand⟨Jν⟩tdo not reach their bulk values\nuntil all these magnons have decayed, which happens at\nr∼l=vg/(αω). Fig. 5(b) is a bit special, as there the\nk= 0 mode is resonantly excited. This k= 0 magnon\nformally has a decay length l= 0, as its group velocity\nvg= 0, but we can see nevertheless that rbulk(ω)> r0.\nAtω=ωKit., the skyrmion is maximally relying on the\nresonant excitation of the ferromagnetic background to\nmove. Certain kinds of “lazy” jellyfish analogously rely\non ocean currents to help them move, instead of wasting\nprecious energy propelling themselves on their own. We\nsummarise the frequency dependence of rbulk(ω) in the\nfollowing expression,\nrbulk(ω)∼(\nr0, ω < ω Kit.,\nl=vg/(αω), ω > ω Kit..(53)\nIn the limit of small damping, rbulk(ω) therefore experi-\nences a large jump as ωcrosses the ferromagnetic gap.\nIt is also perfectly legal (although more complicated,\ndue to non-vanishing boundary terms), to integrate\nEq. (C10) between rmin= 0 and r0< rmax< rbulk(ω).\nBelow the gap, ω < ω Kit., this change makes no differ-\nence to which of the terms in Eq. (C10) dominate. ⟨˙Pm\nν⟩t\nand⟨J(2)\nν⟩twill already have reached their bulk values\nbyr=r0, so they will vanish after integration, leaving\nthe friction term α˙M· ∇νMas the sole contributor toFtrans. However, if ω > ω Kit.,⟨˙Pm\nν⟩tand⟨J(2)\nν⟩twill\nnot yet have reached their bulk values at r=rmax, and\nthey will dominate the friction term α˙M· ∇νMifαis\nvery small. In this case, Ftrans — as calculated in the\nvicinity of the skyrmion — mostly originates from emit-\nted magnons, rather than friction. We can therefore say\nthat in the ω > ω Kit.range, the skyrmion moves by a\nmagnon emission mechanism, in which asymmetrically\nemitted magnons result in a momentum counter kick to\nthe skyrmion, which causes it to move in the opposite\ndirection to ensure the overall conservation of linear mo-\nmentum in the system. We should however keep in mind\nthat if we extend rmaxtorbulk(ω) and above, all magnon\nmomentum eventually gets lost through damping to the\nsurroundings, leaving only the α˙M· ∇νMfriction con-\ntribution in Ftrans.\nWe now wish to obtain some concrete numerical val-\nues for Ftrans and the resulting vtrans. If we were to\nuse Eq. (14) for this purpose, we would have to calcu-\nlate all the a(1)(r),a∗(1)(r) fields up to r=rbulk(ω),\nwhich can get very large at small αforω > ω Kit., see\nEq. (53). In principle, the far-field analytical expressions\na(1),scatt,a(1),consteventually become good approxima-\ntions to the numerically evaluated a(1),a∗(1), but this\nonly happens after r≥2π/k0. The low kscattering\nmodes, excited when ω≳ωKit., therefore continue to\npose a numerical challenge, as the analytical approxi-\nmations of a(1)(r), a∗(1)(r) for these modes only starts\nto be accurate at very large r. We can avoid these\nheadaches if we instead use Eq. (16) to calculate Ftrans.\nThe great advantage of Eq. (16) is that the integrand\nis always bounded to the skyrmion radius, because the13\nterms ∇iˆM(0)and∇i(ˆM(0)·B(0)\neff) both vanish for r≳r0.\nImportantly, this happens independently of the driving\nfrequency ω, as ˆM(0)andB(0)\neffdescribe only the static\ntexture and don’t know anything about the driving. Of\ncourse, there is a price to pay for these numerical advan-\ntages — once evaluated in terms of a(1), a∗(1), Eq. (16)\nis algebraically much uglier and harder to interpret than\nEq. (14), see Eq. (H1) for the full expression. Neverthe-\nless with some help from Mathematica these algebraic dif-\nficulties disappear and we can comfortably use Eq. (H1)\nfor the numerical evaluation of Ftrans.\nLet us now choose some parameters for the drive, while\nalso trying to keep it as general as possible. We need a\ntilted driving field b1(t) to get a finite vtrans, sobzmust\nin any case be non-zero. For the in-plane driving field, we\nhave two degrees of freedom in the choice of bRandbL.\nAs Eq. (48) does not contains any cross-terms ∝bRbL,\nit is sufficient to consider the cases bR= 0 and bL= 0\nseparately to obtain the full vtrans. Any mixed in-plane\ndriving with both bRandbLnon-zero would just result\nin a total velocity\nvtrans(bL, bR) =vtrans(bR, bL= 0) + vtrans(bL, bR= 0),\ni.e., the vector sum of the two velocities generated using\neither bR= 0 or bL= 0. In Fig. 5(a) and (b), we show\nωbr.\n00.050.10.15speedvtrans/10−4ωbr.ωKit.\n00.20.40.60.81\nb0= 1,bR= 0\nbz=bL= 0.01\n0 0.5 1 1.5 20π2π\nωangleβ\nα= 0.03, sim.\nα= 0.09\nα= 0.03\nα= 0.01b0= 1,bL= 0\nbz=bR= 0.01\n0 0.5 1 1.5 20π2π\nω(a) (b)\n(c) (d)\nFIG. 5. Second order skyrmion translational velocity vtrans\nas a function of ωforδ, h= 0,γ <0, a range of αand a tilted\ndriving field with bz= 0.01 and either bL= 0.01, bR= 0 (left\ncolumn) or bR= 0.01, bL= 0 (right column). The top row\n(panels (a) and (b)) shows the speed vtrans while the bottom\nrow (panels (c) and (d)) shows the angle βbetween vtransand\nex. In contrast to left-polarised driving, which only resonantly\nexcites the breathing mode, right-polarised driving resonantly\nexcites both the breathing and Kittel modes. Varying the\nphase δshifts the angle βby +δin panel (c) and −δin panel\n(d). Similarly, setting h=π/2 (Bloch skyrmion) shifts β\ndown by π/2 for both panels (c) and (d). The dotted data\npoints were obtained from numerical simulations with α=\n0.03 on mumax3.\nthe speed vtrans =|vtrans|as function of ωfor the casesbz=bL= 0.01, bR= 0 and bz=bR= 0.01, bL= 0, re-\nspectively. For γ <0, the Kittel mode is only resonantly\nexcited in the linear response when bR>0. This explains\nwhy there is only one resonance due to the breathing\nmode in panel (a), versus two due to the breathing and\nKittel modes in panel (b). Another consequence of this\neffect is that vtrans is an order of magnitude larger in\npanel (b) compared to panel (a). This happens because\nthe skyrmion jellyfish does not get any help from the\nferromagnetic “ocean” in the case of a left-polarised in-\nplane driving, so it moves more slowly. As αis reduced,\nthe resonant peaks in vtransbecome better defined, while\nforω > ω Kit.vtrans(ω) tends to a constant value. Simul-\ntaneously the transitions at ω∼ωbr.between β= 2π\nandβ=πin panel (c), and similarly between β=πand\nβ= 2πin panel (d), happen over a shorter range of ω.\nThus, following the predictions of Eq. (50) for ω < ω Kit.\nand in the limit of small α, the skyrmion always travels\nin the ±ex-directions, except ifω=ωbr.orω=ωKit., in\nwhich case β= 3π/2 and it travels in the −ey-direction.\nWe can change the angle βby changing ω, or alterna-\ntively by tuning the phase shift δbetween the bzand\nbR/Lcomponents in b1(t).\nTo check our analytical calculations we have also per-\nformed numerical simulations of the driven skyrmion us-\ning mumax3 [28, 29]. We defined the skyrmion coordi-\nnateR(t) to be the position where the out-of-plane mag-\nnetisation component ˆMz(r, t) is most negative. Given\nthat we simulate the system on a discrete lattice, a rough\nestimate of the skyrmion coordinate, accurate to half the\nlattice spacing, is provided by the lattice coordinates of\nthe spin with the most negative ˆMz. To improve on this,\nwe fit the ˆMzcomponents of the five spins neighbouring\nthis spin to the right and left ( ±ex-direction), as well\nas up and down ( ±ey-direction), to two parabolas. The\nminima of these parabolas then give us much more ac-\ncurate values for R(t) = (Rx(t), Ry(t))T. Finally, to cal-\nculate the velocity vtrans =˙R, we measure the slope of\nR(t) at stroboscopic time intervals ∆ t= 2π/ω— this en-\nables us to get rid of any periodic oscillations in R(t) and\nconsequently also vtrans. The resulting data, plotted as\nblue dots in Fig. 5, agree very nicely with our analytical\ncalculation for the same damping parameter α= 0.03.\nVI. CONCLUSION AND OUTLOOK\nDriving a chiral magnet with a weak spatially homo-\ngeneous, time-oscillating magnetic field B1(t) universally\nactivates the translational mode(s) of the magnet. We\nhave developed an analytical theory inspired by the ef-\nfective Thiele equation which is able to precisely pre-\ndict the resulting translational velocity for any mag-\nnetic texture. Within this approach, the force causing\nthe motion is quadratic in the amplitude of the driv-\ning field, Ftrans∝B2\n1, and consists of two components,\nˆM(0)·(˙M(1)×∇iM(1)) and α˙M(1)·∇iM(1). Physically,\nthe first term represents the rate of change of the local14\nmomentum density of the magnons while the second term\nis a friction term, explicitly proportional to the Gilbert\ndamping α. We showed that the ˆM(0)·(˙M(1)×∇iM(1))\ncontribution vanishes from Ftrans for a driven skyrmion,\nand in fact generally for any driven localised topologi-\ncal magnetic texture embedded in a ferromagnet. For\ndriven bulk systems, ˆM(0)·(˙M(1)×∇iM(1)) is generally\nexpected to contribute, but might also vanish for other\nreasons. For example, it does vanish in the driven heli-\ncal and conical magnets studied in [20], but only because\nof the preservation of translational symmetry in the di-\nrections perpendicular to the helical pitch q. In a bulk\nsystem with less symmetry, such as a skyrmion lattice,\nwe would generally expect the ˆM(0)·(˙M(1)×∇iM(1)) to\nsurvive. In this case, we would expect Ftrans to be inde-\npendent of αin the limit of low damping at all driving\nfrequencies.\nTheα-dependence of the translational velocity vtrans\ndepends not only on Ftrans, but also on what happens\non the left side of the Thiele equation. If the static tex-\nture is topologically trivial, G= 0, as is the case for\nexample in helical and conical phases, only the dissi-\npation term Dvtrans survives. In this case, both sides\nof the Thiele equation are proportional to α, so overall\nvtrans isindependent ofα, at least when we drive non-\nresonantly. In the case of the skyrmion, G̸= 0, and\nin the limit of low damping it will dominate the dissi-\npation term. However, the na¨ ıve conclusion from this\nthatvtrans∼αis wrong. The reason for this is that de-\nspite only the α˙M(1)· ∇iM(1)contributing, Ftrans∼α\nonly when we drive non-resonantly below the ferromag-\nnetic gap, ω < ω Kit.. At the breathing and Kittel reso-\nnances, or above the ferromagnetic gap ω > ω Kit.,Ftrans\nis in fact independent ofαin the low damping limit.\nConsequently, vtrans is also independent of αat these\ndriving frequencies. In the limit of low damping, we\ncould thus identify two different regimes for the (non-\nresonantly) driven skyrmion: friction-driven if ω < ω Kit.\nand magnon-emission-driven if ω > ω Kit., with vtrans∼α\nin the first case but independent of αin the second. We\nwould expect to see similar behaviour in other driven lo-\ncalised magnetic defects such as skyrmion bubbles (con-\ntaining no DMI), as long as these defects are topologically\nnon-trivial. On the other hand, for defects with Q= 0,\nas is the case for example in skyrmionium [30], the situ-\nation changes dramatically. This is because the absence\nofG= 4πQezon the left side of the Thiele equation\nmeans that only the friction terms αDvtrans survives, so\nthat the velocity of skyrmionium would be enhanced by\na factor 1 /α, making it a much better candidate than a\nskyrmion if the goal is to maximise vtrans. One way to\nenhance the skyrmion’s vtransis to drive it near a wall. In\nthis setting, as the skyrmion moves parallel to the wall,\nthe gyrocoupling force G×vtrans perpendicular to the\nwall gets compensated by a push-back force coming from\nthe wall. Thus the skyrmion’s velocity vtrans obtains the\ndesired 1 /αenhancement from the surviving dissipation\nterm.It is also interesting to ask what happens to the sys-\ntem if we shake it harder, i.e., we increase the strength\nof the driving field B1(t). Generally, we would expect\nour weak driving assumption B1/B0≪1 to break down\nat some critical value of Bcrit\n1, after which the driven\nsystem develops dynamical instabilities. In the driven\nhelical phases studied in [20], we saw that the leading\norder instabilities are Floquet magnon “laser” instabil-\nities, and Bcrit.\n1∼αdirectly depends on the amount\nof damping in the system. In the presence of these in-\nstabilities, the driven system becomes a kind of time\nquasicrystal, with macroscopic occupation of a magnon\nstate whose frequency and momentum are incommensu-\nrate with the driving frequency ωand lattice momen-\ntumq, respectively. This effect is in fact universal to\nany periodically driven bulk system with lattice sym-\nmetry, and we would also expect to see it for example\nin a driven skyrmion lattice. It is less easy to predict\nwhat the leading order instability would be in a localised\nsystem such as the driven single skyrmion. If it were\nstill the Floquet magnon mechanism, we would expect\na range of scattering kmagnons to become unstable,\nrather than just a single one, due to the lack of dis-\ncrete lattice symmetry of the single skyrmion. Another\noption is that the ferromagnetic bulk breaks down, and\nthe skyrmion serves as a kind of “seed” for the forma-\ntion of a skyrmion lattice — see also [31], where the\n“seeds” instead took the form of artificial rectangular\nholes in the sample. To test what happens, we have\nperformed some preliminary numerical simulations, keep-\ning the driving frequency near the breathing resonance,\nω= 0.83, with b0= 1, α= 0.01, bL= 0 also fixed, and\nincreasing bz=bRbetween 0 .001 and 0 .1. The results\nof these simulations show that there is first a transition\nfrom the expected vtrans∼bzbRpredicted by our weak\ndriving field theory to vtrans∼√bzbRatbcrit\nz, bcrit\nR∼0.01.\nThen, at bz=bR∼0.05, the skyrmion disappears, with\nthe system’s topological charge Qgoing from −1 to 0.\nHowever, before making any hasty conclusions about the\nfate of the driven skyrmion as we increase B1, further nu-\nmerical experimentation at different driving frequencies\nand lattice discretisations is required, as these factors can\ngreatly influence what happens to the driven system.\nWe now wish to make an experimental prediction for\nthe speed Vtrans of the driven skyrmion. We do this for\nthe metallic chiral magnet MnSi, whose micromagnetic\nparameters are γ=−1.76×10−11T−1s−1,J= 7.05×\n10−13Jm−1,D= 2.46×10−4Jm−2,M0= 1.52×105\nAm−1[32–34] and α∼0.01. Using the results of our am-\nplitude run, we assume that bcrit\n1∼bcrit\nz∼bcrit\nR∼0.01,\nwhich translates into Bcrit\n1∼5mT in physical units. This\nis within an order of magnitude of Bcrit\n1for driven helical\nmagnets, which we calculated to be 0 .5mT on resonance\n[20]. Using the parameters of a tilted driving field with\nBz∼Bx∼5 mT and Ω ∼100 GHz, we predict the\nskyrmion to reach speeds of Vtrans∼30 mms−1. This\nis over two orders of magnitude larger than the mini-\nmum depinning velocity for skyrmions, estimated from15\nmeasurements of the Hall effect to be 0 .2 mms−1for a\ncritical current density 2 jc[12]. Thus, it should be easy\nto observe the skyrmion jellyfish experimentally, at least\nin materials with low pinning such as MnSi. In the real\nworld all systems will also carry a degree of disorder.\nFar from hindering the skyrmion motion, this might ac-\ntually have an enhancing effect on its speed Vtrans. A\ndriven skyrmion lattice will start to rotate via a mecha-\nnism similar to the one we presented here for the transla-\ntion motion. Recent experiments where skyrmion lattices\nin the insulator Cu2OSeO3were driven by femtosecond\nlasers pulses resulted in rotational speeds up to 2 ×107–\n108deg s−1, which is over six order of magnitude faster\nthan predicted theoretically using our theory for a clean\nsystem. Finding a way to incorporate disorder into our\ntheoretical model is therefore crucial to gaining better\npredictive power for future experiments.\nTo conclude, we have presented a fully analytical the-\nory for realising a skyrmion jellyfish by driving a chiral\nmagnet with oscillating magnetic fields. We hope these\ninsights will be of interest to other fans of nano-scale\nman-made marine life.\nACKNOWLEDGMENTS\nWe thank Joachim Hemberger, Christian Pfleiderer,\nVolodymyr Kravchuk, Markus Garst and especiallyAchim Rosch for useful discussions and guidance. We\nacknowledge the financial support of the DFG via SPP\n2137 (project number 403505545) and CRC 1238 (project\nnumber 277146847, subproject C04). We also thank\nthe Regional Computing Center of the University of\nCologne (RRZK) for providing computing time on the\nDFG-funded (Funding number: INST 216/512/1FUGG)\nHigh Performance Computing (HPC) system CHEOPS\nas well as technical support. The highly accurate shoot-\ning numerics for the finite- kscatttering modes was im-\nplemented using the BigFloat type and the excellent\nDifferentialEquations.jl library [22], which has the\nRK065 algorithm readily available to deal with the singu-\nlarity at r= 0.\nAUTHOR CONTRIBUTIONS\nNdS designed the study, performed the analytical and\nnumerical calculations and wrote the paper. VL provided\ntechnical support with implementing the eigenbasis.\n[1] I. E. Dzyaloshinskii, Theory of helicoidal structures in\nantiferromagnets. i. nonmetals, J. Exp. Theor. Phys. 19,\n960 (1964).\n[2] A. 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K.\nWertheim, Magnetic properties of the monosilicides of\nsome 3d transition elements, Journal of Applied Physics\n37, 1256 (1966), https://doi.org/10.1063/1.1708422.17\nAppendix A: Supplementary videos\nThe supplementary videos first order bZ.mp4 and\nfirst order bR.mp4 show the response of the mag-\nnetisation up to O(ϵ1),ˆM(0)(r) +M(1)(r, t), to driv-\ning fields bzcos(ωt)ezandbR(cos(ωt)ex+ sin( ωt)ey), re-\nspectively. In first order bZ.mp4 , the radial symme-\ntry of the skyrmion is preserved at all times, as only\nthem= 0 angular momentum sector is excited. This\nis no longer the case in first order bR.mp4 , where\nthem=±1 angular momentum sectors are excited.\nInsecond order tilted drive.mp4 , we show the re-\nsponse including the second order translational motion,\nˆM(0)(r−vtranst) +M(1)(r, t) to a tilted driving field\nb1(t) =bR(cos(ωt)ex+ sin( ωt)ey) +bzcos(ωt)ez. The\nskyrmion starts to “swim” as a result of its periodic first\norder asymmetric contractions and relaxations — just\nlike a jellyfish.\nAppendix B: Activation of Translational Modes –\nauxiliary calculations\n1.∇iˆM·(ˆM×LLG )projection\nTaking ˆM×Eq. (7) and multiplying everything by\nsgn(γ)/M0, we obtain\nsgn(γ)ˆM×˙ˆM=|γ|\u0010\n(ˆM·Beff)ˆM−Beff\u0011\n+α˙ˆM,(B1)\nwhere we used ˆM·˙ˆM=1\n2d\ndt\u0010\nˆM·ˆM\u0011\n= 0. Next, we\nproject Eq. (B1) onto ∇iˆM,i={x, y, z}, and then inte-\ngrate over 3D space,\nsgn(γ)Z\nd3r∇iˆM·\u0010\nˆM×˙ˆM\u0011\n=αZ\nd3r∇iˆM·˙ˆM,\n(B2)\nwhere we got rid of the two first terms on the right of\nEq. (B1) using ˆM· ∇iˆM=1\n2∇i\u0010\nˆM·ˆM\u0011\n= 0 and\nZ\nd3rBeff·∇iˆM=−Z\nd3rδF\nδM·dˆM\ndri=−1\nM0∇iF= 0,\n(B3)\nas the free energy is translationally invariant. Finally,\nsubstituting Eq. (11) into Eq. (B2), collecting all terms\nproportional to ϵ2and keeping only the DC component\nof the resulting equation, we obtain Eq. (12).\n2.∇iˆM(0)·(ˆM(0)×LLG )projection\nThis time we act with ∇iˆM(0)·(ˆM(0)×, i.e. only the\nO(ϵ0) components of ˆM·(ˆM×, on Eq. (7). We are still\ninterested in the O(ϵ2) contribution at the end, which\ncan now only come from the LLG terms. For the˙ˆMandαˆM×˙ˆMterms, these contributions are\nZ\nd3r∇iˆM(0)·(ˆM(0)×˙ˆM(2)) =−(G×vtrans)i,\nαZ\nd3r∇iˆM(0)·(ˆM(0)×(ˆM(0)×˙ˆM(2))) =α(Dvtrans)i,\nwhere we used ˆM(0)⊥M(1),˙M(1)to get rid of the ˆM(0)×\n(M(1)×˙M(1)) term in the dissipation term. Thus, the\nO(ϵ2) contributions coming from the gyrocoupling and\ndissipation term of the LLG are actually simpler using\nthis method of projection, compared to App. B.1.\nAs always though, nothing is for free. In return for\nsimpler gyrocoupling and dissipation contributions, we\nget extra terms coming from Beff. While previously this\nterm integrated out to zero (see Eq. (B3)), it now pro-\nduces finite O(ϵ2) contributions,\nZ\nd3r∇iˆM(0)·\u0010\nˆM(0)×(ˆM×Beff)\u0011\n=\nZ\nd3r\"\n(∇iˆM(0)·M(1))(ˆM(0)·B(1)\neff)\n+ (ˆM(0)·B(0)\neff)(∇iˆM(0)·M(2))− ∇ iˆM(0)·B(2)\neff#\n,\n(B4)\nwhere we used ˆM(0)⊥M(1)and ˆM(0)∥B(0)\neff⊥ ∇ iˆM(0).\nWe need to check what happens to the M(2)\nstat.in Eq. (B4),\nas its presence in the integral would make our new ap-\nproach unusable for evaluating vtrans. For this, it is help-\nful to split M(2)\nstat.into components parallel and perpen-\ndicular to ˆM(0),\nM(2)\nstat.=w∥ˆM(0)+w⊥e−+w∗\n⊥e+,\nwhere the w∥,w⊥are complex coefficients to be deter-\nmined. Using the normalisation condition ⟨ˆM·ˆM⟩t= 1\nto all orders of ϵ, we see that the w∥component is already\ndefined as a function of the O(ϵ1) fields,\nw∥=−1\n2⟨M(1)·M(1)⟩t,\nwhere we used the expansion of ˆMgiven in Eq. (9). Next,\nwe will show that the w⊥,±components, which we have\nnot calculated, drop out of Eq. (B4), and only w∥re-\nmains. Integrating the last term on the RHS of Eq. (B4)\nby parts, we have\nZ\nd3r∇iˆM(0)·B(2)\neff\n=Z\nd3r∇iˆM(0)·\u0010\n˜J∇2−2˜D∇×\u0011\nM(2)\n=Z\nd3rM(2)·\u0010\n˜J∇2−2˜D∇×\u0011\n∇iˆM(0)\n=Z\nd3rM(2)· ∇iB(0)\neff\n=Z\nd3rM(2)· ∇i\u0010\n(ˆM(0)·B(0)\neff)ˆM(0)\u0011\n,18\nwhere ˜J=J/M 0,˜D=D/M 0. We assumed that the sur-\nface terms in the integration by parts step vanish. This\nassumption is valid when ∇iM(0)is bounded, as for ex-\nample it is in the case of a single skyrmion, but not in\ngeneral. In the penultimate step we switched the orders\nof the ∇iand ˜J∇2−2˜D∇× operators, and in the last\nstep we used ˆM(0)∥B(0)\neff. Inserting this into Eq. (B4)\nand simplifying, the only remaining term carrying M(2)\nreads\n−Z\nd3rˆM(0)·M(2)∇i\u0010\nˆM(0)·B(0)\neff\u0011\n. (B5)\nSubstituting\nM(2)=−tvtrans· ∇ˆM(0)+M(2)\nstat.,\nvalid for short t, into Eq. (B5), we see that only the w∥\nterm survives, giving\n1\n2Z\nd3r⟨M(1)·M(1)⟩t∇i\u0010\nˆM(0)·B(0)\neff\u0011\n.\nPutting this all together we obtain Eq. (16).\nAppendix C: LLG in the language of Tµν\n1. Lagrangian density L,Tµνand divergence of Tµν\nThe total Lagrangian density of the chiral magnet\nmodelled by Eq. (1) is\nL=Ldyn+Lstat, (C1)\nLdyn= sgn( γ)A·˙ˆM,\nLstat=1\n2∇µˆM· ∇µˆM+ˆM·(∇ × ˆM)−bext·ˆM,\nwhere the gauge field A(ˆM) obeys∂Ak\n∂ˆMj−∂Aj\n∂ˆMk=ϵijkˆMi.\nNote that in Eq. (C1), we are using dimensionless space,\ntime and magnetic field units, defined as ˜ ri= (D/J)ri,\n˜t=D2|γ|/(JM0)tandbi=M0J/D2Bi, respectively,\nand immediately dropping the tildes on ˜ ri,˜tfor a cleaner\nnotation. The stress energy tensor resulting from trans-\nlational symmetry is\nTµν=∂L\n∂(∂µˆM)·∂νˆM−δµνL, (C2)\nwhere µ, ν={t, x, y, z }. To take into account the phe-\nnomenological damping, the divergence of Tµνneeds to\nbe updated to include a term proportional to αon the\nRHS,\n∂µTµν=α˙ˆM·∂νˆM. (C3)\nThe LLG is then obtained from the spatial components\nν={x, y, z}of Eq. (C3). We can use Eq. (C3) to de-\nfine some momentum and current densities. We definethe rate of change of the momentum density purely as a\nfunction of Ldyn,\n˙Pν=∂\n∂t \n∂L\n∂˙ˆMdyn\n· ∇νˆM!\n− ∇ ν\u0000\nLdyn\u0001\n= sgn( γ)ˆM·\u0010˙ˆM×∇νˆM\u0011\n. (C4)\nThis allows us to define the current densities purely in\nterms of Lstatas\n(Jν)i=∂Lstat\n∂∇iˆM· ∇νˆM−δiνLstat(C5)\nIt is not too difficult to show that the divergence of these\ncurrent densities is given by\n∇ ·Jν=beff· ∇νˆM, (C6)\nwhere beffwas defined in Eq. (7). Also, when r≳l\nso that all finite kmagnons have decayed, ˆMbecomes\nconstant in space. The only surviving term in Jνin this\ncase comes from the −bext·ˆMinL, so that\nJν= (bext·ˆM)eν.\nThe continuity equation updated to include Gilbert\ndamping reads\n˙Pν+∇ ·Jν=α˙ˆM· ∇νˆM. (C7)\n2. Eq. (C7) to order O(ϵ2)\nAt second order in ϵ,˙P(2)\nνand (˙ˆM· ∇νˆM)(2)will have\ntwo kinds of terms which either include or don’t include\nvtrans. It is straightforward to show that the terms which\nsurvive after time averaging over Tare\n⟨˙P(2)\nν⟩t=−sgn(γ)(qtop×vtrans)ν+⟨˙Pm\nν⟩t\n⟨(˙ˆM· ∇νˆM)(2)⟩t=−(Dvtrans)ν+⟨˙M(1)· ∇νM(1)⟩t.\nwhere qtopandDare just the local topological charge\nand dissipation matrix densities of the static texture,\nqtop\nα=1\n2ϵαβγˆM(0)·(∇βˆM(0)×∇γˆM(0))\nDαβ=∇αˆM(0)· ∇βˆM(0)(C8)\nand⟨˙Pm\nν⟩is the time-averaged rate of change of the\nmagnon momentum density,\n˙Pm\nν= sgn( γ)ˆM(0)·\u0010\n˙M(1)×∇νM(1)\u0011\n. (C9)\nWe also need to calculate the ∇·Jνto order O(ϵ2). This\nshould not carry any contribution from vtranst, as the\nlack of a time derivative in Jνwould result in such a term\nexploding for large t. For short times, the contribution19\ntoˆM(2)coming from vtrans is−t(vtrans· ∇)ˆM(0). Using\nthis and Eq. (C6), we can write down the contributions\nto∇ ·Jνcoming form vtrans as\n−t(vtrans)α\u0010\nb(0)\neff· ∇ν∇αˆM(0)+∇νˆM(0)· ∇αb(0)\neff\u0011\n=−tvtrans· ∇\u0010\nb(0)\neff· ∇νˆM(0)\u0011\n= 0.\nwhere we used b(0)\neff∥ˆM(0)and ˆM(0)· ∇νˆM(0)= 0 in\nthe last line. While ∇ ·Jνdoes not carry any terms\nproportional to vtrans, it will generally have contributions\nfrom the other non temporally oscillating static second\norder term, M(2)\nstat..\nThe analysis we have performed enables us to separate\nthevtransterms from the other terms in Eq. (C7), giving\n−sgn(γ)\u0000\nqtop×vtrans\u0001\nν+α(Dvtrans)ν=Ftot\nν,(C10)\nFtot\nν=−⟨˙Pm\nν⟩t− ⟨∇ · J(2)\nν⟩t+α⟨˙M(1)· ∇νM(1)⟩t.\nIn its given form, Eq. (C10) is actually useless for cal-\nculating vtrans because of the presence of the unknown\ncomponent M(2)\nstat.in⟨∇ ·Jν⟩t. To obtain vtrans, we\nwould have to integrate Eq. (C10) over all space to make\n∇ ·Jνvanish, which just returns Eq. (12). Neverthe-\nless, Eq. (C10) is useful for investigating which of ⟨˙Pm\nν⟩t,\n⟨∇ ·Jν⟩torα⟨˙M(1)· ∇νM(1)⟩tplays a dominant role in\nthe local force density.\nAppendix D: Equation of motion for a, a∗\nOnly two kinds of terms survive the projection of\nEq. (20) onto e±:e±·e±= 1 and e±·(e3×e∓) =±i.\nThis results in the following equations of motion\nsgn(γ)d\ndt(as) =i{F, as}\n−iα\u0012\n(1−a∗a)d\ndt(as)−d\ndt(a∗a)as\u0013\n,\nsgn(γ)d\ndt(a∗s) =i{F, a∗s}\n+iα\u0012\n(1−a∗a)d\ndt(a∗s)−d\ndt(a∗a)a∗s\u0013\n,(D1)\nwhere s=q\n1−a∗a\n2, correct to all orders in a, a∗.\nAppendix E: Expressions for F(1)and F(2)\nTo find F(1)andF(2), we substitute Eq. (17) into\nEq. (1) and Taylor expand in a, a∗. To avoid clutter-\ning the expressions with too many constants, all the free\nenergies we list below are rescaled via ˜F= (D/J2)F. We\nalso use a dimensionless length scale ˜ ri= (D/J)ri, im-\nplying ˜∇i= (J/D)∇ifor the spatial gradients appearingin the Heisenberg and DMI energy terms. For readability\nwe then drop the tildes on both ˜Fand ˜r.\nIn Eq. (5), we already introduced a rescaled dimension-\nless amplitude b0for the static component of the external\nmagnetic field. We now do the same for the amplitudes of\nthe oscillating components, defining bi= (M0J/D2)Bi\n1,\ni={x, y, z}. It is more natural to rewrite bx, by, the os-\ncillating field components in the plane of the skyrmion, in\nterms of the circularly polarised driving field components\nbR=bx+by,\nbL=bx−by.(E1)\nUsing polar coordinates, the free energy Fand free en-\nergy density Fare related via F=R∞\n0rdrR2π\n0dχF.\nWith this definition, we obtain the free energy densities\nF(1)\ndrive=ϵ√\n2h\n(a+a∗)\u0010\nbzsin(θ0) cos( ωt+δ)\n−1\n2cos(θ0) (bRcos(ϕ−ωt) +bLcos(ϕ+ωt))\u0011\n−i\n2(a−a∗) (bRsin(ϕ−ωt) +bLsin(ϕ+ωt))i\n,(E2)\nF(2)\ndrive=ϵ\n2a∗ah\n2bzcos(θ0) cos( ωt+δ)\n+ sin( θ0) (bRcos(ϕ−ωt) +bLcos(ϕ+ωt))i\n.(E3)\nF(1)\nno drive= 0, otherwise the skyrmion texture would be\nmoving, rather than static, in the absence of a driving\nfieldb1(t). The lowest non-zero contribution is therefore\nquadratic in a, a∗, and given by\nF(2)\nno drive=−1\n4r2h\n+aa∗\u0010\n−4b0r2cos(θ0) + 2r2θ′2\n0+ 4r2θ′\n0\n+ 6rsin(2θ0)−3 cos(2 θ0)−1\u0011\n+ 4i(a∗∂χa−a∂χa∗)(cos( θ0)−rsin(θ0))\n−4\u0010\n∂χa∂χa∗+r2a′a∗′\u0011\n+ (a2+a∗2)\u0010\n−r2θ′2\n0−2r2θ′\n0\n+ sin2(θ0) +rsin(2θ0)\u0011i\n.(E4)\nAppendix F: force fand matrix Hm\nThe force f(t) is given by {F(1)\ndrive, a},\nf(t) =1\n2√\n2 \nbzsin(θ0)\u0010\nei(ωt+δ)+e−i(ωt+δ)\u0011\n(F1)\n−1\n2\u0010\n(cos(θ0)−1)\u0010\nbRei(χ+h−ωt)+bLei(χ+h+ωt)\u0011\u0011\n−1\n2\u0010\n(cos(θ0) + 1)\u0010\nbRe−i(χ+h−ωt)+bLe−i(χ+h+ωt)\u0011\u0011!\n.20\nTo calculate Hm, we evaluate the Poisson brackets\n{F(2)\nno drive, a},{F(2)\nno drive, a∗}, which include a term\nZ\nr dr ∂ ram(r){∂ra∗\nm(r), am(r′)}=\nZ\nr dr ∂ ram(r)∂r{a∗\nm(r), am(r′)}=1\nr′∂r′(r′am(r′)).\nHmcan then be written in the form\nHm=1\u0012\n−∂2\nr−1\nr∂r+m2+ 1\nr2+b0+V0\u0013\n(F2)\n+σz2m\nr2+σzVm\nz+σxVx,\nV0=3 (cos(2 θ0)−1)\n4r2−3 sin(2 θ0)\n2r\n+b0(cos(θ0)−1)−θ′\n0−θ′2\n0\n2,\nVm\nz=2m\nr2(cos(θ0)−1−rsin(θ0)),\nVx=−1\n2r2\u0000\nsin2(θ0) +rsin(2θ0)−r2θ′2\n0−2r2θ′\n0\u0001\n,\nwhere the potentials V0,Vx,Vm\nzvanish for r≫r0.Appendix G: Damped eigenbasis of σzHm\n1. Particle-hole property of\f\f\fm, k(0)E\nTaking the complex conjugate of Eq. (29) and using\nthe property σxHmσx=H−m, we have\nE∗\nm,k,i(sgn( γ)−iασz)|m, k, i ⟩∗=σzσxH−mσx|m, k, i ⟩∗\nPre-multiplying this equation with σx, and using σxσz=\n−σzσx, we obtain\n−E∗\nm,k,i(sgn( γ) +iασz)σx|m, k, i ⟩∗=σzH−mσx|m, k, i ⟩∗.\nThus, σx|m, k, i ⟩∗is also an eigenvector of σzH−mwith\neigenvalue −E∗\nm,k,i.\n2. Inner products between\f\f\fm, k(0)E\nFor each m-sector, the scattering states obey\nD\nm, k(0)\f\f\fσz\f\f\fm, k′(0)E\n=δ(k−k′)\nk,\nD\nm, k(0)\f\f\fσzσx\f\f\fm, k(0)E\n= 0.\nIn addition, we have for the m= 0 sector,\nD\n0,br.(0)\f\f\fσz\f\f\f0,br.(0)E\n= 1,\nD\n0,br.(0)\f\f\fσzσx\f\f\f0,br.(0)E\n= 0,D\n0,br.(0)\f\f\fσz\f\f\f0, k(0)E\n= 0,\nD\n0,br.(0)\f\f\fσzσx\f\f\f0, k(0)E\n= 0.\nAnd finally for the m=±1 sectors,\nD\n±1,trans.(0)\f\f\fσz\f\f\f±1,trans.(0)E\n=±1,\nD\n±1,trans.(0)\f\f\fσz\f\f\f±1, k(0)E\n= 0.\n3. First order perturbation theory in α\nSubstituting Eq. (30) into Eq. (29) and keeping only the linear in αterms, we obtain the equation\n(ϵ(0)\nm,kσz−ϵ(1)\nm,k)\f\f\fm, k(0)E\n+ϵ(0)\nm,k\f\f\fm, k(1)E\n=σzHm\f\f\fm, k(1)E\n. (G1)\nProjecting\nm, k(0)\f\fσzonto Eq. (G1) and using ϵ(0)\nm,k\nm, k(0)\f\fσz=\nm, k(0)\f\fHm, we obtain the O(α) corrections to\nthe energies,\nϵ(1)\nbr.=D\n0,br.(0)\f\f\f0,br.(0)E\nϵ(0)\nbr., ϵ(1)\ntrans. = 0, ϵ(1)\nm,k=ϵ(0)\nm,k. (G2)21\nIf we instead project\nm, k′(0)\f\fσzwith k̸=k′onto Eq. (G1), we obtain the O(α) corrections to the eigenvectors,\n\f\f\f0,br.(1)E\n=1\n2D\n0,br.(0)\f\f\fσx\f\f\f0,br.(0)E\nσx\f\f\f0,br.(0)E\n+Z∞\n0k dkϵ(0)\nbr.\nϵ(0)\nk−ϵ(0)\nbr.D\n0, k(0)\f\f\f0,br.(0)E\f\f\f0, k(0)E\n+Z∞\n0k dkϵ(0)\nbr.\nϵ(0)\nk+ϵ(0)\nbr.D\n0, k(0)\f\f\fσx\f\f\f0,br.(0)E\nσx\f\f\f0, k(0)E\n,\n\f\f\f±1,trans.(1)E\n= 0,\n\f\f\f0, k(1)E\n=Z∞\n0k′dk′ϵ(0)\nk\nϵ(0)\nk+ϵ(0)\nk′⟨0, k′|σx|0, k⟩σx|0, k′⟩\n+ϵ(0)\nk\nϵ(0)\nbr.−ϵ(0)\nkD\n0,br.(0)\f\f\f0, k(0)E\f\f\f0,br.(0)E\n+ϵ(0)\nk\nϵ(0)\nbr.+ϵ(0)\nkD\n0,br.(0)\f\f\fσx\f\f\f0, k(0)E\nσx\f\f\f0,br.(0)E\n,\n\f\f\f±1, k(1)E\n=Z∞\n0k′dk′ϵ(0)\nk\nϵ(0)\nk+ϵ(0)\nk′D\n∓1, k′(0)\f\f\fσx\f\f\f±1, k(0)E\nσx\f\f\f∓1, k′(0)E\n∓D\n±1,trans.(0)\f\f\f±1, k(0)E\f\f\f±1,trans.(0)E\n.(G3)\nThe first order corrections to the steady state coefficients in Eq. (34) are given by\nc(1)\nm,bd.,±ω=c(0)\nm,bd.,±ωD\nm,bd.(0)\f\f\f−m,bd.(0)E\n−c(0)\n−m,bd.,∓ω\u0010D\nm,bd.(0)\f\f\fσx\f\f\f−m,bd.(0)E\n−D\nm,bd.(0)\f\f\fσzσx\f\f\f−m,bd.(1)E\u0011\n+Z∞\n0k dk \nc(0)\nm,k,±ω\u0010D\nm,bd.(0)\f\f\fm, k(0)E\n+D\nm,bd.(0)\f\f\fσz\f\f\fm, k(1)E\u0011\n−c(0)\n−m,k,∓ω\u0010D\nm,bd.(0)\f\f\fσx\f\f\f−m, k(0)E\n−D\nm,bd.(0)\f\f\fσzσx\f\f\f−m, k(1)E\u0011!\n,\nc(1)\nm,k,±ω=c(0)\nm,k,±ω\n−Z∞\n0k′dk′c(0)\n−m,k′,∓ω\u0010D\nm, k(0)\f\f\fσx\f\f\f−m, k′(0)E\n−D\nm, k(0)\f\f\fσzσx\f\f\f−m, k′(1)E\u0011\n+c(0)\nm,br.,±ω\u0010D\nm, k(0)\f\f\fm,bd.(0)E\n+D\nm, k(0)\f\f\fσz\f\f\fm,bd.(1)E\u0011\n−c(0)\n−m,bd.,∓ω\u0010D\nm, k(0)\f\f\fσx\f\f\f−m,bd.(0)E\n−D\nm, k(0)\f\f\fσzσx\f\f\f−m,bd.(1)E\u0011\n.(G4)\nOnly the m= 0 coefficients may be calculated using Eq. (G4) and they are given by\nc(1)\n0,br.,±ω=c(0)\n0,br.,±ωD\n0,br.(0)\f\f\f0,br.(0)E\n−1\n2c(0)\n0,br.,∓ωD\n0,br.(0)\f\f\fσx\f\f\f0,br.(0)E\n−Z∞\n0k dk \nϵ(0)\nbr.\nϵ(0)\nk−ϵ(0)\nbr.c(0)\n0,k,±ωD\n0,br.(0)\f\f\f0, k(0)E\n+ϵ(0)\nbr.\nϵ(0)\nk+ϵ(0)\nbr.c(0)\n0,k,∓ωD\n0,br.(0)\f\f\fσx\f\f\f0, k(0)E!\n,\nc(1)\n0,k,±ω=c(0)\n0,k,±ω−Z∞\n0k′dk′ϵ(0)\nk\nϵ(0)\nk+ϵ(0)\nk′c(0)\n0,k′,∓ωD\n0, k(0)\f\f\fσx\f\f\f0, k′(0)E\n−ϵ(0)\nk\nϵ(0)\nbr.−ϵ(0)\nkc(0)\n0,br.,±ωD\n0, k(0)\f\f\f0,br.(0)E\n−ϵ(0)\nk\nϵ(0)\nbr.+ϵ(0)\nkc(0)\n0,br.,∓ωD\n0, k(0)\f\f\fσx\f\f\f0,br.(0)E\n.(G5)22\nThe first order corrections to the steady state coefficients in the m=±1 sectors using the |m=−1, k= 0⟩mode\nmethod in Sec. IV.2 are given by\n˜c∗(1)\n1,k′,∓ω=1\nϵ(0)\n0+ϵ(0)\nk′ \n2ϵ(0)\n0˜c∗(0)\n1,k′,∓ω+D\n1, k′(0)\f\f\fσx(ϵ(0)\n0σz���H−1σz)\f\f\f\f\u0012f−1,±ω\n−f∗\n1,∓ω\u0013\u001d\n−ϵ(0)\nk′Z∞\nk>0k dkϵ(0)\n0−ϵ(0)\nk\nϵ(0)\nk+ϵ(0)\nk′˜c(0)\n−1,k,±ωD\n1, k′(0)\f\f\fσx\f\f\f−1, k(0)E!\n,\n˜c(1)\n−1,k′,±ω=1\nϵ(0)\nk′−ϵ(0)\n0 D\n−1, k′(0)\f\f\f(ϵ(0)\n0σz−H−1σz)\f\f\f\f\u0012f−1,±ω\n−f∗\n1,∓ω\u0013\u001d\n+ϵ(0)\nk′Z∞\nk>0k dkϵ(0)\nk+ϵ(0)\n0\nϵ(0)\nk+ϵ(0)\nk′˜c∗(0)\n1,k,∓ωD\n−1, k′(0)\f\f\fσx\f\f\f1, k(0)E!\n,\n˜c(1)\n1,trans. ,∓ω= 0.(G6)\nAppendix H: Calculation of ˜Fifor the skyrmion\nThe neatest thing to do is to calculate ˜Fx+i˜Fy, which is\n˜Fx\ntrans+i˜Fy\ntrans=−*Z\nr dr dχ eiχ \n1√\n2\u0012\nθ′\n0(a(1)+a∗(1)) +sin(θ0)\nr(a(1)−a∗(1))\u0013\u0012\n−1√\n2r2\u0010\n2i(rcos(θ0) + sin( θ0))(∂χa∗(1)−∂χa(1))\n+ 2r2(1 +θ′\n0)(∂ra(1)+∂ra∗(1))\n+ (a(1)+a∗(1))(2rcos(2 θ0) +b0r2sin(θ0) + sin(2 θ0))\u0011\n+1\n2sin(θ0) (bRcos(ϕ−ωt) +bLcos(ϕ+ωt)) +bzcos(θ0) cos( ωt+δ)\u0013\n+a∗(1)a(1)∂r\u0012\n2θ′\n0+ (θ′\n0)2+sin2(θ0)\nr2+sin(2θ0)\nr−b0cos(θ0)\u0013!+\nt(H1)\nin terms of the first order a(1), a∗(1)fields for the Bloch/N´ eel skyrmion. Then ˜Fx\ntrans,˜Fy\ntransare the real and imaginary\nparts of Eq. (H1), respectively. Notice how much more algebraically complex the ˜Fx,y\ntranscomponents are, compared to\nFx,y\ntrans given in Eq. (K2)!\nAppendix I: Contour integral for far-field linear\nresponse\nHere, we derive a(1),scatt\n0,±ω, with the understanding that\na(1),scatt\n±1,±ωcan be obtained straightforwardly using the\nsame technique. For γ < 0, the denominator of the\f\fm, k(0)\u000b\nuterm in Eq. (33) reads\n∓ω−Ek=∓ω+ (b0+k2)(1 + iα).\nThe case + ωproduces very strongly damped and thus\nphysically irrelevant roots, so we can immediately set\na(1),scatt\n0,−ωto zero. For the case −ω, we have two complexroots±k∗, with\nkω=k0−iαω/v g+O(α2), (I1)\nwhere k0=√ω−b0,vg= 2k0. Using the far-field limits\nJm(kr) =r\n2\nπkrcos\u0010\nkr−mπ\n2−π\n4\u0011\n,\nYm(kr) =r\n2\nπkrsin\u0010\nkr−mπ\n2−π\n4\u0011\n,23\nvalid for r≫2π/k, we can write a(1),scatt\n0,ω to leading order\ninαas\na(1),scatt\n0,ω =Z∞\n−∞−ik dk c(0)\nk,ω\n(k−kω)(k+kω)s\n1\n2π|k|r·\n\u0010\nei(π\n4−kr−δ0,k)−e−i(π\n4−kr−δ0,k)\u0011\n,(I2)\nwhere we define c(0)\nk,ω= 0 for k < 0. We can convert\nthis to an integral in the complex k-plane, where the\nintegration contour is a semi-circle in the lower/upper\nhalf-plane, depending on the sign in e∓ikr. Applying the\nresidue theorem, we obtain Eq. (41).\nAppendix J: VanishingR\nd2rˆM(0)· ⟨˙M(1)×∇iM(1)⟩t\nWe will now show whyR\nd2rˆM(0)· ⟨˙M(1)×∇iM(1)⟩t\nvanishes in the case of a driven single skyrmion. In polar\ncoordinates, and taking the linear combination ˙Pm\n⊥=\n˙Pm\nx+i˙Pm\nyto simplify the algebra, we have\nZ\nd2r⟨˙Pm\n⊥⟩t=Z\nd2r⟨˙Pm\nx+i˙Pm\ny⟩t=\nZ\nr dr dχ eiχ\u0012\n∂r+i∂χ\nr\u0013X\nm,m′\nω′=±ωω′a(1)\nm,ω′a∗(1)\nm′,ω′eiχ(m−m′).\nIn the above, we can use integration by parts to replace\ni∂χ/r→1/r. Integrating the resulting expression over\nχ, we can write the expression as\nZ\nd2r⟨˙Pm\n⊥⟩t=Z∞\n0dr ∂rX\nm,m′\nω′=±ωrω′a(1)\nm,ω′a∗(1)\nm′,ω′δm+1,m′\n=hX\nm,m′\nω′=±ωrω′a(1)\nm,ω′a∗(1)\nm′,ω′δm+1,m′i∞\n0.\nThe only contribution in the a(1)fields which survives as\nr→ ∞ isa(1),const.\n−1,±ω— everything else decays on a length\nscale l∼1/α, and therefore vanishes at large enough\nr, see also Eq. (40) and (41). Due to the δm+1,m′term,\na(1),const.\n−1,±ωwill always be paired with a(1)\n0,±ω, which vanishes\nforr→ ∞ . Thus, we can conclude thatR\nd2r⟨˙Pm\n⊥⟩t= 0,\nand consequently alsoR\nd2r⟨˙Pm\nx⟩t=R\nd2r⟨˙Pm\ny⟩t= 0, for\na single skyrmion or any other texture where the topolog-\nical charge is localised and embedded in a a ferromagnet.\nAppendix K: Dependence of Ftrans onαforω≤ωKit.\nWe are interested in the behaviour of Ftrans in the\nlimit of low damping, α→0, and at driving frequencies\nat or below the gap, ω≤ωKit.. We have established\nthat only the second term in Eq. (14), which is explicitlyproportional to α, gives a non-zero contribution to Ftrans.\nLet us denote the integrand of this term as Fx,y, such that\nFi=˙M(1)· ∇iM(1). (K1)\nIn Sec. IV, we calculated M(1)as a function of polar\ncoordinates r, χ. For this reason, it is algebraically neater\nto consider the linear combination Fx+iFy. We will\ninvestigate what happens as a function of ωto the time-\nand polar angle- averaged Fx+iFy, which reads\n⟨Fx+iFy⟩χ,t=\u001c\neiχ\u0012\n˙a(1)∂ra∗(1)+ ˙a∗(1)∂ra(1)\n+i\nr(˙a(1)∂χa∗(1)+ ˙a∗(1)∂χa(1)) (K2)\n+cos(θ0)\nr\u0010\n˙a(1)a∗(1)−˙a∗(1)a(1)\u0011\u0013\u001d\nχ,t\nin terms of the linear response fields a(1),a∗(1). For sim-\nplicity, we also set δ, h= 0, which makes all the c(0)\n0,k,±ω,\n˜c(0)\n±1,k,±ωcoefficients real. Alternative choices will have\nan incidence on the force components Fx,y\ntrans, but not on\nits magnitude |Ftrans|. Ifω < ω Kit.,ω̸=ωbr., i.e. we\nare below the gap and away from the resonances, in the\nlimit α→0 all the a(1)\n0,±ω,a(1)\n±1,±ωfields become purely\nreal. After some algebra, we find\nlim\nα→0⟨Fx⟩χ,t(ω̸=ωbr./Kit. ) = 0\nlim\nα→0⟨Fy⟩χ,t(ω̸=ωbr./Kit. ) = 2h\na(1)′\n0,+ω\u0010\na(1)\n−1,+ω−a(1)\n1,+ω\u0011\n−a(1)′\n0,−ω\u0010\na(1)\n−1,−ω−a(1)\n1,−ω\u0011i\n+2\nrcos(θ0)h\na(1)\n0,+ω\u0010\na(1)\n1,+ω+a(1)\n−1,+ω\u0011\n−a(1)\n0,−ω\u0010\na(1)\n−1,−ω+a(1)\n1,−ω\u0011i\nHence, in the limit of low damping and away from the\nbreathing and Kittel resonances, ⟨Fx⟩χ,tvanishes and\nonly the ⟨Fy⟩χ,tcomponent survives and tends to a con-\nstant finite value. Thus, we can conclude that the force\nacting on the skyrmion is purely in the ey-direction, and\ngiven by\nlim\nα→0Ftrans(ω̸=ωbr., ωKit.) =αeyZ\n2πr dr⟨Fy⟩χ,t.\n(K3)\nIf instead ω=ωbr.exactly, then a(1)\n0,±ωwill be purely\nimaginary and grow like 1 /(iα). This changes ⟨Fx,y⟩χ,t24\nto\nlim\nα→0⟨Fx⟩χ,t(ω=ωbr.) = 2 ih\n−a(1)′\n0,+ω\u0010\na(1)\n−1,+ω+a(1)\n1,+ω\u0011\n+a(1)′\n0,−ω\u0010\na(1)\n−1,−ω+a(1)\n1,−ω\u0011i\n+2i\nrcos\u0000\nθ0\u0001h\na(1)\n0,+ω\u0010\na(1)\n1,+ω−a(1)\n−1,+ω\u0011\n+a(1)\n0,−ω\u0010\na(1)\n−1,−ω−a(1)\n1,−ω\u0011i\nlim\nα→0⟨Fy⟩χ,t(ω=ωbr.) = 0 .\nThis time the situation is reversed and ⟨Fy⟩χ,tvanishes.\nThe resulting force is\nlim\nα→0Ftrans(ω=ωbr.) =αexZ\n2πr dr⟨Fx⟩χ,t.(K4)\nImportantly, now we have ⟨Fx⟩χ,t∝1/αin the limit\nα→0 because of the a(1)\n0,±ωcomponents. Hence,\nlim\nα→0Ftrans(ω=ωbr.)∼α·1\nα∼const. (K5)Finally, we consider the case ω=ωKit.=b0. In this\ncase, depending on sgn( γ) either a(1)\n−1,+ωanda(1)\n1,−ωor\na(1)\n1,+ωanda(1)\n−1,−ωwill be purely imaginary and ∝1/(iα),\nsee Eq. (39). If γ <0 and bR>0,a(1)\n−1,+ωanda(1)\n1,−ωare\npurely imaginary and we have\nlim\nα→0⟨Fx⟩χ,t(ω=ωKit.) = 2 i\u0010\na(1)′\n0,+ωa(1)\n−1,+ω−a(1)′\n0,−ωa(1)\n1,−ω\u0011\n+2i\nrcos(θ0)\u0010\na(1)\n0,+ωa(1)\n−1,+ω+a(1)\n0,−ωa(1)\n1,−ω\u0011\nlim\nα→0⟨Fy⟩χ,t(ω=ωKit.) = 0 . (K6)\nThe forces are therefore the same as in Eq. (K4), but\nwith⟨Fx⟩χ,tas defined in Eq. (K6). Thus at the Kittel\nresonance we also have\nlim\nα→0Ftrans(ω=ωKit.)∼const. (K7)"    },    {        "title": "2211.02490v1.Derivation_of_Interacting_Two_Qubit_Dynamics_from_Spin_Boson_Model.pdf",        "content": "arXiv:2211.02490v1  [quant-ph]  4 Nov 2022Derivation of Interacting Two-Qubit Dynamics from\nSpin-Boson Model\nHiroaki Matsueda1,2, Yukiya Ide1, and Sadamichi M aekawa3,4\n1Department of Applied Physics, Graduate School of Engineee ring, Tohoku University, Sendai\n980-8579, Japan\n2Center for Science and Innovation in Spintronics, Tohoku Un iversity, Sendai 980-8577, Japan\n3RIKEN Center for Emergent Matter Science, Wako, Saitama 351 -0198, Japan\n4Kavli Institute for Theoretical Physics, University of Chi nese Sciences, Beijing 100190, China\nE-mail: hiroaki.matsueda.c8@tohoku.ac.jp\n(Received July 31, 2022)\nWe derive damping equations of motion for interacting two-s pin states from a spin-boson model\nin order to examine qubit dynamics in quantum computers. On t he basis of the composite operator\nmethod, we develop the Caldeira-Leggett approach for open q uantum systems so that the entangle-\nment dynamics originated from the two-spin correlation can be taken. We demonstrate numerical\nresults for time dependence on the two-spin dynamics. We find that the relaxation of the total spin is\ndescribed by a quantum version of the Landau-Lifshitz-Gilb ert equation for magnetic materials. We\nalso find that a two-spin composite mode keeps oscillation ev en after the total spin has been fully\nrelaxed. We thus conclude that the two-spin correlation due to the presence of the composite mode\nis stable against dissipation. We consider the mechanism of why the correlation is maintained.\nKEYWORDS: qubit, spin-boson model, relaxation, composite operator, equation of motion,\nCaldeira-Leggett approach\n1. Introduction\nNowadays, exploring quantum technologies such as quantum c omputation, quantum cryptogra-\nphy, and quantum sensing is turning into a realistic goal for current engineering. Ten years have\npassed already after the D-wave machine, a kind of quantum an nealers, was commercially provided,\nand the machine is getting used for various optimization pro blems. Recent flagship research projects\nassociated with quantum technologies aim to construct larg e-scale fault-tolerant universal quantum\ncomputers in the middle of this century. For this purpose, de velopment of noisy intermediate-scale\nquantum (NISQ) computers is an important milestone at the pr esent stage. We believe that future\nsophisticated society will be highly supported from these t echnologies, and thus it is necessary to\npromote basic science behind the technologies. There are ma inly two directions for the advanced re-\nsearch. One is direct treatment of fault tolerance algorith m and implementation as long-term research,\nand the other is deep examination of NISQ itself as short or me dium-term plan. The latter is closely\nrelated to non-equilibrium physics in which the dynamics of our qubit system is highly disturbed\nby the environmental noise and interaction among qubits the mselves when the qubits are massively\nintegrated on the substrate. Since a qubit can be identified w ith a quantum spin, our target model is\nthe so-called spin-boson model in which the interacting qua ntum spins couple with bosonic degrees\nof freedom. Thus, our interest is to understand operational stability of single spin and entanglement\namong multiple spins in this model. This is because quantum c omputation is realized by sequential\nchange of qubit states with external perturbation as unitar y gates.\nMotivated by the abovementioned consideration, we theoret ically examine the spin dynamics in\n1the spin-boson model. Here, the entanglement control of qub its is a key for various quantum tech-\nnologies, and thus we particularly focus on whether dynamic al behavior of non-local correlation or\nentanglement is stably controlled against dissipation due to the presence of the environment. This\ntype of works was recently done as a toy model for the D-wave ma chine [1]. However, this is a very\nspecial case in which we can successfully integrate out boso nic degrees of freedom in terms of the\nSuzuki-Trotter decomposition. We would like to get versati le techniques for more general cases.\nFor this purpose, we first derive damping equations of motion for the total spin from our spin-\nboson model in which two spins interact with each other. We de veloped old approaches such as Feyn-\nman’s influential functional and Caldeira-Leggett model. W e assume Ohmic spectral distribution of\nenvironmental degrees of freedom and Markovian approximat ion to get a closed equation of motion.\nWe then find that the result is equivalent to a quantum version of the Landau-Lifshitz-Gilbert (LLG)\nequation for macroscopic spin precession in magnetic mater ials [2, 3]. Here, the Gilbert constant is\nproportional to the coe fficient of the distribution function. On the basis of the total -spin dynamics, we\nnext focus on the internal dynamics of two spins in order to un derstand the stability of their nonlocal\ncorrelation. The abovementioned theoretical method is als o applied to a composite operator associ-\nated with the correlation. We find that the two-spin correlat ion is maintained even after the total spin\nhas been relaxed and the stability of two-spin correlation i s different from that of the total-spin dy-\nnamics [3]. In the usual spectroscopy in quantum many-body s ystems, low-lying states are dominated\nby composite spins [4]. We expect that the stability of the dy namics strongly depends on the spatial\nsize of correlated spin cloud. We demonstrate numerical res ults and mention why the stability of the\nnonlocal correlation appears.\n2. Model and Equations of Motion for Interacting Two-Spin Dy namics\nWe consider interacting two qubits (quantum spins) coupled to the bosonic environment. We start\nwith the following Hamiltonian\nH=/summationdisplay\nk,αωkbα†\nkbα\nk+/summationdisplay\nαBα\n0Sα+J/summationdisplay\nαSα\n1Sα\n2+/summationdisplay\nk,α/parenleftBig\nνkbα\nk+ν∗\nkbα†\nk/parenrightBig\nSα, (1)\nwhere we consider two S=1/2 spins, Sα=Sα\n1+Sα\n2(α=1,2,3),bα†\nkandbα\nkare boson operators\nwith mode kand index of angular momentum α,Bα\n0is associated with the energy di fference of qubit\nstates,ǫ, and transverse field, h,/vectorB0=(h,0,ǫ), and Jis antiferromagnetic coupling between spins.\nHere we assume a special form of boson operators with index αso that we can find simple damping\nequations for spins. Furthermore, the coupling between qub its is assumed to be of Heisenberg type\nfor simplicity, but this is an ideal situation. We should not e that the coupling actually depends on the\ntype of qubit design. For simplicity we introduce\nAα=Bα\n0+Lα,Lα=/summationdisplay\nk/parenleftBig\nνkbα\nk+ν∗\nkbα†\nk/parenrightBig\n, (2)\nand then the Hamiltonian is simply represented as H=/summationtext\nkωk/vectorb†\nk·/vectorbk+J/vectorS1·/vectorS2+/vectorA·/vectorS.\nThe equations of motion for single spin operators, /vectorS1and/vectorS2, are represented as\n∂\n∂t/vectorS1=J/vectorm−/vectorS1×/vectorA,∂\n∂t/vectorS2=−J/vectorm−/vectorS2×/vectorA, (3)\nwhere the composite spin operator /vectormis defined by\nmα=ǫαβγSγ\n1Sβ\n2, /vectorm=/vectorS2×/vectorS1, (4)\n2and this operator characterizes entanglement between two s pins. To understand the relation between\n/vectormand entanglement, it is useful to remember the definition of t he single spin /vectorS1. The states,|↑∝angbracketright\nand|↓∝angbracketright, are the eigenstates of Sz\n1in the single-spin case. The transition between these state s are\nrepresented by Sx\n1andSy\n1. Here, the eigenstates of mzare the product and entangled states ( |↑↑∝angbracketright,\n|↓↓∝angbracketright, and|↑↓∝angbracketright± i|↓↑∝angbracketright). Then, mxandmycorrespond to the transition from the product state ( |↑↑∝angbracketrightor\n|↓↓∝angbracketright) to the singlet state, and these transition operators play a crucial role on generating or keeping\nentanglement. Thus, we particularly focus on the relaxatio n dynamics of /vectorm. Note that the equation of\nmotion for the total spin /vectorSdoes not include /vectormbecause/vectorScommutes with the Heisenberg coupling:\n∂\n∂t/vectorS=−/vectorS×/vectorA=−1\n2/parenleftBig/vectorS×/vectorA−/vectorA×/vectorS/parenrightBig\n. (5)\nThus, the alternative treatment of /vectormis important for the examination of internal dynamics betwe en\nspins. The equation of motion for /vectormis given by\n∂\n∂t/vectorm=1\n2J/parenleftBig/vectorS2−/vectorS1/parenrightBig\n−/vectorm×/vectorA=1\n2J/parenleftBig/vectorS2−/vectorS1/parenrightBig\n−1\n2/parenleftBig\n/vectorm×/vectorA−/vectorA×/vectorm/parenrightBig\n. (6)\nNote that we have taken symmetrized procedure in Eqs. (5) and (6) in order to avoid technical di ffi-\nculty associated with noncommutativity of quantum operato rs.\nThese equations still contain bosonic operators through /vectorA. Let us remove the bosonic degrees of\nfreedom. For this purpose, the Heisenberg equation of motio n for environmental boson is given by\ni∂\n∂tbα\nk=ωkbα\nk+ν∗\nkSα, (7)\nand the formal solution can be obtained as\nbα\nk(t)=e−iωktbα\nk(0)−iν∗\nk/integraldisplayt\n0dt′e−iωk(t−t′)Sα/parenleftbigt′/parenrightbig. (8)\nWe would like to obtain a closed form of equations of motion fo r/vectorSand/vectorm, and for this purpose we\nsubstitute the bosonic solution into the equations. We assu me the bosonic spectrum as\nJ(ω)=/summationdisplay\nk|νk|2δ(ω−ωk)=ηω, (9)\nwhere this assumption represents the Ohmic process and the c oefficientηplays a central role on the\nrelaxation of spin dynamics.\nBy combining these equations with use of Markovian approxim ation (ωcis the cut-offfrequency\nforJ(ω), and we take it as a large constant), the final form of total sp in dynamics is given by\n∂\n∂t/vectorS(t)=/vectorB(t)×/vectorS(t)−ηsinωct\nt/parenleftBig/vectorS(t)×/vectorS(0)−/vectorS(0)×/vectorS(t)/parenrightBig\n−α(t)/parenleftBigg\n/vectorS(t)×∂\n∂t/vectorS(t)−∂\n∂t/vectorS(t)×/vectorS(t)/parenrightBigg\n, (10)\nwhere the coefficientα(t) is defined by\nα(t)=η/integraldisplayωct\n0dτsinτ\nτ,lim\nωct→∞α(t)=πη\n2, (11)\nand/vectorB(t) is defined by\nBα(t)=Bα\n0+/summationdisplay\nk/parenleftBig\nνke−iωktbα\nk(0)+ν∗\nkeiωktbα†\nk(0)/parenrightBig\n. (12)\n3The result is essentially a quantum version of the LLG equati on with the damping coe fficientα(t). In\nthe right hand side of Eq. (10), the second term represents co rrelation between the initial state and the\nstate at time t. For largeωcvalues, the second term becomes negligible with time. The la st term in\nEq. (10) shows damping, and also produces quantum e ffects that do not contain in the classical LLG\ndynamics. The quantum e ffects are originated in non-commutativity between /vectorSand∂/vectorS/∂t, and then\nthe magnitude of the expectation value of /vectorSmay not be a conserved quantity. We will briefly discuss\nthis point later.\nOn the other hand, /vectormshows the following dynamics\n∂\n∂t/vectorm(t)=/vectorB(t)×/vectorm(t)+1\n2(J−2ηωc)/parenleftBig/vectorS2(t)−/vectorS1(t)/parenrightBig\n−ηsinωct\nt/parenleftBig\n/vectorm(t)×/vectorS(0)−/vectorS(0)×/vectorm(t)/parenrightBig\n−α(t)/parenleftBigg\n/vectorm(t)×∂\n∂t/vectorS(t)−∂\n∂t/vectorS(t)×/vectorm(t)/parenrightBigg\n. (13)\nBy substituting /vectorm×/vectorS=/parenleftBig/vectorS1−/vectorS2/parenrightBig\n/2+i/vectormand/vectorS×/vectorm=/parenleftBig/vectorS2−/vectorS1/parenrightBig\n/2+i/vectorminto Eq. (13), we obtain\n∂\n∂t/vectorm(t)=/vectorB(t)×/vectorm(t)−1\n2(J−2ηωc)/parenleftBig\n/vectorm(t)×/vectorS(t)−/vectorS(t)×/vectorm(t)/parenrightBig\n−ηsinωct\nt/parenleftBig\n/vectorm(t)×/vectorS(0)−/vectorS(0)×/vectorm(t)/parenrightBig\n−α(t)/parenleftBigg\n/vectorm(t)×∂\n∂t/vectorS(t)−∂\n∂t/vectorS(t)×/vectorm(t)/parenrightBigg\n. (14)\nA striking feature of this equation is that the relaxation te rm (the last term in the right hand side)\nturns offwhen/vectorSis in a stationary condition ∂/vectorS/∂t=0. In this case, the operator /vectorSin the second\nterm behaves as a static field. The third term represents quan tum correlation between initial state and\nthe state at time t, but this term is negligible as we have already pointed out. F urthermore, decoupling\nof the nature of /vectorSand/vectormis facilitated by taking J=2ηωc. Therefore, the dynamics of /vectormis very\nstable against the relaxation of /vectorS. More precisely, the stationary condition must be represen ted by\n∝angbracketleftψ|∂/vectorS/∂t|ψ∝angbracketright=∂∝angbracketleft/vectorS∝angbracketright/∂t=0 with the initial quantum state |ψ∝angbracketright. Thus, the abovementioed statement\nwould be too strong. We expect slow damping of the composite s pin/vectorm, and the damping behavior of\n/vectormwould be represented by sophisticated treatment of higher- order equation of motion. We thus think\nthat the energy relaxation time T1is determined by the time scale of relaxation of the total spi n∝angbracketleft/vectorS∝angbracketright,\nand the decoherence time T2between two spins is determined by the time scale of relaxati on of the\ncomposite spin∝angbracketleft/vectorm∝angbracketright. In the next section, we analyze these coupled equations of m otion to examine\nthe feature of the dynamics of ∝angbracketleft/vectorS∝angbracketrightand∝angbracketleft/vectorm∝angbracketright.\nIn the previos works associated with inertial spin dynamics in ferromagnets, higher-order terms\nof the LLG equation have been considered [5, 6]. It is an inter esting future work to examine their\nrelationship with the present result.\n3. Numerical Results\nBefore going into numerical details, we briefly examine the q ubit system without bosons in order\nto find a guideline for determining the magnitudes of Jandǫ(h=0). In this case, the Hamiltonian\nwithout bosons, H0, is transformed into\nH0=J/vectorS1·/vectorS2+ǫSz=J\n2S(S+1)−3J\n4+ǫSz, (15)\nwhere/vectorS·/vectorS=S(S+1)=2/vectorS1·/vectorS2+3/2 and thus H0can be represented by using the total spin\nS. The singlet is characterized by S=0 and Sz=0, while the triplet is characterized by S=1\nandSz=−1,0,+1. Here, we compare the energy of singlet E(S=0,Sz=0)=−3J/4 with\nthe energy of one of triplets E(S=1,Sz=−1)=J/4−ǫ. Then, E(S=1,Sz=−1) becomes\n4lower than E(S=0,Sz=0) forǫ > J. When we take a parameter range in which the triplet\n(disentangled product state) is stabilized, a viewpoint of classical LLG dynamics would be reasonable\nfor describing the dynamics of /vectorS. In this proceeding, we would like to start with such a simple case,\nand then consider the coupling with bosonic environment. We are interested in a parameter region in\nwhich singlet and triplet states are strongly competing wit h each other, but this is a future work. For\ncomparison, we show that the operator /vectormsatisfies the following relation\n/vectorm·/vectorm=3\n8−1\n2/vectorS1·/vectorS2=3\n4−1\n4S(S+1). (16)\nThus the magnitude of this quantity is also characterized by the total spin S. This value for the singlet\nstate with finite amont of entanglement is larger than that fo r the triplet state. This result also supports\nthat/vectormcharacterizes entanglement between two spins.\nFor numerical simulation, we take J=1,ωc=200, and/vectorB0=(h,0,ǫ)=(0,0,2). We intro-\nduce the initial quantum state as a product (disentangled) s tate, and then consider how the two-spin\ncorrelation or entanglement is generated by the time evolut ion:\n|ψ∝angbracketright∝(a|↑∝angbracketright1+(1−a)|↓∝angbracketright1)⊗(b|↑∝angbracketright2+(1−b)|↓∝angbracketright2)⊗|ϕ∝angbracketright, (17)\nwhere we take a=0.7 and b=0.3, and|ϕ∝angbracketrightis a bosonic part. For this initial state, the expectation\nvalues of/vectorSand/vectormare, respectively, given by\n∝angbracketleft/vectorS∝angbracketright=∝angbracketleftψ|/vectorS|ψ∝angbracketright=(0.72,0,0),∝angbracketleft/vectorm∝angbracketright=∝angbracketleftψ|/vectorm|ψ∝angbracketright=(0,−0.25,0). (18)\nNote that the magnitudes of these vectors depend on the selec tion of the initial quantum state |ψ∝angbracketright.\nWe do not take a full polarized state |↓↓∝angbracketright(or the maximally-entangled singlet state) at t=0, since\n∝angbracketleft/vectorm∝angbracketright(or∝angbracketleft/vectorS∝angbracketright) is zero in this case. Our equations of motion, Eqs. (10) and ( 14), are operator relations,\nnot classical vector equations. Thus, we must take expectat ion values by the state |ψ∝angbracketrightin order to\nintroduce graphical representation. In this process, /vectorS×∂/vectorS/∂tin Eq. (10) and /vectorm×∂/vectorS/∂tin Eq. (14)\nare respectively decomposed into two independent terms:\n∝angbracketleft/vectorS×∂\n∂t/vectorS∝angbracketright∼∝angbracketleft/vectorS∝angbracketright×∂\n∂t∝angbracketleft/vectorS∝angbracketright,∝angbracketleft/vectorm×∂\n∂t/vectorS∝angbracketright∼∝angbracketleft/vectorm∝angbracketright×∂\n∂t∝angbracketleft/vectorS∝angbracketright. (19)\nThe dynamics of∝angbracketleft/vectorS∝angbracketrightafter this approximation becomes equivalent to the classic al LLG equation,\nexcept that∝angbracketleft/vectorB∝angbracketrightstill contains information of bosons. Here we neglect time d ependence on /vectorB(t) that\nappears as a result of the second term in Eq. (12). In this case , we can solve the equation of motion,\nand we find that the relaxation time scale T1is proportional to (1 +π2η2|∝angbracketleft/vectorS∝angbracketright|2)/2πηǫ|∝angbracketleft/vectorS∝angbracketright|. Unfor-\ntunately, quantum e ffects originated from non-commutativity between /vectorSand∂/vectorS/∂tand polaronic\neffects are lost in this approximation, and then |∝angbracketleft/vectorS∝angbracketright|is kept. Thus, we suppose that the realistic relax-\nation time scale may change. In the present approximation, t he decoherence time T2becomes infinity\ndue to the stability of the dynamics of /vectorm. The precise estimation of T2is an important future work,\nbut we can say T2>T1even within the present simple analysis.\nWe demonstrate time evolution of ∝angbracketleft/vectorS∝angbracketrightand∝angbracketleft/vectorm∝angbracketrightforη=0.008 (2ηωc=3.2>J=1) in Fig. 1.\nWe find that∝angbracketleft/vectorS∝angbracketrightdecays into the direction of −∝angbracketleft/vectorB∝angbracketright. This feature is consistent with the classical LLG\ndynamics. As we have already discussed, the length of ∝angbracketleft/vectorS∝angbracketright, 0.72, is conserved in the present approx-\nimation, and∝angbracketleft/vectorS∝angbracketrightdoes not become (0 ,0,−1) even after the long time. We particularly focus on the\nentanglement dynamics represented by ∝angbracketleft/vectorm∝angbracketright. In contrast to∝angbracketleft/vectorS∝angbracketright, the coherent oscillation of ∝angbracketleft/vectorm∝angbracketrightis main-\ntained even after∝angbracketleft/vectorS∝angbracketrighthas been relaxed to the stationary point. The coherent oscil lation corresponds to\ncontinuous spin flip ( |↑↓∝angbracketright↔|↓↑∝angbracketright ) between two spins. We find that the phase di fference between∝angbracketleftmx∝angbracketright\nand∝angbracketleftmy∝angbracketrightisπ/2. As we have already mentioned, the equation of motion for /vectormdoes not contain the\ndamping term if ∂S/∂tbecomes zero. This is the origin of the stable oscillation of ∝angbracketleft/vectorm∝angbracketright. Therefore, the\ntwo-spin dynamics is essentially di fferent from total-spin dynamics.\n5Fig. 1. Spin dynamics for η=0.008. (a)∝angbracketleft/vectorS∝angbracketright, (b)/angbracketleftbig/vectorm/angbracketrightbig, and (c) Graphical representation of (a) and (b). In\nfigure (c), the blue curve represents ∝angbracketleft/vectorS∝angbracketright, and the red curve represents/angbracketleftbig/vectorm/angbracketrightbig.\n4. Concluding Remarks\nWe derived the spin dynamics in the spin-boson model in order to examine the entanglement\ncontrol of qubits against dissipation due to the presence of the environment. For the total spin, we\nfound that the result is consistent with the LLG equation for macroscopic spin precession in magnetic\nmaterials although the result also contains some quantum e ffects. Here, the Gilbert constant is propor-\ntional to the coefficient of the distribution function of the bosonic degrees of freedom. However, the\nentanglement dynamics originated from a composite spin sho ws different behavior. We numerically\nshowed the relaxation dynamics of the total spin and the stab ility of the entanglement dynamics. In\nthe present approximation, the decoherence time T2is infinity, and more precise treatment based on\nthe higher-order equation of motion is an interesting futur e work.\nH.M. is supported by JPSJ KAKENHI (Nos. 21K03380, 21H04446, 21H03455) from MEXT\nJapan and CSIS, Tohoku University, Japan. S.M is supported b y JST CREST Grant (Nos. JPMJCR19J4,\nJPMJCR1874, and JPMJCR20C1) and JSPJ KAKENHI (Nos. 17H0292 7 and 20H01865) from MEXT,\nJapan.\nReferences\n[1] Y . Bando and H. Nishimori, Phys. Rev. A 104, 022607 (2021).\n[2] J. Anders, C. R. J. Sait, and S. A. R. Horsley, New. J. Phys. 24, 033020 (2022).\n[3] H. Matsueda, Y . Ide, and S. Maekawa, in preparation.\n[4] T. Otaki, Y . Yahagi, and H. Matsueda, J. Phys. Soc. Jpn. 86, 084709 (2017).\n[5] K. Neeraj et al., Nat. Phys. 17, 245 (2021).\n[6] M.-C. Ciornei, J. M. Rubi, and J.-E. Wegrowe, Phys. Rev. B 83, 020410(R) (2011).\n6"    },    {        "title": "2211.07744v2.Magnetization_Dynamics_in_Synthetic_Antiferromagnets_with_Perpendicular_Magnetic_Anisotropy.pdf",        "content": "1 \n Magnetization Dynamics in Synthetic Antiferromagnets with Perpendicular \nMagnetic Anisotropy  \n \nDingbin Huang1,*, Delin Zhang2, Yun Kim1, 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  \n \n \nABSTRACT:   \nUnderstanding the rich physics of magnetization dynamics in  perpendicular  synthetic \nantiferromagnets  (p-SAFs)  is crucial  for developing next-generation  spintronic devices.  In this \nwork, we systematically investigate  the magnetization dynamics in p-SAFs  combining time-\nresolved magneto -optical Kerr effect (TR -MOKE)  measurements with theoretical modeling . \nThese model analyses, based on a Landau -Lifshitz -Gilbert approach incorporating exchange \ncoupling , provide detail s about the  magnetization dynamic characteristics  including  the amplitude s, \ndirections, and phases of the precession  of p-SAFs under varying magnetic fields . These model -\npredicted characteristics are in excellent quantitative agreement with TR-MOKE measurements \non an asymmetric p -SAF. We further reveal the damping mechanisms of two precession modes  \nco-existing in the p -SAF and successfully identify individual contributions from different sources , \ninclud ing Gilbert damping of each ferromagnetic layer , spin  pumping, and inhomogeneous \nbroadening . Such a comprehensive understanding of magnetization dynam ics in p -SAFs, obtained \n                                                 \n*Author s to whom correspondence should be addressed : huan1746@umn.edu  and wang4940@umn.edu  2 \n by integrating high -fidelity TR -MOKE measurements and theoretical modeling, can guide  the \ndesign of p-SAF-based architectures for spintronic  applications . \n \nKEYWORDS:  Synthetic antiferromagnets; Perpendicular magnetic anisotropy; Magnetization \nDynamics; Time -resolved magneto -optical Kerr effect; Spintronics   3 \n 1 INTRODUCTION  \nSynthetic antiferromagnet ic (SAF) structures have attracted considerable  interest for \napplications in spin mem ory and logic devices  because of their unique magnetic  configuration s [1-\n3]. The SAF structures are composed of two ferromagnetic (FM) layers anti -parallelly coupled \nthrough a non -magnetic (NM) spacer, offer ing great  flexibilit ies for the  manipulat ion of magnetic \nconfigurations  through  external stimuli  (e.g., electric -field and spin-orbit torque , SOT) . This \npermit s the design of new architecture s for  spintronic applications , such as magnetic tunnel \njunct ion (MTJ), SOT  devices, domain wall devices, sky rmion devices, among others  [4-7]. The \nSAF structures  possess many advantages for such applications , including  fast switching speeds \n(potentially in the THz regimes),  low off set fields, small switching current s (and thus low energy \nconsumption) , high thermal stability,  excellent resilience  to perturbations from external magnetic  \nfields, and large turnabilit y of magnetic properties  [3,8-16].  \nA comprehensive study  of the magnetization dynami cs of SAF structures  can facilitate the \nunderstanding of the switching behavior of spintronic  devices , and ultimately guide the design of \nnovel device architectures . Different from a single FM free layer, magnetization dynamics of the \nSAF structures involv es two modes of precession , namely high -frequency (HF) and low -frequency  \n(LF) modes,  that result from the hybridization of magnetizations  precession  in the two FM layers . \nThe relative phase and precession amplitude in two FM layers can significantly affect the spin-\npumping enhancement of magnetic damping  [17], and thus play an important role in determining \nthe magnetization dynamic behaviors in SAFs. Heretofore, the exchange -coupling strength and \nmagnetic damping constant of SAFs have been studied by ferromagnetic resonance (F MR)  [18-\n21] and optical metrolog y [22-25]. Most FMR -based experimental studies were limited to SAFs \nwith in -plane magnetic anisotropy (IM A). For device applications, perpendicular magnetic 4 \n anisotropy (PMA) gives  better scalability  [3,26] . Therefore , the characteristics of magnetization \ndynamics of perpendicular SAF (p-SAF) structures  are of much valu e to investigat e. In addition, \nprior  studies mainly focus ed on the mutual  spin pumping between two FM layers  [22,27,28] . A \nmore thorough understanding of the contribution s from various sources, including inhomogeneous \nbroadening  [29], remains elusive . \nIn this paper, we report  a comprehensive study of the magnetization dynamics of p -SAFs  by \nintegrating high -fidelity experiments and theoretical modeling to detail the characteristic \nparameters. These parameters describe the amplitude, phase, and direction of magnetization \nprecess ion of both the HF and LF modes for the two exchange -coupled FM layers in a p -SAF.  We \nconduct all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) measurements  [30-33] \non an asymmetric p -SAF structure with two different FM layers. The field-dependent  amplitude \nand phase of TR -MOKE signal s can be well captured by our theoretical  model, which in turn \nprovid es comprehensive physical insights into the magnetization dynamics of p -SAF structures.  \nMost importantly, we show that inhomogeneous broadening  plays a  critical  role in determining \nthe effective damping of both HF and LF modes, especially at low fields.  We demonstrate the \nquantification of contributions from inhomogeneous broadening  and mutual spin pumping  (i.e., \nthe exchange of angular momentum between two FM layers via pumped spin currents ) [21] to the \neffect ive damping, enabl ing accurate determination of the Gilbert damping for individual FM \nlayers. Results of this work  are beneficial  for designing  p-SAF-based architectures  in spintronic  \napplication s. Additionally, this work also serves as a successful example demonstrating that  TR-\nMOKE, as an all -optical met rology, is a powerful tool to capture the magnetization dynamics and \nreveal the rich physics of complex structures that involve multilayer coupling . \n 5 \n 2 METHODOLOTY  \n2.1 Sample preparation and characterization  \nOne SAF  structure  was deposited onto thermally oxidize d silicon  wafers with a 300 -nm SiO 2 \nlayer by magnetron sputtering at room  temperature (RT) in a six -target ultra -high vacuum (UHV) \nShamrock sputtering system. The  base pressure is below  5×10−8 Torr. The stacking structure of \nthe SAF is:  [Si/SiO 2]sub/[Ta(5)/Pd(3)] seed/[Co(0.4)/Pd(0.7)/Co(0.4)] FM1/[Ru(0.6)/Ta(0.3)] NM/ \nCoFeB(1) FM2/[MgO(2)/Ta(3)] capping . The numbers in parentheses denote the layer thicknesses in \nnanometers. After deposition, the sample was annealed at 250 ℃ for 20 minutes by a rapid -\nthermal-annealing process.  The two FM layers are CoFeB and Co/Pd/Co layers, separated by a \nRu/Ta spacer, forming an asymmetric p -SAF structure ( i.e., two FM layers having different \nmagnetic properties). The M-Hext loops were characterized by a physical propert y measurement \nsystem (PPMS) with a vibrating -sample magnetometer (VSM) module. The resulting M-Hext loops \nare displayed in Fig. 1(a). Under low out -of-plane fields ( Hext < 500 Oe), the total magnetic \nmoments in two FM layers of the SAF stack perfectly cancel out each other: M1d1 = M2d2 with Mi \nand di being the magnetization and thickness of each FM layer ( i = 1 for the top CoFeB layer and \ni = 2 for the bottom Co/Pd/Co laye r). The spin-flipping field ( Hf  ≈ 500 Oe ) in the out -of-plane \nloop indicates the bilinear interlayer -exchange -coupling (IEC) J1 between the two FM layers : J1 = \n−HfMs,1d1 ≈ −0.062 erg cm-2 [34]. The values of Ms,1, Ms,2, d1, and d2 can be found in Table SI of \nthe Supplemental Material (SM)  [35]. \n \n2.2 Theoretical foundation  of magnetization dynamics for a p -SAF structure  \nThe magnetic free energy per unit area for a p -SAF structure with uniaxial PMA can be \nexpressed as [36]: 6 \n 𝐹=−𝐽1(𝐦1⋅𝐦2)−𝐽2(𝐦1⋅𝐦2)2\n+∑2\n𝑖=1𝑑𝑖𝑀s,𝑖[−1\n2𝐻k,eff,𝑖(𝐧⋅𝐦𝑖)2−𝐦𝑖⋅𝐇ext] (1) \nwhere J1 and J2 are the strength of the bilinear and biquadratic IEC. mi = Mi / Ms,i are the normalized \nmagnetization vectors for individual FM layers ( i = 1, 2). di, Ms,i, and Hk,eff, i denote, respectively, \nthe thickness, saturation magnetization, and the effective anisotropy field of the i-th layer. n is a \nunit vector indicating the sur face normal direction of the film. For the convenience of derivation \nand discussion, the direction of mi is represented in the spherical coordinates by the polar angle θi \nand the azimuthal angle φi, as shown in Fig. 1 (b). \nThe equilibrium direction of magne tization in each layer (𝜃0,𝑖,𝜑0,𝑖) under a given Hext is \nobtained by minimizing F in the (𝜃1,𝜑1,𝜃2,𝜑2) space. The magnetization precession is governed \nby the Landau -Lifshitz -Gilbert  (LLG)  equation considering the mutual spin pumping between two \nFM layers [27,37 -40]: \n𝑑𝐌𝑖\n𝑑𝑡=−𝛾𝑖𝐌𝑖×𝐇eff,𝑖+(𝛼0,𝑖+𝛼sp,𝑖𝑖)\n𝑀s,𝑖𝐌𝒊×𝑑𝐌𝒊\n𝑑𝑡−𝛼sp,𝑖𝑗\n𝑀s,𝑖𝐌𝒊×(𝐦𝐣×𝑑𝐦𝒋\n𝑑𝑡)×𝐌𝒊 (2) \nOn the right -hand side of Eq. (2), the first term describes the precession with the effective field \nHeff,i in each layer, given by the partial derivative of the total free energy in the M space via 𝐇eff,𝑖=\n−∇𝐌𝑖𝐹. The second term represents the relaxation induced by Gilbert damping ( α) of the i-th layer, \nwhich includes the intrinsic ( 𝛼0,𝑖) and spin -pumping -enhanced ( 𝛼sp,𝑖𝑖) damping. For TR -MOKE \nmeasurements, 𝛼0,𝑖 and 𝛼sp,𝑖𝑖 are indistinguishable. Hence, we def ine 𝛼𝑖=𝛼0,𝑖+𝛼sp,𝑖𝑖 to include \nboth terms. The last term in Eq. (2) considers the influence of pumped spin currents from the layer \nj on the magn etization dynamics of the layer  i.  7 \n The time evolution of Mi can be obtained by solving the linearized Eq. (2). Details are provided \nin Note 1  of the SM  [35]. The solutions to Eq. (2) in spherical coordinates are:  \n[𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[    𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF]    \nexp(𝑖𝜔HF𝑡)+\n[    𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF]    \nexp(𝑖𝜔LF𝑡) (3) \nwith Δ𝜃𝑖 and Δ𝜑𝑖 representing the deviation angles of magnetization from its equilibrium direction \nalong the polar and azimuthal directions . The last two terms are the linear combination of two \neigen -solutions, denoted by  superscripts HF (high -frequency mode) and LF (low -frequency mode). \nω is the complex angular frequencies of two modes, with the real and imaginary parts representing \nthe precession angular frequency ( 𝑓/2𝜋) and relaxation rate (1/ τ), respectively. For each mode, \nthe complex prefactor vector [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 contains detailed information about the \nmagnetization dynamics. As illustrated in Fig. 1 (c), the moduli, |𝐶𝜃,𝑖| and |𝐶𝜑,𝑖| correspond to the \nhalf cone angles of t he precession in layer i along the polar and azimuthal directions for a given \nmode immediately after laser heating, as shown by Δ𝜃 and Δ𝜑 in Figs. 1 (b-c). The phase \ndifference between Δ𝜃𝑖 and Δ𝜑𝑖, defined as Arg(Δ𝜃𝑖/Δ𝜑𝑖)=Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) with Arg  \nrepresenting the argument of complex numbers, determines the direction of precession. If Δ𝜃𝑖 \nadvances Δ𝜑𝑖 by 90°, meaning Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°, the precession is counter -clockwise (CCW) \nin the θ-φ space  (from a view against Mi).Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=−90°, on the contrary, suggests \nclockwise (CW) precession [ Fig. 1 (d)]. Further, the argument of 𝐶𝜃,2/𝐶𝜃,1 provides the relative \nphase in two FM layers. Arg(𝐶𝜃,2/𝐶𝜃,1)=0° corresponds to the precession motions in two FM \nlayers that are in -phase (IP) in terms of θ for a given mode. While the out -of-phase (OOP) \nprecession in terms of θ is represented by Arg(𝐶𝜃,2/𝐶𝜃,1)=180°  [Fig. 1 (e)]. Given the precession 8 \n direction in each  layer and the phase difference between the two FM layers in terms of θ, the phase \ndifference in terms of φ can be automatically determined.   \n \n \nFIG. 1 (a) Magnetic hysteresis ( M-Hext) loops of the p -SAF stack. The magnetization is n ormalized \nto the saturation magnetization ( M/Ms). (b) Schematic illustration of the half cone angles (Δ θ and \nΔφ) and precession direction of magnetization. The precession direction is defined from a view \nagainst the equilibrium direction ( 0, φ0) of M. The representative precession direction in the \nschematic is counterclockwise (CCW). (c) The relation between precession half cone angles and \nthe prefactors. (d) The relation between precession direction and the prefactors. (e) The relative \nphase between two FM layers for different prefactor values.  \n \nAs for the effective damping 𝛼eff=1/2𝜋𝑓𝜏, in addition to the intrinsic damping ( α0,i) and the \nspin-pumping contribution ( αsp,ii and αsp,ji) considered in Eq. (2), inhomogeneities can also bring \nsubstantial damping enhancement [32,33,41,42] . Here, we m odel the total relaxation rate as \nfollows:  \n9 \n 1\n𝜏Φ=−Im(𝜔Φ)+1\n𝜏inhomoΦ (4) \nThe superscript Φ = HF or LF, representing either the high -frequency or low -frequency precession \nmodes. 𝜔Φ includes both the intrinsic and spin -pumping contributions. The inhomogeneous \nbroadening is calculated as:  \n1\n𝜏inhomoΦ=∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐻k,eff,𝑖|\n𝑖Δ𝐻k,eff,𝑖+∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐽𝑖|\n𝑖Δ𝐽𝑖 (5) \nwhere the first summation represents the contrib ution from the spatial variation of the effective \nanisotropy field of individual FM layers (Δ Hk,eff, i). The second summation denotes the contribution \nfrom the spatial fluctuations of the bilinear and biquadratic IEC (Δ J1 and Δ J2). According to \nSlonczewski’s “thickness fluctuations” theory, Δ J1 generates J2 [43,44] . Therefore,  the fact that  J2 \n= 0 for our sample suggests that  ΔJ1 is sufficiently small, allowing us to neglect  the inhomogeneous \nbroadening from th e fluctuations of both the bilinear and biquadratic IEC  in the following analyses . \n \n2.3 Detection of magnetization dynamics  \nThe magnetization dynamics of the p -SAF sample is detected by TR -MOKE, which is \nultrafast -laser -based metrology utilizing a pump -probe configuration. In TR -MOKE, pump laser \npulses interact with the sample, initiating magnetization dynamics in magnetic layers via inducing \nultrafast thermal demagnetization.  The laser -induced heating brings a rapid decrease to  the \nmagnetic anisotropy  fields  and IEC [45,46] , which changes 𝜃0,𝑖, 𝜑0,𝑖 and initiates the precession.  \nThe magnetizati on dynamics due to pump excitation is detected by a probe beam through the \nmagneto -optical Kerr effect. In our setup, the incident probe beam is normal to the sample surface \n(polar MOKE); therefore, the Kerr rotation angle ( 𝜃K) of the reflected probe beam  is proportional \nto the z component of the magnetization [47]. More details about the experimental setup can be 10 \n found in Refs. [30,32] . For p -SAF, TR -MOKE signals contain two oscillating frequencies that \ncorrespond to the HF and LF  modes  (𝑓HF>𝑓LF). The signals  are proportional to the change in \n𝜃K and can be analyzed as follows:  \nΔ𝜃K(𝑡)=𝐴+𝐵𝑒−𝑡/𝜏T+𝐶HFcos(2𝜋𝑓HF𝑡+𝛽HF)���−𝑡/𝜏HF+𝐶LFcos(2𝜋𝑓LF𝑡+𝛽LF)𝑒−𝑡/𝜏LF (6) \nwhere the exponential term  𝐵𝑒−𝑡/𝜏T is related to the thermal background  with 𝜏T being the time \nscale of heat dissipation . The rest two terms on the right -hand side are the precession terms  with \nC, f, β, and τ denoting , respectively, the amplitude, frequency, phase, and relaxation time of the \nHF and LF  modes.  \nAfter excluding the thermal background from TR -MOKE signals, the precession is modeled \nwith the initial conditions of step -function de creases in 𝐻k,eff,𝑖 and 𝐽𝑖, following the ultrafast laser \nexcitation [48]. This is a reasonable  approximation since the precession period (~15 -100 ps for \nHext > 5 kOe) is much longer than the time scales of the laser excitation (~1.5 ps) and subsequent \nrelaxations among electrons, magnons, and lattice (~ 1 -2 ps) [49], but much shorter than the time \nscale of heat dissipation -governed recovery (~400 ps). With these  initial conditions , the prefactors \nin Eq. ( 3) can be determined  (see m ore details in Note 1  of the SM  [35]).  \nFor our SAF structure, 𝜃K detected by the probe beam contain s weighted contributions from \nboth the top and  bottom FM layers:  \n𝜃K(𝑡)\n𝜃K,s=𝑤cos𝜃1(𝑡)+(1−𝑤)cos𝜃2(𝑡) (7) \nwhere 𝜃K,s represents the Kerr rotation angle when the SAF s tack is saturated along the positive \nout-of-plane ( z) direction. w is the weighting factor, considering the different contributions to the \ntotal MOKE signals from two FM layers. w can be obtained from static MOKE measurements [50], \nwhich gives 𝑤= 0.457 (see more details in Note 2  of the SM  [35]). 11 \n  \n3 RESULTS AND DISCUSSION  \n3.1 Field -dependent p recession frequencies and equilibrium magnetization directions  \nTR-MOKE signals measured at varying  Hext are depicted in Fig. 2 (a). The external field is \ntilted 15 ° away from in-plane [θH = 75°, as defined by Fig. 2 (c)] to achieve larger amplitdues of  \nTR-MOKE signals [51]. The signals can be fitted to Eq.  (6) to extract the LF and HF precession \nmodes. The field -dependen t precession frequenc ies of both modes  are summarized in Fig. 2 (b). \nFor simplicity, when analyzing precession frequencies, magnetic damping  and mutual spin \npumping  are neglected due to its insignificant impacts on  precession frequencies.  By comparing \nthe experimental data and the prediction of ωHF/2π and ωLF/2π based on E q. (3), the effective \nanisotropy fields and the IEC strength  are fitted as Hk,eff, 1 = 1.23 ± 0.28  kOe, Hk,eff, 2 = 6.18 ± 0.13  \nkOe, J1 = −0.050 ± 0.020  erg cm−2, and J2 = 0. All parameters and their determination methods are \nsummarized in Table SI of the SM  [35]. The fitted J1 is close to that obtained from the M-Hext \nloops (~−0.062 erg cm-2). The inset of Fig 2 (b) shows the zoom ed-in view of field -dependent \nprecession frequencies around Hext = 8 kOe, where a n anti -crossing feature is observed: a  narrow \ngap (~2  GHz) open s in the frequency dispersion curves of the HF and LF modes owing to the weak \nIEC between two FM layers. Without a ny IEC, the precession frequencies of two FM layers would \ncross at Hext = 8 kOe, as indicated by the green dashed line and blue dashed line in the figure. We \nrefer to t hese two sets of crossing frequencies as the single -layer natural frequencies of two FM \nlayers (FM 1 and FM 2) in the following discussions .  12 \n  \nFIG. 2 (a) TR -MOKE signals under varying Hext when θH = 75° [as defined in panel (c)]. Circles \nare the experimental data and black lines are the fitting curves based on Eq. (6). (b) The precession \nfrequencies of the HF and LF modes as functions of Hext. Circles are experimental data and solid \nlines are fitting curves. The  inset highlights the zoomed -in view of the field -dependent frequencies \naround 8 kOe, where the green dashed line and blue dashed line are the single -layer (SL) \nprecession frequencies of FM 1 and FM 2 without interlayer exchange coupling. (c) Schematic \nillustration of the definition of the equilibrium polar angles ( θ0,1 and θ0,2), and the direction of the \nexternal magnetic field ( θH). The illustration is equivalent to Fig. 1(b) due to symmetry. (d) θ0,1 \nand θ0,2 as functions of Hext. The dash -dotted line plots the difference between the two equilibrium \npolar angles.  \n \n13 \n Based on the fitted stack properties ( Hk,eff,1, Hk,eff,2, J1, and J2), the equilibrium magnetization \ndirections in the two layers can be calculated. For SAFs with weak IEC compared with uniaxial \nPMA, the azimuthal angles of the magnetization in two FM layers are always the same as that of \nthe external field at equilibrium status. Therefore, two polar angles will be sufficient to describe \nthe equilibrium magnetization con figuration. Figure 2(c) illustrates the definition of the \nequilibrium polar angles of two FM layers ( θ0,1, θ0,2) and the external field ( θH). The values of θ0,1, \nθ0,2, and the difference between these two polar angles as functions of Hext are shown in Fig. 2(d). \nWhen Hext is low (<  1.6 kOe), magnetic anisotropy and antiferromagnetic coupling are dominant \nand |θ0,1 − θ0,2| is larger than 90 °. As Hext increases, both θ0,1 and θ0,2 approach θH. When Hext is \nhigh (> 15 kOe), the Zeeman energy becomes dominant and both M1 and M2 are almost aligned \nwith Hext. \n \n3.2 Cone angle, direction, and phase of magnetization  precession revealed by modeling  \nBesides the equilibrium configuration, using sample properties extracted from Fig. 2 (b) as \ninput parameters, the LLG -based modeling (described in section 2.2) also provide s information o n \nthe cone angle, direction, and phase of magnetization precession for each mode ( Fig. 1 ). The \ndiscussion in this section  is limited to the case  without damping an d mutual spin pumping . They \nwill be  considered  in Note 4 of the SM  [35], sections 3.3, and 3.4. The calculation results are \nshown in  Fig. 3 , which are categorized into three regions. At high external fields ( Hext > 1.6 kOe, \nregions 2 and 3), both FM layers precess CCW [ Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°], and the polar angles of \nmagnetization in two layers are  in-phase [Arg(𝐶𝜃,2/𝐶𝜃,1)= 0°] for the HF mode and  out-of-phase \n[Arg(𝐶𝜃,2/𝐶𝜃,1) = 180° ] for the LF mode. This is the reason for the HF mode (LF mode) also \nbeing called the acoustic mode ( optical mode) in the literature  [23]. The criterion to differentiate 14 \n region 2 from region 3 is the FM layer that dominat es a given precessional mode  (i.e., the layer \nwith larger precession cone angles) . In region 2  (1.6 kOe < Hext < 8 kOe) , the HF mode is \ndominated by FM 2 because FM 2 has larger cone angles than FM 1. This is reasonable since the \nhigher precession frequency is closer to the natural frequency of FM 2 [see Fig. 2(b)] in region 2. \nSimilarly, in region 3, the HF mode is dominated by FM 1 with larger precession cone angles.  \nWhen Hext is low (region 1), the angle between two magnetizations is larger than 90° [ Fig. 2 (d)] \nowing to the  more  dominan t AF-exchange -coupling  energy as compared with the Zeeman energy . \nIn this region, magnetization dynamics exhibits some unique features.  Firstly, CW [ Arg(𝐶𝜃,𝑖/\n𝐶𝜑,𝑖)=−90°] precession emerges: for each mode, the dominant layer precesses CCW (FM 2 for \nthe HF mode and FM 1 for the  LF mode) and the subservient layer precesses CW (FM 1 for the HF \nmode and FM 2 for the LF mode). This is because the effective field for the subservient layer [ e.g., \nHeff,1 for the HF mode, see Eq.  (2)] precesses CW owing to the CCW precession of the dominant \nlayer when |𝜃0,1−𝜃0,2|>90° [Fig. 2(d)] . In other words, a low Hext that makes |𝜃0,1−𝜃0,2|>\n90° is a necessary condition for the CW precession.  However, it is not a sufficient condition. In \ngeneral, certain degrees  of symmetry breaking ( Hk,eff,1 ≠ Hk,eff,2 or the field is tilted away from the \ndirection normal to the easy axis ) are also needed to generate CW precession.  For example, for \nsymmetric a ntiferromagnets ( Hk,eff,1 = Hk,eff,2) under fields perpendicular to the easy axis, CW \nprecession does not appear even at low fields (Fig. 2(a) in Ref. [52]). See Note 5 of the SM  [35] \nfor more details.  Secondly, as shown in  Fig. 3 , the precession motions in two FM layers are always \nin-phase for both HF and LF modes; thus, there is no longer a clear differentiation between \n“acoustic mode” and “optical mo de”. Instead, the two modes can be differentiated as “right -handed” \nand “left -handed” based on the chirality [53]. Here, we define the chirality with respect to a \nreference direction taken as the projection of Hext or M2 (magnetization direction of the layer with 15 \n a higher Hk,eff) on the easy axis [ -z direction in Fig. 3 (c)]. Lastly, the shape of the precession cone \nalso varies in different regions. Δ θi and Δφi are almost the same for both modes in region 3, \nindic ating the precession trajectories are nearly circular. While in regions 1 and 2, Δ θi and Δφi are \nnot always equal, suggesting the precession trajectories may have high ellipticities.  \n \n \nFIG. 3 The calculated  half cone angle, direction, and phase of magnetization precession for (a) the \nHF mode and (b) the LF mode. In the top row, four curves represent the polar and azimuthal  half \ncone angles of precession in two FM layers. All  half cone angles are normalized with r espect to \nΔθ1. The middle row shows the value of Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) under different Hext. A value of 90°  (−90°) \nrepresents CCW (CW) precession. The bottom row is the phase difference of the polar angles in \ntwo layers. A value of 0° (180°) corresponds to the polar angles of the magnetization in two layers \nare IP (OOP ) during precession.  Dashed lines correspon d to the reference case where damping is \nzero in both layers. (c) Schematic illustrations of the cone angle, direction, and phase of \n16 \n magnetization precession for the HF and LF modes in different regions, and their corresponding \ncharacteristics regarding ch irality and phase difference.  \n \n3.3 Amplitude and phase of TR -MOKE signals  \nActual magnetization dynamics is resolvable  as a linear combination of the two eigenmodes  \n(the HF and  the LF modes ). By taking into account the initial conditions (i.e., laser excitation , see \nNote 1  of the SM  [35]), we can determine the amplitude and phase of the two modes in TR -MOKE \nsignals . Figure 4 (a) summarizes the amplitudes of both HF and LF modes [CHF and CLF in Eq.  (3)] \nunder different  Hext. Noted that  the y-axis represents Kerr angle ( θK) instead of the cone angle of \nprecession.  The LF mode  has a local minimum near 8 kOe, where the two FM layers have similar \nprecession cone angles but opposite phase s for the LF mode [ Fig. 3 (b)]. The amplitude s of both \nmodes decrease with Hext in the high -field region. This is similar to the single -layer case, where \nthe amplitudes of TR -MOKE signals decrease with Hext because the decrease  in Hk,eff induced by \nlaser heating is not able to significantly  alternate  the equilibrium magnetization direction when the \nZeeman energy dominates  [51]. The LF mode also has an amplitude peak at low fields ( Hext < 3 \nkOe), where the dominant layer of FM 1 changes its equilibrium direction dramatically with Hext \n(from ~75° to 170°) as shown in Fig. 2(d).  \nTo directly compare the amplitudes of TR -MOKE signals and the LLG -based calculations , the \nweighting factor w and the initial conditions are needed. The initial conditions are determined by \n𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′, representing the instantan eous effective anisotropy fields and IEC strength \nupon laser heating. These instantaneous properties are different from their corresponding room -\ntemperature values ( Hk,eff,1, Hk,eff,2, and J1). The accurate determination of𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′ \ndemands the modeling of the laser heating process as well as the temperature dependence of stack \nproperties, which are challenging. Here, we treat these three variables as adjustable parameters and 17 \n determine their values by fitting the field -dependent amp litudes of TR -MOKE signals , which \nyields 𝐻k,eff,1′𝐻k,eff,1⁄=0.90±0.01, 𝐻k,eff,2′𝐻k,eff,2⁄=0.95±0.01, and 𝐽1′𝐽1⁄=0.83±0.01. \nIt is apparent that the field dependence of TR -MOKE signal amplitude is in excellent agreement \nwith the theoretical modeling , as s hown in Fig. 4 (a). \nFigure 4 (b) shows the calculated  half polar cone angles for each mode in each FM layer. In \nTR-MOKE signals, the optical mode (the LF mode in regions 2 and 3) tends to be partially \ncanceled out  because the two layers precess out -of-phase.  Therefore, compared with Fig. 4 (a), the \ninformation in Fig. 4 (b) better reflects the actual intensity of both modes in FM 1 and FM 2. In Fig. \n4(b), the precession cone angles of both modes in FM 1 (Δ𝜃1HF,Δ𝜃1LF) have local maxima at the \nanti-crossing field (Hext ≈ 8 kOe). On the contrary, Δ𝜃2LF and Δ𝜃2HF of FM 2 have their maxima \neither above or below  the anti-crossing field. This is because FM 2 has larger precession amplitudes \n(cone angles) than FM 1 at the anti-crossing field if there is no IEC [the dotted  lines of FM 1 (SL) \nand FM 2 (SL) in Fig. 4 (b)]. With IEC, FM 2 with larger cone angles can drive the precession motion \nin FM 1 significantly near the anti-crossing field, where IEC is effective. Subsequently, the \nprecession amplitudes of FM 1 exhibit local maxima as its cone angle peaks at the anti-crossing  \nfield [solid lines in Fig. 4(b)]. Also, compared with the uncoupled case [FM 1 (SL) in Fig. 4(b)], \nFM 1 in the SAF structure has a much larger cone angle at the boundary between regions 1 and  2 \n(Hext ≈ 1.6 kOe).  This corresponds to the case where FM 1 fast switch ing is driven by Hext, as shown \nin Fig. 2( d). The energy valley of FM 1 created by IEC and uniaxial anisotropy is canceled out by \nHext. As a result, any perturbation in Hk,eff,1 or IEC can induce a large change in 𝜃1. \nBesides amplitude, the phase of TR -MOKE signals [ HF and LF in Eq. (6)] also provides \nimportant information about the magnetization dynamics in SAF  [Fig. 4 (c)]. In Fig. 4 (c), the phase \nof the HF mode stays constant around π. However, the LF mode goes through a π-phase shift at 18 \n the transition from region 2 to region 3. Th is phase shift can be explained by the change of the \ndominant layer from region 2 to region 3 for the LF mode [ Fig. 3(c)]. As illustrated in Fig. 4 (d), \nthe LF  mode (optical mode in regions 2 and 3) has opposite phases in FM 1 (~0°) and FM 2 (~180°). \nConsidering the two FM layers have comparable optical contributions to TR -MOKE signals ( w ≈ \n0.5), TR -MOKE signals will reflect the phase of the dominant layer for each mode. In region 3, \nFM 2 has larger p recession cone angles than FM 1 for the LF  mode ; therefore, LF TR-MOKE signals \nhave the same phase as FM 2 (~180°). However, in region 2, the dominant layer shifts from FM 2 to \nFM 1 for the LF mode. Hence, the phase of LF TR-MOKE signals also change s by ~180° t o be \nconsistent with the phase of FM 1 (~0°). As for the HF mode, since the two layers always have  \nalmost  the same phase ( ~180°), the change of the dominant layer does not cause a shift in the phase \nof TR -MOKE signals.   \nBy comparing Fig. 4 (d) and Fig. 3 (a-b), one can notice that the phase difference between two \nFM layers could deviate from 0° or 180° when damping  and mutual spin pumping  is considered \n[Fig. 4(d)]. The deviation of phase allows energy  to be transferred from one FM layer to the other \nduring precession via exchange coupling  [54]. In our sample system, FM 2 has a higher damping \nconstant ( 𝛼1= 0.020 and 𝛼2=0.060); therefore, the net transfer of energy is from FM 1 to FM 2. \nMore details can be found in Note 4 of the SM  [35], which shows the phase of TR -MOKE signals \nis affected by Gilbert damping in both layers and the mutual spin pumping . By fitting the phase \n[Fig. 4(c)] and the damping [ Fig. 5(a) ] of TR -MOKE signals simultaneously, we obtained 𝛼sp,12 \n= 0.010 ± 0.004, 𝛼sp,21=0.007−0.007+0.009, 1= 0.020 ± 0.002, and 2 = 0.060 ± 0.008. Nonreciprocal \nspin pumping damping ( 𝛼sp,12≠𝛼sp,21) has been reported in asymmetric FM 1/NM/FM 2 trilayers \nand attributed to the different spin -mixing conductance ( 𝑔𝑖↑↓) at the two FM/NM interfaces [27], \nfollowing 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑗↑↓/(8𝜋𝑀s,𝑖𝑑𝑖), with 𝑔𝑖 the 𝑔-factor of the i-th layer and 𝜇B the Bohr 19 \n magneton [55]. The above equation neglects the spin -flip scattering in NM and assumes that the \nspin accumulation in the NM spacer equally flows back to FM 1 and FM 2  [37]. However, the \nuncertainties of our 𝛼sp,𝑖𝑗 are too high to justify the nonreciprocity of 𝛼sp,𝑖𝑗 (see Note 3 of the SM  \n[35] for detailed uncertainty analyses). In fact, if the spin backflow to FM i is proportional  to 𝑔𝑖↑↓, \nthen 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑖↑↓𝑔𝑗↑↓/[4𝜋𝑀s,𝑖𝑑𝑖(𝑔𝑖↑↓+𝑔𝑗↑↓)] (Eq. 1.14 in Ref. [56]). In this case, the different \nspin-mixing conductance at two FM/NM interfaces ( 𝑔1↑↓≠𝑔2↑↓) will not lead to nonreciprocal \n𝛼sp,𝑖𝑗. Although differences in 𝑔𝑖 and magnetic moment per area ( 𝑀s,i𝑑𝑖) can potentially  lead to \nnonreciprocal 𝛼sp,𝑖𝑗, the values of 𝑔𝑖 and 𝑀s,i𝑑𝑖 for the two FM layers are expected to be similar \n(the net magnetization of SAF is zero without external fields). Therefore, nearly reciprocal 𝛼sp,𝑖𝑗 \nare plausible for our SAF stack. Assu ming 𝑔𝑖↑↓ values are similar at the two FM/NM interfaces \n(𝑔1↑↓≈𝑔2↑↓=𝑔↑↓), this yields 𝑔↑↓ =8𝜋𝑀s,𝑖𝑑𝑖𝛼sp,𝑖𝑗/(𝑔𝑖𝜇B) = 1.2 ~ 1.7 × 1015 cm−2. 𝑔↑↓ can also \nbe estimated from the free electron density per spin ( n) in the NM layer: 𝑔↑↓ ≈ 1.2𝑛2/3 [57]. With \nn = 5.2 × 1028 m−3 for Ru [58] (the value of n is similar for Ta [59]), 𝑔↑↓ is estimated to be 1.7  × 1015 \ncm−2, the same order as the 𝑔↑↓ value from TR -MOKE measurements, which justifies the 𝛼sp,𝑖𝑗 \nvalues derived from TR -MOKE are within a reasonable range. The values of 𝛼1 and 𝛼2 will be \ndiscussed in section 3.4.  \n 20 \n  \nFIG. 4 (a) Amplitudes of TR -MOKE signals a s functions of Hext. The circles and curves represent \nexperimental data and modeling fitting , respectively. (b) The calculated precession  half cone \nangles at different Hext. Red curves and black curves represent the cone angles of the HF  mode and \nthe LF mode in FM 1 (solid lines) and FM 2 (dash ed lines). Dotted lines are the precession cone \nangles of single -layer (SL) FM 1 and FM 2 without IEC. (c) Phases of TR -MOKE signals at  varying  \nHext. Circles and curves are experimental data and modeling fitting (𝛼sp,12=0.010, 𝛼sp,21=\n0.007, 𝛼1=0.020, 𝛼2=0.060). (d) Simulated precession phase of the HF mode (red curves) and \nthe LF mode (black curves) in FM 1 (solid lines) and FM 2 (dash ed lines).  \n \n3.4 Magnetic damping of the HF and LF precession modes  \nIn addition to the amplitude and phase of TR -MOKE signals for the p -SAF stack, the model \nanalyses also provide a better understanding of magnetic damping. Figure 5 (a) shows the effective \ndamping constant ( 𝛼eff=1/2𝜋𝑓𝜏) measured at different Hext (symbols), in comparison with \n21 \n model ing fitting (solid lines). The general Hext dependence of αeff can be well captured by the \nmodel. The fitted Gilbert damping, 1= 0.020 ± 0.002 and 2 = 0.060 ± 0.008 are close to the \nGilbert damping of Ta/CoFeB(1 nm)/MgO thin films (~0.017) [41,60]  and Co/Pd multilayers with \na similar tCo/tPd ratio (~0.085)  [61]. Other fitted parameters are  Δ𝐻k,eff,1=0.26±0.02 kOe, \nΔ𝐻k,eff,2= 1.42±0.18 kOe, 𝛼12sp=0.010±0.004  𝛼21sp=0.007−0.007+0.009. Δ𝐽1 and Δ𝐽2 are set to be \nzero, as explained in Sec. 2.2. More details regarding the values and determination methods of all \nparameters involved in our data reduction are provided in Note 3 of the SM  [35]. Dashed lines \nshow the calculated 𝛼eff without inhomogeneous broadening. At high Hext, the difference between \nthe solid lines  and dashed lines approaches zero because the inhomogeneous broadening is \nsuppressed. At low Hext, the solid lines are  significantly higher than the dashed lines , indicating \nsubstantial inhomogeneous broadening contributions .  \nThe effective  damping shows interesting features  near the anti-crossing field. As shown in \nFig. 5(b), due to the effective coupling  between two FM layers near the anti-crossing field, the \nhybridization of precession in two FM layers leads to a mix of damping with contributions from \nboth layers. The effective damping  of the FM 1-dominant mode  reaches a maximum within the \nanti-crossing region ( 7  Hext  10 kOe) and is higher than the single -layer (SL)  FM 1 case. \nSimilarly, the hybridized HF and LF  modes at 8.5 kOe exhibit  a lower 𝛼eff (~0.073) compared to \nthe SL FM 2 case. eff consists of contributions from Gilbert damping ( 𝛼𝑖), mutual spin pumping \n(𝛼sp,𝑖𝑗, 𝑖≠𝑗), and inhomogeneous broadening ( Δ𝐻k,eff,𝑖 and Δ𝐽𝑖). To better understand the mixing \ndamping behavior, Fig. 5 (c) shows eff after excluding the inhomogeneous contribution ( 𝛼effinhomo). \nCompared to the SL layer c ase (green and blue dashed lines), the HF and LF modes (red and black \ndashed lines) clearly suggest that IEC effectively mixes the damping in two layers around the anti -\ncrossing field. Without the IEC, precession in FM 2 with a higher damping relaxes faster  than that 22 \n in FM 1. However, the IEC provides a channel to transfer energy from FM 1 to FM 2, such that the \ntwo layers have the same precession relaxation rate for a given mode. Near the anti -crossing field, \ntwo layers have comparable precession cone angles; therefore, the damping values of the \nhybridized modes are roughly the average of two FM layers. In addition to the static IEC, dynamic \nspin pumping can also modify the damping of individual modes. The black and red solid lines \nrepresent the cases with mutu al spin pumping ( 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0.007). Generally, in \nregions 2 & 3, mutual spin pumping reduces the damping of the HF mode and increases the \ndamping of the LF mode because the HF (LF) mode is near in -phase (out -of-phase). Overall, the \nstatic  IEC still plays the essential role for the damping mix near the anti -crossing field.  \n \n \nFIG. 5 (a) Effective damping constant under varying Hext. Circles are experimental data. Solid \nlines are fitting curves based on Eqs. (4 -5). Dashed lines denote eff after the removal of \ninhomogeneous -broadening contribution. (b) A zoomed -in figure of panel (a) between 5 kOe and \n15 kOe. Blue and green circles are measured effective damping of the mode dominated by FM 1 \nand FM 2, respectively. Blu e and green dashed lines are the 𝛼eff of FM 1 and FM 2 single layer \nwithout IEC. (c) Effective damping after excluding the inhomogeneous contribution as a function \nof Hext. The HF mode (red curves) and the LF mode (black curves) are represented by solid (o r \ndashed) curves when the mutual spin pumping terms ( 𝛼sp,12 and 𝛼sp,21) are considered (or \nexcluded). The d ashed green and blue lines are the SL cases for FM 1 and FM 2, respectively.  \n \n \n \n23 \n 4 CONCLUSION  \nWe systematically investigate d the magnetization dynamics excited by ultrafast laser pulses in \nan asymmetric p -SAF sample both theoretically and experimentally. We obtained d etailed \ninformation regarding magnetization dynamics, including the cone angles, directions, and phases \nof spin precession in each layer under different Hext. In particular, the dynamic features in the low -\nfield region  (region 1)  exhibiting  CW precession, were  revealed. The r esonance between the \nprecession of two FM layers occurs at the boundary between regions 2 an d 3, where an anti -\ncrossing feature is present in the frequency vs. Hext profile . The dominant FM layer for a given \nprecession mode also switches from region 2 to region 3. The amplitude and phase of TR -MOKE \nsignals are well captured by theoretical modeling . Importantly , we successfully quantified the \nindividual contributions from various sources to the effective damping , which enables the \ndetermination of Gilbert damping for both FM layers.  At low Hext, the contribution of \ninhomogeneous broadening to the effective damping is significant. Near the anti-crossing field, \nthe effective damping of  two coupled  modes contains substantial contributions from both FM \nlayers owing to the strong hybridization  via IEC . Although the analyses were made for an \nasymme tric SAF sample, this approach can be directly  applied to study magnetization dynamics \nand magnetic properties of  general complex material systems with  coupled multilayers , and thus  \nbenefits the design and optimization of spintronic materials via structural engineering.  \n \nAcknowledgements  \nThis work is  primarily  supported by the National Science Foundation ( NSF, CBET - 2226579). \nD.L.Z gratefully acknowledges the funding support from  the ERI program (FRANC) “Advanced \nMTJs for computation in and near ra ndom access memory” by DARPA, and ASCENT, one of six 24 \n centers in JUMP (a Semiconductor Research Corporation program, sponsored by MARCO and \nDARPA).  J.P.W  and X.J.W also appreciate the partial support from the UMN MRSEC Seed \nprogram (NSF, DMR -2011401 ). D.B.H . would like to thank the support from the UMN 2022 -2023 \nDoctoral Dissertation Fellowship. 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Magn. 48, 3288 (2012).  \n 1 \n Supplement al Material  for \nMagnetization Dynamics in Synthetic Antiferromagnets  with Perpendicular \nMagnetic Anisotropy   \n \nDingbin Huang1,*, Delin Zhang2, Yun Kim1, 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  \n \nSupplement al Note 1: Analyses of the magnetization precession in each  Ferromagnetic  (FM) \nlayer  \nFor the convenience of derivation, mi is represented in  the spherical coordinate s with the polar \nangle θi and the azimuthal angle φi, as shown in Fig. 1(b): \n𝐦𝑖=(sin𝜃𝑖cos𝜑𝑖,sin𝜃𝑖sin𝜑𝑖,cos𝜃𝑖) (S1) \nAccordingly, t he expressi on of  Eq. ( 2) in the spherical coordinate s is:  \n{        𝜃̇1=−𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜑1−𝛼1sin𝜃1𝜑̇1+𝛼sp,12sin𝜃2cos(𝜃2−𝜃1)𝜑̇2\n𝜑̇1=𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜃1+𝛼1\nsin𝜃1𝜃̇1−𝛼sp,12\nsin𝜃1𝜃̇2\n𝜃̇2=−𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜑2−𝛼2sin𝜃2𝜑̇2+𝛼sp,21sin𝜃1cos(𝜃1−𝜃2)𝜑̇1\n𝜑̇2=𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜃2+𝛼2\nsin𝜃2𝜃̇2−𝛼sp,21\nsin𝜃2𝜃̇1 (S2) \n \n*Author s to whom correspondence should be addressed : huan1746@umn.edu  and wang4940@umn.edu  2 \n where, a dot over variables  represents  a derivative with respect to time. When Mi  precesses  around \nits equilibrium direction:  \n{𝜃𝑖=𝜃0,𝑖+Δ𝜃𝑖\n𝜑𝑖=𝜑0,𝑖+Δ𝜑𝑖 (S3) \nwith \ni  and \ni  representing the deviation angles of  Mi from  its equilibrium direction  along the \npolar and azimuthal directions.  Assuming the deviation is small, under the first -order \napproximation, the first -order partial derivative of F in Eq. (S2) can be expanded as:  \n{    ∂𝐹\n∂𝜃𝑖≈∂2𝐹\n∂𝜃𝑖2Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖∂𝜃𝑖Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜃𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜃𝑖Δ𝜑𝑗\n∂𝐹\n∂𝜑𝑖≈∂2𝐹\n∂𝜃𝑖𝜕𝜑𝑖Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖2Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜑𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜑𝑖Δ𝜑𝑗 (S4) \nBy substituting Eq. ( S4), Equation ( S2) is linearized  as [1]: \n[   Δ𝜃̇1\nΔ𝜑̇1\nΔ𝜃̇2\nΔ𝜑̇2]   \n=𝐊[Δ𝜃1\nΔ𝜑1\nΔ𝜃2\nΔ𝜑2] (S5) \nwhere, K is a 4×4 matrix, con sisting of  the properties  of individual FM layers  and the second -\norder derivatives of F in terms of 𝜃1,𝜑1,𝜃2,and𝜑2. Equation (S5) has four eigen -solutions, in the \nform of 𝐶exp(𝑖𝜔𝑡), corresponding to four precession frequencies:  ±𝜔HF and ±𝜔LF. A pair of \neigen -solutions with the same absolute precession frequency are physically equivalent. Therefore, \nonly two eigen -solutions need to be considered:  \n{Δ𝜃𝑖=𝐶𝜃,𝑖HFexp(𝑖𝜔HF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖HFexp(𝑖𝜔HF𝑡) and {Δ𝜃𝑖=𝐶𝜃,𝑖LFexp(𝑖𝜔LF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖LFexp(𝑖𝜔LF𝑡) (S6) \nAfter r earrange ment , the full solutions in the spherical coordinates are expressed as below (also \nEq. (3) in the main paper).  3 \n [𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=\n[   𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]   \n+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[    𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF]    \nexp(𝑖𝜔HF𝑡)+\n[    𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF]    \nexp(𝑖𝜔LF𝑡)    (S7) \nThe prefactors of these  eigen -solutions  provide information about magnetization dynamics of both \nthe HF and LF modes. Directly from solving Eq. (S2), one can obtain the relative ratios of these \nprefactors , which are [𝐶𝜑1HF/𝐶𝜃1HF,𝐶𝜃2HF/𝐶𝜃1HF,𝐶𝜑2HF/𝐶𝜃1HF] and [𝐶𝜑1LF /𝐶𝜃1LF ,𝐶𝜃2LF /𝐶𝜃1LF ,𝐶𝜑2LF /𝐶𝜃1LF ]. \nThese ratios provide  precession information of each mode, as presented in Fig. 3.   \nObtaining the absolute values of [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 for each mode requires the initial \nconditions of precession , which i s necessary for fitting the actual precession amplitudes in TR -\nMOKE signals . In TR -MOKE measurements, magnetization precession is initiated by laser \nheating, which reduces the magnetic anisotropy of each  FM layer and the interlayer exchange \ncoupling streng th between two FM layers  [2]. Considering the laser heating process is ultrafast \ncompared with magnetization precession while the following cooling due to heat dissipation is \nmuch slower than magnetization dynamics, we approximately model the temporal profiles of \neffective anisotropy fields and exchange coupling as step functions.  Owing to the sudden change \nin magnetic properties  induced by laser heating , magnetization in each layer will establish a new \nequilibrium direction (𝜃0,𝑖′,𝜑0,𝑖′). In other words, M i deviates from its new eq uilibrium direction \nby Δ𝜃𝑖=𝜃0,𝑖−𝜃0,𝑖′, Δ𝜑𝑖=𝜑0,𝑖−𝜑0,𝑖′. Substituting 𝑡=0 to Eq. ( S7), one can get the initial \nconditions for magnetization dynamics:  \nΔ𝜃𝑖(𝑡=0)=𝐶𝜃,𝑖HF+𝐶𝜃,𝑖LF=𝜃0,𝑖−𝜃0,𝑖′  \nΔ𝜑𝑖(𝑡=0)=𝐶𝜑,𝑖HF+𝐶𝜑,𝑖LF=𝜑0,𝑖−𝜑0,𝑖′=0 (S8) \nOnce the initial conditions are set, the absolute values of all prefactors can be obtained . \n 4 \n Supplementa l Note 2: Estimation of each layer’s contribution to total TR -MOKE signals  \nThe contribution from each FM layer is estimated by static MOKE measurement. According \nto Ref. [3], the resu lt from this method matches well with that from the optical calculation. The \nsample is perpendicularly saturated before the static MOKE measurement. Then the out -of-plane \nM-Hext loop ( Fig. S1) is measured by static MOKE. As shown in the figure, two different \nantiferromagnetic (AF) configurations have different normalized MOKE signals, indicating the \ndifferent contribution s to the total signals by two layers.  The weighting factor is calculated by:   \n−𝑤+(1−𝑤)=0.085 (S9) \nwhich gives 𝑤=0.457. Considering the relatively small layer thicknesses [FM 1: CoFeB(1), spacer: \nRu(0.6)/Ta(0.3), and FM 2: Co(0.4)/Pd(0.7)/Co(0.4)], it is reasonable that FM 1 and FM 2 make \ncomparable contributions to the total TR -MOKE signals ( i.e., w ≈ 0.5). \n \nFIG. S1 Static MOKE hysteresis loop. Magnetic fields are applied along the out -of-plane direction.  \n \n \n \n5 \n Supplemental Note 3: Summary of the parameters and uncertainties for data reduction  \nGiven that a number of variables are involved in the analysis, TABLE SI summarizes the major \nvariables discussed in the manuscript, along with their values and determinatio n methods.  \nTABLE SI. Summary of the values and determination methods of parameters used in the data \nreduction. The reported uncertainties are one -sigma uncertainties from the mathematical model \nfitting to the TR -MOKE measurement data.  \nParameters  Values  Determination Methods  \nHf ~500 Oe  VSM  \nMs,1 1240 emu cm−3 VSM  \nMs,2  827 emu  cm−3 VSM  \nd1 1 nm  Sample structure  \nd2 1.5 nm  Sample structure  \nHk,eff,1  1.23 ± 0.28 kOe  Fitted from f vs. Hext [Fig. 2(b)]  \nHk,eff,2 6.18 ± 0.13 kOe  Fitted from f vs. Hext [Fig. 2(b)]  \nγ1 17.79 ± 0.04  \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)]  \nγ2 17.85 ± 0.04  \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)]  \nJ1 −0.050 ± 0.020  \nerg cm−2 Fitted from f vs. Hext [Fig. 2(b)]  \nJ2 0 Fitted from f vs. Hext [Fig. 2(b)]  \nw 0.457  Static MOKE  \n𝐻k,eff,1′/𝐻k,eff,1 0.90 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝐻k,eff,2′/𝐻k,eff,2 0.95 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝐽1′/𝐽1 0.83 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝛼1 0.020 ± 0.002  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n𝛼2 0.060 ± 0.008  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \nΔ𝐻k,eff,1 0.26 ± 0.02 kOe  Fitted from eff vs. Hext [Fig. 5(a)]  \nΔ𝐻k,eff,2 1.42 ± 0.18 kOe  Fitted from eff vs. Hext [Fig. 5(a)]  \n𝛼sp,12 0.010 ± 0.004  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n𝛼sp,21 0.007−0.007+0.009 Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n 6 \n Supplemental Note  4: Impacts of 𝜶𝟏, 𝜶𝟐, and mutual spin pumping on the phase  \nWithout damping, the phase difference in the precession polar angles of two FM layers \n[Arg(𝐶𝜃2/𝐶𝜃1)] is always 0° or 180°, as shown in Fig. 3 of the main article. However, this does not \nnecessarily hold if either the damping or mutual spin pumping is considered. The changes in the \nphase difference due to damping are depicted in Fig. S2. When 1 = 2, the phase difference \nbetween two layers stays at 0° or 180° [ Fig. S2(a)], identical to the lossless case ( 1 = 2 = 0) in \nFig. 3. As a result, the initial phase of TR -MOKE signals ( ) also stays at 0° or 180° [ Fig. S2(b)]. \nHowever, when 𝛼1≠𝛼2, Arg(𝐶𝜃2/𝐶𝜃1) deviates from 0° or 180° especially at high fields ( Hext > \n5 kOe) [ Fig. S2(c,e)]. The layer with a higher damping [FM 1 in (c) or FM 2 in (e)] tends to have a \nmore advanced phase at high fields (regions 2 and 3). For example, in Fig. S2(e), 0° < Arg( 𝐶𝜃2/𝐶𝜃1) \n< 180° for both HF and LF modes in regions 2 and 3. The deviation from the perfect in -phase (0°) \nor out -of-phase (180°) condition allows the IEC to transfer energy from the low -damping layer to \nthe high -damping layer, such that the precession in  both layers can damp at the same rate [4]. As \na result, the initial phase of the TR -MOKE signals also changes, which opens a negative or positive  \ngap at high fields (> 10 kOe) for both modes, as shown in Fig. S2(d,f). This enables us to determine \nthe difference between 1 and 2 by analyzing the in itial phase of TR -MOKE signals.  7 \n  \nFIG. S 2 Impact of 𝛼1 and 𝛼2 on the phase without mutual spin pumping. (a,c,e) The phase \ndifference between the polar angles in two layers for HF and LF modes. (b,d,f) The calculated \ninitial phase of TR -MOKE signals for each mode with 1 = 2 = 0.02 (a,b), 1 = 0.06 and 2 = \n0.02 (c,d), and 1 = 0.02 and 2 = 0.06 (e,f). The mutual spin pumping is set as 𝛼sp,12=𝛼sp,21 = \n0 for all three cases. The rest of the parameters used in this calculation can be found in TABLE SI.  \n \nThe impact of mutual spin pumping on the precession phase is illustrated in Fig. S3, where \nthree different cases of either the one -way (𝛼sp,12 or 𝛼sp,21) or two -way (both 𝛼sp,12 and 𝛼sp,21) \nspin pumping are considered. A reference case without the consideration of mutual spin pumping \n(1 = 0.02, 2 = 0.06, and 𝛼sp,12= 𝛼sp,21 = 0) is also plotted (dashed curves) for the ease of \ncomparison. In general, it can be seen that mutual spin pumping could also change the phase \ndifference in the precession polar angles of two layers, and thus the initial phase o f TR -MOKE \nsignals noticeably. This can be explained by the damping modification resulting from spin \npumping. In regions 2 and 3, Eq. (2) can be approximately rearranged as:  \n \n8 \n 𝑑𝐦𝑖\n𝑑𝑡≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+[𝛼𝑖−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)]𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n=−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼̅𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡 (S10) \nwhere 𝐶𝑗/𝐶𝑖 represents the ratio of the cone angles in the j-th FM layer to the i-th FM layer. 𝐶𝑗/𝐶𝑖 \nis positive for the in -phase mode and negative for the out -of-phase mode.  θ0,1 and θ0,2 are the \nequilibrium polar angle s of M1 and M2, as defined in Fig. 2(c).  Therefore, the mutual spin -pumping \nterm either enhances or reduces the damping depending on the mode. 𝛼̅𝑖 = 𝛼𝑖−\n𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2) represents the effective Gilbert damping in the i-th FM layer after \nconsidering the mutual spin -pumping effect. This modification to damping is more significant \nwhen the i-th layer is subservient with a smaller cone angle ( e.g., FM 2 for the HF mode in region \n3), while the j-th layer is dominant with a mu ch larger precession cone angle ( e.g., FM 1 for the LF \nmode in region 3), leading to a large ratio of |𝐶𝑗/𝐶𝑖|. \nIn Fig. S 3(a), only the spin current injected from FM 1 to FM 2 is considered. According to the \nabove analysis, 𝛼sp,21 can only bring noticeable modifications to the damping of FM 2 when FM 1 \nis the dominant layer. Based on Fig. 3 in the main article, the LF mode in region 2 and HF mode \nin region 3 satisfy this condition (FM 1 dominant and FM 2 subservient). As shown in Fig. S 3(a), \nthe phase difference  noticeably deviates from the reference case without mutual spin pumping \n(dashed curves) in region 2 for the LF mode (black curves) and in region 3 for the HF mode (red \ncurves). For the LF mode in region 2, the precession motions in two layers are nearly o ut-of-phase \n(negative C1/C2); therefore, the spin pumping from FM 1 enhances the damping in FM 2. Since 1 \n(0.02) is less than 2 (0.06), the spin pumping from FM 1 to FM 2 further increases |  𝛼̅1 − 𝛼̅2| \nbetween the two layers. Consequently, the phase difference shifts further away from 180°. While 9 \n for the HF mode in region 3, 𝛼sp,21 reduces the damping of FM 2 because C1/C2 is positive resulting \nfrom the near in -phase feature of this mode. Hence, |  𝛼̅1 − 𝛼̅2| becomes smaller and the phase \ndifference gets closer to 0°. In Fig. S 3(c), only 𝛼sp,12 is considered, which requires FM 2 as the \ndominant layer (the HF mode in region 2 and LF mode in region 3) for noticeable changes in |  𝛼̅1 \n− 𝛼̅2|. For the HF mode  in region 2, spin pumping from FM 2 reduces 𝛼̅1 given that the precession \nmotions in two layers are nearly in phase (positive C2/C1). Therefore, |  𝛼̅1 − 𝛼̅2| increases and the \nphase difference in Fig. S 3(c) shifts further away from 0° in region 2. However,  for the LF mode \nin regions 3, the nearly out -of-phase precession in two FM layers (negative C1/C2) increases 𝛼̅1 \nand reduces |  𝛼̅1 − 𝛼̅2|. As a result, the phase difference in Fig. S 3(c) shifts toward 180°. When \nboth 𝛼sp,12 and 𝛼sp,21 are considered [ Fig. S 3(e)], a combined effect is expected for the phase \ndifference with noticeable changes for both the HF and LF modes in regions 2 and 3.  \nThe impacts of mutual spin pumping on the phase difference between the HF and LF modes \nare reflected by the initial phase of TR -MOKE signals [  in Fig. S 3(b,d,f)]. Compared with the \nreference case without mutual spin pumping (dashed curves), the introduction of mutual spin \npumping tends to change the gap in  between the two modes. As shown in Fig. S3(e,f), the values \nof two mutual -spin-pumping induced damping terms are chosen as 𝛼sp,12 = 0.013 and 𝛼sp,21 = \n0.004, such that the  gap of the initial phase of TR -MOKE signals is closed at high fields \n(region  3). Therefore, the initial phase of TR -MOKE signals provides certain measurement \nsensitivities to 𝛼sp,12 and 𝛼sp,21, which enables us to extract the values of 𝛼sp,𝑖𝑗 from \nmeasurement fitting. Here, we acknowledge that the measurement sensitivity to 𝛼sp,𝑖𝑗 from TR -\nMOKE is limited, which subsequently leads to relatively large error bars for 𝛼sp,𝑖𝑗 (see Table SI).  \n 10 \n  \nFIG. S3 Impact of mutual spin pumping on the phase with fixed damping values of 1 = 0.02 and \n2 = 0.06. (a,c,e) The phase difference between the polar angles in two layers for HF and LF modes. \n(b,d,f) The calculated initial phase of TR -MOKE signals ( ) for each mode with 𝛼sp,12= 0 and \n𝛼sp,21 = 0.01 (a,b), 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0 (c,d), and  𝛼sp,12 = 0.013 and 𝛼sp,21 = 0.004 (e,f). \nFor the third case (e,f), the values of mutual spin pumping are chosen to close the  gap in panel \n(f) for Hext > 15 kOe. The rest of the parameters used in this calculation can be found in TABLE \nSI. Dashed lines  represent the reference case without mutual spin pumping ( 1 = 0.02, 2 = 0.06, \nand 𝛼sp,12= 𝛼sp,21 = 0).  \n \nSupplemental Note 5: Region diagram s for p -SAFs with different degrees of asymmetries  \nFigure S4 shows the region diagrams for p -SAFs with different degrees of asymmetries , \nrepresented by the difference of Hk,eff in two FM layers. Hk,eff,1 = Hk,eff,2 corresponds to the \nsymmetric case (lowest asymmetry), as shown by Fig. S4(c). While  the SAF  in Fig. S4(a) has the \nhighest asymmetry: Hk,eff,1 = 2 kOe, Hk,eff,2 = 6 kOe. Figure S4 clearly shows that |𝜃0,1−𝜃0,2|>\n90° is a necessary but not sufficient condition for region 1 (CW precession). Because regions 2 or \n3 also appear to the left of the red cu rve (where |𝜃0,1−𝜃0,2|>90°), especially when θH is close \nto 90° and Hk,eff,1 is close to Hk,eff,2. \n11 \n  \nFIG. S4 Region diagrams of p -SAFs with different  degrees of  asymmetries: Hk,eff,1 = 2 kOe, Hk,eff,2 \n= 6 kOe (a), Hk,eff,1 = 4 kOe, Hk,eff,2 = 6 kOe (b), Hk,eff,1 = 6 kOe, Hk,eff,2 = 6 kOe (c). The blue \nbackground represents region 1. The green background covers regions 2 and 3. The red curve \nshows the conditions where |𝜃0,1−𝜃0,2|=90°. |𝜃0,1−𝜃0,2|>90° to the left of the red curve. 𝛼1, \n𝛼2, 𝛼sp,12, and 𝛼sp,21 are set as zero. 𝛾1=𝛾2=17.8 rad ns−1 kOe−1. Values of the rest parameters are \nthe same as those in Table SI.  \n \nReferences  \n[1] Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Angular dependence of ferromagnetic \nresonance in exchange -coupled Co/Ru/Co trilayer structures, Phys. Rev. B 50, 6094 (1994).  \n[2] W. Wang, P. Li, C. Cao, F. Liu, R. Tang, G. Chai, and C. Jiang, Temperature dependence \nof interlayer exchange coupling and Gilbert damping in synthetic antiferromagnetic \ntrilayers investigated using broadband ferromagnetic resonance, Appl. Phys. Lett. 113, \n042401 (2018).  \n[3] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V. Paluskar, R. Huijink, H. J. M. \nSwagten, and B. Koopmans, Control of speed and efficiency of ultrafast demagnetization \nby direct transfer of spin angular momentum, Nat. Phys. 4, 855 (2008).  \n[4] D. H. Zanette, Energy exchange between coupled mechanical oscil lators: linear regimes, J. \nPhys. Commun. 2, 095015 (2018).  \n \n"    },    {        "title": "2211.08048v2.Nonlinear_sub_switching_regime_of_magnetization_dynamics_in_photo_magnetic_garnets.pdf",        "content": "1 \n Nonlinear s ub-switching regime of magnetization  dynamics in photo -magnetic  garnets  \nA. Frej, I. Razdolski, A. Maziewski, and A. Stupakiewicz  \nFaculty of Physics, University of Bialystok, 1L Ciolkowskiego, 1 5-245 Bialystok, Poland  \nAbstract.  We analyze, both experimentally and numerically, the nonlinear regime of the \nphoto -induced coherent magnetization dynamics in cobalt -doped yttrium iron garnet films. \nPhoto -magnetic excitation with femtosecond laser pulses reveals a strongly nonlinear \nrespo nse of the spin subsystem with a significant increase of the effective Gilbert damping. By \nvarying both laser fluence and the external magnetic field, we show that this nonlinearity \noriginates in the anharmonicity of the magnetic energy landscape. We numer ically map the \nparameter workspace for the nonlinear photo -induced spin dynamics below the photo -\nmagnetic switching threshold. Corroborated by numerical simulations of the Landau -Lifshitz -\nGilbert equation, our results highlight the key role of the cubic sy mmetry of the magnetic \nsubsystem in reaching the nonlinear spin precession regime. These findings expand the \nfundamental understanding of laser -induced nonlinear spin dynamics as well as facilitate the \ndevelopment of applied photo -magnetism.  \n1. INTRODUCTION  \nRecently,  a plethora of fundamental mechanisms for magnetization dynamics induced by \nexternal stimul i at ultrashort time scale s has been actively d iscussed  [1-5]. The main interest \nis not only in the excit ation of  spin precession but in the switching of ma gnetization between \nmultiple stable states, as it open s up rich possibilities for non-volatile magnetic  data  storage \ntechnology . One of t he most intriguing example s is the phenomenon  of ultrafast switching of \nmagnetization with laser pulses. Energy -efficie nt, non -thermal mechanisms of laser -induced \nmagnetization switching require a theoretical understanding of coherent magnetization \ndynamics in a strongly non -equilibrium environment  [6]. This quasiperiodic motion  of \nmagnetization  is often mode led as  an oscillator where the key parameters , such as frequency \nand damping , are considered within the framework of  the Landau -Lifshit z-Gilbert  (LLG)  \nequation  [1, 7] . Although it is inhe rently  designed to describe small -angle spin precession \nwith in the  linear approximation, there are attempts to extend this formalism into the \nnonlinear regime where the precession parameters become angle -dependent  [8]. This is \nparticularly important in light of the discovery of the so -called p recessional switching , where  \nmagnetization , having been  impulsively driven  out of equilibrium, ends its precessional \nmotion in a different minimum of the potential energy  [6, 9 -11]. Obviously,  such \nmagnetization trajectories are characterized by very large precession angles  (usually on the \norder of tens of  degrees ). It is, however, generally believed that the magnetization excursion \nfrom the equilibrium of about 10 -20 degrees is already sufficient for the violat ion of the linear \nLLG approach  [12, 13] . Thus, an intermediate regime under the switching stimulus threshold \nexists, taking a large area in the phase space and presenting an intriguing c hallenge in \nunderstanding  fundamental spin dynamics.  \nAn impulsive optical stimulus often results in a thermal excitation mechanism, inducing \nconcomitant temperature variations , which can impact the parameters of spin precession  [14-\n16]. This highlights the  special role of the non -thermal optical mechanisms  of switching  [17-2 \n 19]. Among those , we outline photo -magnetic excitation , which has been recently \ndemonstrated in dielectric Co -doped YIG  (YIG:Co)  films  [6, 11] . There, laser photons at a \nwavelength of 1300 nm resonantly excite the 5E → 5T2 electron transition s in Co -ions, resulting \nin an emerging photo -induced magnetic anisotropy  and thus in a highly efficient excitation of \nthe magnetic subsystem  [6]. This photo -induced  effective  anisotropy field features  a near ly \ninstant aneous  rise time (within the femtosecond pump laser pulse  duration), shifting the \nequilibrium direction for the magnetization and thus triggering its l arge -amplitude precession.  \nIn the sub -switching regime  (at excitation strengths just below the switching threshold ), the \nfrequency of the photo -induced magnetization precession has been shown to depend on the \nexcit ation wavelength  [20]. However, nonlinearities in magnetization dynamics in the sub-\nswitching regime have  not yet been described in detail, and the underlying mechanism  for the \nfrequency variations  is not understood.  \nIn this work , we systematically examine the intermediate sub -switching regime characterized \nby large angles of magnetization precession and the nonlinear response of the spin system to \nphoto -magnetic excitations. We show a strong increase of the effective Gilbert damping at \nelevated lase r-induced excitation levels and quantify its nonlinearity within the existing \nphenomenological formalism  [8]. We further map the nonlinear regime in the phase space \nformed by the  effective  photo -induced anisotropy field  and the external magnetic field.   \n \nFig. 1. Sketch of m agnetization dynamics at various  stimulus  levels . Owing to the highly nonlinear \nmagnetization dynamics in the switching regime, the nonlinearity onset manifests in the sub -switching \nregime too . \nThis paper is organized in the following order: in the first part, we describe the details of the \nexperiment for laser -induced large -amplitude magnetization precession. Next, we present the \nexperimental results, followed  by the fitting analysis . Then, we complement our findings with \nthe results of  numerical simulation of the photo -magnetic spin dynamics. Afterward , we \ndiscuss the workspace of parameters for the sub-switching regime of laser -induced \nmagnetization precession. The paper ends with c onclusions.  \n \n3 \n 2. EXPERIMENTAL DETAILS  \nThe experiments were done on  a 7.5 μm -thick YIG:Co  film with a composition of \nY2CaFe 3.9Co0.1GeO 12. The Fe ions at the tetrahedral and octahedral sites are replaced by Co -\nions [21]. The sample was grown by liquid -phase epitaxy on a 400  μm-thick  gadolinium gallium \ngarnet (GGG) substrate. It exhibits eight possible magnetization states along the garnet’s cubic \ncell diagonals due to its cubic magnetocrystalline anisotropy ( 𝐾1=−8.4×103 𝑒𝑟𝑔/𝑐𝑚3) \ndominating the energy landscape over the uniaxial anisotropy ( 𝐾𝑢=−2.5×103 𝑒𝑟𝑔/𝑐𝑚3). \nOwing to the 4 ° miscut, additional in -plane anisotropy is introduced, tilting the magnetization \naxes and  resulting in slightly lower energy of half of the magnetiz ation states in comparison \nto the others. In the absence of the external magnetic field, the equilibrium magnetic state \ncorresponds to the magnetization in the domains close to the <111> -type directions in YIG:Co \nfilm.  Measurements of the Gilbert damping  𝛼 using the fe rromagnetic resonance technique \nresulted in  𝛼≈0.2. This relatively high damping is inextricably linked to the C o dopants  [22-\n24]. \nThe n onlinearity  of an oscillator is usually addressed by varying the intensity of the stimulus \nand comparing the response of the system under study. Here , we investigated the nonlinear \nmagnetization dynamics by varying the optical pump fluence and , thus , the strength of the \nphoto -magnetic effective field driving the magnetization out of the equilibrium. We \nperfor med systematic studies in various magnetic states of YIG:Co governed by the magnitude \nof the  external magnetic field s. The magnetic field  𝐻⊥ was applied perpendicular to the sample \nplane and in -plane magnetic field  𝐻 was applied along the [110] direction of the YIG:Co crystal \nby means of an electromagnet. Owing to the introduced miscut, the studied YIG:Co exhibits \nfour magnetic domains at 𝐻=0 [25]. The large jump at an in-plane magnetic f ield close to \nzero shows the magnetization switching in the domain structures between four magnetic \nphases. The optical spot size in this experiment was around 100 μm while the size of smaller \ndomains was around 5 μm, resulting in the spatial averaging of the domains in the \nmeasurements. This behavior of magnetic domains was dis cussed and visualized in detail  by \nmagneto -optical Faraday effect in our previous papers  [6, 25] . With an increase of the \nmagnetic field up to a round 𝐻=0.4 kOe, larger and smaller domains are formed due to the \ndomain wall motion, eventually resulting in a formation of a single domain in a noncollinear \nstate.  Upon further increase, the magnetization rotates towards the direction of the applied \nfield until a collinear state  with in -plane magnetization orientation  is reached  at about 2 kOe \n(see Fig. 2) . 4 \n  \nFig. 2. Magnetization reversal using static magneto -optical Faraday effect under perpendicular H (a) \nand in -plane H (b) magnetic fields. The grey area indicates the magnetization switching in magnetic \ndomain structure  [25]. The green area shows the saturation range with a collinear state of \nmagnetization.  \nDynamic nonlinearities in the magne tic response were studied employing the pump -probe \ntechnique relying on the optical excitation of the spin precession in YIG :Co film. The pumping \nlaser pulse  at 1300 nm , with  a duration of 50 fs and a repetition rate of 500 Hz , induce d spin \ndynamics through  the photo -magnetic mechanism  [6]. The transient Faraday rotation of the  \nweak probe beam  at 625 nm was used to monitor  the dynamics of the  out-of-plane  \nmagnetization  component Mz. The diameter  of the pump spot was around 1 40 μm , while  the \nprobe beam was focused within the pump spot  with a size of around 50 μm . The fluence of \nthe pump beam was varied in the range  of 0.2  6.5 mJ/cm2, below the switching threshold of \nabout 39 mJ/cm2 [20]. At 1300 nm pump wavelength, the optical absorption in our garnet is \nabout 12%. An estimation of the temperature increase ΔT due to the heat load for the laser \nfluence of 6.5 mJ/cm2 results in ΔT <1 K (see Methods of Ref. 6). The polarization of both \nbeams was linear and set along  the [100] crystallographic direction in YIG:Co for the pump and \nthe [010]  direction  for the probe  pulse . The experiments  were done  at room temperature . At \neach magnetic field, we performed a series of laser fluence -dependent pump -probe \nexperiments measuring the transients of an oscillating magnetization component  normal to \nthe sample plane . We then used a phenomenological damped oscillator response function to \n5 \n fit the experimental data and retrieve the fit  parameters such as a mplitude, frequency, \nlifetime and effective damping. In what follows, we analyze the obtained nonlinearities in the \nresponse of the magnetic system and employ numerical simulations to reproduce the \nexperimental findings.  \n \n3. RESULTS  \nA. Time -resolved photo -magnetic dynamics  \nIn order to determine the characteristics of the photo -magnetic precession , we carried out \ntime -resolved measurements  of a transient Faraday rotation  ∆𝜃𝐹 in YIG:Co  film. Fig. 3(a-d) \nexemplifies a few typical datasets obtained for four v arious pump fluences (between 1.7 and \n6.5 mJ/cm2) in magnetic fields of various strength s. A general trend demonstrating a decrease \nof the precession amplitude and an increase of  its frequency is seen  upon the magnetic field \nincrease . To get further insights into the magnetization dynamics, these datasets were fitted \nwith a damped sine function on top of a non -oscillatory, exponentially decaying background : \n∆𝜃𝐹(∆𝑡)=𝐴𝐹sin(2𝜋𝑓∆𝑡+𝜙)exp (−∆𝑡\n𝜏1)+𝐵exp (−∆𝑡\n𝜏2),   (1) \nwhere 𝛥𝑡 is pump and probe time difference, 𝐴𝐹 is the amplitude, 𝑓 is the frequency , 𝜙 is the \nphase, 𝜏1 is the decay time of precession, and 𝜏2 is the decay time of the background with an \namplitude 𝐵. \n \nFig. 3. Time -resolved Faraday rotation  at different  magnetic fields  H (a-d) and laser fluence s (I1-I4 \ncorrespond  to 1.7, 3.2, 5.0,  and 6.5 mJ/cm2, respectively) . The normalized MZ on the vertical axis is \ndefined as  ΔF/max, where max is obtained for saturation magnetization rotation at H (see. Fig. 2a). \nThe curves are offset  vertically without rescaling.  The s olid lines are fittings with the  damped sine \nfunction  (Eq. 1) .  \n6 \n  \nFig. 4. Photo -magnetic  precession parameters as a  function of p ump fluence  in different  external \nmagnetic field  H: a) amplitude  of the Faraday rotation AF, b) frequency  of the precession , and c) \neffective damping. Different colors correspond to different external magnetic fields. The s olid lines are \nthe linear fits  where applicable , while the dashed lines are the visual guides.  Some of the error bars \nare smaller than the data point symbols.  \nAt low applied fields  𝐻<1 kOe, where the photo -magnetic  anisotropy field ( 𝐻𝐿) contribution \nto the total effective magnetic field is the strongest, the largest magnetization precession \namplitude is observed. Figure 4 show s the most important parameters of the magnetiza tion \nprecession, that is, amplitude, frequency and effective damping  (Fig. 4a -c). The latter is \nobtained from the frequency and the lifetime as  (2𝜋𝑓𝜏1)−1. Although the amplitude \ndependence on the pump fluence is mostly linear, the other two parameters exhibit a more \ncomplicated dependence, which is indicative of the noticeable nonlinearity in the magnetic \nsystem. In particular,  at 𝐻=0.4 and 0.5 kOe, we observe d an increase in the effective \ndamping with laser fluence, resulting in a faster decay of the magnetic precession. This is \nfurther corroborated by the frequency decrease seen in Fig. 4b. It is seen that the behavior of \nthe magnetic subsystem is noticeably dissimilar at low ( below 1  kOe) and high  (above 2 kOe)  \nmagnetic fields. At higher magnetic fie lds 𝐻>1 kOe we were unable to observe nonlinear \nmagnetization response at pump fluences up to 10 mJ/cm2. This is indicative of a significant \ndifference in the dynamic response in the collinear and noncollinear states of the magnetic \nsubsystem.  \n \n4. Nonlinear precession of magnetization in anisotropic cubic crystal s \nThe data shown in Fig. 4c clearly indicates the nonlinearity in the magnetic response \nmanifesting in the increase of the effective damping with the excitation (laser) fluence. \nPreviously, similar behavior was found in a number of metallic systems  [26-29] and quickly \nattributed to laser heating. Interestingly, Chen et al . [30] found a decrease of the effective \ndamping with laser fluence in FePt, while invoking the temper ature dependence of magnetic \ninhomogeneities to explain the results. There, the impact of magnetic inhomogeneity -driven \ndamping contribution exhibits a similar response to laser heating and an increase in the static \nmagnetic field. A more complicated mecha nism relying on the temperature -dependent \n7 \n competition between the surface and bulk anisotropy contributions and resulting in the \nmodification of the effective anisotropy field has been demonstrated in ultrathin Co/Pt \nbilayers  [31, 32] . \nNonlinear spin dynamics is a rapidly developing subfield enjoying rich prospects for ultrafast \nspintronics  [33]. Importantly, all those works featured thermal excitation of magnetization \ndynamics in metallic, strongly absorptive systems. In  stark contrast, we argue that the \nmechanism in the Co-doped YIG studied here is essentially non -thermal. This negligible \ntemperature change  ΔT is unable to induce significant variations of the parameters in the \nmagnetic syst em of YIG:Co (T N=450 K), thus ruling out the nonlinearity mechanism discussed \nabove. Rather, we note the work by M üller et al.  [34], where the non -thermal nonlinear \nregime of magnetization dynamics in CrO 2 at high laser fluences was ascribed to the spin -wave \ninstabilities at large precession amplitudes  [35]. We also note the recently debated and \nphysically rich mechanisms of magnetic nonlinearities, such as spin inertia  [36-39] and \nrelativistic effects  [40, 41] . Yet, we argue that in our case of a cubic magnetic anisotropy -\ndominated energy landscape, a much simpler explanation for the nonlinear spin dynamics can \nbe suggested. In particular, we  attribute the amplitude -dependent effective dampin g to the \nanharmonicity of the p otential well for magnetization . \n \nFig. 5. Energy landscape  as a function of the polar angle  𝜃𝜑=45°in the linear (𝐻=2.5 kOe, green) \nand nonlinear (𝐻=0.4 kOe, red) precession regime s. The d ashed lines are the parabolic fits in the \nvicinity of the minima . 𝜃 is the polar angle of magnetization  orientation  measured from the normal to \nthe sample plane  along  the [001] axis in YIG:Co .  \nWe performed  numerical calculations of the energy  density  landscape 𝑊(𝜃,𝜑):  \n𝑊(𝜃,𝜑)=𝑊𝑐+𝑊𝑢+𝑊𝑑+𝑊𝑧    (2) \ntaking into account the following terms in the free energy of the system:  the Zeeman energy  \n𝑊𝑧=−𝑴∙𝑯, demagnetizing field  term 𝑊𝑑=−2𝜋𝑀𝑠2sin2𝜃, cubic 𝑊𝑐=𝐾1∙\n(sin4𝜃sin2𝜑cos2𝜃+sin2𝜃cos2𝜃cos2𝜑+sin2𝜃cos2𝜃sin2𝜑) and uniaxial anisotropy \n𝑊𝑢=𝐾𝑢sin2𝜃 (𝜃 and 𝜑 are the polar and azimuthal angles, respectively ). In the calculations, \nwe assume 𝐾1=−9∙ 103 erg/cm3, 𝐾𝑢=−3∙103 erg/cm3, and 𝑀𝑠 is the saturation \n8 \n magnetization of 7.2 Oe [25]. Then, following  [8] and [42], we calculate the precession \nfrequency 𝑓 and the effective damping 𝛼𝑒𝑓𝑓: \n𝑓=𝛾\n2𝜋𝑀𝑠sin𝜃√𝛿2𝑊\n𝛿𝜃2𝛿2𝑊\n𝛿𝜑2−(𝛿2𝑊\n𝛿𝜃𝛿𝜑)2\n,    (3) \n𝛼𝑒𝑓𝑓=𝛼0𝛾(𝛿2𝑊\n𝛿𝜃2+𝛿2𝑊\n𝛿𝜑2sin−2𝜃)\n8𝜋2𝑓𝑀𝑠,     (4) \nwhere  the 𝛾 is gyromagnetic ratio , and 𝛼0 is the Gilbert damping in YIG:Co  [23, 24] . In Fig. 5, \nwe only show the total energy as a function  of the polar angle 𝜃, to illustrate the \nanharmonicity of the potential at small external in -plane magnetic fields. Experimental data \nand calculations of the energy  𝑊(𝜃,𝜑) have been published in Refs.  [25, 43] . There, it is seen \nthat at relative ly small external magnetic fields  canting  the magnetic state , the proximity of a \nneighboring energy minimum (to the right) effectively modifies the potential well for the \ncorresponding o scillator (on the left) , introducing an anharmonicity . On the other hand, at \nsufficiently large magnetic fields, whic h, owing to the Zeeman energy term, modify the \npotential such that a single minimum emerges (shown in Fig. 5 in green), no nonlinearity is \nexpected. This is also in line with the decreas ing impact of the cubic symmetry in the magnetic \nsystem, which is res ponsible for the anharmonicity of the energy potential.  \nTo get yet another calculated quantity that can be compare d to the experiment, we \nintroduced the photo -magnetically in duced effective anisotropy term  𝐾𝐿. This contribution \ndepends on the  laser fluence  I through the effective light -induced field 𝐻𝐿∝𝐼 as: \n𝐾𝐿=−2𝐻𝐿𝑀𝑠cos2𝜃     (5) \nThe presence of this term displaces the equilibrium for net magnetization. The equilibrium \ndirection s can be obtained by minimizing the total energy with and without the photo -\nmagn etic anisotropy term. Then, k nowing the angle between the perturbed and unperturbed \nequilibrium directions for the magnetization, we calculate d the precession amplitude 𝐴. We \nnote the difference between the amplitudes 𝐴𝐹, which refers to the Faraday rotation of the \nprobe beam, and 𝐴 standing for the opening angle of magnetization precession. Alth ough both \nare measured in degrees, their meaning is different.  \nHaving repeated this for a few levels of optical excitation, we obtain ed a linear slope of the \namplitude vs  excitation  strength  dependence. Figure 6 (a-c) illustrates the amplitude, \nfrequency , and (linear ) effective  damping as a function of the external magnetic field. The \nagreement between the calculated parameters and those obtained from fitting the \nexperimental data is an impressive indication of the validity of our total energy approach. \nFurther, the linear effective damping value of 𝛼≈0.2 obtained in the limit of  strong field s, is \nin good agreement with the values known for our Co -doped YIG from previous works  [6, 24] . \nIn principle, the effective damping  in garnets can increase  towards lower magnetic fields.  \nConventionally attributed to the extrinsic damping contributions, this behavior has been \nobserved in rare -earth iron garnets before as well and ascribed to the generation of the \nbackward volume spin w ave mode by ultrashort laser pulses  [44]. It is worth noting that there \nis no nonlinearity phenomenologically embedded in the approach given above.   9 \n  \nFig. 6. Photo -magnetic precession parameters  at various magnetic fields: amplitude (a), frequency (b) , \nand (linear) effective damping (c). The points are from the experimental data, the solid lines are \ncalculated as described in the text. The dark rectangular points are obtained in the FMR experimen ts. \nThe g rey shaded  area indicates the presence of a  domain st ate (DS). The g reen  shaded  area show s the \nmagnetization saturation state . \nYet, the data presented in Fig. 4c indicates the persistent nonlinear behavior of the effective \ndamping. To clarify the role of the potential anharmonicity, we fitted the potentials 𝑊(𝜃,𝜑) \nusing a parabolic function with an anharmonic term : \n𝑊(𝑥)=𝑊0+𝑘[(𝑥−𝑥0 )2+𝛽𝑥(𝑥−𝑥0)4]    (6) \nHere 𝑥=𝜃 or 𝜑, and 𝛽𝑥 is the anharmonicity parameter. We calculated it independently for \n𝜃 and 𝜑 for each dataset of 𝑊(𝜃,𝜑) obtained at different values of the external magnetic \nfield 𝐻 by fitting the total energy with Eq. (6) in the vicinity of the energy minimum  (Fig. 5) . \nThis anharmonicity should be examined on equal footing with the no nlinear damping \ncontribution. To quantify the latter, we follow the approach by Tiberkevich & Slavin  [8] and \nanalyze the effective damping dependencies on the precession amplitude by means of fitting \na second -order polynomial to them : \n𝛼=𝛼0+𝛼2𝐴2.      (7) \nThe examples of th e fit curves are shown in Fig. 7 a, demonstrating a good quality of the fit \nwithin a  certain range of the amplitudes  𝐴 (below 45 ). It should, however, be noted that the \nmodel in Ref.  [8] has been developed for the in -plane magnetic anisotropy, and thus its \napplicability for our case is limited. This is the re ason why we do not go beyond the amplitude \ndependence of the effective damping and do not analy ze the frequency dependence on 𝐴 in \n10 \n the limit of strong effective fields.  We note that the amplitude  𝐴, the opening angle of the \nprecession, should be understood as a mathematical parameter only, and not as a true \nexcursion angle of magnetization obtained in the real experimental conditions. There, large \neffective Gilbert damping values and a short decay t ime of the photo -magnetic anisotropy \npreclude the excursion of magnetization from its equilibrium to reach these 𝐴 values.  \n \nFig. 7. a) Effective damping in the linear and nonlinear precession regimes  of the precession amplitude \n𝐴. The lines are the second -order polynomial fits with Eq. (7). b ) Magnetic field dependence of the \nnonlinearity parameters: n onlinear damping coefficient 𝛼2 (points, obtained from experiments)  and \nthe 𝑊(𝜃) potential  anharmonicity  normalized  𝛽𝜃 (red line , calculated ).  \nWe note that the anharmonicity parameter 𝛽𝑥 calculated for the  W(θ) profiles was found to \nbe a few orders of magnitude larger than that obtained for W(𝜑). This difference in the \nanharmonicity justifies our earlier decision to focus on the shape of W(θ) potential only (cf. \nFig. 5). This means that the potential for magnetization in the azimuthal plane is muc h closer \nto the parabolic shape  and much larger amplit udes of the magnetization precession are \nrequired for it to start manifesting nonlinearities in dynamics. As such, we only consider the \nanharmonicity 𝛽𝑥 originating in the W(θ) potential energy. I n Fig. 7b, we compare the 𝛽𝜃 (red \nline) and 𝛼2 (points) dependencies on the external in -plane magnetic field. It is seen that its \ngeneral shape is very similar, corroborating our assumption that the potential anharmonicity \nis the main driving force behind the obse rved nonlinearity. We argue that thanks to the c ubic \nmagnetic anisotropy in YIG:Co  film, the potential anharmonicity -related mechanism of \nnonlinearity allows for reaching the nonlinear regime at moderate excitation levels.  \n \n5. Simulation s of laser -induced magnetization dynamics  \n11 \n To further prove that the ob served nonlinearities in magnetization dynamics do not require \nintroducing additional inertial or relativistic terms  [33], we complemented our experimental \nfindings with numerical simulations  of the LLG equation:  \n𝑑𝐌\n𝑑𝑡=−𝛾[𝐌×𝐇eff(𝑡)]+𝛼\n𝑀𝑠(𝐌×𝑑𝐌\n𝑑𝑡),    (8) \nwhere 𝐻𝑒𝑓𝑓 is the effective field derived from Eq. (2) as : \n𝐇eff(𝑡)=−∂𝑊𝐴\n∂𝑴+𝐇L(𝑡),     (9) \nWe employ ed the simulation model  from Ref.  [11] and added a term corresponding to the \nexternal magnetic field  𝐻. Calculations performed for a broad range of laser fluence s and \nexternal field values allowed us to obtain a set of  traces  of the magnetization dynamics . Figure \n8 show s a great deal of similarity  between simulations and experimen tal data (cf.  Fig. 3). It is \nseen that t he frequency increases with  increasing external  field  𝐻 while  the amplitude \ndecreases  (see Fig.  8a). The simulations for various stimulus strengths show the expected \ngrowth of the precession amplitude  (see Fig.  8b). \n \nFig. 8. Photo -magnetic  precession obtained in numerical simulations of the LLG equation  for: a) field \ndependence  at moderate excitation level  and b) power dependence  (I=4, 10, 16, and 22 arb. units ) at \n𝐻=0.4 kOe. \nWe further repeated our fit procedure with Eq.(1) to obtain the precession parameters from \nthese data. Figure 9 show s the values of the amplitude  and frequency of the precession in the \npower regime.  At a low field  𝐻=0.4 kOe (red) , the nonlinearity is clearly visible and \ncomparable with experimental data, as seen in Fig. 4. Similarly, at high field s (green) , the \nbehavior is mostly linear.  Figure 9a shows a great deal of similarity between simulations  \n(amplitude parameter)  and experi mental data  (normalize d value AF/max) (cf. Fig. 4a). The \nanalysis of the damping parameter (Fig. 9c) also confirms the exp erimental findings (as in Fig. \n7a), revealing the existence of two regimes, linear and nonlinear . The results of the s imulations \nconfirm that the observation of  the nonlinear response of the magnetic system  can be \nattributed  to the anharmonicity of the energy landscape.  \n12 \n Notably, in the simulations , as well as in the experimental data, we not only observe a second -\norder co rrection to the effective damping 𝛼2, but also a deviation from Eq.(7)  at even larger \namplitudes  (cf. Fig. 7 a and Fig. 9c). The latter manifests as a reduction of the effective damping \ncompared to the expected 𝛼0+𝛼2𝐴2 dependence shown with dashed lines. This higher -order \neffect is unlikely to originate in the multi -magnon scattering contribution since the latter \nwould only further increase the effective damping  [8]. We rather believe that th is is likely an \nartifact of the used damped oscillator model  where  in the range of  𝛼𝑒𝑓𝑓≈1 the quasiperiodic \ndescription of magnetization precession ceases to be physically justified.    \n \nFig. 9. Power dependence of  the a) amplitude  and b) frequency as obtained in the simulations  for low \n(red dataset) and high (green dataset) external magnetic field s. c) Effective damping  in the linear and \nnonlinear  precession regimes . \n \n6. Photo -induced phase diagram of sub -switching regime  \nIt is seen from both experimental and numerical results above that the cubic symmetry of the \nmagnetic system is key for the observed nonlinear magnetization dynamics. To quantify the \nparameter space for the nonlinearity, we first estimate the realistic values of the effective \nlight -induced magnetic field 𝐻𝐿. Throughout a number of works on photo -magnetism in Co -\ndoped garnets, a single -ion approach to magnetic anisotropy is consistent ly utilized. We note \n13 \n that in YIG:Co, it is the Co ions at tetrahedral sites that are predominantly responsible for the \ncubic anisotropy of  the magnetic energy landscape  [22]. In the near -IR range, these ions are \nresonantly excited at the 1300 nm wavelength, resulting in improved efficiency of the photo -\nmagnetic stimulus , as compared to previous works  [45]. Further, we note that at the \nmagnetization switching threshold, about 90% of the Co3+ ions with a concentration on the \norder of 1020 cm-3 are excited with incident photons  [11, 46] . Taking into account the single -\nion contribution to the anisotropy 𝛥𝐾1~105 erg/cm3 [47], and assuming a linear relation \nbetween the absorbed laser power (or fluence) and the effective photo -magnetic field 𝐻𝐿, for \nthe latter we find that 𝐻𝐿~1 kOe is sufficient for the magnetization switching. This means that \nthe sub -switching regime of magnetiz ation dynamics (cf. Fig. 1) refers to the laser fluences (as \nwell as wavelengths) , resulting in smaller effective fields.  \nWe reiterate that in previous works, the impact of the external magnetic field on the photo -\nmagnetically driven magnetization precess ion has not been given detailed attention.  To \naddress this gap , we plotted  the amplitude of the precession  𝐴 calculated in the same way as \nabove in the  sub-switching regime (Fig. 10) . As expected, the amplitude generally increases  \nwith 𝐻𝐿. However, we n ote a critical external field of about 0.5 kOe at which the desired \namplitudes can be reached at smaller light -induced effective fields 𝐻𝐿. At this field, where the \nsystem enters a single domain state, the potential curvature around the energy minimum \ndecreases, thus facilitating the large -angle precession. In other words, external magnetic field s \ncan a ct as leverage for the effective field of the photo -induced anisotropy, thus reducing the \nmagnetization switching threshold.  An exhaustive study of magneti zation switching across the \nparameter space shown in Fig. 10 remains an attractive perspective for future studies.  \n \nFig. 10. Calculated amplitude m ap of the photo -induced magnetization  precession in YIG:Co film. \nIn our analysis, we only considered a truly photo -magnetic excitation and neglected the laser -\ninduced effects of thermal origi n. It is, however, known that laser -driven heating can introduce \nan additional, long -lasting modification of magnetic anisotropy in iron garnets  [48, 49] . The \n14 \n relatively long relaxation times associated with cooling are responsible for the concomitant \nmodulation of the precession parameters and thus facilitate nonlinearities in the response of \nthe magnetic system. Yet, 1300 nm laser excitation of magnetization dynamics in YIG :Co film  \nwas shown to be highly polarization -dependent  [6], thus indicating the dominant role of the \nnon-thermal excitation mechanism. On the other hand, the unavoid able laser -induced heating \nwith experi mental values of laser fluence in YIG:Co film  has been estimated to not exceed 1 K  \n[6]. As such, we do not expect modification of the Gilbert damping associated with the \nproximity of the ma gnetization compensation or N éel temperature in the ferromagnetic \ngarnet  [50]. However, a detailed investigation of the temperature -dependent nonlinear \nmagnetization dynamics in the vicinity of the compensation point or a magnetic phase \ntransition  [51, 52]  represents another promising research direction.  Further , exploring the \nnonlinear  regime  in the response of the magnetic system to intense THz stimul i along the lines \ndiscussed in  [33] enjoys a rich potentia l for spintronic applications.  \n  \n7. CONCLUSIONS  \nIn summary, we studied, both experimentally and numerically, the nonlinear regime of \nmagnetization dynamics in photo -magn etic Co -doped YIG film. After  excitation with \nfemtosecond laser pulses at fluences below the magnetization switching threshold, there is a \nrange of external magnetic field where the magnetic system demonstrates strongly non linear \nprecession characterized by a significant increase of t he effective Gilbert damping. We \nattribute this nonlinearity to the anharmonicity of the potential for the magnetic oscillator \nenhanced by the dominant role of the cubic magnetocrystalline anisotropy. The effective \ndamping and its nonlinear contribution, a s obtained from numerical simulations, both \ndemonstrate a very good agreement with the experimental findings. Simulations of the \nmagnetization dynamics by means of  the LLG equation further confirm the nonlinearity in the \nmagnetic response below the switchi ng limit. Finally, we provide estimations for the realistic , \neffective photo -magnetic fields 𝐻𝐿 and map the workspace of the parameters in the sub -\nswitching, nonlinear regime of photo -induced magnetization dynamics.  \n \nACKNOWLEDGMENTS  \nThis work has been fu nded by the  Foundation for Polish Science ( Grant No. POIR.04.04.00 -00-\n413C/17)  and the National Science Centre Poland (Grant No. DEC -2017/25/B/ST3/01305) . \n \nREFERENCES  \n \n[1]  A. Kirilyuk, A. V. Kimel and T. Rasing, \"Ultrafast optical manipulation of magnetic order,\" \nRev. Mod. Phys., 82 (3), 2731 (2010).  15 \n [2]  J. Walowski and M. 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The string is modelled by a wave equation in a time-dependent interval\nwith two endpoints moving at a constant speed v. For the undamped case, we obtain a con-\nserved functional equivalent to the energy of the solution. We derive precise upper and lower\nestimates for the exponential decay of the energy with explicit constants. These estimates do\nnot seem to be reported in the literature even for the non-travelling case v= 0.\n1.Introduction\nWe consider small transversal vibrations of a uniform string travelling with a constant speed\nvbetween to two pulleys (inlet and outlet) kept at a \fxed distance L. The mechanical setting\nis sketched in Figure 1 where the inlet is \fxed while the outlet is allowed to move transversely\nand attached to a damping device (a dashpoint with a damping factor \u0011).\n \n \n \n \n \n  \n \n \nv\nL\n\nFigure 1. An axially travelling string with a dash-point at the outlet pulley.\nMany mechanical devices with axially moving continua, such as power transmission chains\nand belts, magnetic tapes, band saws and \fbre winders, see for instance [1, 2, 3, 4, 5], are\nlimited in their e\u000eciency and utility due to unwanted vibrations. As a result, stabilization of\naxially moving systems is necessary to reduce or eliminate these vibrations and improve the\noverall performance and productivity of these mechanical systems.\nThe existing approaches in the literature describe the above problem in \fxed space coordi-\nnates, see for instance [3, 1, 6, 7, 8, 9, 10]. Gaiko and van Horssen [11] considered a simpli\fed\nmathematical model describing the small vibration of the string with a mass-spring-dashpoint\ndamping at the outlet. Under the restriction that the speed vand the damping factor are\nsmall, i.e. 0 <v\u001c1 and\u0011\u001c1 the authors in [11] obtained an asymptotic approximation for\nthe solution of (1.1) using a multiple scale approach.\nDate : November 22, 2022.\n2010 Mathematics Subject Classi\fcation. 35L05, 35C10, 93D15.\nKey words and phrases. Axially travelling strings, Fourier series, energy estimates, boundary stabilization.\n1arXiv:2211.10537v1  [math.AP]  18 Nov 20222 Decay estimates for an axially travelling string\nDenoting the displacement function by u, depending on the position salong the string and\nthe time\u001c, with only a dashpoint damping at the outlet, their model can be stated as follows\n8\n>>>><\n>>>>:u\u001c\u001c+ 2vus\u001c\u0000\u0000\n1\u0000v2\u0001\nuss= 0; fors2(0;L) and\u001c >0;\nu(0;\u001c) = 0; for\u001c >0;\n\u0000\n1\u0000v2\u0001\nus(L;\u001c) + (\u0011\u0000v)u\u001c(L;\u001c) = 0; for\u001c >0;\nu(s;0) =u0(s); u\u001c(s;0) =u1(s); fors2(0;L);(1.1)\nwhere the subscripts \u001candsstand for the derivatives in time and space variables respectively.\nThe functions u0andu1represents the initial shape and the initial transverse speed of the\nstring, respectively.\nIn the present work, we do not consider the magnitudes of vand\u0011as small ones. We only\nassume that \u0011\u00150 and that speed vis strictly less then the speed of propagation of the wave\n(here normalized to c= 1), i.e.\n0\u0014v<1: (1.2)\nIf the speed vapproaches the critical speed c= 1, an instability will occur, as shown by\n[5, 10]. Another key di\u000berence with most of the existing works is that we consider the model\nin a moving space coordinates. We introduce the variables\ns=L\u0000x+vtand\u001c=t;\nhence\nx2It:= (vt;L +vt);fort\u00150;\nwhich is an interval travelling in the positive sense of the real axis (as in [12, 4]). It follows\nthat\n@s=\u0000@xand@\u001c=v@x+@t:\nRewriting Problem (1.1) in the new coordinates, we obtain the following (pure) wave equation\nwith a damping at the moving boundary x=vt;\n8\n>>><\n>>>:\u001ett\u0000\u001exx= 0; forx2Itandt>0;\n(1\u0000\u0011v)\u001ex(vt;t)\u0000(\u0011\u0000v)\u001et(vt;t) = 0;fort>0;\n\u001e(L+vt;t) = 0, for t>0;\n\u001e(x;0) =\u001e0(x); \u001et(x;0) =\u001e1(x); forx2I0;(WP)\nwhere\u001e0=u0and\u001e1=u1\u0000v\u001e0\nx:\nLet us denote\n\rv:=1 +v\n1\u0000v; \r\u0011:=1 +\u0011\n1\u0000\u0011and!n:=8\n><\n>:(2n+ 1)\n2i\u0019\u00001\n2ln\r\u0011;if 0\u0014\u0011<1;\nni\u0019\u00001\n2lnj\r\u0011j; if\u0011>1:\nNote that\n\u000f\rv\u00151 for 0\u0014v<1,\n\u000fj\r\u0011j\u00151 and the real part of !nremains negative, for \u0011>0 with\u00116= 1.\nAs a \frst result, we derive a closed form of the solution of (WP) given by the series formulas\n\u001e(x;t) =X\nn2Zan\u0010\n\r\u0011e1\u0000v\nL!n(t+x)+e1+v\nL!n(t\u0000x)\u0011\n;forx2Itandt\u00150; (1.3)\nwhich is a sum of two waves travelling in opposite directions. The coe\u000ecients anare explicitly\ngiven in function of the initial data \u001e0and\u001e1;see Theorem 1.S. Ghenimi and A. Sengouga 3\nNext, we demonstrate how the series formulas (1.3) can be used to achieve the following\nresults:\n\u000fFor the undamped case, i.e. \u0011= 0, we show that the functional1\nEv(t) =1\n2Z\nIt(\u001et+v\u001ex)2+\u0000\n1\u0000v2\u0001\n\u001e2\nxdx; fort\u00150; (1.4)\ndepending on L;t;v and the solution \u001eof (WP) is conserved in time, see Theorem 2.\nNote that under the assumption (1.2), the functional Evis positive-de\fnite and we will\ncall it the \"energy\" of the solution \u001e.\n\u000fFor the damped case \u0011>0with\u00116= 1, the (usual) energy\nEv(t) =1\n2Z\nIt\u001e2\nt(x;t) +\u001e2\nx(x;t)dx; fort\u00150; (1.5)\ndepending on L;t;v and the solution of (WP) decays exponentially. More precisely\n1\n\r2\u0011\rvEv(0)e\u00001\u0000v2\nLlnj\r\u0011jt\u0014Ev(t)\u0014\r2\n\u0011\rvEv(0)e\u00001\u0000v2\nLlnj\r\u0011jt;fort\u00150: (1.6)\nSee Theorem 3 and it corollaries for more sharper estimates. It follows in particular\nthat there is no decay to zero in \fnite time.\nThe exponential decay estimate given in (1.6), with the sharp rate and explicit constants,\nis new to the best to our knowledge. The approach presented here relays on Fourier series\nand Parseval's identity, and it does not involve semigroup theory as in [10, 13]. Even for a\nnon-travelling string, i.e. when v= 0;the precise estimate\n1\n\r2\u0011E0(0)e\u00001\nLlnj\r\u0011jt\u0014E0(t)\u0014\r2\n\u0011E0(0)e\u00001\nLlnj\r\u0011jt;fort\u00150; (1.7)\nseems to be not reported in the literature. The existing results assure only that\nE0(t)\u0014KE 0(0)e\u00001\nLlnj\r\u0011jt;\nfor a (non-explicitly given) positive constant K, see for instance [14, 15, 16, 17].\nFor the special case \u0011= 1 and 0\u0014v<1;the boundary condition at x=vtreads\n\u001ex(vt;t)\u0000\u001et(vt;t) = 0:\nThis is a transparent condition, i.e. there is no re\rections of waves from the boundary x=vt\nand consequently all the initial disturbances leave the interval ( vt;L +vt) at most after a time\nTv:=L\n1 +v+L\n1\u0000v=2L\n1\u0000v2:\nThat is to say that the linear velocity feedback \u001et(vt;t) steers the solution to the zero state\nin the \fnite time Tv. See for instance [10] for the case 0 <v< 1 and [14, 15] for v= 0.\nAfter the present introduction, we derive the exact solution and the expression for the\ncoe\u000ecients of the series formula (1.3). In Section 3, we show that the energy Evof the\nundamped equation is constant in time. In the last section, we show the exponential stability\nwhen\u0011>0.\n1Here and in the sequel, the subscript vis used to emphasize the dependence on the speed v.4 Decay estimates for an axially travelling string\n2.Exact solution\nTo compute the solution of Problem (WP), given by (1.3), we need to compute the coe\u000e-\ncientsan;n2Z:To this end, we need to know the functions \u001e0and\u001e1on an interval larger\nthanI0= (0;L). As in [18, 12], we introduce\nL2:=2L\n1\u0000v\nand extend \u001eto the interval ( L+vt;L 2+vt) by setting\n~\u001e(x;t) =(\u001e(x;t); x 2(vt;L +vt);\n\u0000\u001e\u0010\n1\n\rv(vt\u0000x) +2L\n1+v+vt;t\u0011\n; x2(L+vt;L 2+vt):(2.1)\nThe obtained function is well de\fned since the \frst variable of \u001eremains in the interval\n(vt;L +vt). In particular, the homogeneous boundary condition \u001e(L+vt;t) = 0 remains\nsatis\fed, for every t\u00150.\nRemark 1. Clearly, 0< L\u0014L2=2for0\u0014v <1:Ifv= 0;thenL2= 2Land the function\n~\u001eon(L;2L)is an odd function on (0;2L), with respect to x=L. If0< v < 1;then\u001eis\nextended as an odd function with an extra dilatation on the added interval (L+vt;L 2+vt),\nsee Figure 2.\n \n0\nL\nL2\nx\n\u001e0\nFigure 2. Example of the extension of an initial data \u001e0when 0<v< 1.\nTaking the derivative of (2.1) with respect to x, we obtain\n~\u001ex(x;t) =(\u001ex(x;t); x 2(vt;L +vt);\n1\n\rv\u001ex\u0010\n1\n\rv(vt\u0000x) +2L\n1+v+vt;t\u0011\n; x2(L+vt;L 2+vt):(2.2)\nOn the other hand, the time derivative is extended as follows\n~\u001et(x;t) =(\u001et(x;t); x 2(vt;L +vt);\n\u00001\n\rv\u001et\u0010\n1\n\rv(vt\u0000x) +2L\n1+v+vt;t\u0011\n; x2(L+vt;L 2+vt):(2.3)\nLet us introduce the following family of Hilbert spaces\nHL+vt(It) :=\b\nw2H1(It) ,w(L+vt) = 0\t\n;fort\u00150\nand we assume that the initial data satis\fes\n\u001e02HL(I0); \u001e12L2(I0): (2.4)\nNow, we are ready to state the following existence and uniqueness result for Problem (WP).\nTheorem 1. Under the assumptions (1.2) and(2.4), the solution of Problem (WP)\n\u001e2C([0;T];HL+vt(It))\\C1\u0000\n[0;T];L2(It)\u0001\n; (2.5)S. Ghenimi and A. Sengouga 5\nis given by the series (1.3) where the coe\u000ecients an2Care given by the following formula\nan=1\n4\r\u0011!nL2Z\n0\u0010\n~\u001e0\nx+~\u001e1\u0011\ne\u00001\u0000v\nL!nxdx, forn2Z (2.6)\nand we have\nX\nn2Zj!nanj2=L\n8 (1\u0000v)\r2\u0011L2+vtZ\nvte1\u0000v\nLlnj\r\u0011j(t+x)\u0010\n~\u001ex+~\u001et\u00112\ndx< +1; (2.7)\nwhere ~\u001e0\nxand~\u001e1are extensions of the initial data \u001e0and\u001e1on the interval (0;L2)given above\nby(2.2) and(2.3) respectively.\nProof. The general solution of (WP) is given by D'Alembert's formula\n\u001e(x;t) =f(t+x) +g(t\u0000x);\nwherefandgare arbitrary continuous functions. Let us check the boundary conditions. On\none hand, at the left endpoint we have\n(1\u0000\u0011v)f0((1 +v)t)\u0000g0((1\u0000v)t) = (\u0011\u0000v)\u0000\nf0((1 +v)t) +g0((1\u0000v)t)\u0001\n:\nSettingz= (1\u0000v)t, we obtain\n(1\u0000\u0011v\u0000\u0011+v)f0(\rvz) = (1\u0000\u0011v+\u0011\u0000v)g0(z): (2.8)\nOn the other hand, at the right endpoint, we infer that\nf((1 +v)t+L) +g((1\u0000v)t\u0000L) = 0:\nDenotingy=z\u0000L, we obtain\nf(\rvy+2L\n1\u0000v) =\u0000g(y): (2.9)\nThen, taking the derivative with respect to y;we have\n\rvf0(\rvy+2L\n1\u0000v) =\u0000g0(y): (2.10)\nThanks to (2.8), (2.10) and taking \u0018=\rvy;we can deduce that f0satis\fes\n(1\u0000\u0011v+\u0011\u0000v)\rvf0\u0012\n\u0018+2L\n1\u0000v\u0013\n=\u0000(1\u0000\u0011v\u0000\u0011+v)f0(\u0018): (2.11)\nAfter few simpli\fcations, this can be written as\nf0\u0012\n\u0018+2L\n1\u0000v\u0013\n=\u0000f0(\u0018)\n\r\u0011: (2.12)\nThis formula suggests that f0(\u0018) =e\f\u0018, for some parameter \fto be determined later.\nSubstituting the form e\f\u0018in (2.12), we infer that e\f(\u0018+2L\n1\u0000v)=\u0000e\f\u0018=\r\u0011:Then we have:\n\u000fIf 0\u0014\u0011<1, then\r\u0011\u00151 and we get\ne\f(\u0018+2L\n1\u0000v)=\u0000e\u0000ln\r\u0011e\f\u0018:\nSolving this equation for \f, we obtain a sequence of values \fn= (1\u0000v)!n=L;n2Z;where\n!n=1\n2(2n+ 1)i\u0019\u00001\n2ln\r\u0011:6 Decay estimates for an axially travelling string\n\u000fIf\u0011>1, we have\r\u0011<\u00001 and we obtain another sequence of values \fn= (1\u0000v)!n=L;n2\nZ;where this time\n!n=ni\u0019\u00001\n2lnj\r\u0011j:\nNote that in both cases we have ln j\r\u0011j\u00151 and the real part of !nis negative.\nDue to the superposition principal, it follows that fcan be written as\nf0(\u0018) =X\nn2Zcne1\u0000v\nL!n\u0018; cn2C;\nwherecnare complex coe\u000ecients to be determined later. The function fcan be written as\nf(\u0018) =c+X\nn2ZLcn\n(1\u0000v)!ne1\u0000v\nL!n\u0018;\nfor some constant c:Using (2.9), we deduce that\ng(\u0018) =\u0000c+X\nn2ZLcn\n(1\u0000v)\r\u0011!ne1+v\nL!n\u0018; (2.13)\nwhere we have used the fact that e2!n=\u00001=\r\u0011whether 0\u0014\u0011<1 or\u0011>1:\nThanks to D'Alembert's formula, the solution of Problem (WP) is given by the series\n\u001e(x;t) =X\nn2ZL\n(1\u0000v)\r\u0011!ncn\u0010\n\r\u0011e1\u0000v\nL!n(t+x)+e1+v\nL!n(t\u0000x)\u0011\n: (2.14)\nTo obtain (1.3), we set\nan:=L\n(1\u0000v)\r\u0011!ncn: (2.15)\nThe coe\u000ecient cnare determined as follows. Going back to (2.14), we infer that\n\u001ex(x;t) =X\nn2Zcn\u0012\ne1\u0000v\nL!n(t+x)\u0000\rv\n\r\u0011e1+v\nL!n(t\u0000x)\u0013\n; (2.16)\n\u001et(x;t) =X\nn2Zcn\u0012\ne1\u0000v\nL!n(t+x)+\rv\n\r\u0011e1+v\nL!n(t\u0000x)\u0013\n; (2.17)\nforx2(vt;L +vt) andt\u00150. It follows from (2.2) and (2.3) that the extensions ~\u001exand ~\u001et\nare given by\n~\u001ex(x;t) =8\n>>>>>>><\n>>>>>>>:X\nn2Zcn\u0012\ne1\u0000v\nL!n(t+x)\u0000\rv\n\r\u0011e1+v\nL!n(t\u0000x)\u0013\n;ifx2(vt;L +vt);\n1\n\rvX\nn2Zcn\u0010\ne1\u0000v\nL!n((1+v)t+vt\u0000x\n\rv+2L\n1+v)\n\u0000\rv\n\r\u0011e1+v\nL!n((1\u0000v)t\u0000vt\u0000x\n\rv\u00002L\n1+v)\u0011\n;ifx2(L+vt;L 2+vt);(2.18)\n~\u001et(x;t) =8\n>>>>>>><\n>>>>>>>:X\nn2Zcn\u0012\ne1\u0000v\nL!n(t+x)+\rv\n\r\u0011e1+v\nL!n(t\u0000x)\u0013\n;ifx2(vt;L +vt);\n\u00001\n\rvX\nn2Zcn\u0010\ne1\u0000v\nL!n((1+v)t+vt\u0000x\n\rv+2L\n1+v)\n+\rv\n\r\u0011e1+v\nL!n((1\u0000v)t\u0000vt\u0000x\n\rv\u00002L\n1+v)\u0011\n;ifx2(L+vt;L 2+vt):(2.19)S. Ghenimi and A. Sengouga 7\nTaking the sum of (2.18) and (2.19), we get\n~\u001ex+~\u001et=8\n>>><\n>>>:2X\nn2Zcne1\u0000v\nL!n(t+x); x2(vt;L +vt);\n\u00002\n\rvX\nn2Zcn\rv\n\r\u0011e\u00002!ne1\u0000v\nL!n(t+x); x2(L+vt;L 2+vt):\nSincee\u00002!n=\u0000\r\u0011, then we have the uni\fed expression\n~\u001ex+~\u001et= 2X\nn2Zcne1\u0000v\nL!n(t+x);forx2(vt;L 2+vt) andt\u00150: (2.20)\nUsing the de\fnition of !n;we get\n~\u001ex+~\u001et=8\n>><\n>>:2e1\u0000v\n2L(i\u0019\u0000ln\r\u0011)(t+x)X\nn2Zcne1\u0000v\nLni\u0019(t+x);if 0\u0014\u0011<1;\n2e\u00001\u0000v\n2Llnj\r\u0011j(t+x)X\nn2Zcne1\u0000v\nLni\u0019(t+x); if\u0011>1:(2.21)\nTaking into account thatn\nen\u0019i(1\u0000v)\nL(t+x)=pL2o\nn2Zis an orthonormal basis for L2(vt;L 2+vt),\nfor everyt\u00150, we rewrite (2.20) as\nX\nn2Zcne1\u0000v\nLni\u0019(t+x)\npL2=8\n>><\n>>:1\n2pL2e\u00001\u0000v\n2L(i\u0019\u0000ln\r\u0011)(t+x)\u0010\n~\u001ex+~\u001et\u0011\n;if 0\u0014\u0011<1;\n1\n2pL2e1\u0000v\n2Llnj\r\u0011j(t+x)\u0010\n~\u001ex+~\u001et\u0011\n; if\u0011>1:(2.22)\nBy consequence,\ncn=8\n>>>>>>><\n>>>>>>>:1\n2L2L2+vtZ\nvte\u00001\u0000v\n2L(i\u0019\u0000ln\r\u0011)(t+x)\u0010\n~\u001ex+~\u001et\u0011\ne\u0000n\u0019i(1\u0000v)\nL(t+x)dx; if 0\u0014\u0011<1;\n1\n2L2L2+vtZ\nvte1\u0000v\n2Llnj\r\u0011j(t+x)\u0010\n~\u001ex+~\u001et\u0011\ne\u0000n\u0019i(1\u0000v)\nL(t+x)dx; if\u0011>1;\nforn2Z. Whether 0\u0014\u0011<1 or\u0011>1;in both cases, we have\ncn=1\n2L2L2+vtZ\nvt\u0010\n~\u001ex+~\u001et\u0011\ne\u0000(1\u0000v)\nL!n(t+x)dx, forn2Z.\nFort= 0 and tacking (2.15) into account, we obtain (2.6) as claimed.\nMoreover, as a consequence of Parseval's equality, it comes that\nX\nn2Zjcnj2=8\n>>>>>>><\n>>>>>>>:1\n4L2L2+vtZ\nvt\f\f\fe\u00001\u0000v\n2L(i\u0019\u0000ln\r\u0011)(t+x)\f\f\f2\u0010\n~\u001ex+~\u001et\u00112\ndx; if 0\u0014\u0011<1;\n1\n4L2L2+vtZ\nvt\f\f\fe1\u0000v\n2Llnj\r\u0011j(t+x)\f\f\f2\u0010\n~\u001ex+~\u001et\u00112\ndx; if\u0011>1:8 Decay estimates for an axially travelling string\nWhether 0\u0014\u0011<1 or\u0011>1;it follows that\nX\nn2Zj!nanj2=L2\n\r2\u0011(1\u0000v)2X\nn2Zjcnj2=L\n8\r2\u0011(1\u0000v)L2+vtZ\nvte1\u0000v\nLlnj\r\u0011j(t+x)\u0010\n~\u001ex+~\u001et\u00112\ndx:\nThanks to (2.4), \u001e0and\u001e1belongs toL2(0;L2). Thus, the integral at the right hand side for\nt= 0 is \fnite and X\nn2Zj!nanj2<+1:\nRecalling thatjwnj2=O\u0000\nn2\u0001\n, for large values of n;then\nX\nn2Zjnanj2<+1: (2.23)\nLetT >0 andt2[0;T]:Due to the continuity of the exponential function, we get\f\f\fan\u0010\n\r\u0011e1\u0000v\nL!n(t+x)+e1+v\nL!n(t\u0000x)\u0011\f\f\f\u0014CTjanj;\nwhereCTis a constant depending only on v;\u0011;L andT:\nGoing back to (2.16), (2.17) and due to (2.15), we can check that\f\f\f\fcn\u0012\ne1\u0000v\nL!n(t+x)\u0006\rv\n\r\u0011e1+v\nL!n(t\u0000x)\u0013\f\f\f\f\u0014C0\nTjnanj;\nfor some constant C0\nT.\nTaking (2.23) into account, we infer that \u001e(x;t),\u001ex(x;t) and\u001et(x;t) belong to L2(It), for\nt\u00150. In particular, \u001e(x;t)2HL+vt(It), fort\u00150. The continuity in time of \u001eand\u001etas\nfunctions of twith values inHL+vt(It) andL2(It), respectively, follows as they are the sums\nof uniformly converging series of continuous functions. This shows (2.5). \u0003\n3.A conserved quantity for the string with no damper\nFor the undamped case, i.e. \u0011= 0 in Problem (WP), we show that the energy Evgiven by\n(1.4) is conserved in time.\nTheorem 2. Under the assumptions (1.2) and(2.4), the solution of Problem (WP) satis\fes\nEv(t) =\u00192\u0000\n1\u0000v2\u0001\n2LX\nn2Zj(2n+ 1)anj2; fort\u00150; (3.1)\nwhere the left hand side is independent of t:\nProof. If\u0011= 0;then!n= (2n+ 1)i\u0019=2 and the identity (2.7) becomes\n1\n1\u0000vL2+vtZ\nvt\u0010\n~\u001ex+~\u001et\u00112\ndx=2\u00192\nLX\nn2Zj(2n+ 1)anj2: (3.2)\nUsing the extensions (2.2), (2.3) and considering the change of variable\nx=\rv(vt\u0000\u0018) +2L\n1\u0000v+vt;\nin (L+vt;L 2+vt);then we have\n1\n1\u0000vL2+vtZ\nL+vt\u0010\n~\u001ex(x;t) +~\u001et(x;t)\u00112\ndx=1\n1 +vZ\nIt(\u001ex(\u0018;t)\u0000\u001et(\u0018;t))2d\u0018:S. Ghenimi and A. Sengouga 9\nTaking (3.2) into account, it comes that\n1\n1\u0000vL2+vtZ\nvt\u0010\n~\u001ex+~\u001et\u00112\ndx=1\n1\u0000vZ\nIt(\u001et+\u001ex)2dx+1\n1 +vZ\nIt(\u001ex\u0000\u001et)2dx\n=2\u00192\nLX\nn2Zj(2n+ 1)anj2:\nExpanding ( \u001ex\u0006\u001et)2and collecting similar terms, we get\n1\n1\u0000v2\u0012Z\nIt\u001e2\nx+\u001e2\nt+ 2v\u001ex\u001etdx\u0013\n=\u00192\nLX\nn2Zj(2n+ 1)anj2;fort\u00150: (3.3)\nThe left hand side is equal to 2 Ev(t)=\u0000\n1\u0000v2\u0001\nand (3.1) follows. \u0003\nRemark 2. Using Leibnitz's rule for di\u000berentiation under the integral sign, we can check\ndirectly thatd\ndtEv(t) = 0 , see the appendix.\nRemark 3. The energy expression Ev(t)is also shown to be conserved in time for the Dirichlet\nboundary conditions at both ends, see [12].\nLet us now compare Ev(t) to the usual expression of energy Ev(t) for the wave equation\nCorollary 1. Under the assumptions (1.2) and (2.4), the energy Ev(t)of the solution of\nundamped Problem (WP) satis\fes\nEv(0)\n1 +v\u0014Ev(t)\u0014Ev(0)\n1\u0000v; for t\u00150 (3.4)\nand\n1\n\rvEv(0)\u0014Ev(t)\u0014\rvEv(0); fort\u00150: (3.5)\nProof. It su\u000eces to argue as in the proof of Corollary 2 in [12]. \u0003\nRemark 4. The solution \u001egiven by (1.3), with \u0011= 0;satis\fes the periodicity relation\n\u001e(x+vTv;t+Tv) =\u0000\u001e(x;t),t\u00150: (3.6)\nIt follows in particular that the energy Evis aTv\u0000periodic function in time.\nRemark 5. The equality in (3.4) holds at least if \u001et(x;t0) =\u0006\u001ex(x;t0), forx2It0and some\nt0\u00150. Indeed, we have\nEv(t0) =Ev(t0)\u0006vZL+vt0\nvt0\u001ex(x;t0)\u001et(x;t0)dx= (1\u0006v)Ev(t0);\ni.e.Ev(t0) =Ev(t0)=(1\u0006v). The + and { signs are used respectively.\nRemark 6. Let0<v< 1. Arguing as in [12, 18] , we can show the identity\n1\nv2TvZ\n0\u001e2\nx(vt;t)dt=Ev(0)\n(1\u0000v2)2=TvZ\n0\u001e2\nx(L+vt;t)dt: (3.7)\nThen, we can derive exact boundary observability results at each endpoint for T\u0015Tv=\n2L=\u0000\n1\u0000v2\u0001\n.10 Decay estimates for an axially travelling string\n4.Exponential stability for the string with a boundary damper\nIn this section, we keep 0 \u0014v<1 and assume that \u0011>0.\nTheorem 3. Under the assumptions (1.2) and(2.4), the solution of Problem (WP) satis\fes\n1\n1 +vZ\nIte1+v\nLlnj\r\u0011j(t\u0000x)(\u001ex\u0000\u001et)2dx\n+1\n\r2\u0011(1\u0000v)Z\nIte1\u0000v\nLlnj\r\u0011j(t+x)(\u001et+\u001ex)2dx=8\nLX\nn2Zj!nanj2;(4.1)\nwhere the left hand side is \fnite and independent of t. Moreover, it holds that\nM1e\u00001\u0000v2\nLlnj\r\u0011jt\u0014Ev(t)\u0014M2e\u00001\u0000v2\nLlnj\r\u0011jt;fort\u00150; (4.2)\nwhere\nM1: =2\nLmin\u0010\n1 +v;j\r\u0011j(1+v)(1\u0000v)\u0011X\nn2Zj!nanj2;\nM2: =2\nLmax\u0010\nj\r\u0011j(1+v)(1 +v);\r2\n\u0011(1\u0000v)\u0011X\nn2Zj!nanj2:\nProof. Let us split the integral in the identity (2.7) to the integrals\nL2+vtZ\nvt=Z\nIt+L2+vtZ\nL+vt; (4.3)\nthen considering the change of variable x=\rv(vt\u0000\u0018) +2L\n1\u0000v+vtin (L+vt;L 2+vt), we\nobtain\n1\n(1\u0000v)\r2\u0011L2+vtZ\nL+vte1\u0000v\nLlnj\r\u0011j(t+x)\u0010\n~\u001ex(x;t) +~\u001et(x;t)\u00112\ndx\n=\u00001\n(1\u0000v)\r2\u0011L2+vtZ\nL+vt\rv\n\r2ve1\u0000v\nLlnj\r\u0011j(t+\rv(vt\u0000\u0018)+vt)e2 lnj\r\u0011j\u0010\n~\u001ex(\u0018;t)\u0000~\u001et(\u0018;t)\u00112\nd\u0018\n=1\n1 +vZ\nIte1+v\nLlnj\r\u0011j(t\u0000\u0018)(\u001ex(\u0018;t)\u0000\u001et(\u0018;t))2d\u0018: (4.4)\nWe used the de\fnition of the extensions (2.2), (2.3) and the fact that e2 lnj\r\u0011j=\r2\n\u0011:Then,\ncombining (4.3) and (4.4) we obtain (4.1).\nExpanding ( \u001ex\u0006\u001et)2and collecting similar terms, we get\nZ\nIt\u00121\n1 +ve1+v\nLlnj\r\u0011j(t\u0000x)+1\n\r2\u0011(1\u0000v)e1\u0000v\nLlnj\r\u0011j(t+x)\u0013\u0000\n\u001e2\nt+\u001e2\nx\u0001\ndx\n\u00002Z\nIt\u00121\n1 +ve1+v\nLlnj\r\u0011j(t\u0000x)\u00001\n\r2\u0011(1\u0000v)e1\u0000v\nLlnj\r\u0011j(t+x)\u0013\n\u001et\u001exdx\n=8\nLX\nn2Zj!nanj2:(4.5)S. Ghenimi and A. Sengouga 11\nForvt\u0014x\u0014L+vtandt\u00150;let us denote\nA(x;t) =1\n1 +ve1+v\nLlnj\r\u0011j(t\u0000x)andB(x;t) =1\n\r2\u0011(1\u0000v)e1\u0000v\nLlnj\r\u0011j(t+x):\nThen, we can rewrite (4.5) as\nZ\nIt(A+B)\u0000\n\u001e2\nt+\u001e2\nx\u0001\ndx\u00002Z\nIt(A\u0000B)\u001et\u001exdx=8\nLX\nn2Zj!nanj2:\nUsing the algebraic inequality\n\u0000jA\u0000Bj\u0000\n\u001e2\nt+\u001e2\nx\u0001\n\u0014\u00062 (A\u0000B)\u001et\u001ex\u0014jA\u0000Bj\u0000\n\u001e2\nt+\u001e2\nx\u0001\n;\nwe get\nZ\nIt((A+B)\u0000jA\u0000Bj)\u0000\n\u001e2\nt+\u001e2\nx\u0001\ndx\u00142\nLX\nn2Zj2!nanj2\n\u0014Z\nIt((A+B) +jA\u0000Bj)\u0000\n\u001e2\nt+\u001e2\nx\u0001\ndx:\nKnowing that ( a+b)\u0000ja\u0000bj= 2 minfa;bgand (a+b) +ja\u0000bj= 2 maxfa;bg, fora;b2R;\nthen the precedent estimation readsZ\nItminfA;Bg\u0000\n\u001e2\nt+\u001e2\nx\u0001\ndx\u00144\nLX\nn2Zj!nanj2\u0014Z\nItmaxfA;Bg\u0000\n\u001e2\nt+\u001e2\nx\u0001\ndx:\nSince lnj\r\u0011j\u00150 andvt\u0014x\u0014L+vt;we have\n1\n1 +ve1+v\nLlnj\r\u0011j(t\u0000L\u0000vt)\u0014A(x;t)\u00141\n1 +ve1+v\nLlnj\r\u0011j(t\u0000vt);\ni.e.\ne\u0000(1+v) lnj\r\u0011j\n1 +ve1\u0000v2\nLlnj\r\u0011jt\u0014A(x;t)\u00141\n1 +ve1\u0000v2\nLlnj\r\u0011jt:\nSimilarly, we obtain\n1\n\r2\u0011(1\u0000v)e1\u0000v2\nLlnj\r\u0011jt\u0014B(x;t)\u0014e\u0000(1+v) lnj\r\u0011j\n1\u0000ve1\u0000v2\nLlnj\r\u0011jt:\nIt follows that\nmin(\n1\nj\r\u0011j(1+v)(1 +v);1\n\r2\u0011(1\u0000v))\nEv(t)\u00142e\u00001\u0000v2\nLlnj\r\u0011jt\nLX\nn2Zj!nanj2\n\u0014max(\n1\n1 +v;1\nj\r\u0011j(1+v)(1\u0000v))\nEv(t);\nhence\n \n2\nLX\nn2Zj!nanj2!\nminn\n1 +v;j\r\u0011j(1+v)(1\u0000v)o\ne\u00001\u0000v2\nLlnj\r\u0011jt\u0014Ev(t)\n\u0014 \n2\nLX\nn2Zj!nanj2!\nmaxn\nj\r\u0011j(1+v)(1 +v);\r2\n\u0011(1\u0000v)o\ne\u00001\u0000v2\nLlnj\r\u0011jt:\nThis shows (4.2) and the theorem follows. \u000312 Decay estimates for an axially travelling string\nRemark 7. Ifv= 0in (4.2);then we get the decay estimate\n \n2\nLX\nn2Zj!nanj2!\ne\u00001\nLlnj\r\u0011jt\u0014E0(t)\u0014\r2\n\u0011 \n2\nLX\nn2Zj!nanj2!\ne\u00001\nLlnj\r\u0011jt;\nwhich is sharper then the estimate (1.7), stated in the introduction.\nRemark 8. The constants in estimation (4.2) are (at least) asymptotically sharp in the sense\nthat if\u0011!0, we recover the estimation (3.4) with its sharp constants, see Remark 5.\nThe next corollary compares Ev(t) to the initial energy Ev(0):\nCorollary 2. Under the assumptions (1.2) and (2.4), the energy of the solution of Problem\n(WP) satis\fes\nminn\n1 +v;j\r\u0011j(1+v)(1\u0000v)o\nmaxn\nj\r\u0011j(1+v)(1 +v);\r2\u0011(1\u0000v)oEv(0)e\u00001\u0000v2\nLlnj\r\u0011jt\u0014Ev(t)\n\u0014maxn\nj\r\u0011j(1+v)(1 +v);\r2\n\u0011(1\u0000v)o\nminn\n1 +v;j\r\u0011j(1+v)(1\u0000v)oEv(0)e\u00001\u0000v2\nLlnj\r\u0011jt;fort\u00150:(4.6)\nProof. Since (4.2) holds also for t= 0, then (4.6) follows by combining the two inequalities\ne1\u0000v2\nLlnj\r\u0011jt\nmaxn\nj\r\u0011j(1+v)(1 +v);\r2\u0011(1\u0000v)oEv(t)\u00142\nLX\nn2Zj!nanj2\n\u00141\nminn\n1 +v;j\r\u0011j(1+v)(1\u0000v)oEv(0)\nand\n1\nmaxn\nj\r\u0011j(1+v)(1 +v);\r2\u0011(1\u0000v)oEv(0)\u00142\nLX\nn2Zj!nanj2\n\u0014e1\u0000v2\nLlnj\r\u0011jt\nminn\n1 +v;j\r\u0011j(1+v)(1\u0000v)oEv(t);\nfort\u00150: \u0003\nThe next corollary gives more simple estimates, but less sharper then (4.2) and (4.6), for\nthe energy Ev:\nCorollary 3. Under the assumptions (1.2) and (2.4), the energy of the solution of Problem\n(WP) satis\fes\n(1\u0000v) \n2\nLX\nn2Zj!nanj2!\ne\u00001\u0000v2\nLlnj\r\u0011jt\u0014Ev(t)\n\u0014\r2\n\u0011(1 +v) \n2\nLX\nn2Zj!nanj2!\ne\u00001\u0000v2\nLlnj\r\u0011jt;fort\u00150 (4.7)S. Ghenimi and A. Sengouga 13\nand\n1\n\r2\u0011\rvEv(0)e\u00001\u0000v2\nLlnj\r\u0011jt\u0014Ev(t)\u0014\r2\n\u0011\rvEv(0)e\u00001\u0000v2\nLlnj\r\u0011jt;fort\u00150: (4.8)\nProof. Since 0\u0014v <1, then it su\u000eces to simplify the constants in (4.2) and (4.6) using the\nfact that 1\u0014j\r\u0011j\u0014j\r\u0011j(1+v)<\r2\n\u0011. \u0003\nRemark 9. This case with a dashpoint damping at the inlet bully can be easily investigated\nby replacing vby\u0000vin the above sections. In Corollary, 1 we have to change vbyjvj;i.e.\nEv(0)\n1 +jvj\u0014Ev(t)\u0014Ev(0)\n1\u0000jvjandEv(0)\n\rjvj\u0014Ev(t)\u0014\rjvjEv(0);fort\u00150:\nMore importantly, we still have the same exponential decay e\u00001\u0000v2\nLlnj\r\u0011jtwhen\u0011 > 0:The\nanalogue of estimations (4.7) is\n2 (1\u0000jvj)\nLX\nn2Zj!nanj2e\u00001\u0000v2\nLlnj\r\u0011jt\u0014Ev(t)\n\u00142\r2\n\u0011(1 +jvj)\nLX\nn2Zj!nanj2e\u00001\u0000v2\nLlnj\r\u0011jt;fort\u00150\nand\rvis replaced by \rjvjin (4.8).\nAcknowledgements\nThe authors have been supported by the General Direction of Scienti\fc Research and Tech-\nnological Development (Algerian Ministry of Higher Education and Scienti\fc Research) PRFU\n# C00L03UN280120220010.\nORCID. Abdelmouhcene Sengouga https://orcid.org/0000-0003-3183-7973.\nReferences\n[1] Marynowski K, Kapitaniak T. Dynamics of axially moving continua. Int J Mech Sci. 2014;81:26{41.\n[2] Hong KS, Pham PT. Control of axially moving systems: A review. Int J Control Autom Syst. 2019;\n17(12):2983{3008.\n[3] Chen LQ. Analysis and control of transverse vibrations of axially moving strings. Appl Mech Rev. 2005;\n58(2):91{116.\n[4] Miranker WL. The wave equation in a medium in motion. IBM J Res Develop. 1960;4(1):36{42.\n[5] Mote CD. A study of band saw vibrations. J Franklin Inst. 1965;279(6):430{444.\n[6] Tan CA, Ying S. Dynamic analysis of the axially moving string based on wave propagation. ASME J Appl\nMech. 1997;64(2):394{400.\n[7] Chen EW, Ferguson NS. Analysis of energy dissipation in an elastic moving string with a viscous damper\nat one end. J Sound Vib,. 2014;333(9):2556{2570.\n[8] Chen E, He Y, Zhang K, et al. A superposition method of re\rected wave for moving string vibration with\nnonclassical boundary. J Chin Inst Engineers. 2019;42(4):327{332.\n[9] Chen E, Yuan J, Ferguson N, et al. A wave solution for energy dissipation and exchange at nonclassical\nboundaries of a traveling string. Mech Syst Sig Process. 2021;150:107272.\n[10] Lee SY, Mote CD. Vibration Control of an Axially Moving String by Boundary Control. J Dyn Sys, Meas,\nControl. 1996;118(1):66{74.\n[11] Gaiko NV, van Horssen WT. On the transverse, low frequency vibrations of a traveling string with boundary\ndamping. J Vib Acoust. 2015;137(4).\n[12] Ghenimi S, Sengouga A. Free vibrations of axially moving strings: Energy estimates and boundary observ-\nability ; 2022. Submitted; http://arxiv.org/abs/2201.01866 .14 Decay estimates for an axially travelling string\n[13] Fung RF, Wu JW, Wu SL. Exponential stabilization of an axially moving string by linear boundary\nfeedback. Automatica. 1999;35(1):177{181.\n[14] Veseli\u0013 c K. On linear vibrational systems with one dimensional damping. Appl Anal. 1988;29(1-2):1{18.\n[15] Cox S, Zuazua E. The rate at which energy decays in a string damped at one end. Indiana Univ Math J.\n1995;:545{573.\n[16] Cherkaoui M. Estimation optimale du taux de d\u0013 ecroissance de l'\u0013 energie pour une \u0013 equation des ondes avec\ncontr^ ole fronti\u0012 ere. INRIA; 1994. Research Report RR-2328. https://hal.inria.fr/inria-00074346 .\n[17] Quinn JP, Russell DL. Asymptotic stability and energy decay rates for solutions of hyperbolic equations\nwith boundary damping. Proc R Soc Edinb A: Math. 1977;77(1-2):97{127.\n[18] Sengouga A. Exact boundary observability and controllability of the wave equation in an interval with two\nmoving endpoints. Evol Equ Control Theory. 2020;9(1):1{25.\nEmail address :amsengouga@gmail.com,abdelmouhcene.sengouga@univ-msila.dz\n(Seyf Eddine Ghenimi, Abdelmouhcene Sengouga) Laboratory of Functional Analysis and Geom-\netry of Spaces, Department of mathematics, Faculty of Mathematics and Computer Sciences,\nUniversity of M'sila, 28000 M'sila, Algeria."    },    {        "title": "2211.12238v1.Generalized_Random_Gilbert_Varshamov_Codes__Typical_Error_Exponent_and_Concentration_Properties.pdf",        "content": "arXiv:2211.12238v1  [cs.IT]  22 Nov 20221\nGeneralized Random Gilbert-Varshamov\nCodes: Typical Error Exponent and\nConcentration Properties\nLan V . Truong and Albert Guill´ en i F` abregas\nAbstract\nWe find the exact typical error exponent of constant composit ion generalized random Gilbert-Varshamov (RGV)\ncodes over DMCs channels with generalized likelihood decod ing. We show that the typical error exponent of the\nRGV ensemble is equal to the expurgated error exponent, prov ided that the RGV codebook parameters are chosen\nappropriately. We also prove that the random coding exponen t converges in probability to the typical error exponent,\nand the corresponding non-asymptotic concentration rates are derived. Our results show that the decay rate of the\nlower tail is exponential while that of the upper tail is doub le exponential above the expurgated error exponent. The\nexplicit dependence of the decay rates on the RGV distance fu nctions is characterized.\nI. I NTRODUCTION\nIntroduced by Shannon [1], random coding is the key techniqu e employed in information theory in order to show\nthat a code with low error probability exists without explic itly constructing it. Codes are constructed at random, and\nthe average error probability over all randomly generated c odes is bounded. Then, it follows that there must exist\na code with error probability at least as low as the ensemble a verage error probability over the codes. In particular,\nfor discrete memoryless channel (DMC), Shannon showed that there exists a code of rate smaller than the channel\ncapacity with vanishing probability of error as the codewor d length increases.\nSince Shannon’s work, random coding has not only been applie d extensively, but has been refined in a number\nof ways. For rates below capacity, Fano [2] characterized th e exponential decay of the error probability defining\nthe random coding exponent (RCE) as the negative normalized logarithm of the ensemble-average error probability.\nIn [3], Gallager derived the RCE in a simpler way and introduc ed the idea of expurgation in order to show the\nexistence of a code with an improved exponent the at low rates . An upper bound to the error exponent for the\nDMC, called sphere-packing bound, was first introduced in [4 ] and it was shown to coincide with the RCE for\nL. V . Truong is with the Department of Engineering, Universi ty of Cambridge, Cambridge CB2 1PZ, U.K. (e-mail: lt407@cam .ac.uk).\nA. Guill´ en i F` abregas is with the Department of Engineerin g, University of Cambridge, Cambridge CB2 1PZ, U.K. and the D epartment of\nInformation and Communication Technologies, Universitat Pompeu Fabra, Barcelona 08018, Spain (e-mail: guillen@iee e.org).\nThis work has been funded in part by the European Research Cou ncil under ERC grant agreement 725411 and by the Spanish Mini stry of\nEconomy and Competitiveness under grant PID2020-116683GB -C22.\nNovember 23, 2022 DRAFT2\nrates higher than a certain critical rate. Nakibo˘ glu in [5] recently derived sphere-packing bounds for some stationar y\nmemoryless channels using Augustin’s method [6].\nMost proofs invoking random coding arguments, assume that c odewords are independent. Random Gilbert-\nVarshamov (RGV) codes were first introduced in [7], and are a f amily of random codes inspired by the basic\nconstruction attaining the Gilbert-Varshamov bound for co des in Hamming spaces. The code construction is based\non drawing codewords recursively from a fixed type class, in s uch a way that a newly generated codeword must be\nat a certain minimum distance from all previously chosen cod ewords, according to some generic distance function.\nFor suitably optimized distance functions, RGV codes attai n Csisz´ ar and K¨ orner’s exponent [8], which is known\nto be at least as high as both the random-coding and expurgate d exponents.\nMost works on random coding and error exponents study the err or exponent of the ensemble-average error\nprobability. In [9], Barg and Forney studied the i.i.d. rand om coding over the binary symmetric channel (BSC) with\nmaximum likelihood decoding and showed that the error expon ent of most random codes is close to the so-called\ntypical random coding (TRC) exponent, strictly larger than the RCE at low rates. Upper and lower bounds on the\nTRC for constant-composition codes and general DMCs were pr ovided in [10]. For the same type of codes and\nchannels, Merhav [11] determined the exact TRC error expone nt for a wide class of stochastic decoders called\ngeneralized likelihood decoders (GLD), of which maximum-l ikelihood is a special case. Merhav derived the TRC\nexponent for spherical codes over coloured Gaussian channe ls [12] and for random convolutional code ensembles\n[13]. Merhav provided a dual expression of the TRC for i.i.d. codes in [14]. Tamir et al. [15] studied the upper and\nlower tails of the error exponent around the TRC exponent for random pairwise-independent constant-composition\ncodes with GLD. It was shown that the tails behave in a non-sym metric way: the lower tail decays exponentially\nwhile the upper tail decays doubly-exponentially; the latt er was first established for a limited range of rates in\n[16]. By studying the behavior of both tails, work in [15] pro ves concentration in probability. The TRC was\nshown to be universally achievable with a likelihood mutual -information decoder in [17]. For pairwise-independent\nensembles and arbitrary channels, Cocco et al. showed in [18] that the probability that a code in the ensembl e has\nan exponent smaller than a lower bound on the TRC exponent is v anishingly small. Recently, Truong et al. showed\nthat, for DMCs, the error exponent of a randomly generated co de with pairwise-independent codewords converges\nin probability to its expectation – the typical error expone nt [19]. For high rates, the result is a consequence of the\nfact that the random-coding error exponent and the sphere-p acking error exponent coincide. For low rates, instead,\nthe convergence is based on the fact that the union bound accu rately characterizes the probability of error. The\npaper also zooms into the behavior at asymptotically low rat es and shows that the error exponent converges in\ndistribution to Gaussian-like distributions.\nA. Contributions\nThis work focusses on the RGV code ensemble and discusses con centration properties of error exponents around\nits TRC. Compared with constant-composition codes, the dep endence among RGV codewords causes standard\nconcentration inequalities such as Hoeffding’s inequalit y not to hold. In this work, we develop new techniques to\novercome the challenges presented by RGV codeword dependen ce. Our main contributions include:\nNovember 23, 2022 DRAFT3\n•We find the exact typical error exponent TRC for the RGV ensemb le by proving matched upper and lower\nbounds on the TRC and show it is equal to its RCE, i.e., to the ma ximum of the expurgated and random-coding\nexponent.\n•We show that the random error exponent converges in probabil ity to the TRC.\n•We characterize the convergence rates of the above converge nce and show that it is exponential for the lower\ntail and double-exponential for the upper tail under some te chnical conditions.\nB. Notation\nRandom variables will be denoted by capital letters, and the ir realizations will be denoted by the corresponding\nlower case letters. Random vectors and their realizations w ill be denoted, respectively, by boldfaced capital and\nlower case letters. Their alphabets will be superscripted b y their dimensions. For a generic joint distribution PXY=\n{PXY(x,y),x∈ X,y∈ Y} , which will often be abbreviated by P, information measures will be denoted in the\nconventional manner, but with a subscript P, that isIP(X;Y)is the mutual information between XandY, and\nsimilarly for other quantities. Natural logarithms are ass umed in the derivations; examples will employ base 2. The\nprobability of an event Ewill be denoted by P(E), the indicator function of event Ewill be denoted by 1{E}, and\nthe expectation operator will be denoted by E[·]. The notation [t]+will stand for max{t,0}.\nFor two positive sequences, {an}and{bn}, the notation an.=bnwill stand for exponential equality, that is\nlimn→∞1\nnlog(an\nbn) = 0 . Exponential inequalities an.\n≤bnandan.\n≥bnare defined as limn→∞1\nnlog(an\nbn)≤0\nandlimn→∞1\nnlog(an\nbn)≥0, respectively. Accordingly, the notation an.=e−n∞means that andecays super-\nexponentialy. For two positive sequences, {an}and{bn}, whose elements are both smaller than one for all large\nenoughn, the notation an˚ =bnwill stand for double-exponential equality, that is\nlim\nn→∞1\nnlog/parenleftbigglogbn\nlogan/parenrightbigg\n= 0. (1)\nSimilarly, an˚≤bnmeans that\nlimsup\nn→∞1\nnlog/parenleftbigglogbn\nlogan/parenrightbigg\n≤0, (2)\nandan˚≥bnstands for\nliminf\nn→∞1\nnlog/parenleftbigglogbn\nlogan/parenrightbigg\n≥0. (3)\nA sequence of random variables {An}∞\nn=1converges to Ain probability, denoted as An(p)−→Aif for all δ >0\n[20, Sec. 2.2],\nlim\nn→∞P[|An−A|> δ] = 0. (4)\nThe empirical distribution, or type, of a sequence x∈ Xn, which will be denoted by ˆPx, is the vector of relative\nfrequencies, ˆPx(x), of each symbol x∈ X inx. The set of all possible empirical distributions of sequenc es of\nlengthnon alphabet Xis denoted by Pn(X). The joint empirical distribution of a pair of sequences, de noted by\nˆPxy, is similarly defined. The set of all possible joint empirica l distributions of sequences of length non alphabets\nXandYis denoted by Pn(X×Y). The type class of QX, denoted by T(QX), is the set of all vectors x∈ Xnwith\nNovember 23, 2022 DRAFT4\nˆPx=QX. The joint type class of PXY, denoted by T(PXY), is the set of pairs of sequences (x,y)∈ Xn×Yn\nwithˆPxy=QXY. In addition, we also define Q(QX)/defines/braceleftbig\nPXX′∈ Pn(X ×X) :PX=PX′=QX/bracerightbig\n.Finally,[M]\ndenotes the set {1,2,···,M}, and[M]2\n∗/defines{(m,m′)∈[M]2:m/\\e}atio\\slash=m′}.\nC. Structure of the Paper\nIn Section II, we introduce error probability and error expo nent of a RGV code. In Section III-A, we introduce\nthe generation of RGV random codebook ensembles. We also men tion about properties of RGV codes and type-\nnumerators in this section. We derive the typical error expo nent for the RGV in Section IV. Finally, we study\nconcentration properties of this ensemble in Section V. Pro ofs of the main results can be found in the corresponding\nsections while the proofs of auxiliary results can be found i n the Appendices.\nII. P RELIMINARIES\nWe assume that the RGV code Cn={x1,x2,...,xM} ∈ Xn,M=enRis employed for transmission over\na DMC channel with channel law W(y|x)forx∈ X,y∈ Y. More specifically, when the transmitter wishes to\nconvey a message m∈ {1,2,···,M}, it sends codeword xm= (xm,1,...,x m,n)∈ Xnover the channel. The\nchannel produces an output vector y= (y1,y2,...,y n)∈ Yn, according to\nW(y|xm) =n/productdisplay\ni=1W(yi|xm,i). (5)\nAt the decoder side, we assume that a GLD [21] is used to infer w hat the transmitted message was. The GLD\n[21] extends the likelihood decoder in [22] and [23], and is a stochastic decoder that randomly selects the message\nestimate ˆmaccording to the posterior probability distribution given the channel output yas follows\nPr(ˆm=m|y) =exp{ng(ˆPxm,y)}/summationtextM\nm=1exp/braceleftbig\nng(ˆPxm′,y)/bracerightbig, (6)\nwhereg(·), henceforth referred to as the decoding metric , is an arbitrary continuous function of a joint distributio n\nPXYonX ×Y . For\ng(PXY) =/summationdisplay\nx∈X/summationdisplay\ny∈YPXY(x,y)logW(y|x), (7)\nwe recover the ordinary likelihood decoder [23]. For\ng(PXY) =β/summationdisplay\nx∈X/summationdisplay\ny∈YPXY(x,y)logW(y|x), (8)\nβ≥0being a free parameter, we extend this to a parametric family of decoders, where βcontrols the skewness\nof the posterior [11]. In particular, β→ ∞ leads to the (deterministic) ML decoder. Other interesting choices are\nassociated with mismatched metrics,\ng(PXY) =β/summationdisplay\nx∈X/summationdisplay\ny∈YPXY(x,y)logW′(y|x), (9)\nW′being different from W, and\ng(PXY) =βIP(X;Y), (10)\nNovember 23, 2022 DRAFT5\nwhich is the stochastic version of the well-known universal maximum mutual information (MMI) decoder [24],\nwhich has been recently proven to be universal in a typical er ror exponent sense [25]. The MMI decoder is\napproached for β→ ∞ .\nThe average probability of error, associated with a given RG V code cnand the GLD, is given by\nPe(cn) =1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\ny∈YnW(y|xm)·exp{ng(ˆPxm′,y)}\n/summationtextM\n˜m=1exp{ng(ˆPx˜m,y)}. (11)\nThe error exponent of code cnis defined as\nEn(cn) =−1\nnlogPe(cn). (12)\nLetR= limn→∞1\nnlogMnbe the rate of the code in bits per channel use. An error expone ntE(R)is said to be\nachievable when there exists a sequence of codes {cn}∞\nn=1such that liminf n→∞En(cn)≥E(R). The channel\ncapacityCis the supremum of the code rates Rsuch that E(R)>0.\nIn the next sections, we introduce RGV codebook ensemble and derive concentration properties of the error\nexponent (12) of sequences of RGV codes Cnin the asymptotic regime.\nIII. RGV R ANDOM CODEBOOK ENSEMBLES AND PROPERTIES\nA. RGV Random Codebook Ensembles\nIn this section, we describe basic RGV codebook constructio n, channel model and GLD. The RGV codebook\nwas first introduced in [7], which extended code constructio ns that attain the Gilbert-Varshamov bound on the\nHamming space [26], [27]. The RGV construction is a randomiz ed constant composition counterpart of such codes\nfor arbitrary DMCs and arbitrary distance functions.\nDefinition 1: LetΩbe the set of bounded, continuous, symmetric, and type-depe ndent functions d(·,·) :Xn×\nXn→R, i.e., bounded functions that satisfy d(x,x′) =d(x′,x)for allx,x′∈Rn, that depend on (x,x′)only\nthrough the joint distribution ˆPxx′, and that are continuous on the probability simplex.\nWe refer to d∈Ωas a distance function, although it need not to be a distance i n the topological space (e.g.,\nit may be negative). Some examples of such distance function include Hamming distance, Bhattacharyya distance,\nand equivocation distance [7].\nThe RGV code Cn={x1,x2,...,xM} ∈ XnwithMcodewords of length nis constructed such that any two\ndistinct codewords x,x′∈ Cnsatisfyd(x,x′)>∆for a given distance function d(·,·)∈Ωand∆∈R. This\nguarantees that the minimum distance of the codebook exceed s the minimum distance ∆. The construction depends\non the input distribution QX∈ Pn(X)and is described by the following steps:\n1) The first codeword, x1, is drawn equiprobably from T(QX);\n2) The second codeword, x2, is drawn equiprobably from\nT(QX,x1)/defines/braceleftbig\n¯x∈ T(QX) :d(¯x,x1)>∆/bracerightbig\n(13)\n=T(QX)\\/braceleftbig\n¯x∈ T(QX) :d(¯x,x1)≤∆/bracerightbig\n, (14)\ni.e., the set of sequences with composition QXwhose distance to x1exceeds∆;\nNovember 23, 2022 DRAFT6\n3) Continuing recursively, the i-th codeword xiis drawn equiprobably from\nT(QX,xi−1\n1)/defines/braceleftbig\n¯x∈ T(QX) :d(¯x,xj)>∆,j= 1,2,...,i−1/bracerightbig\n(15)\n=T(QX,xi−2\n1)\\/braceleftbig\n¯x∈ T(QX,xi−2\n1) :d(¯x,xi−1)≤∆/bracerightbig\n(16)\nwhere for j < k ,xk\nj= (xj,...,xk)is a shorthand notation to denote previously drawn codeword s.\nFor a given RGV code with rate R, typeQX, distance function d, and minimum distance ∆, we define the\nrandom coding exponent (RCE) associated with the decoding m etricgas\nErgv\nrce(R,QX,d,∆)/defineslim\nn→∞−1\nnlogE[Pe(Cn)] (17)\nand the typical random coding (TRC) error exponent associat ed with the decoding metric gas\nErgv\ntrc(R,QX,d,∆)/defineslim\nn→∞−1\nnE[logPe(Cn)], (18)\nprovided that these limits exist1, where the expectation is with respect to the randomness of t he codeCn.\nLetQX∈ P(X),∆∈R,d∈Ω, and define the following quantity\nΓ(PXX′,R)/definesmin\nPY|XX′/braceleftBig\nD(PY|X/⌊ard⌊lW|QX)+IP(X′;Y|X)+[max{g(PXY),α(R,PY)}−g(PX′Y)]+/bracerightBig\n,(19)\nwhere\nα(R,PY)/definesmax\nPX′|Y:PX′=QX,\nIP(X′;Y)≤R/parenleftbig\ng(PX′Y)−IP(X′;Y)/parenrightbig\n+R. (20)\nThe expurgated error exponent for RGV ensemble is defined as\nErgv\nex(R,QX,d,∆)(R,QX,d,∆)/defines min\nPXX′∈Q(QX):d(PXX′)>∆,IP(X;X′)≤R/braceleftbigg\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbigg\n.(21)\nThe main result of [7] is that for ML decoding, and suitably op timized distance function and minimum distance,\nthe constant composition RGV ensemble attains a random codi ng exponent equal to the expurgated exponent, i.e.,\nErgv\nrce(R,QX,d,∆) =Eex(R,QX) (22)\nwhere\nEex(R,QX) = min\nPX′|X:IP(X;X′)≤R,PX′=QX{Γ(PXX′,R)+IP(X;X′)−R}. (23)\nis the expurgated exponent of the independent constant comp osition ensemble with composition QXand GLD.\nIn this paper, we study the TRC of the RGV ensemble Ergv\ntrc(R,QX,d,∆)as well as the concentration of\nthe exponent around the TRC. Specifically, we give a generic e xpression of Ergv\ntrc(R,QX,d,∆)and show that\nErgv\ntrc(R,QX,d,∆) =Eex(R,QX)for suitably optimized minimum distance and distance funct ions. In addition,\nwe provide bounds on the exponential and double-exponentia l concentration rates of the lower and upper tails of\nthe error exponent of RGV codes.\n1For the ensembles and channels considered in this paper, it w ill be shown in Section IV that these limits exist.\nNovember 23, 2022 DRAFT7\nB. Properties of RGV Codebooks\nIn this section, we introduce technical several results cha racterizing the key properties of the generalized RGV\nconstruction. We begin by restating some known properties f rom [7]; we will then introduce a number of other\nproperties that will be helpful in the derivation of our main results.\nLemma 1: [7, Lemma 1] Under the condition:\nR≤ min\nPXX′∈Q(QX):d(PXX′)≤∆I(X;X′)−2δ (24)\nfor some δ >0andxi−1\n1occurring with non-zero probability, we have that\n(1−e−nδ)|T(QX)| ≤ |T(QX,xi−1\n1)| ≤ |T(QX)|,∀i∈[M]. (25)\nLemma 2: [7, Lemma 2] Under the condition (24), for any k,m∈[M],k/\\e}atio\\slash=mandxk,xm∈ T(QX)such that\nd(xk,xm)>∆, then we have\n1−4δ2\nn\n|T(QX)|2e−2δn≤P[Xk=xk,Xm=xm]≤1\n(1−e−nδ)2|T(QX)|2(26)\nwhileP[Xk=xk,Xm=xm] = 0 whenever d(xk,xm)≤∆, where,\nδn/definese−nδ\n1−e−nδ. (27)\nLemma 3: [7, Lemma 4] For any message index m, the marginal distribution of codeword XmisP(xm) =\n1\n|T(QX)|forxm∈ T(QX).\nIn order to derive the TRC and convergence properties of the R GV code ensemble, we need to derive new\nproperties of this random codebook. Some properties of the p airwise independent fixed-composition code ensemble\n[11], [15] are proven to hold for the RGV codebook under some e xtra conditions by other proof techniques. First,\nthe following lemma can be easily proved using the same argum ents as [7].\nLemma 4: Consider the generalized RGV construction with the rate Rsatisfying (24). Then, for any A ⊂[M]\nandδ >0as chosen in [7], under the condition that mink,l∈A:k/\\⌉}atio\\slash=ld(xk,xl)>∆, it holds that\nP/bracketleftbigg/intersectiondisplay\nk∈A{Xk=xk}/bracketrightbigg\n≤1\n(1−e−nδ)|A||T(QX)||A|. (28)\nIn addition, if mink,l∈A:k/\\⌉}atio\\slash=ld(xk,xl)≤∆, it holds that\nP/bracketleftbigg/intersectiondisplay\nk∈A{Xk=xk}/bracketrightbigg\n= 0. (29)\nFurthermore, if mink,l∈[M′]:k/\\⌉}atio\\slash=ld(xk,xl)>∆for anyM′≤M, it holds that\nP/bracketleftbigg/intersectiondisplay\nm∈[M′]{Xm=xm}/bracketrightbigg\n≥1\n|T(QX)|M′. (30)\nIn general, (30) does not hold for any A ⊂[M]as (28), but it holds for the class of subsets {[M′]}M′≤M. Compared\nwith Lemma 2, (30) is tighter at M= 2 if{k,m}={1,2}. However, Lemma 2 is more general, i.e., it holds for\nany subset {k,m}: (k,m)∈[M]×[M], k/\\e}atio\\slash=m}.\nProof: See Appendix A.\nNovember 23, 2022 DRAFT8\nDenote by\nI(m,m′)/defines1{(xm,xm′)∈ T(PXX′)}. (31)\nThen, the following result, whose proof can be found in Appen dix B, holds.\nLemma 5: LetPXX′be a joint-type in Q(QX)such that d(PXX′)>∆. Define\nL(PXX′)/defines|T(PXX′)|\n|T(QX)|2. (32)\nThen, under the condition (24) and d(PXX′)>∆, for any two pairs (i,j),(k,l)∈[M]2\n∗such that (i,j)/\\e}atio\\slash= (k,l),\nit holds that\n(1−4δ2\nn)e−2δnL(PXX′)≤E[I(i,j)]≤1\n(1−e−nδ)2L(PXX′), (33)\nand\nE[I(i,j)I(k,l)]≤1\n(1−e−nδ)4L2(PXX′). (34)\nThis implies that\nE[I(i,j)].= exp{−nIP(X;X′)}, (35)\nE[I(i,j)I(k,l)].\n≤exp{−2nIP(X;X′)}. (36)\nC. Useful Properties of Type Enumerators\nIn this section, we state some important properties of the ty pe enumerator of RGV codebooks. For a given\njoint-type PXX′∈ Q(QX), the type enumerator N(PXX′)is defined as the number of codeword pairs with joint\ntypePXX′, i.e.,\nN(PXX′)/defines/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m1{(xm,xm′)∈ T(PXX′)} (37)\n=/summationdisplay\n(m,m′)∈[M]2∗I(m,m′), (38)\nwhereI(m,m′)is defined in (31).\nLemma 6: Fix arbitrary small positive numbers δ >0andε >0. LetPXX′∈ Q(QX)be a joint distribution\nthat satisfies IP(X;X′)<2R−εandd(PXX′)>∆. Define\nE(PXX′) =/braceleftbigg\nCn:N(PXX′)<(1−4δ2\nn)e−2δnexp{n[2R−IP(X;X′)−ε]}/bracerightbigg\n. (39)\nThen, for any rate Rsatisfying (24), it holds (as nsufficiently large) that\nP/bracketleftbig\nE(PXX′)/bracketrightbig\n≤1\n(1−e−nε/2)2/bracketleftbigge4δn\n/parenleftbig\n1−4δ2n/parenrightbig2/parenleftbig\n1−e−nδ/parenrightbig2e−nε/2+e4δn\n(1−4δ2n)2(1−e−nδ)4−1/bracketrightbigg\n→0 (40)\nasn→ ∞ for any fixed δ >0.\nProof: See Appendix C.\nNovember 23, 2022 DRAFT9\nLemma 7: Letε >0be given and assume that the condition (24) holds. Then, for a nyPXX′∈ Q(QX)such\nthatIP(X;X′)≤2Randd(PXX′)>∆,\nP/bracketleftbig\nN(PXX′)≥en(2R−IP(X;X′)+ε)/bracketrightbig˚≤exp/braceleftbig\n−en(2R−IP(X;X′)+ε)/bracerightbig\n(41)\n.\n≤e−n∞. (42)\nProof: See Appendix D.\nLemma 8: Letε >0be given. Then, for any PXX′∈ Q(QX)such that IP(X;X′)≥2R−εandd(PXX′)>∆\nsuch that the condition (24) holds,\nP/bracketleftbig\nN(PXX′)≥enε/bracketrightbig˚≤exp/braceleftbig\n−enε/bracerightbig\n(43)\n.\n≤e−n∞. (44)\nProof: See Appendix E.\nLemma 9: For anyPXX′∈ Q(QX)such that IP(X;X′)≥2Randd(PXX′)>∆such that the condition (24)\nholds, we have\nP/bracketleftbig\nN(PXX′)≥1/bracketrightbig.= exp{n(2R−IP(X;X′))}. (45)\nProof: See Appendix F.\nThe following lemma is a key result for showing the exponenti ally-decay of the lower tail decay.\nLemma 10: LetPXX′∈ Q(QX)such that d(PXX′)>∆. Then, under the condition (24), we have\nP/bracketleftbig\nN(PXX′)≥ens/bracketrightbig.=e−nE(R,PXX′,s)∀s∈R, (46)\nwhere\nE(R,P,s) =\n\n[IP(X;X′)−2R]+,[2R−IP(X;X′)]+> s\n+∞, [2R−IP(X;X′)]+< s. (47)\nProof: See a detailed proof in Appendix G.\nThe following lemma is a key enabling result to attain the dou ble-exponential bound for the concentration properties\nof the random coding exponent in the RGV codebook. As opposed to the independent fixed-composition ensemble\n[15], a direct application of Suen’s correlation inequalit y as [15, Proof of Lemma 2] does not give the double-\nexponential bound. More specifically, since all RGV codewor ds are correlated, the number of adjacent pairs of a\nfixed pair (m,m′)is nowe2nRwhich causes the term in [15, Eq. (B.18)] to be equal to 1. For the independent\nfixed-composition code ensemble, this term is enR.\nTo overcome this difficulty, we develop another proof for thi s lemma which is not based on the Suen’s correlation\ninequality. See Appendix H for a detailed proof.\nLemma 11: Letε >0andD ⊂ {PXX′∈ Q(QX) :d(PXX′)>∆}be given. Then, under the condition\nmin\nPXX′∈DIP(X;X′)−2δ≤R≤ min\nPXX′∈Q(QX):d(PXX′)≤∆IP(X;X′)−2δ, (48)\nNovember 23, 2022 DRAFT10\nor\nR≤min/braceleftbigg\nmin\nPXX′∈DIP(X;X′)− min\nPXX′∈Q(QX):d(PXX′)≤∆IP(X;X′), min\nPXX′∈Q(QX):d(PXX′)≤∆IP(X;X′)/bracerightbigg\n−2δ\n(49)\nfor some δ >0, we have\nmin\nPXX′∈DP/braceleftbigg\nN(PXX′)≤e−nεE[N(PXX′)]/bracerightbigg\n˚≤exp/braceleftbig\n−min/parenleftbig\nen(R−2δ),en(2R−minPXX′∈DIP(X;X′))/parenrightbig/bracerightbig\n.(50)\nObserve that for d(PXX′) =−IP(X;X′)and∆ =−(R+2δ), the condition (48) holds since\nmin\nPXX′∈DIP(X;X′)−2δ≤max\nPXX′:d(PXX′)>∆IP(X;X′)−2δ (51)\n= max\nPXX′:IP(X;X′)<−∆IP(X;X′)−2δ (52)\n<−(∆+2δ), (53)\nand\nmin\nPXX′:d(PXX′)≤∆IP(X;X′)−2δ= min\nPXX′:IP(X;X′)≥−∆IP(X;X′)−2δ (54)\n=−(∆+2δ). (55)\nHence, the double-exponential expression in (50) holds for this special distance dand∆. The condition (49) also\nholds for many other classes of distances dand different values of ∆.\nFinally, we state the following key lemma, whose proof can be found in Appendix I.\nLemma 12: Recall the definition of Ergv\nex(R,QX,d,∆)2in (21). Let\nA1=/braceleftbigg\nPXX′∈ Q(QX) :d(PXX′)>∆,IP(X;X′)>2R/bracerightbigg\n, (56)\nA2=/braceleftbigg\nPXX′∈ Q(QX) :d(PXX′)>∆,IP(X;X′)≤2R,Γ(PXX′,R−ε)+IP(X;X′)−R≤E0+ε/bracerightbigg\n,(57)\nand define\nF0/defines/intersectiondisplay\nPXX′∈A1∪A2/braceleftbig\nN(PXX′) = 0/bracerightbig\n. (58)\nUnder the conditions that R < Ergv\nex(R,QX,d,∆)and\nmin\nPXX′∈Q(QX):d(PXX′)≤∆IP(X;X′)≥ max\nPXX′∈Q(QX):d(PXX′)>∆IP(X;X′), (59)\nR≤ min\nPXX′∈Q(QX):d(PXX′)≤∆IP(X;X′)−2δ (60)\nfor some δ >0, it holds that\nP(F0)˚≥exp/braceleftbig\n−enmaxPXX′∈A2(2R−IP(X;X′)δ)/bracerightbig\n. (61)\n2Ergv\nex(R,QX,d,∆)is the expurgated error exponent of the RGV code.\nNovember 23, 2022 DRAFT11\nSimilarly to the preceeding discussion, setting d(PXX′)/defines−IP(X;X′), we obtain that\nmin\nPXX′∈Q(QX):d(PXX′)≤∆IP(X;X′)≥ −∆, (62)\nmax\nPXX′∈Q(QX):d(PXX′)>∆IP(X;X′)<−∆, (63)\nso (59) holds. For (60) being hold, it is required that R≤ −(∆+2δ).\nIn connection to Lemma 11, the proof of the related result in [ 15, Prep. 6] cannot be applied here since it uses\nthe Suen’s correlation inequality, i.e. [15, Fact 3]. Since all codewords in RGV ensemble are dependent, the number\nof adjacent nodes in the corresponding adjacency graph is to o big which makes this type of arguments invalid. To\novercome this difficulty, in Appendix I, we develop a new tech nique. However, the double-exponential constant in\n(61) is smaller than the one in [15, Prep. 6] for the fixed-comp osition code ensemble.\nIV. T YPICAL RANDOM CODING EXPONENT OF GILBERT -VARSHAMOV CODES\nIn this section, we show an expression for the TRC of the RGV co de ensemble. The expression, when optimized\nover the distance function d(·,·)and minimum distance ∆, recovers the expurgated exponent for the GLD proposed\nin [21]. The main result, proven in Section IV-A, is stated in the following.\nTheorem 1: LetQX∈ P(X),∆∈R,d∈Ω, and define the following quantity\nΓ(PXX′,R)/definesmin\nPY|XX′/braceleftBig\nD(PY|X/⌊ard⌊lW|QX)+IP(X′;Y|X)+[max{g(PXY),α(R,PY)}−g(PX′Y)]+/bracerightBig\n,(64)\nwhere\nα(R,PY)/definesmax\nPX′|Y:PX′=QX,\nIP(X′;Y)≤R/parenleftbig\ng(PX′Y)−IP(X′;Y)/parenrightbig\n+R. (65)\nThen, for any Rsatisfying the condition in (24), the typical random coding exponent of the RGV code ensemble\nwith the GLD is given by\nErgv\ntrc(R,QX,d,∆) = min\nPX′|X:PX′=QX,\nIP(X;X′)≤2R,d(PXX′)>∆/braceleftbig\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbig\n. (66)\nBefore proceeding with the proof of the result, some discuss ion is in order. Observe that if we remove the\nconstraint d(PXX′)>∆(i.e., no constraint on the distance between each codeword p air), the expression of the\nTRC for the RGV ensemble code in (66) becomes the TRC of the con stant composition code ensemble with\ncomposition QXunder GLD decoding in [11, Eq. (18)]. In addition, as shown be low, when the distance function\nd(·,·)is optimized, and ∆is chosen appropriately, the TRC expression (66) recovers t he expurgated bound in [11,\nTheorem 1] Eex(R,QX).\nThe following results are similar to ones in [7, Section IV].\nCorollary 1: Letε >0be given, and let R,P , andd∈Ωbe given. The TRC of the generalized RGV construction\nwith sufficiently small δ,d(PXX′) =−IP(X;X′),∆ =−(R+2δ), sufficiently large n, and GLD rule is at least\nas high as Eex(R,QX)−ε.\nNovember 23, 2022 DRAFT12\nProof: First, it is easy to see that the choices d(PXX′) =−IP(X;X′)and∆ =−(R+2δ)are valid for all\nRin the sense of satisfying the rate condition in (24) (see pro of of [7, Cor. 2]). Now, under the same choices, we\nhave\nErgv\ntrc(R,QX,d,∆)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nd(PXX′)=−IP(X;X′),∆=−(R+2δ)(67)\n= min\nPX′|X:PX′=QX,\nIP(X;X′)≤2R,IP(X;X′)≤R+2δ/braceleftbig\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbig\n(68)\n= min\nPX′|X:PX′=QX,\nIP(X;X′)≤R+2δ/braceleftbig\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbig\n. (69)\nThe result follows by taking δ→0and using the continuity of Ergv\ntrc(R,QX,d,∆)inR.\nThe following proposition reveals that the above choice of (d,∆)is a choice that maximizes the TRC given in\nTheorem 1.\nLemma 13: Under the setup of Theorem 1 with\nR≤ min\nPXX′∈Q(QX):d(PXX′)≤∆IP(X;X′)−2δ (70)\nfor some δ >0, we have\nErgv\ntrc(R,QX,d,∆)≤Ergv\ntrc(R,QX,d,∆)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nd(PXX′)=−IP(X;X′),∆=−(R+2δ). (71)\nProof: From (70), for all joint type PXX′∈ Q(QX)such that d(PXX′)≤∆, we have R+2δ≤IP(X;X′).\nHence, if R+2δ > IP(X;X′), it holds that d(PXX′)>∆. This means that\n/braceleftbigg\nPXX′∈ Q(QX) :IP(X;X′)< R+2δ/bracerightbigg\n⊂/braceleftbigg\nPXX′∈ Q(QX) :d(PXX′)>∆/bracerightbigg\n. (72)\nIt follows from (72) that for δsufficiently small,\nErgv\ntrc(R,QX,d,∆) = min\nPX′|X:PX′=QX,\nIP(X;X′)≤2R,d(PXX′)>∆/braceleftbigg\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbigg\n(73)\n≤ min\nPX′|X:PX′=QX,\nIP(X;X′)≤2R,IP(X;X′)<R+2δ/braceleftbigg\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbigg\n(74)\n= min\nPX′|X:PX′=QX,\nIP(X;X′)<R+2δ/braceleftbigg\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbigg\n(75)\n=Ergv\ntrc(R,QX,d,∆)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nd(PXX′)=−IP(X;X′),∆=−(R+2δ), (76)\nwhere (76) follows from the continuity of Ergv\ntrc(R,QX,d,∆)inRand (69).\nAs in [7], the choice d(PXX′) =−IP(X;X′)is universally optimal in maximizing the TRC in Theorem 1\n(subject to (24)), in the sense that it does not depend on the c hannel or input distribution.\nIn Fig. 1, we plot various error exponents for the Z-channel with crossover probability 0.001and letQX(0) =\nQX(1) = 1/2. This example was considered in [15], [21]. Specifically, fo r reference we plot the random coding\nexponent Er(R), the expurgated exponent Eex(R), and the TRC Etrc(T)for constant composition codes. For the\nRGV ensemble exponents, we choose d(PXX′) =−IP(X;X′)and∆ =−Rso as to achieve the largest possible\nNovember 23, 2022 DRAFT13\nexponents. We plot the corresponding random coding exponen tErgv\nrce(R)and its corresponding TRC Ergv\ntrc(R)and\nillustrate that they both coincide with the expurgated expo nentEex(R).\n0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .800.20.40.60.811.21.41.61.8\nRError ExponentsEex(R)\nEtrc(R)\nEmin\n0(R)\nEr(R)\nErgv\ntrc(R)\nErgv\nrce(R)\nFig. 1: Error Exponents for the Z-channels with crossover probability 0.001.\nA. Proof of Theorem 1\nThe proofs for both upper and lower bounds follow similar lin es to those in [11]. The main difference is the\ndependence among codeword induced by the RGV ensemble. In or der to analyze this dependence, we developed\nnew concentration inequalities and applied generalized ve rsions of Hoeffding’s inequality.\n1) Lower bound on TRC: First, we prove the following result.\nLemma 14: Recall the definition of α(R,PY)in (65). Fix an ε >0. For any m∈[M], let\nZm(y)/defines/summationdisplay\n˜m/\\⌉}atio\\slash=meng(ˆPX˜m,y). (77)\nand\nAm/defines{Zm(y)≤exp/braceleftbig\nnα(R−ε,ˆPy)}}. (78)\nThen, under the condition (24), it holds that\nP/bracketleftbig\nAm/bracketrightbig.\n≤exp/braceleftbigg\n−enε/bracketleftbigg\n1−e−n(ε+δ)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg\n,∀m∈[M]. (79)\nProof: See Appendix K.\nNovember 23, 2022 DRAFT14\nProposition 1: Under the same assumptions as Theorem 1, the RGV code ensembl e satisfies\nErgv\ntrc(R,QX,d,∆)≥ min\nPXX′∈Q(QX),\nIP(X;X′)≤2R,d(PXX′)>∆/braceleftbigg\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbigg\n. (80)\nProof: Using the GLD, the error probability is\nPe(Cn) =1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\ny∈YnW(y|xm)exp{ng(ˆPxm′,y)}\nexp{ng(ˆPxm,Y)}+/summationtext\n˜m/\\⌉}atio\\slash=mexp{ng(ˆPx˜m,y)}. (81)\nFrom (81), we obtain\nE/bracketleftbig\nPe(Cn)/bracketrightbig\n≤E/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nyW(y|Xm)/summationdisplay\nm′/\\⌉}atio\\slash=mmin/braceleftbigg\n1,eng(ˆPXm′,y)\neng(ˆPXm,y)+/summationtext\n˜m/\\⌉}atio\\slash=meng(ˆPX˜m,y)/bracerightbigg/bracketrightbigg\n(82)\n=E/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nyW(y|Xm)/summationdisplay\nm′/negationslash=m\nd(Xm,Xm′)>∆min/braceleftbigg\n1,eng(ˆPXm′,y)\neng(ˆPXm,y)+/summationtext\n˜m/\\⌉}atio\\slash=meng(ˆPX˜m,y)/bracerightbigg/bracketrightbigg\n, (83)\nwhere (83) follows from the fact that minm′/\\⌉}atio\\slash=md(xm,xm′)>∆for any code Cn= (x1,x2,···,xM)in the RGV\ncodebook ensemble.\nNow, we use similar arguments as [11] with some changes to coo perate the condition d(xm,x˜m)>∆in the\nsum in (83). From (83) and Lemma 14, for any ε >0, we obtain\nE[Pe(Cn)]≤E/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nyW(y|Xm)/summationdisplay\nm′/negationslash=m\nd(Xm,Xm′)>∆min/braceleftbigg\n1,eng(ˆPXm′,y)\neng(ˆPXm,y)+enα(R−ε,ˆPy)/bracerightbigg/bracketrightbigg\n. (84)\nFrom the method of types [28] we have that\nW(y|x˜m) =e−n/bracketleftbig\nH(ˆPx˜m,y)−H(QX)+D/parenleftbig\nˆPx˜m,y/⌊ar⌈⌊lQX×W/parenrightbig/bracketrightbig\n. (85)\nThus, it follows from (85) that\nM/summationdisplay\nm=1/summationdisplay\nyW(y|xm)/summationdisplay\nm′/negationslash=m\nd(xm,xm′)>∆min/braceleftbigg\n1,eng(ˆPxm′,y)\neng(ˆPxm,y)+enα(R−ε,ˆPy)/bracerightbigg\n(86)\n.=M/summationdisplay\nm=1/summationdisplay\nyW(y|xm)/summationdisplay\nm′/negationslash=m\nd(xm,xm′)>∆exp/braceleftBig\n−n/bracketleftbig\nmax{g(ˆPxm,y),α(R−ε,ˆPy)}−g(ˆPxm′,y)/bracketrightbig\n+/bracerightBig\n(87)\n=M/summationdisplay\nm=1/summationdisplay\ny/summationdisplay\nm′/negationslash=m\nd(xm,xm′)>∆exp/braceleftbig/parenleftbig\n−n/bracketleftbig\nH(ˆPxm,y)−H(QX)+D/parenleftbigˆPxm,y/⌊ard⌊lQX×W/parenrightbig/bracketrightbig/parenrightbig/bracerightbig\n×exp/braceleftBig\n−n/bracketleftbig\nmax{g(ˆPxm,y),α(R−ε,ˆPy)}−g(ˆPxm′,y)/bracketrightbig\n+/bracerightBig\n(88)\n.=/summationdisplay\nPXX′∈Q(QX):d(PXX′)>∆N(PXX′)/summationdisplay\nPY|XX′exp/braceleftbig\nnHP(Y|XX′)/bracerightbig\n×exp/braceleftBig/parenleftbig\n−n/bracketleftbig\nH(PXY)−H(QX)+D/parenleftbig\nPXY/⌊ard⌊lQX×W/parenrightbig/bracketrightbig/parenrightbig/bracerightBig\n×exp/braceleftBig\n−n[max{g(PXY),α(R−ε,PY)}−g(PX′Y)]+/bracerightBig\n(89)\n.=/summationdisplay\nPXX′∈Q(QX):d(PXX′)>∆N(PXX′)exp/braceleftBig\n−nmin\nPY|XX′/parenleftBig\n−HP(Y|XX′)+H(PXY)−H(QX)\nNovember 23, 2022 DRAFT15\n+D/parenleftbig\nPXY/⌊ard⌊lQX×W/parenrightbig\n+/bracketleftbig\nmax{g(PXY),α(R−ε,PY)}−g(PX′Y)/bracketrightbig\n+/parenrightBig/bracerightBig\n(90)\n.=/summationdisplay\nPXX′∈Q(QX):d(PXX′)>∆N(PXX′)exp/braceleftBig\n−nmin\nPY|XX′/parenleftbig\nD(PY|X/⌊ard⌊lW|QX)+IP(X′;Y|X)\n+/bracketleftbig\nmax{g(PXY),α(R−ε,PY)}−g(PX′Y)/bracketrightbig\n+/parenrightbig/bracerightBig\n(91)\n=/summationdisplay\nPXX′∈Q(QX):d(PXX′)>∆N(PXX′)exp{−nΓ(PXX′,R−ε)}, (92)\nwhere (88) follows from (85), and (92) follows from (64), whe re the joint type enumerator N(PXX′)has been\ndefined in (38). From (84), (92), and (38), we obtain\nE[logPe(Cn)]≤log/parenleftBig\nE/bracketleftbig\nPe(Cn)/bracketrightbig/parenrightBig\n(93)\n≤log/parenleftbigg/summationdisplay\nPXX′∈Q(QX):d(PXX′)>∆E/bracketleftbig\nN(PXX′)/bracketrightbig\nexp/braceleftbig\n−nΓ(PXX′,R)/bracerightbig/parenrightbigg\n−nR, (94)\nwhere (93) follows from the concavity of logxin(0,∞)and Jensen’s inequality.\nNow, for any PXX′∈ Q(QX)such that d(PXX′)>∆, from Lemma 5,we obtain\nE/bracketleftbig\nN(PXX′)/bracketrightbig\n=M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=mP/bracketleftbig\n(Xm,Xm′)∈ T(PXX′)/bracketrightbig\n(95)\n.=en(2R−IP(X;X′)). (96)\nHence, from (94) and (96), we obtain\nE[logPe(Cn)].\n≤log/parenleftbigg/summationdisplay\nPXX′∈Q(QX):d(PXX′)>∆en(2R−IP(X;X′))exp/braceleftbig\n−nΓ(PXX′,R)/bracerightbig/parenrightbigg\n−nR. (97)\nFrom (97), we finally have\nErgv\ntrc(R,QX,d,∆)≥ min\nPXX′:PX′=PX,\nIP(X;X′)≤2R,d(PXX′)>∆/braceleftbigg\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbigg\n. (98)\nThis concludes the proof of Proposition 1.\n2) Upper bound on TRC:\nProposition 2: Under the same assumptions as Theorem 1, the RGV code ensembl e satisfies\nErgv\ntrc(R,QX,d,∆)≤ min\nPXX′∈Q(QX):\nIP(X;X′)≤2R,d(PXX′)>∆/braceleftbigg\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbigg\n. (99)\nProof: The following proof follows similar lines to the proof in [11 , Sect. 5.2]. However, the same proof cannot\nbe used for the RGV ensemble. In addition to the difference in proofs of Lemmas 6 and (139), we also need to\nmake additional changes in since the decay rate of P[E(PXX′)]in Lemma 6 is not exponential as [11, Eq. (48)].\nGiven a joint-type PXX′∈ Q(QX)such that IP(X;X′)<2R−εandd(PXX′)>∆, let us define\nZmm′(y) =/summationdisplay\n˜m/\\⌉}atio\\slash=m,m′exp{ng(ˆPX˜m,y)}, (100)\nNovember 23, 2022 DRAFT16\nand\nGn(PY|XX′) =/braceleftbigg\nCn:/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=mI(m,m′)/summationdisplay\ny∈T(PY|XX′)1{Zmm′(y)≤exp{n[α(R+2ε,PY)+ε]}}\n≥(1−4δ2\nn)e−2δnexp{n[2R−IP(X;X′)−3ε/2]}.|T(PY|XX′)|/bracerightbigg\n. (101)\nwhereI(m,m′)is defined in (31). Recall the definition of E(PXX′)in Eq. (39) Lemma 6. Then, similarly to [11,\nSect. 5.2] we have\nP/bracketleftbig\nGc\nn(PY|XX′)∩Ec(PXX′)/bracketrightbig\n≤P/bracketleftbigg/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=mI(m,m′)/summationdisplay\ny∈T(PY|XX′)1{Zmm′(y)≤(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}}\n≤exp{n[2R−IP(X;X′)−3ε/2]}.|T(PY|XX′)|,\nN(PXX′)≥(1−4δ2\nn)e−2δnexp{n[2R−IP(X;X′)−ε]}/bracketrightbigg\n(102)\n≤P/bracketleftbigg/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=mI(m,m′)/summationdisplay\ny∈T(PY|XX′)1{Zmm′(y)>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}}\n≥/parenleftbig\nexp{n[2R−IP(X;X′)−ε]}−exp{n[2R−IP(X;X′)−3ε/2]}/parenrightbig\n.|T(PY|XX′)|,\nN(PXX′)≥(1−4δ2\nn)e−2δnexp{n[2R−IP(X;X′)−ε]}/bracketrightbigg\n(103)\n≤P/bracketleftbigg/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=mI(m,m′)/summationdisplay\ny∈T(PY|XX′)1{Zmm′(y)>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PXY)+ε]}}\n≥/parenleftbig\nexp{n[2R−IP(X;X′)−ε]}−exp{n[2R−IP(X;X′)−3ε/2]}/parenrightbig\n.|T(PY|XX′)|/bracketrightbigg\n(104)\n.\n≤E/bracketleftbig/summationtext\nm/summationtext\nm′/\\⌉}atio\\slash=mI(m,m′)/summationtext\ny∈T(PY|XX′)1{Zmm′(y)>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}}/bracketrightbig\nexp{n[2R−IP(X;X′)−ε]}|T(PY|XX′)|\n(105)\n=/summationtext\nm/summationtext\nm′/\\⌉}atio\\slash=m/summationtext\ny∈T(PY|XX′)ζ(m,m′,y)\nexp{n[2R−IP(X;X′)−ε]}|T(PY|XX′)|, (106)\nwhere (105) follows from Markov’s inequality and\nζ(m,m′,y)/definesP/bracketleftbig\n(Xm,Xm′)∈ T(PXX′),Zmm′(y)>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}/bracketrightbig\n(107)\n=/summationdisplay\n(xm,xm′)∈T(PXX′):\nd(xm,xm′)>∆P(xm,xm′)\n×P/bracketleftbig\nZmm′(y)>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n(108)\nwhere (108) follows from the fact that P(xm,xm′) = 0 ifd(xm,xm′)<∆by Lemma 2.\nNow, given a fixed pair (xm,xm′)such that d(xm,xm′)>∆, define\nP∗\nX′|Y/definesargmax\nPX′|YP/bracketleftbig\nN(PX′|Y)>(n+1)−|X||Y|(1−4δ2\nn)e−2δn\n×exp{n[α(R+2ε,PY)+ε−g(PX′Y)]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n, (109)\nNovember 23, 2022 DRAFT17\nwhere\nN(PX′|Y) :=/summationdisplay\n˜m/\\⌉}atio\\slash=m,m′1/braceleftbig\n(X˜m,y)∈ T(P˜XY)/bracerightbig\n. (110)\nThen, we have\nP/bracketleftbig\nZmm′(y)>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n=P/bracketleftbigg/summationdisplay\n˜m/\\⌉}atio\\slash=m,m′exp{ng(ˆPX˜m,y)}\n>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbigg\n(111)\n≤P/bracketleftbigg/summationdisplay\nPX′|YN(PX′|Y)exp{ng(PX′Y)}\n>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbigg\n(112)\n=P/bracketleftbigg/summationdisplay\nPX′|YN(PX′Y)exp{ng(PX′Y)}\n>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbigg\n(113)\n.= max\nPX′|YP/bracketleftbig\nN(PX′|Y)exp{ng(PX′Y)}\n>(n+1)−|X||Y|(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n(114)\n= max\nPX′|YP/bracketleftbig\nN(PX′|Y)>(n+1)−|X||Y|(1−4δ2\nn)e−2δn\n×exp{n[α(R+2ε,PY)+ε−g(PX′Y)]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n(115)\n=P/bracketleftbig\nN(P∗\nX′|Y)>(n+1)−|X||Y|(1−4δ2\nn)e−2δn\n×exp{n[α(R+2ε,PY)+ε−g(P∗\nX′Y)]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n, (116)\nwhere (116) follows from (109).\nNow, for all ˜m∈[M], observe that\nP/bracketleftbig\n(X˜m,y)∈ T(PX′Y)/bracketrightbig\n=/summationdisplay\nx˜m∈T(PX′|Y)P(x˜m) (117)\n=|T(PX′|Y)\n|T(QX)|(118)\n:=p, (119)\nwhere (118) follows from Lemma [7, Lemma 4]. It is easy to see t hatpdoes not depend on ˜m.\nNow, we consider two cases:\nNovember 23, 2022 DRAFT18\nCase 1:IP∗(X′;Y)≤R+2ε. Then, we have\nα(R+2ε,PY)+ε−g(P∗\nX′Y) = max\nPX′|Y:PX′=QX,\nIP(X′;Y)≤R+2ε/parenleftbig\ng(PX′Y)−IP(X′;Y)/parenrightbig\n+R+2ε−g(P∗\nX′Y) (120)\n≥g(P∗\nX′Y)−IP∗(X′;Y)+R+2ε−g(P∗\nX′Y) (121)\n=R+2ε−IP∗(X′;Y). (122)\nOn the other hand, if we let\nγ/definesp\n1−e−nδ, (123)\nwe have\n(M−2)γ.=en(R−IP∗(X′;Y))\n1−e−nδ(124)\nIt follows that\nP/bracketleftbig\nN(P∗\nX′Y)>(n+1)−|X||Y|(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε−g(P∗\nX′Y)]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n.\n≤P/bracketleftbig\nN(P∗\nX′Y)>(M−2)γe2nε/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n(125)\nwhere the last step follows from (124) and (122). Now, let Z˜m/defines1{(X˜m,y)∈ T(PX′Y)}. Then, for all A ⊂\n[M]\\{m,m′}, under the condition (24), by Lemma 4, it holds that\nE/bracketleftbigg/productdisplay\n˜m∈AZ˜m/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbigg\n=/summationdisplay\nx1,x2,···,x|A|P/bracketleftbigg/intersectiondisplay\n˜m∈A{X˜m=x˜m}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbigg/productdisplay\n˜m∈A1{(x˜m,y)∈ T(PX′Y)}. (126)\nNow, observe that\nP/bracketleftbigg/intersectiondisplay\n˜m∈A{X˜m=x˜m}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbigg\n=P/parenleftbig/intersectiontext\n˜m∈A∪{m,m′}{X˜m=x˜m}/parenrightbig\nP/parenleftbig\nXm=xm,Xm′=xm′/parenrightbig (127)\n≤1\n(1−e−δn)|A|+2/parenleftbigg1\n|T(QX)||A|+2/parenrightbigg|T(QX)|2\n1−4δ2ne2δn(128)\nwhere (128) follows from Lemma 4 and Lemma 2 with noting that d(xm,xm′)>���.\nNovember 23, 2022 DRAFT19\nHence, it holds that\nE/bracketleftbigg/productdisplay\n˜m∈AZ˜m/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbigg\n≤/parenleftbigge2δn\n1−4δ2n/parenrightbigg/parenleftbigg1\n(1−e−δn)|A|+2/parenrightbigg/summationdisplay\nx1,x2,···,x|A|/productdisplay\n˜m∈AP[X˜m=x˜m]/productdisplay\n˜m∈A1{(x˜m,y)∈ T(PX′Y)} (129)\n=/parenleftbigge2δn\n1−4δ2n/parenrightbigg/parenleftbigg1\n(1−e−δn)|A|+2/parenrightbigg/productdisplay\n˜m∈A/summationdisplay\nx˜mP[X˜m=x˜m]1{(x˜m,y)∈ T(PX′Y)} (130)\n=/parenleftbigge2δn\n1−4δ2n/parenrightbigg/parenleftbigg1\n(1−e−δn)|A|+2/parenrightbigg/productdisplay\n˜m∈AP/bracketleftbig\n(X˜m,y)∈ T(PX′Y)/bracketrightbig\n(131)\n.=/parenleftbiggp\n1−e−δn/parenrightbigg|A|\n, (132)\nwhere (129) follows from Lemma 4 (under the condition (24)). Hence, by applying Lemma 20, we have\nP/bracketleftbig\nN(P∗\nX′Y)>(M−2)γe2nε/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig.\n≤exp/braceleftbigg\n−enRD(e−na/⌊ard⌊le−nb)/bracerightbigg\n(133)\nwhereD(p/⌊ard⌊lq)is the relative entropy between two Bernouilli distributio ns, with success probability p,q, respectively,\nanda/definesIP∗(X′;Y)−2ε+(1/n)log(1−e−nδ)andb/definesIP∗(X′;Y)+(1/n)log(1−e−nδ). Sinceb−a= 2ε, by\nusing the following fact [29, Sec. 6.3]:\nD(a/⌊ard⌊lb)≥aloga\nb+b−a, (134)\nwe have\nD/parenleftbig\ne−an/⌊ard⌊le−bn/parenrightbig\n≥e−bn/bracketleftbig\n1+e(b−a)n((b−a)n−1)/bracketrightbig\n(135)\n.=e−nIP∗(X′;Y)e2nε2nε. (136)\nFrom (133) and (136), for any pair (xm,xm′)such that d(xm,xm′)>∆, we obtain\nP/bracketleftbig\nN(P∗\nX′Y)>(M−2)γe2nε/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig.\n≤exp/braceleftbigg\n−en(R−IP∗(X′;Y))e2nε2nε/bracerightbigg\n(137)\n≤exp/braceleftbigg\n−e−2nεe2nε2nε/bracerightbigg\n(138)\n= exp/braceleftbig\n−2nε/bracerightbig\n, (139)\nwhere (138) follows from the condition IP∗(X′;Y)≤R+2ε.\nCase 2:IP∗(X′;Y)> R+2ε. For this case, for any pair (xm,xm′)such that d(xm,xm′)>∆, we have\nP/bracketleftbig\nN(P∗\nX′Y)>(n+1)−|X||Y|(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε−g(P∗\nX′Y)]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n≤P/bracketleftbig\nN(P∗\nX′Y)≥1/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n(140)\n≤E[N(P∗\nX′Y)/vextendsingle/vextendsingleXm=xm,Xm′=xm′] (141)\n=/summationdisplay\n˜m/\\⌉}atio\\slash=m,m′P/parenleftbig\n(X˜m,y)∈ T(PX′Y)/vextendsingle/vextendsingleXm=xm,Xm′=xm′/parenrightbig\n, (142)\nwhere (140) follows from the fact that N(P∗\nX′Y)∈Z+, and (141) follows from the Markov’s inequality.\nNovember 23, 2022 DRAFT20\nNow, by using (128) with A={˜m}, we have\nP/bracketleftbigg\n{X˜m=x˜m}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbigg\n≤1\n(1−e−δn)3/parenleftbigg1\n|T(QX)|3/parenrightbigg|T(QX)|2\n1−4δ2ne2δn(143)\n.=1\n|T(QX)|. (144)\nFrom (142) and (144), for any pair (xm,xm′)such that d(xm,xm′)>∆, we obtain\nP/bracketleftbig\nN(P∗\nX′Y)>(n+1)−|X||Y|(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε−g(P∗\nX′Y)]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n.\n≤(M−2)p (145)\n.=en(R−IP∗(X′;Y))(146)\n≤e−2nε, (147)\nwhere (146) follows from (119), and (147) follows from condi tionIP∗(X′;Y)> R+2ε.\nFrom (139) and (147), for any pair (xm,xm′)such that d(xm,xm′)>∆, we have\nP/bracketleftbig\nN(P∗\nX′Y)>(n+1)−|X|(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε−g(P∗\nX′Y)]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig\n≤e−2nε. (148)\nFrom (116) and (148), we obtain\nP/bracketleftbig\nZmm′(y)>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}/vextendsingle/vextendsingleXm=xm,Xm′=xm′/bracketrightbig.\n≤e−2nε(149)\nwhere the constant in.\n≤does not depends on xm,xm′.\nIt follows from (108) and (149) that\nP/bracketleftbig\n(Xm,Xm′)∈ T(PXX′),Zmm′(y)>(1−4δ2\nn)e−2δnexp{n[α(R+2ε,PY)+ε]}/bracketrightbig\n.\n≤/summationdisplay\n(xm,xm′)∈T(PXX′)\nd(xm,xm′)>∆P(xm,xm′)e−2nε(150)\n≤/summationdisplay\n(xm,xm′)∈T(PXX′)\nd(xm,xm′)>∆/parenleftbigg1\n1−e−nδ/parenrightbigg21\n|T(QX)|2e−2nε(151)\n.\n≤e−nIP(X;X′)e−2nε, (152)\nwhere (151) follows from Lemma 4. By combining (106) and (152 ), we obtain\nP/bracketleftbig\nGc\nn(PY|XX′)∩Ec(PXX′)/bracketrightbig.\n≤e−2nε. (153)\nOn the other hand, by Lemma 6, we also have\nPr[Ec(PXX′)]→1. (154)\nNow, for any fixed joint-type PXX′∈ Q(QX)such that IP(X;X′)<2R−ε, define\nFn(PXX′)/defines/intersectiondisplay\nPY|XX′/braceleftbig\nGn(PY|XX′)∩Ec(PXX′)/bracerightbig\n. (155)\nNovember 23, 2022 DRAFT21\nThen, from (153) and (154), for any fixed joint-type PXX′∈ Q(QX)such that IP(X;X′)<2R−ε, we have\nP[Fc\nn(PXX′)] =P/bracketleftbigg/uniondisplay\nPY|XX′{Gc\nn(PY|XX′)∩Ec(PXX′)}∪E(PXX′)/bracketrightbigg\n(156)\n≤P/bracketleftbigg/uniondisplay\nPY|XX′{Gc\nn(PY|XX′)∩Ec(PXX′)}/bracketrightbigg\n+P[E(PXX′)] (157)\n≤/summationdisplay\nPY|XX′P/bracketleftbig\nGc\nn(PY|XX′)∩Ec(PXX′)/bracketrightbig\n+P[E(PXX′)] (158)\n.\n≤ |T(PY|XX′)|e−2nε+o(1) (159)\n→0, (160)\nwhich leads to P[Fn(PXX′)]→1asn→ ∞ .\nNow, for a given code Cn∈ Fn(PXX′), define\nV(Cn,PY|XX′) ={(m,m′,y) :Zmm′(y)≤exp[n(α(R+2ε,PY)+ε)]}, (161)\nand\nVm,m′(Cn,PY|XX′) ={y: (m,m′,y)∈ V(Cn,PY|XX′)}. (162)\nThen, by definition of Gn(PY|XX′)in (101), for any fixed joint type PXX′∈ Q(QX)such that IP(X;X′)<2R−ε\nandd(PXX′)>∆, and for any Cn∈ Fn(PXX′), it holds that\n/summationdisplay\nm,m′1{(xm,xm′)∈ T(PXX′)}|T(PY|XX′)∩Vm,m′(Cn,PY|XX′)|\n|T(PY|XX′)|\n≥(1−4δ2\nn)e−2δnexp/bracketleftbig\nn(2R−IP(X;X′)−3ε/2)/bracketrightbig\n. (163)\nNow, let\nP∗\nXX′:= argmin\nPXX′:PX′=PX,\nIP(X;X′)≤2R,d(PXX′)>∆/braceleftbig\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbig\n. (164)\nThen, for any ρ >1, we have\nE[(Pe(Cn))1/ρ]\n=E/bracketleftbigg/parenleftbigg1\nM/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\nyW(y|Xm)exp{ng(ˆPXm′y)}\nexp{ng(ˆPXmy)}+exp{ng(ˆPXm′y)}+Zmm′(y)/parenrightbigg1/ρ/bracketrightbigg\n(165)\n=/summationdisplay\nCnP[Cn]/parenleftbigg1\nM/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\nyW(y|xm)exp{ng(ˆPxm′y)}\nexp{ng(ˆPxmy)}+exp{ng(ˆPxm′y)}+Zmm′(y)/parenrightbigg1/ρ\n(166)\n=/summationdisplay\nCnP[Cn]/parenleftbigg1\nM/summationdisplay\nPXX′∈Q(QX)/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m1{(xm,xm′)∈ T(PXX′)}\n×/summationdisplay\nPY|XX′/summationdisplay\ny∈T(PY|XX′)W(y|xm)exp{ng(ˆPxm′y)}\nexp{ng(ˆPxmy)}+exp{ng(ˆPxm′y)}+Zmm′(y)/parenrightbigg1/ρ\n(167)\n=/summationdisplay\nCnP[Cn]/parenleftbigg1\nM/summationdisplay\nPXX′∈Q(QX)/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m1{(xm,xm′)∈ T(PXX′)}\nNovember 23, 2022 DRAFT22\n×/summationdisplay\nPY|XX′/summationdisplay\ny∈T(PY|XX′)W(y|xm)exp{ng(ˆPxm′y)}\nexp{ng(ˆPxmy)}+exp{ng(ˆPxm′y)}+Zmm′(y)/parenrightbigg1/ρ\n(168)\n≥/summationdisplay\nCn∈Fn(P∗\nXX′)P[Cn]/parenleftbigg1\nM/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m1{(xm,xm′)∈ T(P∗\nXX′)}\n×/summationdisplay\nPY|XX′/summationdisplay\ny∈T(PY|XX′)∩Vm,m′(Cn,PY|XX′)W(y|xm)exp{ng(ˆPxm′y)}\nexp{ng(ˆPxmy)}+exp{ng(ˆPxm′y)}+Zmm′(y)/parenrightbigg1/ρ\n(169)\n.=/summationdisplay\nCn∈Fn(P∗\nXX′)P[Cn]/parenleftbigg1\nM/summationdisplay\nPY|XX′/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m1{(xm,xm′)∈ T(P∗\nXX′)}\n×|T(PY|XX′)∩Vm,m′(Cn,PY|XX′)|\n|T(PY|XX′)|exp/braceleftbigg\n−n[D(PY|X/⌊ard⌊lW|QX)]+IP(X′;Y|X)\n+[max{g(PXY),α(R+2ε,PY)+ε}−g(PX′Y)]+/bracerightbigg/parenrightbigg1/ρ\n(170)\n≥/summationdisplay\nCn∈Fn(P∗\nXX′)P[Cn]/parenleftbigg1\nM/summationdisplay\nPY|XX′(1−4δ2\nn)e−2δnexp/bracketleftbig\nn(2R−IP∗(X;X′)−3ε/2)/bracketrightbig\n×exp/braceleftbigg\n−n[D(PY|X/⌊ard⌊lW|QX)]+IP(X′;Y|X)+[max{g(PXY),α(R+2ε,PY)+ε}−g(PX′Y)]+/bracerightbigg/parenrightbigg1/ρ\n(171)\n.=P[Fc\nn(P∗\nXX′)]/parenleftbigg/summationdisplay\nPY|XX′(1−4δ2\nn)e−2δnexp/bracketleftbig\nn(R−IP∗(X;X′)−3ε/2)/bracketrightbig\n×exp/braceleftbigg\n−n[D(PY|X/⌊ard⌊lW|QX)]+IP(X′;Y|X)+[max{g(PXY),α(R+2ε,PY)+ε}−g(PX′Y)]+/bracerightbigg/parenrightbigg1/ρ\n(172)\n.=P[Fc\nn(P∗\nXX′)]/parenleftbigg\nexp/bracketleftbig\nn(R−IP∗(X;X′)−3ε/2)/bracketrightbig\nexp[−nΓ(PXX′,R+2ε)]/parenrightbigg1/ρ\n, (173)\nwhere (168) follows from Tonelli’s theorem [30], (170) foll ows from (149), and (171) follows from (163), (173)\nfollows from δn→0and the definition of Γ(PXX′,R).\nFrom (173), it follows that\nErgv\ntrc(R,QX,d,∆) =−1\nnlim\nρ→∞ρlog/parenleftBig\nE[Pe(Cn)1/ρ]/parenrightBig\n≤Γ(P∗\nXX′,R)+IP∗(X;X′)−R+O(ε) (174)\n= min\nPXX′:PX′=PX,\nIP(X;X′)≤2R,d(PXX′)>∆/braceleftbigg\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbigg\n+O(ε) (175)\nfor anyε >0. By taking ε→0, we obtain (99). This concludes the proof of Proposition 2.\nV. C ONCENTRATION PROPERTIES\nIn this section, we study the concentration properties of th e RGV ensemble with GLD. In particular, we study\nthe lower tail P/bracketleftbig\n−1\nnlogPe(Cn)≤E0/bracketrightbig\nand derive both upper and lower bounds. We show that both boun ds exhibit\nNovember 23, 2022 DRAFT23\nan exponential decay. We also derive upper and lower bounds t o the upper tail P/bracketleftbig\n−1\nnlogPe(Cn)≥E0/bracketrightbig\n. We show\nthat the lower tail exhibits a doubly-exponential behavior .\nA. Lower Tail\nIn this section, we derive exponential upper and lower bound s to the lower tail probability. Before proceeding,\nwe define the following sets\nL(R,E0)/defines{PXX′∈ Q(QX) :d(PXX′)>∆,[2R−IP(X;X′)]+≥Γ(PXX′,R)+R−E0}, (176)\nM(R,E0)/defines/braceleftbig\nPXX′∈ Q(QX) :d(PXX′)>∆,[2R−IP(X;X′)]+≥Λ(PXX′,R)+R−E0/bracerightbig\n(177)\nwhere\nΛ(PXX′,R) = min\nPY|XX′/braceleftbig\nD(PY|X/⌊ard⌊lW|QX)+IP(X′;Y|X)+β(R,PY)−g(PX′Y)/bracerightbig\n, (178)\nβ(R,PY) = max\nP˜X|Y:P˜X=QX/braceleftbig\ng(P˜XY)+[R−IP(˜X;Y)]+/bracerightbig\n. (179)\nWe have the following result.\nTheorem 2: Consider the ensemble of RGV codes Cnof rateRand composition QXsatisfying condition (24).\nThen, it holds that\nP/bracketleftbigg\n−1\nnlogPe(Cn)≤E0/bracketrightbigg\n.\n≤exp/braceleftbig\n−nEub\nlt(R,E0)/bracerightbig\n, (180)\nP/bracketleftbigg\n−1\nnlogPe(Cn)≤E0/bracketrightbigg\n.\n≥exp/braceleftbig\n−nElb\nlt(R,E0)/bracerightbig\n. (181)\nwhere\nEub\nlt(R,E0)/definesmin\nPXX′∈L(R,E0)[IP(X;X′)−2R]+, (182)\nElb\nlt(R,E0)/definesmin\nPXX′∈M(R,E0)[IP(X;X′)−2R]+, (183)\nrespectively.\nBefore proceeding with the proof, we discuss an example in Fi gure 2 where the lower tail bounds are shown for the\nZ-channel with crossover probability w= 0.001andR= 0.2. In particular, we show the lower tail upper and lower\nbounds on the tail exponent for constant composition and for the RGV ensemble with d(PXX′) =−IP(X;X′)\nand∆ =−R. The numerical results show that Eub\nlt=Elb\nltfor the both constant composition and RGV ensembles.\nThis can be explained by the fact that there is only one empiri cal channel PX′Yfor each output type PYfor this\ncase [21, p. 5046]. Hence, [max{g(PXY),α(R,PY)}−g(PX′Y)] = [R−I(q)]+=β(R,PY)−g(PX′Y), which\nleads toΛ = Γ for anyRand crossover probability. Fig. 2 illustrates that the lowe r tail for the RGV code ensemble\ndecays faster than that for the constant composition ensemb le. This can be explained by the the fact that at R= 0.2\nthe typical error exponent of the RGV ensemble is higher than that for constant composition (see Figure 1).\nNovember 23, 2022 DRAFT24\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 .1 1.200.20.40.60.811.2\nE0Lower tail exponentsElb\nlt\nElb\nlt(RGV)\nEub\nlt\nEub\nlt(RGV)\nFig. 2: Lower tail exponents for constant composition and RG V codes for the Z-channel.\n1) Proof of the Lower Tail Upper Bound: Let\nBε(m,y) =/braceleftbigg\nCn:Zm(y)≤exp{nα(R−ε,ˆPy)}/bracerightbigg\n, (184)\nand\nBε/definesM/uniondisplay\nm=1/uniondisplay\nyBε(m,y). (185)\nThen, under the condition (24), by Lemma 14, we have\nP/braceleftbig\nBε(m,y)/bracerightbig\n≤exp/braceleftbigg\n−enε/bracketleftbigg\n1−e−n(ε+δ)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg\n. (186)\nHence, by the union bound, we have\nP{Bε} ≤M/summationdisplay\nm=1/summationdisplay\nyP/braceleftbig\nBε(m,y)/bracerightbig\n(187)\n≤M/summationdisplay\nm=1/summationdisplay\nyexp/braceleftbigg\n−enε/bracketleftbigg\n1−e−n(ε+δ)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg\n(188)\n≤enR|Y|nexp/braceleftbigg\n−enε/bracketleftbigg\n1−e−n(ε+δ)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg\n(189)\nwhere (188) follows from (186), which decays double-expone ntially fast.\nNovember 23, 2022 DRAFT25\nNow, by using the same arguments as [15, Proof of Theorem 1], w e have\nP/bracketleftbigg\n−1\nnlogPe(Cn)≤E0/bracketrightbigg\n≤P/bracketleftbigg\nCn∈ Bc\nε,1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=me−nΓ(ˆPXm,Xm′,R−ε)≥e−nE0/bracketrightbigg\n+P{Bε} (190)\n.\n≤P/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=me−nΓ(ˆPXm,Xm′,R−ε)≥e−nE0/bracketrightbigg\n(191)\n=P/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=me−nΓ(ˆPXm,Xm′,R−ε)1/braceleftbig\nd(Xm,Xm′)>∆/bracerightbig\n≥e−nE0/bracketrightbigg\n(192)\n=P/bracketleftbigg/summationdisplay\nPXX′∈Q(QX):d(PXX′)>∆N(PXX′)exp/braceleftbig\n−nΓ(PXX′,R−ε)/bracerightbig\n≥en(R−E0)/bracketrightbigg\n(193)\n.= max\nPXX′∈Q(QX):d(PXX′)>∆P/bracketleftbigg\nN(PXX′)≥exp{n(Γ(PXX′,R−ε)+R−E0)}/bracketrightbigg\n(194)\nwhere (190) follows from [15, Eq. (60)], and (192) follows fr om the fact that all codes Cnin the RGV ensemble\nsatisfyd(xm,xm′)>∆for allm/\\e}atio\\slash=m′.\nNow, define\nSε(R,E0)/defines/braceleftbigg\nPXX′∈ Q(QX) : [2R−IP(X;X′)]+≥Γ(PXX′,R−ε)+R−E0/bracerightbigg\n. (195)\nThen, from (194) and Lemma 10, under the condition 24, we obta in\nP/bracketleftbigg\n−1\nnlogPe(Cn)≤E0/bracketrightbigg\n.\n≤e−nEub\nlt(R,E0,ε), (196)\nwhere\nEub\nlt(R,E0,ε)/defines min\nPXX′∈Q(QX):d(PXX′)>∆\n\n[IP(X;X′)−2R]+, P XX′∈ Sε(R,E0)\n+∞, otherwise(197)\n= min\nPXX′∈Sε(R,E0):d(PXX′)>∆[IP(X;X′)−2R]+, (198)\nwith the convention that the minimum over an empty set is defin ed as infinity. Since εcan take any positive value,\nfrom (196) and (198), we obtain (181). This concludes our pro of of the upper bound in Theorem 2.\n2) Proof of the Lower Tail Lower Bound: The proof follows similar arguments as [15, Section B]. For t he\nRGV ensemble, however, existing techniques to lower bound o n the probability of the lower tail for the constant\ncomposition codes cannot be applied. For example, due to the dependence among codewords, key proposition [15,\nPrep. 4] can no longer be applied. We develop new techniques t o deal with the dependence among codewords.\nFor a given (m,m′)∈[M]2\n∗, andy∈ Yn, define\nZm,m′(y) =/summationdisplay\n˜m∈{1,2,···,M}\\{m,m′}exp/braceleftbig\nng(ˆPX˜m,y)/bracerightbig\n. (199)\nLetσ >0and define the set\nˆBn(σ,m,m′,y) =/braceleftBig\nCn:Zmm′(y)≥exp{n(β(R,ˆPy)+σ)}/bracerightBig\n, (200)\nNovember 23, 2022 DRAFT26\nand its complement ˆGn(σ,m,m′,y) =ˆBc\nn(σ,m,m′,y), whereβ(R,PY)is defined in (179). Let\nˆBn(σ) =M/uniondisplay\nm=1/uniondisplay\nm′/\\⌉}atio\\slash=m/uniondisplay\nyˆBn(σ,m,m′,y), (201)\nand\nˆGn(σ) =ˆBc\nn(σ). (202)\nLetε >0and define\n˜Λ(PXX′,R,ε) = min\nPY|XX′/braceleftbig\nD(PY|X/⌊ard⌊lW|QX)+IP(X′;Y|X)+[max{g(PXY),β(R,PY)+ε}−g(PX′Y)]+/bracerightbig\n.\n(203)\nThen, we have\nP/bracketleftbigg\n−1\nnlogPe(Cn)≤E0/bracketrightbigg\n.\n≥P/bracketleftbigg\nCn∈ˆGn(ε),1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=me−n˜Λ(ˆPXm′,Xm,R,ε)≥e−nE0/bracketrightbigg\n(204)\n=P/bracketleftbigg\nCn∈ˆGn(ε),1\nMM/summationdisplay\nm=1/summationdisplay\nm′/negationslash=m\nd(Xm,Xm′)>∆e−n˜Λ(ˆPXm′,Xm,R,ε)≥e−nE0/bracketrightbigg\n, (205)\nwhere (204) follows from [15, Eq. (83)], and (205) follows fr om the fact that d(xm,xm′)>∆for any RGV code.\nOn the other hand, we also have\n˜Λ(PXX′,R,ε) = Λ(PXX′,R)+ε. (206)\nHence, from (205) and (206), we obtain\nP/bracketleftbigg\n−1\nnlogPe(Cn)≤E0/bracketrightbigg\n.\n≥P/bracketleftbigˆGn(ε)∩G0/bracketrightbig\n, (207)\nwhere\nG0=/braceleftbigg\nCn:M/summationdisplay\nm=1/summationdisplay\nm′/negationslash=m\nd(Xm,Xm′)>∆e−n˜Λ(ˆPXm′,Xm,R,ε)≥en(R−E0)/bracerightbigg\n. (208)\nIt then follows that\nP/bracketleftbigg\n−1\nnlogPe(Cn)≤E0/bracketrightbigg\n.\n≥P/bracketleftbigˆGn(ε)∩G0/bracketrightbig\n(209)\n=P[G0]−P/bracketleftbig\nG0∩ˆBn(ε)/bracketrightbig\n(210)\n≥P[G0]−M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\nyP/bracketleftbigˆBn(ε,m,m′,y)∩G0/bracketrightbig\n. (211)\nNow, observe that\nP[G0] =P/bracketleftbigg/summationdisplay\nPXX′∈Q(QX):\nd(PXX′)>∆N(PXX′)e−n(Λ(PXX′,R)+ε)≥en(R−E0)/bracketrightbigg\n(212)\n.=/summationdisplay\nPXX′∈Q(QX):\nd(PXX′)>∆P/bracketleftbigg\nN(PXX′)≥en(Λ(PXX′,R)+R−E0+ε)/bracketrightbigg\n. (213)\nNovember 23, 2022 DRAFT27\nDefine the set S′\nε(R,E0) ={PXX′: [2R−IP(X;X′)]+≥Λ(PXX′,R)+R−E0+ε}.\nThen, under condition (24), by Proposition 10, it holds that\nP[G0].= exp{−nElb\nlt(R,E0,ε)}, (214)\nwhere\nElb\nlt(R,E0,ε) = min\nPXX′∈Q(QX):\nd(PXX′)>∆\n\n[IP(X;X′)−2R]+PXX′∈ S′\nε(R,E0)\n+∞ PXX′/∈S′\nε(R,E0)(215)\n= min\nPXX′∈{PXX′∈Q(QX):d(PXX′)>∆}∩S′ε(R,E0)[IP(X;X′)−2R]+. (216)\nNow, we study the second term in (211). For any joint type PXY∈ Pn(X ×Y), define\nNy(PXY)/defines/summationdisplay\n˜m∈[M]\\{ˆm,¨m}1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig\n. (217)\nThen, we have\nP/bracketleftbigˆBn(ε,ˆm,¨m,y)∩G0/bracketrightbig\n=P/bracketleftbigg/summationdisplay\n˜m∈[M]\\{ˆm,¨m}eng(ˆPX˜m,y)≥en(β(R,ˆPy)+ε),M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=me−n(Λ(ˆPx˜m,xm,R)+ε)≥en(R−E0)/bracketrightbigg\n(218)\n≤P/bracketleftbigg/summationdisplay\n˜m∈[M]\\{ˆm,¨m}eng(ˆPX˜m,y)≥en(β(R,ˆPy)+ε)/bracketrightbigg\n(219)\n≤P/bracketleftbigg/summationdisplay\nPXY:PX=QXNy(PXY)eng(PXY)≥en(β(R,ˆPy)+ε)/bracketrightbigg\n(220)\n.\n≤/summationdisplay\nPXY:PX=QXP/bracketleftbigg\nNy(PXY)≥en(β(R,ˆPy)−g(PXY)+ε)/bracketrightbigg\n(221)\n≤/summationdisplay\nPXY:PX=QXP/bracketleftbigg\nNy(PXY)≥en([R−IP(X;Y)]++ε)/bracketrightbigg\n(222)\n=/summationdisplay\nPXY:PX=QXP/bracketleftbigg/summationdisplay\n˜m∈[M]\\{ˆm,¨m}1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig\n≥en([R−IP(X;Y)]++ε)/bracketrightbigg\n(223)\n=/summationdisplay\nPXY:PX=QX,IP(X;Y)>0P/bracketleftbigg/summationdisplay\n˜m∈[M]\\{ˆm,¨m}1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig\n≥en([R−IP(X;Y)]++ε)/bracketrightbigg\n, (224)\nwhere (222) follows from (179), and (224) follows from\nP/bracketleftbigg/summationdisplay\n˜m∈[M]\\{ˆm,¨m}1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig\n≥en([R−IP(X;Y)]++ε)/bracketrightbigg\n= 0 (225)\nifIP(X;Y) = 0 .\nNow, in order to bound P/bracketleftbig/summationtext\n˜m∈[M]\\{ˆm,¨m}1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig\n≥en([R−IP(X;Y)]++ε)/bracketrightbig\nfor each joint type\nPXYsuch that PX=QX, we will use the following lemma.\nNovember 23, 2022 DRAFT28\nLemma 15: [31, Lemma 1.8] Suppose that X1,X2,···,Xnare random variables such that 0≤Xi≤1, for\ni= 1,2,···,n. Setp=1\nn/summationtext\niE[Xi]and fix a real number tsuch that np+ 1< t < n . Ifε0>0is such that\nt−1 =np(1+ε0), then\nP/bracketleftbiggn/summationdisplay\ni=1Xi≥t/bracketrightbigg\n≤2e−nD(p(1+ε0)/⌊ar⌈⌊lp). (226)\nMore specifically, for any ˜m∈[M]\\{ˆm,¨m}, observe that\nE/bracketleftbig\n1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig/bracketrightbig\n=P/bracketleftbig\n(X˜m,y)∈ T(PXY)/bracketrightbig\n(227)\n=/summationdisplay\nx˜m∈T(QX):(x˜m,y)∈T(PXY)P(x˜m) (228)\n=/summationdisplay\nx˜m∈T(QX):(x˜m,y)∈T(PXY)1\n|T(QX)|(229)\n.=e−nIP(X;Y), (230)\nwhere (229) follows from 3, and (230) follows from [28].\nIt follows from (230) that\np/defines1\nM−2/summationdisplay\n˜m∈[M]\\{ˆm,¨m}E/bracketleftbig\n1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig/bracketrightbig\n(231)\n.=e−nIP(X;Y). (232)\nNow, there exists a δ(ε)< εsuch that min{IP(X;Y) :IP(X;Y)>0}> δ(ε). Then, we have\n/summationdisplay\nPXY:PX=QX,IP(X;Y)>0P/bracketleftbigg/summationdisplay\n˜m∈[M]\\{ˆm,¨m}1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig\n≥en([R−IP(X;Y)]++ε)/bracketrightbigg\n≤/summationdisplay\nPXY:PX=QX,IP(X;Y)>0P/bracketleftbigg/summationdisplay\n˜m∈[M]\\{ˆm,¨m}1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig\n≥en([R−IP(X;Y)]++δ(ε))/bracketrightbigg\n. (233)\nBy applying Lemma 15 for the sequence of Bernoulli random var iables{/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig\n˜m∈[M]\\{ˆm,¨m}\nwitht=en([R−IP(X;Y)]++δ(ε)), we obtain\nP/bracketleftbigg/summationdisplay\n˜m∈[M]\\{ˆm,¨m}1/braceleftbig\n(X˜m,y)∈ T(PXY)/bracerightbig\n≥en([R−IP(X;Y)]++δ(ε))/bracketrightbigg\n˚≤exp/braceleftbigg\n−(M−2)D/parenleftbig\nen([R−IP(X;Y)]+−R+δ(ε))/⌊ard⌊le−nIP(X;Y)/parenrightbig/bracerightbigg\n(234)\n˚≤exp/braceleftbigg\n−en([R−IP(X;Y)]++δ(ε))/bracerightbigg\n(235)\nfor any joint type PXY∈ Pn(X × Y)such that PX=QX, where (235) follows from the fact that D(a/⌊ard⌊lb)≥\na/parenleftbig\nloga\nb−1/parenrightbig\n[32].\nIt follows from that\nP/bracketleftbigˆBn(ε,ˆm,¨m,y)∩G0/bracketrightbig˚≤max\nPXYexp/braceleftbigg\n−en([R−IP(X;Y)]++δ(ε))/bracerightbigg\n(236)\n˚≤exp/braceleftbig\n−enδ(ε)/bracerightbig\n. (237)\nNovember 23, 2022 DRAFT29\nFrom (211), (214), and (237), we finally obtain\nP/bracketleftbigg\n−1\nnlogPe(Cn)≤E0/bracketrightbigg\n.\n≥exp/braceleftbig\n−nElb\nlt(R,E0,ε)/bracerightbig\n−M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\nyexp/braceleftbig\n−enδ(ε)/bracerightbig\n(238)\n.= exp/braceleftbig\n−nElb\nlt(R,E0,ε)/bracerightbig\n−e2nR|Y|nexp/braceleftbig\n−enδ(ε)/bracerightbig\n(239)\n.= exp/braceleftbig\n−nElb\nlt(E,E0,ε)/bracerightbig\n. (240)\nDue to the arbitrariness of ε >0, it follows that\nP/bracketleftbigg\n−1\nnlogPe(Cn)≤E0/bracketrightbigg\n.\n≥exp{−nElb\nlt(R,E0)}, (241)\nwhich proves the lower bound of Theorem 2.\nB. Upper Tail\nIn this section, we derive double-exponential upper and low er bounds to the upper tail probability. First, we\nintroduce some new notation which will be used throughout th is section. Recall the definitions of A1andA2in\n(56) and (57), respectively. Let\nV(R,E0) =/braceleftbig\nPXX′∈ Q(QX) :d(PXX′)>∆,IP(X;X′)≤2R,Λ(PXX′,R)+IP(X;X′)−R≤E0/bracerightbig\n,(242)\nU(R,E0) =/braceleftbig\nPXX′∈ Q(QX) :d(PXX′)>∆,IP(X;X′)≤2R,Γ(PXX′,R)+IP(X;X′)−R≤E0/bracerightbig\n.(243)\nand\nA3=/braceleftbig\nPXX′∈ Q(QX) :d(PXX′)>∆,IP(X;X′)≤2R,Γ(PXX′,R−ε)+IP(X;X′)−R > E 0+ε/bracerightbig\n.\n(244)\nTheorem 3: Consider the RGV ensemble Cnof rateRand composition QXsatisfying condition (24). Assume\nthat the conditions in Lemma 11 hold for D=˜V(R,E0,σ). Then, the upper tail can be bounded as\nP/bracketleftbigg\n−1\nnlogPe(Cn)≥E0/bracketrightbigg\n˚≤exp/braceleftbig\n−exp/braceleftbig\nnEub\nut(R,E0)/bracerightbig/bracerightbig\n(245)\nwhere\nEub\nut(R,E0) = max\nPXX′∈V(R,E0)min/braceleftbig\n2R−IP(X;X′),E0−Λ(PXX′,R)−IP(X;X′)+R,R/bracerightbig\n. (246)\nIn addition, under the conditions\nmax\nPXX′∈A3IP(X;X′)≤min\nPXX′∈A2IP(X;X′) (247)\nmin\nPXX′:d(PXX′)≤∆IP(X;X′)≥max\nPXX′:d(PXX′)>∆IP(X;X′), (248)\nwe have that\nP/bracketleftbigg\n−1\nnlogPe(Cn)≥E0/bracketrightbigg\n˚≥exp/braceleftbig\n−exp/braceleftbig\nnElb\nut(R,E0)/bracerightbig/bracerightbig\n,∀E0< Eex(R,QX). (249)\nNovember 23, 2022 DRAFT30\nwhere\nElt\nut(R,E0) = max\nPXX′∈U(R,E0)/braceleftbig\n2R−IP(X;X′)}. (250)\nIn Figure 3 we show the double-exponential bounds for the upp er tail for constant composition and the RGV\nensemble with d(PXX′) =−IP(X;X′)and∆ =−RforR= 0.2. We observe that for constant composition\nthe decay is indeed double-exponential even if the bounds on ly coincide for high values of E0(above the TRC\nexponent). Instead, for the RGV ensemble, the bound Elt\nut(R,E0) = 0 for values of E0of interest. This implies\nthat the decay of the upper tail for Etrc≤E0≤Eexis sub-double-exponential; for E0> Eexthe behavior of the\nupper tail is double-exponential as suggested by Eut\nutfor the RGV ensemble. Figure. 3 also shows that the decay\nrate of RGV code is slower than the constant composition code . This can be explained by the the fact that the\nerror probability in RGV code is expected to be smaller than t he constant composition codes since the later is more\nstructured as in the Fig. 2.\n0.4 0.5 Etrc 0.6Eex 0.7 0.8 0.9 102·10−24·10−26·10−28·10−20.10.120.140.160.180.2\nE0Upper tail exponents\nElb\nut\nEub\nut\nElb\nut(RGV)\nEub\nut(RGV)\nFig. 3: Upper Tails\n1) Proof of the Upper Tail Upper Bound: The proof is based on [15, Proof of Theorem 2] with important c hanges\nto account for the dependency among codewords in the RGV code book ensemble. See also the proofs of Lemma\n11 and Lemma 16 below for specific changes.\nLemma 16: For every σ >0, under condition (24) the following holds\nP/braceleftbigˆBn(σ)/bracerightbig˚≤exp{−enσ} (251)\nNovember 23, 2022 DRAFT31\nwhereˆBn(σ)has been defined in (201).\nProof: See Appendix L.\nWe start by defining the following set\n˜V(R,E0,σ)/defines/braceleftbig\nPXX′∈ Q(QX) :d(PXX′)>∆,IP(X;X′)≤2R,˜Λ(PXX′,R,σ)+IP(X;X′)−R≤E0−ε/bracerightbig\n(252)\nforσ >0,ε >0, where˜Λ(PXX′,R,ε)was defined in (203).\nUnder condition (24), we have that\nE[N(PXX′)] =E/bracketleftbigg/summationdisplay\n(m,m′)∈[M]2\n∗1{(Xm,Xm′)∈ T(PXX′)}/bracketrightbigg\n(253)\n=/summationdisplay\n(m,m′)∈[M]2\n∗P/bracketleftbig\n(Xm,Xm′)∈ T(PXX′)/bracketrightbig\n(254)\n=/summationdisplay\n(m,m′)∈[M]2∗/summationdisplay\nxm,xm′∈T(PXX′)P/parenleftbig\nxm,xm′/parenrightbig\n(255)\n.=en(2R−IP(X;X′)), (256)\nwhere (256) follows from Lemma 2.\nFor a given message pair m,m′∈[M]2\n∗, andy∈ Yn, recall the definitions of Zm,m′(y),ˆBn(σ), andˆGn(σ)in\n(199), (201), and (202), respectively. Then, we have\nP/bracketleftbig\nCn∈ˆGn(σ),−1\nnlogPe(Cn)≥E0/bracketrightbigg\n(257)\n≤P/bracketleftbigg\nCn∈ˆGn(σ),1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\nyW(y|Xm)eng(ˆPXm′y)\neng(ˆPXmy)+eng(ˆPXm′y)+Zmm′(y)≤e−nE0/bracketrightbigg\n(258)\n=P/bracketleftbigg\nCn∈ˆGn(σ),1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m:d(Xm,Xm′)>∆/summationdisplay\nyW(y|Xm)eng(ˆPXm′y)\neng(ˆPXmy)+eng(ˆPXm′y)+Zmm′(y)≤e−nE0/bracketrightbigg\n(259)\n˚≤ min\nPXX′∈˜V(R,E0,σ)P/bracketleftbigg\nN(PXX′)≤en(˜Λ(PXX′,R,σ)+R−E0)/bracketrightbigg\n(260)\n˚≤ min\nPXX′∈˜V(R,E0,σ)P/bracketleftbigg\nN(PXX′)≤en/parenleftbig\n2R−IP(X;X′)−ε/parenrightbig/bracketrightbigg\n(261)\n˚≤min\npXX′∈˜V(R,E0,σ)exp/braceleftbig\n−min/parenleftbig\nen(2R−IP(X;X′)),enR/parenrightbig/bracerightbig\n(262)\nwhere (258) follows from (11), (259) follows from the fact th atd(Xm,Xm′)>∆with probability 1by the RGV\nrandom codebook generation, (260) follows the same argumen ts to achieve [15, Eq. (146)], (261) follows from\n(252), and (262) follows from (256) and Lemma 11.\nIt follows from (262) that for σ >0,\nP/bracketleftbigg\nCn∈ˆGn(σ),−1\nnlogPe(Cn)≥E0/bracerightbigg\n˚≤exp/braceleftbig\n−exp/braceleftbig\nnE1(R,E0,σ)/bracerightbig/bracerightbig\n, (263)\nNovember 23, 2022 DRAFT32\nwhere\nE1(R,E0,σ) = max\nPXX′∈˜V(R,E0,σ)min{2R−IP(X;X′),R}. (264)\nTherefore, we have\nP/bracketleftbigg\n−1\nnlogPe(Cn)≥E0/bracketrightbigg\n=P/bracketleftbigg\nCn∈ˆGn(σ),−1\nnlogPe(Cn)≥E0/bracketrightbigg\n+P/bracketleftbigg\nCn∈ˆGc\nn(σ),−1\nnlogPe(Cn)≥E0/bracketrightbigg\n(265)\n=P/bracketleftbigg\nCn∈ˆGn(σ),−1\nnlogPe(Cn)≥E0/bracketrightbigg\n+P/bracketleftbigg\nCn∈ˆBn(σ),−1\nnlogPe(Cn)≥E0/bracketrightbigg\n(266)\n≤P/bracketleftbigg\nCn∈ˆGn(σ),−1\nnlogPe(Cn)≥E0/bracketrightbigg\n+P/bracketleftbig\nCn∈ˆBn(σ)/bracketrightbig\n(267)\n˚≤exp/braceleftbig\n−exp/braceleftbig\nnE1(R,E0,σ)/bracerightbig/bracerightbig\n+exp{−enσ} (268)\nwhere (266) follows from ˆBn(σ) =ˆGc\nn(σ), (268) follows from Lemma 16 and (263).\nFinally, by using the same arguments as to obtain [15, Eq. (17 5)] from [15, Eq. (153)], from (268), we obtain\nP/bracketleftbigg\n−1\nnlogPe(Cn)≥E0/bracketrightbigg\n˚≤exp/braceleftBig\n−enEub\nut(R,E0)/bracerightBig\n, (269)\nwhich concludes our proof of the upper bound on the upper tail .\n2) Proof of the Upper Tail Lower Bound: Let\nBε(m,y) =/braceleftbigg\nCn:Zm(y)≤exp{nα(R−ε,ˆPy)}/bracerightbigg\n, (270)\nand\nBε/definesM/uniondisplay\nm=1/uniondisplay\nyBε(m,y). (271)\nThen, under condition (24), by Lemma 14 and the union bound, w e have\nP{Bε} ≤enR|Y|nexp/braceleftbigg\n−enε/bracketleftbigg\n1−e−n(ε+δ)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg\n. (272)\nNow, define Gε(m,y) =Bc\nε(m,y)andGε=Bc\nε.\nRecall the definition of Zm(y)in (77). We have that\nP/bracketleftbigg\n−1\nnlogPe(Cn)≥E0/bracketrightbigg\n=P/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\nyW(y|Xm)exp{ng(ˆPXm′y)}\nexp{ng(ˆPXmy)}+Zm(y)≤e−nE0/bracketrightbigg\n(273)\n=P/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m:d(Xm,Xm′)>∆/summationdisplay\nyW(y|Xm)exp{ng(ˆPXm′y)}\nexp{ng(ˆPXmy)}+Zm(y)≤e−nE0/bracketrightbigg\n(274)\n≥P/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m:d(Xm,Xm′)>∆/summationdisplay\nyW(y|Xm)exp{ng(ˆPXm′y)}\nexp{ng(ˆPXmy)}+Zm(y)≤e−nE0,Cn∈ Gε/bracketrightbigg\n(275)\n˚≥P/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m:d(ˆPXm,Xm′)>∆exp/braceleftbig\n−nΓ(ˆPXm,Xm′,R−ε)/bracerightbig\n≤e−nE0,Cn∈ Gε/bracketrightbigg\n, (276)\nNovember 23, 2022 DRAFT33\nwhere (274) follows from the fact that mini/\\⌉}atio\\slash=jd(xi,xj)>∆for all RGV code (x1,x2,···,xM), and (276)\nfollows from the same arguments to obtain [15, Eq. (178)].\nNow, define\nE0/defines/braceleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m:d(ˆPXm,Xm′)>∆exp/braceleftbig\n−nΓ(ˆPXm,Xm′,R−ε)/bracerightbig\n≤e−nE0/bracerightbigg\n. (277)\nThen, we have\nP/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m:d(ˆPXm,Xm′)>∆exp/braceleftbig\n−nΓ(ˆPXm,Xm′,R−ε)/bracerightbig\n≤e−nE0,Cn∈ Gε/bracketrightbigg\n=P/bracketleftbigg\nCn∈ E0,Cn∈ Gε/bracketrightbigg\n(278)\n=P/bracketleftbiggM/intersectiondisplay\n˜m=1/intersectiondisplay\nyGε(˜m,y)/vextendsingle/vextendsingleE0/bracketrightbigg\nP(E0) (279)\n=/parenleftbigg\n1−P/bracketleftbiggM/uniondisplay\n˜m=1/uniondisplay\nyGc\nε(˜m,y)/vextendsingle/vextendsingleE0/bracketrightbigg/parenrightbigg\nP[E0] (280)\n≥/parenleftbigg\n1−M/summationdisplay\nm=1/summationdisplay\nyP/bracketleftbig\nGc\nε(˜m,y)/vextendsingle/vextendsingleE0/bracketrightbig/parenrightbigg\nP[E0] (281)\n=P[E0]−M/summationdisplay\nm=1/summationdisplay\nyP/bracketleftbig\nBε(˜m,y)∩E0/bracketrightbig\n. (282)\nNow, observe that\nP[E0] =P/bracketleftbigg1\nMM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m:d(ˆPXm,Xm′)>∆exp/braceleftbig\n−nΓ(ˆPXm,Xm′,R−ε)/bracerightbig\n≤e−nE0/bracketrightbigg\n(283)\n˚ =P/bracketleftbigg/intersectiondisplay\nPXX′∈Q(QX):d(PXX′)>∆/braceleftbigg\nN(PXX′)≤en(Γ(PXX′,R−ε)+R−E0)/bracerightbigg/bracketrightbigg\n, (284)\nwhere (284) follows by using the same arguments to achieve [1 5, Eq. (187)].\nRecall the definition of F0in (58) in Lemma 12, i.e.,\nF0=/intersectiondisplay\nPXX′∈A1∪A2/braceleftbig\nN(PXX′) = 0/bracerightbig\n. (285)\nDefine\nF(PXX′)/defines/braceleftbigg\nN(PXX′)≤en(Γ(PXX′,R−ε)+R−E0)/bracerightbigg\n. (286)\nNovember 23, 2022 DRAFT34\nThen, from (284) and (286), we obtain\nP[E0] ˚ =P/bracketleftbigg/intersectiondisplay\nPXX′∈Q(QX):d(PXX′)>∆F(PXX′)/bracketrightbigg\n(287)\n=P/bracketleftbigg/intersectiondisplay\nPXX′∈A1∪A2∪A3F(PXX′)/bracketrightbigg\n(288)\n=P/bracketleftbigg/intersectiondisplay\nPXX′∈A3F(PXX′)∩/intersectiondisplay\nPXX′∈A1∪A2F(PXX′)/bracketrightbigg\n(289)\n≥P/bracketleftbigg/intersectiondisplay\nPXX′∈A3F(PXX′)∩F0/bracketrightbigg\n(290)\n=P/bracketleftbigg/intersectiondisplay\nPXX′∈A3F(PXX′)/vextendsingle/vextendsingleF0/bracketrightbigg\nP[F0] (291)\n=/parenleftbigg\n1−P/bracketleftbigg/uniondisplay\nPXX′∈A3Fc(PXX′)/vextendsingle/vextendsingleF0/bracketrightbigg/parenrightbigg\nP[F0] (292)\n≥P[F0]−/summationdisplay\nPXX′∈A3P/bracketleftbig\nFc(PXX′)/vextendsingle/vextendsingleF0/bracketrightbig\nP[F0] (293)\n≥P[F0]−/summationdisplay\nPXX′∈A3P/bracketleftbig\nFc(PXX′)∩F0/bracketrightbig\n(294)\n≥P[F0]−/summationdisplay\nPXX′∈A3P/bracketleftbig\nFc(PXX′)/bracketrightbig\n, (295)\nwhere (290) follows from the fact that for each joint type PXX′∈ A1∪ A2, it holds that {N(QXX′) = 0} ⊂\n{N(QXX′)≤en(Γ(PXX′,R−ε)+R−E0)}.\nEquation (295) resembles [15, Eq. (205)] with subtle differ ences in the definition of sets A1,A2andA3. However,\nsince all the codewords in RGV are dependent, [15, Eq. (218)] does not hold. We proceed with different arguments.\nFor anyPXX′∈ A3, we have\nP/bracketleftbig\nFc(PXX′)/bracerightbig\n=P/braceleftbigg\nN(PXX′)≥en(Γ(PXX′,R−ε)+R−E0)/bracketrightbigg\n(296)\n≤P/bracketleftbigg\nN(PXX′)≥enεen(2R−IP(X;X′))/bracketrightbigg\n, (297)\nwhere (297) follows from the definition of the set A3, which implies that\nΓ(PXX′,R−ε)+R−E0>2R−IP(X;X′)+ε. (298)\nOn the other hand, by Lemma 7, we have\n/summationdisplay\nPXX′∈A3P/bracketleftbigg\nN(PXX′)≥enεen(2R−IP(X;X′))/bracketrightbigg\n˚≤max\nPXX′∈A3exp/braceleftbigg\n−enR(2R−IP(X;X′)+ε)/bracerightbigg\n(299)\n= exp/braceleftbigg\n−en(2R−maxPXX′∈A3IP(X;X′)+ε)/bracerightbigg\n(300)\n≤exp/braceleftbigg\n−en(2R−minPXX′∈A2IP(X;X′)+ε)/bracerightbigg\n, (301)\nwhere (301) follows from the condition (247).\nNovember 23, 2022 DRAFT35\nNow, under the condition (248), by Lemma 12, we have\nP{F0}˚≥exp/braceleftbig\n−enmaxPXX′∈A2(2R−IP(X;X′))/bracerightbig\n. (302)\nFrom (295), (301), and (302), we obtain\nP[E0]˚≥exp/braceleftbig\n−enmaxPXX′∈A2(2R−IP(X;X′))/bracerightbig\n−exp/braceleftbigg\n−en(2R−minPXX′∈A2IP(X;X′)+ε)/bracerightbigg\n(303)\n˚ = exp/braceleftbig\n−enmaxPXX′∈A2(2R−IP(X;X′))/bracerightbig\n. (304)\nTo bound P[Bε(˜m,y)∩E0], we use the following arguments. As [15], let\nN2:=/braceleftbigg\n(m,m′) :m/\\e}atio\\slash=m′,m,m′∈ {1,2,···,⌊M/2⌋−1}/bracerightbigg\n. (305)\nDefine\nS:=/braceleftbigg\n(x1,x2,···,x⌊M⌋/2⌋)∈Rn×Rn···×Rn\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\n⌊M/2⌋times: min\ni,j∈{1,2,···,⌊M/2⌋},i/\\⌉}atio\\slash=j{d(xi,xj)}>∆/bracerightbigg\n. (306)\nSince the distance between two codewords in a RGV ensemble is at least∆, we have\nP[Bε(˜m,y)∩E0]\n≤P/bracketleftbigg/summationdisplay\n(m,m′)∈N2e−nΓ(ˆPXm,Xm′,R−ε)≤en(R−E0)/bracketrightbigg\n×P/bracketleftbigg/summationdisplay\nm′∈{⌊M/2⌋,···,M}\\{˜m}eng(ˆPXm′y)≤enα(R−ε,ˆPy)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/braceleftbigg/summationdisplay\n(m,m′)∈N2e−nΓ(ˆPXm,Xm′,R−ε)≤en(R−E0)/bracerightbigg\n∩/braceleftbigg\n(X1,X2,···,X⌊M⌋/2⌋)∈ S/bracerightbigg/bracketrightbigg\n. (307)\nNow, for any tuple (x1,x2,···,x⌊M⌋)such that mini,j∈{1,2,···,⌊M/2⌋},i/\\⌉}atio\\slash=j{d(xi,xj)}>∆, it holds that\nP/parenleftbigg\nX⌊M/2⌋+1=x⌊M/2⌋+1,X⌊M/2⌋+2=x⌊M/2⌋+2,···,XM=xM/vextendsingle/vextendsingleX1=x1,···,X⌊M/2⌋=x⌊M/2⌋/parenrightbigg\n=P(X1=x1,X2=x2,···,XM=xM)\nP(X⌊M/2⌋=x⌊M/2⌋,···,X1=x1)(308)\n.\n≤1\n|T(QX)|⌈M/2⌉, (309)\nwhere (309) follows from Lemma 4. Hence, by using the same arg uments as the proof of Lemma 14, we obtain\nP[Bε(˜m,y)∩E0]≤exp/braceleftbigg\n−enε/bracketleftbigg\n1−e−n(ε+δ)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg\nP(E0). (310)\nFrom (282), (304), and (310), we have\nP/bracketleftbigg\n−1\nnlogPe(Cn)≥E0/bracketrightbigg\n˚≥/parenleftbigg\n1−enR|Y|nexp/braceleftbigg\n−enε/bracketleftbigg\n1−e−n(ε+δ)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg/parenrightbigg\nexp/braceleftbig\n−enmaxPXX′∈A2(2R−IP(X;X′))/bracerightbig\n(311)\n˚ = exp/braceleftbig\n−enmaxPXX′∈A2(2R−IP(X;X′))/bracerightbig\n(312)\nwhich concludes the proof.\nNovember 23, 2022 DRAFT36\nC. Convergence in Probability\nThis section enumerates properties of the tail exponents de rived in Sections V-A and V-B, respectively, and\nestablishes the convergence in probability to the TRC expon ent of the RGV . In particular, the following results can\nbe obtained by using the same arguments as the proofs of [15, P rop. 1], [15, Prop. 3], [15, Prop. 2], respectively,\nand are therefore stated without proof. Define\n˜E(R)/defines min\nPXX′∈Q(QX):IP(X;X′)≤2R,d(PXX′)>∆/braceleftbig\nΛ(PXX′,R)+IP(X;X′)−R/bracerightbig\n. (313)\nProposition 3 (Lower tail): Eub\nlt(R,E0)andElb\nlt(R,E0)have the following properties\n1) For fixed R,Eub\nlt(R,E0)andElb\nlt(R,E0)are decreasing in E0.\n2)Eub\nlt(R,E0)>0if and only if E0< Ergv\ntrc(R,QX,∆,d).\n3)Elb\nlt(R,E0)>0ifE0<˜E(R).\n4)Elb\nlt(R,E0) =∞for anyE0< Emin\n0(R), where\nEmin\n0(R)/defines min\nPXX′∈Q(QX):d(PXX′)>∆/braceleftbig\nΓ(PXX′,R)−[2R−IP(X;X′)]++R/bracerightbig\n(314)\nProposition 4 (Upper tail): Eub\nut(R,E0)andElb\nut(R,E0)have the following properties\n1) For fixed R,Eub\nut(R,E0)andElb\nut(R,E0)are increasing in E0.\n2)Eub\nut(R,E0)>0if and only if E0> Ergv\ntrc(R,QX,∆,d).\n3)Elb\nut(R,E0)>0ifE0>˜E(R).\nFrom Propositions 3 and 4, the following result states the co nvergence in probability to the TRC of the RGV\nensemble.\nCorollary 2: For any RGV ensemble with GLD, under the conditions in Lemma 1 1 and Lemma 12, we have\nthat\n−1\nnlogPe(Cn)(p)−→Ergv\ntrc(R,QX,d,∆). (315)\nRecall that for d(PXX′) =−IP(X;X′)and∆ =−(R+ 2δ), the conditions in Lemma 11 and Lemma 12\nhold. Hence, Corollary 2 holds for this important case for wh ichErgv\nrce(R,QX,d,∆) =Ergv\ntrc(R,QX,d,∆) =\nEex(R,QX).\nVI. C ONCLUSIONS\nWe have studied the RGV code ensemble and have studied the typ ical error exponent and upper and lower\nerror exponent tails. We have shown that the lower tail decay s exponentially while the upper tail exhibits a decay\nthat is between exponential and double-exponential; it is s ub-double-exponential below the expurgated exponent\nand double-exponential above the expurgated exponent. In a ddition, we have shown that the error exponent of a\nsufficiently long RGV code concentrates in probability arou nd the typical error exponent; this is also shown to\ncoincide with the random coding exponent of the RGV ensemble , known to coincide with the maximum of the\nexpurgated and the random-coding exponent. This suggests t hat every code in the ensemble asymptotically attains\nas high an error exponent as it is known from random codes.\nNovember 23, 2022 DRAFT37\nAPPENDIX A\nPROOF OF LEMMA 4\nAssume that A={i1,i2,···,il}where1≤i1< i2<···< il≤Mfor some l∈[M]. First, if\nminj,k∈[l],j/\\⌉}atio\\slash=kd(xij,xik)≤∆, then by the RGV generation, we have\nP(xi1,xi2,···,xil) = 0. (316)\nHence, (29) trivially holds.\nNow, under the condition minj,k∈[l],j/\\⌉}atio\\slash=kd(xij,xik)>∆, we have\nP/bracketleftbigg/intersectiondisplay\nk∈A{Xk=xk}/bracketrightbigg\n=P(xi1,xi2,···,xil) (317)\n=/summationdisplay\nxi1−1\n1,xi2−1\ni1+1,···,xil−1\nil−1+1:d(xk,xl)>∆∀k,l∈[il],k/\\⌉}atio\\slash=lP(xi1−1\n1)P(xi1|xi1−1\n1)P(xi2−1\ni1+1|xi1\n1)P(xi2|xi2−1\n1)\n×P(xi3−1\ni2+1|xi2\n1)P(xi3|xi3−1\n1)×···×P(xil−1\nil−1+1|xil−1\n1)P(xll|xil−1\n1) (318)\n=/summationdisplay\nxi1−1\n1,xi2−1\ni1+1,···,xil−1\nil−1+1:d(xk,xl)>∆∀k,l∈[il],k/\\⌉}atio\\slash=lP(xi1−1\n1)P(xi2−1\ni1+1|xi1\n1)P(xi3−1\ni2+1|xi2\n1)···P(xil−1\nil−1+1|xil−1\n1)\n×l/productdisplay\nj=1P(xij|xij−1\n1) (319)\n=/summationdisplay\nxi1−1\n1,xi2−1\ni1+1,···,xil−1\nil−1+1:d(xk,xl)>∆∀k,l∈[il],k/\\⌉}atio\\slash=lP(xi1−1\n1)P(xi2−1\ni1+1|xi1\n1)P(xi3−1\ni2+1|xi2\n1)···P(xil−1\nil−1+1|xil−1\n1)\n×l/productdisplay\nj=11\n|T(QX,xij−1\n1)|. (320)\nOn the other hand, under the condition 24, by Lemma 1, we have\n|T(QX)| ≥ T(QX,xi−1\n1)| ≥(1−e−nδ)|T(QX)|,∀i∈[M] (321)\nfor allxi−1\n1occurring with non-zero probability.\nFrom (320) and (321), if minj,k∈[l],j/\\⌉}atio\\slash=kd(xij,xik)>∆, we obtain\nP/bracketleftbigg/intersectiondisplay\nk∈A{Xk=xk}/bracketrightbigg\n≤/summationdisplay\nxi1−1\n1,xi2−1\ni1+1,···,xil−1\nil−1+1:d(xk,xl)>∆∀k,l∈[il],k/\\⌉}atio\\slash=lP(xi1−1\n1)P(xi2−1\ni1+1|xi1\n1)P(xi3−1\ni2+1|xi2\n1)\n×···×P(xil−1\nil−1+1|xil−1\n1)1\n(1−e−nδ)l|T(QX)|l(322)\n≤/summationdisplay\nxi1−1\n1,xi2−1\ni1+1,···,xil−1\nil−1+1P(xi1−1\n1)P(xi2−1\ni1+1|xi1\n1)P(xi3−1\ni2+1|xi2\n1)\n×···×P(xil−1\nil−1+1|xil−1\n1)1\n(1−e−nδ)l|T(QX)|l(323)\n=1\n(1−e−nδ)l|T(QX)|l(324)\nNovember 23, 2022 DRAFT38\n=1\n(1−e−nδ)|A||T(QX)||A|. (325)\nIn addition, for any M′≤M, from (320) and (321), if mink,l∈[M′]:k/\\⌉}atio\\slash=ld(xk,xl)>∆, we also have\nP/bracketleftbigg/intersectiondisplay\nk∈[M′]{Xk=xk}/bracketrightbigg\n≥/summationdisplay\nxi1−1\n1,xi2−1\ni1+1,···,xil−1\nil−1+1P(xi1−1\n1)P(xi2−1\ni1+1|xi1\n1)P(xi3−1\ni2+1|xi2\n1)\n×···×P(xil−1\nil−1+1|xil−1\n1)1\n|T(QX)|M′ (326)\n=1\n|T(QX)|M′. (327)\nThis concludes our proof of Lemma 4.\nAPPENDIX B\nPROOF OF LEMMA 5\nFirst, we prove (35). Observe that\nE[I(i,j)] =P/bracketleftbig\n(Xi,Xj)∈ T(PXX′)/bracketrightbig\n(328)\n=/summationdisplay\n(xi,xj)∈T(PXX′)P(xi,xj). (329)\nNow, let\nδn/definese−nδ\n1−e−nδ. (330)\nThen, under the condition (24) and d(PXX′)>∆, by Lemma 2, we have\n(1−4δ2\nn)\n|T(QX)|2e−2δn≤P(xi,xj)≤1\n(1−e−nδ)2|T(QX)|2,∀(xi,xj)∈ T(PXX′) (331)\nsinced(xi,xj) =d(PXX′)>∆. From (329) and (331), we have\n(1−4δ2\nn)e−2δn|T(PXX′)|\n|T(QX)|2≤E[I(i,j)]≤1\n(1−e−nδ)2|T(PXX′)|\n|T(QX)|2. (332)\nRecall the definition of L(PXX′)in (32). From (332), we have\n(1−4δ2\nn)e−2δnL(PXX′)≤E[I(i,j)]≤1\n(1−e−nδ)2L(PXX′). (333)\nNow, we prove (36). We consider three cases:\nNovember 23, 2022 DRAFT39\n•Case 1:i=k,j/\\e}atio\\slash=l. Observe that\nE[I(i,j)I(i,l)]\n=P/bracketleftbig\n(Xi,Xj)∈ T(PXX′),(Xi,Xl)∈ T(PXX′)/bracketrightbig\n(334)\n=/summationdisplay\n(xi,xj,xl)∈T3(QX)P(xi,xj,xl)1{{(xi,xj)∈ T(PXX′)}∩{(xi,xl)∈ T(PXX′)}} (335)\n≤1\n(1−e−nδ)3/summationdisplay\n(xi,xj,xl)∈T3(QX)P(xi)P(xj)P(xl)1{{(xi,xj)∈ T(PXX′)}∩{(xi,xl)∈ T(PXX′)}}\n(336)\n=1\n(1−e−nδ)3/summationdisplay\nxi∈T(QX)P(xi)P/bracketleftbig\n(xi,Xj)∈ T(PXX′)/bracketrightbig\nP/bracketleftbig\n(xi,Xl)∈ T(PXX′)/bracketrightbig\n(337)\n=1\n(1−e−nδ)3/summationdisplay\nxi∈T(QX)P(xi)L2(PXX′) (338)\n=1\n(1−e−nδ)3L2(PXX′), (339)\nwhere (336) follows from Lemma 4 and Lemma 3.\n•i/\\e}atio\\slash=k,j=l. The proof is similar to Case 1.\n•i/\\e}atio\\slash=k,j/\\e}atio\\slash=l. Then, we have\nE[I(i,j)I(k,l)]\n=P/bracketleftbig\n(Xi,Xj)∈ T(PXX′),(Xk,Xl)∈ T(PXX′)/bracketrightbig\n(340)\n=/summationdisplay\n(xi,xj,xk,xl)∈T4(QX)P(xi,xj,xk,xl)1{{(xi,xj)∈ T(PXX′)}∩{(xk,xl)∈ T(PXX′)}} (341)\n≤1\n(1−e−nδ)4/summationdisplay\n(xi,xj,xk,xl)∈T4(QX)P(xi)P(xj)P(xk)P(xl)\n×1{{(xi,xj)∈ T(PXX′)}∩{(xk,xl)∈ T(PXX′)}} (342)\n=1\n(1−e−nδ)4P/bracketleftbig\n(Xi,Xj)∈ T(PXX′)/bracketrightbig\nP/bracketleftbig\n(Xk,Xl)∈ T(PXX′)/bracketrightbig\n(343)\n=1\n(1−e−nδ)4L2(PXX′), (344)\nwhere (342) follows from Lemma 4 and Lemma 3.\nFrom (339) and (344), for any pairs (i,j)∈[M]2\n∗and(k,l)∈[M]2\n∗such that (i,j)/\\e}atio\\slash= (k,l), we have\nE[I(i,j)I(k,l)]≤1\n(1−e−nδ)4L2(PXX��), (345)\nand we obtain (34).\nFinally, by [28], it is easy to see that\nL(PXX′).=e−nIP(X;X′). (346)\nHence, we obtain (35) and (36) from (33) and (34), respective ly.\nThis concludes our proof of Lemma 5.\nNovember 23, 2022 DRAFT40\nAPPENDIX C\nPROOF OF LEMMA 6\nObserve that\nE[N(PXX′)] =E/bracketleftbigg/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)/bracerightbig/bracketrightbigg\n(347)\n=/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m/braceleftbigg/summationdisplay\n(xm,xm′)∈T(PXX′):\nd(xm,xm′)>∆P(xm,xm′)+/summationdisplay\n(xm,xm′)∈T(PXX′):\nd(xm,xm′)≤∆P(xm,xm′)/bracerightbigg\n. (348)\nOn the other hand, by Lemma 2, under the condition 24, it holds that\nP(xm,xm′) = 0 (349)\nifd(xm,xm′)≤∆, and\n1−4δ2\nn\n|T(QX)|2e−2δn≤P(xm,xm′)≤1\n(1−e−nδ)2|T(QX)|2(350)\nifd(xm,xm′)>∆.\nFrom (348), (349) and (350), for any joint type PXX′such that d(PXX′)>∆, we obtain\nE[N(PXX′)]≥/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\n(xm,xm′)∈T(PXX′):\nd(xm,xm′)>∆P(xm,xm′) (351)\n≥/summationdisplay\nm/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\n(xm,xm′)∈T(PXX′):\nd(xm,xm′)>∆1−4δ2\nn\n|T(QX)|2e−2δn(352)\n=M(M−1)/summationdisplay\n(xm,xm′)∈T(PXX′)\nd(PXX′)>∆1−4δ2\nn\n|T(QX)|2e−2δn(353)\n=M(M−1)|T(PXX′)|1−4δ2\nn\n|T(QX)|2e−2δn(354)\n≥(n+1)−3|X|2(1−4δ2\nn)e−2δnen(2R−IP(X;X′)), (355)\nwhere (355) follows from [28].\nThen, as nsufficiently large, we have\nP/bracketleftbig\nE(PXX′)/bracketrightbig\n=P/bracketleftbig\nN(PXX′)<(1−4δ2\nn)e−2δnexp{n[2R−IP(X;X′)−ε]}/bracketrightbig\n(356)\n≤P/bracketleftbig\nN(PXX′)< e−nε/2E[N(PXX′)]/bracketrightbig\n(357)\n=P/bracketleftbiggN(PXX′)\nE[N(PXX′)]−1<−(1−e−nε/2)/bracketrightbigg\n(358)\n≤Var(N(PXX′))\n(1−e−nε/2)2/parenleftbig\nE[N(PXX′)]/parenrightbig2, (359)\nwhere (357) follows from (355), and (359) follows from Cauch y-Schwarz inequality.\nNow, let\nI(m,m′)/defines1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)/bracerightbig\n, (360)\nNovember 23, 2022 DRAFT41\nand\nL(PXX′)/defines|T(PXX′)|\n|T(QX)|2. (361)\nThen, it holds that [28],\nL(PXX′)≥(n+1)−3|X|e−nIP(X;X′). (362)\nHence, as nsufficiently large, we have\nM(M−1)L(PXX′)≥(n+1)−3|X|en(2R−1\nn−IP(X;X′))(363)\n≥(n+1)−3|X|en(ε−1\nn)(364)\n≥enε/2, (365)\nwhere (365) follows from ε >>(logn)/√n.\nIn addition, for any two fixed pairs (m,m′)and(˜m,ˆm)in[M]2\n∗such that (m,m′)/\\e}atio\\slash= (˜m,ˆm), by Lemma 5, we\nhave\n(1−4δ2\nn)e−2δnL(PXX′)≤E[I(m,m′)]≤1\n(1−e−nδ)2L(PXX′), (366)\nand\nE[I(m,m′)I(˜m,ˆm)]≤1\n(1−e−nδ)4L2(PXX′). (367)\nIt follows that\nVar(N(PXX′)) =E[N2(PXX′)]−/parenleftbig\nE[N(PXX′)]/parenrightbig2(368)\n=/summationdisplay\nm,m′,˜m,ˆmE[I(m,m′)I(˜m,ˆm)]−/parenleftbig\nE[N(PXX′)]/parenrightbig2(369)\n=/summationdisplay\nm,m′E[I(m,m′)]+/summationdisplay\n(m,m′)/\\⌉}atio\\slash=(˜m,ˆm)E/bracketleftbig\nI(m,m′)I(˜m,ˆm)/bracketrightbig\n−/parenleftbig\nE[N(PXX′)]/parenrightbig2(370)\n≤M(M−1)1\n(1−e−nδ)2L(PXX′)\n+M(M−1)[M(M−1)−1]1\n(1−e−nδ)4L2(PXX′)−/parenleftbig\nE[N(PXX′)]/parenrightbig2. (371)\nFrom (354), (359), and (371), as nsufficiently large, we have\nP/bracketleftbig\nE(PXX′)/bracketrightbig\n≤Var(N(PXX′))\n(1−e−nε/2)2/parenleftbig\nE[N(PXX′)]/parenrightbig2(372)\n≤1\n(1−e−nε/2)2/bracketleftbiggM(M−1)1\n(1−e−nδ)2L(PXX′)+M(M−1)[M(M−1)−1]1\n(1−e−nδ)4L2(PXX′)\n/parenleftbig\n(1−4δ2n)e−2δnM(M−1)L(PXX′)/parenrightbig2−1/bracketrightbigg\n(373)\n≤1\n(1−e−nε/2)2/bracketleftbigge4δn\n/parenleftbig\n1−4δ2n/parenrightbig2/parenleftbig\n1−e−nδ/parenrightbig2/parenleftbigg1\nM(M−1)L(PXX′)/parenrightbigg\n+e4δn\n(1−4δ2n)2(1−e−nδ)4−1/bracketrightbigg\n(374)\n≤1\n(1−e−nε/2)2/bracketleftbigge4δn\n/parenleftbig\n1−4δ2n/parenrightbig2/parenleftbig\n1−e−nδ/parenrightbig2e−nε/2+e4δn\n(1−4δ2n)2(1−e−nδ)4−1/bracketrightbigg\n, (375)\nNovember 23, 2022 DRAFT42\nwhere (375) follows from (365).\nAPPENDIX D\nPROOF OF LEMMA 7\nIt is clear that (41) holds if IP(X;X′) = 0 since the LHS of this inequality is equal to 0for this case. Now, we\nconsider the case IP(X;X′)>0. Then, we can choose δ(ε)such that 0< δ(ε)<< such that IP(X;X′)> δ(ε).\nWith an abuse of notation, we assume that δ(ε) =ε.\nNow, observe that\nN(PXX′) =M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1{(Xm,Xm′)∈ T(PXX′)}. (376)\nBy Lemma 5, we have\nE[1{(Xm,Xm′)∈ T(PXX′)}].=e−nIP(X;X′),∀(m,m′)∈[M]2\n∗, (377)\nwhich leads to\np/defines1\nM(M−1)E[N(PXX′)] (378)\n.=e−nIP(X;X′). (379)\nBy choosing t=en(2R−IP(X;X′)+ε)+1, then it is clear that\nM(M−1)p≤t−1< M(M−1)−1 (380)\nasnsufficiently large if IP(X;X′)>0and choose εsuch that 0< ε << . Then, by applying Lemma 15, we\nobtain\nP/bracketleftbig\nN(PXX′)≥en(2R−IP(X;X′)+ε)/bracketrightbig˚≤exp/braceleftbig\n−M(M−1)D/parenleftbig\ne−n(IP(X;X′)−ε)/⌊ard⌊le−nIP(X;X′)/parenrightbig/bracerightbig\n. (381)\nNow, by using the fact that D(a/⌊ard⌊lb)≥a/parenleftbig\nloga\nb−1/parenrightbig\n[32], we have\nD/parenleftbig\ne−n(IP(X;X′)−ε)/⌊ard⌊le−nIP(X;X′)/parenrightbig\n≥e−n(IP(X;X′)−ε)/parenleftbig\nnε−1/parenrightbig\n. (382)\nFrom (381) and (382), we obtain (41). Finally, (42) is a strai ghtforward consequence of (41). This concludes our\nproof of Lemma 7.\nAPPENDIX E\nPROOF OF LEMMA 8\nSimilar to the proof of Lemma 8, by applying Lemma 15 with t=enε, we finally have\nP/bracketleftbig\nN(PXX′)≥enε/bracketrightbig˚≤exp/braceleftbig\n−M(M−1)D(en(ε−2R)/⌊ard⌊le−nIP(X;X′))/bracerightbig\n. (383)\nOn the other hand, we have\nD(en(ε−2R)/⌊ard⌊le−nIP(X;X′))≥en(ε−2R)/parenleftbig\nn(ε−2R+IP(X;X′)−1). (384)\nFrom (383) and (384), we obtain (43) and (44).\nNovember 23, 2022 DRAFT43\nAPPENDIX F\nPROOF OF LEMMA 9\nObserve that\nE[N(PXX′)] =M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=mE[I(m,m′)] (385)\n.=en(2R−IP(X;X′))(386)\nwhere (386) follows from Lemma 5. An upper bound in (45) simpl y follows from Markov’s inequality and (386).\nTo show the lower bound, we use Suen’s correlation inequalit y [15, Appendix A]. However, the dependency\ngraph is now different from the one in [15, Proof of Lemma 6]. I n this new dependency graph, each vertex (i,j)\nis connected to all other vertices or M(M−1)−1vertices. Using the results of Lemma 5, we have\nΘ : =1\n2/summationdisplay\n(i,j)∈[M]2∗/summationdisplay\n(k,l)∈[M]2∗,(k,l)/\\⌉}atio\\slash=(i,j)E[I(i,j)I(k,l)] (387)\n.\n≤1\n2e2nRe2nRe−2nIP(X;X′)(388)\n.=en(4R−2IP(X;X′)), (389)\nand\nΩ = max\n(i,j)∈[M]2∗/summationdisplay\n(k,l)∈[M]2∗,(k,l)/\\⌉}atio\\slash=(i,j)E[I(k,l)] (390)\n.=e2nRe−nIP(X;X′)(391)\n.=en(2R−IP(X;X′)). (392)\nIn addition, we have\n∆ =E[N(PXX′)] (393)\n.=en(2R−IP(X;X′)). (394)\nFrom (389), (392), and (394), we obtain\n∆2\n8Θ.\n≥1, (395)\nand\n∆\n6Ω.= 1. (396)\nNow, by [15, Eq. (A.6)], we have\nP[N(PXX′) = 0]≤exp/braceleftbigg\n−min/parenleftbigg∆2\n8Θ,∆\n6Ω,∆\n2/parenrightbigg/bracerightbigg\n(397)\n.\n≤exp/braceleftbigg\n−min/parenleftbigg\n1,1,1\n2en(2R−IP(X;X′))/parenrightbigg/bracerightbigg\n(398)\n= exp/braceleftbigg\n−1\n2en(2R−IP(X;X′))/bracerightbigg\n, (399)\nNovember 23, 2022 DRAFT44\nwhere (399) follows from the assumption IP(X;X′)≥2R.\nFrom (399), by using the same arguments as [15, Proof of Lemma 6], we obtain\nP/bracketleftbig\nN(PXX′)≥1/bracketrightbig.\n≥exp{n(2R−IP(X;X′))}, (400)\nwhich is compatible with the upper bound, proving Lemma 9.\nAPPENDIX G\nPROOF OF LEMMA 10\nFrom Lemma 7 and the fact that 0 =e−n∞, it holds that\nP[N(PXX′)≥ens].= exp(−n∞) (401)\nifs >[2R−IP(X;X′)]+.\nNow, for s <[2R−IP(X;X′)]+and2R≤IP(X;X′), thens≤0. It follows that\nP[N(PXX′)≥ens] =P[N(PXX′)≥1] (402)\n.= exp/braceleftbig\nn(2R−IP(X;X′))/bracerightbig\n(403)\n= exp/braceleftbig\n−n[IP(X;X′)−2R]+/bracerightbig\n, (404)\nwhere (403) follows from Lemma 9.\nOn the other hand, for s <[2R−IP(X;X′)]+and2R > IP(X;X′), then we have\nP[N(PXX′)≥ens]≤1 (405)\n= exp/braceleftbig\n−n[IP(X;X′)−2R]+/bracerightbig\n. (406)\nIn addition, for this case, there exists ε >0such that 2ε≤min{2R−IP(X;X′),[2R−IP(X;X′)]+−s}. Hence,\nby applying Lemma 6, we have\nP/bracketleftbig\nN(PXX′)≥(1−4δ2\nn)e−2δnexp{n[2R−IP(X;X′)−ε]}/bracketrightbig\n→1. (407)\nFurthermore, as nsufficiently large, we also have\nP[N(PXX′)≥ens]≥P/bracketleftbig\nN(PXX′)≥en(2R−IP(X;X′)−2ε)/bracketrightbig\n(408)\n≥P/bracketleftbig\nN(PXX′)≥(1−4δ2\nn)e−2δnexp{n[2R−IP(X;X′)−ε]}/bracketrightbig\n(409)\n= 1+o(1) (410)\n= (1+o(1))exp/braceleftbig\n−n[IP(X;X′)−2R]+/bracerightbig\n(411)\n.= exp/braceleftbig\n−n[IP(X;X′)−2R]+/bracerightbig\n, (412)\nwhere (410) follows from (407), and (411) follows from [IP(X;X′)−2R]+= 0 for2R > IP(X;X′).\nFrom (406) and (412), we obtain\nP[N(PXX′)≥ens].= exp/braceleftbig\n−n[IP(X;X′)−2R]+/bracerightbig\n(413)\nNovember 23, 2022 DRAFT45\nfors <[2R−IP(X;X′)]+and2R > IP(X;X′).\nBy combining (404) and (413), we have\nP[N(PXX′)≥ens].= exp/braceleftbig\n−n[IP(X;X′)−2R]+/bracerightbig\n(414)\nfor alls <[2R−IP(X;X′)]+.\nFinally, from (401) and (414), we obtain\nE(R,P,s) =\n\n[IP(X;X′)−2R]+,[2R−IP(X;X′)]+> s\n+∞, [2R−IP(X;X′)]+< s. (415)\nThis concludes our proof of Lemma 10.\nAPPENDIX H\nPROOF OF LEMMA 11\nFirst, we prove the following auxiliary lemma.\nLemma 17: For anyx∈[0,M−1], the following holds:\n1−(1−x)M<2e−Mx(416)\nasMsufficiently large.\nProof of Lemma 17: Letg(x)/defines1−(1−x)M−2e−Mx. This function has positive first-order derivative,\nhenceg(x)is increasing. Hence, for any x∈[0,M−1], we have\ng(x)≤g(M−1) (417)\n= 1−/parenleftbigg\n1−1\nM/parenrightbiggM\n−2\ne(418)\n→1−3\neasM→ ∞ (419)\n<0, (420)\nwhere (419) follows from/parenleftbig\n1+1\nx/parenrightbig−x→1/easx→ ∞ . This concludes our proof of Lemma 17.\nNow, we return to prove Lemma H. Observe that\nN(PXX′) =M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1{(Xm,Xm′)∈ T(PXX′)}. (421)\nIt follows that\nE[N(PXX′)] =M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=mP{(Xm,Xm′)∈ T(PXX′)} (422)\n.=en(2R−IP(X;X′)), (423)\nwhere (423) follows from Lemma 5. Then, we have\nP/braceleftbigg\nN(PXX′)≤e−nεE[N(PXX′)]/bracerightbigg\n.\n≤P/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)/bracerightbig\n≤en(2R−IP(X;X′)−ε)/bracerightbigg\n. (424)\nNovember 23, 2022 DRAFT46\nWe consider two cases:\n•The condition (48) holds.\nOn the space Xn×Xn···×Xn\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nMtermsdefine a probability measure PΠsuch that\nPΠ(x1,x2,···,xM) =M/productdisplay\nm=1P[Xm=xm],∀(x1,x2,···,xM)∈ Xn×Xn···×Xn\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nMterms. (425)\nThen, for this case, for any PXX′∈ D, we have\nP/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)/bracerightbig\n≤en(2R−IP(X;X′)−ε)/bracerightbigg\n=/summationdisplay\nx1,x2,···,xMP(x1,x2,···,xM)/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(xm,xm′)∈ T(PXX′)/bracerightbig\n≤en(2R−IP(X;X′)−ε)/bracerightbigg\n(426)\n≤1\n(1−e−nδ)M/summationdisplay\nx1,x2,···,xMPΠ(x1,x2,···,xM)/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(xm,xm′)∈ T(PXX′)/bracerightbig\n≤en(2R−IP(X;X′)−ε)/bracerightbigg\n(427)\n=e−enRlog(1−e−nδ)PΠ/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)/bracerightbig\n≤en(2R−IP(X;X′)−ε)/bracerightbigg\n(428)\n˚≤e−enRlog(1−e−nδ)exp/braceleftbigg\n−min/parenleftbigg\nen(2R−IP(X;X′)),enR/parenrightbigg/bracerightbigg\n, (429)\nwhere (427) follows from Lemma 4, and (429) follows from [15, Lemma 2].\nFrom (424) and (429), we obtain\nmin\nPXX′∈DP/braceleftbigg\nN(PXX′)≤e−nεE[N(PXX′)]/bracerightbigg\n˚≤e−enRlog(1−e−nδ)exp/braceleftbigg\n−min/parenleftbigg\nen(2R−minPXX′∈DIP(X;X′)),enR/parenrightbigg/bracerightbigg\n(430)\n˚≤e−enRlog(1−e−nδ)exp/braceleftbig\n−en(R−2δ)/bracerightbig\n(431)\n˚ = exp/braceleftbig\n−en(R−2δ)/bracerightbig\n, (432)\nwhere (431) follows from minPXX′∈DIP(X;X′)≤R+2δfor this case, and (432) follows from −log(1−e−nδ)∼\ne−nδ.\n•Case 2: The condition (49) holds.\nFor this case, observe that\nP/braceleftbigg\nN(PXX′)> e−nεE[N(PXX′)]/bracerightbigg\n≥P/braceleftbigg/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)/bracerightbig\n> en(2R−IP(X;X′)−ε)/bracerightbigg\n∩/braceleftbigg\nmin\n(m,m′)∈[M]2\n∗d(Xm,Xm′)>∆/bracerightbigg/bracerightbigg\n(433)\n=/summationdisplay\nx1,x2,···,xMP(x1,x2,···,xM)\nNovember 23, 2022 DRAFT47\n×1/braceleftbigg/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(xm,xm′)∈ T(PXX′)/bracerightbig\n> en(2R−IP(X;X′)−ε)/bracerightbigg\n∩/braceleftbigg\nmin\n(m,m′)∈[M]2∗d(xm,xm′)>∆/bracerightbigg/bracerightbigg\n(434)\n≥/summationdisplay\nx1,x2,···,xMPΠ(x1,x2,···,xM)\n×1/braceleftbigg/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(xm,xm′)∈ T(PXX′)/bracerightbig\n> e−nεE[N(PXX′)]/bracerightbigg\n∩/braceleftbigg\nmin\n(m,m′)∈[M]2∗d(xm,xm′)>∆/bracerightbigg/bracerightbigg\n(435)\n=PΠ/braceleftbigg/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)/bracerightbig\n> e−nεE[N(PXX′)]/bracerightbigg\n∩/braceleftbigg\nmin\n(m,m′)∈[M]2∗d(Xm,Xm′)>∆/bracerightbigg/bracerightbigg\n,\n(436)\nwhere (435) follows from Lemma 4 with M′=Mand Lemma 3.\nFrom (436), we have\nP/braceleftbigg\nN(PXX′)≤e−nεE[N(PXX′)]/bracerightbigg\n≤Pr\nΠ/braceleftbigg/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)/bracerightbig\n≤e−nεE[N(PXX′)]/bracerightbigg\n∪/braceleftbigg\nmin\n(m,m′)∈[M]2∗d(Xm,Xm′)≤∆/bracerightbigg/bracerightbigg\n(437)\n=PΠ/braceleftbiggM/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)/bracerightbig\n≤e−nεE[N(PXX′)]/bracerightbigg\n+PΠ/braceleftbigg\nmin\n(m,m′)∈[M]2\n∗d(Xm,Xm′)≤∆/bracerightbigg\n.\n(438)\nNow, observe that\n/braceleftbigg\nmin\n(m,m′)∈[M]2\n∗d(Xm,Xm′)≤∆/bracerightbigg\n=/braceleftbiggM/uniondisplay\nm=1/uniondisplay\nm′/\\⌉}atio\\slash=m{d(Xm,Xm′)≤∆}/bracerightbigg\n(439)\n=/braceleftbiggM/uniondisplay\nm=1/uniondisplay\nm′/\\⌉}atio\\slash=m/uniondisplay\n˜PXX′∈Q(QX):d(˜PXX′)≤∆{(Xm,Xm′)∈ T(˜PXX′)}/bracerightbigg\n. (440)\nTherefore, we have\nPΠ/braceleftbigg\nmin\n(m,m′)∈[M]2\n∗d(Xm,Xm′)≤∆/bracerightbigg\n=PΠ/braceleftbiggM/uniondisplay\nm=1/uniondisplay\nm′/\\⌉}atio\\slash=m/uniondisplay\n˜PXX′∈Q(QX):d(˜PXX′)≤∆{(Xm,Xm′)∈ T(˜PXX′)}/bracerightbigg\n(441)\n≤/summationdisplay\n˜PXX′∈Q(QX):d(˜PXX′)≤∆M/summationdisplay\nm=1PΠ/braceleftbigg/uniondisplay\nm′/\\⌉}atio\\slash=m{(Xm,Xm′)∈ T(˜PXX′)}/bracerightbigg\n. (442)\nNovember 23, 2022 DRAFT48\nNow, for any joint-type ˜PXX′∈ Q(QX)such that d(˜PXX′)≤∆, we have\nPΠ/braceleftbigg/uniondisplay\nm′/\\⌉}atio\\slash=m{(Xm,Xm′)∈ T(˜PXX′)}/bracerightbigg\n=E/bracketleftbigg\nPΠ/braceleftbigg/uniondisplay\nm′/\\⌉}atio\\slash=m{(Xm,Xm′)∈ T(˜PXX′)}/vextendsingle/vextendsingle/vextendsingle/vextendsingleXm/bracerightbigg/bracketrightbigg\n(443)\n= 1−E/bracketleftbigg\nPΠ/braceleftbigg/intersectiondisplay\nm′/\\⌉}atio\\slash=m{(Xm,Xm′)/∈ T(˜PXX′)}/vextendsingle/vextendsingle/vextendsingle/vextendsingleXm/bracerightbigg/bracketrightbigg\n(444)\n= 1−E/bracketleftbigg/parenleftbigg\nPΠ/braceleftbig\n(Xm,XmmodM+1)/∈ T(˜PXX′)/vextendsingle/vextendsingle/vextendsingle/vextendsingleXm/bracerightbig/parenrightbiggM/bracketrightbigg\n(445)\n.= 1−/parenleftbig\n1−e−nI˜P(X;X′)/parenrightbigM, (446)\nwhere (446) follows from the standard calculation (eg. [28] ).\nNow, from the condition (49), we have\nR≤ min\n˜PXX′∈Q(QX):d(˜PXX′)≤∆I˜P(X;X′)−2δ, (447)\nwhich leads to\ne−nmin˜PXX′∈Q(QX):d(˜PXX′)≤∆I˜P(X;X′)≤e−nR=M−1. (448)\nFrom (442) and (446), we obtain\nPΠ/braceleftbigg\nmin\n(m,m′)∈[M]2∗d(Xm,Xm′)≤∆/bracerightbigg\n.\n≤M/bracketleftbigg\n1−/parenleftbig\n1−e−nmin˜PXX′∈Q(QX):d(˜PXX′)≤∆I˜P(X;X′)/parenrightbigM/bracketrightbigg\n(449)\n˚≤2Mexp/braceleftbigg\n−Me−nmin˜PXX′∈Q(QX):d(˜PXX′)≤∆I˜P(X;X′)/bracerightbigg\n(450)\n˚≤exp/braceleftbigg\n−en/parenleftbig\nR−min˜PXX′∈Q(QX):d(˜PXX′)≤∆I˜P(X;X′)/parenrightbig/bracerightbigg\n(451)\n˚≤exp/braceleftbigg\n−en/parenleftbig\n2R+2δ−min˜PXX′∈DI˜P(X;X′)/parenrightbig/bracerightbigg\n(452)\nwhere (450) follows from Lemma 17 with (448), (452) follows f rom the condition (49).\nOn the other hand, by [15, Prep. 6], we have\nPΠ/braceleftbigg\nN(PXX′)≤e−nεE[N(PXX′)]/bracerightbigg\n(453)\n.=PΠ/braceleftbigg\nN(PXX′)≤e−nεen(2R−IP(X;X′))/bracerightbigg\n(454)\n˚≤exp/braceleftbigg\n−en(2R−IP(X;X′))/bracerightbigg\n. (455)\nFrom (452) and (455), under the condition (49), we have\nmin\nPXX′∈DP/braceleftbigg\nN(PXX′)≤e−nεE[N(PXX′)]/bracerightbigg\n˚≤exp/braceleftbigg\n−en(2R−minPXX′∈DIP(X;X′))/bracerightbigg\n. (456)\nFinally, we obtain by combining (432) for the case 1 and (456) for the case 2.\nThis concludes our proof of Lemma H.\nNovember 23, 2022 DRAFT49\nAPPENDIX I\nPROOF OF LEMMA 12\nDefine a new probability measure ΠonXn×Xn···×Xn\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nM times:\nPΠ(x1,x2,···,xM) =M/productdisplay\nm=1P[Xm=xm],∀(x1,x2,···,xM). (457)\nObserve that\nP/parenleftbig\nF0/parenrightbig\n=P/braceleftbigg/summationdisplay\nPXX′∈A1∪A2N(PXX′) = 0/bracerightbigg\n(458)\n=P/braceleftbigg/summationdisplay\nPXX′∈A1∪A2M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1{(Xm,Xm′)∈ T(PXX′)}= 0/bracerightbigg\n(459)\n=P/braceleftbigg/intersectiondisplay\nPXX′∈A1∪A2M/intersectiondisplay\nm=1/intersectiondisplay\nm′/\\⌉}atio\\slash=m{(Xm,Xm′)∈ T(PXX′)}c/bracerightbigg\n(460)\n=P/braceleftbigg/intersectiondisplay\nPXX′∈A1∪A2M/intersectiondisplay\nm=1/intersectiondisplay\nm′/\\⌉}atio\\slash=m{{(Xm,Xm′)∈ T(PXX′)}∩{d(Xm,Xm′)>∆}}c/bracerightbigg\n(461)\n=P/braceleftbigg/intersectiondisplay\nPXX′∈A1∪A2M/intersectiondisplay\nm=1/intersectiondisplay\nm′/\\⌉}atio\\slash=m{(Xm,Xm′)/∈ T(PXX′)}∪{d(Xm,Xm′)≤∆}/bracerightbigg\n(462)\n=/summationdisplay\nx1,x2,···,xMP(x1,x2,···,xM)\n×/productdisplay\nPXX′∈A1∪A2M/productdisplay\nm=1/productdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbigg\n{(xm,xm′)/∈ T(PXX′)}∪{d(xm,xm′)≤∆}/bracerightbigg\n(463)\n=/summationdisplay\nx1,x2,···,xMP(x1,x2,···,xM)\n×/productdisplay\nPXX′∈A1∪A2M/productdisplay\nm=1/productdisplay\nm′/\\⌉}atio\\slash=m/parenleftbigg\n1−1/braceleftbig\n{(xm,xm′)∈ T(PXX′)}∩{d(xm,xm′)>∆}/bracerightbig/parenrightbigg\n(464)\n≥/summationdisplay\nx1,x2,···,xMP(x1,x2,···,xM)\n×/productdisplay\nPXX′∈A1∪A2M/productdisplay\nm=1/productdisplay\nm′/\\⌉}atio\\slash=m/parenleftbigg\n1−1/braceleftbig\n(xm,xm′)∈ T(PXX′)}/parenrightbigg\n1{d(xm,xm′)>∆} (465)\n=/summationdisplay\nx1,x2,···,xMP(x1,x2,···,xM)\n×/productdisplay\nPXX′∈A1∪A2M/productdisplay\nm=1/productdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n(xm,xm′)/∈ T(PXX′)}1{d(xm,xm′)>∆} (466)\n≥/summationdisplay\nx1,x2,···,xMPΠ(x1,x2,···,xM)\n×ΠPXX′∈A1∪A2ΠM\nm=1Πm′/\\⌉}atio\\slash=m1/braceleftbig\n(xm,xm′)/∈ T(PXX′)}1{d(xm,xm′)>∆} (467)\nNovember 23, 2022 DRAFT50\n=PΠ/braceleftbigg/intersectiondisplay\nPXX′∈A1∪A2M/intersectiondisplay\nm=1/intersectiondisplay\nm′/\\⌉}atio\\slash=m{(Xm,Xm′)/∈ T(PXX′)}∩{d(Xm,Xm′)>∆}/bracerightbigg\n(468)\n=PΠ/braceleftbigg/summationdisplay\nPXX′∈A1∪A2M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n{(Xm,Xm′)∈ T(PXX′)}∪{d(Xm,Xm′)≤∆}/bracerightbig\n= 0/bracerightbigg\n, (469)\nwhere (461) follows from d(PXX′)>∆for allPXX′∈A1∪A2andd(xm,xm′) =d(ˆPxm,xm′), (465)\nfollows from the fact that 1−1/braceleftbig/braceleftbig\n(xm,xm′)∈ T(PXX′)} ∩ {d(xm,xm′)>∆}/bracerightbig\n=/parenleftbig\n1−1/braceleftbig\n(xm,xm′)∈\nT(PXX′)}/parenrightbig\n1{d(xm,xm′)>∆}ifd(xm,xm′)>∆and1−1/braceleftbig/braceleftbig\n(xm,xm′)∈ T(PXX′)} ∩ {d(xm,xm′)>\n∆}/bracerightbig\n≥0 =/parenleftbig\n1−1/braceleftbig\n(xm,xm′)∈ T(PXX′)}/parenrightbig\n1{d(xm,xm′)>∆}ifd(xm,xm′)≤∆, (467) follows from [7,\nLemma 4] and Lemma 4.\nTo apply Lemma (21), we form a dependency graph as follows. De fine the family of Bernoulli random variables\n{I(m,m′,PXX′)}PXX′∈A1∪A2,(m,m′)∈[M]2\n∗, where\nI(m,m′,PXX′)/defines1/braceleftbig\n(Xm,Xm′)∈ T(PXX′)∪{d(Xm,Xm′)≤∆}/bracerightbig\n. (470)\nThen, we have\nEΠ[I(m,m′,PXX′)] =PΠ/braceleftbig\n(Xm,Xm′)∈ T(PXX′)∪{d(Xm,Xm′)≤∆}/bracerightbig\n(471)\n≤PΠ{(Xm,Xm′)∈ T(PXX′)}+PΠ/braceleftbig\nd(Xm,Xm′)≤∆}. (472)\nOn the other hand, we have\nPΠ/braceleftbig\nd(Xm,Xm′)≤∆}=/summationdisplay\nxm,xm′PΠ(xm,xm′)1{d(xm,xm′)≤∆} (473)\n=/summationdisplay\nxm,xm′P(xm)P(xm′)1{d(xm,xm′)≤∆} (474)\n=/summationdisplay\nPXX′∈Q(QX)/summationdisplay\n(xm,xm′)∈T(PXX′)P(xm)P(xm′)1{d(xm,xm′)≤∆} (475)\n=/summationdisplay\nPXX′∈Q(QX)/summationdisplay\n(xm,xm′)∈T(PXX′)1\n|T(QX)|21{d(xm,xm′)≤∆} (476)\n=/summationdisplay\nPXX′∈Q(QX)/summationdisplay\n(xm,xm′)∈T(PXX′)1\n|T(QX)|21{d(PXX′)≤∆} (477)\n.= max\nPXX′∈Q(QX):d(PXX′)≤∆e−nIP(X;X′)(478)\n=e−nminPXX′∈Q(QX):d(PXX′)≤∆IP(X;X′)(479)\n≤e−nmaxPXX′∈Q(QX):d(PXX′)>∆IP(X;X′), (480)\nwhere (476) follows from 3, and (480) holds by the condition ( 60) under (59).\nNovember 23, 2022 DRAFT51\nIt follows from (472) and (480) that\nEΠ[I(m,m′,PXX′)]≤PΠ{(Xm,Xm′)∈ T(PXX′)}+e−nmaxPXX′∈Q(QX):d(PXX′)>∆IP(X;X′)(481)\n.=e−nIP(X;X′)+e−nmaxPXX′∈Q(QX):d(PXX′)>∆IP(X;X′)(482)\n≤e−nIP(X;X′)+e−nIP(X;X′)(483)\n.=e−nIP(X;X′), (484)\nwhere (483) follows from the fact that d(PXX′)>∆for allPXX′∈ A1∪A2.\nNow, we set\nx(m,m′,PXX′)/defines1−exp/braceleftbig\n−enIP(X;X′)/bracerightbig\n. (485)\nThen, under the condition minPXX′∈A1∪A2IP(X;X′)> R, for all(m,m′,PXX′)∈[M]2\n∗×(A1∪A2), it holds\nthat\nEΠ[I(m,m′,PXX′)] (486)\n.\n≤e−nIP(X;X′)(487)\n.= 1−exp/braceleftbigg\n−e−nIP(X;X′)/bracerightbigg\n(488)\n.=/parenleftbigg\n1−exp/braceleftbigg\n−e−nIP(X;X′)/bracerightbigg/parenrightbigg/parenleftbigg\nexp/braceleftbigg\n−e−nIP(X;X′)/bracerightbigg/parenrightbigg|A1∪A2|enR\n(489)\n=x(m,m′,PXX′)/productdisplay\n(˜m,˜m′,˜PXX′)∼(m,m′,PXX′)/parenleftbigg\n1−x(˜m,˜m′,˜PXX′)/parenrightbigg\n, (490)\nwhere (488) follows from the fact that limx→0e−x\n1−x= 1, (489) follows from |A1∪ A2| ≤ |Q(QX)|which is\nsub-exponential in nandminPXX′∈A1∪A2IP(X;X′)> R.\nThen, by applying Lemma 21 with A= [M]2\n∗×(A1∪A2)andB=∅, under the condition minPXX′∈A1∪A2IP(X;X′)>\nRwe have\nPΠ/braceleftbigg/summationdisplay\nPXX′∈A1∪A2M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m1/braceleftbig\n{(Xm,Xm′)∈ T(PXX′)}∪{d(Xm,Xm′)≤∆}/bracerightbig\n= 0/bracerightbigg\n≥min\nPXX′∈A1∪A2/parenleftbigg\nexp/braceleftbigg\n−enIP(X;X′)/bracerightbigg/parenrightbigg|A1∪A2|M(M−1)\n(491)\n˚ = exp/braceleftbigg\n−enmaxPXX′∈A1∪A2(2R−IP(X;X′))/bracerightbigg\n(492)\n= exp/braceleftbigg\n−enmaxPXX′∈A2(2R−IP(X;X′))/bracerightbigg\n, (493)\nwhere (493) follows from the definition of A1andA2.\nFinally, the condition minPXX′∈A1∪A2IP(X;X′)> R is the same as minPXX′∈A2IP(X;X′)> R , which is\nequivalent to the condition that\nE0< Eg\nex(R,QX,d,∆)/defines min\nPXX′∈Q(QX):d(PXX′)>∆,IP(X;X′)≤R/braceleftbigg\nΓ(PXX′,R)+IP(X;X′)−R/bracerightbigg\n(494)\n=Ergv\nex(R,QX,d,∆), (495)\nNovember 23, 2022 DRAFT52\nwhere (494) is obtained by using the same arguments to achiev e [15, I. (30)]. This concludes our proof of Lemma\n12.\nAPPENDIX J\nCONCENTRATION INEQUALITIES FOR SUMS OF BERNOULLI RANDOM VARIABLES\nTo obtain the TRC or develop concentration inequalities for the random coding exponents, we need to develop\nconcentration inequalities for a sum of Bernoulli random va riables. Since in RGV codebooks, all the codewords are\ncorrelated, standard concentration inequalities such as S uen’s correlation inequality [15], [33] cannot be applied.\nThe main reason is that these standard inequality require a l ocal dependency in the sum of random variables which\nonly holds for the fixed-composition or i.i.d. random ensemb les but not for RGV ones. We develop concentration\ninequalities for a sum of nterms where each term depends on all the n−1other terms. Thanks to the structure\nof all these random variables, some concentration inequali ties in the probability literature can be applied. In this\nsection, we list all these inequalities. For the newly-deve loped inequality, the proof can be found in appendices.\nLemma 18: [31, Lemma 2.1] Fix a positive number nand let{x1,x2,···,xn}be real numbers from the interval\n[0,1]. For every A⊂[n], letζAbe defined as\nζA=/productdisplay\ni∈Axi/productdisplay\ni∈[n]\\A(1−xi). (496)\nThen,\n/summationdisplay\nA⊂[n]ζA=n/summationdisplay\nj=0/summationdisplay\nA∈∂j[n]ζA= 1 (497)\nand\nn/summationdisplay\ni=1xi=n/summationdisplay\nj=0j/summationdisplay\nA∈∂j[n]ζA, (498)\nwhere∂j[n]denotes the family consisting of all subsets of [n]of cardinality j∈ {0,1,2,···,n}.\nThe following result can be also derived from Lemma 15.\nLemma 19: Suppose that X1,X2,···,Xnare random variables such that Xi∈ {0,1}, fori= 1,2,···,n. Set\np=1\nn/summationtextn\ni=1E[Xi]. Then, for any ν∈[0,p), it holds that\nP/bracketleftbiggn/summationdisplay\ni=1Xi≤n(p−ν)−1/bracketrightbigg\n≤2e−nD(p−ν/⌊ar⌈⌊lp). (499)\nProof: Let˜Xi/defines1−Xifor alli∈[n]and set˜p/defines1−p. Then, we have\n˜p=1\nnn/summationdisplay\ni=1E[˜Xi]. (500)\nLett−1 =n(1−p)+n(1−p)ε0for some ε0>0such that (1−p)(1+ε0)<1. Then, by applying Lemma 15\nfor the Bernoulli sequence ˜X1,˜X2,···,˜Xn, we have\nP/bracketleftbiggn/summationdisplay\ni=1˜Xi≥t/bracketrightbigg\n≤2e−nD(˜p(1+ε0)/⌊ar⌈⌊l˜p)(501)\n= 2e−nD((1−p)(1+ε0)/⌊ar⌈⌊l1−p). (502)\nNovember 23, 2022 DRAFT53\nFrom (502) and ˜Xi= 1−Xifor alli∈[n], we obtain\nP/bracketleftbiggn/summationdisplay\ni=1Xi≤n−t/bracketrightbigg\n≤2e−nD((1−p)(1+ε0)/⌊ar⌈⌊l1−p). (503)\nNow, by setting ε0/definesν/(1−p), we have t=n(1−p+ν)+1 . Then, from (503), we have\nP/bracketleftbiggn/summationdisplay\ni=1Xi≤n(p−ν)−1/bracketrightbigg\n≤2e−nD((1−p)(1+ε0)/⌊ar⌈⌊l1−p)(504)\n= 2e−nD(1−p+ν/⌊ar⌈⌊l1−p)(505)\n= 2e−nD(p−ν/⌊ar⌈⌊lp), (506)\nwhere (506) follows from D(a/⌊ard⌊lb) =D(1−a/⌊ard⌊l1−b). Final note is that (1−p)(1+ε0) = 1−p+ν <1for all\nν∈[0,p).\nNow, we recall the following result.\nLemma 20: [31, Theorem 1.2] There exists a universal constant c≥1satisfying the following. Suppose\nX1,X2,···,Xnare random variables such that 0≤Xi≤1, fori= 1,2,···,n. Assume further that there exists\nconstantγ∈(0,1)such that for all A⊂[n]the following condition holds true:\nE/bracketleftbigg/productdisplay\ni∈AXi/bracketrightbigg\n≤γ|A|(507)\nwhere|A|denotes the cardinality of A. Fix a real number νfrom the interval/parenleftbig\n0,1\nγ−1/parenrightbig\nand sett=nγ+nγν.\nThen,\nP/bracketleftbiggn/summationdisplay\ni=1Xi≥t/bracketrightbigg\n≤ce−nD(γ(1+ν)/⌊ar⌈⌊lγ), (508)\nwhereD(γ(1+ν)/⌊ard⌊lγ)is the Kullback-Leibler distance between γ(1+ν)andγ.\nNow, to bound the probability in (469), we recall the followi ng version of Suen’s correlation inequality lemma\nin [33].\nLemma 21: [33, Lemma 1] Let {Uk}k∈K, whereKis a set of multidimensional indexes, be a family of Bernoull i\nrandom variables. Let Gbe a dependency graph for {Uk}k∈K, i.e., a graph with vertex set Ksuch that if Aand\nBare two disjoint subsets of K, andGcontains no edge between AandB, then the families {Uk}k∈Aand\n{Uk}k∈Bare independent. Let SA/defines/summationtext\nk∈AUkfor anyA⊂ K . Moreover, we write k∼lif(k,l)is an edge in\nthe dependency graph G. Suppose further that xk,k∈ K are real numbers such that 0≤xk<1and\nE[Uk]≤xk/productdisplay\nl∼k/parenleftbig\n1−xl), k∈ K. (509)\nThen, for any two subsets A,B⊂ K, it holds that\nP/parenleftbig\nSA= 0|SB= 0/parenrightbig\n≥/productdisplay\ni∈A(1−xi). (510)\nNovember 23, 2022 DRAFT54\nAPPENDIX K\nPROOF OF LEMMA 14\nFix anm∈[M]. For any conditional type PX′Y∈ Pn(X ×Y)such that PX′=QXandPY=ˆPy, define\nNm,y(PX′Y)/defines/vextendsingle/vextendsingle/braceleftbig\nXm′: (Xm′,y)∈ T(PX′Y),m′/\\e}atio\\slash=m/bracerightbig/vextendsingle/vextendsingle (511)\n=/summationdisplay\nm���/\\⌉}atio\\slash=m1/braceleftbig\n(Xm′,y)∈ T(PX′Y)/bracerightbig\n. (512)\nObserve that\nE/bracketleftbig\n1/braceleftbig\n(Xm′,y)∈ T(PX′Y)/bracerightbig/bracketrightbig\n=P/bracketleftbig\n(Xm′,y)∈ T(PX′Y)/bracketrightbig\n(513)\n=/summationdisplay\nx′m∈T(PX′|Y)P(Xm′=xm′) (514)\n=/summationdisplay\nx′m∈T(PX′|Y)1\n|T(QX)|(515)\n.=e−nIP(X′;Y), (516)\nwhere (515) follows from Lemma 3, and (516) follows from [28] . Hence,Nm,y(PX′Y)is a sum of M−1binary-\nvalued random variables, each has the expectation e−nI(X′;Y).\nNow, from (77) and (512), we can express Zm(y)as\nZm(y) =/summationdisplay\nPX′|Y:PX′=QXNm,y(PX′Y)eng(PX′Y). (517)\nHence, by considering the randomness of {Xm′}, we have\nP/bracketleftbig\nZm(y)≤exp/braceleftbig\nnα(R−ε,ˆPy)}/bracketrightbig\n≤P/bracketleftbigg/summationdisplay\nPX′|Y:PX′=QXNm,y(PX′Y)eng(PX′Y)≤exp/braceleftbig\nnα(R−ε,ˆPy)}/bracketrightbigg\n(518)\n≤P/bracketleftbigg\nmax\nPX′|Y:PX′=QXNm,y(PX′Y)eng(PX′Y)≤exp/braceleftbig\nnα(R−ε,ˆPy)}/bracketrightbigg\n(519)\n=P/bracketleftbigg/intersectiondisplay\nPX′|Y:PX′=QX/braceleftbig\nNm,y(P)eng(PX′Y)≤exp/braceleftbig\nnα(R−ε,ˆPy)}/bracerightbig/bracketrightbigg\n(520)\n=P/bracketleftbigg/intersectiondisplay\nPX′|Y:PX′=QX/braceleftbig\nNm,y(PX′Y)≤exp/braceleftbig\nnα(R−ε,ˆPy)−g(PX′Y)}/bracerightbigg/bracketrightbigg\n. (521)\nAs mentioned above, Nm,y(PX′Y)is a sum of M−1binary-valued random variables, each has the expectation\ne−nI(X′;Y). However, different from i.i.d. random codebook ensembles , these random variables are correlated.\nAs [21, Appendix B], we argue that by the definition of α(R−ε,ˆPy), there must exist some P∗\nX′|Ysuch that\nforP∗\nX′Y/definesˆPy×P∗\nX′|Y,IP∗(X′;Y)≤R−εandR−ε−IP∗(X′;Y)≥α(R−ε,ˆPy)−g(P∗\nX′Y). To see why\nthis is true, assume conversely, that for every PX′|Y, which define PX′Y/definesˆPy×PX′|Y, eitherIP(X′;Y)> R−ε\norR−IP(X′;Y)−ε < α(R−ε,ˆPy)−g(PX′Y), which means that for every PX′Y,\nR−ε <max/braceleftbig\nIP(X′;Y),IP(X′;Y)+α(R−ε,ˆPy)−g(PX′Y)/bracerightbig\n(522)\n=IP(X′;Y)+/bracketleftbig\nα(R−ε,ˆPy)−g(PX′Y)]+, (523)\nNovember 23, 2022 DRAFT55\nwhich implies that for every PX′|Y, there exists t∈[0,1]such that\nR−ε <max/braceleftbig\nIP(X′;Y),IP(X′;Y)+α(R−ε,ˆPy)−g(PX′Y)/bracerightbig\n(524)\n=IP(X′;Y)+t/bracketleftbig\nα(R−ε,ˆPy)−g(PX′Y)], (525)\nor equivalently,\nα(R−ε,ˆPy)>max\nPX′|Y:PX′=QXmin\n0≤t≤1g(PX′Y)+R−IP(X′;Y)−ε\nt(526)\n= max\nPX′|Y:PX′=QX\n\ng(PX′Y)+R−IP(X′;Y)−ε I P(X′;Y)≤R−ε\n−∞ IP(X′;Y)> R−ε(527)\n= max\nPX′|XY:PX′=QX,\nIP(X′;Y)≤R−ε/bracketleftbig\ng(PX′Y)−IP(X′;Y)/bracketrightbig\n+R−ε (528)\n=α(R−ε,ˆPy), (529)\nwhich is a contradiction.\nNow, from (521) and the existence of P∗\nX′Yas above, it holds that\nP/bracketleftbig\nZm(y)≤exp/braceleftbig\nnα(R−ε,ˆPy)}/bracketrightbig\n≤P/bracketleftbig\nNm,y(P∗\nX′Y)≤exp/braceleftbig\nn[α(R−ε,ˆPy)−g(P∗\nX′Y)]}/bracketrightbig\n. (530)\nDifferent from [11], Ny(P∗\nX′Y)is now not the sum of i.i.d. Bernoulli random variables but th ese random variables\nare still identically distributed and weakly dependent.\nNow, let\nZm′/defines1/braceleftbig\n(Xm′,y)∈ T(P∗\nX′Y)/bracerightbig\n,∀m′∈M−/defines[M]\\{m}, (531)\nand\np/definesP/bracketleftbig\n(X2,y)∈ T(P∗\nX′Y)/bracketrightbig\n. (532)\nNow, let ν∈(0,p)be chosen such that\n(M−1)(p−ν) = exp/braceleftbig\nn[α(R−ε,ˆPy)−g(P∗\nX′Y)]/bracerightbig\n. (533)\nThe existence of νis guaranteed since (533) is equivalent to\nν=p−exp/braceleftbig\nn[α(R−ε,ˆPy)−g(P∗\nX′Y)]/bracerightbig\nM−1(534)\n≥p−exp/braceleftbig\nn[R−ε−IP∗(X′;Y)]/bracerightbig\nM−1(535)\n=p−exp/braceleftbig\nn[R−ε−IP∗(X′;Y)]/bracerightbig\nexp(nR)−1(536)\n.= exp/braceleftbig\n−nIP∗(X′;Y)/bracerightbig\n−exp/braceleftbig\n−n(IP∗(X′;Y)+ε)/bracerightbig\n>0, (537)\nsoν∈(0,p).\nNovember 23, 2022 DRAFT56\nBy applying Lemma 19 with n=M−1,Xi=Zi,p=P/bracketleftbig\n(X2,y)∈ T(P∗\nX′Y)/bracketrightbig\n, andνsatisfying (533), we\nhave\nP/bracketleftbigg\nNy(P∗\nX′Y)≤exp/braceleftbig\nn[α(R−ε,ˆPy)−g(P∗\nX′Y)]/bracerightbig/bracketrightbigg\n.=P/bracketleftbigg\nNy(P∗\nX′Y)≤exp/braceleftbig\nn[α(R−ε,ˆPy)−g(P∗\nX′Y)]/bracerightbig/bracketrightbigg\n(538)\n≤2exp/parenleftbig\n−(M−1)D(p−ν/⌊ard⌊lp)/parenrightbig\n(539)\n.= exp(−enRD(p−ν/⌊ard⌊lp)). (540)\nNow, since p.= exp(−nIP∗(X′;Y)), from (537), we also have\n(M−1)/bracketleftbig\n(γ−1)(p−ν)/bracketrightbig\n.\n≤exp(nR)/bracketleftbigg/parenleftbigg1\n1−e−nδ−1/parenrightbigg\nexp/braceleftbig\n−n(IP∗(X′;Y)+ε)/bracerightbig/bracketrightbigg\n(541)\n.\n≤e−n(δ+ε)\n1−e−nδexp/bracketleftbig\nn(R−IP∗(X′;Y))/bracketrightbig\n. (542)\nOn the other hand, we have\nexp(−enRD(p−ν/⌊ard⌊lp)) = exp/braceleftbigg\n−enRD(e−an/⌊ard⌊le−bn)/bracerightbigg\n(543)\nwherea/definesR+g(P∗\nX′Y)−α(R−ε,ˆPy)andb/definesIP∗(X′;Y). It is easy to see that\na−b=R+g(P∗\nX′Y)−α(R−ε,ˆPy)−IP∗(X′;Y) (544)\n≥ε. (545)\nHence, by using the following fact [29, Sec. 6.3]:\nD(a/⌊ard⌊lb)≥aloga\nb+b−a, (546)\nwe have\nD(e−an/⌊ard⌊le−bn)≥e−bn/bracketleftbig\n1+e(b−a)n((b−a)n−1)/bracketrightbig\n. (547)\nHence, we obtain\nexp(−enRD(p−ν/⌊ard⌊lp))≤exp/braceleftbigg\n−en(R−IP∗(X′;Y))[1−e−nε(1+nε)]/bracerightbigg\n. (548)\nFrom (540), (542), and (548), we obtain\nP/bracketleftbigg\nNm,y(P∗\nX′Y)≤exp/braceleftbig\nn[α(R−ε,ˆPy)−g(P∗\nX′Y)]/bracerightbig/bracketrightbigg\n.\n≤exp/braceleftbigge−n(δ+ε)\n1−e−nδexp/bracketleftbig\nn(R−IP∗(X′;Y))/bracketrightbig/bracerightbigg\nexp/braceleftbigg\n−en(R−IP∗(X′;Y))[1−e−nε(1+nε)]/bracerightbigg\n(549)\n= exp/braceleftbigg\n−en(R−IP∗(X′;Y))/bracketleftbigg\n1−e−n(δ+ε)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg\n(550)\n≤exp/braceleftbigg\n−enε/bracketleftbigg\n1−e−n(δ+ε)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg\n, (551)\nNovember 23, 2022 DRAFT57\nwhere (551) follows from the fact that IP∗(X′;Y)≤R−ε.\nFrom (530) and (551), we obtain\nPr/bracketleftbig\nZm(y)≤exp/braceleftbig\nnα(R−ε,ˆPy)}/bracketrightbig\n.\n≤exp/braceleftbigg\n−enε/bracketleftbigg\n1−e−n(δ+ε)\n1−e−nδ−e−nε(1+nε)/bracketrightbigg/bracerightbigg\n. (552)\nThis concludes our proof of Lemma 14.\nAPPENDIX L\nPROOF OF LEMMA 16\nThe proof is based on [15, Proof of Prep. 5]. However, there ar e some changes to account for the dependency\namong the codewords. One such an important change is to repla ce the Hoeffding’s inequality in [15, Proof of\nPrep. 5] by a generalized version of this inequality in [34].\nBy using the union bound, we have\nP{ˆBn(σ)}=P/braceleftbiggM/uniondisplay\nm=1/uniondisplay\nm′/\\⌉}atio\\slash=m/uniondisplay\nyˆBn(σ,m,m′,y)/bracerightbigg\n(553)\n≤M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\nyP/braceleftbigg\nˆBn(σ,m,m′,y)/bracerightbigg\n. (554)\nIn addition, for any joint type PXY∈ Pn(X ×Y), let\nN(PXY)/defines/summationdisplay\n˜m∈[M]\\{m,m′}1{(X˜m,y)∈ T(PXY)}, (555)\nthen we also have\nP/braceleftbigˆBn(σ,m,m′,y)/bracerightbig\n.=/summationdisplay\nPXY:PX=QX,IP(X;Y)≤RP/braceleftbigg\nN(PXY)≥en(β(R,PY)+σ−g(PXY))/bracerightbigg\n+/summationdisplay\nPXY:PX=QX,IP(X;Y)>RP/braceleftbigg\nN(PXY)≥en(β(R,PY)+σ−g(PXY))/bracerightbigg\n(556)\nwhere (556) follows from [15, Eq. (H.6)].\nNow, observe that\nP/braceleftbigg\nN(PXY)≥en(β(R,PY)+σ−g(PXY))/bracerightbigg\n≤P/braceleftbigg\nN(PXY)≥en(R+σ−IP(X;Y))/bracerightbigg\n(557)\n=P/braceleftbigg/summationdisplay\n˜m∈[M]\\{m,m′}1{(X˜m,y)∈ T(PXY)} ≥en(R+σ−IP(X;Y))/bracerightbigg\n(558)\nwhere (557) follows from [15, Eq. (H.9)].\nNovember 23, 2022 DRAFT58\nDefine a new probability measure ΠonXn×Xn···×Xn\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\nM times:\nPΠ(x1,x2,···,xM) =M/productdisplay\nm=1P(Xm=xm),∀(x1,x2,···,xM). (559)\nNote that for any A⊂[M]\\{m,m′}, under the condition (24) we have\nE/bracketleftbigg/productdisplay\n˜m∈A1{(X˜m,y)∈ T(PXY)}/bracketrightbigg\n≤1\n(1−e−nδ)|A|EΠ/bracketleftbigg/productdisplay\n˜m∈A1{(X˜m,y)∈ T(PXY)}/bracketrightbigg\n(560)\n=1\n(1−e−nδ)|A|/productdisplay\n˜m∈AP/braceleftbig\n(˜Xm,y)∈ T(PXY)/bracerightbig\n(561)\nwhere (560) follows from the change of measure and Lemma 4.\nNow, we have\nP/braceleftbig\n(˜Xm,y)∈ T(PXY)/bracerightbig\n=/summationdisplay\n˜xm∈T(PXY|y)P(˜xm) (562)\n=/summationdisplay\n˜xm∈T(PXY|y)1\n|T(QX)|(563)\n.=e−nIP(X;Y)(564)\nwhere (563) follows from Lemma 3, and (564) follows from [28] .\nFrom (561) and (564), we obtain\nE/bracketleftbigg/productdisplay\n˜m∈A1{(X˜m,y)∈ T(PXY)}/bracketrightbigg\n.\n≤γ|A|(565)\nwhere\nγ= (1−e−nδ)−1e−nIP(X;Y). (566)\nHence, if R≥IP(X;Y), we have\nP/braceleftbigg/summationdisplay\n˜m∈[M]\\{m,m′}1{(X˜m,y)∈ T(PXY)} ≥en(R+σ−IP(X;Y))/bracerightbigg\n.\n≤exp/braceleftbigg\n−enRD/parenleftbigg\n(1−e−nδ)−1eσ−IP(X;Y)/vextenddouble/vextenddouble/vextenddouble/vextenddouble(1−e−nδ)−1e−nIP(X;Y)/parenrightbigg/bracerightbigg\n(567)\n≤exp/braceleftbigg\n−enR(1−e−nδ)−1e−n(IP(X;Y)−σ)./parenleftbigg\nloge−n(IP(X;Y)−σ)\ne−nIP(X;Y)−1/parenrightbigg/bracerightbigg\n(568)\n= exp/braceleftbigg\n−(1−e−nδ)−1en(R−IP(X;Y)+σ)(nσ−1)} (569)\n˚≤exp{−enσ}, (570)\nwhere (567) follows from Lemma 20, (568) follows from the fac t thatD(a/⌊ard⌊lb)≥a/parenleftbig\nloga\nb−1/parenrightbig\n[32, p. 167], and\n(570) follows from R≥IP(X;Y).\nFrom (558) and (570), we obtain\nP/braceleftbigg\nN(PXY)≥en(β(R,PY)+σ−g(PXY))/bracerightbigg\n˚≤exp{−enσ},ifIP(X;Y)≥R. (571)\nNovember 23, 2022 DRAFT59\nSimilarly, for the case R < IP(X;Y), we have\nP/braceleftbigg\nN(PXY)≥en(β(R,PY)+σ−g(PXY))/bracerightbigg\n≤P/braceleftbig\nN(PXY)≥enσ/bracerightbig\n(572)\n=P/braceleftbigg/summationdisplay\n˜m∈[M]\\{m,m′}1{(X˜m,y)∈ T(PXY)} ≥enσ/bracerightbigg\n≤exp/braceleftbigg\n−enRD/parenleftbig\n(1−e−nδ)−1e−n(R−σ)/vextenddouble/vextenddouble(1−e−nδ)−1e−nIP(X;Y)/parenrightbig/bracerightbigg\n(573)\n= exp/braceleftbigg\n−(1−e−nδ)−1enσ[n(IP(X;Y)−R+σ)−1]/bracerightbigg\n(574)\n˚≤exp{−enσ}, (575)\nwhere (573) is obtained by applying Lemma 20 and the change of measures as the arguments to achieve (570),\nand (575) follows from the same arguments to achieve (568), a nd (575) follows from IP(X;Y)> R.\nFrom (556), (571), and (575), we obtain\nP/braceleftbigg\nˆBn(σ,m,m′,y)/bracerightbigg\n˚≤exp{−enσ}. (576)\nFrom (554) and (576), we finally obtain\nP{ˆBn(σ)}˚≤M/summationdisplay\nm=1/summationdisplay\nm′/\\⌉}atio\\slash=m/summationdisplay\nyexp{−enσ} (577)\n˚ = exp{−enσ}. (578)\nThis concludes our proof of Lemma 16.\nREFERENCES\n[1] C. E. Shannon, “A mathematical theory of communication, ”Bell System Technical Journal , vol. 27, pp. 379–423, 1948.\n[2] R. M. Fano, Transmission of Information . New York: Wiley, 1961.\n[3] R. G. Gallager, “Simple derivation of the coding theorem and some applications,” IEEE Trans. Inf. Theory , vol. 11, pp. 3–18, Jan 2008.\n[4] C. E. Shannon, R. G. Gallager, and E. R. Berlekamp, “Lower bounds to error probability for coding in discrete memoryle ss channels I-II,”\nInformation and Control , vol. 10, pp. 65–103, 522–552, 1967.\n[5] B. Nakibo˘ glu, “The sphere packing bound for memoryless channels,” Problems of Information Transmission , vol. 56, pp. 201–244, 2020.\n[6] B. Nakibo˘ glu, “The Augustin capacity and center,” Problems of Information Transmission , vol. 55, no. 4, pp. 299–342, 2019.\n[7] A. Somekh-Baruch, J. Scarlett, and A. Guill´ en i F` abreg as, “Generalized random Gilbert-Varshamov codes,” IEEE Trans. Inf. Theory ,\nvol. 65, no. 6, pp. 3452–3469, 2019.\n[8] I. Csisz´ ar and J. K¨ orner, “Graph decomposition: A new k ey to coding theorems,” IEEE Trans. Inf. Th. , vol. 27, pp. 5–11, 1981.\n[9] A. Barg and G. D. Forney, “Random codes: minimum distance s and error exponents,” IEEE Trans. Inf. Theory , vol. 48, no. 9, pp. 2568–\n2573, 2002.\n[10] A. Nazari, A. Anastasopoulos, and S. S. Pradhan, “Error exponent for multiple-access channels: Lower bounds,” IEEE Trans. Inf. Theory ,\nvol. 60, no. 9, pp. 5095–5115, 2014.\n[11] N. Merhav, “Error exponents of typical random codes,” IEEE Trans. Inf. Theory , vol. 64, no. 9, pp. 6223–6235, 2018.\n[12] N. Merhav, “Error exponents of typical random codes for the colored Gaussian channel,” IEEE Trans. Inf. Theory , vol. 65, no. 12,\npp. 8164–8179, 2019.\n[13] N. Merhav, “Error exponents of typical random trellis c odes,” IEEE Trans. Inf. Theory , vol. 66, no. 4, pp. 2067–2077, 2019.\nNovember 23, 2022 DRAFT60\n[14] N. Merhav, “A lagrange–dual lower bound to the error exp onent of the typical random code,” IEEE Trans. Inf. Theory , vol. 66, no. 6,\npp. 3456–3464, 2019.\n[15] R. Tamir, N. Merhav, N. Weinberger, and A. Guill´ en i F` a bregas, “Large deviations behavior of the logarithmic erro r probability of random\ncodes,” IEEE Trans. Inf. Theory , vol. 66, no. 11, pp. 6635–6659, 2020.\n[16] R. Ahlswede and G. Dueck, “Good codes can be produced by a few permutations,” IEEE Trans. Inf. Theory , vol. 28, no. 3, pp. 430–443,\n1982.\n[17] R. T. (Averbuch) and N. Merhav, “Universal decoding for the typical random code and for the expurgated code,” IEEE Trans. Inf. Theory ,\n2022.\n[18] G. Cocco, A. Guill´ en i F` abregas, and J. Font-Segura, “ Typical error exponents: A dual domain derivation,” IEEE Trans. Inf. Theory , to\nappear 2022.\n[19] L. V . Truong, G. Cocco, J. Font-Segura, and A. Guill´ en i F` abregas, “Concentration properties of random codes,” ArXiv , vol. abs/2203.07853,\n2022.\n[20] R. 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Th. , vol. 68,\npp. 2156–2168, Apr. 2022.\n[26] E. N. Gilbert, “A comparison of signalling alphabets,” Bell System Technical Journal , vol. 31, pp. 504–522, 1952.\n[27] R. R. Varshamov, “Estimate of the number of signals in er ror correcting codes,” Doklady Akademii Nauk SSSR , vol. 117, no. 5, pp. 739–741,\n1957.\n[28] I. Csisz´ ar, “The method of types,” IEEE Trans. Inf. Th. , vol. 44, no. 6, pp. 2505–23, 1998.\n[29] N. Merhav, “Statistical physics and information theor y,”Foundations and Trends® in Communications and Information Theory , vol. 6,\nno. 1–2, pp. 1–212, 2010.\n[30] H. Royden and P. Fitzpatrick, Real Analysis . Pearson, 4th ed., 2010.\n[31] C. Pelekis and J. Ramon, “Hoeffding’s inequality for su ms of weakly dependent random variables,” Mediterranean Journal of Mathematics\n14 (6), 1-16 , 2017.\n[32] N. Merhav, Statistical Physics and Information Theory , vol. 6 of Foundations and Trends in Communications and Information T heory .\nNow Publishers Inc, 2010.\n[33] S. Janson, “New versions of suen’s correlation inequal ity,” Random Struct. Algorithms , vol. 13, pp. 467–483, 1998.\n[34] R. Impagliazzo and V . Kabanets, “Constructive proofs o f concentration bounds,” Electron. Colloquium Comput. Complex. , vol. 17, p. 72,\n2010.\nNovember 23, 2022 DRAFT"    },    {        "title": "2211.12889v1.The_fractional_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "The fractional Landau-Lifshitz-Gilbert equation\nR.C. Verstraten1, T. Ludwig1, R.A. Duine1,2, C. Morais Smith1\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584CC Utrecht, The Netherlands\n2Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: November 24, 2022)\nThe dynamics of a magnetic moment or spin are of high interest to applications in technology.\nDissipation in these systems is therefore of importance for improvement of efficiency of devices,\nsuch as the ones proposed in spintronics. A large spin in a magnetic field is widely assumed to\nbe described by the Landau-Lifshitz-Gilbert (LLG) equation, which includes a phenomenological\nGilbert damping. Here, we couple a large spin to a bath and derive a generic (non-)Ohmic damping\nterm for the low-frequency range using a Caldeira-Leggett model. This leads to a fractional LLG\nequation, where the first-order derivative Gilbert damping is replaced by a fractional derivative of\norders≥0. We show that the parameter scan be determined from a ferromagnetic resonance\nexperiment, where the resonance frequency and linewidth no longer scale linearly with the effective\nfield strength.\nIntroduction. — The magnetization dynamics of mate-\nrials has attracted much interest because of its techno-\nlogical applications in spintronics, such as data storage\nor signal transfer [1–3]. The right-hand rule of magnetic\nforces implies that the basic motion of a magnetic mo-\nment or macrospin Sin a magnetic field Bis periodic\nprecession. However, coupling to its surrounding (e.g.,\nelectrons, phonons, magnons, and impurities) will lead\nto dissipation, which will align SwithB.\nSpintronics-based devices use spin waves to carry sig-\nnals between components [4]. Contrary to electronics,\nwhich use the flow of electrons, the electrons (or holes)\nin spintronics remain stationary and their spin degrees\nof freedom are used for transport. This provides a sig-\nnificant advantage in efficiency, since the resistance of\nmoving particles is potentially much larger than the dis-\nsipation of energy through spins. The spin waves con-\nsist of spins precessing around a magnetic field and they\nare commonly described by the Landau-Lifshitz-Gilbert\n(LLG) equation [5]. This phenomenological description\nalso includes Gilbert damping, which is a term that\nslowly realigns the spins with the magnetic field. Much\neffort is being done to improve the control of spins for\npractical applications [6]. Since efficiency is one of the\nmain motivations to research spintronics, it is important\nto understand exactly what is the dissipation mechanism\nof these spins.\nAlthough the LLG equation was first introduced phe-\nnomenologically, since then it has also been derived from\nmicroscopic quantum models [7, 8]. Quantum dissipation\nis a topic of long debate, since normal Hamiltonians will\nalways have conservation of energy. It can be described,\nfor instance, with a Caldeira-Leggett type model [9–13],\nwhere the Hamiltonian of the system is coupled to a bath\nof harmonic oscillators. These describe not only bosons,\nbut any degree of freedom of an environment in equilib-\nrium. These oscillators can be integrated out, leading toan effective action of the system that is non-local and ac-\ncounts for dissipation. The statistics of the bath is cap-\ntured by the spectral function J(ω), which determines\nthe type of dissipation. For a linear spectral function\n(Ohmic bath), the first-order derivative Gilbert damping\nis retrieved.\nThe spectral function is usually very difficult to calcu-\nlate or measure, so it is often assumed for simplicity that\nthe bath is Ohmic. However, J(ω) can have any contin-\nuous shape. Hence, a high frequency cutoff is commonly\nput in place, which sometimes justifies a linear expan-\nsion. However, a general expansion is that of an sorder\npower-law, where scould be any positive real number.\nA spectral function with such a power-law is called non-\nOhmic, and we refer to sas the “Ohmicness” of the bath.\nIt is known that non-Ohmic baths exist [14–23] and that\nthey can lead to equations of motion that include frac-\ntional derivatives [24–28]. Because fractional derivatives\nare non-local, these systems show non-Markovian dynam-\nics which can be useful to various applications [29–31].\nHere, we show that a macroscopic spin in contact\nwith a non-Ohmic environment leads to a fractional LLG\nequation, where the first derivative Gilbert damping gets\nreplaced by a fractional Liouville derivative. Then, we ex-\nplain how experiments can use ferromagnetic resonance\n(FMR) to determine the Ohmicness of their environ-\nment from resonance frequency and/or linewidth. This\nwill allow experiments to stop using the Ohmic assump-\ntion, and use equations based on measured quantities\ninstead. The same FMR measurements can also be done\nwith anisotropic systems. Aligning anisotropy with the\nmagnetic field may even aid the realization of measure-\nments, as this can help reach the required effective field\nstrengths. In practice, the determination of the type of\nenvironment is challenging, since one needs to measure\nthe coupling strength with everything around the spins.\nHowever, with the experiment proposed here, one canarXiv:2211.12889v1  [cond-mat.mes-hall]  23 Nov 20222\nessentially measure the environment through the spin it-\nself. Therefore, the tools that measure spins can now also\nbe used to determine the environment. This information\nabout the dissipation may lead to improved efficiency,\nstability, and control of applications in technology.\nDerivation of a generalized LLG equation. — We con-\nsider a small ferromagnet that is exposed to an external\nmagnetic field. Our goal is to derive an effective equa-\ntion of motion for the magnetization. For simplicity, we\nmodel the magnetization as one large spin (macrospin)\nˆS. Its Hamiltonian (note that we set /planckover2pi1andkBto\none) reads ˆHs=B·ˆS−KˆS2\nz, where the first term\n(Zeeman) describes the coupling to the external mag-\nnetic fieldB, and the second term accounts for (axial)\nanisotropy of the magnet. However, since a magnet con-\nsists of more than just a magnetization, the macrospin\nwill be in contact with some environment. Following\nthe idea of the Caldeira-Leggett approach [9–13, 32], we\nmodel the environment as a bath of harmonic oscillators,\nˆHb=/summationtext\nαˆp2\nα/2mα+mαω2\nαˆx2\nα/2, where ˆxαand ˆpαare\nposition and momentum operators of the α-th bath oscil-\nlator with mass mαand eigenfrequency ωα>0. Further-\nmore, we assume the coupling between the macrospin and\nthe bath modes to be linear, ˆHc=/summationtext\nαγαˆS·ˆxα, where\nγαis the coupling strength between macrospin and the\nα-th oscillator. Thus, the full Hamiltonian of macrospin\nand environment is given by ˆH=ˆHs+ˆHc+ˆHb.\nNext, we use the Keldysh formalism in its path-integral\nversion [33, 34], which allows us to derive an effective ac-\ntion and, by variation, an effective quasi-classical equa-\ntion of motion for the macrospin. For the path-integral\nrepresentation of the macrospin, we use spin coherent\nstates [34]|g/angbracketright= exp(−iφˆSz) exp(−iθˆSy) exp(−iψˆSz)|↑/angbracketright,\nwhereφ,θ, andψare Euler angles and |↑/angbracketrightis the eigen-\nstate of ˆSzwith the maximal eigenvalue S. Spin co-\nherent states provide an intuitive way to think about\nthe macrospin as a simple vector S=/angbracketleftg|ˆS|g/angbracketright=\nS(sinθcosφ,sinθsinφ,cosθ) with constant length Sand\nthe usual angles for spherical coordinates θandφ. For\nspins, the third Euler angle ψpresents a gauge freedom,\nwhich we fix as in Ref. [35] for the same reasons explained\nthere.\nAfter integrating out the bath degrees of freedom, see\nSup. Mat. [36] for details, we obtain the Keldysh partition\nfunctionZ=/integraltext\nDgexp[iS], with the Keldysh action\nS=/contintegraldisplay\ndt/bracketleftbig\nS˙φ(1−cosθ)−Beff(Sz)·S/bracketrightbig\n−/contintegraldisplay\ndt/contintegraldisplay\ndt/primeS(t)α(t−t/prime)S(t/prime). (1)\nThe first term, called Berry connection, takes the role\nof a kinetic energy for the macrospin; it arises from\nthe time derivative acting on the spin coherent states\n(−i∂t/angbracketleftg|)|g/angbracketright=S˙φ(1−cosθ). The second term is the po-\ntential energy of the macrospin, where we introduced aneffective magnetic field, Beff(Sz) =B−KSzez, given by\nthe external magnetic field and the anisotropy. The third\nterm arises from integrating out the bath and accounts\nfor the effect of the environment onto the macrospin;\nthat is, the kernel function α(t−t/prime) contains informa-\ntion about dissipation and fluctuations. Dissipation is\ndescribed by the retarded and advanced components\nαR/A(ω) =/summationtext\nα(γ2\nα/2mαω2\nα)ω2/[(ω±i0)2−ω2\nα], whereas\nthe effect of fluctuations is included in the Keldysh com-\nponent,αK(ω) = coth(ω/2T) [αR(ω)−αA(ω)]. This is\ndetermined by the fluctuation-dissipation theorem, as we\nassume the bath to be in a high-temperature equilibrium\nstate [33, 34, 37].\nFrom the Keldysh action, Eq. (1), we can now de-\nrive an equation of motion for the macrospin by taking\na variation. More precisely, we can derive quasi-classical\nequations of motion for the classical components of the\nanglesθandφby taking the variation with respect to\ntheir quantum components [38]. The resulting equations\nof motion can be recast into a vector form and lead to a\ngeneralized LLG equation\n˙S(t) =S(t)×/bracketleftbigg\n−Beff[Sz(t)] +/integraldisplayt\n−∞dt/primeα(t−t/prime)S(t/prime) +ξ(t)/bracketrightbigg\n,\n(2)\nwith the dissipation kernel [39] given by\nα(ω) =/integraldisplay∞\n−∞dε\nπεJ(ε)\n(ω+i0)2−ε2, (3)\nwhere we introduced the bath spectral density J(ω) =/summationtext\nα(πγ2\nα/2mαωα)δ(ω−ωα) [33, 36]. The last term in\nEq. (2) contains a stochastic field ξ(t), which describes\nfluctuations (noise) caused by the coupling to the bath;\nthe noise correlator for the components of ξ(t) is given\nby/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδmnαK(t−t/prime). Next, to get a\nbetter understanding of the generalized LLG equation,\nwe consider some examples of bath spectral densities.\nFractional Landau-Lifshitz-Gilbert equation. — For the\ngeneralized LLG equation (2), it is natural to ask: In\nwhich case do we recover the standard LLG equation?\nWe can recover it for a specific choice of the bath spectral\ndensityJ(ω), which we introduced in Eq. (3). Roughly\nspeaking,J(ω) describes two things: first, in the delta\nfunctionδ(ω−ωα), it describes at which energies ωα\nthe macrospin can interact with the bath; second, in\nthe prefactor πγ2\nα/2mαωα, it describes how strongly the\nmacrospin can exchange energy with the bath at the fre-\nquencyωα. In our simple model, the bath spectral den-\nsity is a sum over δ-peaks because we assumed excitations\nof the bath oscillators to have an infinite life time. How-\never, also the bath oscillators will have some dissipation\nof their own, such that the δ-peaks will be broadened. If,\nfurthermore, the positions of the bath-oscillator frequen-\nciesωαis dense on the scale of their peak broadening, the\nbath spectral density becomes a continuous function in-\nstead of a collection of δ-peaks. In the following, we focus3\non cases where the bath spectral density is continuous.\nSince the bath only has positive frequencies, we have\nJ(ω≤0) = 0. Even though J(ω) can have any pos-\nitive continuous shape, one might assume that it is an\napproximately linear function at low frequencies; that is,\nJ(ω) =α1ωΘ(ω)Θ(Ωc−ω), (4)\nwhere Θ(ω) = 1 forω > 0 and Θ(ω) = 0 forω < 0 and\nΩcis some large cutoff frequency of the bath such that we\nhaveωsystem/lessmuchT/lessmuchΩc. Reservoirs with such a linear\nspectral density are also known as Ohmic baths. Insert-\ning the Ohmic bath spectral density back into Eq. (3),\nwhile sending Ω c→∞ , we recover the standard LLG\nequation,\n˙S(t) =S(t)×/bracketleftBig\n−Beff[Sz(t)] +α1˙S(t) +ξ(t)/bracketrightBig\n,(5)\nwhere the first term describes the macrospin’s precession\naround the effective magnetic field, the second term—\nknown as Gilbert damping—describes the dissipation of\nthe macrospin’s energy and angular momentum into the\nenvironment, and the third term describes the fluctu-\nations with/angbracketleftξm(t)ξn(t/prime)/angbracketright= 4α1Tδmnδ(t−t/prime), which\nare related to the Gilbert damping by the fluctuation-\ndissipation theorem. Note that the same results can\nbe obtained without a cutoff frequency by introducing\na counter term, which effectively only changes the zero-\nenergy level of the bath, see Sup. Mat. [36] for details.\nThe assumption of an Ohmic bath can sometimes be\njustified, but is often chosen out of convenience, as it is\nusually the simplest bath type to consider. To our knowl-\nedge, there has been little to no experimental verification\nwhether the typical baths of magnetizations in ferromag-\nnets are Ohmic or not. To distinguish between Ohmic\nand non-Ohmic baths, we need to know how the mag-\nnetization dynamics depends on that difference. Hence,\ninstead of the previous assumption of a linear bath spec-\ntral density (Ohmic bath), we now assume that the bath\nspectral density has a power-law behavior at low frequen-\ncies,\nJ(ω) = ˜αsωsΘ(ω)Θ(Ωc−ω), (6)\nwhere we refer to sas Ohmicness parameter [40]. It is\nconvenient to define αs= ˜αs/sin(πs/2) and we should\nnote that the dimension of αsdepends on s. Fors= 1\nwe recover the Ohmic bath. Correspondingly, baths with\ns < 1 are called sub-Ohmic and baths with s > 1 are\ncalled super-Ohmic. For 0 <s< 2 and Ωc→∞ , we find\nthe fractional LLG equation\n˙S(t) =S×[−Beff[Sz(t)] +αsDs\ntS(t) +ξ(t)],(7)\nwhereDs\ntis a (Liouville) fractional time derivative of or-\ndersand the noise correlation is given by /angbracketleftξm(t)ξn(t/prime)/angbracketright=\n2αsTδmn(t−t/prime)−s/Γ(1−s); for a detailed calculation,see the Sup. Mat. [36]. Indeed, in the limit of s→1, we\nrecover the regular LLG equation. The fractional LLG\nequation (7) seems quite similar to the standard LLG\nequation (5). However, the first-order time derivative in\nthe Gilbert damping is replaced by a fractional s-order\ntime derivative in the fractional Gilbert damping; this\nhas drastic consequences for the dissipative macrospin\ndynamics.\nFractional Gilbert damping. — Fractional derivatives\nhave a long history [29], and many different definitions\nexist for varying applications [26, 27, 29]. From our mi-\ncroscopic model, we found the Liouville derivative [41],\nwhich is defined as\nDs\ntS(t) =dn\ndtn1\nΓ(n−s)/integraldisplayt\n−∞dt/prime(t−t/prime)n−1−sS(t/prime),(8)\nwherenis the integer such that n≤s<n + 1. This can\nbe interpreted as doing a fractional integral followed by\nan integer derivative. The fractional integral is a direct\ngeneralization from the rewriting of a repeated integral,\nby reversing the order of integration into a single one,\nwhich leads to extra powers of ( t−t/prime).\nTo provide some intuition to the effects of fractional\nfriction, we propose a thought experiment. Suppose an\nobject is traveling at constant speed, then x(t) =vt.\nHence, the friction force acting on the object goes like\nDs\ntx(t)∝vt1−s. Therefore, we find three regimes. For\ns= 1, the friction is constant in time. For s<1, the fric-\ntion force increases with time. Hence, longer movements\nwill be less common. For s > 1, the friction decreases\nwith time, so longer movements will be more likely once\nset in motion.\nWithin the fractional LLG equation, we thus see two\nimportant new regimes. For s < 1 (sub-Ohmic), the\nfriction is more likely to relax (localize) the spin (e.g.\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.10.51510\n[ωd-(B 0-KS)]/[α sS(B 0-KS)s][αsS(B 0-KS)s]2sin2θ/Ω2s0.20.40.60.81.1.21.41.61.8\nFIG. 1. A lin-log plot of the amplitude sin2θas a function of\ndriving frequency ωdplotted in dimensionless units for several\nvalues ofs. The resonance peaks change, depending on s.\nThe resonance frequency ωresand linewidth ∆ H/2have been\noverlayed with crosses. The red dashed crosses have been\ncalculated numerically, whereas the black solid crosses are the\nderived results from Eqs. (10) to (12).4\nsub-diffusion) towards the B-field direction. For small\nmovements, the friction could be very small, whereas it\nwould greatly increase for bigger movements. This could\ndescribe a low dissipation stable configuration. For s >\n1 (super-Ohmic), the friction could reduce as the spin\nmoves further, which in other systems is known to cause\nL´ evy-flights or super-diffusion [24, 25, 42]. This might\nlead the system to be less stable, but can potentially also\ngreatly reduce the amount of dissipation for strong signal\ntransfer: In a similar way to the design of fighter-jets,\nunstable systems can be easily changed by small inputs,\nwhich leads to more efficient signal transfer.\nFerromagnetic Resonance. — FMR is the phenomenon\nwhere the spin will follow a constant precession in a ro-\ntating external magnetic field. The angle θfrom thez-\naxis at which it will do so in the steady state will vary\naccording to the driving frequency ωdof the magnetic\nfield. Close to the natural frequency of the precession,\none generally finds a resonance peak [43]. We assume a\nmagnetic field of the form\nBeff(t) =\nΩ cos(ωdt)\nΩ sin(ωdt)\nB0−KSz\n, (9)\nwhere Ω is the strength of the rotating component, and\nwe will neglect thermal noise. We search for a steady\nstate solution of S(t) in the rotating frame where Beff(t)\nis constant. We will assume a small θapproximation\nwhere the ground state is in the positive zdirection, i.e.\n0<Ω/lessmuchB0−KSandαsS/lessmuch(B0−KS)1−s. Then (see\nSup. Mat. [36] for details of the calculations), we find\nthat the resonance occurs at a driving frequency\nωres≈(B0−KS) + (B0−KS)sαsScos/parenleftBigπs\n2/parenrightBig\n.(10)\nIt should be noted that this is different from what was to\nbe expected from any scaling arguments, since the cosine\nterm is completely new compared to previous results [43],\nand it vanishes precisely when s= 1. However, this new\nnon-linear term scales as ( B0−KS)s, which is an easily\ncontrollable parameter. In the limit where B0−KSis\nsmall (resp. large), the linear term will vanish and the\ns-power scaling can be measured for the sub(resp. super)-\nOhmic case. The amplitude at resonance is found to be\nsin2θres≈Ω2\n/bracketleftbig\nαsS(B0−KS)ssin/parenleftbigπs\n2/parenrightbig/bracketrightbig2, (11)\nand the Full Width at Half Maximum (FWHM) linewidth\nis given by\n∆H/2≈2αsS(B0−KS)ssin/parenleftBigπs\n2/parenrightBig\n. (12)\nDepending on the experimental setup, it might be eas-\nier to measure either the resonance location or the width\n0.0 0.5 1.0 1.5 2.001234\nωres/(B0-KS)ΔH/2/[αsS(B0-KS)s]\n0.01 0.10 1 10 1000.0110s0.20.40.60.81.1.21.41.61.8FIG. 2. A plot of the linewidth in Eq. (13) as a function of res-\nonance frequency for several values of s. The inset shows the\nsame plot in a log-log scale, where the slope of the linewidth\nis precisely the Ohmicness sof the bath.\nof the peak. Nevertheless, both will give the opportu-\nnity to see the sscaling inB0−KS. The presence\nof the anisotropy provides a good opportunity to reach\nweak or strong field limits. In fact, the orientation of the\nanisotropy can help to add or subtract from the magnetic\nfield, which should make the required field strengths more\nreachable for experiments. Some setups are more suit-\nable for measuring the width as a function of resonance\nfrequency. When s= 1, this relation can be directly\nderived from Eqs. (10) and (12). However, when s/negationslash= 1,\nthe relation can only be approximated for strong or weak\ndamping. For small αsS, we see that\n∆H/2≈2αsS(ωres)ssin/parenleftBigπs\n2/parenrightBig\n. (13)\nThe resonance peaks have been calculated numerically\nin FIG. 1 in dimensionless values. The red dashed lines\nshow the location of the numerically calculated peak and\nthe FWHM line width. The black solid lines show the\nlocation of the analytically approximated result for the\npeak location and FWHM line width [Eqs. (10) and (12)].\nFor smallαsSand Ω, we see a good agreement be-\ntween the analytical results and the numerical ones, al-\nthough sub-Ohmic seems to match more closely than\nsuper-Ohmic. This could be due to the greater sta-\nbility of sub-Ohmic systems, since the approximations\nmight affect less a stable system. As one might expect\nfrom the thought experiment presented earlier, we can\nsee in FIG. 1 that sub-Ohmic systems require higher,\nmore energetic, driving frequencies to resonate, whereas\nsuper-Ohmic systems already resonate at lower, less en-\nergetic, driving frequencies. In FIG. 2, we provide a plot\nof Eq. (13) to facilitate further comparison with experi-\nments. If the assumption of Gilbert damping was correct,\nall that one would see is a slope of one in the log-log inset.\nConclusion. — By relaxing the Ohmic Gilbert damp-\ning assumption, we have shown that the low-frequency\nregime of magnetization dynamics can be modeled by a5\nfractional LLG equation. This was done by coupling the\nmacrospin to a bath of harmonic oscillators in the frame-\nwork of a Caldeira-Leggett model. The Keldysh formal-\nism was used to compute the out-of-equilibrium dynam-\nics of the spin system. By analyzing an FMR setup, we\nfound ans-power scaling law in the resonance frequency\nand linewidth of the spin, which allows for a new way to\nmeasure the value of s. This means that experiments in\nmagnetization dynamics and spintronics can now avoid\nthe assumption of Gilbert damping and instead measure\nthe Ohmicness of the environment. This could aid in a\nbetter understanding of how to improve efficiency, stabil-\nity, and control of such systems for practical applications.\nAcknowledgments. — This work was supported by the\nNetherlands Organization for Scientific Research (NWO,\nGrant No. 680.92.18.05, C.M.S. and R.C.V.) and (partly)\n(NWO, Grant No. 182.069, T.L. and R.A.D.).\n[1] I. D. Mayergoyz, G. Bertotti, and C. 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Dalir and M. Bashour, Applied Mathematical Sci-\nences 4, 1021 (2010).\n[31] C. Gardiner and P. Zoller, Quantum noise: a handbook of\nMarkovian and non-Markovian quantum stochastic meth-\nods with applications to quantum optics (Springer Science\n& Business Media, 2004).\n[32] A. O. Caldeira and A. J. Leggett, Phys. Rev. A 31, 1059\n(1985).\n[33] A. Kamenev, Field theory of non-equilibrium systems\n(Cambridge University Press, 2011).\n[34] A. Altland and B. D. Simons, Condensed Matter Field\nTheory (Cambridge University Press, 2010), 2nd ed.\n[35] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N.\nKiselev, and A. Altland, Physical Review Letters 114,\n176806 (2015).\n[36] R. C. Verstraten, T. Ludwig, R. A. Duine, and\nC. Morais Smith, Supplementary material .\n[37] R. Kubo, Reports on Progress in Physics 29, 255 (1966).\n[38] In a straightforward variation with respect to quantum\ncomponents, we would only obtain a noiseless quasi-\nclassical equation of motion because the information\nabout noise (fluctuations) is included in the Keldysh\npart ofα(t−t/prime), which appears in the action only with\neven powers of quantum components. However, there is a\nway [44] that allows us to retain information about noise\nin the quasi-classical equation of motion; see also [33, 34].\nNamely, we perform a Hubbard-Stratonovich transforma-\ntion to linearize the contribution quadratic in quantum\ncomponents. This linearization comes at the cost of in-\ntroducing a new field, which takes the role of noise.\n[39] The dissipation kernel αis closely related to the retarded\nαRand advanced αAcomponents. Namely, it is given by\nα(ω) =−αR(ω)−αA(−ω).\n[40] From a mathematical perspective, any continuous but\nnot smooth function can still be expanded in a power-law\nfor small enough parameters. Hence, the only assumption\nthat we make in this model is that the frequencies in the\nsystem are very small. Then, there will always exist an\n0< s∈Rsuch that this expansion holds. In contrast,6\nthe Ohmic expansion can only be made for smooth con-\ntinuous functions.\n[41] This definition does not have any boundary conditions,\nas they would have to be at −∞ and would dissipate\nbefore reaching a finite time. One can, however, enforce\nboundary conditions by applying a very strong magnetic\nfield for some time such that the spin aligns itself, and\nthen quickly change to the desired field at t= 0.\n[42] A. A. Dubkov, B. Spagnolo, and V. V. Uchaikin, Interna-\ntional Journal of Bifurcation and Chaos 18, 2649 (2008).\n[43] T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman,\nPhysical Review Research 2, 023221 (2020).\n[44] A. Schmid, Journal of Low Temperature Physics 49, 609\n(1982).The fractional Landau-Lifshitz-Gilbert equation\nSupplementary Material\nR.C. Verstraten1, T. Ludwig1, R.A. Duine1,2, C. Morais Smith1\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584CC Utrecht, The Netherlands\n2Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: November 24, 2022)\nCONTENTS\nI. Keldysh microscopic model 1\nA. Hamiltonian 1\nB. Keldysh partition function 2\nC. Quasi-classical equation of motion 4\nD. Generalized Landau-Lifshitz-Gilbert equation 8\nII. Fractional derivative from non-Ohmic spectral function 9\nA. Calculating the effective Greens functions 9\nB. Ohmic spectral function 11\nC. Sub-Ohmic spectral function 13\nD. Super-Ohmic spectral function 15\nE. Comparison Ohmic versus non-Ohmic 17\nIII. FMR powerlaw derivation 18\nA. Ferromagnetic Resonance 18\nB. Resonance frequency and amplitude 20\nC. Calculating the FWHM linewidth 21\nIV. Dimensional analysis 22\nReferences 22\nI. KELDYSH MICROSCOPIC MODEL\nFor pedagogical reasons we start with a microscopic derivation of the usual LLG equation before going into the\nfractional one. In this section, we combine spin coherent states with the Keldysh formalism [1, 2] to derive a stochastic\nLangevin-like equation of motion of a (macro) spin [3].\nA. Hamiltonian\nIn the main text, we introduced a spectral function J(ω) with a cutoff frequency Ω c. This was originally done from\nthe perspective that any spectral function could be expanded to linear order; hence, the model would only be valid\nup to some highest frequency. However, the cutoff is also important for the model to be realistic, since any physical\nspectral function should vanish as ω→∞ . In the main text, we stated that the same results can be obtained by\nintroducing a constant counter term in the Hamiltonian. This is a term which exactly completes the square of the\ncoupling term and the harmonic potential of the bath and can be seen as a normalization of the zero-energy level. If\nwe instead start the model with this counter term and drop the cutoff, we will get a Greens function αct(ω), which is\nprecisely such that the original Greens function can be written as α(ω) =α(0) +αct(ω), i.e., the counter term in the\nHamiltonian removes the zero frequency contribution of the Greens function. This α(ω= 0) generates a term in the\nequation of motion that goes as/integraltext∞\n0d/epsilon1J(/epsilon1)\nπ/epsilon1[S(t)×S(t)]. Since the integral is finite, with a frequency cutoff in J(ω),\nthe entire term is zero due to the cross product. This means that the equation of motion will be identical if we startarXiv:2211.12889v1  [cond-mat.mes-hall]  23 Nov 20222\neither from the regular Hamiltonian with a frequency cutoff, or with a counter term and no cutoff. Here, we choose\nto show the method that includes a counter term, because then we do not need to calculate terms which would have\ncanceled either way.\nThe microscopic system that we describe is a large spin in an external magnetic field, where the spin is linearly\ncoupled to a bath of harmonic oscillators in the same way as in Refs. [4–9]. Therefore, our Hamiltonian has the form\nof a system, coupling, bath, and counter term; H(t) =Hs+Hc+Hb+Hct, where\nHs=B·ˆS−KS2\nz,\nHc=/summationdisplay\nαγαˆS·ˆxα,\nHb=/summationdisplay\nαˆp2\nα\n2mα+mαω2\nα\n2ˆx2\nα,\nHct=/summationdisplay\nαγ2\nα\n2mαω2αˆS2. (1)\nHere,Bis the (effective) magnetic field, ˆSis the spin, Kis thez-axis anisotropy, γαis the coupling strength, and\nαis the index over all harmonic oscillators which have position ˆxα, momentum ˆpα, massmαand natural frequency\nωα. Notice that the counter term is constant, since S2is a conserved quantity, and that we have indeed completed\nthe square, such that\nH(t) =B·ˆS−KS2\nz+/summationdisplay\nαˆp2\nα\n2mα+/bracketleftBigg/radicalbigg\nmαω2α\n2ˆxα+/radicalBigg\nγ2α\n2mαω2αˆS/bracketrightBigg2\n. (2)\nB. Keldysh partition function\nWe will use the Keldysh formalism to derive a quasi-classical equation of motion. Since this is an out-of-equilibrium\nsystem, a common choice would be to use the Lindblad formalism with a master equation [10]. However, Lindblad can\nonly describe Markovian systems, which will not be the case when we introduce a non-Ohmic bath. In the Keldysh\nformalism, one starts with an equilibrium density matrix in the far past (effectively infinite on the relevant time scale).\nThis then gets evolved with the time evolution operator as usual. However, in contrast to ordinary path integrals,\nonce the present has been reached, one evolves back to the infinite past. Since there is infinite time for evolution,\nwe can reach out-of-equilibrium states adiabatically. The benefit of integrating back to the infinite past is that we\nbegin and end with the same in-equilibrium system, which means equilibrium techniques can be used, at the cost of\nhaving both the forward ( O+) and backward ( O−) quantities to take care of. To reach useful results, one can apply\na Keldysh rotation to the classical ( Oc= (O++O−)/2) and quantum ( Oq=O+−O−) components with the added\nnotation/vectorO=/parenleftbigg\nOc\nOq/parenrightbigg\n. To derive a quasi-classical equation of motion, the action can be expanded in all the quantum\ncomponents, after which the Euler-Lagrange equation for the quantum components provides the equation of motion\nin terms of the classical components.\nFIG. 1. Figure extracted from Ref. [2]. The Keldysh contour starts at t=−∞, evolves forward to some time t, and then\nevolves backwards in time to t=−∞.\nTo begin, we write down the Keldysh partition function\nZ= Tr/braceleftbigg\nTKexp/bracketleftbigg\n−i/contintegraldisplay\nKdtH(t)/bracketrightbigg\nρ0/bracerightbigg\n, (3)3\nwhereTKis the Keldysh time ordering, ρ0is the density matrix at t=−∞, and the integral runs over the Keldysh\ncontour, as shown in FIG. 1. After discretizing the Keldysh time integral in the way of FIG. 1, we can rewrite the\ntrace as path-integrals over the spin coherent state |g/angbracketrightand the oscillators |ˆxα/angbracketrightand|ˆpα/angbracketright. This yields\nZ=/integraldisplay\nDg/productdisplay\nα/integraldisplay\nDˆxα/integraldisplay\nDˆpαeiS[g,{ˆxα},{ˆpα}], (4)\nwith the Keldysh action\nS[g,{ˆxα},{ˆpα}] =/contintegraldisplay\nKdt/bracketleftBigg\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g\n+/summationdisplay\nα/parenleftBigg\n−γαSg·ˆxα+ˆpα·˙ˆxα−γ2\nαS2\ng\n2mαω2α−ˆp2\nα\n2mα−mαω2\nα\n2ˆx2\nα/parenrightBigg/bracketrightBigg\n, (5)\nwhere we defined Sg=/angbracketleftg|ˆS|g/angbracketright.\nThe continuous path-integral seems to miss the boundary term /angbracketleftˆx1,α,g1|ρ0|ˆx2N,α,g2N/angbracketright/angbracketleftˆp2N,α|ˆx2N,α/angbracketright, but it is\nincluded in the Keldysh contour, as it connects the beginning and final contour time at t=−∞; see Ref. [2].\nNow, we will integrate out the bath degrees of freedom, beginning by completing the square and performing the\nGaussian integral over ˆpα. The Gaussian contribution in ˆpαwill act as a constant prefactor, so it will drop out of\nany calculation of an observable due to the normalization. Hence, we can effectively set it to one to find\n/integraldisplay\nDˆpαexp/bracketleftbigg\n−i/contintegraldisplay\nKdt/parenleftbiggˆp2\nα\n2mα−ˆpα·˙ˆxα/parenrightbigg/bracketrightbigg\n= exp/bracketleftbigg\ni/contintegraldisplay\nKdt/parenleftBig\n−mα\n2ˆxα∂2\ntˆxα/parenrightBig/bracketrightbigg\n, (6)\nwhere we also did a partial integration in ˆxα. Next we will perform a similar approach for the positions, but it is\nuseful to apply the Keldysh rotation first. Note that we can directly rewrite the integral over the Keldysh contour as\na regular time integral over the quantum components. However, one must still rewrite the contents of the integral in\nterms of the quantum and classical parts of the variables, since the Keldysh rotation does not immediately work for\nproducts. The action can first be written as\niS[g,{ˆxα}] =i/integraldisplay\ndt/parenleftBig\n[(−i∂t/angbracketleftg|)|g/angbracketright]q−[B·Sg]q+K/bracketleftbig\nS2\nz,g/bracketrightbigq\n−/summationdisplay\nα/braceleftBigg\n[γαSg·ˆxα]q+γ2\nα[S2\ng]q\n2mαω2α+/bracketleftBigmα\n2ˆxα/parenleftbig\n∂2\nt+ω2\nα/parenrightbigˆxα/bracketrightBigq/bracerightBigg/parenrightBigg\n. (7)\nWe can then derive that\n−[γαSg·ˆxα]q=−γα/bracketleftbig\nS+\ng·ˆx+\nα−S−\ng·ˆx−\nα/bracketrightbig\n=−γα/bracketleftbigg/parenleftbigg\nSc\ng+1\n2Sq\ng/parenrightbigg\n·/parenleftbigg\nˆxc\nα+1\n2ˆxq\nα/parenrightbigg\n−/parenleftbigg\nSc\ng−1\n2Sq\ng/parenrightbigg\n·/parenleftbigg\nˆxc\nα−1\n2ˆxq\nα/parenrightbigg/bracketrightbigg\n=−γα/bracketleftbig\nSc\ngˆxq\nα+Sq\ngˆxc\nα/bracketrightbig\n=−γα/bracketleftbigg/parenleftbigSc\ngSq\ng/parenrightbig\nτx/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg/bracketrightbigg\n, (8)\nwhere we introduced τx=/parenleftbigg\n0 1\n1 0/parenrightbigg\nin the Keldysh (classical, quantum) space represented by an upper index candq\nrespectively. Next, we want to derive a similar form for the part of the action that is quadratic in ˆxα. Since these are\nharmonic oscillators in equilibrium, we can refer the reader to Ref. [2], noting that a unit mass was used there, and\nconclude that\n/bracketleftBig\n−mα\n2ˆxα/parenleftbig\n∂2\nt+ω2\nα/parenrightbigˆxα/bracketrightBigq\n=/parenleftbigˆxc\nαˆxq\nα/parenrightbig/parenleftBigg\n0/bracketleftbig\nG−1\nα/bracketrightbigA\n/bracketleftbig\nG−1\nα/bracketrightbigR/bracketleftbig\nG−1\nα/bracketrightbigK/parenrightBigg/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg\n, (9)\nwhere the retarded and advanced Greens functions read\n[G−1\nα]R/A(t−t/prime) =δ(t−t/prime)mα\n2[(i∂t±i0)2−ω2\nα]. (10)\nThe±i0 is introduced because we need an infinitesimal amount of dissipation on the bath for it to remain in equilibrium\nand the sign is tied to causality. This is because there is also an infinitesimal amount of energy transfer from the4\nmacroscopic spin to each of the oscillators. This results in an extra first-order derivative term, which is found by\nmultiplying out the square with i0. One might want to set these terms to zero immediately, but as it turns out,\nthese are very important limits, which shift away poles from integrals that we need to compute later. Once that is\ndone, the limits are no longer important for the final result, and they may finally be put to zero. Since the bath is in\nequilibrium, we can use the fluctuation dissipation theorem to compute the Keldysh component using\nGK\nα(ω) =/bracketleftbig\nGR\nα(ω)−GA\nα(ω)/bracketrightbig\ncoth/parenleftBigω\n2T/parenrightBig\n.\nThe ˆxdependent part of the action is now given by\niSX=i/integraldisplay\ndt/bracketleftbigg\n−γα/parenleftbigSc\ngSq\ng/parenrightbig\nτx/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg\n+/parenleftbigˆxc\nαˆxq\nα/parenrightbig\nG−1\nα/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg/bracketrightbigg\n, (11)\nwhich we can compute by completing the square to find\niSX=i/integraldisplay\ndt/bracketleftbigg\n−γ2\nα\n4/vectorST\ng/parenleftbigg\n0GA\nα\nGR\nαGK\nα/parenrightbigg\n/vectorSg/bracketrightbigg\n. (12)\nBefore we write down the final effective action, we also have to rewrite the quadratic part in Sin a similar vector\nform, which is\n−γ2\nα[S2\ng]q\n2mαω2α=−γ2\nα\n2mαω2α/parenleftbigSc\ngSq\ng/parenrightbig\nτx/parenleftbiggSc\ng\nSq\ng/parenrightbigg\n. (13)\nCombining everything together, we find that the partition function of the system is given by Z=/integraltext\nDg eiS[g], with\nthe effective action\niS[g] =i/integraldisplay\ndt/braceleftBigg\n/bracketleftbig\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g/bracketrightbigq−/integraldisplay\ndt/prime/vectorST\ng(t)/parenleftbigg\n0αA\nαRαK/parenrightbigg\n(t−t/prime)/vectorSg(t/prime)/bracerightBigg\n, (14)\nwhereαA/R(t−t/prime) =/summationtext\nα/parenleftBig\nγ2\nα\n4GA/R\nα(t−t/prime) +γ2\nα\n2mαω2αδ(t−t/prime)/parenrightBig\nandαK(t−t/prime) =/summationtext\nαγ2\nα\n4GK\nα(t−t/prime).\nC. Quasi-classical equation of motion\nIn the quasi-classical regime, we are interested in solutions where the quantum components ( q) are small compared\nto the classical components ( c). We can thus neglect terms of O[(q)3], but we must be careful with ( q)2. We can\nuse a Hubbard-Stratonovich transformation to convert ( q)2terms into an expression with just ( q), but with a new\nfieldξadded to the path integral [3]. The action will then contain only terms of linear order in ( q), which means the\npartition function has the form Z∼/integraltext\nDcDq exp[if(c)q] =/integraltext\nDc1\n2πδ[f(c)]. Hence, only solutions that satisfy f(c) = 0\ncontribute to the path integral. Within that subset, we want to minimize the action.\nIn order to derive the equation of motion of the system, we must understand the relation between |g/angbracketrightandSg=\n/angbracketleftg|S|g/angbracketright. Using the Euler angle representation [1], we can describe |g/angbracketrightas\n|g/angbracketright=g|↑/angbracketright=e−iφSze−iθSye−iψSz|↑/angbracketright=e−iφSze−iθSy|↑/angbracketrighte−iψS(15)\nand similarly\n/angbracketleftg|=eiψS/angbracketleft↑|eiθSyeiφSz. (16)\nNote that the ψangle is now independent of the quantum state |↑/angbracketright, since this angle is describing the rotation of the\nvector pointing in the spin direction, which is symmetric. Hence, this will yield a gauge symmetry.\nUsing the Euler angle representation in the first terms of Eq. (14), we see that\n(−i∂t/angbracketleftg|)|g/angbracketright=/parenleftBig\n˙ψSeiψS/angbracketleft↑|eiθSyeiφSz+eiψS/angbracketleft↑|˙θSyeiθSyeiφSz+eiψS/angbracketleft↑|eiθSy˙φSzeiφSz/parenrightBig\ne−iφSze−iθSy|↑/angbracketrighte−iψS\n=˙ψS+˙θ/angbracketleft↑|Sy|↑/angbracketright+˙φ/angbracketleft↑|eiθSySze−iθSy|↑/angbracketright. (17)5\nWe note that/angbracketleft↑|Sy|↑/angbracketright= 0, while the last term includes a rotation of the spin up state by θdegrees in the ydirection\nand then measures the Szcomponent of that state, which is Scosθ. Hence,\n(−i∂t/angbracketleftg|)|g/angbracketright=˙ψS+˙φScosθ. (18)\nWe now define a new variable χsuch thatψ=χ−φ, which results in\n(−i∂t/angbracketleftg|)|g/angbracketright= ˙χS−˙φ(1−cosθ)S. (19)\nMaking use of the Euler angle representation, we also see that\nSg=S\nsinθcosφ\nsinθsinφ\ncosθ\n. (20)\nWe see thatB·Sg=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]. Similarly, KS2\nz,g=KS2cos2θ. Now, we still have to\ncompute the quantum parts of these quantities. We first note that\nSq\ng,x/S= [sinθcosφ]q= 2 cosθcsinθq\n2cosφccosφq\n2−2 sinθccosθq\n2sinφcsinφq\n2;\nSq\ng,y/S= [sinθsinφ]q= 2 sinθccosθq\n2cosφcsinφq\n2+ 2 cosθcsinθq\n2sinφccosφq\n2;\nSq\ng,z/S= [cosθ]q=−2 sinθcsinθq\n2;\n[cos2θ]q=−2 sinθccosθcsinθq. (21)\nNext, we will choose a gauge for χas in Ref. [11], which is\n˙χc=˙φc(1−cosθc)\nχq=φq(1−cosθc). (22)\nDefiningp= 1−cosθ, we see that [(−i∂t/angbracketleftg|)|g/angbracketright]q=/bracketleftBig\n˙χS−˙φpS/bracketrightBig\nq=S/bracketleftBig\nφq˙pc−˙φcpq/bracketrightBig\n. Now,pq= 2 sinθcsinθq\n2and\n˙pc=˙θcsinθccosθq\n2+˙θq\n2cosθcsinθq\n2,which leads to\n[(−i∂t/angbracketleftg|)|g/angbracketright]q=S/bracketleftBig\nφq˙pc−˙φcpq/bracketrightBig\n=S/bracketleftBigg\nφq˙θcsinθccosθq\n2+φq˙θq\n2cosθcsinθq\n2−2˙φcsinθcsinθq\n2/bracketrightBigg\n. (23)\nNext, we want to express B·Sq\ngin terms of Euler angles. We see that\nB·Sq\ng=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]q\n= 2S/bracketleftBig\nBx/parenleftbigg\ncosθcsinθq\n2cosφccosφq\n2−sinθccosθq\n2sinφcsinφq\n2/parenrightbigg\n+By/parenleftbigg\nsinθccosθq\n2cosφcsinφq\n2+ cosθcsinθq\n2sinφccosφq\n2/parenrightbigg\n−Bzsinθcsinθq\n2/bracketrightBig\n, (24)\nwhere we used the results from Eq. (21). Similarly, we have\nK/bracketleftbig\nS2\nz,g/bracketrightbigq=KS2[cos2θ]q=−2KS2sinθccosθcsinθq. (25)\nCombining these results, we conclude that\n/bracketleftbig\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g/bracketrightbigq=S/bracketleftBigg\nφq˙θcsinθccosθq\n2+φq˙θq\n2cosθcsinθq\n2−2(−Bz+KScosθc+˙φc) sinθcsinθq\n2\n−2Bx/parenleftbigg\ncosθcsinθq\n2cosφccosφq\n2−sinθccosθq\n2sinφcsinφq\n2/parenrightbigg\n−By/parenleftbigg\nsinθccosθq\n2cosφcsinφq\n2+ cosθcsinθq\n2sinφccosφq\n2/parenrightbigg/bracketrightBigg\n. (26)6\nRemark that this expression only contains odd powers of ( q), so that we can neglect all higher-order terms to get\n/bracketleftBig\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g/bracketrightBigq\n=S/bracketleftBig\n−θqsinθc(−Bz+KScosθc+˙φc)\n−θqcosθc(Bxcosφc+Bysinφc) +φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig\n. (27)\nNow, we focus on the part of the action in Eq. (14) that comes from the bath, given by\niSb[g] =−i/integraldisplay\ndt/integraldisplay\ndt/prime/vectorST\ng(t)/parenleftbigg\n0αA\nαRαK/parenrightbigg\n(t−t/prime)/vectorSg(t/prime). (28)\nLet us first consider what Sq\ngandSc\ngare in terms of φandθ. By performing some trigonometric operations on each\nof the components, we find that\nSc\ng=S\nsinθccosθq\n2cosφccosφq\n2−cosθcsinθq\n2sinφcsinφq\n2\nsinθccosθq\n2sinφccosφq\n2+ cosθcsinθq\n2cosφcsinφq\n2\ncosθccosθq\n2\n (29)\nand\nSq\ng= 2S\ncosθcsinθq\n2cosφccosφq\n2−sinθccosθq\n2sinφcsinφq\n2\nsinθccosθq\n2cosφcsinφq\n2+ cosθcsinθq\n2sinφccosφq\n2\n−sinθcsinθq\n2\n. (30)\nBy expanding in the quantum components of Sc\ngandSq\ng, we see that\nSc\ng= (q)0+O/parenleftbig\n(q)2/parenrightbig\n,\nSq\ng= (q)1+O/parenleftbig\n(q)3/parenrightbig\n.\nSince the action only contains terms with at least one Sq\ng, we know that the only way to obtain a term of order ( q)2\nis from (Sq\ng)2. Hence, we may neglect all terms beyond linear ( q) inS(c/q)\ng in the quasi-classical regime. This results\nin\nSc\ng=S\nsinθccosφc\nsinθcsinφc\ncosθc\n, (31)\nSq\ng=S\nθqcosθccosφc−φqsinθcsinφc\nφqsinθccosφc+θqcosθcsinφc\n−θqsinθc\n. (32)\nA useful remark for later is that this shows that\nSq\ng=θq∂\n∂θcSc\ng+φq∂\n∂φcSc\ng. (33)\nGoing back to iSb[g], we can rewrite this as a convolution, in the sense that\niSb[g] =−i/integraldisplay\ndt/bracketleftbig\nSc\ng(t)·/parenleftbig\nαA∗Sq\ng/parenrightbig\n(t) +Sq\ng(t)·/parenleftbig\nαR∗Sc\ng/parenrightbig\n(t) +Sq\ng(t)·/parenleftbig\nαK∗Sq\ng/parenrightbig\n(t)/bracketrightbig\n, (34)\nwhere (f∗g)(t) =/integraltext∞\n−∞dt/primef(t−t/prime)g(t/prime). We see that the first two terms contain precisely one quantum component,\nbut the last term has two quantum components. When writing down the Euler-Lagrange equation of motion, it is\nimportant to realize that the convolution operation will act as if it is a simple multiplication, since the convolution\nobeys\nd\ndx(f(x)∗g)(t) =/parenleftbiggdf\ndx∗g/parenrightbigg\n(t). (35)7\nWe now concentrate on the ( q)2part of this action, for which we would like to use a Hubbard-Stratonovich transfor-\nmation in order to reduce this to linear in ( q). Recall that a Hubbard-Stratonovich transformation is given by\nexp/bracketleftBig\n−a\n2x2/bracketrightBig\n=/radicalbigg\n1\n2πa/integraldisplay\nDξexp/bracketleftbigg\n−ξ2\n2a−ixξ/bracketrightbigg\n. (36)\nHowever, we see that our action does not contain any purely quadratic terms, but rather a Greens functional shape\nasSq\ng(t)αK(t−t/prime)Sq\ng(t/prime). Hence, to use a Hubbard-Stratonovich like transformation, we must derive it from a Greens\nfunction exponential, similarly to Ref. [3]. Assuming that this is renormalizable and that αKcan be rewritten into a\ndistribution, we have\n1 =/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/prime/parenleftbigg\nξ(t)−2/integraldisplay\ndt/prime/primeSq\ng(t/prime/prime)αK(t/prime/prime−t)/parenrightbigg\n[−2iαK]−1(t−t/prime)/parenleftbigg\nξ(t/prime)−2/integraldisplay\ndt/prime/prime/primeαK(t/prime−t/prime/prime/prime)Sq\ng(t/prime/prime/prime)/parenrightbigg/bracketrightbigg\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)\n−iSq\ng(t)δ(t−t/prime)ξ(t/prime)−iξ(t)δ(t−t/prime)Sq\ng(t/prime)−2iSq\ng(t)αK(t−t/prime)Sq\ng(t/prime)/bracketrightbig\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)−2iSq\ng(t)δ(t−t/prime)ξ(t/prime)−2iSq\ng(t)αK(t−t/prime)Sq\ng(t/prime)/bracketrightbigg\n,\nwhere we used that/integraltext\ndt/primeαK(t−t/prime)[αK]−1(t/prime−t/prime/prime) =δ(t−t/prime/prime) and that 2 iαKis positive real. Therefore, we find that\nexp/bracketleftbigg\n−i/integraldisplay\ndt/integraldisplay\ndt/primeSq\ng(t)αK(t−t/prime)Sq\ng(t/prime)/bracketrightbigg\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg\n·exp/bracketleftbigg\ni/integraldisplay\ndtSq\ng(t)ξ(t)/bracketrightbigg\n. (37)\nThe double integral in the first exponential signifies the statistical properties of ξ. For instance, if αKis delta-like,\nthenξwould have Gaussian statistics (e.g. white noise), but in general we will have time correlated noise defined by\nαK[3], such that\n/angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime). (38)\nSince there is no gdependence in the double ξexponential, we will leave it out of S[g] and only remember these\nstatistics. Our partition function is then given by\nZ=/integraldisplay\nDξexp (iSn[ξ])/integraldisplay\nDgexp (iSsc[g,ξ]), (39)\nwhere the noise action is given by\niSn[ξ] =−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime) (40)\nand the semi-classical action is given by\niSsc[g,ξ] =i/integraldisplay\ndtS/bracketleftBig\n−θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc)\n+φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig\n+i/integraldisplay\ndt/bracketleftbig\nξ(t)Sq\ng(t)/bracketrightbig\n−i/integraldisplay\ndt/bracketleftbig\nSc\ng(t)·/parenleftbig\nαA∗Sq\ng/parenrightbig\n(t) +Sq\ng(t)·/parenleftbig\nαR∗Sc\ng/parenrightbig\n(t)/bracketrightbig\n, (41)\nwhereSc\ng(t) andSq\ng(t) include only up to first-order corrections in quantum components. Assuming that αA/Rcan\nbe written in terms of distributions, we can define the distribution αdiss(t) =−αR(t)−αA(−t) and rewrite the\nsemi-classical action as\niSsc[g,ξ] =i/integraldisplay\ndtS/bracketleftBig\n−θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc)\n+φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig\n+i/integraldisplay\ndt/bracketleftbig/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\nSq\ng(t). (42)8\nRecall that, using the Euler angles, we have/integraltext\nDg=/integraltext\nDθDφ sin(θ). Technically, the factor of sin( θ) would end up\nin the action. However, since one could define ρ= cos(θ) as a new variable in order to avoid this, we know that this\nterm is not relevant to the physics. Hence, we can disregard it.\nSince all terms in iSsc[g,ξ] are either linear in θqorφq, we find two Euler-Lagrange equations of the form\nδLsc\nδθq= 0 andδLsc\nδφq= 0. (43)\nRemembering Eq. (33), we see thatδSq\ng(t)\nδθq=δSc\ng(t)\nδθcandδSq\ng(t)\nδφq=δSc\ng(t)\nδφc. Hence, the e.o.m. can be rearranged to yield\n˙φc=1\nSsinθc/bracketleftbig\n−B(Sc\nz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδθc(44)\nand\n˙θc=−1\nSsinθc/bracketleftbig\n−B(Sc\nz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδφc, (45)\nwhereB(Sc\nz) =\nBx\nBy\nBz−KSc\nz\n.\nD. Generalized Landau-Lifshitz-Gilbert equation\nWe want to show that the equations found by the microscopic model are in fact precisely of the LLG form. For\nthis, we will have to start from the LLG equation, and introduce the same two Euler angles θandφfor the spin, and\nshow that this gives rise to the same set of equations as previously deduced.\nWe begin with the generalized LLG equation\n˙S(t) =S(t)×[−B(Sz) + (αdiss∗S) (t) +ξ(t)], (46)\nwhereαdiss(t) =−αR(t)−αA(−t),/angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime) andB(Sz) = (Bx,By,Bz−KSz)T. Since the velocity\nofSis always perpendicular to S, we know that the magnitude of Sis constant. Hence, we can go to spherical\ncoordinates, such that\nS=S\nsinθcosφ\nsinθsinφ\ncosθ\n. (47)\nInserting this into the LLG equation, we firstly see that\n˙S=˙θ∂S\n∂θ+˙φ∂S\n∂φ=˙θS\ncosθcosφ\ncosθsinφ\n−sinθ\n+˙φS\n−sinθsinφ\nsinθcosφ\n0\n.\nNow, we notice that the RHS of the LLG equation can, without loss of generality, be written as S(t)×rwith\nr= (x,y,z )T. Working this out explicitly, we find that the LLG equation ˙S=S×rbecomes\nS\n˙θcosθcosφ−˙φsinθsinφ\n˙θcosθsinφ+˙φsinθcosφ\n−˙θsinθ\n=S\nzsinθsinφ−ycosθ\nxcosθ−zsinθcosφ\nysinθcosφ−xsinθsinφ\n. (48)\nWe note that the equation corresponding to the zcomponent can be written as\n˙θ=−1\nsinθr·\n−sinθsinφ\nsinθcosφ\n0\n=−1\nSsinθr·∂S\n∂φ. (49)9\nNow, we add up the ˆ xand ˆyequations, such that the ˙θcancels (i.e.−ˆxsinφ+ ˆycosφ). This yields\n˙φsinθ(sin2φ+ cos2φ) =−zsinθ(sin2φ+ cos2φ) +ycosθsinφ+xcosθcosφ,\nwhich simplifies to\n˙φ=1\nsinθr·\ncosθcosφ\ncosθsinφ\n−sinθ\n=1\nSsinθr·∂S\n∂θ. (50)\nBy inserting r=−B(Sz) + (αdiss∗S) (t) +ξ(t), we see that this is identical to the equations derived from the\nmicroscopic model\n˙φc=1\nSsinθc/bracketleftbig\n−B(Sz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδθc; (51)\n˙θc=−1\nSsinθc/bracketleftbig\n−B(Sz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδφc. (52)\nTherefore, we may conclude that our microscopic model is described by the generalized LLG equation.\nFor the fractional LLG equation, we are in particular interested in the case where αdiss∗S=αsDs\ntS, whereDs\ntis\na fractional derivative. For instance, assuming 0 <s< 1, the Liouville fractional derivative is given by\nDs\ntf(t) =1\nΓ(1−s)/integraldisplayt\n−∞(t−t/prime)−sf/prime(t/prime)dt/prime. (53)\nSo, ifαdiss=αsΘ(t)\nΓ(1−s)t−s∂t, then, because of the convolution with S, we would find a fractional LLG equation, whereas\nαdiss=α1δ(t)∂twould give the regular LLG equation.\nII. FRACTIONAL DERIVATIVE FROM NON-OHMIC SPECTRAL FUNCTION\nHere, we will compute the type of dissipation which comes from the spectral function. We will first derive the\nspectral function from microscopic quantities to see how it ends up in the Greens function. Then, we will calculate\nthe dissipation for three different cases.\nA. Calculating the effective Greens functions\nWe recall that\nαA/R(t−t/prime) =/summationdisplay\nα/parenleftbiggγ2\nα\n4GA/R\nα(t−t/prime) +δ(t−t/prime)γ2\nα\n2mαω2α/parenrightbigg\n,\nwhere\n[G−1\nα]R/A(t−t/prime) =δ(t−t/prime)mα\n2[(i∂t±i0)2−ω2\nα].\nBy the fluctuation dissipation theorem, we also have\nαK(ω) =/bracketleftbig\nαR(ω)−αA(ω)/bracketrightbig\ncoth/parenleftBigω\n2T/parenrightBig\n.\nWe are interested in finding closed forms for αR/A/K(t−t/prime). Using the relation\n/integraldisplay\ndt/primeG−1(t−t/prime)G(t/prime−t/prime/prime) =δ(t−t/prime/prime), (54)\nwe note that\nmα\n2[(i∂t±i0)2−ω2\nα]GR/A\nα(t−t/prime/prime) =δ(t−t/prime/prime). (55)10\nThe Fourier transform1yields\nGR/A\nα(ω) =1\nmα\n2[(ω±i0)2−ω2α]. (56)\nWe therefore find that\nαR/A(ω) =/summationdisplay\nα/parenleftbiggγ2\nα\n4GA/R\nα(ω) +γ2\nα\n2mαω2α/parenrightbigg\n=/summationdisplay\nαγ2\nα\n2mα/bracketleftbigg1\n(ω±i0)2−ω2α+1\nω2α/bracketrightbigg\n=/summationdisplay\nαγ2\nα\n2mαω2αω2\n(ω±i0)2−ω2α.(57)\nThe spectral function is given by the imaginary part of the Fourier transform of the dynamical susceptibility\nχ(ω) =δ\nδS(ω)/summationdisplay\nαγαˆxα(ω). (58)\nThe Hamiltonian e.o.m. for the bath can be found by going back to our starting Hamiltonian\nH=B·S+/summationdisplay\nαγαS·ˆxα+/summationdisplay\nαˆp2\nα\n2mα+mαω2\nα\n2ˆx2\nα+/summationdisplay\nαγ2\nα\n2mαω2αS2.\nThe e.o.m. reads\n˙ˆxα=ˆpα\nmαand ˙ˆpα=−γαS−mαω2\nαˆxα. (59)\nCombining both equations and taking the Fourier transform, we find\nˆxα(ω) =γα\nmα[(ω+i0)2−ω2α]S(ω), (60)\nwhere we have taken an infinitesimal amount of dissipation + i0 on the oscillators into account. This leads to\nχ(ω) =/summationdisplay\nαγ2\nα\nmα[(ω+i0)2−ω2α]=/summationdisplay\nαγ2\nα\n2mαωα/parenleftbigg1\nω+i0−ωα−1\nω+i0 +ωα/parenrightbigg\n. (61)\nWe remark that\nIm1\nx+i0=−πδ(x), (62)\nwhich leads to\nJ(ω) = Imχ(ω) =−/summationdisplay\nαπγ2\nα\n2mαωα[δ(ω−ωα)−δ(ω+ωα)] =−π\n2/summationdisplay\nαγ2\nα\nmαωαδ(ω−ωα), (63)\nwhere we used that all oscillator frequencies are positive. We can identify the spectral function in αR/Aas\nαR/A(ω) =/summationdisplay\nαγ2\nα\n2mαω2αω2\n(ω±i0)2−ω2α=−/integraldisplay∞\n0dε\nπω2ε−1J(ε)\n(ω±i0)2−ε2. (64)\nNow, we will assume a particular shape for J(ε). This can be either Ohmic or non-Ohmic, but in general we may\nassume a power-law behavior as some J(ε) =αsεs.\n1We use the convention f(ω) =/integraltext∞\n−∞dteiωtf(t) andf(t) =1\n2π/integraltext∞\n−∞dωe−iωtf(ω), withδ(t) =1\n2π/integraltext∞\n−∞dωeiωt.11\nB. Ohmic spectral function\nBeginning with the Ohmic case J(ε) =α1ε, we see that\n2iImαR/A(ω) =αR/A(ω)−/bracketleftBig\nαR/A/bracketrightBig∗\n(ω)\n=−α1/integraldisplay∞\n0dε\nπ/bracketleftbiggω2\n(ω±i0)2−ε2−ω2\n(ω∓i0)2−ε2/bracketrightbigg\n=−α1/integraldisplay∞\n−∞dε\n2π/bracketleftbiggω2\n(ω±i0)2−ε2−ω2\n(ω∓i0)2−ε2/bracketrightbigg\n=−α1ω2/integraldisplay∞\n−∞dε\n2π/bracketleftbigg1\n(ω±i0 +ε)(ω±i0−ε)−1\n(ω∓i0 +ε)(ω∓i0−ε)/bracketrightbigg\n=±4i0ω3α1/integraldisplay∞\n−∞dε\n2π1\n(ω±i0 +ε)(ω±i0−ε)(ω∓i0 +ε)(ω∓i0−ε), (65)\nwhich has four poles at ε=±1(ω±2i0). Since the integral scales as <1/|ε|, we can add an infinite radius half circle\nto complete a complex contour integral. Notice from symmetry that we will always have one of each of the four poles.\nWe can thus drop the ±signs inside since this only changes the notation order in the fraction. We thus find that\n2iImαR/A(ω) =±4i0ω3α1/integraldisplay∞\n−∞dε\n2π1\n(ε+ω+i0)(ε−ω−i0)(ε+ω−i0)(ε−ω+i0). (66)\nCompleting the contour along the top, we find poles at ε=±ω+i0, which yields\n2iImαR/A(ω) =∓0·4ω3α1/bracketleftbigg1\n(ω+i0 +ω+i0)(ω+i0 +ω−i0)(ω+i0−ω+i0)\n+1\n(−ω+i0 +ω+i0)(−ω+i0−ω−i0)(−ω+i0−ω+i0)/bracketrightbigg\n=∓0·4ω3α1/bracketleftbigg1\n2(ω+i0)(2ω)(2i0)+1\n(2i0)(−2ω)2(−ω+i0)/bracketrightbigg\n=±i\n2ω3α1/bracketleftbigg1\nω2+i0ω+1\nω2−i0ω/bracketrightbigg\n=±i\n2ω3α1/bracketleftbiggω2−i0ω+ω2+i0ω\nω4/bracketrightbigg\n=±iωα1. (67)\nHence, Im αR/A(ω) =±α1ω/2. Similarly,\n2 ReαR/A(ω) =αR/A(ω) +/bracketleftBig\nαR/A/bracketrightBig∗\n(ω)\n=−α1/integraldisplay∞\n0dε\nπ/bracketleftbiggω2\n(ω±i0)2−ε2+ω2\n(ω∓i0)2−ε2/bracketrightbigg\n=−α1/integraldisplay∞\n−∞dε\n2π/bracketleftbiggω2\n(ω±i0)2−ε2+ω2\n(ω∓i0)2−ε2/bracketrightbigg\n=−α1/integraldisplay∞\n−∞dε\n2π/bracketleftbiggω2\n(ω±i0 +ε)(ω±i0−ε)+ω2\n(ω∓i0 +ε)(ω∓i0−ε)/bracketrightbigg\n=α1/integraldisplay∞\n−∞dε\nπω2(ε2−ω2)\n(ε+ω+i0)(ε−ω−i0)(ε+ω−i0)(ε−ω+i0). (68)12\nSince the integral scales as 1 /|ε|, we can freely add the infinite circular contour along the top. Applying the residue\ntheorem, we find\n2 ReαR/A(ω) = 2iα1/bracketleftbiggω2((ω+i0)2−ω2)\n(ω+i0 +ω+i0)(ω+i0 +ω−i0)(ω+i0−ω+i0)\n+ω2((−ω+i0)2−ω2)\n(−ω+i0 +ω+i0)(−ω+i0−ω−i0)(−ω+i0−ω+i0)/bracketrightbigg\n= 2α1i/bracketleftbiggω2(2ωi0)\n2(ω+i0)(2ω)(2i0)+ω2(−2ωi0)\n(2i0)(−2ω)2(−ω+i0)/bracketrightbigg\n=α1\n2i/bracketleftbiggω2\nω+i0−ω2\nω−i0/bracketrightbigg\n=α1\n2i/bracketleftbiggω2(−2i0)\nω2/bracketrightbigg\n=α10, (69)\nwhich means that Re αR/A(ω) =α1\n2·0 = 0. Hence,\nαR/A(ω) =±α1iω\n2(70)\nand sinceαdiss(t) =−αR(t)−αA(−t) we have\nαdiss(ω) =−αR(ω)−αA(−ω) =−α1iω. (71)\nIn the LLG equation, we thus find\n(αdiss∗S) (t) =1\n2π/integraldisplay\ndωe−iωt(αdiss∗S) (ω)\n=1\n2π/integraldisplay\ndωe−iωtαdiss(ω)S(ω)\n=1\n2π/integraldisplay\ndωe−iωt(−α1iω)S(ω)\n=α1∂\n∂t1\n2π/integraldisplay\ndωe−iωtS(ω)\n=α1˙S(t). (72)\nFurthermore, we have that\nαK(ω) = [αR(ω)−αA(ω)] coth/parenleftBigω\n2T/parenrightBig\n=iα1ωcoth/parenleftBigω\n2T/parenrightBig\n. (73)\nWe have two regimes from the cotangent which crossover around ω≈2T. If the temperature is large enough that we\ncan approximate coth/parenleftbigω\n2T/parenrightbig\n≈2T\nω, then we have αK(ω)≈2iα1T. Therefore, we find\nαK(t) = 2iα1T1\n2π/integraldisplay∞\n−∞dωe−iωt= 2iα1Tδ(t), (74)\nand thus\n/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 4α1Tδm,nδ(t−t/prime). (75)\nWe see that Eq. (72) and Eq. (75) combine to give the regular LLG equation with first-order dissipation and white\nnoise fluctuation:\n˙S(t) =S(t)×/bracketleftBig\n−B+α1˙S(t) +ξ(t)/bracketrightBig\n. (76)13\nC. Sub-Ohmic spectral function\nWe now consider the case where\nJ(ε) =αssin/parenleftBigπs\n2/parenrightBig\nεs, (77)\nwith 0<s< 1. In this case,\nαR/A(ω) =−αssin/parenleftBigπs\n2/parenrightBig/integraldisplay∞\n0dε\nπω2εs−1\n(ω±i0)2−ε2. (78)\nConsidering the LLG equation, the relevant dissipation term is ( αdiss∗S)(t). In terms of the Fourier transform of\nαdiss(t), we find that\n(αdiss∗S)(t) =/integraldisplay∞\n−∞dt/primeαdiss(t−t/prime)S(t/prime)\n=−/integraldisplay∞\n−∞dt/prime[αR(t−t/prime) +αA(t/prime−t)]S(t/prime)\n=−1\n2π/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n−∞dω/bracketleftBig\ne−iω(t−t/prime)αR(ω) +eiω(t−t/prime)αA(ω)/bracketrightBig\nS(t/prime)\n=−1\n2π/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n−∞dωe−iω(t−t/prime)/bracketleftbig\nαR(ω) +αA(−ω)/bracketrightbig\nS(t/prime)\n=αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n−∞dω/integraldisplay∞\n0dεe−iω(t−t/prime)/bracketleftbiggω2εs−1\n(ω+i0)2−ε2+(−ω)2εs−1\n(−ω−i0)2−ε2/bracketrightbigg\nS(t/prime)\n=−αssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n0dε/integraldisplay∞\n−∞dω/bracketleftbigg\ne−iω(t−t/prime)ω2εs−1\nε2−(ω+i0)2S(t/prime)/bracketrightbigg\n. (79)\nNotice that we have two poles at ω=±ε−i0, below the real axis. If t−t/prime<0, then the exponential will go to zero as\nω→+i∞. Hence we could close the ωintegration with a complex contour as an infinite half-circle along the top, and\nget zero from the Cauchy theorem. If t−t/prime>0, however, we see that the exponential goes to zero when ω→−i∞.\nHence, we can close the ωintegration along the bottom. Thus, using the residue theorem (reversing the integration\ndirection), we find\n(αdiss∗S)(t) =−αssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n0dε2πiΘ(t−t/prime)\n/bracketleftBig\ne−i(ε−i0)(t−t/prime)(ε−i0)2εs−1\n(ε−i0 +i0 +ε)+e−i(−ε−i0)(t−t/prime)(−ε−i0)2εs−1\n(−ε−i0 +i0−ε)/bracketrightBig\nS(t/prime)\n=−iαssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dε/bracketleftbigg\ne−iε(t−t/prime)εs+1\nε+eiε(t−t/prime)εs+1\n−ε/bracketrightbigg\nS(t/prime)\n=−iαssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dε/bracketleftBig\ne−iε(t−t/prime)−eiε(t−t/prime)/bracketrightBig\nεsS(t/prime)\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dεsin[ε(t−t/prime)]εsS(t/prime)\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg/bracketleftbig\nεs−1cos[ε(t−t/prime)]S(t/prime)/bracketrightbigt/prime=t\nt/prime=t0−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg\nεs−1S(t)−εs−1cos[ε(t−t0)]S(t0)−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n.(80)\nThe first term vanishes because of the cross product with S(t) in the LLG equation. The second term is where we\nhad to be careful. Here, we should realize that the −∞ is physically only indicating that it is a time very far in the\npast. So, to avoid unphysical infinities, we introduced a finite initial time t0and we will take t0→−∞ later. For\nthis, we need to introduce some fractional derivative notation. We define the Riemann-Liouville (RL) and Caputo (C)14\nderivatives of order s, with an integer nsuch thatn≤s<n + 1, as\nRL\nt0Ds\ntf(t) =dn\ndtn1\nΓ(n−s)/integraldisplayt\nt0dt/prime(t−t/prime)n−1−sf(t/prime), (81)\nC\nt0Ds\ntf(t) =1\nΓ(n−s)/integraldisplayt\nt0dt/prime(t−t/prime)n−1−sf(n)(t/prime), (82)\nwhere we reserve the simpler Ds\ntnotation for the Liouville derivative that was used in the main text.\nNow, rescaling ε→ε/(t−t0) andε→ε/(t−t/prime) in the second and third terms of Eq. (80) respectively, we have\n(αdiss∗S)(t) = 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dεcos(ε)εs−1/bracketleftbigg\n(t−t0)−sS(t0) +/integraldisplayt\nt0dt/prime(t−t/prime)−s˙S(t/prime)/bracketrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/bracketleftBig\ncos/parenleftBigπs\n2/parenrightBig\nΓ(s)/bracketrightBig/bracketleftbigg\n(t−t0)−sS(t0) +/integraldisplayt\nt0dt/prime(t−t/prime)−s˙S(t/prime)/bracketrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s)/bracketleftbig\n(t−t0)−sS(t0) + Γ(1−s)C\nt0Ds\ntS(t)/bracketrightbig\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s)Γ(1−s)/bracketleftbigg(t−t0)−s\nΓ(1−s)S(t0) +C\nt0Ds\ntS(t)/bracketrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBigπ\nsin(πs)RL\nt0Ds\ntS(t)\n=αsRL\nt0Ds\ntS(t) =αsDs\ntS(t), (83)\nwhere we used several identities from Sec. 6 in the Sup. Mat. of Ref. [12] and in the last line we sent t0→−∞ .\nFor the noise correlation, we have to compute the Keldysh component. This is\nαK(t) =1\n2π/integraldisplay∞\n−∞dωe−iωtαK(ω)\n=1\n2π/integraldisplay∞\n−∞dωe−iωt[αR(ω)−αA(ω)] coth/parenleftBigω\n2T/parenrightBig\n=αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωe−iωt/bracketleftbiggω2εs−1\nε2−(ω+i0)2−ω2εs−1\nε2−(ω−i0)2/bracketrightbigg\ncoth/parenleftBigω\n2T/parenrightBig\n(now sendω→−ωin the advanced part)\n=αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dω/bracketleftbigge−iωtω2εs−1\nε2−(ω+i0)2+eiωtω2εs−1\nε2−(−ω−i0)2/bracketrightbigg\ncoth/parenleftBigω\n2T/parenrightBig\n=αssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ωt)ω2εs−1\nε2−(ω+i0)2coth/parenleftBigω\n2T/parenrightBig\n. (84)\nNow, we send ω→ω/tandε→ε/t, which yields\nαK(t) =αst−1−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ω)ω2εs−1\nε2−(ω+i0)2coth/parenleftBigω\n2tT/parenrightBig\n. (85)\nTaking the high temperature limit, we get\nαK(t) =αst−1−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ω)ω2εs−1\nε2−(ω+i0)22tT\nω\n= 2Tαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ω)ωεs−1\nε2−(ω+i0)2\n=−2Tαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ω)ωεs−1\n(ω+i0−ε)(ω+i0 +ε). (86)15\nNow, we want to close the integral over ωwith an infinite half-circle. For this, we need fast enough convergence of\nthe integrand to zero. Splitting the cosine into two exponential parts cos( ω) = (eiω+e−iω)/2, we see that the first\nterm goes to zero when ω→i∞and that the second term goes to zero when ω→−i∞. Since we have poles at\nω=±ε−i0, the integral along the top half-plane vanishes. The integral along the bottom is then computed as\nαK(t) =−Tαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωe−iωωεs−1\n(ω+i0−ε)(ω+i0 +ε)\n=−Tαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε−2πi/bracketleftbigge−i(ε−i0)(ε−i0)εs−1\nε−i0 +i0 +ε+e−i(−ε−i0)(−ε−i0)εs−1\n−ε−i0 +i0−ε/bracketrightbigg\n= 2iTαst−ssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/bracketleftbigge−iεεs\n2ε+eiεεs\n2ε/bracketrightbigg\n= 2iTαst−ssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dεcos(ε)εs−1\n= 2iTαst−ssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s)\n=iTαst−ssin (πs)\nπΓ(s)\n=iTαst−s\nΓ(1−s). (87)\nWe therefore find that\n/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 2αsTδm,n(t−t/prime)−s\nΓ(1−s), (88)\nwhere we assumed that t≥t/prime. Therefore, we have now found the fractional LLG equation\n˙S(t) =S(t)×[−B+αsDs\ntS(t) +ξ(t)]. (89)\nD. Super-Ohmic spectral function\nWe now consider the case where\nJ(ε) =αssin/parenleftBigπs\n2/parenrightBig\nεs, (90)\nwith 1< s < 2. In this case, everything is equivalent to the sub-Ohmic case, up to Eq. (80), where we wanted to\nrewrite the dissipation into a fractional derivative. We had to introduce a finite initial time t0, which lead to\n(αdiss∗S)(t) =−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dεsin[ε(t−t/prime)]εsS(t/prime)\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg/bracketleftbig\nεs−1cos[ε(t−t/prime)]S(t/prime)/bracketrightbigt/prime=t\nt/prime=t0−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg\nεs−1S(t)−εs−1cos[ε(t−t0)]S(t0)−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n.(91)\nThe first term vanishes because of the cross product with S(t) in the LLG equation. However, the second term is\nmore problematic compared to the sub-Ohmic case, since the identity used to rewrite it only holds for s <1. We\nsolve this by writing it as a time derivative\nεs−1cos[ε(t−t0)]S(t0) =d\ndtεs−2sin[ε(t−t0)]S(t0), (92)16\nand then switching the ordering of the derivative and integral. Performing also one more partial integration in t/prime, we\nget\n(αdiss∗S)(t)\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbiggd\ndtεs−2sin[ε(t−t0)]S(t0) +/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/parenleftbiggd\ndtεs−2sin[ε(t−t0)]S(t0) +/bracketleftBig\nεs−2sin[ε(t−t/prime)]˙S(t/prime)/bracketrightBigt/prime=t\nt/prime=t0−/integraldisplayt\nt0dt/prime/braceleftBig\n−sin[ε(t−t/prime)]εs−2¨S(t/prime)/bracerightBig/parenrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbiggd\ndtεs−2sin[ε(t−t0)]S(t0) +εs−2sin[ε(t−t0)]˙S(t0) +/integraldisplayt\nt0dt/primesin[ε(t−t/prime)]εs−2¨S(t/prime)/bracerightbigg\n. (93)\nNow, rescaling ε→ε/(t−t0) andε→ε/(t−t/prime) respectively, we have\n(αdiss∗S)(t)\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dεsin(ε)εs−2/bracketleftbiggd\ndt(t−t0)1−sS(t0) + (t−t0)1−s˙S(t0) +/integraldisplayt\nt0dt/prime(t−t/prime)1−s¨S(t/prime)/bracketrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/bracketleftbigg\nsin/parenleftbiggπ(s−1)\n2/parenrightbigg\nΓ(s−1)/bracketrightbigg/bracketleftbigg\n(1−s)(t−t0)−sS(t0) + (t−t0)1−s˙S(t0) +/integraldisplayt\nt0dt/prime(t−t/prime)1−s¨S(t/prime)/bracketrightbigg\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s−1)/bracketleftBig\n(1−s)(t−t0)−sS(t0) + (t−t0)1−s˙S(t0) + Γ(2−s)C\nt0Ds\ntS(t)/bracketrightBig\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s−1)Γ(1−(s−1))/bracketleftbigg(t−t0)−s\nΓ(1−s)S(t0) +(t−t0)1−s\nΓ(2−s)˙S(t0) +C\nt0Ds\ntS(t)/bracketrightbigg\n=−αssin (πs)\nππ\nsin[π(s−1)]RL\nt0Ds\ntS(t)\n=αsDs\ntS(t), (94)\nwhere we used several identities from Sec. 6 in the Sup. Mat. of Ref. [12], Ref. [13, p.893], and in the last line we\nsentt0→−∞ .\nFor the noise correlation, we have to compute the Keldysh component. This is\nαK(t) =1\n2π/integraldisplay∞\n−∞dωe−iωtαK(ω)\n=1\n2π/integraldisplay∞\n−∞dωe−iωt[αR(ω)−αA(ω)] coth/parenleftBigω\n2T/parenrightBig\n=αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωe−iωt/bracketleftbiggω2εs−1\nε2−(ω+i0)2−ω2εs−1\nε2−(ω−i0)2/bracketrightbigg\ncoth/parenleftBigω\n2T/parenrightBig\n. (95)\nNow, we send ω→−ωin the advanced part\nαK(t) =αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dω/bracketleftbigge−iωtω2εs−1\nε2−(ω+i0)2+eiωtω2εs−1\nε2−(−ω−i0)2/bracketrightbigg\ncoth/parenleftBigω\n2T/parenrightBig\n=αssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ωt)ω2εs−1\nε2−(ω+i0)2coth/parenleftBigω\n2T/parenrightBig\n. (96)\nThen, we insert cos( ωt) =d\ndtsin(ωt)\nωand we send ω→ω/tandε→ε/t, which yields\nαK(t) =d\ndtαssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ωt)ωεs−1\nε2−(ω+i0)2coth/parenleftBigω\n2T/parenrightBig\n=d\ndtαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ω)ωεs−1\nε2−(ω+i0)2coth/parenleftBigω\n2tT/parenrightBig\n. (97)17\nTaking the high-temperature limit, we get\nαK(t) =d\ndtαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ω)ωεs−1\nε2−(ω+i0)22tT\nω\n=d\ndt2Tαst1−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ω)εs−1\nε2−(ω+i0)2\n=−2Tαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ω)εs−1\n(ω+i0−ε)(ω+i0 +ε). (98)\nNow, we want to close the integral over ωwith an infinite half-circle. For this, we need fast enough convergence of\nthe integrand to zero. Splitting the cosine into two exponential parts sin( ω) = (eiω−e−iω)/2i, we see that the first\nterm goes to zero when ω→i∞and that the second term goes to zero when ω→−i∞. Since we have poles at\nω=±ε−i0, the integral along the top half-plane vanishes. The integral along the bottom is then computed as\nαK(t) =−iTαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωe−iωεs−1\n(ω+i0−ε)(ω+i0 +ε)\n=−iTαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε−2πi/bracketleftbigge−i(ε−i0)εs−1\nε−i0 +i0 +ε+e−i(−ε−i0)εs−1\n−ε−i0 +i0−ε/bracketrightbigg\n=−2Tαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/bracketleftbigge−iεεs−2\n2−eiεεs−2\n2/bracketrightbigg\n=i2Tαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dεsin(ε)εs−2\n=−i2Tαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s−1)\n=iTαst−ssin (πs)\nπΓ(s)\n=iTαst−s\nΓ(1−s). (99)\nWe therefore find the same expression as in the sub-Ohmic case, which leads to\n/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 2αsTδm,n(t−t/prime)−s\nΓ(1−s). (100)\nTherefore, we have now also found the fractional LLG equation in the super-Ohmic case.\nE. Comparison Ohmic versus non-Ohmic\nIn the Ohmic case, we started with J(ω) =α1ωand ended up with a α1˙S(t) friction term, and noise correlation\n4α1kBTδ(t−t/prime). On the other hand, in the non-Ohmic case, we started with J(ε) =αssin/parenleftbigπs\n2/parenrightbig\nεsand ended up with\na frictionαsDs\ntS(t) and noise correlation 2 αskBT(t−t/prime)−s\nΓ(1−s). Although it is clear that both the non-Ohmic J(ω) and\nthe friction term will go to the Ohmic case when s→1, the noise is less straightforward. We can see that, as s→1,\nthe Gamma function will blow up, hence sending the correlation to zero, with the exception of t=t/prime. In this case,\nthe numerator blows up even before taking the limit of s→1. Hence, we can expect this function to behave as a\ndelta function. To find the correct prefactor, we integrate the distribution with a test-function f(t/prime) = 1 to find\nlim\ns→1/integraldisplay∞\n−∞d(t−t/prime)(t−t/prime)−s\nΓ(1−s)·1 = lim\ns→1/bracketleftbigg(t−t/prime)1−s\n(1−s)Γ(1−s)/bracketrightbigg∞\n(t−t/prime)=−∞\n= lim\ns→1/bracketleftbigg(t−t/prime)1−s\nΓ(2−s)/bracketrightbigg∞\n(t−t/prime)=−∞\n=/bracketleftbigg(t−t/prime)\n|t−t/prime|Γ(1)/bracketrightbigg∞\n(t−t/prime)=−∞\n= 2.\nHence, we see that the limit of the noise correlation becomes 4 α1kBTδ(t−t/prime), which is precisely as in the Ohmic case.18\nIII. FMR POWERLAW DERIVATION\nHere, we calculate the response of this spin-bath system to a rotating magnetic field. We will find the steady state\nsolutions and calculate several quantities that experiments could measure.\nA. Ferromagnetic Resonance\nWe will study the effects of a rotating magnetic field on the fractional LLG (FLLG) equation\n˙S(t) =S(t)×/braceleftbig\n−Beff[t,S(t)] +αsDs\ntS(t) +ξ(t)/bracerightbig\n. (101)\nFor simplicity, we assume that the temperature of the bath is low compared to the energy of the external fields, such\nthat the thermal noise ξ(t) may be neglected. We apply a rotating magnetic field\nBeff[t,S(t)] =\nΩ cos(ωdt)\nΩ sin(ωdt)\nB0−KSz(t)\n (102)\nand use spherical coordinates\nS=S\nsinθcosφ\nsinθsinφ\ncosθ\n. (103)\nWe will assume a small θapproximation, where the ground state is in the positive zdirection, i.e. 0 <Ω/lessmuchB0−KS\nandαsS/lessmuch(B0−KS)1−s. As shown in Section I D, we may rewrite the FLLG equation of this form in spherical\ncoordinates as\n˙φ=1\nsinθ[−Beff[t,S(t)] +αsDs\ntS(t)]·\ncosθcosφ\ncosθsinφ\n−sinθ\n; (104)\n˙θ= [−Beff[t,S(t)] +αsDs\ntS(t)]·\nsinφ\n−cosφ\n0\n. (105)\nIn the rotating frame, where B(t) is constant, we could expect the system to go to a steady state after some time.\nHence, we introduce a new coordinate such, that φ=ωdt−ϕ. We may then set ˙ ϕ=˙θ= 0 to find the steady state in\nthe rotating frame, where\nωdsinθ= [−Beff[t,S(t)] +αsDs\ntS(t)]·\ncosθcos(ωdt−ϕ)\ncosθsin(ωdt−ϕ)\n−sinθ\n; (106)\n0 = [−Beff[t,S(t)] +αsDs\ntS(t)]·\nsin(ωdt−ϕ)\n−cos(ωdt−ϕ)\n0\n. (107)\nWe note that S(t) is now only time dependent in the rotating-frame term, which means that we can explicitly calculate\nthe fractional derivative. The Liouville derivative works well with Fourier transforms, hence the fractional derivative\nof a trigonometric function is given by\nDs\ntsin(ωt) =|ω|ssin/parenleftBig\nωt+ sign(ω)πs\n2/parenrightBig\n, (108)\nand similarly for a cosine. We remark that the Liouville derivative of a constant can only be described by setting the\ninitial time to some finite t0, in which case it becomes zero2. Combining this with the steady state expression for S,\n2Since physically the infinite time only models a long time in the past, we will apply it as such. Formally, there are some restrictions on\nthe functions that the Liouville derivative can be applied to. These include restrictions such as its integral over the whole domain being\nfinite. Since this is not the case for a non-zero constant on an infinite interval, we have to regularize the lower integral boundary −∞\nas a finitet0.19\nwe find\nαsDs\ntS(t) =αsDs\ntS\nsinθcos(ωdt−ϕ)\nsinθsin(ωdt−ϕ)\ncosθ\n=αsSsinθ|ωd|s\ncos/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\nsin/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\n0\n. (109)\nHence, we find that\nωdtanθ= [−Beff[t,S(t)] +αsDs\ntS(t)]·\ncos(ωdt−ϕ)\nsin(ωdt−ϕ)\n−tanθ\n\n=\n−\nΩ cos(ωdt)\nΩ sin(ωdt)\nB0−KScosθ\n+αsSsinθ|ωd|s\ncos/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\nsin/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\n0\n\n·\ncos(ωdt−ϕ)\nsin(ωdt−ϕ)\n−tanθ\n\n=−Ω cosϕ+B0tanθ−KSsinθ+αsSsinθ|ωd|scos/parenleftBig\nsign(ωd)πs\n2/parenrightBig\n(110)\nand\n0 = [−Beff[t,S(t)] +αsDs\ntS(t)]·\nsin(ωdt−ϕ)\n−cos(ωdt−ϕ)\n0\n\n=\n−\nΩ cos(ωdt)\nΩ sin(ωdt)\nB0−KScosθ\n+αsSsinθ|ωd|s\ncos/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\nsin/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\n0\n\n·\nsin(ωdt−ϕ)\n−cos(ωdt−ϕ)\n0\n\n= Ω sinϕ−αsSsinθ|ωd|ssin/parenleftBig\nsign(ωd)πs\n2/parenrightBig\n. (111)\nWith some rearranging, we have\nϕ= arcsin/bracketleftbiggαsS\nΩ|ωd|ssin/parenleftBig\nsign(ωd)πs\n2/parenrightBig\nsinθ/bracketrightbigg\n(112)\nand\nΩ cosϕ= Ω cos arcsin/bracketleftbiggαsS\nΩ|ωd|ssin/parenleftBig\nsign(ωd)πs\n2/parenrightBig\nsinθ/bracketrightbigg\n=/radicalbigg\nΩ2−/bracketleftBig\nαsS|ωd|ssin/parenleftBig\nsign(ωd)πs\n2/parenrightBig\nsinθ/bracketrightBig2\n= (B0−KScosθ−ωd) tanθ+αsSsinθ|ωd|scos/parenleftBig\nsign(ωd)πs\n2/parenrightBig\n. (113)\nSquaring this, we get\nΩ2= (B0−KScosθ−ωd)2tan2θ+ (αsS|ωd|s)2sin2θ+ 2αsS|ωd|s(B0−KScosθ−ωd) cos/parenleftBigπs\n2/parenrightBigsin2θ\ncosθ,\nand multiplying by cos2θ= 1−sin2θ, we have\n(1−sin2θ)Ω2= (B0−KS/radicalbig\n1−sin2θ−ωd)2sin2θ+/bracketleftBig\n(αsS|ωd|s)2−2αsS|ωd|sKScos/parenleftBigπs\n2/parenrightBig/bracketrightBig\nsin2θ(1−sin2θ)\n+ 2αsS|ωd|s(B0−ωd) cos/parenleftBigπs\n2/parenrightBig\nsin2θ/radicalbig\n1−sin2θ. (114)\nWe could go further and make this into (effectively) a 4th order equation for sin2θ. However, since we are in a small\nθlimit, we will solve this equation up to first order in sin2θ, which yields\nΩ2= sin2θ/braceleftBig\nΩ2(B0−KS−ωd)2+/bracketleftBig\n(αsS|ωd|s)2−2αsS|ωd|sKScos/parenleftBigπs\n2/parenrightBig/bracketrightBig\n+ 2αsS|ωd|s(B0−ωd) cos/parenleftBigπs\n2/parenrightBig/bracerightBig\n. (115)20\nHence, we find that\nsin2θ=Ω2\nΩ2+ (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs\n2/parenrightbig\n=Ω2\n(B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs\n2/parenrightbig+O(Ω4). (116)\nNote that theO(Ω4) should formally be dimensionless, but we explain what we mean with terms being small in\nSection IV, as this is more subtle with fractional dimensions.\nB. Resonance frequency and amplitude\nSince we are studying ferromagnetic resonance, we want to find the driving frequency for which we get the largest\nresponse from the magnetic system. Since we know that the resonance for an Ohmic system is at ωd=B0−KS, we\nwill expand the formula around this point to find the new maximum. To compute the resonance frequency ωres, we\nthus first assume that ωres≈(B0−KS)(1+y) withysmall, such that|ωres|s≈(B0−KS)s(1+sy) (forB0−KS > 0).\nThis results in\nsin2θ=Ω2\n(B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs\n2/parenrightbig\n≈Ω2\n(B0−KS)2y2+ (αsS)2(B0−KS)2s(1 + 2sy)−2αsS(B0−KS)s+1(1 +sy)ycos/parenleftbigπs\n2/parenrightbig. (117)\nNow, we put the derivative with respect to yequal to zero, to get\n(B0−KS)2y+s(αsS)2(B0−KS)2s−αsS(B0−KS)s+1(1 + 2sy) cos/parenleftBigπs\n2/parenrightBig\n= 0. (118)\nHence, we find that\ny=−s(αsS)2(B0−KS)2s+αsS(B0−KS)s+1cos/parenleftbigπs\n2/parenrightbig\n(B0−KS)2−2sαsS(B0−KS)s+1cos/parenleftbigπs\n2/parenrightbig\n=αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig\n+O(αsS)2, (119)\nwhich results in\nωres≈(B0−KS)/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBig\n= (B0−KS) +αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig\n. (120)\nWe see that the resonance frequency gets shifted by a small amount, depending on s, which scales non-linearly.\nInserting this result into Eq. (116), we can now also find an approximation for the amplitude at resonance:\nsin2θres=Ω2\n(B0−KS−ωres)2+ (αsS|ωres|s)2+ 2αsS|ωres|s(B0−KS−ωres) cos/parenleftbigπs\n2/parenrightbig\n≈Ω2/braceleftBig/bracketleftBig\nαsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/bracketrightBig2\n+/parenleftBig\nαsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingles/parenrightBig2\n−2αsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingles\nαsS(B0−KS)scos2/parenleftBigπs\n2/parenrightBig/bracerightBig−1\n=Ω2\n[αsS(B0−KS)s]2/bracketleftBig\ncos2/parenleftBigπs\n2/parenrightBig\n+/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2s\n−2/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingles\ncos2/parenleftBigπs\n2/parenrightBig/bracketrightBig−1\n=Ω2\n[αsS(B0−KS)s]2/bracketleftBig\n1−cos2/parenleftBigπs\n2/parenrightBig\n+O(αsS)/bracketrightBig−1\n≈Ω2\n/bracketleftbig\nαsS(B0−KS)ssin/parenleftbigπs\n2/parenrightbig/bracketrightbig2. (121)\nSince the sine function decreases as smoves away from one, we see that the amplitude actually increases for non-Ohmic\nenvironments.21\nC. Calculating the FWHM linewidth\nNext, we are interested not only in the location of the resonance, but also how sensitive the resonance is to the\ndriving frequency. One way to describe this is by using the Full Width at Half Maximum measure. This provides a\nwell-defined line width independently of the shape of the peak. It is found by measuring the width of the peak at\nhalf the height of its maximum. This can be measured in the laboratories, but it can also be computed. Since our\nfunction of interest is of the form sin2θ(ωd) = Ω2/g(ωd), it makes sense to approximate the inverse function instead\nof the regular one. To this end, we will translate the FWHM measurement to the inverse function, and then Taylor\nexpandg(ωd) near resonance as a parabola to solve for the new condition of this inverse function. Notice that from\nEq. (116), we have\ng(ωd) = (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftBigπs\n2/parenrightBig\n. (122)\nThe FWHM condition is\nΩ2\ng(ωd)= sin2θ(ωd) =sin2θ(ωres)\n2=Ω2\n2g(ωres), (123)\nhence we must solve for 2 g(ωres) =g(ωd). To this end, let us assume that ωd=ωres+yand expand g(ωd) iny. We\nwill use that\n|a+y|n≈an+nan−1y+1\n2n(n−1)an−2y2\nfor smallyanda>0. Then,\ng(ωres+y)\n= (B0−KS−ωres−y)2+ (αsS|ωres+y|s)2+ 2αsS|ωres+y|s(B0−KS−ωres−y) cos/parenleftBigπs\n2/parenrightBig\n≈(B0−KS−ωres)2+ (αsSωs\nres)2+ 2αsSωs\nres(B0−KS−ωres) cos/parenleftBigπs\n2/parenrightBig\n+y/parenleftBig\n−2(B0−KS��ωres) + 2s(αsS)2ω2s−1\nres−2αsScos/parenleftBigπs\n2/parenrightBig/braceleftbig\nωs\nres+sωs−1\nres[ωres−(B0−KS)]/bracerightbig/parenrightBig\n+y2/bracketleftBig\n1 +s(2s−1)(αsS)2ω2s−2\nres−2sαsSωs−1\nrescos/parenleftBigπs\n2/parenrightBig\n+s(s−1)αsSωs−2\nres(B0−KS−ωres) cos/parenleftBigπs\n2/parenrightBig/bracketrightBig\n=g(ωres) +y/parenleftBigg\n2αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig\n+ 2s(αsS)2(B0−KS)2s−1/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBig2s−1\n−2αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/braceleftBigg/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs\n+sαsScos/parenleftBigπs\n2/parenrightBig\n(B0−KS)s−1/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs−1/bracerightBigg/parenrightBigg\n+y2/braceleftBigg\n1 +s(2s−1)(αsS)2(B0−KS)2s−2/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBig2s−2\n−2sαsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs−1\n−s(s−1)(αsS)2(B0−KS)2s−2cos2/parenleftBigπs\n2/parenrightBig/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs−2/bracerightBigg\n≈g(ωres) +y/braceleftBig\n2s(αsS)2(B0−KS)2s−1/bracketleftBig\n1−2 cos2/parenleftBigπs\n2/parenrightBig/bracketrightBig/bracerightBig\n+y2/braceleftBig\n1−2s(αsS)(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig\n+ (αsS)2(B0−KS)2s−2/bracketleftBig\ns(2s−1)−3s(s−1) cos2/parenleftBigπs\n2/parenrightBig/bracketrightBig/bracerightBig\n+O(αsS)3.\n(124)\nNow, we set 2 g(ωres) =g(ωres+y) =g(ωres) +by+ay2, and remark that g(ωres) = (αsS)2(B0−KS)2ssin2/parenleftbigπs\n2/parenrightbig\n, in\norder to find that\ny=−b±/radicalbig\nb2+ 4ag(ωres)\n2a=⇒∆FWHM =/radicalbig\nb2+ 4ag(ωres)\na. (125)22\nHence, we find that the lowest-order contribution to the linewidth is given by\n∆FWHM≈/radicalBig\n4(αsS)2(B0−KS)2ssin2/parenleftbigπs\n2/parenrightbig\n+O(αsS)3\n1 +O(αsS)\n= 2(αsS)(B0−KS)ssin/parenleftBigπs\n2/parenrightBig\n+O(αsS)2. (126)\nIV. DIMENSIONAL ANALYSIS\nThe fractional derivative in the LLG equation has an impact on the dimensions of quantities. We can firstly see\nthat in the chosen units, we have [ B0−KS] = [ωd] = time−1. Assuming Sto be dimensionless, then [ ωd] = [αsDs\ntS] =\n[αs][ωd]s, hence [αs] = [ωd]1−s. We can now start to understand what we mean when we say that certain quantities\nare small, since this has to be relative to something else. For instance, when we say αsSis small, we understand this\nasαsS/lessmuch(B0−KS)1−s. For Ω it is simpler, since there is no fractional derivative acting with it. Hence, for Ω small\nwe simply mean Ω /lessmuchB0−KS. We can now also define some dimensionless variables, such as α/prime\ns=αsS(B0−KS)s−1\nand Ω/prime= Ω/(B0−KS). We have used these variables in the figures to show the general behavior of the quantities.\n[1] A. Altland and B. D. Simons, Condensed Matter Field Theory (Cambridge University Press, 2010), 2nd ed.\n[2] A. Kamenev, Field theory of non-equilibrium systems (Cambridge University Press, 2011).\n[3] A. Schmid, Journal of Low Temperature Physics 49, 609 (1982).\n[4] A. O. Caldeira and A. J. Leggett, Physical Review Letters 46, 211 (1981).\n[5] A. O. Caldeira and A. J. Leggett, Physica A: Statistical Mechanics and its Applications 121, 587 (1983).\n[6] A. O. Caldeira and A. J. Leggett, Annals of Physics 149, 374 (1983).\n[7] A. O. Caldeira, An introduction to macroscopic quantum phenomena and quantum dissipation , vol. 9780521113755 (Cam-\nbridge University Press, 2012).\n[8] A. O. Caldeira and A. J. Leggett, Phys. Rev. A 31, 1059 (1985).\n[9] U. Weiss, Quantum dissipative systems (World scientific, 2012).\n[10] C. Gardiner and P. Zoller, Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with\napplications to quantum optics (Springer Science & Business Media, 2004).\n[11] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N. Kiselev, and A. Altland, Physical Review Letters 114, 176806\n(2015).\n[12] R. C. Verstraten, R. F. Ozela, and C. M. Smith, Physical Review B 103, L180301 (2021).\n[13] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, 2014)."    },    {        "title": "2211.13486v1.Influence_of_non_local_damping_on_magnon_properties_of_ferromagnets.pdf",        "content": "In\ruence of non-local damping on magnon properties of ferromagnets\nZhiwei Lu,1,\u0003I. P. Miranda,2,\u0003Simon Streib,2Manuel Pereiro,2Erik Sj oqvist,2\nOlle Eriksson,2, 3Anders Bergman,2Danny Thonig,3, 2and Anna Delin1, 4\n1Department of Applied Physics, School of Engineering Sciences, KTH Royal\nInstitute of Technology, AlbaNova University Center, SE-10691 Stockholm, Swedeny\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden\n3School of Science and Technology, Orebro University, SE-701 82, Orebro, Sweden\n4SeRC (Swedish e-Science Research Center), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden\n(Dated: November 28, 2022)\nWe study the in\ruence of non-local damping on magnon properties of Fe, Co, Ni and Fe 1\u0000xCox\n(x= 30%;50%) alloys. The Gilbert damping parameter is typically considered as a local scalar\nboth in experiment and in theoretical modelling. However, recent works have revealed that Gilbert\ndamping is a non-local quantity that allows for energy dissipation between atomic sites. With\nthe Gilbert damping parameters calculated from a state-of-the-art real-space electronic structure\nmethod, magnon lifetimes are evaluated from spin dynamics and linear response, where a good\nagreement is found between these two methods. It is found that non-local damping a\u000bects the\nmagnon lifetimes in di\u000berent ways depending on the system. Speci\fcally, we \fnd that in Fe, Co,\nand Ni the non-local damping decreases the magnon lifetimes, while in Fe 70Co30and Fe 50Co50an\nopposite, non-local damping e\u000bect is observed, and our data show that it is much stronger in the\nformer.\nINTRODUCTION\nIn recent years, there has been a growing interest in\nmagnonics, which uses quasi-particle excitations in mag-\nnetically ordered materials to perform information trans-\nport and processing on the nanoscale. Comparing to the\nconventional information device, the magnonics device\nexhibits lower energy consumption, easier integrability\nwith complementary metal-oxide semiconductor (CMOS)\nstructure, anisotropic properties, and e\u000ecient tunability\nby various external stimuli to name a few [1{10]. Yttrium\niron garnet (YIG) [11] as well as other iron garnets with\nrare-earth elements (Tm, Tb, Dy, Ho, Er) [12] are very\npromising candidates for magnonics device applications\ndue to their low energy dissipation properties and, thus,\nlong spin wave propagation distances up to tens of \u0016m.\nContrary, the damping of other materials for magnonics,\nlike CoFeB, is typically two orders of magnitude higher\ncompared to YIG [12], leading to much shorter spin wave\npropagation distances. A clear distinction can be made\nbetween materials with an ultra-low damping parame-\nter, like in YIG, and those with a sign\fciantly larger,\nbut still small, damping parameter. Materials like YIG\nare insulating, which hinders many of the microscopic\nmechanisms for damping, resulting in the low observed\ndamping parameter. In contrast, materials like CoFeB\nare metallic. In research projects that utilize low damp-\ning materials, YIG and similar non-metallic low damping\nsystems are typically favored. However, metallic systems\nhave an advantage, since magnetic textures can easily by\nin\ruenced by electrical currents. Hence, there is good\n\u0003These two authors contributed equally\nyCorresponding author: zhiweil@kth.sereason to consider metallic systems for low damping ap-\nplications, even though their damping typically is larger\nthan in YIG. One can conclude that Gilbert damping\nis one of the major bottlenecks for the choice of mate-\nrial in magnonics applications and a detailed experimen-\ntal as well as theoretical characterisation is fundamen-\ntal for this \feld of research, especially for metallic sys-\ntems. Thus, a more advanced and detailed understand-\ning of Gilbert damping is called for, in order to overcome\nthis obstacle for further development of magnonics-based\ntechnology.\nWhereas most studies consider chemical modi\fcations\nof the materials in order to tune damping [13, 14], only a\nfew focus on the fundamental physical properties as well\nas dependencies of the Gilbert damping. Often Gilbert\ndamping is considered as a phenomenological scalar pa-\nrameter in the equation of motion of localized atom-\nistic magnetic moments, i.e. the Landau-Lifshitz-Gilbert\n(LLG) equation [15]. However, from using the general\nRayleigh dissipation function in the derivation proposed\nby Gilbert [16], it was theoretically found that the Gilbert\ndamping should be anisotropic, a tensor, and non-local.\nFurthermore, it depends on the temperature and, thus,\non underlying magnon as well as phonon con\fgurations\n[17{20]. This is naturally built into the multiple theoret-\nical methods developed to predict the damping parame-\nter, including breathing Fermi surface model [21], torque\ncorrelation model [22], and linear response formulation\n[23]. For instance, the general Gilbert damping tensor\nas a function of the non-collinear spin con\fguration has\nbeen proposed in Ref. 24.\nNonetheless, an experimental veri\fcation is still miss-\ning due to lacking insights into the impact of the gen-\neralised damping on experimental observables. In a re-\ncent experiment, however, the anisotropic behavior of the\ndamping has been con\frmed for Co 50Fe50thin \flms andarXiv:2211.13486v1  [cond-mat.mtrl-sci]  24 Nov 20222\nwas measured to be of the order of 400% [25], with respect\nto changing the magnetization direction. Changes of\nGilbert damping in a magnetic domain wall and, thus, its\ndependency on the magnetic con\fguration was measured\nin Ref. [26] and \ftted to the Landau-Lifshitz-Baryakhtar\n(LLBar) equation, which includes non-locality of the\ndamping by an additional dissipation term proportional\nto the gradient of the magnetisation [27{29]. However,\nthe pair-wise non-local damping \u000bijhas not yet been\nmeasured.\nThe most common experimental techniques of evaluat-\ning damping are ferromagnetic resonance (FMR) [30] and\ntime-resolved magneto-optical Kerr e\u000bect (TR-MOKE)\n[31]. In these experiments, Gilbert damping is related\nto the relaxation rate when (i)slightly perturbing the\ncoherent magnetic moment out of equilibrium by an ex-\nternal magnetic \feld [32] or (ii)when disordered mag-\nnetic moments remagnetise after pumping by an ultrafast\nlaser pulse [33]. Normally, in case (i)the non-locality is\nsuppressed due to the coherent precession of the atomic\nmagnetic moments. However, this coherence can be per-\nturbed by temperature, making non-locality in principle\nmeasurable. One possible other path to link non-local\ndamping with experiment is magnon lifetimes. Theoret-\nically, the magnon properties as well as the impact of\ndamping on these properties can be assessed from the\ndynamical structure factor, and atomistic spin-dynamics\nsimulations have been demonstrated to yield magnon dis-\npersion relations that are in good agreement with exper-\niment [34]. In experiment, neutron scattering [35] and\nelectron scattering [36] are the most common methods for\nprobing magnon excitations, where the linewidth broad-\nening of magnon excitations is related to damping and\nprovides a way to evaluate the magnon lifetimes [37]. It is\nfound in ferromagnets that the magnon lifetimes is wave\nvector (magnon energy) dependent [38{40]. It has been\nreported that the magnon energy in Co \flms is nearly\ntwice as large as in Fe \flms, but they have similar magnon\nlifetimes, which is related to the intrinsic damping mech-\nanism of materials [41]. However, this collective e\u000bect of\ndamping and magnon energy on magnon lifetimes is still\nan open question. The study of this collective e\u000bect is of\ngreat interest for both theory and device applications.\nHere, we report an implementation for solving the\nstochastic Landau-Lifshitz-Gilbert (SLLG) equation in-\ncorporating the non-local damping. With the dynamical\nstructure factor extracted from the spin dynamics sim-\nulations, we investigate the collective e\u000bect of non-local\ndamping and magnon energy on the magnon lifetimes.\nWe propose an e\u000ecient method to evaluate magnon life-\ntimes from linear response theory and verify its validity.\nThe paper is organized as follows. In Sec. I, we give\nthe simulation details of the spin dynamics, the adiabatic\nmagnon spectra and dynamical structure factor, and the\nmethodology of DFT calculations and linear response.\nSec. II presents the non-local damping in real-space, non-\nlocal damping e\u000bects on the spin dynamics and magnon\nproperties including magnon lifetimes of pure ferromag-nets (Fe, Co, Ni), and Fe 1\u0000xCox(x= 30%;50%) alloys.\nIn Sec. III, we give a summary and an outlook.\nI. THEORY\nA. Non-local damping in atomistic spin dynamics\nThe dynamical properties of magnetic materials at \f-\nnite temperature have been so far simulated from atom-\nistic spin dynamics by means of the stochastic Landau-\nLifshitz-Gilbert equation with scalar local energy dissipa-\ntion. Here, the time evolution of the magnetic moments\nmi=mieiat atom site iis well described by:\n@mi\n@t=mi\u0002\u0012\n\u0000\r[Bi+bi(t)] +\u000b\nmi@mi\n@t\u0013\n;(1)\nwhere\ris the gyromagnetic ratio. The e\u000bective \feld Bi\nacting on each magnetic moment is obtained from:\nBi=\u0000@H\n@mi: (2)\nThe here considered spin-Hamiltonian Hconsists of a\nHeisenberg spin-spin exchange:\nH=\u0000X\ni6=jJijei\u0001ej: (3)\nHere,Jij{ the Heisenberg exchange parameter { cou-\nples the spin at site iwith the spin at site jand is cal-\nculated from \frst principles (see Section I C). Further-\nmore,\u000bis the scalar phenomenological Gilbert damp-\ning parameter. Finite temperature Tis included in\nEq. (1) via the \ructuating \feld bi(t), which is modeled\nby uncorrelated Gaussian white noise: hbi(t)i= 0 and\nb\u0016\ni(t)b\u0017\nj(t0)\u000b\n= 2D\u000eij\u000e\u0016\u0017\u000e(t\u0000t0), where\u000eis the Kro-\nnecker delta, i;jare site and \u0016;\u0017=fx;y;zgCartesian\nindices. Furthermore, the \ructuation-dissipation theo-\nrem givesD=\u000bkBT\n\rmi[42], with the Boltzman constant\nkB.\nA more generalized form of the SLLG equation that\nincludes non-local tensorial damping has been reported\nin previous studies [20, 43, 44] and is:\n@mi\n@t=mi\u00020\n@\u0000\r[Bi+bi(t)] +X\nj\u000bij\nmj@mj\n@t1\nA;(4)\nwhich can be derived from Rayleigh dissipation func-\ntional in the Lagrange formalism used by Gilbert [16].\nIn the presence of non-local damping, the Gaussian \ruc-\ntuating \feld ful\flls [43, 45, 46]\n\nb\u0016\ni(t)b\u0017\nj(t0)\u000b\n= 2D\u0016\u0017\nij\u000e(t\u0000t0); (5)\nwithD\u0016\u0017\nij=\u000b\u0016\u0017\nijkBT\n\rmi. The damping tensor \u000b\u0016\u0017\nijmust be\npositive de\fnite in order to be physically-de\fned. Along3\nwith spatial non-locality, the damping can also be non-\nlocal in time, as discussed in Ref. [47]. To prove the\n\ructuation-dissipation theorem in Eq. (5), the Fokker-\nPlanck equation has to be analysed in the presence of\nnon-local damping, similar to Ref. [15]. This is, however,\nnot the purpose of this paper. Instead, we will use the\napproximation \u000b\u0016\u0017\nij=1\n3Trf\u000biig\u000eij\u000e\u0016\u0017within the di\u000busion\nconstantD. Such an approximation is strictly valid only\nin the low temperature limit.\nTo solve this SLLG equation incorporating the non-\nlocal damping, we have implemented an implicit mid-\npoint solver in the UppASD code [48]. This iterative\n\fx-point scheme converges within an error of 10\u000010\u0016B,\nwhich is typically equivalent to 6 iteration steps. More\ndetails of this solver are provided in Appendix A. The\ninitial spin con\fguration in the typical N= 20\u000220\u000220\nsupercell with periodic boundary conditions starts from\ntotally random state. The spin-spin exchange interac-\ntions and non-local damping parameters are included up\nto at least 30 shells of neighbors, in order to guarantee\nthe convergence with respect to the spatial expansion of\nthese parameters (a discussion about the convergence is\ngiven in Section II A). Observables from our simulations\nare typically the average magnetisation M=1\nNPN\nimi\nas well as the magnon dispersion.\nB. Magnon dispersion\nTwo methods to simulate the magnon spectrum are\napplied in this paper: i)the dynamical structure factor\nandii)frozen magnon approach.\nFor the dynamical structure factor S(q;!) at \fnite\ntemperature and damping [34, 49], the spatial and time\ncorrelation function between two magnetic moments iat\npositionrandjat positionr0as well as di\u000berent time 0\nandtis expressed as:\nC\u0016(r\u0000r0;t) =hm\u0016\nr(t)m\u0016\nr0(0)i\u0000hm\u0016\nr(t)ihm\u0016\nr0(0)i:(6)\nHereh\u0001idenotes the ensemble average and \u0016are Carte-\nsian components. The dynamical structure factor can be\nobtained from the time and space Fourier transform of\nthe correlation function, namely:\nS\u0016(q;!) =1p\n2\u0019NX\nr;r0eiq\u0001(r\u0000r0)Z1\n\u00001ei!tC\u0016(r\u0000r0;t)dt:\n(7)\nThe magnon dispersion is obtained from the peak\npositions of S(q;!) along di\u000berent magnon wave vectors\nqin the Brillouin zone and magnon energies !. It\nshould be noted that S(q;!) is related to the scattering\nintensity in inelastic neutron scattering experiments [50].\nThe broadening of the magnon spectrum correlates to\nthe lifetime of spin waves mediated by Gilbert damping\nas well as intrinsic magnon-magnon scattering processes.\nGood agreement between S(q;!) and experiment hasbeen found previously [34].\nThe second method { the frozen magnon approach\n{ determines the magnon spectrum directly from the\nFourier transform of the spin-spin exchange parameters\nJij[51, 52] and non-local damping \u000bij. At zero tempera-\nture, a time-dependent external magnetic \feld is consid-\nered,\nB\u0006\ni(t) =1\nNX\nqB\u0006\nqeiq\u0001Ri\u0000i!t; (8)\nwhereNis the total number of lattice sites and B\u0006\nq=\nBx\nq\u0006iBy\nq. The linear response to this \feld is then given\nby\nM\u0006\nq=\u001f\u0006(q;!)B\u0006\nq: (9)\nWe obtain for the transverse dynamic magnetic suscep-\ntibility [53, 54]\n\u001f\u0006(q;!) =\u0006\rMs\n!\u0006!q\u0007i!\u000bq; (10)\nwith saturation magnetization Ms, spin-wave frequency\n!q=E(q)=~and damping\n\u000bq=X\nj\u000b0je\u0000iq\u0001(R0\u0000Rj): (11)\nWe can extract the spin-wave spectrum from the imagi-\nnary part of the susceptibility,\nIm\u001f\u0006(q;!) =\rMs\u000bq!\n[!\u0006!q]2+\u000b2q!2; (12)\nwhich is equivalent to the correlation function S\u0006(q;!)\ndue to the \ructuation-dissipation theorem [55]. We\n\fnd that the spin-wave lifetime \u001cqis determined by the\nFourier transform of the non-local damping (for \u000bq\u001c1),\n\u001cq=\u0019\n\u000bq!q: (13)\nThe requirement of positive de\fniteness of the damping\nmatrix\u000bijdirectly implies \u000bq>0, since\u000bijis diago-\nnalized by Fourier transformation due to translational\ninvariance. Hence, \u000bq>0 is a criterion to evaluate\nwhether the damping quantity in real-space is physically\nconsistent and whether \frst-principles calculations are\nwell converged. If \u000bq<0 for some wave vector q, energy\nis pumped into the spin system through the correspon-\ndent magnon mode, preventing the system to fully reach\nthe saturation magnetization at su\u000eciently low temper-\natures.\nThe e\u000bective damping \u000b0of the FMR mode at q=\n0 is determined by the sum over all components of the\ndamping matrix, following Eqn.11,\n\u000btot\u0011\u000b0=X\nj\u000b0j: (14)\nTherefore, an e\u000bective local damping should be based\non\u000btotif the full non-local damping is not taken into\naccount.4\nC. Details of the DFT calculations\nThe electronic structure calculations, in the framework\nof density functional theory (DFT), were performed us-\ning the fully self-consistent real-space linear mu\u000en-tin\norbital in the atomic sphere approximation (RS-LMTO-\nASA) [56, 57]. The RS-LMTO-ASA uses the Haydock\nrecursion method [58] to solve the eigenvalue problem\nbased on a Green's functions methodology directly in\nreal-space. In the recursion method, the continued frac-\ntions have been truncated using the Beer-Pettifor termi-\nnator [59], after a number LLof recursion levels. The\nLMTO-ASA [60] is a linear method which gives precise\nresults around an energy E\u0017, usually taken as the center\nof thes,panddbands. Therefore, as we calculate \fne\nquantities as the non-local damping parameters, we here\nconsider an expression accurate to ( E\u0000E\u0017)2starting\nfrom the orthogonal representation of the LMTO-ASA\nformalism [61].\nFor bcc FeCo alloys and bcc Fe we considered LL= 31,\nwhile for fcc Co and fcc Ni much higher LLvalues (51 and\n47, respectively), needed to better describe the density of\nstates and Green's functions at the Fermi level.\nThe spin-orbit coupling (SOC) is included as a l\u0001s\n[60] term computed in each variational step [62]. All\ncalculations were performed within the local spin den-\nsity approximation (LSDA) exchange-functional (XC) by\nvon Barth and Hedin [63], as it gives general magnetic\ninformation with equal or better quality as, e.g., the\ngeneralized gradient approximation (GGA). Indeed, the\nchoice of XC between LSDA and GGA [64] have a mi-\nnor impact on the onsite damping and the shape of the\n\u000bqcurves, when considering the same lattice parame-\nters (data not shown). No orbital polarization [65] was\nconsidered here. Each bulk system was modelled by a\nbig cluster containing \u001855000 (bcc) and\u0018696000 (fcc)\natoms located in the perfect crystal positions with the re-\nspective lattice parameters of a= 2:87\u0017A (bcc Fe and bcc\nFe1\u0000xCox, su\u000eciently close to experimental observations\n[66]),a= 3:54\u0017A (fcc Co [20, 67]), and a= 3:52\u0017A (fcc\nNi [68]). To account for the chemical disorder in the\nFe70Co30and Fe 50Co50bulks, the electronic structure\ncalculated within the simple virtual crystal approxima-\ntion (VCA), which has shown to work well for the fer-\nromagnetic transition metals alloys (particularly for el-\nements next to each other in the Periodic Table, such\nas FeCo and CoNi) [69{76], and also describe in a good\nagreement the damping trends in both FeCo and CoNi\n(see Appendix C).\nAs reported in Ref. [77], the total damping of site\ni, in\ruenced by the interaction with neighbors j, can\nbe decomposed in two main contributions: the onsite\n(fori=j), and the non-local (for i6=j). Both can be\ncalculated, in the collinear framework, by the followingexpression,\n\u000b\u0016\u0017\nij=\u000bCZ1\n\u00001\u0011(\u000f)Tr\u0010\n^T\u0016\ni^Aij(^T\u0017\nj)y^Aji\u0011\nd\u000fT!0K\u0000\u0000\u0000\u0000!\n\u000bCTr\u0010\n^T\u0016\ni^Aij(\u000fF+i\u000e)(^T\u0017\nj)y^Aji(\u000fF+i\u000e)\u0011\n;\n(15)\nwhere we de\fne ^Aij(\u000f+i\u000e) =1\n2i(^Gij(\u000f+i\u000e)\u0000^Gy\nji(\u000f+i\u000e))\nthe anti-Hermitian part of the retarded physical Green's\nfunctions in the LMTO formalism, and \u000bC=g\nmti\u0019a\npre-factor related to the i-th site magnetization. The\nimaginary part, \u000e, is obtained from the terminated con-\ntinued fractions. Also in Eq. 15, ^T\u0016\ni= [\u001b\u0016\ni;Hso] is the\nso-called torque operator [20] evaluated in each Cartesian\ndirection\u0016;\u0017=fx;y;zgand at site i,\u0011(\u000f) =\u0000@f(\u000f)\n@\u000fis\nthe derivative of the Fermi-Dirac distribution f(\u000f) with\nrespect to the energy \u000f,g= 2\u0010\n1 +morb\nmspin\u0011\ntheg-factor\n(not considering here the spin-mixing parameter [78]),\n\u001b\u0016are the Pauli matrices, and mtiis the total magnetic\nmoment of site i(mti=morbi+mspini). This results\nin a 3\u00023 tensor with terms \u000b\u0016\u0017\nij. In the real-space bulk\ncalculations performed in the present work, the \u000bij(with\ni6=j) matrices contain o\u000b-diagonal terms which are can-\ncelled by the summation of the contributions of all neigh-\nbors within a given shell, resulting in a purely diagonal\ndamping tensor, as expected for symmetry reasons [15].\nTherefore, as in the DFT calculations the spin quanti-\nzation axis is considered to be in the z([001]) direction\n(collinear model), we can ascribe a scalar damping value\n\u000bijas the average \u000bij=1\n2(\u000bxx\nij+\u000byy\nij) =\u000bxx\nijfor the\nsystems investigated here. This scalar \u000bijis, then, used\nin the SLLG equation (Eq. 1).\nThe exchange parameters Jijin the Heisenberg\nmodel were calculated by the Liechtenstein-Katsnelson-\nAntropov-Gubanov (LKAG) formalism [79], according to\nthe implementation in the RS-LMTO-ASA method [61].\nHence all parameters needed for the atomistic LLG equa-\ntion have been evaluated from ab-initio electronic struc-\nture theory.\nII. RESULTS\nA. Onsite and non-local dampings\nTable I shows the relevant ab-initio magnetic prop-\nerties of each material; the TCvalues refer to the Curie\ntemperature calculated within the random-phase approx-\nimation (RPA) [80], based on the computed Jijset. De-\nspite the systematic \u000btotvalues found in the lower limit\nof available experimental results (in similar case with,\ne.g., Ref. [81]), in part explained by the fact that we\nanalyze only the intrinsic damping, a good agreement\nbetween theory and experiment can be seen. When the\nwhole VCA Fe 1\u0000xCoxseries is considered (from x= 0%\ntox= 60%), the expected Slater-Pauling behavior of5\nthe total magnetic moment [73, 82] is obtained (data not\nshown).\nFor all systems studied here, the dissipation is domi-\nnated by the onsite ( \u000bii) term, while the non-local pa-\nrameters (\u000bij,i6=j) exhibit values at least one order of\nmagnitude lower; however, as it will be demonstrated in\nthe next sections, these smaller terms still cause a non-\nnegligible impact on the relaxation of the average magne-\ntization as well as magnon lifetimes. Figure 1 shows the\nnon-local damping parameters for the investigated ferro-\nmagnets as a function of the ( i;j) pairwise distance rij=a,\ntogether with the correspondent Fourier transforms \u000bq\nover the \frst Brillouin Zone (BZ). The \frst point to no-\ntice is the overall strong dependence of \u000bon the wave\nvectorq. The second point is the fact that, as also re-\nported in Ref. [20], \u000bijcan be an anisotropic quantity\nwith respect to the same shell of neighbors, due to the\nbroken symmetry imposed by a preferred spin quantiza-\ntion axis. This means that, in the collinear model and for\na given neighboring shell, \u000bijis isotropic only for equiva-\nlent sites around the magnetization as a symmetry axis.\nAnother important feature that can be seen in Fig. 1\nis the presence of negative \u000bijvalues. Real-space neg-\native non-local damping parameters have been reported\npreviously [20, 77, 97]. They are related to the decrease\nof damping at the \u0000-point, but may also increase \u000bqfrom\nthe onsite value in speci\fc qpoints inside the BZ; there-\nfore, they cannot be seen as ad hoc anti-dissipative con-\ntributions. In the ground-state, these negative non-local\ndampings originate from the overlap between the anti-\nHermitian parts of the two Green's functions at the Fermi\nlevel, each associated with a spin-dependent phase factor\n\b\u001b(\u001b=\";#) [20, 80].\nFinally, as shown in the insets of Fig. 1, a long-range\nconvergence can be seen for all cases investigated. An\nillustrative example is the bcc Fe 50Co50bulk, for which\nthe e\u000bective damping can be \u001860% higher than the con-\nverged\u000btotif only the \frst 7 shells of neighbors are con-\nsidered in Eq. 14. The non-local damping of each neigh-\nboring shell is found to follow a1\nr2\nijtrend, as previously\nargued by Thonig et al. [20] and Umetsu et al. [97].\nExplicitly,\n\u000bij/sin(k\"\u0001rij+ \b\") sin(k#\u0001rij+ \b#)\njrijj2; (16)\nwhich also qualitatively justi\fes the existence of negative\n\u000bij's. Thus, the convergence in real-space is typically\nslower than other magnetic quantities, such as exchange\ninteractions ( Jij/1\njrijj3) [80], and also depends on the\nimaginary part \u000e(see Eq. 15) [20]. The di\u000berence in the\nasymptotic behaviour of the damping and the Heisenberg\nexchange is distinctive; the \frst scales with the inverse of\nthe square of the distance while the latter as the inverse\nof the cube of the distance. Although this asymptotic\nbehaviour can be derived from similar arguments, both\nusing the Greens function of the free electron gas, the\nresults are di\u000berent. The reason for this di\u000berence issimply that the damping parameter is governed by states\nclose to the Fermi surface, while the exchange parameter\ninvolves an integral over all occupied states [20, 79].\nFrom bcc Fe to bcc Fe 50Co50(Fig. 1(a-f)), with in-\ncreasing Co content, the average \frst neighbors \u000bijde-\ncreases to a negative value, while the next-nearest neigh-\nbors contributions reach a minimum, and then increase\nagain. Similar oscillations can be found in further shells.\nAmong the interesting features in the Fe 1\u0000xCoxsystems\n(x= 0%;30%;50%), we highlight the low \u000bqaround the\nhigh-symmetry point H, along the H\u0000PandH\u0000N\ndirections, consistently lower than the FMR damping.\nBoth\u000bvalues are strongly in\ruenced by non-local con-\ntributions &5 NN. Also consistent is the high \u000bqob-\ntained forq=H. For long wavelengths in bcc Fe, some\n\u000bqanisotropy is observed around \u0000, which resembles the\nsame trait obtained for the corresponding magnon dis-\npersion curves [80]. This anisotropy changes to a more\nisotropic behavior by FeCo alloying.\nFar from the more noticeable high-symmetry points,\n\u000bqpresents an oscillatory behavior along BZ, around the\nonsite value. It is noteworthy, however, that these oscil-\nlatory\u000bqparameters exhibit variations up to \u00182 times\n\u000bii, thus showing a pronounced non-local in\ruence in\nspeci\fcqpoints.\nIn turn, for fcc Co (Fig. 1(g,h)) the \frst values are\ncharacterized by an oscillatory behavior around zero,\nwhich also re\rects on the damping of the FMR mode,\n\u000bq=0. In full agreement with Ref. [20], we compute a\npeak of\u000bijcontribution at rij\u00183:46a, which shows\nthe long-range character that non-local damping can ex-\nhibit for speci\fc materials. Despite the relatively small\nmagnitude of \u000bij, the multiplicity of the nearest neigh-\nbors shells drives a converged \u000bqdispersion with non-\nnegligible variations from the onsite value along the BZ,\nspecially driven by the negative third neighbors. The\nmaximum damping is found to be in the region around\nthe high-symmetry point X, where thus the lifetime of\nmagnon excitations are expected to be reduced. Simi-\nlar situation is found for fcc Ni (Fig. 1(i,j)), where the\n\frst neighbors \u000bijare found to be highly negative, con-\nsequently resulting in a spectrum in which \u000bq> \u000bq=0\nfor everyq6= 0. In contrast with fcc Co, however, no\nnotable peak contributions are found.\nB. Remagnetization\nGilbert damping in magnetic materials determines the\nrate of energy that dissipates from the magnetic to other\nreservoirs, like phonons or electron correlations. To ex-\nplore what impact non-local damping has on the energy\ndissipation process, we performed atomistic spin dynam-\nics (ASD) simulations for the aforementioned ferromag-\nnets: bcc Fe 1\u0000xCox(x= 0%;30%;50%), fcc Co, and\nfcc Ni, for the (i)fully non-local \u000bijand (ii)e\u000bective\n\u000btot(de\fned in 14) dissipative case. We note that, al-\nthough widely considered in ASD calculations, the adop-6\nTABLE I. Spin ( mspin) and orbital ( morb) magnetic moments, onsite ( \u000bii) damping, total ( \u000btot) damping, and Curie temper-\nature (TC) of the investigated systems. The theoretical TCvalue is calculated within the RPA. In turn, mtdenotes the total\nmoments for experimental results of Ref. [82].\nmspin(\u0016B)morb(\u0016B)\u000bii(\u000210\u00003) \u000btot(\u000210\u00003) TC(K)\nbcc Fe (theory) 2.23 0.05 2.4 2.1 919\nbcc Fe (expt.) 2.13 [68] 0 :08 [68] \u0000 1:9\u00007:2 [33, 83{89] 1044\nbcc Fe 70Co30(theory) 2.33 0.07 0.5 0.9 1667\nbcc Fe 70Co30(expt.) mt= 2:457 [82] \u0000 0:5\u00001:7a[33, 83, 90] 1258 [92]\nbcc Fe 50Co50(theory) 2.23 0.08 1.5 1.6 1782\nbcc Fe 50Co50(expt.) mt= 2:355 [82] \u0000 2:0\u00003:2b[25, 33, 83] 1242 [93]\nfcc Co (theory) 1.62 0 :08 7.4 1.4 1273\nfcc Co (expt.) 1 :68(6) [94] \u0000 \u0000 2:8(5) [33, 89] 1392\nfcc Ni (theory) 0 :61 0 :05 160.1 21.6 368\nfcc Ni (expt.) 0 :57 [68] 0 :05 [68] \u0000 23:6\u000064 [22, 83, 87{89, 95, 96] 631\naThe lower limit refers to polycrystalline Fe 75Co2510 nm-thick \flms from Ref. [33]. Lee et al. [90] also found a low Gilbert damping in\nan analogous system, where \u000btot<1:4\u000210\u00003. For the exact 30% of Co concentration, however, previous results [33, 84, 91] indicate\nthat we should expect a slightly higher damping than in Fe 75Co25.\nbThe upper limit refers to the approximate minimum intrinsic value for a 10 nm-thick \flm of Fe 50Co50jPt (easy magnetization axis).\ntion of a constant \u000btotvalue (case (ii)) is only a good ap-\nproximation for long wavelength magnons close to q= 0.\nFirst, we are interested on the role of non-local damp-\ning in the remagnetization processes as it was already\ndiscussed by Thonig et al. [20] and as it is important\nfor,e.g., ultrafast pump-probe experiments as well as all-\noptical switching. In the simulations presented here, the\nrelaxation starts from a totally random magnetic con-\n\fguration. The results of re-magnetization simulations\nare shown in Figure 2. The fully non-local damping (i)\nin the equation of motion enhances the energy dissipa-\ntion process compared to the case when only the e\u000bective\ndamping (ii)is used. This e\u000bect is found to be more pro-\nnounced in fcc Co and fcc Ni compared to bcc Fe and bcc\nFe50Co50. Thus, the remagnetization time to 90% of the\nsaturation magnetisation becomes \u00185\u00008 times faster\nfor case (i)compared to the case (ii). This is due to\nthe increase of \u000bqaway from the \u0000 point in the whole\nspectrum for Co and Ni (see Fig. 1), where in Fe and\nFe50Co50it typically oscillates around \u000btot.\nFor bcc Fe 70Co30, the e\u000bect of non-local damping on\nthe dynamics is opposite to the data in Fig. 2; the re-\nlaxation process is decelerated. In this case, almost the\nentire\u000bqspectrum is below \u000bq=0, which is an interest-\ning result given the fact that FMR measurements of the\ndamping parameter in this system is already considered\nan ultra-low value, when compared to other metallic fer-\nromagnets [33]. Thus, in the remagnetization process of\nFe70Co30, the majority of magnon modes lifetimes is un-\nderestimated when a constant \u000btotis considered in the\nspin dynamics simulations, which leads to a faster overall\nrelaxation rate.\nAlthough bcc Fe presents the highest Gilbert damp-ing obtained in the series of the Fe-Co alloys (see Table\nI) the remagnetization rate is found to be faster in bcc\nFe50Co50. This can be explained by the fact that the ex-\nchange interactions for this particular alloy are stronger\n(\u001880% higher for nearest-neighbors) than in pure bcc\nFe, leading to an enhanced Curie temperature (see Table\nI). In view of Eq. 13 and Fig. 1, the di\u000berence in the\nremagnetization time between bcc Fe 50Co50and elemen-\ntal bcc Fe arises from \u000bqvalues that are rather close,\nbut where the magnon spectrum of Fe 50Co50has much\nhigher frequencies, with corresponding faster dynamics\nand hence shorter remagnetization times.\nFrom our calculations we \fnd that the sum of non-local\ndamping\u0010P\ni6=j\u000bij\u0011\ncontributes with \u000013%,\u000081%,\n\u000087%, +80%, and +7% to the local damping in bcc Fe,\nfcc Co, fcc Ni, bcc Fe 70Co30, and bcc Fe 50Co50, respec-\ntively. The high positive ratio found in Fe 70Co30indi-\ncates that, in contrast to the other systems analyzed, the\nnon-local contributions act like an anti-damping torque,\ndiminishing the local damping torque. A similar anti-\ndamping e\u000bect in antiferromagnetic (AFM) materials\nhave been reported in theoretical and experimental in-\nvestigations ( e.g., [98, 99]), induced by electrical current.\nHere we \fnd that an anti-damping torque e\u000bect can have\nan intrinsic origin.\nTo provide a deeper understanding of the anti-damping\ne\u000bect caused by a positive non-local contribution, we an-\nalytically solved the equation of motion for a two spin\nmodel system, e.g. a dimer. In the particular case when\nthe onsite damping \u000b11is equal to the non-local con-\ntribution\u000b12, we observed that the system becomes un-\ndamped (see Appendix B). As demonstrated in Appendix\nB, ASD simulations of such a dimer corroborate the re-7\nFIG. 1. Non-local damping ( \u000bij) as a function of the nor-\nmalized real-space pairwise ( i;j) distance computed for each\nneighboring shell, and corresponding Fourier transform \u000bq\n(see Eq. 11) from the onsite value ( \u000bii) up to 136 shells of\nneighbors (136 NN) for: (a,b) bcc Fe; (c,d) bcc Fe 70Co30;\n(e,f) bcc Fe 50Co50in the virtual-crystal approximation; and\nup to 30 shells of neighbors (30 NN) for: (g,h) fcc Co; (i,j) fcc\nNi. The insets in sub\fgures (a,c,e,g,i) show the convergence\nof\u000btotin real-space. The obtained onside damping values are\nshown in Table I. In the insets of the left panel, green full\nlines are guides for the eyes.\nsult of undamped dynamics. It should be further noticed\nthat this proposed model system was used to analyse\nthe stability of the ASD solver, verifying whether it can\npreserve both the spin length and total energy. Full de-\ntail of the analytical solution and ASD simulation of a\nspin-dimer and the anti-damping e\u000bect are provided inAppendix B.\nFIG. 2. Remagnetization process simulated with ASD, con-\nsidering fully non-local Gilbert damping ( \u000bij, blue sold lines),\nand the e\u000bective damping ( \u000btot, red dashed lines), for: (a) fcc\nNi; (b) fcc Co; and (c) bcc Fe 1\u0000xCox(x= 0%;30%;50%).\nThe dashed gray lines indicate the stage of 90% of the satu-\nration magnetization.\nC. Magnon spectra\nIn order to demonstrate the in\ruence of damping on\nmagnon properties at \fnite temperatures, we have per-\nformed ASD simulations to obtain the excitation spectra\nfrom the dynamical structure factor introduced in Sec-\ntion I. Here, we consider 16 NN shells for S(q;!) calcula-\ntions both from simulations that include non-local damp-\ning as well as the e\u000bective total damping (see Appendix\nD for a focused discussion). In Fig. 3, the simulated\nmagnon spectra of the here investigated ferromagnets are\nshown. We note that a general good agreement can be\nobserved between our computed magnon spectra (both\nfrom the the frozen magnon approach as well as from the\ndynamical structure factor) and previous theoretical as\nwell as experimental results [34, 52, 80, 100{103], where\ndeviations from experiments is largest for fcc Ni. This\nexception, however, is well known and has already been\ndiscussed elsewhere [104].\nThe main feature that the non-local damping causes to\nthe magnon spectra in all systems investigated here, is in\nchanges of the full width at half maximum (FWHM) 4q\nofS(q;!). Usually,4qis determined from the super-\nposition of thermal \ructuations and damping processes.\nMore speci\fcally, the non-local damping broadens the\nFWHM compared to simulations based solely on an e\u000bec-\ntive damping, for most of the high-symmetry paths in all\nof the here analyzed ferromagnets, with the exception of\nFe70Co30. The most extreme case is for fcc Ni, as \u000bqex-\nceeds the 0:25 threshold for q=X, which is comparable\nto the damping of ultrathin magnetic \flms on high-SOC\nmetallic hosts [105]. As a comparison, the largest di\u000ber-\nence of FWHM between the non-local damping process\nand e\u000bective damping process in bcc Fe is \u00182 meV, while\nin fcc Ni the largest di\u000berence can reach \u0018258 meV. In\ncontrast, the di\u000berence is \u0018\u00001 meV in Fe 70Co30and the8\nlargest non-local damping e\u000bect occurs around q=N\nand in the H\u0000Pdirection, corroborating with the dis-\ncussion in Section II A. At the \u0000 point, which corresponds\nto the mode measured in FMR experiments, all spins in\nthe system have a coherent precession. This implies that\n@mj\n@tin Eq. 4 is the same for all moments and, thus, both\ndamping scenarios discussed here (e\u000becive local and the\none that also takes into account non-local contributions)\nmake no di\u000berence to the spin dynamics. As a conse-\nquence, only a tiny (negligible) di\u000berence of the FWHM\nis found between e\u000bective and non-local damping for the\nFMR mode at low temperatures.\nThe broadening of the FWHM on the magnon spec-\ntrum is temperature dependent. Thus, the e\u000bect of non-\nlocal damping to the width near \u0000 can be of great in-\nterest for experiments. More speci\fcally, taking bcc Fe\nas an example, the di\u000berence between width in e\u000bective\ndamping and non-local damping process increases with\ntemperature, where the di\u000berence can be enhanced up to\none order of magnitude from T= 0:1 K toT= 25 K.\nNote that this enhancement might be misleading due to\nthe limits of \fnite temperature assumption made here.\nThis temperature dependent damping e\u000bect on FWHM\nsuggests a path for the measurement of non-local damp-\ning in FMR experiments.\nWe have also compared the di\u000berence in the imaginary\npart of the transverse dynamical magnetic susceptibility\ncomputed from non-local and e\u000bective damping. De\fned\nby Eq. 12, the imaginary part of susceptibility is re-\nlated to the FWHM [15]. Similar to the magnon spectra\nshown in Fig. 3, the susceptibility di\u000berence is signi\f-\ncant at the BZ boundaries. Taking the example of fcc\nCo, Im\u001f\u0006(q;!) for e\u000bective damping processes can be\n11:8 times larger than in simulations that include non-\nlocal damping processes, which is consistent to the life-\ntime peak that occurs at high the symmetry point, X,\ndepicted in Fig. 4. In the Fe 1\u0000xCoxalloy, and Fe 70Co30,\nthe largest ratio is 1 :7 and 2:7 respectively. The intensity\nat \u0000 point is zero since \u000bqis independent on the coupling\nvector and equivalent in both damping modes. The ef-\nfect of non-local damping on susceptibility coincides well\nwith the magnon spectra from spin dynamics. Thus, this\nmethod allows us to evaluate the magnon properties in a\nmore e\u000ecient way.\nD. Magnon lifetimes\nBy \ftting the S(q;!) curve at each wave vector with\na Lorentzian curve, the FWHF and hence the magnon\nlifetimes,\u001cq, can be obtained from the simple relation\n[15]\n\u001cq=2\u0019\n4q: (17)\nFigure 4 shows the lifetimes computed in the high-\nsymmetry lines in the BZ for all ferromagnets here in-vestigated. As expected, \u001cqis much lower at the qvec-\ntors far away from the zone center, being of the order\nof 1 ps for the Fe 1\u0000xCoxalloys (x= 0%;30%;50%),\nand from\u00180:01\u00001 ps in fcc Co and Ni. In view of\nEq. 13, the magnon lifetime is inversely proportional to\nboth damping and magnon frequency. In the e\u000bective\ndamping process, \u000bqis a constant and independent of\nq; thus, the lifetime in the entire BZ is dictated only by\n!q. The situation becomes more complex in the non-\nlocal damping process, where the \u001cqis in\ruenced by the\ncombined e\u000bect of changing damping and magnon fre-\nquency. Taking Fe 70Co30as an example, even though\nthe\u000bqis higher around the \u0000, the low magnon frequency\ncompensates the damping e\u000bect, leading to an asymp-\ntotically divergent magnon lifetime as !q!0. However,\nthis divergence becomes \fnite when including e.g. mag-\nnetocrystalline anisotropy or an external magnetic \feld\nto the spin-Hamiltonian. In the H\u0000Npath, the magnon\nenergy of Fe 70Co30is large, but \u000bqreaches\u00184\u000210\u00004\natq=\u00001\n4;1\n4;1\n2\u0001\n, resulting in a magnon lifetime peak of\n\u001810 ps. This value is not found for the e\u000bective damping\nmodel.\nIn the elemental ferromagnets, as well as for Fe 50Co50,\nit is found that non-local damping decreases the magnon\nlifetimes. This non-local damping e\u000bect is signi\fcant in\nboth Co and Ni, where the magnon lifetimes from the \u000bij\nmodel di\u000ber by an order of magnitude from the e\u000bective\nmodel (see Fig. 4). In fact, considering \u001cqobtained from\nEq. 13, the e\u000bective model predicts a lifetime already\nhigher by more than 50% when the magnon frequencies\nare\u001833 meV and\u001814 meV in the K\u0000\u0000 path ( i.e.,\nnear \u0000) of Ni and Co, respectively. This di\u000berence mainly\narises, in real-space, from the strong negative contriu-\ntions of\u000bijin the close neighborhood around the refer-\nence site, namely the NN in Ni and third neighbors in Co.\nIn contrast, due to the \u000bqspectrum composed of almost\nall dampings lower than \u000btot, already discussed in Section\nII A, the opposite trend on \u001cqis observed for Fe 70Co30:\nthe positive overall non-local contribution guide an anti-\ndamping e\u000bect, and the lifetimes are enhanced in the\nnon-local model.\nAnother way to evaluate the magnon lifetimes is from\nthe linear response theory. As introduced in Section I B,\nwe have access to magnon lifetimes at low temperatures\nfrom the imaginary part of the susceptibility. The \u001cq\ncalculated from Eq. 13 is also displayed in Fig. 4. Here\nthe spin-wave frequency !qis from the frozen magnon\nmethod. The magnon lifetimes from linear response have\na very good agreement with the results from the dynam-\nical structure factor, showing the equivalence between\nboth methods. Part of the small discrepancies are re-\nlated to magnon-magnon scattering induced by the tem-\nperature e\u000bect in the dynamical structure factor method.\nWe also \fnd a good agreement on the magnon lifetimes\nof e\u000bective damping in pure Fe with previous studies\n[106]. They are in the similar order and decrease with\nthe increasing magnon energy. However, their results\nare more di\u000bused since the simulations are performed at9\nFIG. 3. Magnon spectra calculated with non-local Gilbert damping and e\u000bective Gilbert damping in: (a) bcc Fe; (b) bcc\nFe70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni. The black lines denote the adiabatic magnon spectra calculated from\nEq. 7. Full red and open blue points denote the peak positions of S(q;!) at each qvector for\u000btotand\u000bijcalculations,\nrespectively, at T= 0:1 K. The width of transparent red and blue areas corresponds to the full width half maximum (FWHM)\non the energy axis \ftted from a Lorentzian curve, following the same color scheme. To highlight the di\u000berence of FWHM\nbetween the two damping modes, the FWHMs shown in the magnon spectrum of Fe 1\u0000xCox, Co, and Ni are multiplied by 20,\n5, and1\n2times, in this order. The triangles represent experimental results: in (a), Fe at 10 K [102] (yellow up) and Fe with\n12% Si at room-temperature [101] (green down); in (d), Co(9 ML)/Cu(100) at room-temperature [103] (green down); in (e) Ni\nat room-temperature (green down) [100]. The standard deviation of the peaks are represented as error bars.\nroom-temperature.\nIII. CONCLUSION\nWe have presented the in\ruence of non-local damping\non spin dynamics and magnon properties of elemental fer-\nromagnets (bcc Fe, fcc Co, fcc Ni) and the bcc Fe 70Co30\nand bcc Fe 50Co50alloys in the virtual-crystal approxima-\ntion. It is found that the non-local damping has impor-\ntant e\u000bects on relaxation processes and magnon prop-\nerties. Regarding the relaxation process, the non-local\ndamping in Fe, Co, and Ni has a negative contribution\nto the local (onsite) part, which accelerates the remagne-\ntization. Contrarily, in\ruenced by the positive contribu-\ntion of\u000bij(i6=j), the magnon lifetimes of Fe 70Co30and\nFe50Co50are increased in the non-local model, typically\nat the boundaries of the BZ, decelerating the remagneti-\nzation.\nConcerning the magnon properties, the non-local\ndamping has a signi\fcant e\u000bect in Co and Ni. More\nspeci\fcally, the magnon lifetimes can be overestimated\nby an order of magnitude in the e\u000bective model for these\ntwo materials. In real-space, this di\u000berence arises as a\nresult of strong negative non-local contributions in theclose neighborhood around the reference atom, namely\nthe NN in Ni and the third neighbors in Co.\nAlthough the e\u000bect of non-local damping to the\nstochastic thermal \feld in spin dynamics is not included\nin this work, we still obtain coherent magnon lifetimes\ncomparing to the analytical solution from linear response\ntheory. Notably, it is predicted that the magnon lifetimes\nat certain wave vectors are higher for the non-local damp-\ning model in some materials. An example is Fe 70Co30, in\nwhich the lifetime can be \u00183 times higher in the H\u0000N\npath for the non-local model. On the other hand, we\nhave proposed a fast method based on linear response\nto evaluate these lifetimes, which can be used to high-\nthroughput computations of magnonic materials.\nFinally, our study provides a link on how non-local\ndamping can be measured in FMR and neutron scat-\ntering experiments. Even further, it gives insight into\noptimising excitation of magnon modes with possible\nlong lifetimes. This optimisation is important for any\nspintronics applications. As a natural consequence of\nany real-space ab-initio formalism, our methodology and\n\fndings also open routes for the investigation of other\nmaterials with preferably longer lifetimes caused by non-\nlocal energy dissipation at low excitation modes. Such\nmaterials research could also include tuning the local10\nFIG. 4. Magnon lifetimes \u001cqof: (a) bcc Fe; (b) bcc Fe 70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni as function of q,\nshown in logarithmic scale. The color scheme is the same of Fig. 3, where blue and red represents \u001cqcomputed in the e\u000bective\nand non-local damping models. The transparent lines and opaque points depict the lifetimes calculated with Eq. 13 and by\nthe FWHM of S(q;!) atT= 0:1 K (see Eq. 17). The lifetime asymptotically diverges around the \u0000-point due to the absence\nof anisotropy e\u000bects or external magnetic \feld in the spin-Hamiltonian.\nchemical environments by doping or defects.\nIV. ACKNOWLEDGMENTS\nFinancial support from Vetenskapsr\u0017 adet (grant num-\nbers VR 2016-05980 and VR 2019-05304), and the\nKnut and Alice Wallenberg foundation (grant number\n2018.0060) is acknowledged. Support from the Swedish\nResearch Council (VR), the Foundation for Strategic Re-search (SSF), the Swedish Energy Agency (Energimyn-\ndigheten), the European Research Council (854843-\nFASTCORR), eSSENCE and STandUP is acknowledged\nby O.E. . Support from the Swedish Research Coun-\ncil (VR) is acknowledged by D.T. and A.D. . The\nChina Scholarship Council (CSC) is acknowledged by\nZ.L.. The computations/data handling were enabled by\nresources provided by the Swedish National Infrastruc-\nture for Computing (SNIC) at the National Supercom-\nputing Centre (NSC, Tetralith cluster), partially funded\nby the Swedish Research Council through grant agree-\nment No. 2016-07213.\n[1] A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye,\nM. Krawczyk, J. Gr afe, C. Adelmann, S. Cotofana,\nA. Naeemi, V. I. Vasyuchka, et al. , J. Phys. Condens.\nMatter 33, 413001 (2021).\n[2] P. Pirro, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Nat. Rev. Mater 6, 1114 (2021).\n[3] B. Rana and Y. 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Sakuma, J. Appl. Phys. 117, 013912 (2015).13\nAppendix A: Numerical solver\nIn this Appendix, the numerical method to solve Eq.\n1 is described. In previous studies, several numerical\napproaches have been proposed to solve the local LLG\nequations, including HeunP method, implicit midpoint\nmethod, Depondt-Merten's method [107], semi-implicit\nA (SIA) and semi-implicit B (SIB) methods [42]. To solve\nthis non-local LLG equation, we use the \fxed-point iter-\nation midpoint method. We have done convergence tests\non this method and \fnd that it preserve the energy and\nspin length of the system, which is demonstrated in Fig.\n5 for the case of a dimer. With stable outputs, the solver\nallows for a relatively large time step size, typically of\nthe order of \u0001 t\u00180:1\u00001 fs.\nFollowing the philosophy of an implicit midpoint\nmethod, the implemented algorithm can be described as\nfollows. Let mt\nibe the magnetic moment of site iat a\ngiven time step t. Then we can de\fne the quantity mmidand the time derivative of mi, respectively, as\nmmid=mt+1\ni+mt\ni\n2;\n@mi\n@t=mt+1\ni\u0000mt\ni\n\u0001t:(A1)\nUsing this de\fnition in Eq. 4, the equation of motion\nof thei-th spin becomes:\n@mi\n@t=mmid\u00020\n@\u0000\r[Bi(mmid) +bi(t)] +X\nj\u000bij\nmj@mj\n@t1\nA:\n(A2)\nThus, with a \fxed-point scheme, we can do the follow-\ning iteration\nmt+1(k+1)\ni =mt\ni+ \u0001t0\n@ \nmt+1(k)\ni +mt\ni\n2!\n\u00020\n@\u0000\r\"\nBi \nmt+1(k)\ni +mt\ni\n2!\n+bi(t)#\n+X\nj\u000bij\nmjmt+1(k)\nj\u0000mt\nj\n\u0001t1\nA1\nA:\n(A3)\nIfmt+1(k+1)\ni\u0019mt+1(k)\ni , the self-consistency con-\nverges. Typically, about 6 iteration steps are needed.\nThis solver was implemented in the software package Up-\npASD [48] for this work.\nAppendix B: Analytical model of anti-damping in\ndimers\nIn the dimer model, there are two spins on site 1 and\nsite 2 denoted by m1andm2, which are here supposed\nto be related to the same element { so that, naturally,\n\u000b11=\u000b22>0. Also, let's consider a su\u000eciently low\ntemperature so that bi(t)!0, which is a reasonable\nassumption, given that damping has an intrinsic origin\n[108]. This simple system allows us to provide explicit\nexpressions for the Hamiltonian, the e\u000bective magnetic\n\felds and the damping term. From the analytical solu-\ntion, it is found that the dimer spin system becomes an\nundamped system when local damping is equal to non-\nlocal damping, i.e.the e\u000bective damping of the system\nis zero.\nFollowing the de\fnition given by Eq. 4 in the main\ntext, the equation of motion for spin 1 reads:\n@m1\n@t=m1\u0002\u0012\n\u0000\rB1+\u000b11\nm1@m1\n@t+\u000b12\nm2@m2\n@t\u0013\n;(B1)\nand an analogous expression can be written for spin 2.\nFor sake of simplicity, the Zeeman term is zero and thee\u000bective \feld only includes the contribution from Heisen-\nberg exchange interactions. Thus, we have B1= 2J12m2\nandB2= 2J21m1. Withj\u000bijj\u001c 1, we can take the\nLL form@mi\n@t=\u0000\rmi\u0002Bito approximate the time-\nderivative on the right-hand side of the LLG equation.\nLetm1=m2and\u000b12=\u0015\u000b11. SinceJ12=J21and\nm1\u0002m2=\u0000m2\u0002m1, then we have\n@m1\n@t=\u00002\rJ12m1\u0002\u0014\nm2+ (1\u0000\u0015)\u000b11\nm1(m1\u0002m2)\u0015\n:\n(B2)\nTherefore, when \u000b12=\u000b21=\u000b11(i.e.,\u0015= 1), Eq. B1\nis reduced to:\n@m1\n@t=\u00002\rJ12m1\u0002m2; (B3)\nand the system becomes undamped. It is however\nstraightforward that, for the opposite case of a strong\nnegative non-local damping ( \u0015=\u00001), Eq. B2 describes\na common damped dynamics. A side (and related) con-\nsequence of Eq. B2, but important for the discussion in\nSection II B, is the fact that the e\u000bective onsite damp-\ning term\u000b\u0003\n11= (1\u0000\u0015)\u000b11becomes less relevant to the\ndynamics as the positive non-local damping increases\n(\u0015!1), or, in other words, as \u000btot= (\u000b11+\u000b12) strictly\nincreases due to the non-local contribution. Exactly the\nsame reasoning can be made for a trimer, for instance,\ncomposed by atoms with equal moments and exchange\ninteractions ( m1=m2=m3,J12=J13=J23), and\nsame non-local dampings ( \u000b13=\u000b12=\u0015\u000b11).14\nThe undamped behavior can be directly observed from\nASD simulations of a dimer with \u000b12=\u000b11, as shown in\nFig. 5. Here the magnetic moment and the exchange are\ntaken the same of an Fe dimer, m1= 2:23\u0016BandJ12=\n1:34 mRy. Nevertheless, obviously the overall behavior\ndepicted in Fig. 5 is not dependent on the choice of\nm1andJ12. Thezcomponent is constant, while the x\nandycomponents of m1oscillate in time, indicating a\nprecessing movement.\nIn a broader picture, this simple dimer case exempli\fes\nthe connection between the eigenvalues of the damping\nmatrix\u000b= (\u000bij) and the damping behavior. The occur-\nrence of such undamped dynamics has been recently dis-\ncussed in Ref. [109], where it is shown that a dissipation-\nfree mode can occur in a system composed of two sub-\nsystems coupled to the same bath.\n0.00 0.02 0.04 0.06 0.08 0.10\nt(ps)0.2\n0.00.20.40.60.81.0Magnetization\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0\nEnergy(mRy)mxmymzmEnergy\nFIG. 5. Spin dynamics at T= 0 K of an undamped dimer\nin which\u000b12=\u000b21=\u000b11(see text). The vector m1is\nnormalized and its Cartesian components are labeled in the\n\fgure asmx,myandmz. The black and grey lines indicate\nthe length of spin and energy (in mRy), respectively.\nAppendix C: E\u000bective and onsite damping in the\nFeCo and CoNi alloys\nAs mentioned in Section I, the simple VCA model al-\nlows us to account for the disorder in 3 d-transition-metal\nalloys in a crude but e\u000ecient way which avoids the use\nof large supercells with random chemical distributions.\nWith exactly the same purpose, the coherent potential\napproximation (CPA) [110] has also been employed to\nanalyze damping in alloys ( e.g., in Refs. [84, 111, 112]),\nshowing a very good output with respect to trends, when\ncompared to experiments [33, 81]. In Fig. 6 we show\nthe normalized calculated local (onsite, \u000bii) and e\u000bec-\ntive damping ( \u000btot) parameters for the zero-temperature\nVCA Fe 1\u0000xCoxalloy in the bcc structure, consistent with\na concentration up to x\u001960% of Co [33]. The computed\nvalues in this work (blue, representing \u000bii, and red points,\nrepresenting \u000btot) are compared to previous theoretical\nCPA results and room-temperature experimental data.\nThe trends with VCA are reproduced in a good agree-ment with respect to experiments and CPA calculations,\nshowing a minimal \u000btotwhen the Co concentration is\nx\u001930%. This behavior is well correlated with the local\ndensity of states (LDOS) at the Fermi level, as expected\nby the simpli\fed Kambersk\u0013 y equation [113], and the on-\nsite contribution. Despite the good agreement found, the\nvalues we have determined are subjected to a known error\nof the VCA with respect to the experimental results.\nThis discrepancy can be partially explained by three\nreasons: ( i) the signi\fcant in\ruence of local environ-\nments (local disorder and/or short-range order) to \u000btot\n[25, 77]; ( ii) the fact that the actual electronic lifetime\n(i.e., the mean time between two consecutive scattering\nevents) is subestimated by the VCA average for random-\nness in the FeCo alloy, which can have a non-negligible\nimpact in the damping parameter [22, 114]; and ( iii) the\nin\ruence on damping of noncollinear spin con\fgurations\nin \fnite temperature measurements [54, 115]. On top of\nthat, it is also notorious that damping is dependent on\nthe imaginary part of the energy (broadening) [22, 114],\n\u000e, which can be seen as an empirical quantity, and ac-\ncounts for part of the di\u000berences between theory and ex-\nperiments.\n 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035\n 0  10  20  30  40  50  60 0 5 10 15 20 25Damping value\nDOS at EF (states/Ry−atom)\nCo concentration (%)onsite  (αii)\ntotal  (αtot)\nTurek et al.  (αtot)\nMankovsky  et al.  [2013]  (αtot)\nMankovsky  et al.  [2018]  (αtot)\nSchoen et al.\nn(EF)\nFIG. 6. (Color online) Left scale : Computed Gilbert e\u000bec-\ntive (\u000btot, red circles) and onsite ( \u000bii, blue squares) damping\nparameters as a function of Co the concentration ( x) for bcc\nFe1\u0000xCoxbinary alloy in the virtual-crystal approximation.\nThe values are compared with previous theoretical results us-\ning CPA, from Ref. [84] (gray full triangles), Ref. [54] (black\nopen rhombus), Ref. [112] (yellow open triangles), and room-\ntemperature experimental data [33]. Right scale : The calcu-\nlated density of states (DOS) at the Fermi level as a function\nofx, represented by the black dashed line.\nIn the spirit of demonstrating the e\u000bectiveness of the\nsimple VCA to qualitatively (and also, to some extent,\nquantitatively) describe the properties of Gilbert damp-\ning in suitable magnetic alloys, we also show in Fig. 7 the\nresults obtained for Co xNi1\u0000xsystems. The CoNi alloys15\nare known to form in the fcc structure for a Ni concen-\ntration range of 10% \u0000100%. Therefore, here we mod-\neled CoxNi1\u0000xby a big fcc cluster containing \u0018530000\natoms in real-space with the equilibrium lattice parame-\nter ofa= 3:46\u0017A. The number of recursion levels consid-\nered isLL= 41. A good agreement with experimental\nresults and previous theoretical calculations can be no-\nticed. In particular, the qualitative comparison with the-\nory from Refs. [81, 84] indicates the equivalence between\nthe torque correlation and the spin correlation models\nfor calculating the damping parameter, which was also\ninvestigated by Sakuma [116]. The onsite contribution\nfor each Co concentration, \u000bii, is omitted from Fig. 7\ndue to an absolute value 2 \u00004 times higher than \u000btot,\nbut follows the same decreasing trend. Again, the over-\nall e\u000bective damping values are well correlated with the\nLDOS, and re\rect the variation of the quantity1\nmtwith\nCo concentration (see Eq. 15).\n 0 0.005 0.01 0.015 0.02 0.025\n 0  10  20  30  40  50  60  70 10 15 20 25 30Damping value\nDOS at EF (states/Ry−atom)\nCo concentration (%)total  (αtot)\nMankovsky  et al.  [2013]  (αtot)\nStarikov et al.  (αtot)\nSchoen [2017] et al.\nn(EF)\nFIG. 7. (Color online) Left scale : Computed Gilbert e\u000bective\n(\u000btot, red circles) damping parameters as a function of the Co\nconcentration ( x) for fcc Co xNi1\u0000xbinary alloy in the virtual-\ncrystal approximation. The values are compared with previ-\nous theoretical results using CPA, from Ref. [84] (gray full\ntriangles), Ref. [81] (gold full circles), and room-temperature\nexperimental data [89]. Right scale : The calculated density of\nstates (DOS) at the Fermi level as a function of x, represented\nby the black dashed line.\nAppendix D: E\u000bect of further neighbors in the\nmagnon lifetimes\nWhen larger cuto\u000b radii ( Rcut) of\u000bijparameters are\nincluded in ASD, Eq. A3 takes longer times to achieve a\nself-consistent convergence. In practical terms, to reach a\nsizeable computational time for the calculation of a given\nsystem,Rcutneeds to be chosen in order to preserve the\nmain features of the magnon properties as if Rcut!1 .\nA good quantity to rely on is the magnon lifetime \u001cq,as it consists of both magnon frequency and q-resolved\ndamping (Eq. 13). In Section II C, we have shown the\nequivalence between Eq. 13 and the inverse of FWHM\non the energy axis of S(q;!) for the ferromagnets inves-\ntigated here. Thus, the comparison of two \u001cqspectra for\ndi\u000berentRcutcan be done directly and in an easier way\nusing Eq. 13.\nFIG. 8. (Color online) Magnon lifetimes calculated using Eq.\n13 for: (a) bcc Fe; and (b) bcc Fe 50Co50, using a reduced set\nof 16 NN shells (opaque lines), and the full set of 136 NN\nshells (transparent lines).\nAn example is shown in Figure 8 for bcc Fe and bcc\nFe50Co50. Here we choose the \frst 16 NN ( Rcut\u00183:32a)\nand compare the results with the full calculated set of\n136 NN (Rcut= 10a). It is noticeable that the reduced\nset of neighbors can capture most of the features of the\n\u001cqspectrum for a full NN set. However, long-range in-\n\ruences of small magnitudes, such as extra oscillations\naround the point q=Hin Fe, can occur. In particu-\nlar, these extra oscillations arise mainly due to the pres-\nence of Kohn anomalies in the magnon spectrum of Fe,\nalready reported in previous works [52, 80]. In turn, for\nthe case of Fe 50Co50, the long-range \u000bijreduces\u000btot, and\ncauses the remagnetization times for non-local and e\u000bec-16\ntive dampings to be very similar (see Fig. 2). For the\nother ferromagnets considered in the present research,comparisons of the reduced Rcutwith analogous quality\nwere reached."    },    {        "title": "2212.12673v1.Anatomy_of_ultrafast_quantitative_magneto_acoustics_in_freestanding_nickel_thin_films.pdf",        "content": "arXiv:2212.12673v1  [cond-mat.mtrl-sci]  24 Dec 2022Anatomy of ultrafast quantitative magneto-acoustics in fr eestanding nickel thin films\nAntonia Ghita1, Tudor-Gabriel Mocioi1, Alexey M. Lomonosov2, Jiwan\nKim3, Oleksandr Kovalenko1, Paolo Vavassori4,5, and Vasily V. Temnov1∗\n1LSI, Ecole Polytechnique, CEA/DRF/IRAMIS, CNRS,\nInstitut Polytechnique de Paris, F-91128, Palaiseau, Fran ce\n2B+W Department, Offenburg University of Applied Sciences, 77 652 Offenburg, Germany\n3Department of Physics, Kunsan National University, 54150 K unsan, Korea\n4CIC nanoGUNE BRTA, E-20018 Donostia-San Sebastian, Spain a nd\n5IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao , Spain\n(Dated: December 27, 2022)\nWe revisit the quantitative analysis of the ultrafast magne to-acoustic experiment in a freestand-\ning nickel thin film by Kim and Bigot [1] by applying our recent ly proposed approach of magnetic\nand acoustic eigenmodes decomposition by Vernik et al. [2]. We show that the application of our\nmodeling to the analysis of time-resolved reflectivity meas urements allows for the determination\nof amplitudes and lifetimes of standing perpendicular acou stic phonon resonances with unprece-\ndented accuracy. The acoustic damping is found to scale as ∝ω2for frequencies up to 80 GHz\nand the peak amplitudes reach 10−3. The experimentally measured magnetization dynamics for\ndifferent orientations of an external magnetic field agrees w ell with numerical solutions of magneto-\nelastically driven magnon harmonic oscillators. Symmetry -based selection rules for magnon-phonon\ninteractions predicted by our modeling approach allow for t he unambiguous discrimination between\nspatially uniform and non-uniform modes, as confirmed by com paring the resonantly enhanced\nmagneto-elastic dynamics simultaneously measured on oppo site sides of the film. Moreover, the\nseparation of time scales for (early) rising and (late) decr easing precession amplitudes provide ac-\ncess to magnetic (Gilbert) and acoustic damping parameters in a single measurement.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nSince early experimental studies [3–5], ultrafast\nmagneto-elastic interactions driven by femtosecond light\npulsesareconvenientlydescribedinthetime-domain: the\ndynamics of magnetization driven by single or multiple\nacoustic pulses with picosecond duration are monitored\nusing the magneto-optical pump-probe technique. This\nintuitive picture allows for an elegant description of mag-\nnetization precession amplified by a sequence of acoustic\npulseswithanappropriatetimeintervalviathe magneto-\nacoustic coherent control mechanism [1, 6]. Moreover,\nthe time-domain picture of ultrafast magneto-acoustics\nfacilitates the interpretation of magnetization switching\n[7, 8], where the duration of acoustic pulses is shorter\nthan the period of ferromagnetic resonance (FMR) pre-\ncession.\nAn alternative view on ultrafast magneto-acoustics\nis provided in magneto-optical transient grating exper-\niments [9–12]. Here, the spectrally separated quasi-\nmonochromatic acoustic excitations allow for observing\nresonant amplification of FMR precession induced by\neach acoustic mode. The dependence of FMR frequency\non the external magnetic field makes it possible to tune\nthe FMR precession in resonance with an acoustic mode\nofinterest. Veryrecentlywehaveextended thisapproach\nto ultrafast magneto-acoustic dynamics in free-standing\nthin films and multilayers [2]. Our theoretical approach\n∗vasily.temnov@cnrs.fris based on eigenmodes decomposition of both acoustic\nand magnetization dynamics, which allows for a more in-\nsightful analysis of ultrafast magneto-acoustic dynamics\nexperimentsin termsofresonantmagneto-elasticinterac-\ntions between individual modes of longitudinal acoustic\nphononsand perpendicularstanding spin wave(magnon)\nmodes [13]. For instance, the application of such rigor-\nous theoretical analysis to resonant phonon-magnon in-\nteractions in freestanding multilayer structures predicts\nthe key role of the symmetry of magnetic and acoustic\nmodes in prescribing well-defined selection rules for in-\ndividual phonon-magnon interactions. One of the most\nrelevant conclusions was that in symmetric structures in-\nteractionsbetween magnon and phonon eigenmodes with\ndifferent symmetries were forbidden.\nIn this paper, with the purpose of bench-marking the\npower of our improved approach, we apply it to reinter-\npreting the experimental results by Kim and Bigot [1]\nobtained for a 300 nm freestanding nickel thin film. We\nshow that even for such thick structures, where frequen-\ncies of spatially uniform (FMR) and non-uniform (spin\nwave or magnon) modes cannot be distinguished using\nconventional approaches employed so far, our approach\nenables the detection of their excitation thanks to the\nsymmetry-dependentselectionrulesthatgoverntheirres-\nonant interaction with acoustic modes. The results of\nthis work are multiple: from the one side they demon-\nstrate the ability of our modeling to retrieve fundamen-\ntal parametersgoverningthe complex physics involved in\nultrafast magneto-acousticexperiments with an unprece-\ndented accuracy, from the other side corroboratethat the\nphysical picture embodied in our model is particularly2\nzs(t)Ultrafast \noptical \npump Ultrashort \noptical probe \n    FMR FMR+magnons \n  Time-resolved \nreflectivity, MOKE \n(rotation/ellipticity) Acoustic \n  pulseTime (~ps) \nLength (~ nm) \n4\n3\n2\n1\n0\n-1−∆ψ  (°) \n800 600 400 200 0\nTime delay (ps)-2 -1 01\n∆0 1 (\n R / R4 -)\n  =  65 ° \n 46.5 ° \n 35 ° \n 26 ° \n 15.5 ° (a) \n(b) \nFIG. 1. (a) Schematic picture of the experiment and acous-\ntic pulse propagating inside the sample. The shaded expo-\nnentially decaying functions illustrate the optical penet ration\ndepth of pump and probe pulses, respectively. (b) Experi-\nmental data for reflectivity and Kerr rotation.\ninsightful, for example by highlighting the importance of\nsymmetries in magneto-acoustics.\nII. EXPERIMENT\nFreestanding nickel membranes in the experiment in\nRef. [1] had a thickness L=300 nm and were obtained by\ndepositing Ni on a glass substrate with a layer of sodium\nchloride in between them. The layer was subsequently\ndissolved in water to leave the Ni film stretched on a\nsample holder with a hole. The film was stretched later-\nally by gluing a silver paste around the edges of the film,\nwhich created a static strain in Nickel upon drying out.\nThe nickel thin film was optically excited at the front\nside by a femtosecond pump pulse (400 nm wavelength,45 fs pulse duration, 10 kHz repetition rate, 1.5 mJ/cm2\nfluence), launching pulses of coherent longitudinal acous-\ntic phonons with a duration of a few picoseconds propa-\ngating inside the sample at a constant speed cs=6 nm/ps\n(see Fig. 1(a)). Due to the inverse magnetostrictive ef-\nfect, these acoustic pulses drove the magnetization dy-\nnamics inside the Ni film. Time-delayed probe pulses of\n800nm detected transient changes in the reflectivity and\nmagneto-opticalKerreffect(MOKE)rotationbothatthe\nfront and back sides of the sample. A rotating perma-\nnent magnet positioned on top of the sample produced a\nmagneticfieldwith reportedmagnitude µ0H∼0.4Tat a\nvariable angle ξwith respect to the surface normal. Due\nto the magnetic anisotropy the equilibrium direction of\nmagnetization was non-collinear with the external mag-\nnetic field and made an angle θwith the surface normal.\nFig. 1(b) shows the measured differential reflectivity\n∆R\nRand Kerr rotation ψat the back side of the film for\nfive different orientations of the external magnetic field:\nξ= 15.5◦, 26◦, 35◦, 46.5◦and 65◦. The slowly varying\nthermal background in reflectivity and Kerr rotation sig-\nnals originated from heat diffusion from the front to the\nback side of the film and will be subtracted throughout\nthemanuscriptinordertofacilitatethequantitativecom-\nparison with simulations of rapidly varying elastic and\nmagneto-elastic transients. Complementary Kerr rota-\ntion and reflectivity measurements have been performed\nat the front side ofthe film: these data will be introduced\nand discussed in Fig. 2(a) for reflectivity and Fig. 4(a)\nfor Kerr rotation.\nIII. PHYSICAL MODEL\nExcitation of acoustic and magnetic transients in fer-\nromagnetic nickel with femtosecond laser pulses can\nbe adequately described by the phenomenological two-\ntemperature model (TTM) [14], which governs the phe-\nnomena of ultrafast demagnetization on a deeply sub-\npicosecond time scale [15, 16] and generation of ultra-\nshort acoustic pulses on a picosecond time scale [17]. In\nthe currentpaper, we aregoing to disregardthe transient\nultrafast demagnetization and focus on the interaction\nbetween fs-laser-generated acoustic pulses and magneti-\nzation dynamics. Within the framework of the TTM,\nthe non-equilibrium hot electrons are initially generated\nthrough the absorption of an optical pump pulse within\nits skin depth. Subsequently, they transport energy into\nthe depth of the sample via electron diffusion and heat\nup the cold lattice via electron-phonon scattering. These\ncomplex spatio-temporal dynamics result in the emission\nof picosecond acoustic pulses caused by the thermal ex-\npansionofrapidlyheatedlattice. Incaseofafreestanding\nnickel film, these acoustic pulses generated at the front\nside of the sample, propagate through the film, are re-\nflected at the back Ni/air interface (with a reflection co-\nefficient equal to -1), and keep bouncing back and forth\nbetween these two interfaces before they decay due to\nvarious phonon scattering mechanisms.3\nThe magnetization dynamics induced by such ultra-\nshort acoustic pulses can be adequately described by\na phenomenological approach using magneto-elastically\ndriven Landau-Lifshitz-Gilbert (LLG) equations [3–5].\nAdapted to the experimental geometry in Fig. 1, the\nphenomenological free energy density F=FZ+Fd+\nFex+Fmetakes into account the Zeeman term FZ=\n−µ0M0m·Hdue tothepresenceoftheexternalmagnetic\nfieldH, the anisotropy energy Fd=/parenleftbig1\n2µ0M2\n0+K/parenrightbig\nm2\nz\nconsisting of the thin-film shape anisotropy and the phe-\nnomenological anisotropy constant Kdue to static built-\nin strains in a stretched nickel membrane, the exchange\nenergyFex=1\n2M2\n0/summationtext3\ni=1D/parenleftBig\n∂m\n∂xi/parenrightBig2\nand the magneto-\nelastic energy Fme(t) =b1m2\nzεzz(z,t) (b1≃107J/m3for\nNickel [18]) due to the interaction with an acoustic pulse\nεzz(z,t). The relation between the angle ξof the mag-\nnetic field and θmagnetization at equilibrium is given\nby:\nsin(θ−ξ) =˜M\n2Hsin2θ, (1)\nwhere˜M=M0+2K\nµ0M0andM0is the saturation magne-\ntization for Ni. The length of the magnetization vector\nstays constant in our model (assuming constant temper-\nature), so the magnetization dynamics can be describedwith the unit magnetization vector mand its precession\ns(z,t):\nm=m0+s(z,t), (2)\nwhich can be represented as a sum of magnetic eigen-\nmodes:\ns(z,t) =∞/summationdisplay\nn=0s(n)(t)cos(knz). (3)\nwherekn=πn/Lis the wavevector of the n-th magnetic\neigenmode and free boundary conditions for magnetiza-\ntion dynamics are assumed. The n= 0 magnetic eigen-\nmode with uniform spatial profile corresponds to ferro-\nmagnetic resonance (FMR), while higher-order n≥1\nmodes describe spatially non-uniform modes of exchange\nmagnons.\nIt has been shown [2, 19] that in the linear approxima-\ntion when the acoustic strains are small, the magneto-\nelasticallydriven dynamicsforeach magnonmode satisfy\nthe equation of a damped driven harmonic oscillator,\nd2s(n)\nz\ndt2+2αωnds(n)\nz\ndt+ω2\nns(n)\nz=fn(t),(4)\nwhereαis the Gilbert damping parameter and magnon\neigenfrequencies ωnobey\nωn=γµ0/radicalbigg/parenleftBig\nHcosξ−/parenleftBig\n˜M−˜Dk2n/parenrightBig\ncosθ/parenrightBig2\n+/parenleftBig\nHsinξ+˜Dk2nsinθ/parenrightBig/parenleftBig\nHsinξ+/parenleftBig\n˜M+˜Dk2n/parenrightBig\nsinθ/parenrightBig\n.(5)\nHere˜D=D/(/planckover2pi1γµ0) is the exchange stiffness ( D=\n430meV˚A2from Ref. [13]) and γdenotes the gyro-\nmagnetic ratio.\nThe external magneto-elastic driving force\nfn(t) =Pn(H)/integraldisplayL\n0εzz(z,t)cos(knz)dz, (6)\nis proportional to the overlap integral between the\nmagnon eigenmode with the acoustic strain pulse\nεzz(z,t). For our experimental geometry the prefactor\nPn(H) =µ0γ2b1sin(2θ)/parenleftBig\n˜Dk2\nnsinθ+Hsinξ/parenrightBig\nM0L(7)\nis proportional to the magnetostriction coefficient b1and\ndepends both on the magnitude and orientation of an\nexternal magnetic field H.\nUnderstanding the magneto-elasticdynamics governed\nby Eq. (4) is facilitated by decomposing the acoustic\nstrain pulse in its eigenmodes according to\nεzz(z,t) =∞/summationdisplay\np=1ε(p)\nzz(z)e−γptcos(ωpt+ϕp).(8)We assume acoustic eigenmodes to oscillate at frequen-\nciesωp=cskpand decay with damping constants γp;\nϕpdenote their initial phases. In a freestanding film,\nthe acoustic eigenmodes obey the free boundary condi-\ntionsfortheacousticdisplacement(correspondingtozero\nstrains at both Ni/air interfaces) resulting in\nε(p)\nzz(z) =apsin(kpz), (9)\nwherekp=πp/Lis the wavevector of the p-th acous-\ntic eigenmode. Using the decomposition of the acoustic\nstrain in its respective eigenmodes, the expression of the\nmagneto-elastic driving force becomes\nfn(t) =Pn(H)∞/summationdisplay\np=1Inpape−γptcos(ωpt+ϕp).(10)\nHere we have introduced the overlap integral\nInp=/integraldisplayL\n0cos(knz)sin(kpz)dz (11)\nbetweenthe n-thmagneticand p-thacousticeigenmodes.\nTo sum up this section, after having quantified the\nacoustic strain and decomposed it in its eigenmodes, we4\nTime (ns) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Reflectivity (a.u.) x10-4 \n-2 02\nExperiment \nFit \n100\n200\n300\nStrain x10 -3 \n-2 02Time (ns) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Reflectivity (a.u.) x10-4 \n-2 02\nExperiment \nFit z (nm) (a)\n(b) \n(c) \nFIG. 2. (a) Experimentally measured reflectivity at the fron t side of the Ni film, superimposed with its fit as a superpositi on\nof 10 decaying sinusoidal eigenmodes. (b) Color map showing the reconstituted strain inside the film as a function of posi tion\nand time. (c) Experimental reflectivity measurement at the b ack side of the Ni film, together with its fit.\ncan use equations (4 - 5) and (10 - 11) to simulate the\ntime evolution of the magnetization precession. To solve\nEq. (4) numerically, we use the 4th-order Runge-Kutta\nmethod.\nIV. ANALYSIS OF ACOUSTIC DATA\nThe change in reflectivity measured by the probe pulse\nafter excitation with the pump pulse is related to the\nstrain inside the film through the sensitivity function\nf(z), which dependsonthe opticalconstantsofNi. Their\nrelation (as in [20]) is given by\n∆R(t)\nR= 2Re/parenleftBig∆r(t)\nr/parenrightBig\n=/integraldisplayL\n0εzz(z,t)f(z)dz,(12)\nwhere we used the expression for the sensitivity function\nin the complex notation [17]:\nf(z) =16π\nλRe/parenleftbigg\ni∂˜n\n∂εzz˜n\n˜n2−1ei4π˜n\nλz/parenrightbigg\n.(13)\nHere ˜n= 2.48 + 4.38idenotes the complex index of\nrefraction of Ni at the probe wavelength λ= 800nm\nand its derivative with respect to the applied strain\n∂˜n\n∂εzz= 0.6−1.8i[17] is called the photo-elastic coeffi-\ncient. The spatial dependence of the sensitivity function\nis dominated by the exponential decay ∝exp(−z/δskin)within the optical penetration depth of the probe pulse\nδskin=λ\n4πIm(˜n)= 14.5nm.\nUsing the previous decomposition of strain into eigen-\nmodes, we obtain the following expression for the mea-\nsured reflectivity\n∆R(t)\nR=∞/summationdisplay\np=1apJpe−γptcos(ωpt+ϕp),(14)\nwhere we can interpret Jp=/integraltextL\n0f(z)sin(kpz)dzas the\ndetection integral of the p-th mode. This expression for\nthe transient reflectivity shows that it can be represented\nas a sum of damped harmonic oscillations, suggesting\nthat the Fourier transform of the signal could be useful\nincharacterizingtheacousticeigenmodes. Panels(a)and\n(c) in Fig. 2 show the reflectivity signal at the front and\nback side of the film, while Fig. 3(a) displays the Fourier\ntransformof the backside reflectivity signal. The data al-\nlows us to distinguish ten peaks in the Fourier transform,\nso we carry out the analysis using the first ten acoustic\nmodes.\nTo obtain the amplitudes, lifetimes, and phases of the\nrespective acoustic modes, we performed a non-linear\nleast squares fitting (using the Levenberg-Marquardt al-\ngorithm) of the reflectivity data at the back side with\nEq. (14), where ap,γp, andϕpare taken as fit param-\neters and frequencies ωpare extracted from the Fourier\nspectrum. As an initial guess for the amplitudes ap, we5\n020 40 |fft(Reflectivity)| (a.u.) Experiment \nFi t\n020 40 \nData\nQuadratic fit \nPolynomial fit \n0 20 40 60 80 100\nFrequency (GHz) 00.5110 -3 \nExperiment \nh = 20 nm \nh = 60 nm \nTTM (a)\n(b) \n(c)\nFIG. 3. (a) Fourier Transform of the experimental reflectiv-\nity and its fit, for the backside of the film. (b) Damping as a\nfunction of frequency, as extracted from the fitting procedu re.\nThe result follows a quadratic law to a very good approxima-\ntion. (c) Amplitudes of the acoustic modes, as extracted fro m\nthe fitting procedure. The dashed line represents the acous-\ntic frequency spectrum according to Eq. (5) in Ref. [17] for\nNickel thin films excited by weak fs-pump pulses. , ampli-\ntudes obtained with Eq. (15) assuming exponential heating\nprofiles with h=20 nm and 60 nm are shown for comparison\n(continuous lines).\n.\nused the integrated intensity of each peak, and for damp-\ning constants γp- the width of spectral lines from the\nFourier spectrum. It is important to choose a rather pre-\ncise initial guess for amplitudes and damping constants\nas the algorithm is highly sensitive to them. The results\nof such fitting in Fig. 2(a,c) and Fig. 3(a) appear to be\nin an excellent agreement with experimental data.\nFurthermore, panels (b) and (c) in Fig. 3 show the de-\npendence of damping and amplitudes for the first eight\neigenmodesontheirfrequencies. Thesecond-degreepoly-\nnomial fit of damping γpas a function of frequency shows\nthat the quadratic term dominates. Thus, we can con-\nclude that damping scales quadratically with frequency\nup to around 80GHz. This result is consistent withRef. [21], suggesting that the attenuation mechanism is\ndue to the phonon-phonon scattering.\nThe straightforward attempt to understand the fitted\namplitudes (Fig. 3(c)) within the framework of the TTM\nfailed. Using Eq. (5) and the set of experimental fit pa-\nrameters in the low-fluence excitation regime (pump flu-\nence∼0.01mJ/cm2)inNickelthinfilms[17]resultsinthe\ninitial heat penetrationdepth h=20nm that onlyslightly\nexceeds the optical skin depth of our pump pulses. In\nterms of the acoustic amplitudes the results of the TTM\nare well-approximated by a simplified phenomenological\nmodel assuming an instantaneous heating with an expo-\nnential profile ∝exp(−z/h) giving rise to\nap∝/integraldisplayL\n0e−z/hsin(kpz)dz, (15)\nThe strong disagreement between the theory and the ex-\nperimental data indicates that this modeling cannot be\napplied. In the strong-excitation regime used in this ex-\nperiment the parameters of the two-temperature model\ndisplay strong dependence on the pump fluence [22] re-\nsulting in larger electronic heat capacity and weaker\nelectron-phononcoupling. Botheffectsfavoralargerheat\npenetrationdepthmediatedbyhotelectrondiffusiondur-\ning the increaseelectron-phononrelaxationtime. We can\naccount for this effect by assuming a larger heat penetra-\ntion depth h=60 nm, which provides a better approxi-\nmation to the experimental data. However, it is clear\nthat the discussed theoretical models represent oversim-\nplifications and a further systematic study of the strong\nexcitation regime of picosecond acoustic pulses is neces-\nsary.\nUsing the obtained amplitudes, phases, and damping\nparameters, we can reconstruct the strain inside the film\nas a function of space and time. Figure 2(b) shows an\nevolution of spatial strain that is in accordance with the\nintuitive image of an acoustic echo propagating back and\nforth, undergoing reflections at both ends of the film and\ndamping in time. But, in addition to this intuitive pic-\nture, the eigenmode decomposition also helps explaining\nthe broadening of acoustic echo in time domain, which is\ndue to the frequency-dependent damping.\nV. ANALYSIS OF MAGNETIZATION DYNAMICS\nMagnetization dynamics in the Ni film are analyzed by\nmeasuring the Kerr rotation angle. The depth sensitivity\nfunctionofMOKEbecomesimportantincaseofultrafast\nmagnetization dynamics, varying within the skin depth\noflight dueto the presenceofspatiallynon-uniformhigh-\nfrequency magnons. The relation between the detected\nKerr rotation and the magnetization precession sz(z,t)\ninside the film is given by\n∆ψ(t)\nψs=/integraldisplayL\n0sz(z,t)g(z)dz (16)\nwhereψsis the static Kerr rotation angle, ∆ ψ(t) is its\nchange due to magnetization precession, and g(z) is the6\n0 0.2 0.4 0.6 0.8 \nTime  (ns) (a) Experiment\nFrontside\nBackside \n0 0.2 0.4 0.6 0.8 \nTime (ns) (b) Theory\nFrontside \nBackside m (t) (a.u.) z 46.5 ° \n 35 ° \n 26 ° \n 15.5 °   =  65 ° \nFIG. 4. Comparison of magnetization dynamics at the front\nand back side of the film, as obtained (a) experimentally and\n(b) from our simulations.\ndepth sensitivity function for the polar MOKE [23]:\ng(z) =4π\nλRe/parenleftBig\niQMO˜n2\n1+ ˜n2e−i4π˜n\nλz/parenrightBig\n.(17)\nHere, unlike the acoustical sensitivity function, the\nmagneto-optical response is valued by the complex\nmagneto-optical(Voigt) constant QMO=i˜ǫxy\n˜ǫxx=−(4.9+\n10.5i)×10−3[24].\nUsing our previous decomposition in magnon eigen-\nmodes, we get an expression that ties the dynamics of\nindividual magnon modes to the Kerr rotation:\n∆ψ(t)\nψs=∞/summationdisplay\nn=0˜Jns(n)\nz(t), (18)\nwhere˜Jn=/integraltextL\nz=0g(z)cos(knz)dzisthedetectionintegral\nof then-th magnon mode. Using this expression, we can\nfit the results of our magnetization dynamics simulation\nwith those of the experiment in the next section.\nVI. RESONANT PHONON-MAGNON INTERACTIONS\nThe experimental data for the back- and front side-\nKerr rotation are presented in Fig. 4(a) for different ori-\nentations of the external magnetic field. We simulate\nthe Kerr rotation by solving Eq. (4) for each magnon\nand using the sensitivity function defined in the previ-\nous section to obtain the Kerr rotation from Eq. (18).\nIn order to reach an agreement with experimental datafor all angles ξin Fig. 4, we have used the values for\nthe anisotropy constant K, magnitude of the magnetic\nfieldHand Gilbert damping αas fit parameters. Us-\ning the magnetic field of 0 .3T, the anisotropy constant\nK= 2.05·105J/m3and the Gilbert damping α= 0.04,\nwe achieve the quantitative agreement between the ex-\nperimental data and simulations (Fig. 4). The value of\nthe magnetic field stays within the expected error bar for\na permanent magnet placed on top of the sample. The\nvalue of the Gilbert damping is equal to the one obtained\nin a recent study of ultrafast magnetization dynamics in\nnickel nanomagnets [25]. The simulated Kerr rotation at\nthe front and back sides are represented in Fig. 4(a,b).\nWe observe an excellent agreement between the experi-\nmental data and simulations, except for the initial ther-\nmal excitation of magnetization at the front side, which\nwe did not account for in our model.\nGiven this agreement between experimental data and\nsimulations, we analyze the peculiarities of the magne-\ntization dynamics at different angles. For 15 .5◦, the\nmagnetization dynamics at the back side is in-phase with\nthat at the front-side, while for 65◦they areπ-shifted.\nMoreover,we notice that the magnetizationprecessionat\n15.5 and 65◦lasts longer and is stronger than at other\nangles. While at 15 .5◦and 65◦there are some slowly\nvarying long-lived dynamics, at angles 26, 35, and 46 .5◦\nwe observe weak, somewhat irregular beating patterns.\nA complementary perspective is presented in Fig. 5\nwhich show the reconstructed magnetization dymnamics\ninside the sample (Fig. 5(b-c)) as a function of position\nand time. After the transient regime dies out, the mag-\nnetization profile for 15 .5◦is approximately uniform in\nspace, suggesting that the dynamics is dominated by the\nFMR (n= 0) mode. On the other hand, the dynamics\nat 65◦follow the spatially antisymmetric profile with re-\nspect to the middle of the film, which suggests that in\nthis configuration the n= 1 magnon dominates. This\nconclusion is inline with the observed π-phase shift be-\ntween the data at the front and the backside at 65◦, see\nFig. 4. At an intermediate angle of 35◦, the magnetiza-\ntion dynamics with a much smaller amplitude are mainly\nvisible at early delays times. This suggests that in this\nintermediate regime no magnon modes are resonantlyex-\ncited.\nAll these observations can be explained by a simple\ntheory for a driven harmonic oscillator. The main result\nis that the oscillation amplitude is resonantly enhanced\nwhen the natural frequency (here, that of magnons ωn)\nequalsthe drivingfrequency(in ourcase, thatofphonons\nωp). Away from resonance, the transient regime is char-\nacterized by a beating pattern because of the difference\nbetween the natural and driving frequencies.\nFigure 5(a) shows the amplitude of magnetization pre-\ncession as a function of its frequency and the angle of\nthe external magnetic field. Two bright spots are visible:\none at the point where the frequency of the first phonon\n(p= 1) matches that of FMR ( n= 0) and another\none where the frequency of the second phonon ( p= 2)\nmatches that of the first magnon ( n= 1).7\nFIG. 5. (a) Fourier transform of simulated Kerr rotation as a function of magnetic field angle. The dashed lines indicate t he\nfrequencies of phonons and the continuous white lines are th e dispersion curves of magnons. Vertical cross-sections in to the\nheat map, corresponding to the three angles in panel b, are sh own. (b) Magnetization dynamics inside the film, as a functio n\nof position and time, for three experimental angles. (c) Mag netization profiles, taken at the times indicated in panel (b ) by\ndashed lines.\nFIG. 6. Representation of magnonic eigenmodes (along the\nhorizontal axis) and acoustic eigenmodes (along the axis),\nwith their corresponding overlap integrals.\nHowever, for our symmetric freestanding membrane\nthe overlap integral Inp(Fig. 6) is zero when the acous-\ntic modes possess a different symmetry from that of the\nmagnon modes. This means that symmetric (antisym-\nmetric) acoustic modes will interact only with symmet-ric (antisymmetric) magnon modes, respectively. There-\nfore, the symmetry-based selection rules become as im-\nportant for resonant phonon-magnon interaction as the\npreviously mentionned frequency matching condition.\nThese considerations enable us to identify the driv-\ning forces of the magnetization dynamics observed at the\nthree angles shown in Fig. 5. At 15 .5◦, the frequency\nof the first phonon ( p= 1) matches that of the first few\nmagnonsbutthesymmetryofthemodesallowsonlyeven\nmagnons to be excited. Since the overlap integral decays\nwith increasing magnon number, the dominant magnetic\nmode at 15◦is the FMR ( n= 0). At 65◦, the second\nphonon (p= 2), whose frequency matches the frequen-\ncies of the first few magnons, interacts only with odd\nmagnons. Thus, the dominant mode in this case is the\nfirst magnon ( n= 1). At 35◦, the magnon frequencies\nare between the frequency of the first ( p= 1) and sec-\nond (p= 2) phonon and exhibit no resonant interac-\ntion. Hence, the magnetization precession is significantly\nweaker.\nFigure7illustratesthequantitativeagreementbetween\ntheory and experiment for the two resonantly driven\nmagnetization dynamics. There are two timescales in-\nvolved in such phonon-magnon interactions: excitation\n(the transient regime) and decay (the driven regime).\nWhen the driving frequency is equal to the eigenfre-\nquency, the response in the transient regime follows\n(1−e−t/τexc)cosωpt, whereωpis the frequency of the\nacoustic driving mode and τexcis a characteristic relax-\nation time of magnetization dynamics, which is related\nto Gilbert damping as τexc= (αωn)−1, whereωnis the8Magnetization (degrees) 0 0.5 1 1.5 2-1 01Magnetization dynamics at 15.5° \nτdecay = 1.2 ns τexc  = 0.45 ns\nTime (ns) \n0 0.2 0.4 0.6 0.8 1\nTime (ns) -1 01Magnetization dynamics at 65° \nτdecay = 0.57 ns τexc  = 0.23 ns(a)\n(b) \nFIG. 7. Long-scan magnetization dynamics at the backside of\nthe film, under the magnetic field angles (a) ξ= 15.5◦and (b)\nξ= 65◦. Experimental data is represented by dots, the simu-\nlated curve is a black continuous line, while the dashed curv e\ndenotes the fit envelope A(t). The separation of timescales\nis clearly visible in both graphs: the initial growth govern ed\nby the Gilbert damping is followed by the decay due to the\nacoustic decay.\nfrequency of the dominant magnon. On the other hand,\nthe decay of magnetization is a driven regime when mag-\nnetization dynamics follow the acoustic driving force as\n∝e−t/τdecaycos(ωpt), whereτdecay= 1/γpis the acous-\ntic mode lifetime. Thus, to a good approximation, the\noverall magnetization dynamics fit inside an envelope of\nthe form\nA(t)∝(1−e−t/τexc)e−t/τdecay. (19)\nWe can assign these characteristic relaxation times to\nthe correspondingacoustic and magnetic eigenmodes, re-\nspectively, as shown in Fig. 7. For the magnetization dy-\nnamics at 15 .5◦, the characteristic time of the excitation\nphase isτ(1)\nexc= 0.45ns and the decay time is τ(1)\ndecay=\n1.2ns. We can see a clear correlation between these two\ntimes and the corresponding lifetimes of the acoustic and\nmagnetic eigenmodes,1\nγ(0)\nG= 0.44ns and1\nγ(1)\nac= 1.19ns.\nSimilarly, for the case of magnetization dynamics at 65◦,\nwe extract the excitation time τ(2)\nexc= 0.23ns and a value\nof the decay time τ(2)\ndecay= 0.57ns. Again, this value is\nconsistent with the lifetimes of the first magnon mode\n1\nγ(1)\nG= 0.22ns and of the second acoustic eigenmode,\n1\nγ(2)\nac= 0.56ns.\nThis model also allows for extracting quality factors of\nmagnons and phonon resonances. The magnon quality\nfactorofQG= 13≃1/(2α)doesnotdependonthemode\nnumber, in agreement with previously reported resultson frequency-independent Gilbert damping α[26, 27].\nAcoustic quality factors Q(1)\nac= 37 andQ(2)\nac= 29 are\nslightly different due to the observed nonlinear depen-\ndence of acoustic damping on frequency, converging to\nthe approximate scaling Qac(ω)∝ω−1for higher-order\nacoustic modes. The observed high values of acoustic\nquality factors Qac>QGenable the separation of time-\nscales in the excitation and decay phases in the magne-\ntization dynamics.\nQualitatively similar magnetization dynamics have\nbeen observed earlier in transient grating experiments\n[9]. However, in the latter case the conspicuous decay of\nmagnetization dynamics was explained by the complex\nspatio-temporal dynamics of the magnitude of the mag-\nnetization vector Mz(x,t) on the temperature T(x,t) in\nthe periodically demagnetized nickel film [10, 12]. In this\ncontext, our experimental geometry provides an advan-\ntage of isolating resonant phonon-magnon interactions\nfrom thermal effects and extracting their properties from\nthe same measurement.\nVII. SUMMARY AND CONCLUSIONS\nIn this manuscript, we reported on the quantitative\nanalysis of experimental data by Kim and Bigot in a\nfree-standing nickel thin film [1] based on the decom-\nposition of magnetic and acoustic dynamics in phonon\nand magnon eigenmodes, respectively. The time-domain\nfitting of transient reflectivity data on both sides of the\nnickel film provides frequencies, lifetimes and phases of\nindividual acoustic eigenmodes. The latter is shown\nto drive the magnetization dynamics, to be in quan-\ntitative agreement with time-resolved MOKE measure-\nments. Notably, the comparison of MOKE signals on\nboth sides of the sample evidence the in-phase FMR dy-\nnamics (n= 0, with minor contributions of symmetric\nmagnon eigenmodes n= 2,4,...) induced by the lowest\norder (p=1) symmetric acoustic mode and the opposite-\nsign magnetization oscillation of antisymmetric magnon\nmodes (n= 1,3,...). Being in a quantitative agreement\nwith a simple theoretical model with tabulated material\nparameters, the experimental data clearly evidence the\nresonantly enhanced excitation of nonuniform magnon\nmodes. Moreover, accurate fitting of the magnetization\ndynamics driven by long-lived p= 1 (9.8 GHz) and p= 2\n(19.1 GHz) acoustic modes delivers the correct value for\nmagnetic Gilbert damping α= 0.04, corresponding to\nthe quality factor Qm= 13 for magnon modes. Be-\ning smaller than the quality factors of acoustic modes,\nthis magnetic quality factor assures optimum conditions\nforresonantphonon-magnonexcitation, the phenomenon\nto be further explored in the ultrahigh THz-frequency\nregime [2, 28].9\nACKNOWLEDGMENTS\nThis article is dedicated to the memory of Jean-Yves\nBigot, in whose labs the reported measurements have\nbeen performed. The support of the Physics Depart-\nment of ´Ecole Polytechnique and Institut Polytechnique\ndePariswithintheframeworkofa Projet de Recherche en\nLaboratoire and by the ANR-21-MRS1-0015-01 ”IRON-\nMAG” is gratefully acknowledged. P.V. acknowledgessupport from the Spanish Ministry of Science and In-\nnovation and the European Union under the Maria\nde Maeztu Units of Excellence Programme (CEX2020-\n001038-M) and the project PID2021-123943NB-I00\n(MICINN/FEDER). J.K. acknowledges support from\nBasic Science Research Program through the National\nResearchFoundationofKorea(NRF) fundedbytheMin-\nistryofEducation(2022R1I1A3072023)andbytheMSIT\n(2021R1A4A1031920).\n[1] J.-W. Kim and J.-Y. Bigot,\nPhys. Rev. B 95, 144422 (2017).\n[2] U. Vernik, A. M. Lomonosov, V. S. Vlasov, L. N. Kotov,\nD. A. Kuzmin, I. V. Bychkov, P. Vavassori, and V. V.\nTemnov, Physical Review B 106, 144420 (2022).\n[3] A. V. Scherbakov, A. S. Salasyuk, A. V. Akimov,\nX. Liu, M. Bombeck, C. Br ¨uggemann, D. R. 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Parpiiev, T. Pez-\neril, D. Makarov, R. Abrudan, R. Meckenstock,\nF. Radu, M. Farle, H. Zabel, and V. V. Temnov,\nPhys. Rev. B 99, 104412 (2019).\n[28] A. Kimel, A. Zvezdin, S. Sharma, S. Shallcross,\nN. De Sousa, A. Garc´ ıa-Mart´ ın, G. Salvan, J. Hamrle,\nO. Stejskal, J. McCord, et al. , Journal of Physics D: Ap-\nplied Physics 55, 463003 (2022)."    },    {        "title": "2301.02042v1.Improved_Gilbert_Varshamov_bounds_for_hopping_cyclic_codes_and_optical_orthogonal_codes.pdf",        "content": "arXiv:2301.02042v1  [cs.IT]  5 Jan 20231\nImproved Gilbert-Varshamov bounds for hopping\ncyclic codes and optical orthogonal codes\nChenyang Zhang, Chong Shangguan, and Gennian Ge\nAbstract\nHopping cyclic codes (HCCs) are (non-linear) cyclic codes w ith the additional property that the ncyclic shifts of\nevery given codeword are all distinct, where nis the code length. Constant weight binary hopping cyclic co des are also\nknown as optical orthogonal codes (OOCs). HCCs and OOCs have various practical applications and have been studied\nextensively over the years.\nThe main concern of this paper is to present improved Gilbert -Varshamov type lower bounds for these codes, when\nthe minimum distance is bounded below by a linear factor of th e code length. For HCCs, we improve the previously best\nknown lower bound of Niu, Xing, and Yuan by a linear factor of t he code length. For OOCs, we improve the previously\nbest known lower bound of Chung, Salehi, and Wei, and Yang and Fuja by a quadratic factor of the code length. As\nby-products, we also provide improved lower bounds for freq uency hopping sequences sets and error-correcting weakly\nmutually uncorrelated codes. Our proofs are based on tools f rom probability theory and graph theory, in particular the\nMcDiarmid’s inequality on the concentration of Lipschitz f unctions and the independence number of locally sparse grap hs.\nKeywords. Gilbert-Varshamov bound; non-linear cyclic codes; hoppin g cyclic codes; optical orthogonal codes;\nfrequency hopping sequences sets; error-correcting weakl y mutually uncorrelated codes\nI. I NTRODUCTION\nGiven integers q,nandd, estimating the maximum size of q-ary codes of length nand minimum distance dis a\nfundamental problem in coding theory. The Gilbert–Varsham ov bound (GV bound for short) is a classic lower bound\non the size of codes. For fixed q,n→ ∞ , anddbounded below by a linear factor of n, improving upon the GV bound\nsubstantially is a well-known difficult task. In this paper w e will present improved GV-type bounds for several classes\nof non-linear cyclic codes.\nTo move forward let us begin with some needed definitions. For a positive integer q, let[q] ={0,1,...,q−1}.\nA vector x∈[q]nis denoted by x= (x1,...,x n). For two vectors x,y∈[q]n, the Hamming distance d(x,y)is the\nnumber of coordinates where they differ, namely, d(x,y) =|{1≤i≤n:xi/\\e}atio\\slash=yi}|. Acode of length nand alphabet\nsizeqis a subset of [q]n, whose elements are called codewords . The minimum distance d(C)of a code C⊆[q]nis\ndefined to be d(C) := min{d(x,y) :x,y∈C,x/\\e}atio\\slash=y}. Aq-ary code with length n, sizeM, and minimum distance d\nis denoted as an (n,M,d)q-code .\nNext, let us briefly recall some known GV-type bounds.\nA. GV-type bounds\na) Generic codes.: For an integer 1≤t≤nand a vector x∈[q]n, letB(x,t) :={y∈[q]n:d(x,y)≤t}\ndenote the Hamming ball of radius tcentered at x. Given a radius t, it can be easily seen that for every x∈[q]n, the\nvolume of the Hamming ball B(x,t)is independent of x, which satisfies that\n|B(x,t)|=t/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(q−1)i=:Volq(n,t).\nThis project is supported by the National Key Research and De velopment Program of China under Grant Nos. 2020YFA0712100 and\n2018YFA0704703, the National Natural Science Foundation o f China under Grant Nos. 11971325, 12231014, and 12101364, t he Natural Science\nFoundation of Shandong Province under Grant No. ZR2021QA00 5, and the Beijing Scholars Program.\nC. Zhang is with the Research Center for Mathematics and Inte rdisciplinary Sciences, Shandong University, Qingdao 266 237, China (e-mail:\nchener@mail.sdu.edu.cn).\nC. Shangguan is with the Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingda o 266237, China, and also\nwith the Frontiers Science Center for Nonlinear Expectatio ns, Ministry of Education, Qingdao 266237, China (e-mail: t heoreming@163.com).\nG. Ge is with the School of Mathematical Sciences, Capital No rmal University, Beijing 100048, China (e-mail: gnge@zju. edu.cn).2\nThe GV bound, proved independently by Gilbert [7] and Varsha mov [22], states that there exist (n,M,d)q-codes\nwith\nM≥qn\nVolq(n,d−1). (1)\nFor binary codes and d/n≤0.499, Jiang and Vardy [11] improved the lower bound in (1) by a line ar factor of n. Based\non their work, Vu and Wu [23] showed that for every q≥2andτ≤d/n≤1−1/q−ǫ, whereτ,ǫare some absolute\nreals, there exist (n,M,d)q-codes with\nM= Ω/parenleftbiggnqn\nVolq(n,d−1)/parenrightbigg\n. (2)\nb) Non-linear cyclic codes.: For every x= (x1,...,x n)∈[q]nand0≤i≤n−1, we call the vector\nπi(x) := (xi+1,xi+2,...,x i+n)1thei-thcyclic shift ofx. LetC(x) :={πi(x) : 0≤i≤n−1}denote the set\nconsisting of all cyclic shifts of x. Note that C(x)is viewed as a multi-set. A code C⊆[q]nis said to be a cyclic code\nif for every x∈C,C(x)⊆C. In the literature, there are a number of works devoting to th e constructions of linear\ncyclic codes with minimum distance ranging from a constant t o a sublinear function of n, say,O(n\nlnn)(see, e.g. [19,\nChapter 8]). However, when minimum distance is bounded belo w by a linear factor of n, understanding whether there\nexist asymptotically good linear cyclic codes is a longstanding open question in codin g theory (see [6], [8]).\nFornon-linear cyclic codes, the question above has been solved only recent ly. Haviv, Langberg, Schwartz, and\nYaakobi [8] proved that for every prime code length, there ex ist binary cyclic codes asymptotically attaining the GV\nbound. Later, Niu, Xing, and Yuan [16] showed that for every qandnthere exist (n,M,d)q-cyclic codes with\nM≥qn(1−n2e−ǫ2(√n−2)\n2)\nVolq(n,d−1)−1. (3)\nc) Constant weight codes.: For a vector x∈ {0,1}n, the weightwt(x)is defined to be the number of its\nnon-zero coordinates, namely, wt(x) =|{1≤i≤n:xi/\\e}atio\\slash= 0}|. A code is said to be a constant weight code , if all\ncodewords of it have the same weight. An (n,M,d)2-code with constant weight wis denoted as an (n,M,d;w)-code,\nwhere we omitted the q= 2 in the subscript. For x∈ {0,1}n,wt(x) =w, and0≤t≤2w, letB(x,t;w) :={y∈\n{0,1}n:d(x,y)≤t,wt(y) =w}denote the constant weight Hamming ball of radius tcentered at x. Giventandw,\nit is known that for every x∈ {0,1}nwithwt(x) =w, the volume of B(x,t;w)is independent of x, which satisfies\nthat\n|B(x,t;w)|=⌊t/2⌋/summationdisplay\ni=0/parenleftbiggw\ni/parenrightbigg/parenleftbiggn−w\ni/parenrightbigg\n=:Vol(n,t;w).\nLevenshtein [14] proved a GV-type bound for constant weight codes, showing that there exist (n,M,d;w)-codes\nwith\nM≥/parenleftbign\nw/parenrightbig\nVol(n,d−1;w). (4)\nSimilarly to the improvement of (2) upon (1), Kim, Liu, and Tr an [12] improved (4) by a linear factor of n, showing\nthat forτ≤d/n≤(1−ǫ)p(1−p), whereτ,ǫ,p are some absolute reals, there exist (n,M,d;pn)-codes with\nM= Ω/parenleftBigg\nn/parenleftbign\npn/parenrightbig\nVol(n,d−1;pn)/parenrightBigg\n. (5)\nd) Constant weight non-linear cyclic codes.: In the literature, there is also a GV-type lower bound for con stant\nweight non-linear cyclic codes. More precisely, Chung, Sal ehi, and Wei [1], [3] and Yang and Fuja [24] implicitly showed\nthat there exist (n,M,d;w)-cyclic codes with\nM≥/parenleftbign\nw/parenrightbig\n−f(n,w,d)\nn·Vol(n,d−1;w), (6)\nwheref(n,w,d)is some function of n,w,d (see [1, Theorem 2] and [24, Theorem 3] for details).\n1Throughout this paper, the addition in the subscripts of xis calculated modulo n.3\nGiven the discussion above, it is natural to ask whether one c an improve (3) and (6) by a linear or polynomial\nfactor of n, similarly to the improvements of (2) and (5) made upon (1) an d (4). This is one of the motivating questions\nof this paper. Note that the method which proves (2) and (5) ca nnot be applied directly to improve (3) and (6), as one\nhas to take into account the property of cyclic codes. Based o n some probabilistic and graph theoretic tools and some\nideas from [11], [12], [16], [23], we will answer the above qu estion in a strong sense in Theorems II.1 and II.2 below.\nThe main concern of this paper is to present improved GV-type bounds for two special classes of non-linear cyclic\ncodes, namely, hopping cyclic codes and its constant weight counterpart, optical orthogonal codes. As by-products,\nwe also provide improved lower bounds for frequency hopping sequences sets and error-correcting weakly mutually\nuncorrelated codes as well.\nNext, we will introduce these codes and our results in the det ail.\nII. M AIN RESULTS\nA. Hopping cyclic codes\nHopping cyclic codes are (non-linear) cyclic codes with the additional property that the ncyclic shifts of every\ngiven codeword are all distinct. Formally speaking, a code C⊆[q]nis called a hopping cyclic code (HCC for short) if\nfor every x∈C,C(x)⊆C, and moreover C(x)consists of ndistinct elements. We will call a hopping cyclic code\nC⊆[q]nan(n,M,d)q-HCC, if it is itself an (n,M,d)q-code. HCCs were originally designed to construct frequenc y\nhopping sequences sets (see [4], [16] and Section III-A belo w), but have found their own interest as an intriguing class\nof codes. The reader is referred to [16] for more background o n HCCs.\nFor any absolute constant ǫ >0andd <(n−2√n)(1−1/q−ǫ), Niu, Xing, and Yuan (see Theorem III.5 in [16])\nshowed that there exist (n,M,d)q-HCCs with Mbounded below by (3).\nOur first main result improves the above GV-type bound for HCC s (and hence for non-linear cyclic codes) by\nasymptotically a linear factor of n.\nTheorem II.1. For positive integers n,q,d and absolute reals τ,ǫ∈(0,1−1/q)satisfying τ≤d/n≤1−1/q−ǫ,\nthere exist (n,M,d)q-HCCs with\nM≥cnqn\nVolq(n,d−1),\nwherecis a positive real depending only on τ,ǫ,q .\nB. Optical orthogonal codes\nConstant weight binary hopping cyclic codes are also known a s optical orthogonal codes (OOCs for short). OOCs\nenable a large number of asynchronous users to transmit info rmation efficiently and reliably. They have been widely\nused in various practical scenarios like code-division mul tiple-access systems and spread spectrum communication. A\nlarge number of existing papers were devoted to the construc tions of OOCs, see for example [1], [2], [9], [13], [17],\n[24].\nWe will denote an (n,M,d)2-HCC with constant weight wby an(n,M,d;w)-OOC, where we omitted the q= 2\nin the subscript. Chung, Salehi, and Wei [1], [3] and Yang and Fuja [24] showed that there exist (n,M,d;w)-OOCs\nwithMbounded below by (6).\nOur second main result improves the above GV-type bound for O OCs (and hence for constant weight non-linear\ncyclic codes) by asymptotically a quadratic factor of n.\nTheorem II.2. For positive integers n,q,d and absolute reals p∈(0,1), ǫ∈(0,1/10), τ∈(0,p(1−p))satisfying\npn∈Z, τ≤d/n≤(1−ǫ)p(1−p), there exist (n,M,d;pn)-OOCs with\nM≥cn/parenleftbign\npn/parenrightbig\nVol(n,d−1;pn),\nwherecis a positive real depending only on p,ǫ,τ .4\nC. Sketch of the proofs and comparison with related works\nSince the work of Jiang and Vardy [11], it is now well-known th at one can lower-bound the size of a code via\nthe lower bound of the independence number of a locally spars e graph defined approximately. Loosely speaking, let\nGbe a graph with vertex set [q]n, where two vertices (vectors) x,y∈[q]nare connected by an edge if and only if\nd(x,y)≤d−1. Then, an independent set IofGcorresponds to an (n,|I|,d)q-code. In fact, there is a one-to-one\ncorrespondence between independent sets of Gand codes in [q]nwith minimum distance at least d. Therefore, given\nthe minimum distance d, finding a large code in [q]nis equivalent to finding a large independent set in G.\nSince for every x∈[q]n, there are exactly Volq(n,d−1)−1vectorsy∈[q]n\\{x}such that d(x,y)≤d−1,\nevery vertex of Gis connected to exactly Volq(n,d−1)−1vertices in [q]n. So, one can construct an independent set\ninGof sizeqn\nV olq(n,d−1), or equivalently, an (n,qn\nVolq(n,d−1),d)q-code, by greedily picking vertices and throwing their\nneighbors. This gives the GV bound (1).\nJiang and Vardy [11] showed that the graph Gdefined above is in fact “locally sparse” (this will be clarifi ed later\nin Section IV-A). Therefore, there is a more economic way tha n the greedy algorithm to find a large independent set in\nG(see Lemma IV .1 below). This leads to their improvement on th e GV bound of generic codes (1) by a linear factor of\nn. Kim, Liu, and Tran [12] used a similar idea in their proof of ( 5), which improves the GV bound of constant weight\ncodes (4) by a linear factor of n.\nThe high level idea in the proofs of Theorems II.1 and II.2 wil l follow the above framework. However, as the codes\nconsidered in this paper need not only to have large minimum d istance but also to be hopping cyclic, the corresponding\ngraphs are not as neat as the graph Gdescribed above.\nThe proofs of Theorems II.1 and II.2 can be summarized as foll ows:\nStep 1. define an appropriate graph GHCC (resp.GOOC ) whose independent sets correspond to HCCs (resp. OOCs);\nStep 2. show that GHCC (resp.GOOC ) has sufficiently many vertices;\nStep 3. show that GHCC (resp.GOOC ) is in fact locally sparse;\nStep 4. use known lower bound on the independence number of lo cally sparse graphs to show that GHCC (resp.GOOC )\nhas sufficiently large independent sets, and hence there exi st sufficiently large HCCs (resp. OOCs).\nNext, we would like to compare our work with some previous pap ers [11], [12], [16], [23]. On one hand, in\n[11], [12], [23], Gand its constant weight counterpart were used to prove impro ved GV-type bounds for generic codes\nand constant weight codes, respectively. GHCC (resp.GOOC ) defined in this paper is quite different from G(resp. its\nconstant weight counterpart). To be more precise, for x∈[q]nlet\nd(x) = min{dH(πi(x),πj(x)) : 0≤i < j≤n−1}= min{dH(x,πi(x)) : 1≤i≤n−1} (7)\ndenote the minimum distance d(C(x)). By definition, C⊆[q]nis an HCC only if for every x∈C,d(x)≥1and\nmoreover, Cis an(n,M,d)q-HCC only if for every x∈C,d(x)≥d. Therefore, compared with Gwhose vertex set\nis[q]n, in the vertex set of GHCC we have to rule out all vectors xwithd(x)< d. In fact, the vertex set of GHCC is\nnot vectors in [q]nbut some well-defined subsets of vectors in [q]n. Hence, to execute Steps 1 and 2, we have to show\nthat for many vectors x∈[q]n,d(x)is quite large. Similar discussion works for GOOC . The details can be found in\nSections V and VI below.\nOn the other hand, to show that GHCC andGOOC are locally sparse, we will use some well-developed tools th at\nwere continually improved in [11], [12], [23]. In particula r, we will use two upper bounds on the intersection volume\nof Hamming balls, which were proved in [12] to show that Gand its constant weight counterpart are locally sparse.\nLastly, let us compare our work with [16]. Although [16] is no t graph theoretic, an important step in their proof of\n(3) was also to show that for qn−o(qn)vectorsx∈[q]n,d(x)is quite large (see [16, Lemma III.4]). To do so, they\nused some standard concentration inequality for martingal es. In this paper, we used the McDiarmid’s inequality instea d,\nwhich makes our proof easier and gives a better control (uppe r bound) on the lower order term o(qn)(see Remark V .2\nbelow).\nA summary of previous mentioned GV-type bounds is presented in Table I.\nThe rest of this paper is organized as follows. In Section III we will mention two applications of Theorem II.1,\nwhich gives new lower bounds for frequency hopping sequence s sets and error-correcting weakly mutually uncorrelated\ncodes. In Section IV we will collect the tools that are used in Steps 1-4. In Section IV-A we will formally define locally\nsparse graphs (this will be used in Step 1) and state a lower bo und on its independence number (this will be used in Step\n4, see Lemma IV .1 below). In Section IV-B we will introduce th e McDiarmid’s inequality (this will be used in Step 2,\nsee Lemma IV .2 below). In Section IV-C we will state two upper bounds on the intersection volume of Hamming balls5\nTABLE I: A summary of GV-type bounds\nGV-type lower bounds of M Improved GV-type lower bounds of M\nGeneric(n,M,d)q-codesqn\nV olq(n,d−1)(1) ( see [7], [22]) Ω/parenleftBig\nnqn\nV olq(n,d−1)/parenrightBig\n(2) (see [11], [23])\nNon-linear (n,M,d)q-cyclic codes(1−o(1))qn\nV olq(n,d−1)(3) (see [9], [16])cnqn\nV olq(n,d−1)Theorem II.1 (this paper)\nConstant weight (n,M,d;pn)-codes/parenleftBign\npn/parenrightBig\nV ol(n,d−1;pn)(4) ( see [14]) Ω/parenleftBigg\nn/parenleftBign\npn/parenrightBig\nV ol(n,d−1;pn)/parenrightBigg\n(5) (see [12])\nConstant weight non-linear (n,M,d;pn)-cyclic codes/parenleftBign\nw/parenrightBig\n−f(n,d,w)\nn·V ol(n,d−1;w)(6) (see [1], [3], [24])cn/parenleftBign\npn/parenrightBig\nV ol(n,d−1;pn)Theorem II.2 (this paper)\n(this will be used in Step 3, see Lemmas IV .3 and IV .4 below). T he proofs of Theorems II.1 and II.2 will be presented\nin Sections V and VI respectively. We will conclude this pape r in Section VII.\nIII. A PPLICATIONS OF THE MAIN RESULTS\nA. Frequency hopping sequences\nIn this subsection, we will present an application of Theore m II.1 to frequency hopping sequences. We will need\nthe following easy lemma.\nLemma III.1. (i)For every x,y∈[q]n,x∈C(y)if and only if y∈C(x); moreover, C(x) =C(y)if and only if\nx∈C(y).\n(ii)LetCbe an(n,M,d)q-HCC. Then n|M, andCcan be partitioned into a pairwise disjoint union of M/n C(x)’s,\nwherex∈C.\nProof. Note that (i) follows straightforwardly from the definition ofC(x). To prove (ii), it is not hard to check that the\nrelation∼on the set Cdefined by x∼yifC(x) =C(y)is an equivalence relation. By (i) and the definition of an\nHCC, each equivalent class equals to C(x)for some x∈C, and hence consists of ndistinct vectors. Therefore, (ii)\nfollows from the fact that the equivalent classes given by ∼form a partition of C.\nLetCbe an(n,M,d)q-HCC. Given the lemma above, consider the partition of Cformed by the equivalent classes\ngiven by ∼. If we choose a representative element from each of these equ ivalent classes, then these representatives\nform a set of frequency hopping sequences (FHSs for short). An FHS set obtained by an (n,M,d)q-HCC in the above\nmanner is denoted as an (n,M/n,n −d)q-FHS, which is a set F⊆[q]nwith|F|=M/n such that\n•for every x∈Fand0< i≤n−1,Hx,x(i) :=n−dH(x,πi(x))≤n−d;\n•for every distinct x,y∈Fand0≤i≤n−1,Hx,y(i) :=n−dH(x,πi(y))≤n−d;\nwhereHx,x(i)andHx,y(i)are known as the auto- and cross-Hamming correlation functions at time delay i(see [4],\n[5], [16]). FHSs were designed for the transmission of radio signals and have been studied extensively. In fact, the work\nof Niu et al. [16] was motivated by the study of FHSs.\nNote that one can also define FHSs directly using the Hamming c orrelation functions. For the sake of saving space,\nwe will not formally state such a definition. It was observed i n [4], [16] that the two definitions of FHSs are equivalent,\nsince there is a one-to-one correspondence between HCCs and FHSs. Based on such a correspondence and (3), Niu et\nal. (see [16, Corollary III.6]) showed that for any absolute constantǫ >0andλ≥n−(n−2√n)(1−1/q−ǫ)there\nexist(n,M,λ)q-FHSs with\nM≥qn(1−n2e−ǫ2(√n−2)\n2)\nn(Volq(n,n−λ−1)−1). (8)\nThe following result improves (8) by a linear factor of n.\nProposition III.2. For positive integers n,q,λ and absolute reals τ,ǫ∈(0,1−1/q)satisfying 1/q+ǫ≤λ/n≤1−τ,\nthere exist (n,M,λ)q-FHSs with\nM≥cqn\nVolq(n,n−λ−1),6\nwherecis a positive real depending only on τ,ǫ,q .\nProof. This is an easy consequence of Lemma III.1 (ii), Theorem II.1 , and the definition of FHSs.\nB. Error-correcting weakly mutually uncorrelated codes\nFor1≤κ≤n, a code C∈[q]nis said to be a κ-weakly mutually uncorrelated code (WMUC for short) if for\nallκ≤ℓ≤n−1, no proper prefix of length ℓof a codeword in Cappears as a suffix of another codeword in C,\nincluding itself. Formally speaking, Cis aκ-WMUC if for every x,y∈C(possibly identical) and every κ≤ℓ≤n−1,\n(x1,...,x ℓ)/\\e}atio\\slash= (yn−ℓ+1,...,y n). A WMUC is said to be error-correcting if it has some sufficiently large minimum\ndistance.\nWMUCs and error-correcting WMUCs were introduced by Yazdi, Kiah, Gabrys, and Milenkovic [20] as a technique\nthat allows random access to encoded DNA strands in DNA-base d storage systems. The reader is referred to [20] for\nmore background.\nThe next result shows that HCCs have the bonus of being also WM UC.\nProposition III.3. For positive integers n,q,κ and absolute reals τ,ǫ∈(0,1−1/q)satisfying 1/q+ǫ≤κ/n≤1−τ\nthere exist κ-WMU(n,M,n−κ+1)q-codes with\nM≥cqn\nVolq(n,n−κ),\nwherecis a positive real depending only on τ,ǫ,q .\nProof. By Theorem II.1, under the assumption of the proposition, th ere exists an (n,M′,n−κ+1)q-HCCC′⊆[q]n\nwithM′≥cnqn\nV olq(n,n−κ),wherecis a positive real depending only on τ,ǫ,q . By Lemma III.1 (ii), C′can be partitioned\ninto a pairwise disjoint union of M′/n C(x)’s. Just as the proof of Proposition III.2, let Cbe a subcode of C′formed\nby picking exactly one codeword from every C(x)in the partition. Then, Cis an(n,M,n−κ+ 1)q-code with\nM≥cqn\nVolq(n,n−κ).\nIt remains to show that Cis alsoκ-WMU. Assume to the contrary that there exist x,y∈C(possibly identical)\nsuch that for some κ≤ℓ≤n−1, the prefix (x1,...,x ℓ)is identical to the suffix (yn−ℓ+1,...,y n). Note that {x,y} ⊆\nC⊆C′. AsCis an HCC, πn−ℓ(y) = (yn−ℓ+1,...,y n,y1,...,y n−ℓ)∈C′. AsdH(x,πn−ℓ(y))≤n−ℓ≤n−κ,\nwhich is strictly less than the minimum distance of C′, we must have x=πn−ℓ(y). It implies that {x,y} ⊆C(x)and\n|C∩C(x)| ≥2, a contradiction.\nIV. C OLLECTION OF SOME USEFUL LEMMAS\nIn this section, we will collect the tools that are used in thi s paper.\nA. Locally sparse graphs and their independent sets\nIn this subsection, we will introduce locally sparse graphs and a lower bound on their independence number. A\ngraphG= (V,E)consists of a vertex set Vand an edge set E, whereVis a finite set and Eis a family of 2-subsets\nofV. Two vertices u,v∈Vare said to be connected by an edge if {u,v} ∈E. Two connected vertices are called\nneighbors . The set of neighbors of a vertex uinGis denoted by NG(u). The degree of a vertex u∈VinG, denoted\nbydegG(u), is the size of its neighborhood, i.e., degG(u) =|NG(u)|. Note that we will omit the subscript GinNG(u)\nanddegG(u)when the underlying graph Gis clear from the context. The maximum degree of a graph is the maximum\ndegree among all of its vertices. For a subset S⊆V, the set of edges induced bySconsists of all edges with both\nendpoints in S, namely the edges {{u,v} ∈E:u,v∈S}. A graph is called locally sparse if for every vertex, its\nneighborhood induces only a limit number of edges.\nGiven a graph G= (V,E), a subset I⊆Vis called an independent set ofGif every two vertices in Iare not\nconnected. The independence number ofG, denoted by α(G), is the size of the maximum independent set of G. Using\nthe greedy algorithm mentioned in Section II-C, it is not har d to see that every graph with maximum degree Dhas an\nindependent set of size at least|V|\nD+1. Lemma IV .1 below, which is a direct consequence of [18, Coro llary 1], presents\na better lower bound for the independence number of locally s parse graphs.7\nLemma IV .1. [18, Corollary 1] Let Gbe a graph on nvertices with maximum degree at most D, whereD→ ∞ as\nn→ ∞ . Suppose that for every vertex of G, its neighborhood induces at mostD2\nKedges, where 1≤K≤D2+1. Then\nα(G)≥(1−o(1))·|V|\nD·ln(min{D,K}),\nwhereo(1)→0asK→ ∞ .\nB. McDiarmid’s inequality\nWe will make use of the powerful McDiarmid’s inequality, whi ch, roughly speaking, states that a Lipschitz function\nof random variables is concentrated around its expectation .\nLemma IV .2. [15] (McDiarmid’s inequality). Given sets X1,...,Xn, letX1,...,X nbe independent random variables\ntaking values in X1,...,Xnrespectively. Let f:X1×···×X n→Rbe a mapping. If there exist constants c1,...,c n∈R\nsuch that for each 1≤i≤n,\nsup\nx′\ni∈Xi,x1∈X1,...,xn∈Xn|f(x1,...,x i,...,x n)−f(x1,...,x′\ni,...,x n)| ≤ci,\n(i.e., changing the value of the i-th coordinate xichange the absolute value of fby at most ci), then for any t >0,\nPr[f(X1,X2,...,X n)−E[f(X1,X2,...,X n)]≤ −t]≤exp/parenleftbigg\n−2t2\n/summationtextn\ni=1c2\ni/parenrightbigg\n.\nC. The intersection volume of two Hamming balls\nWe will need the following estimation on the intersection vo lume of two Hamming balls.\nLemma IV .3. [12, Lemma 4.2] For positive integers n,q,d and two absolute reals τ,ǫ∈(0,1−1/q)satisfying\nτ≤d/n≤1−1/q−ǫ, we have that\nVolq(n,d−τn/2)\nVolq(n,d−1)≤e−Ωτ,q(1)·n.\nMoreover, for every x,y∈[q]n,\n|B(x,d−1)∩B(y,d−1)|\nVolq(n,d−1)=e−Ωτ,q(1)·d(x,y).\nSimilar upper bounds can also be proved for the intersection volume of two constant weight Hamming balls.\nLemma IV .4. [12, Lemma 4.3] Given positive integers n,d, and reals p∈(0,1), ǫ∈(0,1/10), τ∈(0,p(1−p))\nsatisfying τ≤d/n≤(1−ǫ)p(1−p). Then\nVol(n,d−τn/2;pn)\nVol(n,d−1;pn)≤2e−Ωǫ,τ(1)n.\nMoreover, for every x,y∈ {0,1}nwith Hamming weight pn,\n|B(x,d−1;pn)∩B(y,d−1;pn)|\nVol(n,d−1;pn)= 2e−Ωǫ(1)·(d(x,y)+d(x,y)2/(d−1)).\nV. P ROOF OF THEOREM II.1\nGiven two reals τ,ǫ∈(0,1−1/q), the goal of Theorem II.1 is to show the existence of large (n,M,d)q-HCCs\nwhenever τ≤d/n≤1−1/q−ǫ. The proof of Theorem II.1 will follow the steps outlined in S ection II-C. First of\nall, note that an (n,M,d)q-codeCis an(n,M,d)q-HCC only if for every x∈C,d(x)≥d(recall (7)). Therefore, to\nconstruct large HCCs, we need to show the existence of a large setA⊆[q]nsuch that for each x∈A,d(x)≥d. Such\na setAin fact exists, as shown by the next lemma.\nLemma V .1. Letn,q be positive integers and ǫ∈(0,1−1/q)be a real. Let\nA={x∈[q]n:d(x)> n(1−1/q−ǫ)}.8\nThen\n|A| ≥qn/parenleftbigg\n1−(n−1)exp(−ǫ2n\n2)/parenrightbigg\n.\nProof. LetX= (X1,...,X n)be a uniformly chosen random element of [q]n. To prove the lemma, it is enough to\nshow that\nPr[d(X)≤n(1−1/q−ǫ)]≤(n−1)exp(−ǫ2n\n2). (9)\nRecall that d(X) = min{d(X,πi(X)) : 1≤i≤n−1}, whereπi(X) = (Xi+1,...,X i+n). By the union bound,\nPr[d(X)≤n(1−1/q−ǫ)] = Pr[∃1≤i≤n−1s.t.d(X,πi(X))≤n(1−1/q−ǫ)]\n≤n−1/summationdisplay\ni=1Pr[d(X,πi(X))≤n(1−1/q−ǫ)].\nAccording to the discussion above, to prove (9) it suffices to show that for every fixed 1≤i≤n−1,\nPr[d(X,πi(X))≤n(1−1/q−ǫ)]≤exp(−ǫ2n\n2).\nNote that choosing X= (X1,...,X n)∈[q]nuniformly at random is equivalent to choosing X1,...,X n∈[q]uniformly\nand independently at random. Let\nδ: [q]×[q]→ {0,1}\ndenote the Kronecker function such that for a,b∈[q],δ(a,b) = 1 ifa=bandδ(a,b) = 0 ifa/\\e}atio\\slash=b. It is straightforward\nto check by definition that for every x∈[q]n,\nd(x,πi(x)) =n−n/summationdisplay\nj=1δ(xj,xj+i). (10)\nWe will apply Lemma IV .2 with f(x1,...,x n) :=d(x,πi(x)). Observe that for every 1≤j≤n, changing the\nvalue ofxjcould only change the values of δ(xj−i,xj)andδ(xj,xj+i)in the right hand side of (10), which in turn\nchanges the absolute value of d(x,πi(x))and hence fby at most 2. In other words, for every 1≤j≤n,\nsup\nx′\nj,x1,...,xn∈[q]|f(x1,...,x j,...,x n)−f(x1,...,x′\nj,...,x n)| ≤2. (11)\nMoreover, by the linearity of expectation, it is easy to see t hat for every 1≤i≤n−1and1≤j≤n,E(δ(Xj,Xj+i)) =\n1/q, which implies that E[f(X1,...,X n) =n(1−1/q). Applying Lemma IV .2 to f, we obtain\nPr[d(X,πi(X))−n(1−1/q)≤ −ǫn]≤exp(−ǫ2n\n2).\nPlugging the above inequality to the aforementioned union b ound proves (9), and hence Lemma V .1.\nRemark V .2. Using a standard concentration inequality for martingales , [16, Lemma III.4] proved a slightly weaker\nresult\n|{x∈[q]n:d(x)>(n−2√n)(1−1/q−ǫ)}| ≥qn/parenleftbigg\n1−n2e−ǫ2(√n−2)\n2/parenrightbigg\n.\nRemark V .3. It is not hard to check by definition that Ais an HCC with minimum distance, say 1. Therefore, similarly\nto the proof of Lemma III.1 (ii), one can show that with the equ ivalence relation ∼,Acan be partitioned into a family\nVof equivalent classes such that every equivalent class cons ists of exactly ndistinct vectors of [q]n. It follows by\nLemma V .1 that\n|V|=|A|\nn≥1\nn·qn/parenleftbigg\n1−(n−1)exp(−ǫ2n\n2)/parenrightbigg\n. (12)\nMoreover, by the definition of Awe have that for every C(x)∈V,\nd(x)> n(1−1/q−ǫ), (13)9\nand for every distinct C(x),C(y)∈V,\nC(x)∩C(y) =∅. (14)\nTo prove Theorem II.1, we will construct an auxiliary graph o n the vertex set V, and then apply Lemma IV .1 to\nthis graph, as detailed below.\nLemma V .4. With the notation of Theorem II.1, fixing some d∈[τn,n(1−1/q−ǫ)], letGHCC= (V,E)be a graph\nconstructed as below, where Vis the family of equivalent classes that partition A, and two distinct vertices (or equivalent\nclasses)C(x),C(y)∈Vform an edge in Eif and only if\nd(C(x),C(y)) = min {d(xi,yj) : 0≤i,j≤n−1} ≤d−1.\nThen the following holds:\n(i)for every independent set I⊆V, the set of vectors C:=/uniontext\nC(x)∈IC(x)forms an (n,n|I|,d)q-HCC;\n(ii)the maximum degree of GHCC is at most D:=Volq(n,d−1);\n(iii) for every vertex of GHCC , its neighborhood induces at mostD2\nKedges, where K=eΩτ,q(1)·n.\nProof. To prove (i), note first that for every C(x)∈V,C(x)consists of ndistinct vectors, which implies that Cis\nindeed an HCC. Moreover, it follows by (14) that |C|=n|I|. Lastly, to prove that d(C)≥d, letc1,c2∈Cbe two\ndistinct codewords. On one hand, if {c1,c2} ⊆C(x)for some C(x)∈I⊆V, then by (13) we have\nd(c1,c2)≥d(x)> n(1−1/q−ǫ)≥d.\nOn the other hand, if c1∈C(x)andc2∈C(y)for distinct C(x),C(y)∈I, then as Iis an independence set in G,\nwe have\nd(c1,c2)≥d(C(x),C(y))≥d,\ncompleting the proof of (i).\nTo prove (ii), note that for every C(x)∈V,\ndeg(C(x)) =|{C(y)∈V\\{C(x)}:d(C(x),C(y))≤d−1}|\n=|{C(y)∈V\\{C(x)}: min\n0≤i≤n−1d(x,πi(y))≤d−1}|\n=|{C(y)∈V\\{C(x)}:∃i∈[n]s.t.πi(y)∈B(x,d−1)}|\n≤ |B(x,d−1)|=D,\nas needed.\nTo prove (iii), fixing a vertex C(x)∈V, letΓdenote the subgraph induced by the neighborhood of C(x). Partition\nV(Γ) =S∪T, where\nS={C(y)∈V(Γ) :d(C(x),C(y))≤d−τn/2}\nand\nT={C(y)∈V(Γ) :d−τn/2< d(C(x),C(y))≤d−1}.\nWe have the following claim.\nClaim V .5. There exists some K=eΩτ,q(1)·nsuch that |S| ≤D/K , and for every vertex C(y)∈T,degΓ(C(y))≤\nD/K .\nGiven the correctness of the claim, it follows that\n|E(Γ)|=1\n2\n/summationdisplay\nC(y)∈SdegΓ(C(y))+/summationdisplay\nC(y)∈TdegΓ(C(y))\n\n≤1\n2/parenleftbigg\n|S|·D+|T|·D\nK/parenrightbigg\n≤D2\nK,\ncompleting the proof of (iii).\nIt remains to prove Claim V .5.10\nProof of Claim V .5. To prove the first inequality, observe that\n|S|=|{C(y)∈V(Γ) :d(C(x),C(y))≤d−τn/2}|\n=|{C(y)∈V(Γ) : min\n0≤i≤n−1d(x,πi(y))≤d−τn/2}|\n=|{C(y)∈V(Γ) :∃i∈[n]s.t.πi(y)∈B(x,d−τn/2)}|\n≤ |B(x,d−τn/2)|=Volq(n,d−τn/2).\nTherefore,\n|S|\nD≤Volq(n,d−τn/2)\nVolq(n,d−1)≤e−Ωτ,q(1)·n, (15)\nwhere the last inequality follows by the first equation of Lem ma IV .3.\nTo prove the second inequality, note that for every C(y)∈T, we have\nNΓ(C(y)) =N(C(x))∩N(C(y))\n={C(v)∈V\\{C(x),C(y)}:d(C(v),C(x))≤d−1, d(C(v),C(y))≤d−1}\n={C(v)∈V\\{C(x),C(y)}: min\n0≤i≤n−1d(v,πi(x))≤d−1,min\n0≤j≤n−1d(v,πj(y))≤d−1}\n={C(v)∈V\\{C(x),C(y)}:∃0≤i,j≤n−1s.t.v∈B(πi(x),d−1)∩B(πj(y),d−1)},\n⊆/braceleftbig\nC(v)∈V:v∈/uniondisplay\n0≤i,j≤n−1B(πi(x),d−1)∩B(πj(y),d−1)/bracerightbig\n,\nwhich implies that\n|NΓ(C(y))| ≤/summationdisplay\n0≤i,j≤n−1|B(πi(x),d−1)∩B(πj(y),d−1)|.\nIt thus follows that\ndeg��(C(y))\nD=|NΓ(C(y))|\nVolq(n,d−1)≤/summationtext\n0≤i,j≤n−1|B(πi(x),d−1)∩B(πj(y),d−1)|\nVolq(n,d−1)\n=/summationdisplay\n0≤i,j≤n−1B(πi(x),d−1)∩B(πj(y),d−1)\nVolq(n,d−1)\n≤/summationdisplay\n0≤i,j≤n−1e−Ωτ,q(1)·d(πi(x),πj(y))\n≤n2·e−Ωτ,q(1)·n=e−Ωτ,q(1)·n,(16)\nwhere the last two inequalities follows from the second equa tion of Lemma IV .3 and the assumption that d(πi(x),πj(y))≥\nd(C(x),C(y))≥d≥τn.\nThe claim follows straightforwardly by combining (15) and ( 16).\nWe proceed to present the proof of Theorem II.1.\nProof of Theorem II.1. LetGHCC= (V,E)be the graph defined in Lemma V .4. Applying Lemma IV .1 in conce rt with\n(12) and Lemma V .4 (ii), (iii) yields that\nα(GHCC)≥(1−o(1))·|V|\nD·ln(min{D,K})\n≥(1−o(1))·qn\nn·1\nVolq(n,d−1)·ln(min{Volq(n,d−1),eΩτ,q(n)})\n≥cqn\nVolq(n,d−1),\nwherecis a constant independent of n. Therefore, according to Lemma V .4 (i), there exist (n,M,d)q-HCCs with\nM≥n·α(GHCC), completing the proof of the theorem.11\nVI. P ROOF OF THEOREM II.2\nGiven reals p∈(0,1), ǫ∈(0,1/10), τ∈(0,p(1−p), the aim of Theorem II.2 is to show the existence of large\n(n,M,d;pn)-OOCs whenever τ≤d/n≤(1−ǫ)p(1−p). Forpn∈Z+, let/parenleftbig[n]\npn/parenrightbig\n={x∈ {0,1}n:wt(x) =pn}. Note\nthat an(n,M,d;pn)-codeCis an(n,M,d;pn)-OOC only if for every x∈C,d(x)≥d. Similarly to the discussion\nat the beginning of Section V, to construct large OOCs we need to show the existence of a large set B⊆/parenleftbig[n]\npn/parenrightbig\nsuch\nthat for each x∈B,d(x)≥d. The proof of this result is in the spirit similar to that of Le mma V .1 but technically a\nbit more involved.\nLemma VI.1. Letǫ >0,0< p <1be reals and nbe a positive integer. Let\nB={x∈/parenleftbigg[n]\npn/parenrightbigg\n:d(x)>(1−ǫ)np(1−p)}.\nThen\n|B| ≥/parenleftbiggn\npn/parenrightbigg/parenleftbigg\n1−n3/2exp/parenleftbig\n−Ωp,ǫ(1)·n\n2/parenrightbig/parenrightbigg\n.\nProof. LetX= (X1,...,X n)∈ {0,1}nbe a random vector, where X1,...,X narenindependent Bernoulli random\nvariables such that for 1≤i≤n,Pr[Xi= 1] =pandPr[Xi= 0] = 1 −p. For the ease of notation, let µ=np(1−p).\nBy the law of conditional probability,\nPr[d(X)≤(1−ǫ)µ|wt(x) =pn] =Pr[d(X)≤(1−ǫ)µ∧wt(x) =pn]\nPr[wt(x) =pn]=/parenleftbign\npn/parenrightbig\n−|B|\n/parenleftbign\npn/parenrightbig.\nTo prove the lemma, it is enough to show that\nPr[d(X)≤(1−ǫ)µ|wt(x) =pn]≤n3/2exp/parenleftbig\n−Ωp,ǫ(1)·n\n2/parenrightbig\n.\nRecall that d(X) = min{1≤i≤n−1 :d(X,πi(X))}, whereπi(X) = (Xi+1,...,X i). Then,\nPr[d(X)≤(1−ǫ)µ|wt(X) =pn]\n=Pr[∃1≤i≤n−1s.t.d(X,πi(X))≤(1−ǫ)µ|wt(X) =pn]\n≤n−1/summationdisplay\ni=1Pr[d(X,πi(X))≤(1−ǫ)µ|wt(X) =pn]\n=n−1/summationdisplay\ni=1Pr[d(X,πi(X))≤(1−ǫ)µ∧wt(x) =pn]\nPr[wt(x) =pn]\n≤n−1/summationdisplay\ni=1Pr[d(X,πi(X))≤(1−ǫ)µ]\nPr[wt(X) =pn]\n≤/parenleftBiggn−1/summationdisplay\ni=1Pr[d(X,πi(X))≤(1−ǫ)µ]/parenrightBigg\n·/radicalbig\n2πnp(1−p)·ℓ(n),\nwithℓ(n) = exp( −1\n12n+1+1\n12pn+1\n12(1−p)n), where the first inequality follows by the union bound, and th e last\ninequality follows from the fact that\nPr[wt(X) =pn] =/parenleftbiggn\npn/parenrightbigg\nppn(1−p)(1−p)n\nand Stirling’s approximation\n/parenleftbiggn\npn/parenrightbigg\nppn(1−p)(1−p)n≥1/radicalbig\n2πnp(1−p)·exp(1\n12n+1−1\n12pn−1\n12(1−p)n).\nAccording to the discussion above, to prove the lemma it suffi ces to show that for every fixed 1≤i≤n−1,\nPr[d(X,πi(X))≤(1−ǫ)µ]≤exp(−Ωp,ǫ(1)·n\n2).12\nSimilarly to the proof of Lemma V .1, we will apply Lemma IV .2 w ithf(x1,...,x n) :=d(x,πi(x)) =n−/summationtextn\nj=1δ(xj,xj+i).\nAs for every i/\\e}atio\\slash=j,E[δ(Xi,Xj)] =p2+(1−p)2, by the linearity of expectation we have that E[f(X1,X2,...,X n)] =\n2np(1−p) = 2µ. Given (11), applying Lemma IV .2 to fyields that\nPr[d(X,πi(X))−2µ≤ −(1+ǫ)µ]≤exp/parenleftbig\n−(1+ǫ)2p2(1−p)2n\n2/parenrightbig\n= exp/parenleftbig\n−Ωp,ǫ(1)·n\n2/parenrightbig\n,\nand then we have arrived at the desired conclusion.\nSimilarly to the discussion in Remark V .3, the set Bobtained by Lemma VI.1 can be partitioned into a family V\nof pairwise disjoint equivalent classes, where\n|V|=|B|\nn≥1\nn·/parenleftbiggn\npn/parenrightbigg/parenleftbigg\n1−n3/2exp(−Ωp,ǫ(1)n\n2)/parenrightbigg\n. (17)\nTo prove Theorem II.2, we will construct an auxiliary graph o n the vertex set V, as detailed below.\nLemma VI.2. Givend∈[τn,(1−ǫ)np(1−p)], letGOOC= (V,E)be a graph constructed as below, where Vis the\nfamily of equivalent classes that partition B, and two distinct vertices C(x),C(y)∈Vare connected if and only if\nd(C(x),C(y))≤d−1. Then the following holds:\n(i)for every independent set I⊆V, the set of vectors C:=/uniontext\nC(x)∈IC(x)forms an (n,n|I|,d;pn)-OOC;\n(ii)the maximum degree of GOOC is at most D:=Vol(n,d−1;pn);\n(iii) for every vertex of GOOC , its neighborhood induces at mostD2\nKedges, where K=eΩτ,ǫ(1)·n.\nAs the proof of Lemma VI.2 is very similar to that of Lemma V .4, we will put it in Appendix.\nNext we present the proof of Theorem II.2.\nProof of Theorem II.2. LetGOOC= (V,E)be the graph defined in Lemma VI.2. Then, applying Lemma IV .1 i n concert\nwith (17) and Lemma VI.2 (ii), (iii) yields that\nα(GOOC)≥(1−o(1))·|V|\nD·ln(min{D,K})\n≥(1−o(1))·1\nn·/parenleftbiggn\npn/parenrightbigg\n·1\nVol(n,d−1;pn)·ln(min{Vol(n,d−1;pn),eΩτ,ǫ(n)})\n≥c/parenleftbign\npn/parenrightbig\nVol(n,d−1;pn),\nwherecis a constant independent of n. Therefore, according to Lemma VI.2 (i), there exist (n,M,d;pn)-OOCs with\nM≥n·α(GOOC), completing the proof of the theorem.\nVII. C ONCLUSIONS\nIn this paper we present improved GV-type bounds for hopping cyclic codes and optical orthogonal codes, which in\nturn give improved GV-type bounds for non-linear cyclic cod es and non-linear constant weight cyclic codes, respective ly.\nSeveral interesting problems remain open.\nQuestion VII.1. It is known that for prime power q≥49, there exist a class of codes, called algebraic geometry cod es,\nthat can significantly improve the GV bound (1)by an exponential power exp(Ω(n))( see [10], [21]). Does such an\nimprovement hold also for (non-linear) cyclic codes? What a bout constant weight cyclic codes?\nQuestion VII.2. All of the results mentioned in this paper are non-construct ive. Can we explicitly construct a class of\nasymptotically good (non-linear) cyclic codes? Or more pre cisely, can we present Zyablov-type bounds for (non-linear )\ncyclic codes? 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Cambridge University Press, 2006.\n[20] S. M. H. Tabatabaei Yazdi, H. M. Kiah, R. Gabrys, and O. Mi lenkovic, “Mutually uncorrelated primers for dna-based da ta storage,” IEEE Trans.\nInform. Theory , vol. 64, no. 9, pp. 6283–6296, 2018.\n[21] M. A. Tsfasman, S. G. Vl˘ adut ¸, and T. Zink, “Modular cur ves, Shimura curves, and Goppa codes, better than Varshamov -Gilbert bound,” Math.\nNachr. , vol. 109, pp. 21–28, 1982.\n[22] R. Varshamov, “Estimate of the number of signals in erro r correcting codes,” Doklady Akadamii Nauk , vol. 117, pp. 739–741, 1957.\n[23] V . Vu and L. Wu, “Improving the Gilbert-Varshamov bound forq-ary codes,” IEEE Trans. Inform. Theory , vol. 51, no. 9, pp. 3200–3208, 2005.\n[24] G.-C. Yang and T. Fuja, “Optical orthogonal codes with u nequal auto- and cross-correlation constraints,” IEEE Trans. Inform. Theory , vol. 41,\nno. 1, pp. 96–106, 1995.\nVIII. A PPENDIX :PROOF OF LEMMA VI.2\nProof of Lemma VI.2. Note that given the proofs of the first two items of Lemma V .4, L emma VI.2 (i) and (ii) can be\nproved easily, so we omit both proofs for clarity. To prove (i ii), letΓ,S,andTbe defined as in the proof of Lemma V .4.\nThen, it suffices to show that Claim V .5 holds also under the se tting of Lemma VI.2.\nClaim VIII.1 (restatement of Claim V .5) .There exists some K=eΩτ,ǫ,p(1)·nsuch that |S| ≤D/K , and for every\nvertexC(y)∈T,degΓ(C(y))≤D/K .\nTo prove the first part of the claim, note that\n|S|\nD=|{C(y)∈V(Γ) :d(C(x),C(y))≤d−τn/2}|\nD≤Vol(n,d−τn/2;pn)\nVol(n,d−1;pn)≤2e−Ωτ,ǫ(1)n, (18)\nwhere the first inequality follows from the same technique th at proves the first inequality in (15), and the second\ninequality follows by the first equation of Lemma IV .4.\nAs for the second part of the claim, using the same strategy th at proves (16) one can show that for every C(y)∈T,14\ndegΓ(C(y))\nD=|NΓ(C(y))|\nVol(n,d−1;pn)≤/summationtext\n0≤i,j≤n−1|B(πi(x),d−1)∩B(πj(y),d−1)|\nVol(n,d−1;pn)\n=/summationdisplay\n0≤i,j≤n−1B(πi(x),d−1;pn)∩B(πj(y),d−1)\nVolq(n,d−1;pn)\n≤/summationdisplay\n0≤i,j≤n−1e−Ωǫ(1)·(d(πi(x),πj(y))+d(πi(x),πj(y))2/(d−1))\n≤n2·e−Ωτ,ǫ,p(1)·n=e−Ωτ,ǫ,p(1)·n,(19)\nwhere the last two inequalities follows from the second equa tion of Lemma IV .4 and the assumption that d(πi(x),πj(y))≥\nd(C(x),C(y))≥d≥τn.\nThe claim follows straightforwardly by combining (18) and ( 19)."    },    {        "title": "2301.09086v1.Boundary_stabilization_of_a_vibrating_string_with_variable_length.pdf",        "content": "BOUNDARY STABILIZATION OF A VIBRATING STRING WITH\nVARIABLE LENGTH\nSEYF EDDINE GHENIMI AND ABDELMOUHCENE SENGOUGA\nAbstract. We study small vibrations of a string with time-dependent length `(t) and bound-\nary damping. The vibrations are described by a 1-d wave equation in an interval with one\nmoving endpoint at a speed `0(t) slower than the speed of propagation of the wave c=1. With\nno damping, the energy of the solution decays if the interval is expanding and increases if the\ninterval is shrinking. The energy decays faster when the interval is expanding and a constant\ndamping is applied at the moving end. However, to ensure the energy decay in a shrinking\ninterval, the damping factor \u0011must be close enough to the optimal value \u0011= 1, correspond-\ning to the transparent condition. In all cases, we establish lower and upper estimates for the\nenergy with explicit constants.\n1.Introduction\nWe consider small transversal vibrations of a uniform string, with a time dependent length.\nThe mechanical setting is sketched in Figure 1 where the left end of the string is \fxed while\nthe right moving end is also allowed to move transversely and attached to a damping device\n(a dash-pot with a damping factor \u0011\u00150).\nFigure 1. A string with one moving end subject to a dash-pot damping.\nDenoting the displacement function by u, depending on the position xalong the string and\nthe timet, the model can be stated as follows\n8\n>>><\n>>>:utt\u0000uxx= 0; for 0<x<` (t) andt>0;\n(1 +\u0011`0(t))ux(`(t);t) + (\u0011+`0(t))ut(`(t);t) = 0;fort>0;\nu(0;t) = 0, for t>0;\nu(x;0) =u0(x); ut(x;0) =u1(x); for 0<x<L:(WP)\nDate : January 24, 2023.\n2010 Mathematics Subject Classi\fcation. 35L05, 35C10, 93D15.\nKey words and phrases. Wave equation, time-dependent domains, generalized Fourier series, boundary\nstabilization.\n1arXiv:2301.09086v1  [math.AP]  22 Jan 20232 Boundary stabilization of a vibrating string\nThe subscripts tandxin (WP) stand for the derivatives in time and space variables respec-\ntively. The functions u0andu1represents the initial shape and the initial transverse speed of\nthe string, respectively. The initial length of the string is denoted by L=`(0).\nWe assume that `2C([0;+1[) and that\n\f\f`0(t)\f\f<1;fort\u00150; (1.1)\nwhich means that the speed of variation of the length of the string `0(t) is strictly less then\nthe speed of propagation of the wave (here simpli\fed to c= 1).\nIn this paper, we are mainly interested in the asymptotic behaviour in time of the energy\nof the solution, de\fned as\nE`(t) :=1\n2Z`(t)\n0u2\nt(x;t) +u2\nx(x;t)dx; fort\u00150: (1.2)\nFor the time-independent interval, i.e. when `(t) =Lfort\u00150, it is well known that:\n\u000fIf\u0011= 0, then it is the energy is constant, i.e. EL(t) =EL(0);fort\u00150.\n\u000fIf\u0011\u00150 with\u00116= 1 , then the energy decays exponentially\nEL(t)\u0014CEL(0)e\u00001\nLlnj\r\u0011jt;for some constant C > 0;\nwhere\r\u0011:=1+\u0011\n1\u0000\u0011, see [15, 4, 3, 11]. In [5], the present authors showed that\n1\n\r2\u0011EL(0)e\u00001\nLlnj\r\u0011jt\u0014EL(t)\u0014\r2\n\u0011EL(0)e\u00001\nLlnj\r\u0011jt;fort\u00150: (1.3)\nThe question of E`(t) behaviour in time is more delicate if the interval depends on time.\nFor instance, for the Dirichlet boundary conditions (which corresponds to \u0011= +1in (WP)),\nthe energy decays if the interval is expanding and increases if the interval is shrinking, see [2].\nSee also [12, 13, 14] when the variation of the length is uniform in time.\nRegarding the case with a velocity feedback at the moving endpoint x=`(t):\n\u000fGugat [6] considered the case ux(`(t);t) +cut(`(t);t) = 0 where cis as constant. See\nalso [8] for the particular `(t) = 1 +vtwherevis a constant, 0 <v< 1.\n\u000fAmmari et al. [1] considered the case ux(`(t);t) +f(t)ut(`(t);t) = 0;wheref2\nL1(0;+1) and`(t) is periodic. See also [7] where `(t) is not necessarily periodic.\nMokhtari [9] considered the case with two moving endpoints.\nFor the special case \u0011= 1, the boundary condition at x=`(t) reads\nux(`(t);t) +ut(`(t);t) = 0:\nThis is a transparent condition, i.e. there is no re\rections of waves from the moving endpoint\nand consequently all the initial disturbances leave the interval (0 ;`(t)) at most after a time\nT`:=\f\u00001(L);\nwhere\f(t) :=t\u0000`(t);see Figure 2. Hence, wether the interval is expanding or shrinking,\nthe linear velocity feedback \u0000ut(`0(t);t) steers the solution to the zero state in the \fnite time\nT`. See for instance [6, 7], and for the particular case `(t) =Lsee [15, 4]. In the remaining of\nthis paper, we will assume that \u0011\u00150 and\u00116= 1:\nThe approach used in the paper at hands is based on generalized Fourier series. This is\npossible since the closed form for the solution of (WP) is given by the series\nu(x;t) =X\nn2Zcn\u0010\ne!n'(t+x)\u0000e!n'(t\u0000x)\u0011\n;for 0<x<` (t) andt\u00150; (1.4)S. Ghenimi and A. Sengouga 3\nFigure 2. An initial disturbance with a small support leaves the interval at\nmost after time T`.\nwhere the coe\u000ecients cncan be explicitly computed in function of the initial data u0;u1:We\ndenoted by !na sequence of complex number given by\n!n:=\u00001\n2lnj\r\u0011j+8\n<\n:2n+ 1\n2i\u0019; if 0\u0014\u0011<1;\nni\u0019; if\u0011>1;(1.5)\nObserve that since j\r\u0011j\u00151 for every \u0011\u00150;the real part of !nis nonpositive.\nBy';we denoted a real function satisfying the functional equation\n'(t+`(t))\u0000'(t\u0000`(t)) = 2: (1.6)\nThis equation often called Moor's equation following his paper [10]. Some pairs of solutions\n(`;\u001e) can be found in [17].\nWe will assume that 'is di\u000berentiable and increasing. More precisely,\n'2C1([\u0000L;+1[) and'0>0;fort\u0015\u0000L: (1.7)\nThe assumption '0>0 is needed in the sequel since '0will serve as a weight for an L2space.\nBesides, a deacreasing \u001ecan not satisfy (1.6) since `(t)>0. See [7] for further discussions on\nthe regularity of the solutions of (1.6).\nIn this work, we demonstrate how the series formulas (1.4) can be used to achieve the\nfollowing results:\n\u000fFor the undamped case, i.e. \u0011= 0, the energy of the solution satis\fes\nm(t)\nM(t0)E`(t0)\u0014E`(t)\u0014M(t)\nm(t0)E`(t0);for 0\u0014t0<t; (1.8)\nwhere\nm(t) := min\nx2[0;`(t)]\b\n'0(t\u0000x);'0(t+x)\t\nandM(t) := max\nx2[0;`(t)]\b\n'0(t\u0000x);'0(t+x)\t\n:(1.9)\nSee Theorem 2 and its corollaries for sharper estimates.\n\u000fFor the damped case \u0011>0;with\u00116= 1, the energy E`(t) satis\fes\n \nelnj\r\u0011j'(t0\u0000`(t0))\nM(t0)!\nm(t)e\u0000lnj\r\u0011j'(t+`(t))E`(t0)\u0014E`(t)\n\u0014 \nelnj\r\u0011j'(t0+`(t0))\nm(t0)!\nM(t)e\u0000lnj\r\u0011j'(t\u0000`(t))E`(t0);for 0\u0014t0<t: (1.10)4 Boundary stabilization of a vibrating string\nSee Theorem 3 and it corollaries for more estimates. The estimate given in (1.10), with\nexplicit constants, is new to the best to our knowledge.\nAfter the present introduction, we derive the exact solution and the expression for the\ncoe\u000ecients of the series formula (1.4). In Section 3, we establish upper and lower estimates\nfor the energy E`(t) of the undamped equation. In Section 4, we deal with the damped case.\nSome examples will be included as a last section.\n2.Exact solution\nLet us introduce the following family of Hilbert spaces\nV0(0;`(t)) :=\b\nw2H1(0;`(t)) ,w(0) = 0\t\n;fort\u00150\nand assume that the initial data satis\fes\nu02V0(0;L); u12L2(0;L): (2.1)\nLetT >0:Then, we have the following existence result for Problem (WP).\nTheorem 1. Under the assumptions (1.1), (1.7) and (2.1), Problem (WP) has a unique\nsolution satisfying\nu2C([0;T];V0(0;`(t)))\\C1\u0000\n[0;T];L2(0;`(t))\u0001\n; (2.2)\ngiven by the series (1.4) where the coe\u000ecients cn2Care computed as follows\ncn=1\n4!nZL\n\u0000L\u0000\n~u0\nx+ ~u1\u0001\ne\u0000!n'(x)dx, forn2Z; (2.3)\nwhere ~u0\nxis an even (resp. ~u1is an odd )extension of the initial data u0(resp.u1)de\fned on\nthe interval (\u0000L;L). Moreover,\nX\nn2Zj!ncnj2=1\n8ZL\n\u0000L\u0000\n~u0\nx+ ~u1\u00012elnj\r\u0011j'(x)dx\n'0(x)<+1: (2.4)\nProof.\u000fThe exact solution: This part of solution is slightly di\u000berent from the approach in\n[16] where the author considered 1 =\u0011instead of\u0011in the boundary condition at x=`(t). We\ninclude it here for the sake clarity. The general solution of (WP) is given by D'Alembert's\nformula\nu(x;t) =f(t+x) +g(t\u0000x); (2.5)\nwherefandgare arbitrary continuous functions. The boundary conditions at the endpoint\nx= 0, we have\nf(t) =\u0000g(t):\nThe condition at x=`(t) implies that\n\u0000\n1 +\u0011`0(t)\u0001\u0002\nf0(t+`(t))\u0000g0(t\u0000`(t))\u0003\n=\u0000\u0000\n\u0011+`0(t)\u0001\u0002\nf0(t+`(t)) +g0(t\u0000`(t))\u0003\n;\nhence\u0002\n1 +\u0011`0(t) +\u0011+`0(t)\u0003\nf0(\u000b(t)) =\u0000\u0002\n1 +\u0011`0(t)\u0000\u0011\u0000`0(t)\u0003\nf0(\f(t)); (2.6)\nwhere\f(t) :=t\u0000`(t) and\u000b(t) :=t+`(t). Then, noting that\n1 +\u0011`0(t)\u0000\u0011\u0000`0(t)\n1 +\u0011`0(t) +\u0011+`0(t)=1\n\r\u0011\f0(t)\n\u000b0(t); (2.7)\nwe can rewrite (2.6) as\n\u000b0(t)f0(\u000b(t)) =\u00001\n\r\u0011\f0(t)f0(\f(t)): (2.8)S. Ghenimi and A. Sengouga 5\nBy integration, it follows that\nf(\u000b(t)) =\u00001\n\r\u0011f(\f(t)) +C: (2.9)\nLet us assume for the moment that C= 0:Then, it is convenient to search for fin the form\nf(\u0018) =e!'(\u0018), for a constant !and some function '. Substituting e!'(\u0018)in (2.9), we get\ne!['(\u000b(t))\u0000'(\f(t))]=\u00001=\r\u0011:\nAssuming that 'satis\fes (1.6), we are led to the following cases:\n- If 0\u0014\u0011<1, then\r\u0011\u00151 and we get\ne!['(\u000b(t))\u0000'(\f(t))]=e(2n+1)i\u0019\u0000ln\r\u0011:\nSolving this equation for !, we obtain a sequence of values !n;n2Z;where\n!n=2n+ 1\n2i\u0019\u00001\n2ln\r\u0011:\n- If\u0011>1, we have \r\u0011<\u00001 and we obtain this time\n!n=ni\u0019\u00001\n2lnj\r\u0011j; n2Z:\nThus, if\u0011\u00150 and\u00116= 1, we always have ln j\r\u0011j\u00151 and!ngiven by (1.5).\nDue to the superposition principal, it follows that fcan be written as\nf(\u0018) =X\nn2Zcne!n'(\u0018); cn2C;\nwherecnare complex coe\u000ecients to be determined later. Since f(\u0018) =\u0000g(\u0018);then D'Alembert's\nformula for the solution yields the series\nu(x;t) =X\nn2Zcn\u0010\ne!n'(t+x)\u0000e!n'(t\u0000x)\u0011\n;for 0<x<` (t) andt\u00150: (2.10)\nIfC6= 0 in (2.9), then we can check that\nf(\u0018) =C\r\u0011\n1 +\r\u0011+X\nn2Zcne!n'(\u0018);\nsolves (2.9). However, this will not a\u000bect the solution of (WP) since f(\u0018) =\u0000g(\u0018) and thus\nthe constant parts of fandgwill be cancelled in the expression (2.10).\n\u000fComputing the coe\u000ecients cn:We extend u(\u0001;t) to an odd function ~ u(\u0001;t) on the interval\n(\u0000`(t);`(t)):This ensures in particular that the boundary condition at x= 0 is satis\fed for\nt\u00150:It follows also that ~ ut(\u0001;t) and ~ux(\u0001;t) are respectively an odd and an even function.\nGoing back to (2.10), we infer that\n~ux+ ~ut= 2'0(t+x)X\nn2Z!ncne!n'(t+x);\nforx2(\u0000`(t);`(t)) andt\u00150. Using the de\fnition of !n;we get\n~ux+ ~ut=8\n>>><\n>>>:2e1\n2(i\u0019\u0000ln\r\u0011)'(t+x)'0(t+x)X\nn2Z!ncneni\u0019' (t+x);if 0\u0014\u0011<1;\n2e\u00001\n2lnj\r\u0011j'(t+x)'0(t+x)X\nn2Z!ncneni\u0019' (t+x); if 1<\u0011< +1;(2.11)6 Boundary stabilization of a vibrating string\nwhich implies that\nX\nn2Z!ncneni\u0019' (t+x)=8\n<\n:1\n2'0(t+x)e1\n2(\u0000i\u0019+ln\r\u0011)'(t+x)(~ux+ ~ut);if 0\u0014\u0011<1;\n1\n2'0(t+x)e1\n2lnj\r\u0011j'(t+x)(~ux+ ~ut); if 1<\u0011< +1:(2.12)\nTaking into account that\b\neni\u0019' (t+x)=p\n2\t\nn2Zis an orthonormal basis of the weighted space\nL2(\u0000`(t);`(t);'0(t+x)dx);we deduce that\n!ncn=8\n>>><\n>>>:1\n4Z`(t)\n\u0000`(t)e\u00001\n2(i\u0019\u0000ln\r\u0011)'(t+x)(~ux+ ~ut)e\u0000ni\u0019' (t+x)dx; if 0\u0014\u0011<1;\n1\n4Z`(t)\n\u0000`(t)e1\n2lnj\r\u0011j'(t+x)(~ux+ ~ut)e\u0000ni\u0019' (t+x)dx; if 1<\u0011< +1;\nforn2Z. Whether 0\u0014\u0011<1 or 1<\u0011< +1;in both cases, we have\ncn=1\n4!nZ`(t)\n\u0000`(t)(~ux+ ~ut)e\u0000!n'(t+x)dx, forn2Z. (2.13)\nTakingt= 0, we obtain (2.3) as claimed.\n\u000fRegularity of the solution: As a consequence of Parseval's equality, we get\nX\nn2Zj!ncnj2=8\n>>><\n>>>:1\n8Z`(t)\n\u0000`(t)\f\f\fe\u00001\n2(i\u0019\u0000ln\r\u0011)'(t+x)\f\f\f2\n(~ux+ ~ut)2dx\n'0(t+x);if 0\u0014\u0011<1\n1\n8Z`(t)\n\u0000`(t)\f\f\fe1\n2lnj\r\u0011j'(t+x)\f\f\f2\n(~ux+ ~ut)2dx\n'0(t+x); if 1<\u0011< +1:\nWhether 0\u0014\u0011<1 or 1<\u0011< +1;the two cases of the precedent identity can be written as\nX\nn2Zj!ncnj2=1\n8Z`(t)\n\u0000`(t)(~ux+ ~ut)2elnj\r\u0011j'(t+x)dx\n'0(t+x): (2.14)\nDue to (1.7) and (2.1), we have for t= 0;\n1p\n'0(x)elnj\r\u0011j'(t+x)\u0000\n~u0\nx+ ~u1\u0001\n2L2(\u0000L;L)\nand (2.4) follows.\nDue to the continuity and di\u000berentiability of '(t+x) and the exponential function, and\nsincejwnj=O(n) for large values of n, the regularity result (2.2) follows from the convergences\nof the series of the solution (2.10) and its derivatives. \u0003\n3.The undamped case\nIn this section, we show some results for the undamped case, i.e. \u0011= 0 in Problem (WP). To\nknow wether the energy is increasing or in creasing, we compute E0\n`(t):Thus, using Leibniz's\nrule for di\u000berentiation under the integral sign, we get\nE0\n`(t) =1\n2`0(t)\u0002\nu2\nx(`(t);t) +u2\nt(`(t);t)\u0003\n+Z`(t)\n0ututt+uxutxdx:\nSinceutt=uxx, thenututt+uxutx= (utux)xand it follows\nE0\n`(t) =1\n2`0(t)\u0002\nu2\nx(`(t);t) +u2\nt(`(t);t)\u0003\n+utux(`(t);t): (3.1)\nThen, we have the following result.S. Ghenimi and A. Sengouga 7\nLemma 1. The energy of solution of Problem (WP) , with\u0011= 0, satis\fes\nE0\n`(t) =\u0000`0(t)\n2\u0010\n1\u0000\f\f`0(t)\f\f2\u0011\nu2\nt(`(t);t);fort>0:\nProof. The boundary condition at x=`(t);with\u0011= 0;reads\nux(`(t);t) =\u0000ut(`(t);t);fort>0:\nReporting this in (3.1), we get\nE0\n`(t) =1\n2`0(t)\u0010\f\f`0(t)\f\f2+ 1\u0011\nu2\nt(`(t);t)\u0000`0(t)u2\nt(`(t);t)\nand the lemma follows. \u0003\nRemark 1. The above lemma means that\nkE`(t)is nondeacreasing if \u00001<`0(t)\u00140;and nonincreasing if 0\u0014`0(t)<1:(3.2)\nThe same result holds for the multidimensional wave equation, in a time-dependent domain,\nwith homogenous Dirichlet boundary conditions, see [2].\nThe next theorem show that the asymptotic behaviour of E`(t) is dictated by '0:\nTheorem 2. Under the assumptions (1.1), (1.7) and (2.1), the solution of Problem (WP)\nsatis\fes\nZ`(t)\n0\u00121\n'0(t+x)+1\n'0(t\u0000x)\u0013\u0000\nu2\nx+u2\nt\u0001\n+ 2\u00121\n'0(t+x)\u00001\n'0(t\u0000x)\u0013\nuxutdx= 4S0;fort\u00150;(3.3)\nwhereS0:=\u00192\n2P\nn2Zj(2n+ 1)cnj2. Moreover, it holds that\nS0m(t)\u0014E`(t)\u0014S 0M(t);fort\u00150; (3.4)\nwherem(t)andM(t)are de\fned in (1.9) .\nProof. If\u0011= 0 then\r\u0011= 1 and!n= (2n+ 1)i\u0019=2. The identity (2.14) becomes\n1\n8Z`(t)\n\u0000`(t)(~ux+ ~ut)2dx\n'0(t+x)=\u00192\n4X\nn2Zj(2n+ 1)cnj2=1\n2S0;fort\u00150: (3.5)\nSince ~uxis an even function of xand that ~utis an odd one, then changing xby\u0000xin the last\nformula, we also obtain\nZ`(t)\n\u0000`(t)(~ux\u0000~ut)2dx\n'0(t\u0000x)= 4S0;fort\u00150: (3.6)\nTaking the sum of (3.5) and (3.6), we obtain\nZ`(t)\n\u0000`(t)(~ux+ ~ut)2dx\n'0(t+x)+Z`(t)\n\u0000`(t)(~ux\u0000~ut)2dx\n'0(t\u0000x)= 8S0:\nExpanding ( ux\u0006ut)2and collecting similar terms, we get\nZ`(t)\n\u0000`(t)\u00121\n'0(t+x)+1\n'0(t\u0000x)\u0013\u0000\n~u2\nx+ ~u2\nt\u0001\n+ 2\u00121\n'0(t+x)\u00001\n'0(t\u0000x)\u0013\n~ux~utdx= 8S0;fort\u00150:(3.7)8 Boundary stabilization of a vibrating string\nAs the function under the integral sign is even, then (3.3) follows.\nNext, we use the algebraic inequality \u00062uxut\u0014u2\nt+u2\nxto obtain\nZ`(t)\n0\u00121\n'0(t+x)+1\n'0(t\u0000x)\u0000\f\f\f\f1\n'0(t+x)\u00001\n'0(t\u0000x)\f\f\f\f\u0013\u0000\nu2\nx+u2\nt\u0001\ndx\u00144S0\n\u0014Z`(t)\n0\u00121\n'0(t+x)+1\n'0(t\u0000x)+\f\f\f\f1\n'0(t+x)\u00001\n'0(t\u0000x)\f\f\f\f\u0013\u0000\nu2\nx+u2\nt\u0001\ndx;\nfort\u00150:Recalling that\n(a+b)\u0000ja\u0000bj= 2 minfa;bgand (a+b) +ja\u0000bj= 2 maxfa;bg; (3.8)\nfora;b2R;then\nZ`(t)\n0min\u001a1\n'0(t+x);1\n'0(t\u0000x)\u001b\u0000\nu2\nx+u2\nt\u0001\ndx\u00142S0\n\u0014Z`(t)\n0max\u001a1\n'0(t+x);1\n'0(t\u0000x)\u001b\u0000\nu2\nx+u2\nt\u0001\ndx;\nfort\u00150:Recalling that E`(t) is de\fned by (1.2), we deduce that\nmin\nx2[0;`(t)]\u001a1\n'0(t+x);1\n'0(t\u0000x)\u001b\nE`(t)\u0014S 0\u0014max\nx2[0;`(t)]\u001a1\n'0(t+x);1\n'0(t\u0000x)\u001b\nE`(t);\nhence\nmin\nx2[0;`(t)]\b\n'0(t+x);'0(t\u0000x)\t\nS0\u0014E`(t)\u0014max\nx2[0;`(t)]\b\n'0(t+x);'0(t\u0000x)\t\nS0;\nwhich is (3.4). \u0003\nRemark 2. An estimation analogue to (3.4) was obtained for the case of homogeneous bound-\nary conditions at both ends, see [7].\nIf'0is monotone then the asymptotic behaviour is dictated by `(t) and we have the following\nre\fnements.\nCorollary 1. Under the assumption of Theorem 2, assume that\n'0is monotone for t2[t0;t1];0\u0014t0<t1: (3.9)\n\u000fIf\u00001<`0(t)\u00140 on [t0;t1], then'0andE`(t) are nondeacreasing and\nS0'0(t\u0000`(t))\u0014E`(t)\u0014S 0'0(t+`(t));fort2[t0;t1]: (3.10)\n\u000fIf 0\u0014`0(t)<1 on [t0;t1], then'0andE`(t) are nonincreasing and\nS0'0(t+`(t))\u0014E`(t)\u0014S 0'0(t\u0000`(t));fort2[t0;t1]: (3.11)\nProof. We already have (3.2). The derivation of the identity (1.6) yields\n1 +`0(t)\n1\u0000`0(t)'0(t+`(t)) ='0(t\u0000`(t)): (3.12)\nTaking into consideration the variation of the function s7!(1 +s)=(1\u0000s) on the interval\n(\u00001;1), it follows that\n0<1 +`0(t)\n1\u0000`0(t)\u00141 if\u00001<`0(t)\u00140 and 1\u00141 +`0(t)\n1\u0000`0(t)if 0\u0014`0(t)<1;S. Ghenimi and A. Sengouga 9\nhence\nif\u00001<`0(t)\u00140;thenm(t) ='0(t\u0000`(t))\u0014'0(t+`(t)) =M(t); (3.13)\nif 0\u0014`0(t)<1;thenM(t) ='0(t\u0000`(t))\u0015'0(t+`(t)) =m(t): (3.14)\nOf course, if '0is monotone and \u00001< `0(t)\u00140 then (3.13) means that '0is necessarily\nnondecreasing, and so is '0(t\u0006`(t)) andE`(t):The same argument can be made when\n0\u0014`0(t)<1. This shows the corollary. \u0003\nRemark 3. The assumption (3.9) is satis\fed in all the examples of the last section.\nRecall thatS0can be computed using the initial data by setting t= 0 in (3.4). If one needs\nto compare E`(t) withE`(t0) for 0\u0014t0<t;then we have the next result.\nCorollary 2. Under the assumption of Theorem 2, we have\nm(t)\nM(t0)E`(t0)\u0014E`(t)\u0014M(t)\nm(t0)E`(t0);for0\u0014t0<t. (3.15)\nMoreover, if 'satis\fes (3.9), then:\n\u000fIf\u00001<`0(t)\u00140 on [t0;t1], then'0andE`(t) are nondecreasing and satisfy\n'0(t\u0000`(t))\n'0(t0+`(t0))E`(t0)\u0014E`(t)\u0014'0(t+`(t))\n'0(t0\u0000`(t0))E`(t0);fort2[t0;t1]: (3.16)\n\u000fIf 0\u0014`0(t)<1 on [t0;t1], then'0andE`(t) are nonincreasing and\n'0(t+`(t))\n'0(t0\u0000`(t0))E`(t0)\u0014E`(t)\u0014'0(t\u0000`(t))\n'0(t0+`(t0))E`(t0);fort2[t0;t1]: (3.17)\nProof. Since (3.4) holds also for t=t0, then the corollary follows from the inequalities\nE`(t)\nM(t)\u0014S 0\u0014E`(t0)\nm(t0)andE`(t0)\nM(t0)\u0014S 0\u0014E`(t)\nm(t):\n\u0003\n4.The damped case\nIn this section, we investigate the case with boundary damping when\n\u0011>0 and\u00116= 1\nin Problem (WP). Let us recall that we still have (3.1), i.e.\nE0\n`(t) =1\n2`0(t)\u0002\nu2\nx(`(t);t) +u2\nt(`(t);t)\u0003\n+utux(`(t);t);fort\u00150:\nbut now the boundary condition at x=`(t) is\n\u0000\n1 +\u0011`0(t)\u0001\nux(`(t);t) +\u0000\n\u0011+`0(t)\u0001\nut(`(t);t) = 0;fort\u00150\nLet us discuss the sign of E0\n`(t) for di\u000berent values of `0(t):\n\u000fIf the interval is shrinking \u00001< `0(t)<0;fort2[t1;t2] where 0\u0014t1< t2;then we\nhave the following cases:\n- If`0(t) =\u00001=\u0011;then the boundary condition at x=`(t) readsut(`(t);t) = 0 and\nthus\nE0\n`(t) =\u00001\n2\u0011u2\nx(`(t);t)\u00140;fort2[t1;t2]:10 Boundary stabilization of a vibrating string\n- If`0(t) =\u0000\u0011;thenux(`(t);t) = 0 and thus\nE0\n`(t) =\u00001\n2\u0011u2\nt(`(t);t)\u00140;fort2[t1;t2]:\n- Assume that `0(t)=2f\u0000 1=\u0011;0;\u0000\u0011g;then after some computation we can rewrite\n(3.1) as\nE0\n`(t) =\u00001\n2\u0000\n`0(t)\u00112+ 2\u0011+`0(t)\u00011\u0000j`0(t)j2\n(1 +\u0011`0(t))2u2\nt(`(t);t);fort2[t1;t2]: (4.1)\nThe sign of E0\n`(t) is opposite to the sign of\nP`(\u0011) =`0(t)\u00112+ 2\u0011+`0(t):\nDue to (1.1), the polynomial P`(\u0011) has a discriminant \u0001 = 4(1 \u0000j`0(t)j2)>0 and thus\nP`(\u0011) has two real roots\n\u00111:=\u0012\n\u00001 +q\n1\u0000j`0(t)j2\u0013\n=`0(t) and\u00112:=\u0012\n\u00001\u0000q\n1\u0000j`0(t)j2\u0013\n=`0(t): (4.2)\nFigure 3. Variation of \u00111and\u00112in function of `0when\u00001<`0<0.\nBoth\u00111and\u00112have the opposite sign of `0(t);fort2[t1;t2] and in particular\n0<\u00111<1<\u00112:\nWe deduce from (4.1) that:\n{If\u00111<\u0011<\u0011 2;fort2[t1;t2];thenP`(\u0011)>0 and by consequence E0\n`(t)<0:\n{If\u0011=\u00111or\u0011=\u00112;fort2[t1;t2];thenP`(\u0011) =E0\n`(t) = 0;i.e. the energy is\nconstant.\n{If\u00112]0;\u00111[[]\u00112;1[;fort2[t1;t2];thenP`(\u0011)<0 and we have E0\n`(t)>0:\n\u000fIf the interval is independent of time, i.e. `0(t) = 0 fort2[t1;t2];thenux(`(t);t) =\n\u0000\u0011ut(`(t);t) and thus\nE0\n`(t) =\u0000\u0011u2\nt(`(t);t)\u00140;fort2[t1;t2]:\n\u000fIf the interval is expanding, i.e. 0 <`0(t)<1 fort2[t1;t2];thenP`(\u0011)>0 for\u0011>0:\nTaking into account (1.1) and (4.1), we deduce that E0\n`(t)\u00140;fort2[t1;t2]:\nRemark 4. To summarize, under the assumption (1:1);\n\r\r\r\r\r\rIf the interval is expanding, then E`(t)is nonincreasing for any \u0011>0.\nIf the interval is shrinking, then E`(t)is nonincreasing if the damping\nfactor can be taken close enough to the optimal value \u0011= 1:(4.3)S. Ghenimi and A. Sengouga 11\nLet us now estimate E`(t) is using'0and exp (\u0000lnj\r\u0011j').\nTheorem 3. Under the assumptions (1.1), (1.7) and (2.1), the solution of Problem (WP)\nsatis\fes\nZ`(t)\n0 \nelnj\r\u0011j'(t+x)\n'0(t+x)+elnj\r\u0011j'(t\u0000x)\n'0(t\u0000x)!\n\u0000\nu2\nx+u2\nt\u0001\n+ 2 \nelnj\r\u0011j'(t+x)\n'0(t+x)\u0000elnj\r\u0011j'(t\u0000x)\n'0(t\u0000x)!\nuxutdx= 4S\u0011;(4.4)\nfort\u00150, whereS\u0011:= 2P\nn2Zj!ncnj2. Moreover, it holds that\nS\u0011~m(t)\u0014E`(t)\u0014S\u0011~M(t);fort\u00150; (4.5)\nwhere\n~m(t) : = min\nx2[0;`(t)]n\n'0(t\u0000x)e\u0000lnj\r\u0011j'(t\u0000x);'0(t+x)e\u0000lnj\r\u0011j'(t+x)o\n;\n~M(t) : = max\nx2[0;`(t)]n\n'0(t\u0000x)e\u0000lnj\r\u0011j'(t\u0000x);'0(t+x)e\u0000lnj\r\u0011j'(t+x)o\n:\nProof. We argue as in the proof of Theorem 2. Noting that ~ uxis an even function of xand\nthat ~utis an odd one, then the identity (2.14) yields\nZ`(t)\n\u0000`(t)elnj\r\u0011j'(t\u0006x)(~ux\u0006~ut)2dx\n'0(t\u0006x)= 4S\u0011;fort\u00150: (4.6)\nHence, summing, we get\nZ`(t)\n\u0000`(t)elnj\r\u0011j'(t+x)(~ux+ ~ut)2dx\n'0(t+x)+Z`(t)\n\u0000`(t)elnj\r\u0011j'(t\u0000x)(~ux\u0000~ut)2dx\n'0(t\u0000x)= 8S\u0011:\nExpanding squares, we get\nZ`(t)\n\u0000`(t) \nelnj\r\u0011j'(t+x)\n'0(t+x)+elnj\r\u0011j'(t\u0000x)\n'0(t\u0000x)!\n\u0000\n~u2\nx+ ~u2\nt\u0001\n+ 2 \nelnj\r\u0011j'(t+x)\n'0(t+x)\u0000elnj\r\u0011j'(t\u0000x)\n'0(t\u0000x)!\n~ux~utdx= 8S\u0011;fort\u00150:(4.7)\nAs the function under the integral sign is even, then (4.4) follows.\nFor 0\u0014x\u0014`(t) andt\u00150;let us denote\nA(x;t) =elnj\r\u0011j'(t\u0000x)\n'0(t\u0000x)andB(x;t) =elnj\r\u0011j'(t+x)\n'0(t+x):\nThen, we can rewrite (4.4) as\nZ`(t)\n0(A(x;t) +B(x;t))\u0000\n~u2\nt+ ~u2\nx\u0001\ndx+ 2 (A(x;t)\u0000B(x;t)) ~ut~uxdx= 4S\u0011:\nUsing the algebraic inequality\n\u0000jA\u0000Bj\u0000\nu2\nt+u2\nx\u0001\n\u00142 (A\u0000B)utux\u0014jA\u0000Bj\u0000\nu2\nt+u2\nx\u0001\n;12 Boundary stabilization of a vibrating string\nwe get\nZ`(t)\n0((A+B)\u0000jA\u0000Bj)\u0000\n~u2\nt+ ~u2\nx\u0001\ndx\u00144S\u0011\u0014Z`(t)\n0((A+B) +jA\u0000Bj)\u0000\n~u2\nt+ ~u2\nx\u0001\ndx:\nThanks to (3.8), the precedent estimation yields\nZ`(t)\n0min(\nelnj\r\u0011j'(t\u0000x)\n'0(t\u0000x);elnj\r\u0011j'(t+x)\n'0(t+x))\n\u0000\n~u2\nt+ ~u2\nx\u0001\ndx\u00142S\u0011\n\u0014Z`(t)\n0max(\nelnj\r\u0011j'(t\u0000x)\n'0(t\u0000x);elnj\r\u0011j'(t+x)\n'0(t+x))\n\u0000\n~u2\nt+ ~u2\nx\u0001\ndx,\nfort\u00150:By consequence\nmin\nx2[0;`(t)](\nelnj\r\u0011j'(t\u0000x)\n'0(t\u0000x);elnj\r\u0011j'(t+x)\n'0(t+x))\nE`(t)\u0014S\u0011\n\u0014max\nx2[0;`(t)](\nelnj\r\u0011j'(t\u0000x)\n'0(t\u0000x);elnj\r\u0011j'(t+x)\n'0(t+x))\nE`(t):\nThis implies (4.5). \u0003\nSince lnj\r\u0011j \u0015 0 for\u0011\u00150;and'is nondecreasing, we have the following immediate\ncorollary.\nCorollary 3. Under the assumption of Theorem 3, it holds that\nS\u0011m(t)e\u0000lnj\r\u0011j'(t+`(t))\u0014E`(t)\u0014S\u0011M(t)e\u0000lnj\r\u0011j'(t\u0000`(t));fort\u00150; (4.8)\nm(t)andM(t)are given by (1.9) .\nIf'0is monotone, then (4.8) can be replaced by more explicit estimation.\nCorollary 4. Under the assumption of Theorem 3, assume that 'satis\fes (3.9) on [t0;t1],\nthen:\n\u000fIf\u00001<`0(t)\u00140 on [t0;t1], then'0is nondecreasing and E`(t) satis\fes\nS\u0011'0(t\u0000`(t))e\u0000lnj\r\u0011j'(t+`(t))\u0014E`(t)\u0014S\u0011'0(t+`(t))e\u0000lnj\r\u0011j'(t\u0000`(t));fort2[t0;t1]:(4.9)\n\u000fIf 0<`0(t)<1 on [t0;t1], then'0andE`(t) are nonincreasing and\nS\u0011'0(t+`(t))e\u0000lnj\r\u0011j'(t+`(t))\u0014E`(t)\u0014S\u0011'0(t\u0000`(t))e\u0000lnj\r\u0011j'(t\u0000`(t));fort2[t0;t1]:(4.10)\nProof. It su\u000eces to argue as in the proof of Corollary 1. \u0003\nRemark 5. If the interval is shrinking, there is competition between the nondecreasing '0and\ne\u0000lnj\r\u0011j'in estimation (4.9). The behaviour of E`(t)depends on the value of the damping \u0011\nas stated in Remark 4.\nTo compare E`(t) with the energy E`(t0) for 0\u0014t0<t;we have the following result.\nCorollary 5. Under the assumption of Theorem 3, we have\nm(t)e\u0000lnj\r\u0011j'(t+`(t))\nM(t0)e\u0000lnj\r\u0011j'(t0\u0000`(t0))E`(t0)\u0014E`(t)\n\u0014M(t)e\u0000lnj\r\u0011j'(t\u0000`(t))\nm(t0)e\u0000lnj\r\u0011j'(t0+`(t0))E`(t0);for0\u0014t0<t: (4.11)S. Ghenimi and A. Sengouga 13\nMoreover, if '0satis\fes (3.9) on [t0;t1], then:\n\u000fIf\u00001<`0(t)\u00140 on [t0;t1], then'0is nondecreasing and E`(t) satis\fes\n'0(t\u0000`(t))e\u0000lnj\r\u0011j'(t+`(t))\n'0(t0+`(t0))e\u0000lnj\r\u0011j'(t0\u0000`(t0))E`(t0)\u0014E`(t)\n\u0014'0(t+`(t))e\u0000lnj\r\u0011j'(t\u0000`(t))\n'0(t0\u0000`(t0))e\u0000lnj\r\u0011j'(t0+`(t0))E`(t0);fort2[t0;t1]:(4.12)\n\u000fIf 0<`0(t)<1 on [t0;t1], then'0andE`(t) are nonincreasing and\n'0(t+`(t))e\u0000lnj\r\u0011j'(t+`(t))\n'0(t0\u0000`(t0))e\u0000lnj\r\u0011j'(t0\u0000`(t0))E`(t0)\u0014E`(t)\n\u0014'0(t\u0000`(t))e\u0000lnj\r\u0011j'(t\u0000`(t))\n'0(t0+`(t0))e\u0000lnj\r\u0011j'(t0+`(t0))E`(t0);fort2[t0;t1]:(4.13)\nProof. Since (4.5) holds also for t=t0, then the corollary follows by combining the inequalities\nelnj\r\u0011j'(t\u0000`(t))\nM(t)E`(t)\u0014S\u0011\u0014elnj\r\u0011j'(t0+`(t0))\nm(t0)E`(t0)\nand\nelnj\r\u0011j'(t0\u0000`(t0))\nM(t0)E`(t0)\u0014S\u0011\u0014elnj\r\u0011j'(t+`(t))\nm(t)E`(t):\n\u0003\nAcknowledgements\nThe authors have been supported by the General Direction of Scienti\fc Research and Tech-\nnological Development (Algerian Ministry of Higher Education and Scienti\fc Research) PRFU\n# C00L03UN280120220010.\nORCID. Abdelmouhcene Sengouga https://orcid.org/0000-0003-3183-7973.\nReferences\n[1] K. Ammari, A. Bchatnia, and K. El Mufti. Stabilization of the wave equation with moving boundary. Eur.\nJ. Control , 39:35{38, 2018. ISSN 0947-3580.\n[2] C. Bardos and G. Chen. Control and stabilization for the wave equation. III: Domain with moving boundary.\nSIAM J. Control Optim. , 19:123{138, 1981.\n[3] M. Cherkaoui. 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Exact controllability for a string equation in domains with moving boundary in\none dimension. Electron. J. Di\u000b. Equations , 2015(98):1{7, 2015.\n[15] K. Veseli\u0013 c. On linear vibrational systems with one dimensional damping. Appl. Anal. , 29(1-2):1{18, 1988.\n[16] A. I. Vesnitskii. The inverse problem for a one-dimensional resonator the dimensions of which vary with\ntime. Radiophysics and Quantum Electronics , 14(10):1209{1215, 1971.\n[17] A. I. Vesnitskii and A. I. Potapov. Some general properties of wave processes in one-dimensional mechanical\nsystems of variable length. Sov. App. Mechanics , 11(4):422{426, 1975. ISSN 1573-8582.\nEmail address :amsengouga@gmail.com,abdelmouhcene.sengouga@univ-msila.dz\n(Seyf Eddine Ghenimi, Abdelmouhcene Sengouga) Laboratory of Functional Analysis and Geom-\netry of Spaces, Department of mathematics, Faculty of Mathematics and Computer Sciences,\nUniversity of M'sila, 28000 M'sila, Algeria."    },    {        "title": "2301.12797v1.Investigation_of_Ultrafast_Demagnetization_and_Gilbert_Damping_and_their_Correlation_in_Different_Ferromagnetic_Thin_Films_Grown_Under_Identical_Conditions.pdf",        "content": "1  Investigation  of Ultrafast  Demagnetization  and Gilbert  Damping  and their  Correlation  in Different  \nFerromagnetic  Thin  Films  Grown  Under  Identical  Conditions1 \nSuchetana Mukhopadhyay, Sudip Majumder, Surya Narayan Panda, Anjan Barman*  \nDepartment  of Condensed  Matter  and Materials  Physics,  S.N. Bose  National  Center  for Basic  \nSciences,  Block -JD, Sector  III, Salt Lake,  Kolkata  700106,  India  \nEmail:  abarman@bose.res.in  \n \n \nAbstract  \nFollowing the demonstration of laser -induc ed ultrafast demagnetization in ferromagnetic nickel,  several  \ntheoretical  and phenomenological  propositions  have sought  to uncover  its underlying  physics.  In this work \nwe revisit the three temperature model (3TM) and the microscopic three  temperature  model  (M3TM)  to \nperform  a comparative  analysis  of ultrafast  demagnetization  in 20-nm-thick  cobalt,  nickel  and permalloy  \nthin films  measured  using  an all-optical  pump - probe  technique.  In addition  to the ultrafast  dynamics  at \nthe femtosecond  timescales,  the nano second  magnetization  precession  and damping  are recorded  at various  \npump  excitation  fluences  revealing  a fluence -dependent  enhancement  in both the demagnetization  times  \nand the damping  factors.  We confirm  that the Curie  temperature  to magnetic  moment  ratio of a given \nsystem acts as a figure of merit for the demagnetization time, while the demagnetization  times  and damping  \nfactors  show  an apparent  sensitivity  to the density  of states  at the Fermi  level for a given system.  \nFurther, from numerical simulations of the ultrafast demagnetization  based on both the 3TM and the M3TM, \nwe extract the reservoir coupling parameters that best  reproduce  the experimental  data and estimate  the \nvalue  of the spin flip scattering  probability  for each system. We discuss how the f luence -dependence of \ninter-reservoir coupling parameters  so extracted  may reflect  a role played  by nonthermal  electrons  in the \nmagnetization  dynamics  at low laser  fluences.  \n \n1. Introduction  \nOptical  excitation  of a magnetic  material  with a short  and intense  laser pulse  sets in motion  a chain  \nof microscopic  events  that result  in a macroscopic,  measurable  quenching  of the net magnetization.  \nSince  the processing  speed  of magnetic  storage  devices  is limited  by the maximum  speeds  at which  the \nmagnetization  can be manipulated,  the possibility  of controlling  magnetization  at sub-picosecond  timescales  \nby employing  ultrashort  laser pulses  holds  tremendous  potential  for applications  in spin-based  memory  and \nstorage  devices  with ultrafast  processing  speeds  including  laser -induced  opto-magnetism  and all-optical  \nswitching  [1–4] as well as THz spintronic  devices  [5–7]. Meanwhile,  the relaxation  timescales  of the laser -\ninduced  magnetization  precession  in ferromagnetic  thin films  associated  with the magnetic  damping  set \nfundamen tal limits  on magnetization  switching  and data transfer  rates  in spintronic  devices.     \nPicosecond  laser -induced  magnetization  quenching  in ferromagnetic  gadolinium  was reported  by \nVaterlaus  et al. in 1991,  who also reported  for the first time a character istic timescale  of 100±80  ps \n[8] associated  with the magnetization  loss. In this early  work,  the timescale  was identified  with the \ntimescale  of the spin-lattice  interactions  mediating  the disruption of magnetic ordering due to laser \nheating of the lattice,  though later works  contradicted  this interpretation  [9, 10]. In fact, by the mid-\n1990s,  it had begun  to be recognized that finer time resolution was necessary to probe this phenomenon \nto uncover a  fuller  picture  of the associated  relaxation  processes  occu rring  at sub-picosecond  timescales.  \nIn 1996,  Beaurepaire  et al. demonstrated  in a seminal  work  that faster  demagnetization  occurring at sub -\npicosecond timescales could be triggered by femtosecond laser pulses in a  nickel thin film [11].  The \nultrafast demag netization phenomenon went on to be demonstrated  in a wide  array  of ferromagnetic  \nsystems  [12–19], triggering  a flurry  of research  that continues  till date. A few years after the pioneering \nexperiments, Koopmans et al. characterized for the  first time the full magneto -optical response to \nfemtosecond laser pulses in a ferromagnet by  time-resolved measurements of Kerr ellipticity and \nrotation [13].  However, the observation of  nonmagnetic contributions to the Kerr rotation signal \nnaturally led some to question  whether  the ultrafast quenching in response to the laser excitation was \nindeed a magnetic phenomenon  [13, 14]. In 2003, Rhie et al.  used time -resolved photoemission  \n \n1First  submitted  in March  2022  2  spectroscopy to probe the collapse of the 3d exchange s plitting in nickel as an unambiguous signature of a \nphotoinduced demagnetization occurring over 300±70 fs [15]. This was soon followed by the first reports \nof an estimate of the characteristic timescale of femtosecond laser -induced demagnetization derived from \nquantum -mechanical principles [16].  \nOne of the major reasons behind this sustained interest is the need to achieve a complete understanding of \nthe microscopic mechanisms that underlie the ultrafast demagnetization phenomenon that have so far \nremained elusive. Much inquiry has been focused on how the laser -induced loss of magnetic order is \ncompensated for via the transfer of angular momentum of the spin system at the associated timescales. \nOver the years, several mechanisms have been proposed for explai ning the angular momentum \nconservation associated with the ultrafast demagnetization, which may be broadly categorized in two \ndistinct classes. One of these argues in favor of a dominant contribution to the demagnetization arising \nfrom nonlocal transport p rocesses driven by the laser pulse, such as superdiffusive transport of spin -\npolarized hot electrons [20, 21] or heat currents [22]. The other narrative relies on local spin -flip scattering \nprocesses occurring by the collision of excited electrons with imp urities, phonons, and magnons [17, 23, \n24]. In this category, electron -lattice and electron -spin relaxation mechanisms are accepted as the major \ndemagnetization channels in the picoseconds following the laser excitation [25], with several works \nattributing  a pivotal role to the material -specific spin -orbit interaction [23, 25 –27]. On the other hand, it is \npossible to model the demagnetization by adopting a thermal description, considering the laser excitation \nas a “heat” source that excites electrons close to the Fermi level instantaneously to very high energies, \nwhose eventual relaxation processes drive the magnetization loss. The laser excitation energy controlled \nby the applied laser pump fluence has a direct influence on the maximum electron temperatures  reached \nand thus plays a pivotal role in the ultrafast magnetization dynamics that follow as a result.  \nIn this work, employing an all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) technique, \nwe investigate the ultrafast spin dynamics along w ith the nanosecond magnetization precession and \ndamping at different laser pump fluences in three ferromagnetic thin films: 20 -nm-thick cobalt, nickel and \npermalloy grown under uniform deposition conditions. Cobalt and nickel are elementary 3d ferromagneti c \ntransition metals well studied in the literature in both their elementary forms and as constituting elements \nin multilayered structures where their strong spin -orbit coupling mediates interfacial effects such as spin \npumping. Permalloy, an alloy comprise d of approximately 80% nickel and 20% iron, is a prototype \nmaterial for spintronic applications due to several desirable qualities such as low coercivity,   large  \npermeability  and  in  particular,   low  Gilbert  damping,   which  aids  in minimizing the  maximum power \nconsumption of devices and allows spin wave propagation on length scales of the order of device size. The \nsimultaneous investigation of the fluence -dependent modulation of ultrafast demagnetization and damping \nin all the above ferromagnetic thin films is motivated by the primary objectives of exploring the dominant \nmicroscopic processes underlying the demagnetization in these systems, correlating the observed \nmagnetization dynamics to the material properties of each system, and in the process  exploring their \ntunability with laser pump fluence. We set out to achieve this in two ways: (a) by modelling the ultrafast \ndemagnetization at various pump fluences using two theoretical models sharing a common link and (b) by \ncorrelating the laser -induced  changes in ultrafast demagnetization to the changes in the Gilbert damping \nfactor which characterizes the magnetization precession. TR -MOKE is a well established local  and  non -\ninvasive all -optical measurement technique tunable to an extremely fine time resolution limited only by \nthe laser pulse width and is thus well suited to our purpose.  \nFor the detailed analysis of ultrafast demagnetization in our samples, we extract the values of \ndemagnetization time and fast relaxation time for each sample using a p henomenological fitting function \nand record its systematic variation with the pump fluence. We subsequently model the demagnetization \ndata using two well known theoretical models: the phenomenological three -temperature model (3TM) and \nthe microscopic three  temperature model (M3TM), and thereby extract values for the microscopic \nparameters relevant for the demagnetization process and calculate the temporal evolution of the simulated \ntemperatures of the electron, spin and lattice systems within the first few picoseconds of the laser \nexcitation. Both these models assume a thermal picture to explain the initial electron temperature rise and \nsubsequent relaxation but differ significantly in their approach with regards to the treatment of the \nmagnetization. A syst ematic study implementing both models to analyze TR -MOKE data recorded under \nuniform experimental conditions on identically prepared samples is absent in the literature and will prove \ninstructive. Further, a thorough investigation and characterization of t he ultrafast demagnetization, \nmagnetic damping and their intercorrelation in three different systems deposited under identical deposition \nconditions has not been carried out before. For the purposes of a comparative analysis, it is  vital to study \nsamples g rown under the same deposition conditions.  The conductivity of the  substrate  can also directly  \ninfluence  demagnetization  time by promoting  or hindering  ultrafast spin transport processes [20, 28] while \nthe deposition technique determines the structural  properties of the samples that may indirectly affect the 3  demagnetization time.  Moreover,  since the experimentally obtained demagnetization timescale has been \nshown to be quite  sensitive to extrinsic factors such as the laser pulse duration and the spectral ba ndwidth \n[29],  it is useful to measure the demagnetization times for various thin films from experiments  performed  \nin the same  experimental  setup  under  near-identical  experimental  conditions.  \n \n2. Materials  and Methods  \n2.1 Sample  fabrication  \n20 nm -thick cobalt,  nickel and permalloy thin films were deposited by electron beam  evaporation \nunder an average base pressure of 1 ×10−7 Torr and at a very low deposition  rate of 0.2 Å/s chosen to \nachieve uniform deposition.  Each film was deposited on an  insulating  8 mm × 8 mm silicon  substrate  \ncoated  with  285-nm-thick  SiO 2. Subsequently, the films were capped in-situ with a 5 -nm-thick \nprotective gold layer (base  pressure  ~1 × 10−7 Torr,  deposition  rate 0.1 Å/s) to prevent  surface  \noxidation  of the ferromagnetic layer and protect it from possible degradation due to high power laser  \nexposure during the optical pump -probe measurements [ 30, 31]. The thickness of the capping  layer  was \nkept more  than three  times  smaller  than the optical  penetration  depth  of 400 nm light in gold.  After  \nthe deposition  of the capping  layer,  the surface  topography  of the samples  was investigated  by atomic  \nforce  microscopy  (AFM)  using  a Tap190Al -G tapping  mode  AFM  probe as shown in Figure 1 ( a). The \naverage surface roughness  Ra of the cobalt,  nickel and  permalloy  films  were  obtained  as 0.91 nm, 0.36 nm \nand 0.41 nm respectively.  Thus,  reasonably  low surface  roughn ess in the sub-nm range  was obtained  \nwhich  is comparable  for all the samples.  In addition,  the static  magneto -optical  Kerr effect  was used to \nstudy  the magnetic  hysteresis  of the deposited  samples  to confirm  their ferromagnetic  nature  (Figure  \n1(b)). \n \n 2.2 All-optical  measurement  of ultrafast  spin dynamics  \nMeasurement of laser -induced magnetization dynamics is carried out using a TR -MOKE  technique \nin two -color optical pump -probe arrangement using a 400 nm pump beam and 800  nm probe beam \nhaving a pump -probe cr oss-correlation width of ~100 fs [30] shown  schematically in Figure 1( c). A \ntypical TR -MOKE trace is shown in Figure 1( d), comprising  ultrafast  demagnetization  (Regime  I), fast \nremagnetization  (Regime  II) and damped  magnetization precession (Regime III).Th is technique enables \nthe direct observation of the  spin dynamics in the femto - and picosecond time domain. Femtosecond \npulses are generated  by an amplified laser system (Libra, Coherent Inc.)  employing a chirped -pulse \nregenerative  amplifier and a Ti:sapphi re laser oscillator (Coherent Inc.)  pumped by a neodymium -doped  \nyttrium lithium fluoride (Nd -YLF) laser.  The second harmonic of the amplified laser output  \n(wavelength = 400 nm, repetition rate = 1 kHz, pulse width >40 fs), generated through a  lithium  \ntribo rate (LBO)  nonlinear  crystal,  is used  for laser  excitation  of the ferromagnetic  thin films.  The \ntime-delayed fundamental beam (wavelength = 800 nm, repetition rate = 1 kHz, pulse width ~40 fs) is \nused to probe the ensuing magnetization dynamics.  In our  setup, different wavelengths are employed \nfor the pump and the probe pulse to eliminate the  possibility of state blocking effects arising from the \nuse of identical wavelengths for pumping  and probing [32]. A computer -controlled variable delay \ngenerator offers  precise control of the  delay time between pump and probe. Before commencing \nmeasurements on any sample, the  zero delay was carefully estimated by maximizing the transient \nreflectivity signal of a bare  silicon  substrate  placed  adjacent  to the sample  on the same  sample  holder.  \nTR-MOKE  experiments are performed with a non -collinear pump -probe geometry.  The pump beam,  \nfocused to a spot size of ~300 µm, is incident obliquely on the sample while the probe  beam, with a \nspot size of ~100 µm, is incident normal to the sample surface and aligned  to the center of the pump \nspot. The pump -probe spatial overlap on the sample was carefully  maintained. The choice of a \nrelatively smaller spot size of the probe beam as compared to the  pump beam facilitates optical \nalignment and ensures that the probe beam detects the local  magnetization changes from a part of the \nsample uniformly irradiated by the pump.  Before  reflection on the samples, the probe beam is polarized \northogonally to the linearly polarized  pump beam. After reflec tion, the Kerr -rotated probe beam is split \nand one part is fed directly  into a Si -photodiode to measure the time -resolved reflectivity signal.  The \nother part is fed  into an identical photodiode after passing through a Glan -Thompson polarizer adjusted \nto a small  angle  from  extinction  to minimize  optical  artifacts  in the Kerr  rotation  signal.  In this way,  \nsimultaneous  measurement  of the time-resolved  total reflectivity  and Kerr  rotation  signals is possible.  \nAn optical chopper operating at 373 Hz placed in th e path of the pump  beam provides a reference signal \nfor the digital signal processing lock -in amplifiers (Stanford  Research  Systems,  SR830)  which  record  \nthe modulated  signal  in a phase  sensitive  manner.  All experiments  were  carried  out under  ambient  \ncondit ions of temperature  and pressure.  4  3. Results  and Discussion  \n 3.1 Theoretical  models  for ultrafast  demagnetization  \nThe phenomenon  of optically  induced  ultrafast  demagnetization  starts  with the irradiation  of the \nmagnetic  sample  with  a brief  and intense  optical  laser  pulse,  exciting  electrons  momentarily a few \nelectron volts above the Fermi level.  Though the exact sequence of events  following  the initial  excitation  \nis difficult  to trace  due to the highly  nonequilibrium  conditions  created  by it, a qualitative  overview  of \nthe complete  demagnetization  process  is fairly  well   established.  The laser  excitation  generates  a \nnonthermal  pool of excited  electrons  which  thermalize  rapidly  within  several  femtoseconds  via \nelectron -electron  interactions.  Spin -dependent  scatter ing events  taking  place  during  this transient  regime  \nlead to a sharp  drop  in magnetization  observable  around  a few hundred  femtoseconds  in the \nexperimental  Kerr  rotation  signal.  Subsequently,  the thermalized  electrons  may release  their  excess  \nenergy  via a variety  of relaxation  channels,  such  as by excitation  of phonons  or magnons.  This results  in a \npartial  recovery  of the magnetization  beyond  which  heat dissipation  into the environment  promotes  further  \nrecovery  on a longer  timescale.  The 3TM  posits  that the thermodynamics  of the demagnetization  \nphenomenon  can be described  simply  by considering  energy  exchange  between  three  thermal  reservoirs  \n[33],  each of which  is assigned  a temperature:  the electrons  at temperature  Te, the lattice  at temperature  \nTl and the electronic  spin reservoir  at temperature  Ts. Since  the reservoirs  are in thermal  contact  and \nthe overall  process  is adiabatic,  equilibration  of the excited  electrons  with the spin and lattice  reservoirs  \nvia energy  transfer  may be described  by coupled  rate equations  in the following  manner:   \n \n \n \n   (1) \n \n \n \nwhere,  Ce, Cl and Cs denote the specific heats of the electron,  lattice and spin reservoirs  respectively,  \nwhile  Gel, Ges, and Gsl denote  the inter -reservoir  coupling  parameters.  The term  P (t) describes  the action  \nof the laser  pulse  as a source  term driving  the excitation  of the electron  reservoir  to high temperatures.  The \nthermal  diffusion  term  \n  describes  heat dissipation  occurring  via thermal  conduction  \nalong  the sample  thicknes s. Under  this description,  the observed  demagnetization  is attributed  to a \nrise in the spin temperature  Ts occurring  shortly  after  the electron  temperature  rise. The coupling  \nstrengths  between  the electron -spin and electron -lattice  subsystems  qualitatively  determine  the efficiency  \nof energy  transfer  between  them  and hence  influence  the timescales  associated  with the demagnetization \nand fast relaxation.  However, the 3TM is purely phenomenological and does  not explicitly consider any \nmicroscopic mechanisms un derlying the phenomenon it describes.  On the other  hand,  the M3TM  \nproposed  by Koopmans  et al. [23] provides  a “spin -projected”  perspective  [27] to explain  the ultrafast  \ntransfer  of angular  momentum  highlighting  the role of Elliot -Yafet  type ultrafast  spin-flip scattering  in \nthe demagnetization  process.  The initial  excitation  by the laser  pulse  disturbs  the electronic  subsystem  \nfrom  equilibrium  which  leads  to an imbalance  in spin-up and spin-down  scattering  rates,  resulting  in the \nobserved  loss of magnetic  order.  The process  is mediated  by spin-orbit  interactions  leading  to the \nformation  of hot spots  in the band  structure  where  spin-up and spin-down  channels  are intermixed.  An \nelectron  scattered  into these  hot spots  via a phonon - or impurity -mediated  scatteri ng will flip its spin with  \na finite  probability.  The individual  scattering  events  are characterized  by a parameter asf across  the  \nsample,  identified  with the probability  of spin-flip due to  electron -phonon  scattering.  The magnitude  \nof this parameter  will directly  depend  on the extent of  spin-orbit  coupling and  hence  is expected  to be \ncomparable  in materials with  similar  spin-orbit  coupling  strengths.  The M3TM  retains  the coupled  rate \nequations  for the electron  and lattice  temperatures,  similar  to the thermal description  provided  by the \n3TM.  However  the fundamental  difference  from  the 3TM  is that in the framework  of the M3TM,  the \nspin bath is formed  by a collection  of two-level  systems  obeying  Boltzmann  statistics.  Instead  of \nassigning  a temperature  to the spin bath,  the normalized  magnetization  is directly  calculated  from  the \nassociated  exchange  splitting.  The rate of change  of the magnetization,  derived  analytically \nconsidering an Elliot -Yafet scattering -driven demagnetization, is parametrized by  asf and c oupled to the \nelectron and the lattice subsystem temperatures. The assignment of a  characteristic temperature to the spin \nsubsystem is replaced in the M3TM by an evolution  equation  for the magnetization:   \n 5  \n        (2) \n \nThe quantity  R is a mate rial-specific  factor  which  influences  the demagnetization  rate and is \nproportional  to asfTc2/μat where  TC is the material Curie temperature and µat is the atomic magnetic \nmoment. Though the two models  differ in their approaches, one can immediately discern certain \nsimilarities in their domains of  validity.  Both models are appropriate only when th e nonlocal \nmechanisms driving ultrafast  demagnetization such as superdiffusive spin transport can be neglected. \nWe note here that it  has been  reported  that spin transport  is not a major  contributor  to the ultrafast  \ndemagnetization in transition metals [34] . We nevertheless use an insulating SiO 2-coated Si  substrate \nfor our samples to minimize spin transport effects such that analysis with the local  models described \nabove should suffice in our case. Any additional contributions arising from  the gold capping  layer  \nwould  be uniform  across  all samples  investigated  and therefore  unlikely  to impact the main results of our \ncomparative study.  Moreover, since the thickness of the  capping layer is much smaller than the \npenetration depth of both 400 nm and 800 nm light  in gold [35], the pump excitation fully penetrates \ndown to the magnetic layer ensuring that the  effect of direct laser excitation of the ferromagnet is \nprobed in our case.  Thus, we set up  numerical calculations based on the models described above in \norder  to extract microscopic  information  from  the experimentally  obtained  demagnetization  traces.  \n 3.2 Ultrafast  demagnetization  in cobalt,  nickel,  and permalloy  thin films  \nWe proceed by performing time -resolved measurements of the polar magneto -optical Kerr  effect in \nthe cobalt, nickel and permalloy thin film samples as a function of the laser fluence.  Measurements are \ncarried out under a strong enough external magnetic field kept constant at  around 2 kOe tilted at a small \nangle from the sample plane to saturate  the magnetization of  the samples. The pump fluence is varied \nbetween 0.8 -8.7 mJ/cm2 by varying the power of the  pump pulse.   The results are presented in Figure 2.   \nTo ascertain that the measured Kerr  signal  reflects  the true magnetization  dynamics  without any \nspurious  contribution  from  optical effects triggered in the initial stages of laser excitation, we also \nexamine the transient  reflectivity signal for each sample. The fluence dependent variation in the \nreflectivity can be  found in Figure S1 of the  Supplementary Materials, demonstrating that at any given \nfluence  the amplitude  of the reflectivity  signal  is negligibly  small  compared  to that of the  Kerr  rotation.  \nWe nevertheless restrict ourselves to the low fluence regime to avoid nonlinear  effects a nd sample \ndamage.  For all our experiments, the probe fluence is kept constant at a  value about half that of the \nlowest pump fluence used to prevent additional contribution to  the spin dynamics by probe excitation.  \nAs seen in Figure 2, the ultrafast demagne tization  completes within 1 ps for all three samples \nconsidered which is followed by a fast recovery of  the magnetization,  all observed  within  the \nexperimental  time window  of 4 ps. These  experimental traces clearly exhibit the “Type -I” or “one -step” \ndemagn etization expected for  transition metal thin films at room temperature and under low -to-\nmoderate pump fluence  [23]. The amplitude of the maximum quenching of the Kerr rotation signal \nincreases with the  laser  fluence,  allowing  us to rule out nonlinear  effec ts [36].  Closer  inspection  of the \ntraces  also reveals an increase of the time taken to demagnetize the samples with increasing fluence for  \nall three  samples.  To quantify  this increase,  we fit our demagnetization  traces  to a phenomenological  \nexpression  based on the 3TM  and valid  in the low laser  fluence  regime  [37]:   \n \n(3) \n \nwhere  Θ(t) is the Heaviside  step function,  δ(t) is the Dirac  delta  function  and Γ(t) is the \nGaussian laser pulse.  The constant A1 represents the value of the normalized  magnetization after \nremagnetization has completed and equilibrium between the electron,  spin and lattice  reservoirs  has \nbeen  re-established.  A2 is proportional  to the initial  rise in electron  temperature  and hence  to the \nmaximum  magnetization  quenching.  A3 represents  the magnitude of state -filling effects present during \nthe onset of the demagnetiz ation response,  which is negligible in our case.  τM and τE are the \ndemagnetization time and fast relaxation  time,  respectively.  Prior  to the fitting,  all the experimental  \ntraces  were  normalized  by hysteresis measurements of the Kerr rotation signal under the saturating \nmagnetic field in the  absence of la ser excitation.  We find that within the range of fluence values \nconsidered,  permalloy  exhibits  the largest  magnetization  quenching  of 54.6%,  followed  by a 23.7%  \nquenching achieved in nickel, while the magnetization of cobalt the least, only about 8% for  the largest  \napplied  fluence.  The demagnetization  occurs  at a characteristic  timescale  of 230-280 fs for cobalt, 160 -6  210 fs for nickel, and 220 -250 fs for permalloy, increasing with the  laser fluence.  This effect can be \nattributed to enhanced spin -fluctuation s at elevated spin  temperatures  for  higher  fluences  [38].      \nAt  a  fluence  of  4.8  mJ/cm2,   the  extracted  demagnetization  times  are 276.6  ± 3.41 fs for cobalt,  \n187.3  ± 2.89 fs for nickel  and 236.8  ± 2.45 fs for permalloy.  The timescale  for the magnetization  \nrecovery  τE also increases  with increasing pump fluence. The variation of these characteristic \ntimescales with laser fluence is  shown  in Figure  3. These  fluence -dependent  trends  in τM and τE hint at a \nspin-flip process -dominated ultrafast demagnetization in our stu died systems [23, 39, 40]. The values  of \nτM extracted  from  our experiments  lie within  the typical  range  of 100-300 fs consistent  with \nprevious reports of the ultrafast demagnetization times in these metals [17, 23], and are  too large  to \nrepresent  a superdi ffusive  transport -driven  demagnetization  [41].  \n For the 3TM and M3TM simulations, we choose a laser pump term given by \nproportional to  the pump fluence F and following a Gaussian temporal profile.  \nThe maximum rise of the  electron temperature a nd thus also the extent of demagnetization depends \nsensitively on this  term which is hence adjusted to reproduce the maximum quenching observed \nexperimentally.  We use a pulse width τp = 100 fs determined by the pump -probe cross -correlation in all  \ncalculations.  Intrinsic  to both the models  we consider  is the assumption  that electron  thermalization occurs \nextremely fast.  The thermal  diffusion term  can be neglected in our case since th e thicknesses of the films \nwe study are kept slightly greater than the optical penetration depth of 400 nm pump beam in  those films.  \nThis ensures uniform heating of the films in the vertical direction while also  avoiding laser penetration \ninto the substrat e in which case heat dissipation into the substrate  would have to be taken into account. \nBesides, the timescales associated with heat dissipation  are generally  tens to hundreds  of picoseconds,  \nmuch  longer  than the demagnetization  and fast relaxation  times,  and hence  unlikely  to significantly  \ninfluence  our observations  at these  timescales.  Since both models we consider are thermal in their \napproach, choosing correct  values for the reservoir specific heats is vital for a proper simulation of the \ndemagnetizati on. For the electronic  specific  heat Ce, we assume  a linear  dependence  on the electronic  \ntemperature Ce(Te) = γTe  derived from the Sommerfeld free -electron approximation where γ is determined \nby the electronic density of states at the Fermi level [42].  The value of γ for permalloy is approximated as \na weighted average of the individual γ values of nickel and iron  in permalloy. The lattice specific heat Cl is \ncalculated at each value of the lattice temperature  according to the following relation derived from Debye \ntheory:  \n(4) \n \nwhere NA is the  Avogadro’s  number,  kB is the Boltzmann’s  constant  and θD  is  the  Debye  temperature.  \nFinally,  we fix the spin specific  heat Cs to its value  at room  temperature  for the 3TM  calculations,  \nobtained  by subtracting  the electronic  and lattice  contributions  from  the experimental values of the total \nspecific heat found in the literature [43].  Considering a  spin-temperature -dependent form of Cs was not \nfound to significantly affect our conclusions  as described in Section IV of the Supplementary Materials.  \nThe fixed parameter set used in  our calculations have been listed in Table 1. To relate the experimental \ndemagnetization to  the temperature  of the spin subsystem  under  the 3TM  framework,  the spin \ntemperature  Ts is  mapped to the magnetization of the system via the Weiss’ mean field theory [44], \nwhich is  then fitted with the experimental magnetization traces to obtain the empirical inter -reservoir  \ncoupling parameters Gel and Ges consistent with the observed dynamics.  We neglect the  spin-lattice \ncoupling parameter Gsl for the 3TM simulations since in ferromagnetic transitio n metals the energy \ntransfer between electrons and lattice is far greater than that between  lattice  and spins  [45].  \n \n Table 1 . Fixed parameter set used in the calculations. Literature values have been used for all parameters listed   \n[30, 31, 34, 35].  \n \n For the 3TM simulations, we proceeded to extract Gel and Ges by first fitting the  demagnetization \ndata at the lowe st fluence to the model. However, fitting the higher fluence  data using identical values \nof the coupling parameters as extracted at the lowest fluence did  not result in a good match to our \nexperimental results.  The coupling parameters extracted  from the lo w fluence data led to an \noverestimation of the demagnetization time at the higher  fluences. It was seen that a 5 -10% increase in Sampl e Tc (K) θD (K) γ (Jm-3K-2) Cs (Jm-3K-1) at (B) \nNi 627  450  1065  3.07 × 105 0.62  \nCo 1388  445  714  1.59 × 105 1.72  \nPy 860  454  992  2.67 × 105 1.00  7  Ges from its value at its adjacent lower fluence  value rectifies the overestimation of the demagnetization \ntime.  On the other h and, the remagnetization dynamics is most sensitive to the Gel parameter so that the \noverall dynamics  is best reproduced  only  by adjusting  both  Gel and Ges. As shown  in Figure  4, the \nresulting  fit shows excellent agreement with the experimental data.  This exercise reveals the crucial role  \nplayed by the electron -spin relaxation channels in determining the timescale associated with  the initial \ndemagnetization while the magnetization recovery is primarily mediated by the  electron -lattice \ninteraction.  We also f ind that the mismatch between model and experiment  can be resolved by \nconsidering an increasing trend of Gel and Ges with pump fluence arising  from a faster demagnetization \nprocess for the same percentage quenching as compared to the  model predictions with in the studied \nfluence range. The values of the microscopic parameters  extracted from the least -squares fits with their \ncorresponding error bounds can be found in  Supplementary Tables S1 -S3. Since the exact values of the \ncoupling parameters extracted  from  the fits naturally  depend  on the values  chosen  for the fixed  \nparameters,  the interpretation  of the results  from  these  fits is best limited  to a comparative  one. \nFor the M3TM  simulations,  the demagnetization  traces  are fitted  directly  to Equation  2 yielding  \nGel and asf as fit parameters.  In this case,  asf plays  a role  in  determining  the  maximum extent and the \nassociated timescale of the demagnetization via the scaling factor R while Gel continues to influence \nmainly the magnetization recovery process.  How ever, the  demagnetization  time is less sensitive  to \nchanges  in asf than it is to Ges in the 3TM  case.  This results in somewhat higher values of Gel and a \nsharper rise with pump fluence than those  extracted  from  the 3TM  simulations,  in order  to compensate  \nfor the overestimation  of demagnetization time that results from the model if Gel  at the lowest fluence is \nused for all  the fits.  We have obtained an asf of ~0.02 for cobalt,  ~0.05-0.06 for nickel, and  ~0.03-0.06 \nfor permalloy.  The value of asf we have ext racted for nickel is an order of  magnitude lower than the value \nasf = 0.185 first reported by Koopmans et al. [23] but quite  close to the value of 0.08 reported by Roth et \nal. [39].  This discrepancy is expected, as the  artificially high value of 0.185 aros e due to an \noverestimation of the electronic specific heat in  the original  work,  avoided  here by considering  \nexperimentally  determined  γ values  reported  in the literature.  The observation  that asf [Co] <asf [Ni] is \nconsistent  with previous  reports  [23, 40]. \nWith  regard  to thermal  treatments  of ultrafast  demagnetization  like the 3TM  and M3TM,  there is \nsome contention centered on the role o f the nonthermal electrons generated within the  first 50-100 \nfemtoseconds  after laser excitation  [9, 48, 49]. Within  this time frame,  the excited  hot electrons  have not \nyet thermalized  and their energies  cannot  be described  by a Fermi -Dirac  distribution.  However,  both \nthe 3TM  and the M3TM  completely  neglect  the finite  electron  thermalization  time and consider  only the \nenergy  exchange  between  the thermalized  electron  population  with the lattice  and/or  the spin subsystem  as the \ninitial  nonequilibrium  distribu tion is assumed  to last for much  shorter  durations  (≳ 10 fs) than the \ntimescale  over which  demagnetization  typically  occurs  (≳ 100 fs). However,  strictly  speaking,  this \nassumption  is valid  only when  the thermalization  proceeds  very rapidly  such as can be ensured with  high \nfluence  excitations  [50].  In the low  fluence  regime,  electron thermalization  is slower,  hence  the influence  \nof nonthermal  electrons  on the measured  dynamics  may become  more  important  with decreasing  pump  \nfluence.  The increasing  of Gel with fluence  deduced  from our calculations may be an indication of a \ndecrease in the efficiency of electron -lattice  interactions  at low laser  fluences  when  nonthermal  electrons  \nprevail.  However,  the specific  values of Gel and Ges values we have obtained for th e different systems \nremain comparable to  values  previously  reported  for these  metals  [11, 23, 37]. \nAfter completion of the fitting routine,  we set up simulations using the optimized values of  the free \nparameters to obtain the temporal evolution of Te, Tl and Ts as a function of the  pump -probe delay.   The \ncalculated profiles of Te, Tl and Ts for four different values of the  pump fluence are presented in Figure \n5 for nickel, showing good agreement between the  results predicted by the 3TM and the M3TM. Simil ar \nprofiles for cobalt and permalloy are  given in Figs.  S2 and S3 of the Supplementary Materials.  As the \npump fluence is increased,  laser excitation deposits a greater amount of energy in the electronic \nreservoir which leads to  a proportionate increase in the initial electron temperature rise.  However, the \nmaximum  electron temperature reached for the highest applied pump fluence remains well below the  \nFermi  temperature  for the respective  material,  ensuring  the assumption  of the linear  temperature  \ndependence  of the electronic  specific  heat remains  valid  [42].  Given  the fundamentally different origins \nof the 3TM and M3TM, the close agreement between the  temperature profiles calculated using the two \nmodels and among the extracted values of the  electron -lattice coupling parameter Gel is remarkable. \nThis observation reflects that, given the  strength of the laser excitation determining the initial \ntemperature excursion of the electron  system,  the Gel  parameter common to both the 3TM and the \nM3TM framework plays a  pivotal  role in determining  the profile  of the ensuing  magnetization  changes.  8  This  also  explains why Gel is seen to follow a similar fluence -dependent trend for a given system in \nboth models.  While the electron temperature is coupled to the spin temperat ure via the Ges parameter  in \nthe 3TM,  it enters  into the evolution  equation  for the magnetization  via  Equation 2 in the M3TM case. \nFrom Figure 5, a qualitative difference in the temporal profiles  of the electron,  spin and lattice \ntemperatures is also appa rent.  Referring to the figures  corresponding  to the 3TM,  while  Te exhibits  a \nsharp  peak  within  the first few hundred  femtoseconds, Ts exhibits a shorter and broader peak at slightly \nlonger time delays.  The delayed peak in Ts occurs when excited electrons t ransfer energy to the spin \ndegrees of  freedom  [51] through  scattering  events  mediated  by the coupling  parameter  Ges, which  \nqualitatively determines the maximum spin temperature value [11].  These scattering events  can result in \nthe generation of magnons, St oner excitations and spin fluctuations which may  play a part in the \nobserved demagnetization [52].  Lastly, the lattice temperature gradually  increases over the first few \npicoseconds after excitation, reaching an equilibrium value at  which  it stays  nearly  constant  over \nseveral  picoseconds.  \nIn Figure  6 we present  a comparison  between  the experimental  demagnetization  curves  and calculated \nelectron temperature profiles for each of our three samples. Under the same applied  pump  fluence,  different  \namplitudes  of magnetization  quenching  and electron  temperature  rise are observed. It is clearly seen that \nnickel quenches more strongly than cobalt, an observation  which  can be explained  by band  structure  \neffects  and a more  efficient  laser -induced  electronic  excitation i n nickel owing to its much lower Curie \ntemperature [34, 40, 53], reflected in the  M3TM as a higher extracted value of the spin -flip probability  \nasf  in nickel.   On the other  hand, a lower electronic specific heat capacity in cobalt as compared to \nnickel [42] predictably  leads  to a much  greater  rise in its electron  temperature  for the same  pump  fluence.  \nIn the case of permalloy,  alloying  with iron can reduce  the electronic  density  of states  at the Fermi  level  \n as compared to nickel, decreasing γ. Because o f the larger electron temperatures reached as a  result, a \nvalue of asf comparable to that of nickel is sufficient to drive a much larger reduction  in the \nmagnetization.  Permalloy also shows a somewhat larger demagnetization time at a  given pump fluence \nas compared to nickel.  Since a standard TR -MOKE optical pump -probe  measurement  cannot  be used  to \nprobe  the demagnetization  response  in alloys  in an element -specific manner, interpreting this finding is \ndifficult. It has been reported previously  using element -specific measurements in a nickel -iron alloy that \nthe demagnetization of nickel  proceeds  faster  than that of iron [54].  Our observation  of a slower  \ndemagnetization  in permalloy may be reflective of this result.  Finally, the demagnetization times for the \nthree systems  at any given  pump  fluence  is seen to sensitively  depend  on the ratio  of TC/µat, \nconfirming  its role as a ‘figure  of merit’  (FOM)  for the demagnetization  time [23].  As shown  in \nFigure 8 ( a), the demagnetization proceeds slower for a lower value  of the FOM. Thus at a  given \nincident laser pump fluence, the demagnetization times are found to correlate with the  values  of this \nquantifying  parameter,  rather  than the material  Curie  temperature  directly.  \n \n 3.3 Precessional  dynamics  and correlation  of Gilbert damping  parameter  with ultrafast  demagnetization  \ntime \nTo gain further  insight  into the microscopic  mechanism  underlying  the ultrafast  \ndemagnetization,  we study also the precessional magnetization dynamics in our thin film  systems. A \ntypical time -resol ved Kerr rotation data showing the different temporal regimes of  laser -induced \nmagnetization dynamics is given in Figure 1( d). Regime I and II represent the  ultrafast  demagnetization  \nand magnetization  recovery  processes  respectively,  while  the oscillatory  signal  of regime  III represents  \nthe magnetization  precession  with the Gilbert  damping  characteristic  of the ferromagnetic  material.  To \nextract  the Gilbert  damping  factor  α, the background -subtracted oscillatory Kerr signal is first fitted \nwith a damped sinusoidal  function  given  by:  \n \n  \n (5) \n   \nwhere  f is the precessional frequency, τ is the relaxation time, and φ is the initial phase of oscillation. \nThe depende nce of f on applied bias magnetic field H is fitted to the Kittel formula relevant for \nferromagnetic systems:  \n \n \n (6) \n \nwhere  γ0 = gµB/��� is  the  gyromagnetic  ratio  and  Meff  is the effective  saturation  magnetization  \nof the probed  volume.  From the Kittel  fit, the values  of g and Meff can be extracted  as fit \nparameters.  While the value  of g is found  to be 2 ± 0.1 for all the samples,  the extracted  values  9  of Meff are 1280.2  ± 1.69 emu/cm3 for cobalt,  400.5  ± 11.46  emu/cm3 for nickel  and 737.1  ± 5.32 \nemu/cm3 for permalloy  (Section  V of the Supplementary  Materials).  \nSubsequently,  the effective  damping  factor  α is calculated  depending  on the extracted  value  of τ and Meff \nas:  \n \n(7) \n \nAlthough  the timescale  of precessional  dynamics  and magnetic  damping  in ferromagnetic  thin films differ  \nby many orders of magnitude,  similar underlying  phenomena may oft en determine  the characteristics of \neither process.  Through several independent investigations relating these  two distinct processes, the \ncorrelation between the demagnetization time τM and the Gilbert  damping  factor  α emerged  as a critical  \nindicator  of the dominant  microscopic  contribution  to the ultrafast  demagnetization.  Koopmans  et al. in \n2005  analytically  derived  an inversely  proportional  relationship  between  the Gilbert  damping  factor  and \nultrafast  demagnetization  time based  on quantum  mechanical  considerations  [16].  However,  the validity  of \nthe model  could  not be confirmed  experimentally  when  τM and α were  tuned  by transition  metal  and rare \nearth  doping  [18].  Later  it was proposed  that the Gilbert  damping  parameter  and ultrafast  \ndemagnetization  time can be either  directly  or inversely  proportional  to each other  depending on the \npredominant micr oscopic mechanism contributing to the magnetic damping  [55].  A proportional  dependence  \nwould  suggest  a dominating  conductivity -like or “breathing  Fermi  surface”  contribution  to the damping  \ndue to the scattering  of intraband  electrons  and holes,  while  an inverse  dependence  may arise in materials  \nwith a dominant  resistivity -like damping  arising  from  inter-band  scattering  processes.  Zhang  et al. later \nsuggested  that an inverse  correlation  between  Gilbert  damping  factor  and ultrafast  demagnetization  time \nmay arise in the presence  of spin transport,  which  provides  an additional  relaxation  channel  resulting  in an \nenhancement  of the magnetic  damping  along  with an acceleration  of the demagnetization  process  at the \nfemtosecond  timescale  [56].  In cases  where  local  spin-flip scattering  processes  are expected  to dominate  the \ndemagnetization  process,  the breathing  Fermi  surface  model  is valid  predicting  a proportional  relationship  \nbetween  τM and α. \nTo study  the correlation  between  ultrafast  demagnetization  and damping  in a system,  both τM and α \nhave to be systematically  varied.  For a given  material,  the exciting  pump  fluence  can be used to \nmodulate the demagnetization time.  As discussed in the previous section, we  observe  an increment  of the \ndemagnetization  time with the pump  fluence  for the systems  under  investigation.  An enhancement  in α \nis also observed  as the pump  fluence  is increased  from  0.8 mJ/cm2 to 8.7 mJ/cm2 for each sample,  \nassoci ated with a corresponding  decrease  in the precessional  relaxation  time.  Within  the experimental  \nfluence  range,  α is found  to be enhanced from 0.0089 to 0.0099 for cobalt, from 0.0346 to 0.0379 for nickel \nand from 0.0106 to  0.0119  for permalloy.  This enhanc ement  of the damping  can be attributed  to the \ntransient  rise in the system temperature as the laser pump fluence is increased [57].  As a laser -induced  \nincrease in electron temperature also drives the demagnetization process at the femtosecond  timescale,  one \ncan qualitatively  correlate  the demagnetization  time with the magnetic  damping at various excitation \nenergies even for a single sample.  Clearly,  a direct correlation  between demagnetization time and the Gilbert \ndamping factor is obtained in our samples b y measuring the magnetization dynamics at various pump \nfluences, as shown in Figure 7 for the  nickel thin film.  Thus, we infer that similar relaxation processes likely \ndrive the laser induced  transient  magnetization  dynamics  at the two different  timescales .   At a low \nfluence  value  of 2.1 mJ/cm2, the extracted  value  of α is the highest  for the nickel  film,  and relatively  \nlow for cobalt  and permalloy.  The observed  trend  in Figure  8 (b) can be directly  correlated  with the \nvalues  of the Sommerfeld  coefficient  γ given  in Table  1. This observation  is in accordance  with \nKamber sky’s  spin flip scattering  theory  [58],  wherein  α is directly  proportional  to the density  of states  at \nthe Fermi  level,  which  in turn governs  the value  of γ. \n \n \n4. Summary  \nTo summarize,  we have investigated  fluence -dependent  ultrafast  demagnetization  in identi cally -grown  \ncobalt,  nickel  and permalloy  thin films  using  an all-optical  time-resolved  magneto -optical Kerr effect \ntechnique.  We study both the femtosecond laser -induced ultrafast  demagnetization  and the magnetization  \nprecession  and damping  in each of our samples  for various  pump  excitation  fluences.  Using  a \nphenomenological  expression,  we extract  the values  of demagnetization  time and fast relaxation  time 10  for each of our samples  to show  that they exhibit  an increasing  trend  with applied  laser fluence  consi stent  \nwith a spin-flip scattering -dominated  demagnetization  in our samples.  Two distinct  ultrafast  demagnetization  \nmodels  can each reproduce  our experimental  demagnetization  traces  remarkably  well.  By directly  fitting  \nour experimental  data to numerical  solutions  of these  models,  we have extracted  values  for the electron -\nlattice  and electron -spin coupling  parameters  and an empirical  estimate  of the spin flip scattering  \nprobability  for each of our samples  and demonstrated  the fluence -dependent  variation  in calculated  \nelectron,  spin,  and lattice  temperatures  as a function  of the pump -probe  delay.  We conjecture  that the \nincreasing  trend  of the extracted  electron -lattice  coupling  parameter  with fluence  may indicate  a decrease  \nin the efficiency  of electron -lattice  interactions  in the low laser fluences  when  nonthermal  electrons may \nplay a role in influencing the magnetization dynamics.  We have compared the  experimental  and theoretical  \nresults  obtained  from  each of the systems  under  investigation  and confirmed  that the ratio of the Curie  \ntemperature  to the atomic  magnetic  moment  acts as a figure  of merit  sensitive  to the demagnetization  time \nfor these  systems.  Our study  demonstrates the mutual agreement between the phenomenological three \ntemperature model  and the microscopic  three  temperature  model  for the systems  we study  and highlights  the \npossibility  of a fluence -dependent  enhancement  in the inter-reservoir  interactions  in the low laser  fluence  \nregime.  Finally,  we have reported  from  our experiments  a direct  correla tion between  the ultrafast  \ndemagnetization  time and the enhancement  of the magnetic  damping  that transiently appears as a result of \nthe laser excitation, reflecting that both phenomena are  sensitive  to similar  underlying  microscopic  \nprocesses.  \n \n5. Acknowledge ment  \nAB gratefully acknowledges the financial support from the S. N. Bose National Centre for  Basic \nSciences (SNBNCBS) under Project No.  SNB/AB/18 -19/211 and Department of Science  and \nTechnology  (DST),  Govt.  of  India  under  Grant  No.  DST/NM/TUE/QM -3/2019-1C-SNB.  SM \nacknowledges  DST,  India  for financial  support  from  INSPIRE  fellowship,  while  SM and SNP  \nacknowledge  SNBNCBS  for senior  research  fellowship.  \n \nReferences  \n[1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, T. Rasing,  All-\noptical  magnetic  recording  with  circularly  polarized  light,  Phys.  Rev.  Lett.  99 (2007)  047601.  \n[2] T. A. Ostler,  J. Barker,  R. F. L. Evans,  R. W. Chantrell,  U. Atxitia,  O. Chubykalo - \nFesenko,  S. El Moussaoui,  L. Le Guyader,  E. Mengotti,  L. J. Heyderman,  F. Nolting,  \nA. Tsukamoto,  A. Itoh,  D. Afanasiev,  B. A. Ivanov,  A. M. Kalashnikova,  K. Vahaplar,  \nJ. Mentink,  A. Kirilyuk,  T. 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The downward arrows represent \nincreased pump fluence.  The symbols  represent  experimental  data points  and the solid  lines  represent  \nfit to Equation  3. 16  \n \n \nFigure 3:  Variation of ( a) demagnetization time τM and ( b) fast relaxation time τE with the pump fluence \nfor 20-nm-thick cobalt, nickel and permalloy films. Both timescales show an increasing trend with \nincreasing pump  fluence.  \n \n \n \n \nFigure 4:  Sensitivity of the demagnetization traces to the various adju stable fit parameters.  \nSymbols  represent  data points  and solid  lines  represent  fits to the 3TM.  Data  shown  is for the \ncobalt  thin film.  For F =0.8  mJ/cm2, Gel=8.2  × 1017 Js−1m−3K−1 and Ges=6.9  × 1016 Js−1m−3K−1; \nfor F =2.1  mJ/cm2, Gel=1.1  × 1018 Js−1m−3K−1 and Ges=8 × 1017 Js−1m−3K−1. \n17   \n \n \nFigure  5: Temporal  evolution  of the reservoir  temperatures  of nickel  Te (blue),  Tl (red),  and Ts (green)  \ncalculated  using  the (a) phenomenological  three  temperature  model  and (b) the microscopic  three  \ntemperature  model.  \n \n \n \n \n \n \n \n \nFigure  6: Comparison  between  (a) the experimental  demagnetization  and (b) the simulated  electron  \ntemperature  profiles  for cobalt,  nickel  and permalloy  at a pump  fluence  of 4.8 mJ/cm2. \n18   \n \n \nFigure  7: (a) Fluence -dependent  variation  of precessional  dynamics  for the nickel  thin film.  \nSymbols  are experimental data points and the solid lines are fits to Equation 5.  (b) Variation of f \nwith pump fluence.  (c) Variation  of α with pump  fluence.  (d) Direct  correlation  between  τM and \nα. \n \n \n \n \n \n \n \n \nFigure 8:  (a) Inverse variation of the demagnetization time with the ratio of Curie temperature to \natomic  magnetic moment and ( b) inverse correlation of the demagnetization time with ma gnetic \ndamping across three  different  materials.  \n19   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nSupplementary  Materials  S1  I. Transient  reflectivity  measurements  \nTime -resolved  transient  reflectivity  signal  is recorded  by TR-MOKE  measurements  \nsimultaneously with the Kerr rotation signal  quantifying the magnetic response.At any  given \nfluence, the maximum amplitude of the transient reflectivity change is about a  factor of 100 \nsmaller as compared to that of the Kerr rotation.  This validates that the  Kerr rotation signal \nreflects the genuine  magnetic response with negligible nonmagnetic  contribution.  \n \n \nFigure S1:  Time -resolved transient reflectivity measured by TR -MOKE for cobalt, nickel, and \npermalloy.  The upward  arrows  represent  increased  pump  fluence.  \n \n \n \nII. Calculation of electron, spin, an d lattice temperature profiles  \nThe calculated  temporal  evolution  of the reservoir  temperatures  for the 20-nm-thick  cobalt  and \npermalloy  thin films  are represented  in Fig. S2 and Fig. S3, respectively.  The electron  \ntemperature  shows  a rapid  and dramatic  increase  within  one picosecond  of the laser excitation,  \nbeyond  which  it decays  to an equilibrium  value  over a few picoseconds.  Over  this period,  the lattice  \nsubsystem  is gradually  heated  above  the equilibrium  temperature  while  the heating  of the spin \nsubsyste m directly  leads  to the demagnetization.  The maximum  electron,  spin and lattice  temperatures  \nincrease  with increasing  laser  fluence.  \nS2  \n \n \nFigure  S2: Electron,  lattice,  and spin temperature  profiles  for the cobalt  thin film calculated  \nusing  (a) the three tem perature model and (b) the microscopic three temperature model for \nvarious laser fluences  (blue:  Te, dashed,  red: Tl and dotted,  green:  Ts). \n \n \nFigure S3: Electron, lattice, and spin temperature profiles for the permalloy thin film calculated \nusing (a)  the three  temperature  model  and (b) the microscopic  three  temperature  model  for various  \nlaser  fluences).  \n \nIII. Extraction  of inter -reservoir  coupling  constants  \nThe least -squares fitting routines set up based on the three temperature model and the  microscopic  \nthree  temperature  model  return  the values  of the inter-reservoir  coupling  \nS3  constants  that best reproduce  the experimental  demagnetization  traces.  The microscopic  three  \ntemperature  model  additionally  returns  the value  of spin-flip probability asf associated with  the \nspin-flip scattering events which directly lead to the  magnetization quenching.  The extracted \nvalues of these microscopic parameters are  given in Tables S1 -S3. Fig. S4 shows a comparison of \nthe values of the electron -lattice  coupling parameter Gel as extracted from the fits to the three \ntemperature model and  the microscopic three temperature model.  For reasons discussed in the \nmain text, the  spin-lattice coupling parameter Gsl is taken to be zero for the three temperature \nmodel  simulations.  Fig. S5 shows  that,  when  Gsl  is  set  equal  to  3  ×  1016 Jm−1s−1m−3K−1 as \nconsidered in Phys.  Rev.  Lett.  76, 4250 (1996), the simulated trace  and calculated electron \ntemperature profile are nearly indistinguishable from the case in  which  Gsl is neglected.  \n \nFigure  S4: Values  of the electron -lattice  coupling  parameter  Gel extracted  from  fitting  to the three  \ntemperature  model  and microscopic  three  temperature  model.  \n \n \n \n \nFigure  S5:  (a)  Results  of three  temperature  modelling  and (b) calculated  reservoir  temperature  \nprofiles  for two values  of Gsl. \n \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8 0.82±0.081  0.85±0.097  0.69±0.037  0.19±0.023  \n2.1 1.08±0.063  1.18±0.105  0.8±0.029  0.16±0.014  \n3.5 1.26±0.057  1.48±0.106  0.91±0.028  0.16±0.009  \n4.8 1.2±0.037  1.57±0.096  0.97±0.015  0.17±0.008  \n6.0 1.21±0.042  1.49±0.072  1.02±0.022  0.17±0.008  \n7.3 1.56±0.076  1.98±0.087  1.15±0.041  0.16±0.004  \n8.7 1.65±0.045  2.18±0.066  1.28±0.036  0.17±0.003  \nTabl e S1:  Microscopic parameters extracted from the 3TM and M3TM s imulations for cobalt.  \n \n \n \nS4   \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8  1.29±0.154  1.41±0.172  2.1±0.182  0.53±0.03  \n2.1 1.52±0.061  1.74±0.077  2.84±0.087  0.55±0.012  \n3.5 1.53±0.0 51 1.77±0.069  2.93±0.076  0.56±0.011  \n4.8 1.65±0.047  1.9±0.06  2.86±0.064  0.51±0.008  \n6.0 1.49±0.036  1.73±0.039  2.74±0.048  0.57±0.007  \n7.3 1.58±0.028  1.88±0.033  2.87±0.038  0.59±0.006  \n8.7 1.58±0.048  1.77±0.039  2.77±0.062  0.62±0.008  \nTabl e S2:  Microscopic pa rameters extracted from the 3TM and M3TM simulations for nickel . \n \n \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8 1.62±0.144  1.86±0.184  2.16±0.145  0.31±0.016  \n2.1 1.79±0.074  2.14± 0.084  2.34±0.073  0.34±0.007  \n3.5 2.05±0.081  2.46±0.065  2.65±0.084  0.39±0.005  \n4.8 2.08±0.068  2.55±0.038  2.81±0.071  0.45±0.006  \n6.0 2.02±0.053  2.5±0.065  2.72±0.058  0.49±0.007  \n7.3 2.07±0.083  2.67±0.052  2.87±0.089  0.56±0.009  \n8.7 2.11±0.084  2.71±0.069  2.87±0 .086 0.64±0.01  \nTabl e S1:  Microscopic parameters extracted from the 3TM and M3TM simulations for permalloy . \n \nIV. Temperature dependence of spin specific heat  \nConsidering the theory of Manchon et al. ( Phys.  Rev.  B 85, 064408 (2012)) as adopted  in Sci. \nRep. 10:6355 (2020), the specific heat Cs of the spin subsystem in the 3TM  framework  can be \ncalculated  as a function  of the spin temperature  Ts as: \n \n      (S1) \n \nFig. S6 below  shows  the trends  in the inter-reservoir  coupling  parameters  for two scenar ios: \nconsidering the  room temperature  value  of Cs(T0) or an explicit temperature  (and hence  time)  \ndependence  of Cs. \n \n \nFigure  S6: Fluence -dependent  variation  of (a) Gel and (b) Ges considering  two different  forms  \nfor the spin specific  heat Cs. \n \n \n \nS5  ∼ V. Bias fiel d dependence of precessional frequency and Gilbert damping  \nThe bias  magnetic  field-dependent variation of precessional dynamics in cobalt,  nickel and permalloy  \nis shown in Fig.  S7 - Fig. S9 for a pump fluence of 4.8 mJ/cm2. The oscillatory Kerr  signal is \nfitted with Eq. 5 of the main text to extract the precessional frequency f and relaxation time τ.   \nThe bias field dependence of the extracted frequency,  shown in Fig.  S7(b) for cobalt, is fitted to the \nKittel formula (Eq. 6) to extract the effective saturation  magnetization  Meff . Once  Meff have  been  \nderived,  the Gilbert  damping  factor  α is calcula ted using Eq. 7 of the main text. The extracted α \ndoes not show any systematic  variation  with  bias field.  \n \n \n \nFigure  S7: (a) Bias field -dependent  variation  of precessional  dynamics  for the cobalt  thin film.  \nSymbols  are experimental  data points  and the solid lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ with \nH. (d) Variation  of α with H. The solid  line is a guide  to the eye. \n \nVI. Pump fluence dependence of precessional frequency and Gilbert \ndamping  \nThe variation of precessional dynamics in cobalt, nickel and permalloy with the laser  pump \nfluence is shown in Fig. S10 for the cobalt and in Fig. S11 for the permalloy thin  film at a \nsaturating  field  of   2 kOe.  The oscilla tory Kerr  signal  is fitted  with Eq.  5 of the main  text to \nextract  the precessional  frequency  f and relaxation  time τ. With  increasing  pump  fluence,  the \nprecessional  frequency  decreases  and the magnetic  damping  is enhanced.  A direct  correlation  \nbetween  the demagnetization  time τM and α is observed.  S6  \n \n \nFigure  S8: (a) Bias field -dependent  variation  of precessional  dynami cs for the nickel  thin film.  \nSymbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ with \nH. (d) Variation  of α with H. The solid  line is a guide  to the eye. \n \n \nFigure  S9: (a) Bias field -dependent  variation  of precessional  dynamics  for the  permalloy  thin  \nfilm.  Symbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. \n(b) Variation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ  \nwith H.  (d) Variation  of α with H. The solid  line is a guide  to the eye. \nS7  \n \n \nFigure S10: (a) Fluence -dependent variation of precessional dynamics for the cobalt thin film. \nSymbo ls are experimental data points and the solid lines are fits to Eq.  5 of the main text.  (b) \nVariation of f with pump  fluence.  (c) Variation  of α with pump  fluence.  (d) Direct  correlation  \nbetween  τM and α. \n \n \n \nFigure S11:  (a) Fluence -dependent variation of precessional dynamics for the permalloy thin film.  \nSymbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with pump  fluence.  (c) Variation  of α with  pump  fluence.  (d) Direct  correlation  \nbetween  τM and α. \n"    },    {        "title": "2302.02556v3.Global_solutions_of_the_Landau__Lifshitz__Baryakhtar_equation.pdf",        "content": "arXiv:2302.02556v3  [math.AP]  29 Jun 2023GLOBAL SOLUTIONS OF THE LANDAU–LIFSHITZ–BARYAKHTAR\nEQUATION\nAGUS L. SOENJAYA AND THANH TRAN\nAbstract. The Landau–Lifshitz–Baryakhtar (LLBar) equation is a generalisa tion of the\nLandau–Lifshitz–Gilbertand the Landau–Lifshitz–Blochequations which takesinto account\ncontributions from nonlocal damping and is valid at moderate temper ature below the Curie\ntemperature. Therefore, it is used to explain some discrepancies b etween the experimen-\ntal observations and the known theories in various problems on mag nonics and magnetic\ndomain-wall dynamics. In this paper, the existence and uniqueness of global weak, strong,\nand regular solutions to LLBar equation are proven. H¨ older contin uity of the solution is\nalso discussed.\n1.Introduction\nThe theory of micromagnetism deals with the study of magnetic phen omena occurring\nin ferromagnetic materials at sub-micrometre length scales. A bett er understanding of the\nmagnetisation dynamics at elevated temperature would contribute to the development of\nultrahigh-density storage technology based on heat-assisted ma gnetic recording; see [ 11,29,\n30,32] and the references therein.\nA widely-studied equation which describes the evolution of magnetic s pin field in ferro-\nmagnetic material is the Landau–Lifshitz–Gilbert (LLG) equation [ 18,24]. According to this\ntheory, the magnetisation of a magnetic body Ω ⊂Rd,d∈ {1,2,3}, denoted by u(t,x)∈R3\nfort >0 andx∈Ω, is described by\n∂u\n∂t=−γu×Heff−λu×(u×Heff), (1.1)\nwhereγ >0 andλ >0 are the gyromagnetic ratio and a phenomenological damping param e-\nter, respectively, and Heffistheeffectivefield(consistingoftheexchangefield, demagnetising\nfield, external magnetic field and others). It is known that far belo w the Curie temperature,\nthe magnetisation of a ferromagnetic material preserves its magn itude. This property is\nreflected in equation ( 1.1) (by taking the dot product of both sides of the equation with u).\nMathematically, the LLG equation has been extensively studied eithe r on bounded or\nunbounded domains where various existence, uniqueness and regu larity properties were dis-\ncussed. A non-exhaustive list includes [ 1,9,10,12,16,20,21,22,33]. Since then, various\ngeneralisations and improvements to the LLG equation have been ma de in the physical and\nmathematical literatures. Awidely-used physical model formicrom agnetism abovetheCurie\ntemperature is the Landau–Lifshitz–Bloch (LLB) equation [ 17]. This equation interpolates\nbetween the LLG equation at low temperatures and the Ginzburg–L andau theory of phase\ntransitions, and is known not to preserve the magnitude of the mag netisation. Mathemati-\ncally, the existence and regularity properties for LLB equation hav e been studied [ 25,27].\nDate: June 30, 2023.\n12 AGUS L. SOENJAYA AND THANH TRAN\nTheLLGandLLBequations, nevertheless, cannotaccountforso meexperimentaldataand\nmicroscopic calculations. These include the nonlocal damping in magne tic metals and crys-\ntals [14,36], or the higher-than-expected spin wave decrement for short-w ave magnons [ 5].\nTheLandau–Lifshitz–Baryakhtar (LLBar)equationproposedby Baryakhtar [ 3,4,5]isbased\nonOnsager’srelationsandgeneralisestheLLGandLLBequations[ 14,15,35]. Thisequation\nhas also been implemented on several commonly-used micromagnetic simulation software,\nsuch asMuMax [2,26] andFidimag [34,35]. Moreover, various micromagnetic simulations\nprovide evidence that the LLBar equation agrees with some of the o bserved experimental\nfindings in micromagnetics, especially those related to ultrafast mag netisation at an elevated\ntemperature; see [ 2,14,28,34,35,36] and the references therein.\nThe LLBar equation in its most general form [ 5,35] reads\n∂u\n∂t=−γu×Heff+Λr·Heff−Λe,ij∂2Heff\n∂xi∂xj,\nwhereurepresents themagnetisation vector, ΛrandΛedenote therelaxation tensor and the\nexchange tensor, respectively. Here, Einstein’s summation notat ion is used. For a polycrys-\ntalline, amorphous soft magnetic materials and magnetic metals at mo derate temperature\n(where nonlocal damping and longitudinal relaxationare significant) , this equation simplifies\n[14,35] to\n∂u\n∂t=−γu×Heff+λrHeff−λe∆Heff.\nwhere the positive scalars γ,λr, andλeare the electron gyromagnetic ratio, relativistic\ndamping constant, and exchange damping constant, respectively . The effective field Heffis\ngiven by\nHeff= ∆u+1\n2χ(1−|u|2)u+lower order terms ,\nwithχ >0 being the magnetic susceptibility of the material.\nIftheexchange interactionisdominant (asisthecaseforordinary ferromagneticmaterial),\nthenu: [0,T]×Ω→R3solves the following problem:\n∂u\n∂t−β1∆u+β2∆2u=β3(1−|u|2)u−β4u×∆u+β5∆(|u|2u) in (0,T)×Ω,(1.2a)\nu(0,·) =u0 in Ω, (1.2b)\n∂u\n∂n=∂(∆u)\n∂n=0 on (0,T)×∂Ω,(1.2c)\nwhereβ1=λr−λe/(2χ) is a real constant (which may be positive or negative), while\nβ2,...,β 5are positive constants. Here, ∂Ω is the boundary of Ω with exterior unit normal\nvector denoted by n.\nTypically, the constant β1will be positive since λe/(2χ) is much smaller than λr. How-\never, in certain situations occurring in spintronics or magnonics whe re the wavelength of\nthe magnons is approaching the exchange length of the ferromagn etic material, λecan be\nsignificant [ 14]. Therefore, we allow β1to take positive or negative values in ( 1.2).\nTo the best of our knowledge, mathematical analysis of the LLBar e quation does not\nexist in the literature. In this paper, we prove the existence, uniqu eness, and regularity\nof a weak and strong solution to problem ( 1.2) in one, two and three spatial dimensions\n(see Theorem 2.2), by using the Faedo–Galerkin approximation and compactness met hod.GLOBAL SOLUTIONS OF LLBAR EQUATION 3\nWe also prove H¨ older continuity properties of the solution (Theore m2.3). This gives a\nmathematical foundation for the rigorous theory of LLBar equat ion which is not currently\navailable in the literature.\nAnother advantage of studying the LLBar equation is for a given init ial datau0, the weak\nsolution to the LLBar equation generally has better regularity comp ared to that of the LLG\nor the LLB equation. Moreover, it is known that the existence of glo bal solutions to the\nLLG equation in 2-D is only guaranteed for sufficiently small initial data [9,16], whereas for\ngeneral initial data, solutions in 2-D could blow-up in finite time [ 23]. As we show in this\npaper, the solution to the LLBar equation exists globally.\nWe note that a related model of magnetisation dynamics in the frame work of frustrated\nmagnets(whichtakesintoaccountlocalandnonlocalinteractions )hasrecentlybeenexplored\nin [13]. The model is based on the LLG equation involving the bilaplacian opera tor, where\nthe magnitude of the magnetisation is conserved in that case.\nThepaper is organisedasfollows. InSection 2, we introduce some notationsandformulate\nthe main results. In Section 3, we establish some a priori estimates that are needed for the\nproof of the main theorems. Section 4is devoted to the proof of the main results. Finally,\nwe collect in the appendix some essential mathematical facts that a re used throughout the\npaper.\n2.Formulation of the main results\n2.1.Notation. We begin by defining some notations used in this paper. The function\nspaceLp:=Lp(Ω;R3) denotes the space of p-th integrable functions taking values in R3and\nWk,p:=Wk,p(Ω;R3) denotes the Sobolev space of functions on Ω ⊂Rdtaking values in R3.\nAlso, we write Hk:=Wk,2. Here, Ω ⊂Rdford= 1,2,3 is an open domain with smooth\nboundary. The operator ∆ denotes the Neumann Laplacian. The pa rtial derivative ∂/∂xi\nwill be written by ∂ifor short.\nIfXisanormedvector space, thespaces Lp(0,T,X)andWk,p(0,T,X)denoterespectively\nthe usual Lebesgue and Sobolev spaces of functions on (0 ,T) taking values in X. The\nspaceC([0,T],X) denotes the space of continuous function on [0 ,T] taking values in X.\nThroughout this paper, we denote the scalar product in a Hilbert sp aceHby/an}b∇acketle{t·,·/an}b∇acket∇i}htHand\nits corresponding norm by /ba∇dbl · /ba∇dblH. We will not distinguish between the scalar product of\nL2vector-valued functions taking values in R3and the scalar product of L2matrix-valued\nfunctions taking values in R3×3, and still denote them by /an}b∇acketle{t·,·/an}b∇acket∇i}htL2.\nThe following frequently-used notations are collected here for the reader’s convenience.\nFirstly, for any vector z∈R3and matrices A,B∈R3×d, we define\nz·A:=/bracketleftbig\nz·A(1)···z·A(d)/bracketrightbig\n∈R1×d,A·B:=d/summationdisplay\nj=1A(j)·B(j)∈R,\nz×A:=/bracketleftbig\nz×A(1)···z×A(d)/bracketrightbig\n∈R3×d,A×B:=d/summationdisplay\nj=1A(j)×B(j)∈R3,(2.1)\nwhereA(j)andB(j)denote the jthcolumn of AandB, respectively.4 AGUS L. SOENJAYA AND THANH TRAN\nNext, for any vector-valued function v= (v1,v2,v3) : Ω⊂Rd→R3, we define\n\n\n∇v: Ω→R3×dby∇v:=/bracketleftbig∂1v···∂dv/bracketrightbig\n=\n∂1v1···∂dv1\n∂1v2···∂dv2\n∂1v3···∂dv3\n,\n∂v\n∂n:∂Ω→R3×1by∂v\n∂n:= (∇v)n=/bracketleftbigg\n∂v1\n∂n∂v2\n∂n∂v3\n∂n/bracketrightbigg⊤\n,\n∆v: Ω→R3×1by ∆v:=/bracketleftbig∆v1∆v2∆v3/bracketrightbig⊤,\n∆∇v: Ω→R3×dby ∆∇v:=\n∆∂1v1···∆∂dv1\n∆∂1v2···∆∂dv2\n∆∂1v3···∆∂dv3\n=∇∆v.(2.2)\nAs a consequence, if uandvsatisfy suitable assumptions and ∂u/∂n= 0 (where nis the\noutward normal vector to ∂D), then\n−/an}b∇acketle{t∆u,v/an}b∇acket∇i}htL2=−3/summationdisplay\ni=1/an}b∇acketle{t∆ui,vi/an}b∇acket∇i}htL2=3/summationdisplay\ni=1/an}b∇acketle{t∇ui,∇vi/an}b∇acket∇i}htL2=/an}b∇acketle{t∇u,∇v/an}b∇acket∇i}htL2,\n/an}b∇acketle{tu×∆u,v/an}b∇acket∇i}htL2=/an}b∇acketle{t∆u,v×u/an}b∇acket∇i}htL2=−/an}b∇acketle{tu×∇u,∇v/an}b∇acket∇i}htL2.(2.3)\nFinally, throughout this paper, the constant Cin the estimate denotes a generic constant\nwhichtakesdifferent valuesatdifferent occurrences. Ifthedepe ndence of Consomevariable,\ne.g.T, is highlighted, we often write C(T). The notation A/lessorsimilarBmeansA≤CBwhere the\nspecific form of the constant Cis not important to clarify.\n2.2.Main results. In the following, we define the notion of weak solutions to ( 1.2). We\nfirst multiply ( 1.2a) (dot product) with a test function φ, integrate over Ω, and (formally)\nuse integration by parts, noting ( 1.2c), to obtain\n/angbracketleftBig∂u(t)\n∂t,φ/angbracketrightBig\nL2+β1/an}b∇acketle{t∇u(t),∇φ/an}b∇acket∇i}htL2+β2/an}b∇acketle{t∆u(t),∆φ/an}b∇acket∇i}htL2\n=β3/angbracketleftbig\n(1−|u(t)|2)u(t),φ/angbracketrightbig\nL2+β4/an}b∇acketle{tu(t)×∇u(t),∇φ/an}b∇acket∇i}htL2−β5/angbracketleftbig\n∇/parenleftbig\n|u(t)|2u(t)/parenrightbig\n,∇φ/angbracketrightbig\nL2.\n(2.4)\nWe next find sufficient conditions for the terms on the right-hand sid e to be well-defined for\nd= 1,2,3. Ifu∈H1, thenu∈L4so that|u|2u∈L4/3. Therefore, the term /an}b∇acketle{t|u|2u,φ/an}b∇acket∇i}htL2is\nwell defined if u∈H1andφ∈H1. Moreover, if u∈H2, thenu∈L∞. Thus, the second\nterm/an}b∇acketle{tu(t)×∇u(t),∇φ/an}b∇acket∇i}htL2and the third term\n/angbracketleftbig\n∇/parenleftbig\n|u(t)|2u(t)/parenrightbig\n,∇φ/angbracketrightbig\nL2=d/summationdisplay\ni=13/summationdisplay\nj=1/angbracketleftbig\n∂i/parenleftbig\n|v|2vj/parenrightbig\n,∂iφj/angbracketrightbig\nL2\non the right-hand side of the above equation are also well defined if u∈H2andφ∈H1.\nThis motivates the following definition of solutions to problem ( 1.2).\nDefinition 2.1. GivenT >0 andu0∈L2(Ω), a function u: [0,T]→H2is aweak solution\nto the problem ( 1.2) ifubelongs to C([0,T];L2)∩L2(0,T;H2) and satisfies\n/an}b∇acketle{tu(t),φ/an}b∇acket∇i}htL2+β1/integraldisplayt\n0/an}b∇acketle{t∇u(s),∇φ/an}b∇acket∇i}htL2ds+β2/integraldisplayt\n0/an}b∇acketle{t∆u(s),∆φ/an}b∇acket∇i}htL2dsGLOBAL SOLUTIONS OF LLBAR EQUATION 5\n=/an}b∇acketle{tu0,φ/an}b∇acket∇i}htL2+β3/integraldisplayt\n0/angbracketleftbig\n(1−|u(s)|2)u(s),φ/angbracketrightbig\nL2ds\n+β4/integraldisplayt\n0/an}b∇acketle{tu(s)×∇u(s),∇φ/an}b∇acket∇i}htL2ds−β5/integraldisplayt\n0/angbracketleftbig\n∇/parenleftbig\n|u(s)|2u(s)/parenrightbig\n,∇φ/angbracketrightbig\nL2ds,(2.5)\nfor allφ∈H2andt∈[0,T].\nA weak solution u: [0,T]→H2is called a strong solution if it belongs to C([0,T];H2)∩\nL2(0,T;H4). In this case, it satisfies ( 1.2) almost everywhere in [0 ,T]×Ω.\nWenowstatethemaintheoremsofthepaper, theproofsofwhichw ill begiveninSection 4\nand Section 5. The first theorem gives the existence, uniqueness, and regularit y of the\nsolution.\nTheorem 2.2. Let Ω⊂Rd,d= 1,2,3, be a bounded domain with Cr+1,1-boundary and let\nu0∈Hr,r∈ {0,1,2,3}, be a given initial data. For any T >0, there exists a global weak\nsolution to ( 1.2) such that\nu∈C([0,T];Hr)∩L2(0,T;Hr+2). (2.6)\nFurthermore, this solution depends continuously on the Hr-norm of the initial data, which\nimplies uniqueness. More precisely, if uandvare solutions corresponding to the initial\ndatau0andv0, respectively, then the following estimates hold\n/ba∇dblu(t)−v(t)/ba∇dbl2\nL2/lessorsimilar/ba∇dblu0−v0/ba∇dbl2\nL2exp/parenleftbigg/integraldisplayt\n0/parenleftBig\n1+/ba∇dblu(s)/ba∇dbl4\nL∞+/ba∇dblv(s)/ba∇dbl4\nL∞/parenrightBig\nds/parenrightbigg\n(2.7)\nand\n/ba∇dblu(t)−v(t)/ba∇dblHr/lessorsimilar/ba∇dblu0−v0/ba∇dblHr, (2.8)\nwhere the constant depends on T. In particular, if the solution uto problem ( 1.2) satisfies\n/integraldisplayT\n0/ba∇dblu(s)/ba∇dbl4\nL∞ds <∞, (2.9)\nthen it is unique.\nMoreover, if r= 2,3 then the solution is a strong solution in the sense of Definition 2.1.\nThe next theorem shows that the strong solution is H¨ older continu ous in time.\nTheorem 2.3. LetT >0 andube the unique strong solution of ( 1.2) with initial data\nu0∈H2. Then\nu∈C0,α(0,T;L2)∩C0,β(0,T;L∞),\nwhereα∈/parenleftbig\n0,1\n2/bracketrightbig\nandβ∈/parenleftbig\n0,1\n2−d\n8/bracketrightbig\n.\nRemark2.4.It can be seen that assumption ( 2.9) holds in many different cases.\n(i) Ifr= 0, assumption ( 2.9) holds for d= 1 ord= 2. Indeed, in case d= 1, we\nhave by using the Gagliardo–Nirenberg inequality (Theorem 6.2) and the fact that\nu∈L2(0,T;H2),\n/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL∞ds/lessorsimilar/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nL2/ba∇dblu(s)/ba∇dbl2\nH1ds≤/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH1ds/lessorsimilar1,\nand in case d= 2,\n/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL∞ds/lessorsimilar/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nL2/ba∇dblu(s)/ba∇dbl2\nH2ds≤/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH2ds/lessorsimilar1.6 AGUS L. SOENJAYA AND THANH TRAN\n(ii) Ifr∈ {1,2,3}, assumption ( 2.9) holds for d= 1,2,3. In this case, with d= 3, we\nhave\n/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL∞ds/lessorsimilar/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH1/ba∇dblu(s)/ba∇dbl2\nH2ds≤/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH2ds/lessorsimilar1.\n3.Faedo–Galerkin Approximation\nLet{ei}∞\ni=1denote an orthonormal basis of L2consisting of eigenvectors for −∆ such that\n−∆ei=λieiin Ω and∂ei\n∂n=0on∂Ω,\nwhereλi>0 are the eigenvalues of −∆, associated with ei. By elliptic regularity results, ei\nis smooth up to the boundary, and we also have\n∆2ei=λ2\nieiin Ω and∂ei\n∂n=∂∆ei\n∂n=0on∂Ω.\nLetVn:= span{e1,...,en}and Π n:L2→Vnbe the orthogonal projection defined by\n/an}b∇acketle{tΠnv,φ/an}b∇acket∇i}htL2=/an}b∇acketle{tv,φ/an}b∇acket∇i}htL2for allφ∈Vnand allv∈L2.\nNote that Π nis self-adjoint and satisfies\n/ba∇dblΠnv/ba∇dblL2≤ /ba∇dblv/ba∇dblL2for allv∈L2. (3.1)\nTo prove the existence of a weak solution to ( 1.2), we will use the Faedo–Galerkin method.\nWe first prove the following two lemmas.\nLemma 3.1. For any vector-valued function v: Ω→R3, we have\n∇(|v|2v) = 2v(v·∇v)+|v|2∇v, (3.2)\n∂/parenleftbig\n|v|2v/parenrightbig\n∂n= 2v/parenleftBig\nv·∂v\n∂n/parenrightBig\n+|v|2∂v\n∂n, (3.3)\n∆(|v|2v) = 2|∇v|2v+2(v·∆v)v+4∇v(v·∇v)⊤+|v|2∆v, (3.4)\nprovided that the partial derivatives are well defined.\nProof.Recall the notations introduced in ( 2.1) and (2.2). Also note that\n∇(|v|2) = 2(v·∇v)⊤and ∆(|v|2) = 2|∇v|2+2v·∆v.\nHence, it follows from the product rule that\n∇(|v|2v) =v/parenleftbig\n∇(|v|2)/parenrightbig⊤+|v|2∇v= 2v(v·∇v)+|v|2∇v,\nproving ( 3.2). Identity ( 3.3) then follows from ( 3.2) and the definition of normal derivatives.\nFinally, the product rule gives\n∆(|v|2v) = ∆(|v|2)v+2∇v∇(|v|2)+|v|2∆v\n= 2|∇v|2v+2(v·∆v)v+4∇v(v·∇v)⊤+|v|2∆v,\nproving ( 3.4). /squareGLOBAL SOLUTIONS OF LLBAR EQUATION 7\nLemma 3.2. For each n∈Nandv∈Vn, define\nF1\nn(v) = ∆v,\nF2\nn(v) = ∆2v,\nF3\nn(v) = Πn(|v|2v),\nF4\nn(v) = Πn(v×∆v),\nF5\nn(v) = Πn∆(|v|2v).\nThenFj\nn,j= 1,...,5, are well-defined mappings from VnintoVn. Moreover, F1\nnandF2\nn\nare globally Lipschitz while F3\nn,F4\nn, andF5\nnare locally Lipschitz.\nProof.For anyv∈Vn, sincev=/summationtextn\ni=1/an}b∇acketle{tv,ei/an}b∇acket∇i}htL2ei, we have\n−∆v=n/summationdisplay\ni=1λi/an}b∇acketle{tv,ei/an}b∇acket∇i}htL2ei∈Vnand ∆2v=n/summationdisplay\ni=1λ2\ni/an}b∇acketle{tv,ei/an}b∇acket∇i}htL2ei∈Vn.\nTherefore, F1\nnandF2\nnmapVnintoVn. Moreover, if the boundary of Ω is sufficiently smooth,\nthen the eigenfunctions ei,i∈N, are smooth functions, and so is v∈Vn. This implies that\n|v|2v, ∆(|v|2v), andv×∆vall belong to L2(Ω), so that F3\nn,F4\nn, andF5\nnare well defined.\nWe now prove the Lipschitz property of these mappings. Using the t riangle inequality, the\northonormality of {ei}and H¨ older’s inequality, for any v,w∈Vnand forj= 1,2, we have\n/vextenddouble/vextenddoubleFj\nn(v)−Fj\nn(w)/vextenddouble/vextenddouble2\nL2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\ni=1λj\ni/an}b∇acketle{tv−w,ei/an}b∇acket∇i}htL2ei/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2\n=n/summationdisplay\ni=1λ2j\ni|/an}b∇acketle{tv−w,ei/an}b∇acket∇i}htL2|2≤/parenleftBign/summationdisplay\ni=1λ2j\ni/parenrightBig\n/ba∇dblv−w/ba∇dbl2\nL2.\nHence,F1\nnandF2\nnare globally Lipschitz.\nNext, it follows from ( 3.1) that\n/ba∇dblF3\nn(v)−F3\nn(w)/ba∇dblL2≤ /ba∇dbl|v|2v−|w|2w/ba∇dblL2\n≤ /ba∇dbl|v|2(v−w)/ba∇dblL2+/ba∇dbl(v−w)·(v+w)w/ba∇dblL2\n≤/parenleftbig\n/ba∇dblv/ba∇dbl2\nL∞+/ba∇dblv+w/ba∇dblL∞/ba∇dblw/ba∇dblL∞/parenrightbig\n/ba∇dblv−w/ba∇dblL2,\nwhere we used the fact that all norms are equivalent in the finite dime nsional subspace Vn.\nThis shows that F3\nnis locally Lipschitz.\nSimilarly, it follows from ( 3.1) that\n/ba∇dblF4\nn(v)−F4\nn(w)/ba∇dblL2≤ /ba∇dblv×∆v−w×∆w/ba∇dblL2\n≤ /ba∇dblv×(∆v−∆w)/ba∇dblL2+/ba∇dbl(v−w)×∆w/ba∇dblL2\n≤ /ba∇dblv/ba∇dblL∞/ba∇dblF2\nn(v)−F2\nn(w)/ba∇dblL2+/ba∇dblv−w/ba∇dblL2/ba∇dbl∆w/ba∇dblL∞.\nSinceF2\nnis Lipschitz, we deduce that F4\nnis locally Lipschitz.\nFinally, note that if v∈Vn, then∂v/∂n= 0. Thus ( 3.3) implies ∂/parenleftbig\n|v|2v/parenrightbig\n/∂n= 0, which\nallows us to use integration by parts to obtain\n/angbracketleftbig\n∆/parenleftbig\n|v|2v/parenrightbig\n,w/angbracketrightbig\n=−/angbracketleftbig\n∇/parenleftbig\n|v|2v/parenrightbig\n,∇w/angbracketrightbig\n=/angbracketleftbig\n|v|2v,∆w/angbracketrightbig\n∀v,w∈Vn.8 AGUS L. SOENJAYA AND THANH TRAN\nTherefore, for any v,w∈Vn, we can use the definition of Π nand integration by parts again\nto have\n/angbracketleftbig\nΠn∆/parenleftbig\n|v|2v/parenrightbig\n,w/angbracketrightbig\n=/angbracketleftbig\n∆/parenleftbig\n|v|2v/parenrightbig\n,w/angbracketrightbig\n=/angbracketleftbig\n|v|2v,∆w/angbracketrightbig\n=/angbracketleftbig\nΠn/parenleftbig\n|v|2v/parenrightbig\n,∆w/angbracketrightbig\n=/angbracketleftbig\n∆Πn/parenleftbig\n|v|2v/parenrightbig\n,w/angbracketrightbig\n.\nThis means ∆ and Π ncommute, so that F5\nn(v) =F1\nn◦F3\nn(v). SinceF1\nnis Lipschitz and F3\nnis\nlocally Lipschitz, we have that F5\nnis locally Lipschitz as well. This completes the proof. /square\nThe Faedo–Galerkin method seeks to approximate the solution to ( 1.2) byun(t)∈Vn\nsatisfying the equation\n\n\n∂un\n∂t−β1∆un+β2∆2un−β3Πn((1−|un|2)un)\n+β4Πn(un×∆un)−β5Πn(∆(|un|2un)) =0in (0,T)×Ω,\nun(0) =u0n in Ω,(3.5)\nwhereu0n∈Vnis an approximation of u0such that if u0∈Hr, then\n/ba∇dblu0n/ba∇dblHr/lessorsimilar/ba∇dblu0/ba∇dblHr (3.6)\nLemma3.2assures us that all the terms in ( 3.5) are well defined. Moreover, the existence\nof solutions to the above ordinary differential equation in Vnis guaranteed by this lemma\nand the Cauchy–Lipschitz theorem.\nWe now prove some a priori estimates for the solution of ( 3.5). First we need the following\nresults.\nLemma 3.3. Let Ω⊂Rdbe an open bounded domain with C∞-boundary and ǫ >0 be\ngiven. Then there exists a positive constant Csuch that the following inequalities hold:\n(i) for any v∈L2(Ω) such that ∆ v∈L2(Ω) satisfying∂v\n∂n= 0 on∂Ω,\n/ba∇dblv/ba∇dbl2\nH2≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∆v/ba∇dbl2\nL2/parenrightbig\n, (3.7)\n/ba∇dbl∇v/ba∇dbl2\nL2≤C/ba∇dblv/ba∇dbl2\nL2+ε/ba∇dbl∆v/ba∇dbl2\nL2, (3.8)\n(ii) for any v∈H1(Ω) such that ∆ v∈H1satisfying∂v\n∂n= 0 on∂Ω,\n/ba∇dbl∆v/ba∇dbl2\nL2≤ /ba∇dbl∇v/ba∇dblL2/ba∇dbl∇∆v/ba∇dblL2, (3.9)\n/ba∇dblv/ba∇dbl2\nH3≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dbl∇∆v/ba∇dbl2\nL2/parenrightbig\n, (3.10)\n(iii) for any v∈H2(Ω) such that ∆2v∈L2(Ω) satisfying∂v\n∂n=∂∆v\n∂n= 0 on∂Ω,\n/ba∇dbl∇∆v/ba∇dbl2\nL2≤ /ba∇dbl∆v/ba∇dblL2/vextenddouble/vextenddouble∆2v/vextenddouble/vextenddouble\nL2, (3.11)\n/ba∇dblv/ba∇dbl2\nH4≤C/parenleftBig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∆v/ba∇dbl2\nL2+/vextenddouble/vextenddouble∆2v/vextenddouble/vextenddouble2\nL2/parenrightBig\n, (3.12)\n(iv) for any v∈H3(Ω) such that ∆2v∈H1(Ω) satisfying∂v\n∂n=∂∆v\n∂n= 0 on∂Ω,\n/ba∇dblv/ba∇dbl2\nH5≤C/parenleftBig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dbl∇∆v/ba∇dbl2\nL2+/vextenddouble/vextenddouble∇∆2v/vextenddouble/vextenddouble2\nL2/parenrightBig\n. (3.13)GLOBAL SOLUTIONS OF LLBAR EQUATION 9\nProof.Inequality ( 3.7) follows from the standard elliptic regularity result with Neumann\nboundary data [ 19, Corollary 2.2.2.6]. Next, using integration by parts, H¨ older’s inequa lity,\nand Young’s inequality, we obtain\n/ba∇dbl∇v/ba∇dbl2\nL2=/an}b∇acketle{t∇v,∇v/an}b∇acket∇i}htL2=−/an}b∇acketle{tv,∆v/an}b∇acket∇i}ht ≤ /ba∇dblv/ba∇dblL2/ba∇dbl∆v/ba∇dblL2≤C/ba∇dblv/ba∇dbl2\nL2+ε/ba∇dbl∆v/ba∇dbl2\nL2,\nproving ( 3.8).\nSimilarly, we have\n/ba∇dbl∆v/ba∇dbl2\nL2=−/an}b∇acketle{t∇∆v,∇v/an}b∇acket∇i}ht ≤ /ba∇dbl∇∆v/ba∇dblL2/ba∇dbl∇v/ba∇dblL2\nand\n/ba∇dbl∇∆v/ba∇dbl2\nL2=−/angbracketleftbig\n∆v,∆2v/angbracketrightbig\n≤ /ba∇dbl∆v/ba∇dblL2/vextenddouble/vextenddouble∆2v/vextenddouble/vextenddouble\nL2,\nproving ( 3.9) and (3.11).\nNext, applying the higher-order regularity result [ 19, Remark 2.5.1.2] to the elliptic oper-\nator (−∆+I) yields\n/ba∇dblv/ba∇dbl2\nH3≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nH1+/ba∇dbl∆v/ba∇dbl2\nH1/parenrightbig\n≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dbl∇∆v/ba∇dbl2\nL2/parenrightbig\n,\nproving ( 3.10). Furthermore, by the same elliptic regularity result,\n/ba∇dblv/ba∇dbl2\nH4≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∆v/ba∇dbl2\nH2/parenrightbig\n≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∆v/ba∇dbl2\nL2+/vextenddouble/vextenddouble∆2v/vextenddouble/vextenddouble2\nL2/parenrightbig\n,\nwhere we applied ( 3.7) to ∆v, noting that the assumptions are satisfied. This proves ( 3.12).\nSimilarly, we have\n/ba∇dblv/ba∇dbl2\nH5≤C/parenleftbig\n/ba∇dbl∆v/ba∇dbl2\nH3+/ba∇dblv/ba∇dbl2\nH3/parenrightbig\n≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dbl∇∆v/ba∇dbl2\nL2+/vextenddouble/vextenddouble∇∆2v/vextenddouble/vextenddouble2\nL2/parenrightbig\n,\nwhere we applied ( 3.10) to ∆v, thus proving ( 3.13). /square\nRemark 3.4.Elliptic regularity result in H2(3.7) holds more generally for a domain with\nC1,1-boundary[ 19, Remark 2.2.2.6]or a convex polygonal domain[ 19, Theorem 4.3.1.4]. The\nHr-regularity results also hold for a domain with Cr−1,1-boundary (see [ 19, Remark 2.5.1.2]).\nLemma 3.5. Letk= (k1,···,kd) be a multi-index and define the operator\nDk:=∂|k|\n∂k1x1···∂kdxd\nwhere|k|=k1+···+kd. Then the following inequalities hold:\n(i) for any u,v∈Hs(Ω), where s > d/2,\n/ba∇dbl|u||v|/ba∇dblHs≤C/ba∇dblu/ba∇dblHs/ba∇dblv/ba∇dblHs. (3.14)\n(ii) for any u,v∈H2∩H|k|,\n/vextenddouble/vextenddoubleDk(|u|2u−|v|2v)/vextenddouble/vextenddouble\nL2/lessorsimilar\n\n/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblu−v/ba∇dblL2 if|k|= 0,\n/parenleftbig\n/ba∇dblu/ba∇dblH1+/ba∇dblv/ba∇dblH1/parenrightbig/parenleftbig\n/ba∇dblu/ba∇dblH2+/ba∇dblv/ba∇dblH2/parenrightbig\n/ba∇dblu−v/ba∇dblH1if|k|= 1,\n/parenleftbig\n/ba∇dblu/ba∇dbl2\nH|k|+/ba∇dblv/ba∇dbl2\nH|k|/parenrightbig\n/ba∇dblu−v/ba∇dblH|k| if|k| ≥2.\n(3.15)\n(iii) for any u,v∈H|k|∩L∞,\n/vextenddouble/vextenddoubleu×Dku−v×Dkv/vextenddouble/vextenddouble\nL2/lessorsimilar/ba∇dblu/ba∇dblL∞/vextenddouble/vextenddoubleDk(u−v)/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble/parenleftbig\nu−v/parenrightbig\nDkv/vextenddouble/vextenddouble\nL2.(3.16)10 AGUS L. SOENJAYA AND THANH TRAN\nProof.Firstly, (3.14)followsfrom /ba∇dbl|v||w|/ba∇dblHs/lessorsimilar/ba∇dblv/ba∇dblL∞/ba∇dblw/ba∇dblHs+/ba∇dblv/ba∇dblHs/ba∇dblw/ba∇dblL∞andtheSobolev\nembedding. Next, we prove ( 3.15). For the case |k|= 0, by H¨ older’s inequality we have/vextenddouble/vextenddouble|u|2u−|v|2v/vextenddouble/vextenddouble\nL2≤/vextenddouble/vextenddouble|u|2(u−v)/vextenddouble/vextenddouble\nL2+/ba∇dbl|u+v||u−v||v/ba∇dblL2\n≤/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dblL∞/ba∇dblu+v/ba∇dblL∞/parenrightbig\n/ba∇dblu−v/ba∇dblL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblu−v/ba∇dblL2.\nFor the case |k|= 1, note that/vextendsingle/vextendsingleDk(|u|2u−|v|2v)/vextendsingle/vextendsingle≤/vextendsingle/vextendsingleDk(|u|2)(u−v)/vextendsingle/vextendsingle+/vextendsingle/vextendsingle|u|2Dk(u−v)/vextendsingle/vextendsingle+/vextendsingle/vextendsingleDk(|u|2−|v|2)v/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle(|u|2−|v|2)Dkv/vextendsingle/vextendsingle\n≤2|u|/vextendsingle/vextendsingleDk(u)/vextendsingle/vextendsingle/vextendsingle/vextendsingleu−v/vextendsingle/vextendsingle+|u|2/vextendsingle/vextendsingleDk(u−v)/vextendsingle/vextendsingle+/vextendsingle/vextendsingleDku+Dkv/vextendsingle/vextendsingle|u−v||v|\n+|u+v|/vextendsingle/vextendsingleDku−Dkv/vextendsingle/vextendsingle|v|+|u+v||u+v|/vextendsingle/vextendsingleDkv/vextendsingle/vextendsingle.\nUsing this, we have/vextenddouble/vextenddoubleDk(|u|2u−|v|2v)/vextenddouble/vextenddouble\nL2\n≤2/vextenddouble/vextenddouble|u|/vextendsingle/vextendsingleDk(u)/vextendsingle/vextendsingle|u−v|/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble|u|2Dk(u−v)/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble|Dku+Dkv| |u−v| |v|/vextenddouble/vextenddouble\nL2\n+/vextenddouble/vextenddouble|u+v| |Dku−Dkv| |v|/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble|u+v| |u−v| |Dkv|/vextenddouble/vextenddouble\nL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dblL6/vextenddouble/vextenddoubleDku/vextenddouble/vextenddouble\nL6+/ba∇dblu/ba∇dblL6/vextenddouble/vextenddoubleDkv/vextenddouble/vextenddouble\nL6+/ba∇dblv/ba∇dblL6/vextenddouble/vextenddoubleDku/vextenddouble/vextenddouble\nL6+/ba∇dblv/ba∇dblL6/vextenddouble/vextenddoubleDkv/vextenddouble/vextenddouble\nL6/parenrightbig\n/ba∇dblu−v/ba∇dblL2\n+/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig/vextenddouble/vextenddoubleDku−Dkv/vextenddouble/vextenddouble\nL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dblH1/vextenddouble/vextenddoubleDku/vextenddouble/vextenddouble\nH1+/ba∇dblu/ba∇dblH1/vextenddouble/vextenddoubleDkv/vextenddouble/vextenddouble\nH1+/ba∇dblv/ba∇dblH1/vextenddouble/vextenddoubleDku/vextenddouble/vextenddouble\nH1+/ba∇dblv/ba∇dblH1/vextenddouble/vextenddoubleDkv/vextenddouble/vextenddouble\nH1/parenrightbig\n/ba∇dblu−v/ba∇dblH1\n+/parenleftbig\n/ba∇dblu/ba∇dblH1/ba∇dblu/ba∇dblH2+/ba∇dblv/ba∇dblH1/ba∇dblv/ba∇dblH2/parenrightbig\n/ba∇dblu−v/ba∇dblH1\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dblH1/ba∇dblu/ba∇dblH2+/ba∇dblu/ba∇dblH1/ba∇dblv/ba∇dblH2+/ba∇dblv/ba∇dblH1/ba∇dblu/ba∇dblH2+/ba∇dblv/ba∇dblH1/ba∇dblv/ba∇dblH2/parenrightbig\n/ba∇dblu−v/ba∇dblH1\n=/parenleftbig\n/ba∇dblu/ba∇dblH1+/ba∇dblv/ba∇dblH1/parenrightbig/parenleftbig\n/ba∇dblu/ba∇dblH2+/ba∇dblv/ba∇dblH2/parenrightbig\n/ba∇dblu−v/ba∇dblH1,\nwhere in the third step we used Gagliardo–Nirenberg’s inequality /ba∇dblu/ba∇dbl2\nL∞/lessorsimilar/ba∇dblu/ba∇dblH1/ba∇dblv/ba∇dblH2\n(valid for d∈ {1,2,3}) and the Sobolev embedding H1⊂L6.\nFor the case |k| ≥2, we use H¨ older’s inequality and ( 3.14) to obtain\n/vextenddouble/vextenddoubleDk(|u|2u−|v|2v)/vextenddouble/vextenddouble\nL2≤/vextenddouble/vextenddoubleDk(|u|2u−|v|2v)/vextenddouble/vextenddouble\nL2≤/vextenddouble/vextenddouble|u|2u−|v|2v/vextenddouble/vextenddouble\nH|k|\n≤/parenleftbig\n/ba∇dblu/ba∇dbl2\nH|k|+/ba∇dblv/ba∇dblH|k|/ba∇dblu+v/ba∇dblH|k|/parenrightbig\n/ba∇dblu−v/ba∇dblH|k|\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH|k|+/ba∇dblv/ba∇dbl2\nH|k|/parenrightbig\n/ba∇dblu−v/ba∇dblH|k|.\nThis completes the proof of ( 3.15). The proof of ( 3.16) is obvious and is omitted. /square\nWe now use the above lemmas to derive a priori estimates on the Galer kin solution un.\nProposition 3.6. LetT >0 be arbitrary and assume that u0∈L2. For each n∈Nand\nallt∈[0,T],\n/ba∇dblun(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∇un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dbl∆un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl4\nL4ds\n+/integraldisplayt\n0/ba∇dbl|un(s)||∇un(s)|/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dblun(s)·∇un(s)/ba∇dbl2\nL2ds/lessorsimilar/ba∇dblu0/ba∇dbl2\nL2,(3.17)\nwhere the constant depends on T, but is independent of n.GLOBAL SOLUTIONS OF LLBAR EQUATION 11\nProof.Taking the inner product of ( 3.5) withun(t), integrating by parts with respect to x\n(noting ( 3.3)) and using the identity ( 3.2), we obtain, for any ǫ >0,\n1\n2d\ndt/ba∇dblun/ba∇dbl2\nL2+β2/ba∇dbl∆un/ba∇dbl2\nL2+β3/ba∇dblun/ba∇dbl4\nL4+2β5/ba∇dblun·∇un/ba∇dbl2\nL2+β5/ba∇dbl|un||∇un|/ba∇dbl2\nL2\n≤β3/ba∇dblun/ba∇dbl2\nL2+|β1|/ba∇dbl∇un/ba∇dbl2\nL2≤C/ba∇dblun/ba∇dbl2\nL2+ǫ/ba∇dbl∆un/ba∇dbl2\nL2,\nwhere in the last step we used ( 3.8). Rearranging the above equation, choosing sufficiently\nsmallǫ, and integrating over (0 ,t), we deduce\n/ba∇dblun(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∆un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl4\nL4ds+/integraldisplayt\n0/ba∇dbl|un(s)||∇un(s)|/ba∇dbl2\nL2ds\n+/integraldisplayt\n0/ba∇dblun(s)·∇un(s)/ba∇dbl2\nL2ds/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl2\nL2ds.\nInvoking Gronwall’s inequality yields the required estimate for all the t erms on the left-\nhand side of ( 3.17), except/integraltextt\n0/ba∇dbl∇un(s)/ba∇dbl2\nL2ds. The bound for this term follows from ( 3.8),\ncompleting the proof of the proposition. /square\nProposition 3.7. Under the assumption of Proposition 3.6, we have\n/ba∇dbl∂tun/ba∇dblL2(0,T;H−2)/lessorsimilar/ba∇dblu0/ba∇dbl2\nL2,\nwhere the constant depends on Tbut is independent of n.\nProof.Taking the inner product of ( 3.5) withϕ∈H2such that /ba∇dblϕ/ba∇dblH2≤1 and integrating\nby parts with respect to x, we have\n/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2=−β1/an}b∇acketle{t∇un,∇ϕ/an}b∇acket∇i}htL2−β2/an}b∇acketle{t∆un,∆ϕ/an}b∇acket∇i}htL2+β3/an}b∇acketle{tun,ϕ/an}b∇acket∇i}htL2−β3/angbracketleftbig\n|un|2un,ϕ/angbracketrightbig\nL2\n+β4/an}b∇acketle{tun×∇un,∇ϕ/an}b∇acket∇i}htL2+β5/angbracketleftbig\n∇(|un|2un),∇ϕ/angbracketrightbig\nL2.\nIt follows from this equation and H¨ older’s inequality that/vextendsingle/vextendsingle/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dblun/ba∇dblH1/ba∇dblϕ/ba∇dblH1+/ba∇dblun/ba∇dblH2/ba∇dblϕ/ba∇dblH2+/ba∇dblun/ba∇dblL2/ba∇dblϕ/ba∇dblL2+/ba∇dblun/ba∇dblL2/ba∇dblun/ba∇dbl2\nL6/ba∇dblϕ/ba∇dblL6\n+/ba∇dblun/ba∇dblL4/ba∇dbl∇un/ba∇dblL2/ba∇dbl∇ϕ/ba∇dblL4+/ba∇dblun/ba∇dblL2/ba∇dblun/ba∇dblL6/ba∇dbl∇un/ba∇dblL6/ba∇dbl∇ϕ/ba∇dblL6,\nwhere we used the Sobolev embedding H1⊂L6. Therefore, integrating over (0 ,t) and noting\nthat/ba∇dblϕ/ba∇dblH2≤1, we obtain\n/integraldisplayt\n0/vextendsingle/vextendsingle/an}b∇acketle{t∂sun(s),ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle2ds/lessorsimilar/integraldisplayt\n0/ba∇dblun/ba∇dbl2\nH2ds/lessorsimilar1\nby Proposition 3.6. Taking supremum over the set {ϕ∈H2:/ba∇dblϕ/ba∇dblH2≤1}and noting that\n/ba∇dbl∂tun/ba∇dblH−2≤sup\nϕ∈H2/vextendsingle/vextendsingle/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/ba∇dblϕ/ba∇dblH2,\nwe obtain the required estimate. /square\nProposition 3.8. LetT >0 be arbitrary and assume that u0∈H1. Then there exists\nT∗>0 such that for n∈Nandt∈[0,T∗], we have\n/ba∇dbl∇un(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∇∆un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dbl|un(s)||∆un(s)|/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dblun(s)·∆un(s)/ba∇dbl2\nL2ds\n/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nH1/lessorsimilar/ba∇dblu0/ba∇dbl2\nH1.12 AGUS L. SOENJAYA AND THANH TRAN\nThe constant depends on T∗, but is independent of n. Here,\n/braceleftBigg\nT∗=Tford= 1,2,\nT∗≤Tford= 3,\nwhereT∗=T∗(/ba∇dblu0/ba∇dblH1).\nProof.Taking the inner product of ( 3.5) with−∆un(t) andintegrating by parts with respect\ntox, we have\n1\n2d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+β1/ba∇dbl∆un/ba∇dbl2\nL2+β2/ba∇dbl∇∆un/ba∇dbl2\nL2−β3/ba∇dbl∇un/ba∇dbl2\nL2\n+β3/angbracketleftbig\n∇(|un|2un),∇un/angbracketrightbig\nL2+β5/angbracketleftbig\n∆(|un|2un),∆un/angbracketrightbig\nL2= 0.(3.18)\nIt follows from ( 3.2) and (3.4) that\n/angbracketleftbig\n∇(|un|2un),∇un/angbracketrightbig\nL2= 2/ba∇dblun·∇un/ba∇dbl2\nL2+/ba∇dbl|un||∇un|/ba∇dbl2\nL2\nand\n/angbracketleftbig\n∆(|un|2un),∆un/angbracketrightbig\nL2= 2/angbracketleftbig\n|∇un|2un,∆un/angbracketrightbig\nL2+2/ba∇dblun·∆un/ba∇dbl2\nL2\n+4/angbracketleftbig\n∇un(un·∇un)⊤,∆un/angbracketrightbig\nL2+/ba∇dbl|un||∆un|/ba∇dbl2\nL2.\nTherefore, after rearranging the terms in ( 3.18), we obtain\n1\n2d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+β2/ba∇dbl∇∆un/ba∇dbl2\nL2+2β3/ba∇dblun·∇un/ba∇dbl2\nL2+β3/ba∇dbl|un||∇un|/ba∇dbl2\nL2\n+2β5/ba∇dblun·∆un/ba∇dbl2\nL2+β5/ba∇dbl|un||∆un|/ba∇dbl2\nL2\n=−β1/ba∇dbl∆un/ba∇dbl2\nL2+β3/ba∇dbl∇un/ba∇dbl2\nL2−2β5/angbracketleftbig\n|∇un|2un,∆un/angbracketrightbig\nL2−4β5/angbracketleftbig\n∇un(un·∇un)⊤,∆un/angbracketrightbig\nL2\n≤ |β1|/ba∇dbl∆un/ba∇dbl2\nL2+β3/ba∇dbl∇un/ba∇dbl2\nL2+6β5/integraldisplay\nΩ|un||∆un||∇un|2dx. (3.19)\nUsing H¨ older’s inequality and Young’s inequality, we can estimate the la st term on the\nright-hand side by\n6β5/integraldisplay\nΩ|un||∆un||∇un|2dx≤6β5/ba∇dbl|un||∆un|/ba∇dblL2/ba∇dbl∇un/ba∇dbl2\nL4\n≤ǫ/ba∇dbl|un||∆un|/ba∇dbl2\nL2+C/ba∇dbl∇un/ba∇dbl4\nL4,\nwhereε >0 is sufficiently small. This inequality together with ( 3.19) yields\nd\ndt/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∇∆un/ba∇dbl2\nL2+/ba∇dblun·∇un/ba∇dbl2\nL2+/ba∇dbl|un||∇un|/ba∇dbl2\nL2+/ba∇dblun·∆un/ba∇dbl2\nL2+/ba∇dbl|un||∆un|/ba∇dbl2\nL2\n/lessorsimilar/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∆un/ba∇dbl2\nL2+/ba∇dbl∇un/ba∇dbl4\nL4. (3.20)\nWeestimate thelast term onthe right-handside of ( 3.20) byinvoking Gagliardo–Nirenberg’s\ninequality ( 6.2).\nCase 1:d= 1. Applying inequality ( 6.2) withv=un,q= 4,r= 1,s1= 0, and s2= 3 gives\n/ba∇dbl∇un/ba∇dbl4\nL4/lessorsimilar/ba∇dblun/ba∇dbl7/3\nL2/ba∇dblun/ba∇dbl5/3\nH3/lessorsimilar/ba∇dblun/ba∇dbl5/3\nH3GLOBAL SOLUTIONS OF LLBAR EQUATION 13\nwhere in the last step we used Proposition 3.6and the assumption that un(0) =u0n∈Vn\nwhich approximates u0. Young’s inequality implies, for any ǫ >0,\n/ba∇dbl∇un/ba∇dbl4\nL4≤C+ǫ/ba∇dblun/ba∇dbl2\nH3/lessorsimilar1+/ba∇dbl∇un/ba∇dbl2\nL2+ǫ/ba∇dbl∇∆un/ba∇dbl2\nL2,\nwhere in the last step we used ( 3.10). Therefore, by choosing ǫ >0 sufficiently small, we\ndeduce from ( 3.20)\nd\ndt/ba∇dbl∇un(t)/ba∇dbl2\nL2/lessorsimilar1+/ba∇dbl∇un(t)/ba∇dbl2\nL2+/ba∇dbl∆un(t)/ba∇dbl2\nL2.\nIntegrating over (0 ,t) and using Proposition 3.6, we obtain\n/ba∇dbl∇un(t)/ba∇dbl2\nL2≤ /ba∇dbl∇un(0)/ba∇dbl2\nL2+C+C/integraldisplayt\n0/ba∇dbl∇un(s)/ba∇dbl2\nL2ds.\nGronwall’s inequality yields the required estimate.\nCase 2:d= 2. Applying inequality ( 6.2) withv=∇un,q= 4,r= 0,s1= 0, and s2= 1\ngives\n/ba∇dbl∇un/ba∇dbl4\nL4/lessorsimilar/ba∇dbl∇un/ba∇dbl2\nL2/ba∇dbl∇un/ba∇dbl2\nH1/lessorsimilar/parenleftbig\n1+/ba∇dbl∆un/ba∇dbl2\nL2/parenrightbig\n/ba∇dbl∇un/ba∇dbl2\nL2,\nwhere in the last step we used ( 3.7) and Proposition 3.6. Therefore, inequality ( 3.20) gives\nd\ndt/ba∇dbl∇un(t)/ba∇dbl2\nL2/lessorsimilar/ba∇dbl∆un(t)/ba∇dbl2\nL2+/parenleftbig\n1+/ba∇dbl∆un(t)/ba∇dbl2\nL2/parenrightbig\n/ba∇dbl∇un(t)/ba∇dbl2\nL2.\nIntegrating over (0 ,t), using Gronwall’s inequality and Proposition 3.6, we deduce\n/ba∇dbl∇un(t)/ba∇dbl2\nL2/lessorsimilar1+exp/parenleftBig/integraldisplayT\n0/parenleftbig\n1+/ba∇dbl∆un(t)/ba∇dbl2\nL2/parenrightbig\ndt/parenrightBig\n/lessorsimilar1,\nproving the result for d= 2.\nCase 3:d= 3. Applying inequality ( 6.2) withv=∇un,q= 4,r= 0,s1= 0,s2= 2, and\nusing (3.10), we infer\n/ba∇dbl∇un/ba∇dbl4\nL4/lessorsimilar/ba∇dbl∇un/ba∇dbl5/2\nL2/ba∇dbl∇un/ba∇dbl3/2\nH2/lessorsimilar/ba∇dbl∇un/ba∇dbl5/2\nL2/parenleftbig\n1+/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∇∆un/ba∇dbl2\nL2/parenrightbig3/4\n/lessorsimilar/ba∇dbl∇un/ba∇dbl5/2\nL2+/ba∇dbl∇un/ba∇dbl4\nL2+/ba∇dbl∇un/ba∇dbl5/2\nL2/ba∇dbl∇∆un/ba∇dbl3/2\nL2.\nYoung’s inequality yields, for ǫ >0 sufficiently small,\n/ba∇dbl∇un/ba∇dbl4\nL4≤C/parenleftBig\n/ba∇dbl∇un/ba∇dbl5/2\nL2+/ba∇dbl∇un/ba∇dbl4\nL2+/ba∇dbl∇un/ba∇dbl10\nL2/parenrightBig\n+ǫ/ba∇dbl∇∆un/ba∇dbl2\nL2.\nInserting this estimate into ( 3.20) and rearranging the terms, we deduce\nd\ndt/ba∇dbl∇un/ba∇dbl2\nL2/lessorsimilar/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∆un/ba∇dbl2\nL2+/ba∇dbl∇un/ba∇dbl5/2\nL2+/ba∇dbl∇un/ba∇dbl4\nL2+/ba∇dbl∇un/ba∇dbl10\nL2\n/lessorsimilar/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∆un/ba∇dbl2\nL2+/ba∇dbl∇un/ba∇dbl10\nL2.\nIntegrating over (0 ,t) and using Proposition 3.6give\n/ba∇dbl∇un(t)/ba∇dbl2\nL2≤ /ba∇dbl∇un(0)/ba∇dbl2\nL2+C+C/integraldisplayt\n0/ba∇dbl∇un(s)/ba∇dbl10\nL2ds.\nBy using Gronwall–Bihari’s inequality (Theorem 6.1) withf(x) =x5(so thatF(x) =x−4/4\nandF−1(x) =x−1/4/√\n2), and noting ( 3.6), we obtain the required estimate for any t∈14 AGUS L. SOENJAYA AND THANH TRAN\n[0,T∗], where T∗=/parenleftbig\n/ba∇dbl∇un(0)/ba∇dbl2\nL2+C/parenrightbig−4/4. This completes the proof of the proposition.\n/square\nRemark3.9.Sinceun(0) =u0napproximates u0, we have T∗≈ /ba∇dbl∇u0/ba∇dbl−8\nL2.\nProposition 3.10. Under the assumption of Proposition 3.8, we have\n/ba∇dbl∂tun/ba∇dbl2\nL2(0,T∗;H−1)/lessorsimilar/ba∇dblu0/ba∇dbl2\nH1,\nwhere the constant depends on T∗but is independent of n.\nProof.Taking the inner product of ( 3.5) withϕ∈H1such that /ba∇dblϕ/ba∇dblH1≤1 and integrating\nby parts with respect to x, we have\n/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2=−β1/an}b∇acketle{t∇un,∇ϕ/an}b∇acket∇i}htL2+β2/an}b∇acketle{t∇∆un,∇ϕ/an}b∇acket∇i}htL2+β3/an}b∇acketle{tun,ϕ/an}b∇acket∇i}htL2−β3/angbracketleftbig\n|un|2un,ϕ/angbracketrightbig\nL2\n+β4/an}b∇acketle{tun×∇un,∇ϕ/an}b∇acket∇i}htL2+β5/angbracketleftbig\n∇(|un|2un),∇ϕ/angbracketrightbig\nL2.\nIt follows from this equation and H¨ older’s inequality that\n/vextendsingle/vextendsingle/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dblun/ba∇dblH1/ba∇dblϕ/ba∇dblH1+/ba∇dblun/ba∇dblH3/ba∇dblϕ/ba∇dblH1+/ba∇dblun/ba∇dblL2/ba∇dblϕ/ba∇dblL2+/ba∇dblun/ba∇dblL2/ba∇dblun/ba∇dbl2\nL6/ba∇dblϕ/ba∇dblL6\n+/ba∇dblun/ba∇dblL4/ba∇dbl∇un/ba∇dblL4/ba∇dbl∇ϕ/ba∇dblL2+/ba∇dblun/ba∇dblL6/ba∇dblun/ba∇dblL6/ba∇dbl∇un/ba∇dblL6/ba∇dbl∇ϕ/ba∇dblL2.\nTherefore, integrating over (0 ,t), using the Sobolev embedding H1⊂L6, and noting that\n/ba∇dblϕ/ba∇dblH1≤1, we obtain\n/integraldisplayt\n0/vextendsingle/vextendsingle/an}b∇acketle{t∂sun(s),ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle2ds/lessorsimilar/integraldisplayt\n0/ba∇dblun/ba∇dbl2\nH3ds/lessorsimilar1\nby Proposition 3.6. Taking supremum over the set {ϕ∈H1:/ba∇dblϕ/ba∇dblH1≤1}and noting that\n/ba∇dbl∂tun/ba∇dblH−1≤sup\nϕ∈H1/vextendsingle/vextendsingle/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/ba∇dblϕ/ba∇dblH1,\nwe obtain the required estimate. /square\nProposition 3.11. Assume that u0∈H2. LetT >0 be arbitrary and T∗be defined as in\nProposition 3.8. For each n∈Nand allt∈[0,T∗],\n/integraldisplayt\n0/ba∇dbl∂sun(s)/ba∇dbl2\nL2ds+/ba∇dbl∆un(t)/ba∇dbl2\nL2+/ba∇dblun(t)/ba∇dbl4\nL4+/ba∇dblun(t)·∇un(t)/ba∇dbl2\nL2+/ba∇dbl|un(t)| |∇un(t)|/ba∇dbl2\nL2\n/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nH2/lessorsimilar/ba∇dblu0/ba∇dbl2\nH2,\nwhere the constant depends on T∗, but is independent of n. Here,∂sun:=∂un\n∂s.\nProof.Taking the inner product of ( 3.5) with∂tunand integrating by parts with respect\ntox, we obtain\n/ba∇dbl∂tun/ba∇dbl2\nL2+β1\n2d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+β2\n2d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+β3\n4d\ndt/ba∇dblun/ba∇dbl4\nL4+β4/an}b∇acketle{tun×∆un,∂tun/an}b∇acket∇i}htL2\n+β5/angbracketleftbig\n∇(|un|2un),∂t∇un/angbracketrightbig\nL2=β3\n2d\ndt/ba∇dblun/ba∇dbl2\nL2.\nFor the last term on the left-hand side, it follows from ( 3.2) that\nβ5/angbracketleftbig\n∇(|un|2un),∂t∇un/angbracketrightbig\nL2= 2β5/angbracketleftbig\nun/parenleftbig\nun·∇un/parenrightbig\n,∂t∇un/angbracketrightbig\nL2+β5/angbracketleftbig\n|un|2∇un,∂t∇un/angbracketrightbig\nL2GLOBAL SOLUTIONS OF LLBAR EQUATION 15\n=β5d\ndt/ba∇dblun·∇un/ba∇dbl2\nL2−β5/an}b∇acketle{tun·∇un,∂tun·∇un/an}b∇acket∇i}htL2\n+β5\n2/angbracketleftbig\n|un|2,∂t/parenleftbig\n|∇un|2/parenrightbig/angbracketrightbig\nL2\n=β5d\ndt/ba∇dblun·∇un/ba∇dbl2\nL2−β5/an}b∇acketle{tun·∇un,∂tun·∇un/an}b∇acket∇i}htL2\n+β5\n2d\ndt/ba∇dbl|un| |∇un|/ba∇dbl2\nL2−β5/angbracketleftbig\n|∇un|2,un·∂tun/angbracketrightbig\nL2.\nTherefore,\n/ba∇dbl∂tun/ba∇dbl2\nL2+β1\n2d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+β2\n2d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+β3\n4d\ndt/ba∇dblun/ba∇dbl4\nL4\n+β5d\ndt/ba∇dblun·∇un/ba∇dbl2\nL2+β5\n2d\ndt/ba∇dbl|un| |∇un|/ba∇dbl2\nL2\n=β3\n2d\ndt/ba∇dblun/ba∇dbl2\nL2−β4/an}b∇acketle{tun×∆un,∂tun/an}b∇acket∇i}htL2\n+β5/an}b∇acketle{tun·∇un,∂tun·∇un/an}b∇acket∇i}htL2+β5/angbracketleftbig\n|∇un|2,un·∂tun/angbracketrightbig\nL2\n≤β3\n2d\ndt/ba∇dblun/ba∇dbl2\nL2+β4/ba∇dblun/ba∇dblL∞/ba∇dbl∆un/ba∇dblL2/ba∇dbl∂tun/ba∇dblL2+2β5/ba∇dblun/ba∇dblL∞/ba∇dbl∇un/ba∇dbl2\nL4/ba∇dbl∂tun/ba∇dblL2\n≤β3\n2d\ndt/ba∇dblun/ba∇dbl2\nL2+C/ba∇dblun/ba∇dbl2\nL∞/ba∇dbl∆un/ba∇dbl2\nL2+C/ba∇dblun/ba∇dbl2\nL∞/ba∇dbl∇un/ba∇dbl4\nL4+ǫ/ba∇dbl∂tun/ba∇dbl2\nL2,\nfor anyǫ >0, where in the last step we used Young’s inequality. Rearranging the inequality,\nwe obtain\n/ba∇dbl∂tun/ba∇dbl2\nL2+d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+d\ndt/ba∇dblun/ba∇dbl4\nL4+d\ndt/ba∇dblun·∇un/ba∇dbl2\nL2+d\ndt/ba∇dbl|un| |∇un|/ba∇dbl2\nL2\n/lessorsimilard\ndt/ba∇dblun/ba∇dbl2\nL2+/ba∇dblun/ba∇dbl2\nL∞/ba∇dbl∆un/ba∇dbl2\nL2+/ba∇dblun/ba∇dbl2\nL∞/ba∇dbl∇un/ba∇dbl4\nL4. (3.21)\nWe now estimate the last two terms on the right-hand side of ( 3.21).\nCase 1:d= 1. It follows from the Sobolev embedding, Proposition 3.6, and Proposition 3.8\nthat\n/ba∇dblun(t)/ba∇dbl2\nL∞/lessorsimilar/ba∇dblun(t)/ba∇dbl2\nH1/lessorsimilar1, t∈[0,T].\nMoreover, the Gagliardo–Nirenberg inequality (Theorem 6.2withv=un,q= 4,r= 1,\ns1= 1, and s2= 2) together with ( 3.7) implies\n/ba∇dbl∇un(t)/ba∇dbl4\nL4/lessorsimilar/ba∇dblun(t)/ba∇dbl3\nH1/ba∇dblun(t)/ba∇dblH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dbl2\nL2, t∈[0,T].\nTherefore, inequality ( 3.21) yields the required result, after integrating over (0 ,t) and using\nProposition 3.6.\nCase 2:d= 2. The Gagliardo–Nirenberg inequality (respectively with v=un,q=∞,\nr=s1= 0,s2= 2, and with v=∇un,q= 4,r=s1= 0,s2= 1) implies\n/ba∇dblun(t)/ba∇dbl2\nL∞/lessorsimilar/ba∇dblun(t)/ba∇dblL2/ba∇dblun(t)/ba∇dblH2/lessorsimilar/ba∇dblun(t)/ba∇dblH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dblL2, t∈[0,T],\n/ba∇dbl∇un(t)/ba∇dbl4\nL4/lessorsimilar/ba∇dbl∇un(t)/ba∇dbl2\nL2/ba∇dbl∇un(t)/ba∇dbl2\nH1/lessorsimilar/ba∇dblun(t)/ba∇dbl2\nH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dbl2\nL2, t∈[0,T],16 AGUS L. SOENJAYA AND THANH TRAN\nwhere we also used ( 3.7) and Proposition 3.8. Inserting these estimates into ( 3.21) and\nintegrating over (0 ,t) yield\n/integraldisplayt\n0/ba∇dbl∂sun(s)/ba∇dbl2\nL2ds+/ba∇dbl∆un(t)/ba∇dbl2\nL2+/ba∇dblun(t)/ba∇dbl4\nL4+/ba∇dblun(t)·∇un(t)/ba∇dbl2\nL2+/ba∇dbl|un(t)| |∇un(t)|/ba∇dbl2\nL2\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dblL2+/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∆un(s)/ba∇dbl3\nL2/parenrightBig\nds\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∆un(s)/ba∇dbl3\nL2/parenrightBig\nds, (3.22)\nwhere in the last step we used Young’s inequality for the term /ba∇dbl∆un(s)/ba∇dblL2. For the last\nterm on the right-hand side, we use ( 3.9) to obtain\n/ba∇dbl∆un/ba∇dbl3\nL2≤ /ba∇dbl∇un/ba∇dbl3/2\nL2/ba∇dbl∇∆un/ba∇dbl3/2\nL2/lessorsimilar/ba∇dbl∇∆un/ba∇dbl3/2\nL2/lessorsimilar1+/ba∇dbl∇∆un/ba∇dbl2\nL2,\nwhere in the penultimate step we used Proposition 3.8, and in the last step we used Young’s\ninequality. Therefore the right-hand side of ( 3.22) is bounded independent of ndue to\nProposition 3.8, proving the proposition for this case.\nCase 3:d= 3. The Gagliardo–Nirenberg inequality (respectively with v=un,q=∞,\nr= 0,s1= 1,s2= 2, and with v=∇un,q= 4,r=s1= 0,s2= 1) implies\n/ba∇dblun(t)/ba∇dbl2\nL∞/lessorsimilar/ba∇dblun(t)/ba∇dblH1/ba∇dblun(t)/ba∇dblH2/lessorsimilar/ba∇dblun(t)/ba∇dblH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dblL2,\n/ba∇dbl∇un(t)/ba∇dbl4\nL4/lessorsimilar/ba∇dbl∇un(t)/ba∇dblL2/ba∇dbl∇un(t)/ba∇dbl3\nH1/lessorsimilar/ba∇dblun(t)/ba∇dbl3\nH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dbl3\nL2,\nfor allt∈[0,T∗] whereT∗is given in Proposition 3.8. Inserting these estimates into ( 3.21),\nintegrating over (0 ,t), and using ( 3.6) yield\n/integraldisplayt\n0/ba∇dbl∂sun(s)/ba∇dbl2\nL2ds+/ba∇dbl∆un(t)/ba∇dbl2\nL2+/ba∇dblun(t)/ba∇dbl4\nL4+/ba∇dblun(t)·∇un(t)/ba∇dbl2\nL2+/ba∇dbl|un(t)| |∇un(t)|/ba∇dbl2\nL2\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dblL2+/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∆un(s)/ba∇dbl3\nL2+/ba∇dbl∆un(s)/ba∇dbl4\nL2/parenrightBig\nds\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∆un(s)/ba∇dbl4\nL2/parenrightBig\nds\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∇∆un(s)/ba∇dbl2\nL2/parenrightBig\nds\n/lessorsimilar1,\nwhere in the last step we used Proposition 3.8, completing the proof of the proposition. /square\nProposition 3.12. Assume that u0∈H2. LetT >0 be arbitrary and T∗be defined as in\nProposition 3.8. For each n∈Nand allt∈[0,T∗],\n/integraldisplayt\n0/vextenddouble/vextenddouble∆2un(s)/vextenddouble/vextenddouble2\nL2ds/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nH2/lessorsimilar/ba∇dblu0/ba∇dbl2\nH2,\nwhere the constant depends on T∗but is independent of n.\nProof.Taking the inner product of ( 3.5) with ∆2unand integrating by parts with respect\ntox, we obtain\n1\n2d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+β1/ba∇dbl∇∆un/ba∇dbl2\nL2+β2/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2GLOBAL SOLUTIONS OF LLBAR EQUATION 17\n=β3/ba∇dbl∆un/ba∇dbl2\nL2−β3/angbracketleftbig\n|un|2un,∆2un/angbracketrightbig\nL2\n−β4/angbracketleftbig\nun×∆un,∆2un/angbracketrightbig\nL2−β5/angbracketleftbig\n∆(|un|2un),∆2un/angbracketrightbig\nL2. (3.23)\nEach term on the right-hand side can be estimated as follows. For th e first term, by Young’s\ninequality, Sobolev embedding and Proposition 3.8,\n/vextendsingle/vextendsingle/angbracketleftbig\n|un|2un,∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle≤C/vextenddouble/vextenddouble|un|2un/vextenddouble/vextenddouble2\nL2+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\n=C/ba∇dblun/ba∇dbl6\nL6+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\n≤C/ba∇dblun/ba∇dbl6\nH1+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\n/lessorsimilar1+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\nfor anyǫ >0. For the second term, by H¨ older’s inequality, Young’s inequality, S obolev\nembedding, and Proposition 3.11, we have\n/vextendsingle/vextendsingle/angbracketleftbig\nun×∆un,∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle≤ /ba∇dblun/ba∇dblL∞/ba∇dbl∆un/ba∇dblL2/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble\nL2\n≤C(/ba∇dblun/ba∇dbl2\nH2/ba∇dbl∆un/ba∇dbl2\nL2)+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\n/lessorsimilar1+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2.\nFinally, by H¨ older’s and Young’s inequalities, we have\n/vextendsingle/vextendsingle/angbracketleftbig\n∆(|un|2un),∆2un/angbracketrightbig/vextendsingle/vextendsingle≤/vextenddouble/vextenddouble∆(|un|2un)/vextenddouble/vextenddouble\nL2/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble\nL2\n≤C/vextenddouble/vextenddouble∆(|un|2un)/vextenddouble/vextenddouble2\nL2+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2. (3.24)\nFor the first term on the right-hand side, it follows from ( 3.4), H¨ older’s inequality, and\nSobolev embedding that\n/vextenddouble/vextenddouble∆(|un|2un)/vextenddouble/vextenddouble2\nL2/lessorsimilar/ba∇dbl∇un/ba∇dbl4\nL6/ba∇dblun/ba∇dbl2\nL6+/ba∇dbl∆un/ba∇dbl2\nL6/ba∇dblun/ba∇dbl4\nL6\n≤ /ba∇dbl∇un/ba∇dbl4\nH1/ba∇dblun/ba∇dbl2\nH1+/ba∇dbl∆un/ba∇dbl2\nH1/ba∇dblun/ba∇dbl4\nH1\n/lessorsimilar/ba∇dblun/ba∇dbl6\nH2+/ba∇dbl∆un/ba∇dbl2\nH1/ba∇dblun/ba∇dbl4\nH1\n/lessorsimilar1+/ba∇dbl∇∆un/ba∇dbl2\nL2, (3.25)\nwhere in the last step we also used Proposition 3.11. Altogether, we deduce from ( 3.23) after\nintegrating over (0 ,t) that\n/ba∇dbl∆un(t)/ba∇dbl2\nL2+/integraldisplayt\n0/vextenddouble/vextenddouble∆2un(s)/vextenddouble/vextenddouble2\nL2ds/lessorsimilar1+/integraldisplayt\n0/ba∇dbl∇∆un(s)/ba∇dbl2\nL2ds/lessorsimilar1,\nwhere in the last step we used Proposition 3.8. This completes the proof of the proposition.\n/square\nProposition 3.13. Assume that u0∈H3. LetT >0 be arbitrary and T∗be defined as in\nProposition 3.8. For each n∈Nand allt∈[0,T∗],\n/ba∇dbl∇∆un(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∇∆2un(s)/ba∇dbl2\nL2ds/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nH3/lessorsimilar/ba∇dblu0/ba∇dbl2\nH3,\nwhere the constant depends on T∗but is independent of n.18 AGUS L. SOENJAYA AND THANH TRAN\nProof.Taking the inner product of ( 3.5) with ∆3unand integrating by parts with respect\ntox, we have\n1\n2d\ndt/ba∇dbl∇∆un/ba∇dbl2\nL2+β1/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2+β2/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble2\nL2−β3/ba∇dbl∇∆un/ba∇dbl2\nL2\n=β3/angbracketleftbig\n∆/parenleftbig\n|un|2un/parenrightbig\n,∆2un/angbracketrightbig\nL2−2β4/angbracketleftbig\n∇un×∇∆un,∆2un/angbracketrightbig\nL2\n−β5/angbracketleftbig\n∇∆(|un|2un),∇∆2un/angbracketrightbig\nL2. (3.26)\nEach term on the right-hand side can be estimated as follows. For th e first term, by ( 3.24)\nand (3.25) we have\n/vextendsingle/vextendsingle/angbracketleftbig\n∆/parenleftbig\n|un|2un/parenrightbig\n,∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar1+/ba∇dbl∇∆un/ba∇dbl2\nL2+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2. (3.27)\nFor the second term, Sobolev embedding and H¨ older’s inequality give\n/vextendsingle/vextendsingle/angbracketleftbig\n∇un×∇∆un,∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle≤ /ba∇dbl∇un/ba∇dblL3/ba∇dbl∇∆un/ba∇dblL6/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble\nL2\n≤ /ba∇dblun/ba∇dblH2/ba∇dblun/ba∇dblH4/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble\nL2\n/lessorsimilar1+/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2, (3.28)\nwhere in the last step we used ( 3.7), (3.12), and Proposition 3.11. For the last term on the\nright-hand side of ( 3.26), by H¨ older’s and Young’s inequalities, and ( 3.2), we deduce\n/vextendsingle/vextendsingle/angbracketleftbig\n∇∆(|un|2un),∇∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle≤/vextenddouble/vextenddouble∇(|un|2un)/vextenddouble/vextenddouble\nH2/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble\nL2\n≤C/vextenddouble/vextenddoubleun/parenleftbig\nun·∇un/parenrightbig/vextenddouble/vextenddouble2\nH2+C/vextenddouble/vextenddouble|un|2|∇un|/vextenddouble/vextenddouble2\nH2+ǫ/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble2\nL2\n≤C/ba∇dblun/ba∇dbl4\nH2/ba∇dbl∇un/ba∇dbl2\nH2+ǫ/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble2\nL2\n/lessorsimilar1+/ba∇dbl∇∆un/ba∇dbl2\nL2+ǫ/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble2\nL2 (3.29)\nfor anyǫ >0, where in the penultimate step we used ( 3.14) and in the last step we used\n(3.10), Proposition 3.8, and Proposition 3.11. Inserting the estimates ( 3.27), (3.28), and\n(3.29) into (3.26) and integrating over (0 ,t) yield\n/ba∇dbl∇∆un(t)/ba∇dbl2\nL2+/integraldisplayt\n0/vextenddouble/vextenddouble∇∆2un(s)/vextenddouble/vextenddouble2\nL2ds/lessorsimilar1+/integraldisplayt\n0/ba∇dbl∇∆un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/vextenddouble/vextenddouble∆2un(s)/vextenddouble/vextenddouble2\nL2ds\n/lessorsimilar1,\nwhere in the last step we used Proposition 3.8and3.12. This completes the proof. /square\nProposition 3.14. Under the assumption of Proposition 3.13, we have\n/ba∇dbl∂tun/ba∇dbl2\nL2(0,T∗;H1)/lessorsimilar/ba∇dblu0/ba∇dbl2\nH3,\nwhere the constant depends on T∗, but is independent of n.\nProof.Taking the inner product of ( 3.5) with−∆∂tunand integrating by parts with respect\ntox, we have\n/ba∇dbl∇∂tun/ba∇dbl2\nL2+β1d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+β2d\ndt/ba∇dbl∇∆un/ba∇dbl2\nL2\n=β3d\ndt/ba∇dbl∇un/ba∇dbl2\nL2−β3/angbracketleftbig\n∇/parenleftbig\n|un|2un/parenrightbig\n,∇∂tun/angbracketrightbig\nL2−β4/an}b∇acketle{t∇un×∆un,∇∂tun/an}b∇acket∇i}htL2\n−β4/an}b∇acketle{tun×∇∆un,∇∂tun/an}b∇acket∇i}htL2+β5/angbracketleftbig\n∇∆(|un|2un),∇∂tun/angbracketrightbig\nL2. (3.30)GLOBAL SOLUTIONS OF LLBAR EQUATION 19\nEach inner product on the right-hand side can be estimated as follow s. For the first in-\nner product, by H¨ older’s inequality, Proposition 3.6,3.8and3.11, and Sobolev embedding\nH1⊂L6, we have\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n∇/parenleftbig\n|un|2un/parenrightbig\n,∇∂tun/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dbl∇un/ba∇dblL6/ba∇dblun/ba∇dbl2\nL6/ba∇dbl∇∂tun/ba∇dblL2\n≤ /ba∇dblun/ba∇dblH2/ba∇dblun/ba∇dbl2\nH1/ba∇dbl∇∂tun/ba∇dblL2\n/lessorsimilar1+ǫ/ba∇dbl∇∂tun/ba∇dbl2\nL2, (3.31)\nfor anyǫ >0, where in the last step we used Young’s inequality. For the second in ner\nproduct, H¨ older’s inequality, Proposition 3.6,3.8,3.13, and Sobolev embedding H2⊂L∞\ngive\nβ4/vextendsingle/vextendsingle/an}b∇acketle{t∇un×∆un,∇∂tun/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dbl∇un/ba∇dblL∞/ba∇dbl∆un/ba∇dblL2/ba∇dbl∇∂tun/ba∇dblL2\n≤ /ba∇dbl∇un/ba∇dblH2/ba∇dbl∆un/ba∇dblL2/ba∇dbl∇∂tun/ba∇dblL2\n/lessorsimilar1+ǫ/ba∇dbl∇∂tun/ba∇dbl2\nL2, (3.32)\nfor anyǫ >0. Similarly, for the third inner product, we have\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tun×∇∆un,∇∂tun/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dblun/ba∇dblL∞/ba∇dbl∇∆un/ba∇dblL2/ba∇dbl∇∂tun/ba∇dblL2\n≤ /ba∇dblun/ba∇dblH2/ba∇dbl∇∆un/ba∇dblL2/ba∇dbl∇∂tun/ba∇dblL2\n/lessorsimilar1+ǫ/ba∇dbl∇∂tun/ba∇dbl2\nL2. (3.33)\nFor the last inner product on ( 3.30), by H¨ older’s and Young’s inequality, we obtain\nβ5/vextendsingle/vextendsingle/angbracketleftbig\n∇∆(|un|2un),∇∂tun/angbracketrightbig\nL2/vextendsingle/vextendsingle≤/vextenddouble/vextenddouble|un|2un/vextenddouble/vextenddouble\nH3/ba∇dbl∇∂tun/ba∇dblL2\n≤C/ba∇dblun/ba∇dbl3\nH3/ba∇dbl∇∂tun/ba∇dblL2\n/lessorsimilar1+ǫ/ba∇dbl∇∂tun/ba∇dbl2\nL2, (3.34)\nfor anyǫ >0, where in the penultimate step we used ( 3.14), and in the last step we used\n(3.10), Proposition 3.6,3.8,3.13, and Young’s inequality. Inserting the estimates ( 3.31),\n(3.32), (3.33) and (3.34) into (3.30), integrating over (0 ,t), and choosing ǫsufficiently small\nyield\n/integraldisplayt\n0/ba∇dbl∇∂sun(s)/ba∇dbl2\nL2ds/lessorsimilar1+/ba∇dbl∇un/ba∇dbl2\nL2/lessorsimilar1,\nwhere in the last step we used Proposition 3.8. This completes the proof. /square\nAs a consequence of Proposition 3.6–Proposition 3.14, we have the following result.\nCorollary 3.15. For anyT >0, letT∗be defined by Proposition 3.8. Assume that the\ninitial data u0satisfiesu0∈Hrforr∈ {0,1,2,3}. Assume further that\n/ba∇dblu0n−u0/ba∇dblHr→0 asn→ ∞,\nwhereu0,nis defined in ( 3.5). Then\n/ba∇dblun/ba∇dblL∞(0,T;Hr)+/ba∇dblun/ba∇dblL4(0,T;L4)+/ba∇dblun/ba∇dblL2(0,T;Hr+2)+/ba∇dbl∂tun/ba∇dblL2(0,T;Hr−2)/lessorsimilar1,(3.35)\nwhere\nT=/braceleftBigg\nTifr= 0,\nT∗ifr >0,(3.36)20 AGUS L. SOENJAYA AND THANH TRAN\nProof.First we recall from ( 3.6) that the given assumption yields /ba∇dblun(0)/ba∇dblHr/lessorsimilar/ba∇dblu0/ba∇dblHr/lessorsimilar1.\nTherefore, Proposition 3.6and Proposition 3.7imply (3.35) whenr= 0, while Proposi-\ntion3.8, Proposition 3.11, Proposition 3.10and inequality ( 3.10) give the result when r= 1.\nNext, Proposition 3.6, Proposition 3.8, Proposition 3.11, Proposition 3.12and inequal-\nity (3.10) give the required estimate for the case r= 2. Finally, Proposition 3.6, Proposi-\ntion3.8, Proposition 3.11, Proposition 3.12, Proposition 3.13, Proposition 3.14and inequal-\nity (3.13) give the result for the case r= 3, completing the proof of the corollary. /square\n4.Proof of Theorem 2.2\nLetr∈ {0,1,2,3}. It follows from ( 3.35) and the Banach-Alaoglu theorem that there\nexists a subsequence of {un}, which is still denoted by {un}, such that\n\n\nun⇀uweakly* in L∞(0,T;Hr),\nun⇀uweakly in L2(0,T;Hr+2),\nun⇀uweakly in L4(0,T;L4),\n∂tun⇀ ∂tuweakly in L2(0,T;Hr−2),(4.1)\nwhereTwas defined in ( 3.36). By the Aubin–Lions–Simon lemma (Theorem 6.3), a further\nsubsequence then satisfies\nun→ustrongly in L2(0,T;H1). (4.2)\nThe next proposition shows the convergence of the nonlinear term s in (3.5).\nProposition 4.1. LetT >0 be arbitrary. Let {φn}be a sequence in Vnsuch that φn→φ\ninH2. For all t∈[0,T], we have\nlim\nn→∞/integraldisplayt\n0/angbracketleftbig\nΠn/parenleftbig\n(1−|un(s)|2)un(s)/parenrightbig\n,φn/angbracketrightbig\nL2ds=/integraldisplayt\n0/angbracketleftbig/parenleftbig\n(1−|u(s)|2)u(s)/parenrightbig\n,φ/angbracketrightbig\nL2ds,(4.3)\nlim\nn→∞/integraldisplayt\n0/an}b∇acketle{tΠn(un(s)×∆un(s)),φn/an}b∇acket∇i}htL2ds=−/integraldisplayt\n0/an}b∇acketle{t(u(s)×∇u(s)),∇φ/an}b∇acket∇i}htL2ds,(4.4)\nlim\nn→∞/integraldisplayt\n0/angbracketleftbig\nΠn/parenleftbig\n∆/parenleftbig\n|un(s)|2un(s)/parenrightbig/parenrightbig\n,φn/angbracketrightbig\nL2ds=/integraldisplayt\n0/angbracketleftbig\n∇/parenleftbig\n|u(s)|2u(s)/parenrightbig\n,∇φ/angbracketrightbig\nL2ds.(4.5)\nProof.By the definition of Π n, in order to prove ( 4.3) it suffices to show\nlim\nn→∞/integraldisplayt\n0/angbracketleftbig\n|un(s)|2un(s),φn/angbracketrightbig\nL2ds=/integraldisplayt\n0/angbracketleftbig\n|u(s)|2u(s),φ/angbracketrightbig\nL2ds. (4.6)\nTo this end, note that H¨ older’s inequality implies/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n|un(s)|2un(s),φn/angbracketrightbig\nL2ds−/integraldisplayt\n0/angbracketleftbig\n|u(s)|2u(s),φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n|un(s)|2un(s),φn−φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n|un(s)|2(un(s)−u(s)),φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n(|un(s)|2−|u(s)|2)u(s),φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ /ba∇dblφn−φ/ba∇dblL6/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl2\nL6/ba∇dblun(s)/ba∇dblL2ds+/ba∇dblφ/ba∇dblL∞/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl2\nL4/ba∇dblun(s)−u(s)/ba∇dblL2dsGLOBAL SOLUTIONS OF LLBAR EQUATION 21\n+/ba∇dblφ/ba∇dblL∞/integraldisplayt\n0/ba∇dblun(s)−u(s)/ba∇dblL2/ba∇dblun(s)+u(s)/ba∇dblL4/ba∇dblu(s)/ba∇dblL4ds\n≤ /ba∇dblφn−φ/ba∇dblH1/ba∇dblun/ba∇dblL2(0,T;H1)+/ba∇dblφ/ba∇dblH2/ba∇dblun/ba∇dblL4(0,T;L4)/ba∇dblun−u/ba∇dblL2(0,T;L2)\n+/ba∇dblφ/ba∇dblH2/ba∇dblun−u/ba∇dblL2(0,T;L2)/ba∇dblun+u/ba∇dblL4(0,T;L4)/ba∇dblu/ba∇dblL4(0,T;L4).\nBy using the Sobolev embedding H1⊂L6andH2⊂L∞, (3.35), and (4.2) we deduce ( 4.6).\nSimilarly, to show ( 4.4), it suffices to show\nlim\nn→∞/integraldisplayt\n0/an}b∇acketle{t(un(s)×∇un(s)),∇φn/an}b∇acket∇i}htL2ds=/integraldisplayt\n0/an}b∇acketle{t(u(s)×∇u(s)),∇φ/an}b∇acket∇i}htL2ds.\nTo this end, note that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{tun(s)×∇un(s),∇φn/an}b∇acket∇i}htL2ds−/integraldisplayt\n0/an}b∇acketle{tu(s)×∇u(s),∇φ/an}b∇acket∇i}htL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{tun(s)×∇un(s),∇φn−∇φ/an}b∇acket∇i}htL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{t(un(s)−u(s))×∇un(s),∇φ/an}b∇acket∇i}htL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{tu(s)×(∇un(s)−∇u(s)),∇φ/an}b∇acket∇i}htL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilar/ba∇dblun/ba∇dblL2(0,T;L4)/ba∇dbl∇un/ba∇dblL2(0,T;L4)/ba∇dbl∇φn−∇φ/ba∇dblL2\n+/ba∇dblun−u/ba∇dblL2(0,T;L4)/ba∇dbl∇un/ba∇dblL2(0,T;L4)/ba∇dbl∇φ/ba∇dblL2\n+/ba∇dblu/ba∇dblL2(0,T;L4)/ba∇dbl∇un−∇u/ba∇dblL2(0,T;L4)/ba∇dbl∇φ/ba∇dblL2.\nBy using the Sobolev embedding H1⊂L4, (3.35), and (4.2), we deduce the required conver-\ngence.\nFor the last convergence ( 4.5), it suffices to show\nlim\nn→∞/integraldisplayt\n0/angbracketleftbig\n∇/parenleftbig\n|un(s)|2un(s)/parenrightbig\n,∇φn/angbracketrightbig\nL2ds=/integraldisplayt\n0/angbracketleftbig\n∇/parenleftbig\n|u(s)|2u(s)/parenrightbig\n,∇φ/angbracketrightbig\nL2ds.\nTo this end, note that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n∇(|un(s)|2un(s)),∇φn/angbracketrightbig\nL2ds−/integraldisplayt\n0/angbracketleftbig\n∇(|u(s)|2u(s)),∇φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n∇(|un(s)|2un(s)),∇φn−∇φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n∇(|un(s)|2un(s))−∇(|un(s)|2u(s)),∇φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n∇(|un(s)|2u(s))−∇(|u(s)|2u(s)),∇φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nThe arguments follow along the line of the previous convergence sta tements and are omitted.\n/square\nWe are now ready to prove Theorem 2.2.22 AGUS L. SOENJAYA AND THANH TRAN\nProof that usatisfies ( 2.5) and (2.6): For any φ∈H2, take a sequence {φn}inVnsuch\nthatφn→φinH2. It follows from ( 3.5) that\n/an}b∇acketle{tun(t),φn/an}b∇acket∇i}htL2+β1/integraldisplayt\n0/an}b∇acketle{t∇un(s),∇φn/an}b∇acket∇i}htL2ds+β2/integraldisplayt\n0/an}b∇acketle{t∆un(s),∆φn/an}b∇acket∇i}htL2ds\n=/an}b∇acketle{tu0n,φn/an}b∇acket∇i}htL2+β3/integraldisplayt\n0/angbracketleftbig\nΠn/parenleftbig\n(1−|un(s)|2)un(s)/parenrightbig\n,φn/angbracketrightbig\nL2ds\n+β4/integraldisplayt\n0/an}b∇acketle{tΠn(un(s)×∆un(s)),φn/an}b∇acket∇i}htL2ds−β5/integraldisplayt\n0/angbracketleftbig\nΠn/parenleftbig\n∆/parenleftbig\n|un(s)|2un(s)/parenrightbig/parenrightbig\n,φn/angbracketrightbig\nL2ds.\nHence, letting n→ ∞and using Proposition 4.1we deduce ( 2.5). Noting ( 4.1), we have\nu∈L∞(0,T;Hr)∩L2(0,T;Hr+2)∩L4(0,T;L4) and ∂tu∈L2(0,T;Hr−2).\nTherefore, applying Theorem 6.4, and noting that [ Hr−2,Hr+2]1/2≡Hr, we obtain ( 2.6).\nProof of ( 2.7): Letuandvbe weak solutions to ( 2.4) with initial data u0andv0∈L2,\nrespectively. Let w=u−v. Then, for all φ∈H2,\n/an}b∇acketle{t∂tw,φ/an}b∇acket∇i}htL2+β1/an}b∇acketle{t∇w,∇φ/an}b∇acket∇i}htL2+β2/an}b∇acketle{t∆w,∆φ/an}b∇acket∇i}htL2\n=β3/an}b∇acketle{tw,φ/an}b∇acket∇i}htL2−β3/angbracketleftbig\n|u|2u−|v|2v,φ/angbracketrightbig\nL2+β4/an}b∇acketle{tu×∇u−v×∇v,∇φ/an}b∇acket∇i}htL2\n−β5/angbracketleftbig\n∇(|u|2u−|v|2v),∇φ/angbracketrightbig\nL2. (4.7)\nBy using integration by parts for the terms with coefficient β4andβ5, we obtain\n/an}b∇acketle{t∂tw,φ/an}b∇acket∇i}htL2+β1/an}b∇acketle{t∇w,∇φ/an}b∇acket∇i}htL2+β2/an}b∇acketle{t∆w,∆φ/an}b∇acket∇i}htL2\n=β3/an}b∇acketle{tw,φ/an}b∇acket∇i}htL2−β3/angbracketleftbig\n|u|2u−|v|2v,φ/angbracketrightbig\nL2−β4/an}b∇acketle{tu×∆u−v×∆v,φ/an}b∇acket∇i}htL2\n+β5/angbracketleftbig\n|u|2u−|v|2v,∆φ/angbracketrightbig\nL2. (4.8)\nBoth equations above will be used at our convenience. Letting φ=win (4.8), we have\n1\n2d\ndt/ba∇dblw/ba∇dbl2\nL2+β2/ba∇dbl∆w/ba∇dbl2\nL2≤ |β1|/ba∇dbl∇w/ba∇dbl2\nL2+β3/ba∇dblw/ba∇dbl2\nL2+β3/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,w/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle\n+β4/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tu×∆u−v×∆v,w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/vextendsingle\n+β5/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle. (4.9)\nWewillnowestimatetheinnerproductsontheright-handside. Fort hefirstinnerproduct,\napplying ( 3.15) yields\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,w/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2. (4.10)\nFor the second inner product on the right-hand side of ( 4.9), we have by using H¨ older’s\ninequality and Young’s inequality\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tu×∆u−v×∆v,w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle=β4/vextendsingle/vextendsingle/an}b∇acketle{tu×∆w+w×∆v,w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle=β4/vextendsingle/vextendsingle/an}b∇acketle{tu×∆w,w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/lessorsimilar/ba∇dblu/ba∇dblL∞/ba∇dbl∆w/ba∇dblL2/ba∇dblw/ba∇dblL2\n≤C/ba∇dblu/ba∇dbl2\nL∞/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2, (4.11)GLOBAL SOLUTIONS OF LLBAR EQUATION 23\nfor anyǫ >0. For the last inner product in ( 4.9), applying ( 3.15), then using Young’s\ninequality yield\nβ5/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2 (4.12)\nfor anyǫ >0. Inserting ( 4.10), (4.11) and (4.12) into (4.9), and choosing ǫsufficiently small,\nwe obtain\nd\ndt/ba∇dblw/ba∇dbl2\nL2+/ba∇dbl∆w/ba∇dbl2\nL2/lessorsimilar/ba∇dbl∇w/ba∇dbl2\nL2+/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞+/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2\n/lessorsimilar/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞+/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2\n/lessorsimilar/parenleftbig\n1+/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2, (4.13)\nwhere we used ( 3.8) and Young’s inequality. Choosing ǫsufficiently small, rearranging the\nabove equation, integrating over (0 ,t), and using Gronwall’s inequality, we infer ( 2.7). Under\nassumption ( 2.9), uniqueness then follows.\nProof of ( 2.8):\nThe case r= 1: Taking φ=−∆win (4.7), and integrating by parts (for the terms with\ncoefficient β1andβ2) we have (after rearranging the terms)\n1\n2d\ndt/ba∇dbl∇w/ba∇dbl2\nL2+β2/ba∇dbl∇∆w/ba∇dbl2\nL2\n≤ |β1|/ba∇dbl∆w/ba∇dbl2\nL2+β3/ba∇dbl∇w/ba∇dbl2\nL2+β3/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle\n+β4/vextendsingle/vextendsingle/an}b∇acketle{tu×∇u−v×∇v,∇∆w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle+β5/vextendsingle/vextendsingle/angbracketleftbig\n∇(|u|2u−|v|2v),∇∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle.(4.14)\nWe will estimate the inner products on the right-hand side. For the fi rst inner product, it\nfollows successively from ( 4.12), (3.9), and Young’s inequality that\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+/ba∇dbl∇w/ba∇dbl2\nL2+ǫ/ba∇dbl∇∆w/ba∇dbl2\nL2,(4.15)\nfor anyǫ >0. For the second inner product, applying ( 3.16) then using Young’s inequality\nyield\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tu×∇u−v×∇v,∇∆w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dblu/ba∇dbl2\nL∞/ba∇dbl∇w/ba∇dbl2\nL2+/ba∇dblw/ba∇dbl2\nL2/ba∇dbl∇v/ba∇dbl2\nL∞+ǫ/ba∇dbl∇∆w/ba∇dbl2\nL2,\n(4.16)\nfor anyǫ >0. For the third inner product on the right-hand side of ( 4.14), applying ( 3.15)\nthen using Young’s inequality give\nβ5/vextendsingle/vextendsingle/angbracketleftbig\n∇(|u|2u−|v|2v),∇∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH1+/ba∇dblv/ba∇dbl2\nH1/parenrightbig/parenleftbig\n/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH1\n+ǫ/ba∇dbl∇∆w/ba∇dbl2\nL2, (4.17)\nfor anyǫ >0. Inserting ( 4.15), (4.16) and (4.17) into (4.14), integrating over (0 ,t), and\nchoosing ǫsufficiently small, we obtain\n/ba∇dbl∇w(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∇∆w(s)/ba∇dbl2\nL2ds\n/lessorsimilar/ba∇dbl∇w(0)/ba∇dbl2\nL2+/integraldisplayt\n0α(s)/ba∇dblw(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0α(s)/ba∇dbl∇w(s)/ba∇dbl2\nL2ds\n/lessorsimilar/ba∇dbl∇w(0)/ba∇dbl2\nL2+/ba∇dblw(0)/ba∇dbl2\nL2/integraldisplayt\n0α(s)exp/parenleftbigg/integraldisplayt\n0α(τ)dτ/parenrightbigg\nds+/integraldisplayt\n0α(s)/ba∇dbl∇w(s)/ba∇dbl2\nL2ds,24 AGUS L. SOENJAYA AND THANH TRAN\nwhere\nα(s) :=/parenleftbig\n/ba∇dblu(s)/ba∇dbl2\nH1+/ba∇dblv(s)/ba∇dbl2\nH1/parenrightbig/parenleftbig\n/ba∇dblu(s)/ba∇dbl2\nH2+/ba∇dblv(s)/ba∇dbl2\nH2/parenrightbig\n+/ba∇dblu(s)/ba∇dbl2\nH3+/ba∇dblv(s)/ba∇dbl2\nH3\nand where in the last step we used ( 2.7) which was proved above in this proof. Note that for\nu0,v0∈H1, we have u,v∈L∞(0,T;H1)∩L2(0,T;H3), and thus (by Gagliardo–Nirenberg\ninequality)/integraldisplayt\n0α(s)ds/lessorsimilar/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH3+/ba∇dblv(s)/ba∇dbl2\nH3ds/lessorsimilar1.\nGronwall’s inequality then yields\n/ba∇dbl∇w(t)/ba∇dbl2\nL2≤C/ba∇dblw(0)/ba∇dbl2\nH1,\nwhich implies ( 2.8).\nThe case r= 2: Taking φ=∂twin (4.8) and using integration by parts for the term with\ncoefficient β5give\n/ba∇dbl∂tw/ba∇dbl2\nL2+β2d\ndt/ba∇dbl∆w/ba∇dbl2\nL2≤β1d\ndt/ba∇dbl∇w/ba∇dbl2\nL2+β3d\ndt/ba∇dblw/ba∇dbl2\nL2+β3/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle\n+β4/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tu×∆u−v×∆v,∂tw/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/vextendsingle\n+β5/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n∆(|u|2u−|v|2v),∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle. (4.18)\nNote that when using integration by parts, the integrals on the bou ndary vanish due to the\nboundary property of wand (3.3). Each inner product on the right-hand side of ( 4.18) can\nbe estimated as follows. For the first inner product, similarly to ( 4.12) we have\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∂tw/ba∇dbl2\nL2 (4.19)\nfor anyǫ >0. For the second inner product, applying ( 3.16) and Young’s inequality yield\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tu×∆u−v×∆v,∂tw/an}b∇acket∇i}htL2/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH2+ǫ/ba∇dbl∂tw/ba∇dbl2\nL2.(4.20)\nFor the last inner product in ( 4.18), by (3.15) and Young’s inequality, we have\nβ5/vextendsingle/vextendsingle/angbracketleftbig\n∆(|u|2u−|v|2v),∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl4\nH2+/ba∇dblv/ba∇dbl4\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH2+ǫ/ba∇dbl∂tw/ba∇dbl2\nL2.(4.21)\nInserting ( 4.19), (4.20) and (4.21) into (4.18), integrating over (0 ,t), and choosing ǫ >0\nsufficiently small, we obtain\n/integraldisplayt\n0/ba∇dbl∂tw(s)/ba∇dbl2\nL2ds+/ba∇dbl∆w(t)/ba∇dbl2\nL2/lessorsimilar/ba∇dblw(0)/ba∇dbl2\nH2+/integraldisplayt\n0β(s)/ba∇dblw(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0β(s)/ba∇dbl∆w(s)/ba∇dbl2\nL2ds,\nwhere\nβ(s) := 1+/ba∇dblu(s)/ba∇dbl4\nH2+/ba∇dblv(s)/ba∇dbl4\nH2.\nGronwall’s inequality then yields\n/ba∇dbl∆w(t)/ba∇dbl2\nL2≤C/ba∇dblw(0)/ba∇dbl2\nH2.\nThis, together with ( 2.7) shown for r= 0 and ( 2.8) forr= 1, gives the required inequal-\nity (2.8) forr= 2.GLOBAL SOLUTIONS OF LLBAR EQUATION 25\nThe case r= 3: We now take φ=−∆∂twin (4.8). Using integration by parts for all\nterms, we have\n/ba∇dbl∂t∇w/ba∇dbl2\nL2+β1d\ndt/ba∇dbl∆w/ba∇dbl2\nL2+β2d\ndt/ba∇dbl∇∆w/ba∇dbl2\nL2\n=β3d\ndt/ba∇dbl∇w/ba∇dbl2\nL2−β3/angbracketleftbig\n∇(|u|2u−|v|2v),∇∂tw/angbracketrightbig\nL2\n+β4/an}b∇acketle{t∇u×∆u−∇v×∆v,∇∂tw/an}b∇acket∇i}htL2+β4/an}b∇acketle{tu×∇∆u−v×∇∆v,∇∂tw/an}b∇acket∇i}htL2\n−β5/angbracketleftbig\n∇∆(|u|2u−|v|2v),∇∂tw/angbracketrightbig\nL2. (4.22)\nEach inner product on the right-hand side of ( 4.22) can be estimated as follows. For the\nfirst inner product, similarly to ( 4.17) we have\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n∇(|u|2u−|v|2v),∇∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH1+/ba∇dblv/ba∇dbl2\nH1/parenrightbig/parenleftbig\n/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH1\n+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2 (4.23)\nfor anyǫ >0. For the second inner product, applying H¨ older’s and Young’s ineq uality yields\nβ4/vextendsingle/vextendsingle/an}b∇acketle{t∇u×∆u−∇v×∆v,∇∂tw/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/lessorsimilar/ba∇dbl∇u×∆u−∇v×∆v/ba∇dbl2\nL2+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2\n/lessorsimilar/ba∇dbl∇u/ba∇dbl2\nL∞/ba∇dbl∆w/ba∇dbl2\nL2+/ba∇dbl∇w/ba∇dbl2\nL∞/ba∇dbl∆v/ba∇dbl2\nL2+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH3+/ba∇dblv/ba∇dbl2\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH3+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2 (4.24)\nfor anyǫ >0, where in the last step we used the Sobolev embedding H2⊂L∞. For the\nthird inner product we have by H¨ older’s inequality, Young’s inequality , and (3.16)\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tu×∇∆u−v×∇∆v,∇∂tw/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/lessorsimilar/ba∇dblu/ba∇dbl2\nL∞/ba∇dbl∇∆w/ba∇dbl2\nL2+/ba∇dblw/ba∇dbl2\nH2/ba∇dbl∇∆v/ba∇dbl2\nL2+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH3/parenrightbig\n/ba∇dblw/ba∇dbl2\nH3+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2 (4.25)\nfor anyǫ >0. For the last inner product in ( 4.22), by using H¨ older’s and Young’s inequality,\nand (3.15), we have\nβ5/vextendsingle/vextendsingle/angbracketleftbig\n∇∆(|u|2u−|v|2v),∇∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl4\nH3+/ba∇dblv/ba∇dbl4\nH3/parenrightbig\n/ba∇dblw/ba∇dbl2\nH3+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2.(4.26)\nDefine\nγ(s) := 1+/ba∇dblu(s)/ba∇dbl4\nH3+/ba∇dblv(s)/ba∇dbl4\nH3.\nInserting ( 4.23), (4.24), (4.25) and (4.26) into (4.22), integrating over (0 ,t), and choosing\nǫ >0 sufficiently small, we obtain\n/ba∇dbl∇∆w(t)/ba∇dbl2\nL2/lessorsimilar/ba∇dblw(0)/ba∇dbl2\nH3+/integraldisplayt\n0γ(s)/parenleftbig\n/ba∇dblw(s)/ba∇dbl2\nL2ds+/ba∇dbl∇w(s)/ba∇dbl2\nL2ds+/ba∇dbl∇∆w(s)/ba∇dbl2\nL2/parenrightbig\nds\n/lessorsimilar/ba∇dblw(0)/ba∇dbl2\nH3+/integraldisplayt\n0γ(s)/ba∇dbl∇∆w(s)/ba∇dbl2\nL2ds,\nwhere we used ( 3.10), (2.7) and (2.8) withr= 1. Gronwall’s inequality yields\n/ba∇dbl∇∆w(t)/ba∇dbl2\nL2≤C/ba∇dblw(0)/ba∇dbl2\nH3\nand the required estimate ( 2.8) forr= 3 then follows.26 AGUS L. SOENJAYA AND THANH TRAN\nExtension from [0 ,T∗] to [0,T] ford= 3: Recall that T∗≤Tford= 3. We will now show\nthat in this case, we also have T∗=T. First, it follows from ( 3.35) that, for r= 1,2,3,\nu∈L∞(0,T∗;Hr)∩L2(0,T∗;Hr+2). (4.27)\nAssume that the following estimate holds (which will be shown in Propos ition4.2later).\n/ba∇dbl∇u(t)/ba∇dbl2\nL2+/ba∇dblu(t)/ba∇dbl4\nL4+/integraldisplayt\n0/ba∇dbl∇∆u(s)/ba∇dbl2\nL2ds\n+/integraldisplayt\n0/vextenddouble/vextenddouble∇(|u(s)|2u(s))/vextenddouble/vextenddouble2\nL2ds+/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl6\nL6ds/lessorsimilar/ba∇dblu0/ba∇dbl2\nH1, t∈[0,T∗].(4.28)\nThen we can repeat the arguments leading to the proof of ( 3.35) withunreplaced by u\nto obtain similar estimates for u, with constant depending on T. Proposition 4.2and (3.35)\nimply that this weak solution uoriginally defined on [0 ,T∗] belongs to C([0,T∗];Hr)∩\nL2(0,T∗;Hr+2), and that u(t,x) remains bounded in this norm as t→T∗from the left.\nTherefore, the technique of continuation of solutions can be applie d and thus the solution u\nexists on the whole interval [0 ,T] for any T >0.\nIt remains to prove ( 4.28).\nProposition 4.2. LetT >0 be arbitrary and T∗be defined as in Proposition 3.8. Letu\nbe the unique weak solution of ( 1.2). Then ( 4.28) holds with a constant depending on T.\nProof.We aim to choose φ=α|u(t)|2u(t) in (2.5), for some positive constant α. Hence,\nwe first consider the nonlinear terms in the resulting equation with th at choice of φ. For the\nterm with coefficient β1, we use ( 3.2) to have\n/angbracketleftbig\n∇u(s),∇/parenleftbig\n|u(s)|2u(s)/parenrightbig/angbracketrightbig\nL2= 2/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2+/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2.(4.29)\nFor the term with coefficient β2, we use integration by parts to have\n/angbracketleftbig\n∆u(s),∆/parenleftbig\n|u(s)|2u(s)/parenrightbig/angbracketrightbig\nL2=−/angbracketleftbig\n∇∆u(s),∇/parenleftbig\n|u(s)|2u(s)/parenrightbig/angbracketrightbig\nL2. (4.30)\nFor the terms involving β3andβ5, it is straightforward to have\n/angbracketleftbig/parenleftbig\n1−|u(s)|2/parenrightbig\nu(s),|u(s)|2u(s)/angbracketrightbig\nL2=/ba∇dblu(s)/ba∇dbl4\nL4−/ba∇dblu(s)/ba∇dbl6\nL6,\n/angbracketleftbig\n∇/parenleftbig\n|u(s)|2u(s)/parenrightbig\n,∇/parenleftbig\n|u(s)|2u(s)/parenrightbig/angbracketrightbig\nL2=/vextenddouble/vextenddouble∇/parenleftbig\n|u(s)|2u(s)/parenrightbig/vextenddouble/vextenddouble2\nL2.\nThe term involving β4vanishes. Altogether, we deduce from choosing φ= 4α|u(t)|2u(t)\nin (2.5) that\nα/ba∇dblu(t)/ba∇dbl4\nL4+8αβ1/integraldisplayt\n0/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2ds+4αβ1/integraldisplayt\n0/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2ds\n−4αβ2/integraldisplayt\n0/angbracketleftbig\n∇∆u(s),∇(|u(s)|2u(s))/angbracketrightbig\nL2ds\n= 4α/angbracketleftbig\nu0,|u(t)|2u(t)/angbracketrightbig\nL2+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL4ds−4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl6\nL6ds\n−4αβ5/integraldisplayt\n0/vextenddouble/vextenddouble∇(|u(s)|2u(s))/vextenddouble/vextenddouble2\nL2ds. (4.31)GLOBAL SOLUTIONS OF LLBAR EQUATION 27\nNext, we choose φ=−2∆u(t) in (2.5) and use integration by parts, noting ( 2.3) so that\nthe term involving β4vanishes. We then have, noting ( 4.29),\n/ba∇dbl∇u(s)/ba∇dbl2\nL2+2β1/integraldisplayt\n0/ba∇dbl∆u(s)/ba∇dbl2\nL2ds+2β2/integraldisplayt\n0/ba∇dbl∇∆u(s)/ba∇dbl2\nL2ds\n= 2/an}b∇acketle{t∇u0,∇u(t)/an}b∇acket∇i}htL2+2β3/integraldisplayt\n0/ba∇dbl∇u(s)/ba∇dbl2\nL2ds\n−4β3/integraldisplayt\n0/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2ds−2β3/integraldisplayt\n0/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2ds\n+2β5/integraldisplayt\n0/angbracketleftbig\n∇∆u(s),∇(|u(s)|2u(s))/angbracketrightbig\nds. (4.32)\nAdding ( 4.31) and (4.32) gives\nα/ba∇dblu(t)/ba∇dbl4\nL4+/ba∇dbl∇u(t)/ba∇dbl2\nL2+2β2/integraldisplayt\n0/ba∇dbl∇∆u(s)/ba∇dbl2\nL2ds+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl6\nL6ds\n−(4αβ2+2β5)/integraldisplayt\n0/angbracketleftbig\n∇∆u(s),∇(|u(s)|2u(s))/angbracketrightbig\nL2ds+4αβ5/integraldisplayt\n0/vextenddouble/vextenddouble∇(|u(s)|2u(s))/vextenddouble/vextenddouble2\nL2ds\n=α/angbracketleftbig\nu0,|u(t)|2u(t)/angbracketrightbig\nL2+/an}b∇acketle{t∇u0,∇u(t)/an}b∇acket∇i}htL2\n−2β1/integraldisplayt\n0/ba∇dbl∆u(s)/ba∇dbl2\nL2ds+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL4ds+2β3/integraldisplayt\n0/ba∇dbl∇u(s)/ba∇dbl2\nL2ds\n−(8αβ1+4β3)/integraldisplayt\n0/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2ds−(4αβ1+2β3)/integraldisplayt\n0/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2ds.(4.33)\nNote that if α=β5/2β2, then the third, ��fth, and sixth terms on the left-hand side add up\nto\n2β2/ba∇dbl∇∆u(s)/ba∇dbl2\nL2−(4αβ2+2β5)/angbracketleftbig\n∇∆u(s),∇(|u(s)|2u(s))/angbracketrightbig\n+4αβ5/vextenddouble/vextenddouble∇(|u(s)|2u(s))/vextenddouble/vextenddouble2\nL2\n=/vextenddouble/vextenddouble/vextenddouble/radicalbig\n2β2∇∆u(s)−/radicalbig\n4αβ5∇(|u(s)|2u(s))/vextenddouble/vextenddouble/vextenddouble2\nL2≥0.\nHence, with this value of αand the use of Young’s inequality, ( 4.33) becomes\nα/ba∇dblu(t)/ba∇dbl4\nL4+/ba∇dbl∇u(t)/ba∇dbl2\nL2+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl6\nL6ds+/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddouble/radicalbig\n2β2∇∆u−/radicalbig\n4αβ5∇(|u|2u)/vextenddouble/vextenddouble/vextenddouble2\nL2ds\n≤C/ba∇dblu0/ba∇dbl4\nL4+ǫ/ba∇dblu(t)/ba∇dbl4\nL4+C/ba∇dbl∇u0/ba∇dbl2\nL2+ǫ/ba∇dbl∇u(t)/ba∇dbl2\nL2\n+2|β1|/integraldisplayt\n0/ba∇dbl∆u(s)/ba∇dbl2\nL2ds+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL4ds+2β3/integraldisplayt\n0/ba∇dbl∇u(s)/ba∇dbl2\nL2ds\n+(8α|β1|+4β3)/integraldisplayt\n0/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2ds+(4α|β1|+2β3)/integraldisplayt\n0/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2ds\n≤C/ba∇dblu0/ba∇dbl2\nH1+ǫ/ba∇dblu(t)/ba∇dbl4\nL4+ǫ/ba∇dbl∇u(t)/ba∇dbl2\nL2,\nwhere in the last step we used the Sobolev embedding H1⊂L4(foru0) and Proposition 3.6\nto boundall the integrals onthe right-handside. Choosing ǫ >0 sufficiently small, we obtain\nthe required estimate. /square28 AGUS L. SOENJAYA AND THANH TRAN\n5.Proof of Theorem 2.3\nProof.For any Banach space X, sinceC0,α2([0,T];X)⊂C0,α1([0,T];X) for 0< α1< α2, it\nsuffices to prove the theorem for α= 1/2 andβ= 1/2−d/8.\nLetT >0 andτ,t∈[0,T] be such that τ < t. Performing integration by parts on ( 2.5)\n(and noting the regularity of the solution ugiven by Theorem 2.2), we have for any φ∈H2,\n/an}b∇acketle{tu(t)−u(τ),φ/an}b∇acket∇i}htL2−β1/integraldisplayt\nτ/an}b∇acketle{t∆u(s),φ/an}b∇acket∇i}htL2ds+β2/integraldisplayt\nτ/angbracketleftbig\n∆2u(s),φ/angbracketrightbig\nL2ds\n=β3/integraldisplayt\nτ/angbracketleftbig\n(1−|u(s)|2)u(s),φ/angbracketrightbig\nL2ds−β4/integraldisplayt\nτ/an}b∇acketle{tu(s)×∆u(s),φ/an}b∇acket∇i}htL2ds\n+β5/integraldisplayt\nτ/angbracketleftbig\n∆(|u(s)|2u(s)),φ/angbracketrightbig\nL2ds.\nTherefore, by H¨ older’s inequality,\n/vextendsingle/vextendsingle/an}b∇acketle{tu(t)−u(τ),φ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle≤ |β1|/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/ba∇dbl∆u(s)/ba∇dblL2ds+β2/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/vextenddouble/vextenddouble∆2u(s)/vextenddouble/vextenddouble\nL2ds\n+β3/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/ba∇dblu(s)/ba∇dblL2ds+β3/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/ba∇dblu(s)/ba∇dbl3\nL6ds\n+β4/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/ba∇dblu(s)×∆u(s)/ba∇dblL2ds\n+β5/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/vextenddouble/vextenddouble∆(|u(s)|2u(s))/vextenddouble/vextenddouble\nL2ds.\nTakingφ=u(t)−u(τ), we obtain\n/ba∇dblu(t)−u(τ)/ba∇dblL2/lessorsimilar/integraldisplayt\nτ/ba∇dbl∆u(s)/ba∇dblL2ds+/integraldisplayt\nτ/vextenddouble/vextenddouble∆2u(s)/vextenddouble/vextenddouble\nL2ds+/integraldisplayt\nτ/ba∇dblu(s)/ba∇dblL2ds\n+/integraldisplayt\nτ/ba∇dblu(s)/ba∇dbl3\nL6ds+/integraldisplayt\nτ/ba∇dblu(s)×∆u(s)/ba∇dblL2ds\n+/integraldisplayt\nτ/vextenddouble/vextenddouble∆(|u(s)|2u(s))/vextenddouble/vextenddouble\nL2ds. (5.1)\nWe will now estimate each term on the right-hand side of ( 5.1). For the linear terms, by\nH¨ older’s inequality and Corollary 3.15,\n/integraldisplayt\nτ/ba∇dbl∆u(s)/ba∇dblL2ds≤ |t−τ|1\n2/ba∇dbl∆u/ba∇dblL2(0,T;L2)/lessorsimilar|t−τ|1\n2,\n/integraldisplayt\nτ/vextenddouble/vextenddouble∆2u(s)/vextenddouble/vextenddouble\nL2ds≤ |t−τ|1\n2/vextenddouble/vextenddouble∆2u/vextenddouble/vextenddouble\nL2(0,T;L2)/lessorsimilar|t−τ|1\n2,\n/integraldisplayt\nτ/ba∇dblu(s)/ba∇dblL2ds≤ |t−τ|1\n2/ba∇dblu/ba∇dblL2(0,T;L2)/lessorsimilar|t−τ|1\n2.\nFor the nonlinear terms on the right-hand side of ( 5.1), by H¨ older’s inequality, Corollary 3.15\nand the Sobolev embedding,\n/integraldisplayt\nτ/ba∇dblu(s)/ba∇dbl3\nL6ds≤ |t−τ|1\n2/ba∇dblu/ba∇dbl3\nL6(0,T;L6)/lessorsimilar|t−τ|1\n2/ba∇dblu/ba∇dbl3\nL∞(0,T;H1)GLOBAL SOLUTIONS OF LLBAR EQUATION 29\n/lessorsimilar|t−τ|1\n2,\n/integraldisplayt\nτ/ba∇dblu(s)×∆u(s)/ba∇dblL2ds≤/integraldisplayt\nτ/ba∇dblu(s)/ba∇dblL∞/ba∇dbl∆u(s)/ba∇dblL2ds\n≤ /ba∇dbl∆u/ba∇dblL∞(0,T;L2)|t−τ|1\n2/ba∇dblu/ba∇dblL2(0,T;L∞)\n≤ |t−τ|1\n2/ba∇dblu/ba∇dblL2(0,T;H2)/ba∇dbl∆u/ba∇dblL∞(0,T;L2)\n/lessorsimilar|t−τ|1\n2,\n/integraldisplayt\nτ/vextenddouble/vextenddouble∆(|u(s)|2u(s))/vextenddouble/vextenddouble\nL2ds≤/integraldisplayt\nτ/vextenddouble/vextenddouble|u(s)|2u(s)/vextenddouble/vextenddouble\nH2ds≤/integraldisplayt\nτ/ba∇dblu(s)/ba∇dbl3\nH2ds\n≤ |t−τ|1\n2/ba∇dblu/ba∇dbl3\nL6(0,T;H2)≤ |t−τ|1\n2/ba∇dblu/ba∇dbl3\nL∞(0,T;H2)\n/lessorsimilar|t−τ|1\n2,\nwhere for the last nonlinear term, we also used ( 3.14). Altogether, we derive from ( 5.1) that\nu∈C0,α(0,T;L2) forα∈(0,1/2].\nFinally, by the Gagliardo–Nirenberg inequality (Theorem 6.2withv=u(t)−u(τ),r= 0,\nq=∞,s1= 0,s2= 2),\n/ba∇dblu(t)−u(τ)/ba∇dblL∞/lessorsimilar/ba∇dblu(t)−u(τ)/ba∇dbl1−d\n4\nL2/ba∇dblu(t)−u(τ)/ba∇dbld\n4\nH2\n/lessorsimilar/ba∇dblu(t)−u(τ)/ba∇dbl1−d\n4\nL2/ba∇dblu/ba∇dbld\n4\nC([0,T];H2)\n/lessorsimilar|t−τ|1\n2−d\n8,\nwhere in the penultimate step we used Theorem 2.2and in the last step we used the previous\npart of this theorem. /square\n6.Appendix\nWe collect in this section a few results which were extensively used in th is paper.\nTheorem 6.1 (Gronwall–Bihari’s inequality [ 6,7]).Letfbe a non-decreasing continuous\nfunction which is non-negative on [0 ,∞) such that/integraltext∞\n11/f(x)dx <∞. LetFbe the anti-\nderivative of −1/fwhich vanishes at ∞. Lety: [0,∞)→[0,∞) be a continuous function\nand letgbe a locally integrable non-negative function on [0 ,∞). Suppose that there exists\ny0>0 such that for all t≥0,\ny(t)≤y0+/integraldisplayt\n0g(s)ds+/integraldisplayt\n0f(y(s))ds.\nLetT∗be the unique solution of the equation\nT∗=F/parenleftbigg\ny0+/integraldisplayT∗\n0g(s)ds/parenrightbigg\n.\nThen for any T′∈(0,T∗), we have\nsup\n0≤t≤T′y(t)≤F−1/parenleftBigg\nF/parenleftBig\ny0+/integraldisplayT′\n0g(s)ds/parenrightBig\n−T′/parenrightBigg\n. (6.1)\nNote that the expression on the right-hand side of ( 6.1) tends to ∞asT′→T∗.30 AGUS L. SOENJAYA AND THANH TRAN\nThe following theorem is a special case of a more general result in [ 8].\nTheorem 6.2 (Gagliardo–Nirenberg inequalities) .Let Ω be a bounded domain of Rdwith\nLipschitz boundary, and let v: Ω→R3. Then\n/ba∇dblv/ba∇dblWr,q≤C/ba∇dblv/ba∇dblθ\nHs1/ba∇dblv/ba∇dbl1−θ\nHs2 (6.2)\nfor allv∈Hs2(Ω), where s1,s2,rare non-negative real numbers satisfying\n0≤s1< s2, θ∈(0,1),0≤r < θs 1+(1−θ)s2,\nandq∈(2,∞] satisfies\n1\nq=1\n2+(s2−s1)θ\nd−s2−r\nd.\nMoreover, when 2 < q <∞, we have\nθ=2q(s2−r)−d(q−2)\n2q(s2−s1).\nTheorem 6.3 (Aubin–Lions–Simon lemma [ 31]).LetX0֒→X ֒→X1be three Banach\nspaces such that the inclusion X0֒→Xis compact and the inclusion X ֒→X1is continuous.\nFor 1≤p,q≤ ∞, let\nWp,q:={v∈Lp(0,T;X0) :vt∈Lq(0,T;X1)}.\n(1) Ifp <∞, thenWp,qis compactly embedded into Lp(0,T;X).\n(2) Ifp=∞andq= 1, then Wp,qis compactly embedded into C([0,T];X).\nTheorem 6.4 (Theorem II.5.14 in [ 7]).LetVandWbe Hilbert spaces. Then the space\n{v∈L2(0,T;V) :∂tv∈L2(0,T;W)}\nis continuously embedded into C([0,T];[V,W]1/2). Here, [ V,W]1/2is the interpolation space\nofVandWwith order1\n2.\nAcknowledgements\nThe authors gratefully thank the anonymous reviewer for the help ful remarks and sugges-\ntions, which greatly improved the clarity of the paper.\nThe first author is supported by the Australian Government Resea rch Training Program\nScholarship awarded at the University of New South Wales, Sydney. The second author is\npartially supported by the Australian Research Council under gran t number DP190101197\nand DP200101866.\nReferences\n[1] F. Alouges and A. Soyeur. On global weak solutions for Landau-L ifshitz equations: existence and\nnonuniqueness. Nonlinear Anal. ,18(1992), 1071–1084.\n[2] Y. Au, M. Dvornik, T. Davison, E. Ahmad, P. S. Keatley, A. Vanst eenkiste, B. Van Waeyenberge, and\nV. V. Kruglyak. Direct excitation of propagating spin waves by focu sed ultrashort optical pulses. Phys.\nRev. Lett. ,110(2013), 097201.\n[3] V. G. Baryakhtar. 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Micro magnetic modeling of anisotropic damp-\ning in magnetic nanoelements. Phys. Rev. B ,88(2013), 054427.\n[15] M. Dvornik, A. Vansteenkiste, and B. Van Waeyenberge. Ther modynamically self-consistent non-\nstochasticmicromagneticmodelfortheferromagneticstate. Applied Physics Letters ,105(2014),162411.\n[16] M. Feischl and T. Tran. Existence of regular solutions of the La ndau–Lifshitz–Gilbert equation in 3d\nwith natural boundary conditions. SIAM J. Math. Anal. ,49(2017), 4470–4490.\n[17] D. A. Garanin. Fokker-Planck and Landau-Lifshitz-Bloch equa tions for classical ferromagnets. Phys.\nRev. B,55(1997), 3050–3057.\n[18] T. Gilbert. A Lagrangian formulation of the gyromagnetic equat ion of the magnetic field. Phys Rev ,\n100(1955), 1243–1255.\n[19] P. Grisvard. Elliptic problems in nonsmooth domains , volume 69 of Classics in Applied Mathematics .\nSociety for Industrial and Applied Mathematics (SIAM), Philadelphia , PA, 2011.\n[20] B. Guo and S. Ding. Landau-Lifshitz equations , volume 1 of Frontiers of Research with the Chinese\nAcademy of Sciences . World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.\n[21] B. L. Guo and M. C. Hong. The Landau-Lifshitz equation of the f erromagnetic spin chain and harmonic\nmaps.Calc. Var. Partial Differential Equations ,1(1993), 311–334.\n[22] S. Guti´ errez and A. de Laire. The Cauchy problem for the Lan dau-Lifshitz-Gilbert equation in BMO\nand self-similar solutions. Nonlinearity ,32(2019), 2522–2563.\n[23] P. Harpes. Uniqueness and bubbling of the 2-dimensional Landa u-Lifshitz flow. Calc. Var. Partial Dif-\nferential Equations ,20(2004), 213–229.\n[24] L. Landau and E. Lifshitz. On the theory of the dispersion of ma gnetic permeability in ferromagnetic\nbodies.Phys. Z. Sowjetunion ,8(1935), 153–168.\n[25] K. N. Le. Weak solutions of the Landau-Lifshitz-Bloch equation .J. Differential Equations ,261(2016),\n6699–6717.\n[26] J. Leliaert, M. Dvornik, J. Mulkers, J. D. Clercq, M. V. Miloˇ sevi ´ c, and B. V. Waeyenberge. Fast\nmicromagnetic simulations on gpu—recent advances made with mumax .Journal of Physics D: Applied\nPhysics,51(2018), 123002.\n[27] Q. Li, B. Guo, F. Liu, and W. Liu. Weak and strong solutions to Lan dau-Lifshitz-Bloch-Maxwell\nequations with polarization. J. Differential Equations ,286(2021), 47–83.\n[28] Y. Li, M. Chen, J. Berakdar, and C. Jia. Gate-controlled magno n-assisted switching of magnetization\nin ferroelectric/ferromagnetic junctions. Phys. Rev. B ,96(2017), 054444.\n[29] A. Meo, W. Pantasri, W. Daeng-am, S. E. Rannala, S. I. Ruta, R . W. Chantrell, P. Chureemart, and\nJ. Chureemart. Magnetization dynamics of granular heat-assiste d magnetic recording media by means\nof a multiscale model. Phys. Rev. B ,102(2020), 174419.\n[30] S. Rannala, A. Meo, S. Ruta, W. Pantasri, R. Chantrell, P. Chur eemart, and J. Chureemart. Models\nof advanced recording systems: A multi-timescale micromagnetic co de for granular thin film magnetic\nrecording systems. Computer Physics Communications ,279(2022), 108462.32 AGUS L. SOENJAYA AND THANH TRAN\n[31] J. Simon. Compact sets in the space Lp(0,T;B).Ann. Mat. Pura Appl. (4) ,146(1987), 65–96.\n[32] M. Strungaru, S. Ruta, R. F. Evans, and R. W. Chantrell. Mode l of magnetic damping and anisotropy\nat elevated temperatures: Application to granular FePt films. Phys. Rev. Appl. ,14(2020), 014077.\n[33] A. Visintin. On Landau-Lifshitz’ equations for ferromagnetism .Japan J. Appl. Math. ,2(1985), 69–84.\n[34] W. Wang. Computer simulation studies of complex magnetic materials . PhD thesis, University of\nSouthampton, October 2015.\n[35] W. Wang, M. Dvornik, M.-A. Bisotti, D. Chernyshenko, M. Beg, M . Albert, A. Vansteenkiste, B. V.\nWaeyenberge, A. N. Kuchko, V. V. Kruglyak, and H. Fangohr. Phe nomenological description of the\nnonlocal magnetization relaxation in magnonics, spintronics, and do main-wall dynamics. Phys. Rev. B ,\n92(2015), 054430.\n[36] T. Weindler, H. G. Bauer, R. Islinger, B. Boehm, J.-Y. Chauleau, and C. H. Back. Magnetic damping:\nDomain wall dynamics versus local ferromagnetic resonance. Phys. Rev. Lett. ,113(2014), 237204.\nSchool of Mathematics and Statistics, The University of New South Wales, Sydney 2052,\nAustralia\nEmail address :a.soenjaya@unsw.edu.au\nSchool of Mathematics and Statistics, The University of New South Wales, Sydney 2052,\nAustralia\nEmail address :thanh.tran@unsw.edu.au"    },    {        "title": "2302.08910v1.Control_of_magnon_photon_coupling_by_spin_torque.pdf",        "content": "Control of magnon-photon coupling by spin torque\nAnish Rai1,\u0003and M. Benjamin Jung\reisch1,y\n1Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, United States\n(Dated: February 20, 2023)\nWe demonstrate the in\ruence of damping and \feld-like torques in the magnon-photon coupling\nprocess by classically integrating the generalized Landau-Lifshitz-Gilbert equation with RLC equa-\ntion in which a phase correlation between dynamic magnetization and microwave current through\ncombined Amp\u0012 ere and Faraday e\u000bects are considered. We show that the gap between two hybridized\nmodes can be controlled in samples with damping parameter in the order of 10\u00003by changing the\ndirection of the dc current density Jif a certain threshold is reached. Our results suggest that an\nexperimental realization of the proposed magnon-photon coupling control mechanism is feasible in\nyttrium iron garnet/Pt hybrid structures.\nI. INTRODUCTION\nCoherent magnon-photon coupling in hybrid cavity-\nspintronics contributed to the advancement of magnon-\nbased quantum information and technologies [1{15]. The\ncollective excitations of an electron spin system in mag-\nnetically ordered media called magnons can couple to mi-\ncrowave photons via dipolar interaction, demonstrating\nlevel repulsion and Rabi oscillations [3]. Strongly cou-\npled magnon-photon systems have been explored to bring\nmany exotic e\u000bects into the limelight, some of which in-\nclude the manipulation of spin currents [16], and bidi-\nrectional microwave-to-optical transduction [17, 18]. In\naddition to the coherent magnon-photon coupling, there\nexists an exciting domain of dissipative magnon-photon\ncoupling where level attraction can be observed, which is\ncharacterized by a coalescence of the hybridized magnon-\nphoton modes [19{25].\nThe theoretical framework of magnon-photon coupling\nis given by the following dispersion relation [26] of the\nhybridized modes:\ne!\u0006=1\n2\u0014\n(e!m+e!c)\u0006q\n(e!m\u0000e!c)2+ 4g2\u0015\n;(1)\nwheree!m=!m\u0000i\u000b!mande!c=!c\u0000i\f!care the\ncomplex resonance frequencies of the magnon and pho-\nton (cavity) modes, respectively. gis the coupling be-\ntween the two modes. \u000band\fare the intrinsic damping\nrates of the magnon and photon modes, respectively. The\nreal and imaginary parts of e!\u0006represent the dispersion\nshape and the linewidth of the coupled modes, respec-\ntively. The second term of the square root in Eq. (1) not\nonly gives the strength of the coupling but also reveals\nthe nature of the coupling. Harder and co-workers [19]\ncarefully introduced a coupling term based on the cavity\nLenz e\u000bect to mitigate the Amp\u0012 ere e\u000bect. However, the\non-demand manipulation of the magnon-photon polari-\nton by spin torques has not been addressed so far.\n\u0003arai@udel.edu\nymbj@udel.eduIn this work, we examine the in\ruence of damping-\nand \feld-like torques in the magnon-photon coupling pro-\ncess. Our results indicate that the magnitude of the\nlevel repulsion (manifested by the frequency gap of the\nhybridized modes) and, hence, the magnon-photon cou-\npling strength can e\u000eciently be controlled by varying the\nmagnitude and the direction of dc current density Jfor\nrealistic parameters of the magnetic properties. By cou-\npling the generalized Landau-Lifshitz Gilbert equation\nwith the RLC equation of the cavity, we show that an\non-demand manipulation of the magnon-photon coupling\nstrength can be achieved for current densities of the order\nas small as 105A/cm2.\nThis article is structured in the following fashion. In\nsection II, we discuss the classical description to model\nour system, in which the ferromagnetic resonance of the\nmagnetic system is strongly coupled to photon resonator\nmode of the microwave cavity. In section III, we intro-\nduce the parameters used for the analysis followed by\na detailed discussion of our \fndings. In section IV, we\nsummarize our work.\nFIG. 1. The schematic of the experimental setup. A pat-\nterned YIG/platinum(Pt) bilayer is the sample under consid-\neration. The dc current is passed through the platinum layer.\nThe microwave current is passed through the cavity and an-\nalyzed using a Vector Network Analyzer (VNA). Here, the\nexternal magnetic \feld is applied along bzdirection.arXiv:2302.08910v1  [cond-mat.mes-hall]  17 Feb 20232\nFIG. 2. Magnon-photon coupling control for an intermediate value of the Gilbert damping parameter \u000band a continuous, low\ncurrent density J: The dispersion ( !\u0000!c) in (a-c) and the linewidth (\u0001 !) in (d-f) are plotted as a function of the \feld detuning\n(!m\u0000!c) for\u000b= 2:27\u000210\u00003. The hybridization of magnon and photon modes is compared for di\u000berent dc current densities\nJ: (a), (d) J = \u00005\u0002105A=cm2, (b), (e) J = 0 A =cm2and (c), (f) J =5 \u0002105A=cm2. The blue and red line represent the two\nhybridized modes. The inset in (b) shows that for larger \feld detuning ( !m\u0000!c), the uncoupled photon mode approaches !c\nmaking (!\u0000!c) approach zero.\nII. CLASSICAL DESCRIPTION\nThe magnetization dynamics in ferromagnetic systems\ncan be described by the generalized Landau-Lifshitz-\nGilbert equation [27{29] given by:\nd~M\ndt=\r~M\u0002~He\u000b\u0000\u000b\nMs \n~M\u0002d~M\ndt!\n+\n\raJ\nMs\u0010\n~M\u0002\u0010\n~M\u0002~ p\u0011\u0011\n\u0000\rbJ\u0010\n~M\u0002~ p\u0011\n;(2)\nwhere~Mis the magnetization vector, Msis the satu-\nration magnetization, ~He\u000bis the e\u000bective magnetic \feld\nincluding external \feld ~H, anisotropy, microwave, and\ndemagnetization \felds, \ris the gyromagnetic ratio, \u000bis\nthe Gilbert damping parameter, ~ pis the spin polarization\nunit vector. Furthermore, the terms proportional to aJ\nandbJare the damping-like torque and \feld-like torque,\nrespectively. The coe\u000ecients aJandbJare de\fned as\n[30]:\naJ=\u0011aJ~\n2eMsd;bJ=\u0011bJ~\n2eMsd; (3)where\u0011aand\u0011bare the damping-like torque e\u000eciency\nand \feld-like torque e\u000eciency, respectively. Jis the\ndc current density, whose polarity determines the direc-\ntions of \feld-like and damping-like torque terms through\nEqs. (2) and (3), ~is the reduced Planck's constant, e\nis the electron charge, and dis the thickness of the fer-\nromagnetic sample. We de\fne the magnetic \feld, mag-\nnetization, and spin polarization unit vectors as ~Ht=\nhx(t)bx+hy(t)by+Hbz,~M=mx(t)bx+my(t)by+Msbzand\n~ p=bz, whereHandMsare the dc magnetic \feld and\nsaturation magnetization, respectively, and hx;y(t) and\nmx;y(t) are the dynamic magnetic \feld and magnetiza-\ntion.\nIf we de\fne the dynamic components, h=hx+ihy\nandm=mx+imy, then Eq. (2) can be reduced to:\n(!\u0000~!m+\r~cJ)m+!sh= 0; (4)\nwhere ~!mis the complex ferromagnetic resonance fre-\nquency de\fned by ~ !m=!m\u0000i\u000b!(where!m'\rHis\nthe ferromagnetic resonance frequency), !s=\rMs, and\n~cJ=bJ\u0000iaJ. The e\u000bective RLC circuit for the cavity3\nFIG. 3. Magnon-photon coupling control for an intermediate value of the Gilbert damping parameter \u000band a pulsed, high\ncurrent density J: The dispersion ( !\u0000!c) in (a-c) and the linewidth (\u0001 !) in (d-f) are plotted as a function of the \feld detuning\n(!m\u0000!c) for\u000b= 2:5\u000210\u00003. The hybridization of magnon and photon modes is compared for di\u000berent dc current densities\nJ: (a), (d) J = \u00005\u0002106A=cm2, (b), (e) J = 0 A =cm2and (c), (f) J =5 \u0002106A=cm2. The blue and red line represent the two\nhybridized modes.\ncan be written as [19]:\nRjx;y(t) +1\nCZ\njx;y(t)dt+Ldjx;y(t)\ndt=V0x;y(t);(5)\nwhere R, L, and C represent the resistance, inductance,\nand capacitance, respectively. V0x;yis the voltage that\ndrives the microwave current. For j=jx+ijyandV0=\nV0x+iV0y, we have [19]\n\u0000\n!2\u0000!2\nc+i2!!c\f\u0001\nj=i!\nLV0; (6)\nwhere!c= 1=p\nLCis the cavity resonance frequency and\n\f= (R=2)p\nC=L is the intrinsic damping of the cavity-\nphoton mode.\nThe microwave magnetic \feld will exert a torque on\nthe magnetization through Amp\u0012 ere's law. The relation\ncan be expressed as:\nhx=KAjy;hy=\u0000KAjx; (7)\nwhereKAis the positive coupling term associated with\na phase relation between jx;yandhx;y. In a similar way,\nthe precessional magnetization will induce a voltage inthe RLC circuit through Faraday induction:\nVx=\u0000KFLdmy\ndt;Vy=KFLdmx\ndt; (8)\nwhereKFis the positive coupling term associated with\na phase relation between Vx;yandmx;y. Combining\nEqs. (4)-(8) gives us the coupled equations of the form:\n\u0012\n!2\u0000!2\nc+i2\f!c! i!2KF\n\u0000i!sKA!\u0000~!m+\r~cJ\u0013\u0012\nj\nm\u0013\n=\u0012\ni!!cj0\n0\u0013\n;(9)\nwherej0=V0p\nC=L. The hybridized eigenmodes are\ncalculated by solving the determinant of Eq. (9). This\nyields the following analytical form [see Supplemental\nMaterial (SM)]:\n~!\u0006=\u0010\n!c\n1+i\f+!m\u0000\u000e\n1+i\u000b\u0011\n\u0006r\u0010\n!c\n1+i\f\u0000!m\u0000\u000e\n1+i\u000b\u00112\n+2!c!sKFKA\n(1+i\u000b)(1+i\f)\n2;\n(10)\nwhere\u000e=\r~cJ. Here,\ris the gyromagnetic ratio and ~ cJ\nis a complex term associated with bJandaJde\fned by\n~cJ=bJ\u0000iaJ.4\nFIG. 4. Variation of coherent magnon-photon coupling (minimum frequency gap between two hybridized modes) for di\u000berent\nvalues of\u000bandJ. For (a), (b), and (c) \u000bis varied from 3 \u000210\u00003to 5\u000210\u00005andJ(continuous) is varied from \u00005\u0002105A=cm2\nto 5\u0002105A=cm2and for (d), (e), and (f) \u000bis varied from 4 \u000210\u00003to 5\u000210\u00005andJ(pulsed) is varied from \u00005\u0002106A=cm2to\n5\u0002106A=cm2. Based on our model, we can distinguish between \feld-like contribution (a) and (d), damping-like contribution\n(b) and (e), and a combination of \feld-like and damping-like contribution to the manipulation of the anticrossing gap (c) and\n(f). For (a) and (d) \u0011a= 0 and\u0011b= 0:05 (pure \feld-like torque e\u000bect), for (b) and (e) \u0011a= 0:2 and\u0011b= 0 (pure damping-like\ntorque e\u000bect), and for (c) and (f) \u0011a= 0:2 and\u0011b= 0:05 (combination of damping-like and \feld-like torque e\u000bects).\nIII. RESULTS AND DISCUSSION\nFor our model we choose the following realistic pa-\nrameters [18, 31{34]. The frequency of the cavity mode\nis selected at !c=2\u0019= 10 GHz with a cavity damping\n\f= 1\u000210\u00004(corresponding to quality factor Q\u00195000).\nThe reduced gyromagnetic ratio ( \r=2\u0019), damping-like\ntorque e\u000eciency ( \u0011a), and \feld-like torque e\u000eciency ( \u0011b)\nare taken as 2 :8\u0002106Hz=Oe, 0:2, and 0:05, respectively.\nFor a Pt/FM bilayer, the typical range of damping-like\ntorque e\u000eciency ( \u0011a) is 0.10 to 0.20 [32, 35, 36] and the\ntypical value of \feld-like torque e\u000eciency ( \u0011b) is\u00190.05\n[33, 37{39]. Due to its low Gilbert damping parame-\nter and high spin density, we choose yttrium iron garnet\n(YIG) as magnetic material. In particular, we consider\na YIG \flm with a thickness t= 2\u000210\u00005cm (smallest\nthickness available commercially) and saturation magne-\ntization,Ms= 144 emu =cm3[18]. For the calculation,\nthe termKFKAis taken as 5\u000210\u00006[19]. For YIG \flms,\ndepending upon the thickness and preparation method, \u000b\nvaries from order 10\u00003to 10\u00005[34, 38, 40{47]. Therefore,\nwe vary\u000bin our model from 3 \u000210\u00003to 5\u000210\u00005. Further-more, we vary Jfrom\u00005\u0002105A=cm2to 5\u0002105A=cm2.\nThe maximum value of chosen current density is at least\none order of magnitude smaller than what is used for\nmagnetic tunnel junctions (MTJs) [48, 49]. Note that, a\ncurrent density of this order of magnitude has previously\nbeen reported for YIG/Pt systems [50] to thermally con-\ntrol magnon-photon coupling in experiment. Reference\n[50] reports that such current density leads to a rise of the\nsystem temperature above 40\u000eC. Negative e\u000bects of heat-\ning on the magnetic properties can be drastically reduced\nby using a pulsed dc current through the Pt layer [51] in-\nstead of using a continuous current. For instance, using a\npulsed current with duty cycle of 50%, heating e\u000bects can\nbe mitigated while reaching reasonable high levels of cur-\nrent density between \u00005\u0002106A=cm2to 5\u0002106A=cm2\n. Such a high value of current density will create an\nOersted \feld and, hence, modify the resonance condi-\ntion. The generated Oersted \feld can be considered as a\ncontribution to the e\u000bective magnetic \feld presented in\nEq. (2). Hence, this \feld will modify the resonance posi-\ntion of the magnon modes in the following way: for one\npolarity of the current density (J), the resonance \feld5\nshifts up, while for the other, it shifts down. Experi-\nmentally, this a\u000bect can be compensated by tuning the\nbiasing magnetic \feld so the resonance frequency remains\nthe same. In the following analysis, we consider two sce-\nnarios: (1) a relatively low continuous current density\nand (2) a higher pulsed current density. The e\u000bects of\nboth conditions on the magnon-photon coupling process\nare compared below. The proposed experimental set up\nand measurement con\fguration is shown in Fig. 1.\nA. Dispersion and Linewidth\nIn Fig. 2 (intermediate value of Gilbert damping pa-\nrameter\u000band continuous, low value of current density\nJ) and Fig. 3 (intermediate value of \u000band pulsed, high\nvalue ofJ), the hybridized mode frequency ( !\u0000!c) and\nlinewidth (\u0001 !) are plotted as a function of the \feld de-\ntuning (!m\u0000!c).\nWe \frst focus on the former, shown in Fig. 2, top pan-\nels: For\u000b= 2:27\u000210\u00003and forJ= 0 A=cm2, we observe\na level attraction of the real part of the eigenvalues [Fig.\n2 (b)] in a small region. For J=\u00005\u0002105A=cm2, a\nsimilar behavior is found [Fig. 2(a)]. However, the be-\nhavior drastically changes for reversed current polarity:\nforJ= 5\u0002105(A=cm2), a gap (a prominent level repul-\nsion) is seen between the hybridized modes. This clearly\nshows that depending upon the strength and direction of\nthe dc current density Jone can tune the gap between\nthe hybridized modes, i.e., transitioning the system into\nthe strong coupling regime. Let us now consider Fig. 3,\ntop panels: A similar but enhanced behavior can be ob-\nserved for higher values of J(i.e.,jJj= 5\u0002106A/cm2)\nand\u000b= 2:5\u000210\u00003[Fig. 3]. As is obvious from Figs. 2\nand 3, there is a shift in the position where the coher-\nent coupling occurs. For negative and positive values of\nJ, the resonance shifts towards the negative and positive\nsides of the \feld detuning ( !m\u0000!c), respectively. This\nshift can be understood by the fact that di\u000berent mag-\nnitudes of \feld-like torques directly a\u000bect the resonance\ncondition as will be discussed in Sec. III B.\nNext, we discuss the lower panels of Figs. 2 and 3. The\nlinewidths of the two hybridized modes distinctly cross\neach other for J= 5\u0002105A=cm2andJ= 5\u0002106A=cm2\nas is expected for a broad coupling region [Fig. 2(f) and\nFig. 3(f)]. This e\u000bect is less distinct for the cases J=\n\u00005\u0002105A=cm2,J=\u00005\u0002106A=cm2andJ= 0 A=cm2.\nHowever, despite the lower number of region in the cross-\ning regime (the crossing is less spread), we emphasize that\nlevel crossings in the linewidths of the hybridized modes\nare are also observed here. We note that the coupling re-\ngion broadens as the current density increases from neg-\native values to positive values [Figs. 2(d,e,f) and Figs. 3\n(d,e,f)] and \fnally a distinct crossing of linewidths is ob-\nserved over a broad range [Fig. 2(f) and Fig. 3 (f)].\nFor special cases discussed in Sec. II of the SM, a\nlevel attraction [Fig. S1 (a,b)] in the real part and level\nrepulsion [Fig. S1 (d,e)] in the imaginary part of theeigenvalues are observed along with exceptional points\n(EPs) [52{55]. For more details on the observed EP we\nrefer to the SM.\nB. Anticrossing gap between hybridized modes\nFigure 4 shows the variation of the anticrossing gap\nbetween the hybridized modes for di\u000berent values of \u000b\nandJ. It is clear that the variation is nonlinear in nature.\nAs is evident from the \fgure, the gap between the two\nhybridized modes becomes smaller for a larger value of\n\u000b. On the other hand, the gap also depends on the dc\ncurrent density J: the value of \u000bfor which the gap is very\nsmall increases if we go from from negative to positive\nvalue of the dc current density. For a positive value of\nJ, we also observe the gap between the hybridized modes\nslowly increases as \u000bincreases and becomes maximum for\na particular value of \u000b, and then decreases if we further\nincrease the value of \u000b, as shown in the inset of Fig 4(f).\nFor the low \u000bregime, the anticrossing gap remains nearly\nthe same for di\u000berent orders of magnitude and directions\nof current density, as is shown in Figs. S2, S3 and S4\nof the SM. However, for the high \u000bregime, we observe\na level repulsion in the real part and a level crossing in\nthe imaginary part of the eigenvalues for di\u000berent orders\nof magnitude and directions of the current density, as is\nshown in Figs. S5, S6 and S7 of the SM. For a di\u000berent\ncoupling strength ( KFKA), we observe a similar trend.\nA positive current density is needed to increase the gap\nbetween the two hybridized modes as is shown in Fig. S8\n(SM).\nIn the following discussion, we chose Gilbert damping\nparameters of \u000b= 2:27\u000210\u00003and\u000b= 2:5\u000210\u00003for\ndi\u000berent orders of magnitude of Jwhere a very small\nanticrossing gap for zero current density is seen, as illus-\ntrated in Fig. 4. For large \u000b(= 4\u000210\u00003) and for very low\n\u000b(= 5\u000210\u00005), the hybridized mode frequency ( !\u0000!c)\nand the linewidth (\u0001 !) plotted as a function of the \feld\ndetuning (!m\u0000!c) are shown in the Fig. S1 and Fig.\nS2 of the SM. Figure 5 shows the variation of the magni-\ntude of the gap between the two hybridized modes with\nrespect to \feld detuning ( !m\u0000!) for\u000b= 2:27\u000210\u00003\nand\u000b= 2:5\u000210\u00003. For a pure \feld-like torque e\u000bect,\nthere is a horizontal shift [as shown in Figs. 5(a) and\n5(d)] of the gap between the hybridized modes towards\nthe positive value of \feld detuning as we go from negative\nto positive values of the current density J. However, for\na damping-like torque e\u000bect, there is a vertical shift [as\nshown in Figs. 5(b) and 5(e)] of the minimum gap (an-\nticrossing gap): the anticrossing gap increases if we go\nfrom negative to positive values of J. For the combined\n\feld-like and damping-like torques e\u000bect, there are both\nhorizontal and vertical shifts as can be seen in Figs. 5(c)\nand 5(f). However, the horizontal shift due to the e\u000bect of\n\feld-like torque, vertical shift due to damping-like torque\nand the combined shift due to both \feld and damping-\nlike torques are more pronounced for higher magnitudes6\nFIG. 5. Variation of the gap between two hybridized modes with respect to the \feld detunings ( !m\u0000!c) for\u000b= 2:27\u000210\u00003\nand continuous low Jfrom \u00005\u0002105A=cm2to 5\u0002105A=cm2[(a),(b) and (c)] and for \u000b= 2:5\u000210\u00003and pulsed high J\nfrom \u00005\u0002106A=cm2to 5\u0002106A=cm2[(d),(e) and (f)]. There is (a),(d) a horizontal shift in the location of the gap for\n\u0011a= 0 and\u0011b= 0:05 (pure \feld-like torque e\u000bect), (b), (e) vertical shift in the location of gap for \u0011a= 0:2 and\u0011b= 0\nand (pure damping-like torque e\u000bect), and (c),(f) horizontal and vertical shifts for \u0011a= 0:2 and\u0011b= 0:05 (combined e\u000bect of\ndamping-like and \feld-like torques).\nofJleading to an unusual behavior as is shown in panel\n(f).\nFinally, we note that introducing the \feld-like and\ndamping-like torques in the Landau-Lifshitz-Gilbert\nequation and coupling it with cavity mode through\ncombined Amp\u0012 ere and Faraday e\u000bects do not produce\nlevel attraction. The coupling term in our analysis\nis not a\u000bected by the \u000eterm [see Eq. (10)], which is\nthe parameter governed by spin torque. This means\nthat transitioning the system from strong coupling to\ndissipative coupling and vice versa cannot be achieved\nby spin-transfer torques.\nIV. SUMMARY\nBy coupling of the generalized LLG equation with the\nRLC equation of the cavity, we revealed the coupling be-\ntween magnon and photon modes under the in\ruence of\ndamping and \feld-like torques. Our results indicate that\nthe magnitude of the level repulsion (manifested by the\nfrequency gap of the hybridized modes) and, hence, themagnon-photon coupling strength can e\u000eciently be con-\ntrolled by varying the magnitude and the direction of dc\ncurrent density Jfor realistic parameters of the magnetic\nproperties. Our model suggests that an on-demand ma-\nnipulation of the magnon-photon coupling strength can\nbe achieved for current densities of the order as small\nas 105A/cm2and an intermediate Gilbert damping of\nthe order 10\u00003. Higher values of Jcan de\fnitely en-\nhance the e\u000bect of damping and \feld-like torques on the\nmagnon-photon coupling provided we use pulses of dc\ncurrent to reduce possible heating e\u000bects. 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Martin6\n1Institute for Solid State Physics, The University of Tokyo, Kashiwa, 277-8581, Japan\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China\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\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan\n6Aix Marseille Univ, Universit´ e de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France\n(Dated: October 31, 2023)\nWe theoretically study spin pumping from a ferromagnetic insulator (FI) into a carbon nanotube\n(CNT) . By employing the bosonization method, we formulate the Gilbert damping induced by the\nFI/CNT junction, which can be measured by ferromagnetic resonance. We show that the increase\nin the Gilbert damping has a temperature dependence characteristic of a Luttinger liquid and is\nhighly sensitive to the Luttinger parameter of the spin sector for a clean interface. We also discuss\nthe experimental relevance of our findings based on numerical estimates, using realistic parameters.\nI. INTRODUCTION\nSpin pumping induced by ferromagnetic resonance\n(FMR) [1, 2] is a fundamental technique in spintronics\nfor generating spin current from a ferromagnet to an ad-\njacent material [3, 4]. While spin pumping has been used\nfor injecting spin into various materials, it can also be uti-\nlized for detecting spin excitations in various systems [5–\n17]. Compared with bulk measurement techniques, such\nas nuclear magnetic resonance (NMR) and neutron scat-\ntering experiments, spin pumping has an advantage in\nsensitivity for nanostructured systems such as surfaces,\nthin films and atomic-layer compounds [5].\nThe study of exotic spin excitations which emerge in\nspecific materials is one of the forefront topics of con-\ndensed matter physics. A typical example is spin exci-\ntation in quasi-one-dimensional interacting electron sys-\ntems, whose low-energy excitation can be described by\nthe Tomonaga-Luttinger liquid [18–20]. Spin excitations\ninherent to the Tomonaga-Luttinger liquid have been\nstudied in carbon nanotubes (CNTs) by using NMR [21–\n23]. While NMR can detect the local spin susceptibility\nin CNTs, the use of spin pumping to detect spin excita-\ntions is expected to provide useful information reflecting\nthe exotic character of the Luttinger liquid, which can-\nnot be captured by NMR. It is thus important to clarify\nwhat kind of information about the Luttinger liquid can\nbe obtained from a spin pumping experiment.\nIn this work, we theoretically formulate the increase\nin the Gilbert damping due to spin pumping in a setup\nin which spin is injected into CNTs. We consider a\nmagnetic junction composed of a ferromagnetic insula-\ntor (FI) and a single-wall CNT (see Fig. 1) and take in-\nterfacial randomness into account with a simple model.\nWe derive an analytic expression for the increase in the\nGilbert damping by utilizing the bosonization method\nand second-order perturbation with respect to the inter-\nfacial exchange coupling.\nWe will focus on the two limiting cases, i.e., a clean in-\nterface and a dirty interface. We show that for both cases\nFIG. 1. Magnetic junction composed of a ferromagnetic in-\nsulator (FI) and a single-wall carbon nanotube (CNT). The\ndimension of the FI is W×W′×d′.\nthe temperature dependence of the increase of the Gilbert\ndamping shows a power-law behavior, with an exponent\nreflecting the Luttinger parameters. For a clean inter-\nface, the exponent includes information on the Luttinger\nparameters in the spin sector and is shown to be sensitive\nto small deviations from unity (which is the value of the\nSU(2) symmetric model in the spin sector). For a dirty\ninterface, the exponent depends on the Luttinger param-\neters of both the spin and charge sectors as in an NMR\nmeasurement. We estimate the increase of the Gilbert\ndamping using realistic parameters and discuss the ex-\nperimental feasibility.\nOur paper is organized as follows. We introduce the\nmicroscopic model of the FI/CNT magnetic junction in\nSec. II. We analytically calculate the increase in the\nGilbert damping in Sec. III and subsequently estimate\nit with realistic parameters in Sec. IV. Finally, we briefly\ndiscuss the experimental relevance of our findings in\nSec. V and summarize our results in Sec. VI. A detailedarXiv:2303.01343v2  [cond-mat.mes-hall]  28 Oct 20232\nderivation of the analytic expressions is given in the two\nAppendices.\nII. MODEL\nLet us consider a junction composed of a CNT and\nFI, whose Hamiltonian is given by H=HCNT+HFI+\nHint. Here, HCNTandHFIdescribe electrons in the CNT\nand FI, respectively, and Hintrepresents the interfacial\nexchange interaction between the CNT and FI. We will\ngive their explicit forms in the subsections that follow.\nA. Carbon nanotube\nThe low-energy Hamiltonian of electrons in CNTs is\ngiven by\nHCNT=HK+HC, (1)\nwhere HKandHCrepresent the kinetic energy and\nthe forward scattering potential due to the screened\nCoulomb interaction, respectively. Using standard con-\nventions [24], the Hamiltonians describing these energies\nof electrons in CNTs are given by\nHK=−ivFZ\ndxX\nrασrψ+\nrασ(x)∂xψrασ(x), (2)\nHC=1\n2Z\ndx dy ρ (x)V(x−y)ρ(y), (3)\nwhere ψrασ(x) is the slowly varying part of the\nfield operator of electrons, vFis the Fermi velocity,\nV(x) is the screened Coulomb potential, and ρ(x) =P\nrασψ†\nrασ(x)ψrασ(x) is the electron density operator.\nThe subscripts, r(=±),α(=±), and σ(=±), repre-\nsent the direction of propagation, the nanotube branch\n(the valley), and the spin orientation, respectively. Using\nthe bosonization method [19, 24], the annihilation oper-\nator describing fermions in the CNT can be expressed in\nterms of bosonic fields, θασ(x) and ϕασ(x), as\nψrασ(x) =ηrασ√\n2πaei(−rθασ(x)+ϕασ(x)), (4)\nwhere ηrασis the Klein factor, and ais a short-length\ncutoff which can be identified with the lattice constant of\nthe CNT. To diagonalize the Hamiltonian, we introduce\nnew bosonic fields for the charge and spin sectors, θjδ(x)\nandϕjδ(x) as\nθασ(x) =1\n2X\njδhjδ(α, σ)θjδ(x), (5)\nϕασ(x) =1\n2X\njδhjδ(α, σ)ϕjδ(x), (6)\nwhere δ(=±) represents symmetric/antisymmetric\nmodes, j(=c, s) indicates the charge/spin mode, hc+=1,hc−=α,hs+=σ, and hs−=ασ. The Hamiltonian\nof the CNTs can be written as\nHCNT=X\nj,δvjδ\n2πZ\ndx[K−1\njδ(∂xθjδ)2+Kjδ(∂xϕjδ)2],(7)\nwhere Kjδis the Luttinger parameter and vjδ=vF/Kjδ.\nB. Ferromagnetic insulator\nWe consider a bulk FI described by the quantum\nHeisenberg model and employ the spin-wave approxi-\nmation assuming that the temperature is much lower\nthan the magnetic transition temperature and the mag-\nnitude of the localized spin, S0, is much larger than\none [8, 9, 11, 15–17, 25]. In this situation, the Hamil-\ntonian for the FI is approximately written as a superpo-\nsition of magnon modes:\nHFI=X\nkℏωkb†\nkbk, (8)\nwhere bkis the annihilation operator of magnons, ℏωk=\nDk2+ℏγghdcis the magnon dispersion, Dis spin stiffness,\nγgis the gyromagnetic ratio, and hdcis the static mag-\nnetic field. We will only focus on uniform spin precession\ninduced by external microwaves. For this purpose, it is\nsufficient to consider the magnon mode of k=0with the\nsimplified Hamiltonian\nHFI=ℏω0b†\n0b0. (9)\nMicrowave absorption in FMR can be related to the\nimaginary part of the retarded spin correlation function,\nwhich is defined as\nGR(ω) =−i\nℏZ∞\n0dt ei(ω+iδ)t⟨[S+\n0(t), S−\n0]⟩, (10)\nwhere S+\n0=√2S0b0and S−\n0=√2S0b†\n0are spin\nladder operators of the FI for k=0andS+\n0(t) =\neiHt/ℏS+\n0e−iHt/ℏ. For an isolated bulk FI, the spin sus-\nceptibility is calculated as:\nGR\n0(ω) =2S0/ℏ\nω−ω0+iδ. (11)\nIn real experiments, the FMR linewidth is finite due to\nthe the Gilbert damping. To represent this finite spin\nrelaxation in the bulk FI, we introduce a phenomeno-\nlogical dimensionless parameter αGand express the spin\ncorrelation function as\nGR\n0(ω) =2S0/ℏ\nω−ω0+iαGω. (12)3\nC. Interfacial exchange interaction\nNow let us consider the interfacial exchange interaction\nbetween the FI and the CNT with the Hamiltonian,\nHint=S+\n0s−+S−\n0s+, (13)\nwhere s±is the spin ladder operator of the CNT, defined\nas\ns−=r\n1\nNFIX\nr,r′X\nα,α′ZW\n0dx J(x)\n×e−i(α−α′)kFx−i(r−r′)qFxψ†\nrα−(x)ψr′α′+(x),(14)\nands+= (s−)†. Here, Wis the length of the interface,\nJ(x) is the interfacial exchange coupling, NFIis the num-\nber of unit cells in the FI, kFis the Fermi wavenumber,\nandqF(≪kF) is the momentum mismatch associated\nwith the two modes. Because the interfacial exchange\ncoupling J(x), which is induced by quantum mechanical\nmixing between CNT and FI, is sensitive to distances of\natoms across the junction, we assumed that it depends\non the position xdue to random atomic configuration\nnear the interface. A simplified model for randomness in\nJ(x) will be accounted for in the next section.\nIII. FORMULATION\nA. Gilbert damping\nUsing second-order perturbation with respect to the\ninterfacial exchange coupling, the spin susceptibility is\ncalculated as\nG(iωn) =1\nG0(iωn)−1−Σ(iωn)(15)\nΣ(iωn) =−1\nℏZℏβ\n0dτeiωnτ⟨Tτs+(τ)s−(0)⟩ (16)\nwhere s±(τ) = eHCNTτ/ℏs±e−HCNTτ/ℏ. The retarded\nspin correlation function is obtained by analytic contin-\nuation iωn→ω+iδas\nGR(ω) =2S0/ℏ\nω−(ω0+δω0) +i(αG+δαG)ω0,(17)\nδω0\nω0≃2S0\nℏω0Re ΣR(ω0), (18)\nδαG≃ −2S0\nℏω0Im ΣR(ω0), (19)\nwhere ΣR(ω) is the retarded self-energy defined by\nΣR(ω) =Z\ndt eiωtΣR(t), (20)\nΣR(t) =−iθ(t)\nℏ⟨[s+(t), s−(0)]⟩, (21)θ(t) is the step function, and αG+δαG≪1 has been\nassumed. In our work, we focus on the increase in the\nGilbert damping due to the junction, δαG, which is writ-\nten in terms of the dynamic spin susceptibility of CNTs.\nB. Self-energy of electrons in CNTs\nBy substituting Eq. (14) into Eq. (21), we obtain\nΣR(t) =−i\nℏθ(t)2S0\nNFIX\nr,r′X\nα,α′ZW\n0dxZW\n0dy⟨J(x)J(y)⟩imp\n×e−i(kF(α−α′)+qF(r−r′))(x−y)Crαr′α′(x, y, t ),(22)\nCrαr′α′(x, y, t ) =⟨[ψ†\nrα,+(x, t)ψr′α′,−(x, t),\nψ†\nr′α′,−(y,0)ψrα,+(y,0)]⟩0. (23)\nHere, ⟨···⟩ impindicates a random average for the in-\nterfacial exchange coupling. For simplicity, we assume\nthat the exchange coupling follows a Gaussian distribu-\ntion whose average and variance are given by\n⟨J(x)⟩imp=J1, (24)\n⟨δJ(x)δJ(y)⟩imp=J2\n2aδ(x−y), (25)\nwhere δJ(x) =J(x)− ⟨J(x)⟩imp. Here, J1andJ2repre-\nsent respectively the average and the standard deviation\nof the distribution. The ratio J2/J1reflects the random-\nness of the interfacial exchange coupling. In particular,\nthe case of J2/J1= 0 corresponds to a clean interface\nwithout randomness.\nAccordingly, the self-energy is calculated as\nΣR(t) = ΣR\n1(t) + ΣR\n2(t), (26)\nΣR\n1(t) =−iθ(t)2S0J2\n1\nℏNFIX\nr,r′,α,α′ZW\n0dxZW\n0dy\n×e−i(kF(α−α′)+qF(r−r′))(x−y)Crαr′α′(x, y, t ),(27)\nΣR\n2(t) =−iθ(t)2S0J2\n2a\nℏNFIX\nr,r′,α,α′ZW\n0dx C rαr′α′(x, x, t ).\n(28)\nSince the integrand of ΣR\n1(t) includes a rapidly oscillating\npart as a function of ( x−y), the integral is negligibly\nsmall except for the case of α=α′andr=r′. There,\nwe obtain\nΣR\n1(t) =−iθ(t)2S0J2\n1\nℏNFIX\nr,αZW\n0dxZW\n0dy C rαrα(x, y, t ).\n(29)\nWe should note that ΣR\n1(t) corresponds to the process\nof electron creation and annihilation in the same branch\nand represents momentum-conserving spin relaxation for\na clean junction. In contrast, ΣR\n2(t) represents spin re-\nlaxation for a “dirty” junction that is independent of the4\nelectron momentum. Here, the word “dirty” means that\nduring spin exchange process the momentum of electrons\nin the CNT is not conserved and transitions between dif-\nferent branches of valleys and propagation directions are\nallowed. The following discussion will consider two lim-\niting cases for the interface. For the clean interface limit\n(J1≫J2), the magnon self-energy is represented with\nΣR\n1(t), while in the dirty interface limit ( J1≪J2), it is\nrepresented with ΣR\n2(t).\nC. Clean interface\nSince the correlation function Crαr′α′(x, y, t ) can be\ncalculated using the bosonization method (see Ap-\npendix A), the self-energy ΣR\n1(t) can be obtained ana-\nlytically. Therefore, the corresponding increase in the\nGilbert damping is obtained as\nδαG,1=−2S0\nℏω0Im ΣR\n1(ω0)\n=−4S0J2\n1\nℏ2ω0(2πa)2NFIZW\n0dxZW\n0dyZ∞\n0dtsinω0t\n×Im\"\u0012sinh(iπa/β ℏvF)\nsinh(π(ia−(x−y)−vFt)/βℏvF)\u0013γ−1\n×\u0012sinh(iπa/β ℏvF)\nsinh(π(ia+ (x−y)−vFt)/βℏvF)\u0013γ+1#\n,(30)\nγ≡Ks+\n4+Ks−\n4+1\n4Ks++1\n4Ks−. (31)\nAfter analytic integration with respect to t(see Ap-\npendix B for details), we obtain\nδαG,1=2\nπΓ(γ)2\nΓ(2γ)S0J2\n1Wa\nℏ2v2\nFNFI\u00122πa\nβℏvF\u00132γ−3\n×I(πW/β ℏvF, γ), (32)\nI(w, γ) =1\nwZw\n0dz′Zz′\n0dz e−2(γ−1)z\n×F(γ−1, γ,2γ; 1−e−4z), (33)\nwhere F(a, b, c ;x) is the hypergeometric function.\nD. Dirty interface\nThe self-energy ΣR\n2(t) can be obtained in a similar\nway as above. The corresponding increase in the Gilbertdamping is given by\nδαG,2=−2S0\nℏω0Im ΣR\n2(ω0)\n=−S0J2\n2aW\nℏ2ω0(πa)2NFIX\nr,r′,α,α′Z∞\n0dtsinω0t\n×Im\"\u0012sinh(iπa/β ℏvF)\nsinh(π(ia−vFt)/βℏvF)\u00132γrαr ′α′#\n,(34)\nγrαr′α′=γ1δr,r′δα,α′+γ2δr,−r′δα,α′\n+γ3δr,r′δα,−α′+γ4δr,−r′δα,−α′,(35)\nγ1= (Ks++Ks−+ 1/Ks++ 1/Ks−)/4, (36)\nγ2= (Kc++Kc−+ 1/Ks++ 1/Ks−)/4, (37)\nγ3= (Ks++Kc−+ 1/Kc++ 1/Ks−)/4, (38)\nγ4= (Kc++Ks−+ 1/Ks++ 1/Kc−)/4. (39)\nWe should note that δαG,2is proportional to W, since the\nspin relaxation rate is determined through spatially-local\nspin exchange in the dirty interface and is proportional\nto the number of spin-exchange channels. After analytic\nintegration with respect to t(see Appendix B for details),\nwe obtain\nδαG,2=1\n2πS0J2\n2aW\nℏ2v2\nFNFI\n×X\nr,r′,α,α′Γ(γrαr′α′)2\nΓ(2γrαr′α′)\u00122πa\nβℏvF\u00132γrαr ′α′−2\n.(40)\nIV. NUMERICAL ESTIMATE\nNext, we evaluate numerically the increase in the\nGilbert damping by using realistic experimental parame-\nters. While the increase was formulated for a single CNT\nin the previous section, to increase the signal, it would be\nmore useful if we considered a junction with a bundle of\nCNTs. Thus, in the following, we will consider a junction\ncomposed of a FI and a bundle of CNTs with an area of\nW×W′(see Fig. 1) and multiply δαG,1andδαG,2by\nthe number of CNTs in the junction, NCNT=W′/d(d:\nthe diameter of CNTs).\nThe parameters are given in Table I. The Fermi ve-\nlocity vF, lattice constant a, diameter d, Luttinger pa-\nrameters of CNTs, Kc+,Kc−, and Ks−, are taken from\nRefs. [20, 24, 26]. The value of Ks+is an experiment\nresult [22] under a magnetic field of 3 .6 T[27]. The spin\namplitude S0and the lattice constant a′are determined\nby assuming that the FI is made from yttrium iron gar-\nnet (YIG). The interfacial exchange coupling ( J1orJ2)\nis roughly estimated to be 2 K [28]. The number of unit\ncells is estimated as NFI=WW′d′/a′3, where d′is the\nthickness of the FI.5\nTABLE I. Parameters used for the numerical estimate.\nMicrowave frequency ω0 1 GHz\nFermi velocity of CNT vF 106m/s\nLattice constant of CNT a 2.46˚A\nDiameter of CNT d 1.5 nm\nAmplitude of spins of FI S0 10\nLattice constant of FI a′12.376˚A\nThickness of FI d′10 nm\nInterfacial exchange couplings J1,J2\nclean interface J1= 2 K, J2= 0\ndirty interface J1= 0 K, J2= 1,2,3 K\nLuttinger parameters Kc+ 0.20\nKs+ 1.07\nKc−,Ks−1\n100 1010-2\n10-3\n10-4\n3 300\nFIG. 2. Temperature dependence of increase in the Gilbert\ndamping, δαG,1, for a clean interface ( J1≫J2).\nA. Clean interface\nThe estimated increase in the Gilbert damping for a\nclean interface ( J1= 2 K ≫J2) is shown in Fig. 2 as\na function of temperature. While δαG,1is proportional\nto 1/Tat high temperatures, it is almost constant at\nlow temperatures. The crossover temperature for a fixed\nlength Wis given by T∗=g(γ)ℏvF/(kBW) (kB: Boltz-\nmann constant), which is proportional to 1 /W. The fac-\ntorg(γ), which depends only on γ, is explicitly shown\nlater. The increase in the Gilbert damping is shown as a\nfunction of the junction length Win Fig. 3. While δαG,1\nis proportional to Wfor a short junction, it is almost\nconstant for a long junction. The crossover length for a\nfixed temperature Tis given by W∗=g(γ)ℏvF/(kBT).\nIn the present estimate, the condition Lth≪vF/ω0\nalways holds, where Lth=ℏvF/kBTis a thermal length.\nUnder this condition, the increase in the Gilbert damping\nbecomes independent of ω0and is approximately given\n3K\n10K\n30K\n100K\n300K\n10-310-410-510-610-2\n10-510-3\n10-4\nFIG. 3. Junction-length dependence of increase in the Gilbert\ndamping, δαG,1, for a clean interface ( J1≫J2).\nby\nδαG,1=Γ(γ)2\nΓ(2γ)S0J2\n1a′3a\n(ℏvF)2dd′\u00122πa\nLth\u00132γ−3\nf(γ, πW/L th),\n(41)\nf(γ, w) =(\nw/π, (w/π≪g(γ)),\ng(γ),(w/π≫g(γ)),(42)\ng(γ) =2\nπZ∞\n0dz e−2(γ−1)zF(γ−1, γ,2γ; 1−e−4z).\n(43)\nFrom this analytic expression, we obtain\nδαG,1∝(\nT2γ−2W, (W≪g(γ)Lth),\nT2γ−3g(γ),(W≫g(γ)Lth).(44)\nThe exponent γ= (Ks++Ks−+K−1\ns++K−1\ns−)/4 cor-\nresponds to unity when Ks+=Ks−= 1. Even in the\npresent estimate employing Ks+= 1.07, the exponent is\nalmost unity ( γ= 1.00114). By setting γ= 1, we can\nreproduce the power in the temperature and junction-\nlength dependence of δαG,1shown in Figs. 2 and 3.\nFinally, let us discuss the factor g(γ). If γis slightly\nlarger than 1 as in the present estimate, the geometric\nfunction is approximated as F(γ−1, γ,2γ;x)≃1. Then,\nthe factor g(γ) is approximately given as\ng(γ) =1\nπ(γ−1). (45)\nThis expression indicates that the increase in the Gilbert\ndamping in the high-temperature limit ( T≫T∗) or the\nlong-junction limit ( W≫W∗) is highly sensitive to the\ndeviation of γfrom unity. The crossover temperature\nT∗and the crossover length W∗also include the factor6\n100 1010-5\n10-810-6\n10-7\n3 300\nFIG. 4. Temperature dependence of increase in the Gilbert\ndamping, δαG,2, for a dirty interface ( J2≫J1). The three\nlines correspond to J2= 1, 2, and 3 K, respectively.\ng(γ)∝(γ−1)−1. Thus, the increase in the Gilbert damp-\ning can be used to investigate small deviations of γfrom\nunity. Then, the Luttinger parameter Ks,+in the spin\nsector can also be determined from Eq. (31) if we know\nwhether it is greater or less than unity. We note that in\nthe NMR measurement [22] Ks,+decreases as the mag-\nnetic field increases. Using this experimental tendency,\nwe expect that Ks,+can be determined uniquely.\nB. Dirty interface\nNext, we consider a dirty interface ( J2≫J1). Figure 4\nshows the increase of the Gilbert damping, δαG,2, as a\nfunction of the temperature for J2= 1, 2, and 3 K. In\nthis case, δαG,2is proportional to T−0.43in the whole\ntemperature range and shows a nontrivial exponent in-\nherent to the Tomonaga-Luttinger liquid.\nThe condition Lth≪vF/ω0also holds for a dirty in-\nterface. Therefore, δαG,2can be approximated as\nδαG,2=1\n2πS0J2\n2aa′3\n(ℏvF)2dd′\n×X\nr,r′,α,α′Γ(γrαr′α′)2\nΓ(2γrαr′α′)\u00122πa\nLth\u00132γrαr ′α′−2\n.(46)\nNoting that a≪Lth, the factor (2 πa/L th)2γrαr ′α′in\nEq. (46) is largely reduced as γrαr′α′increases. There-\nfore, in the sum of Eq. (46), it is sufficient to keep the\nterms in which γrαr′α′takes a minimum value. In the\npresent estimate, γrαr′α′is given by Eq. (35) with\n(γ1, γ2, γ3, γ4) = (1 .001,0.784,1.001.0.784). (47)\nUpon setting the minimum exponent to be γmin= 0.784,\nwe obtain δαG,2∝T2γmin−2=T−0.432, which is consis-tent with the numerical results shown in Fig. 4. There-\nfore, the nontrivial exponent inherent to the Tomonaga-\nLuttinger liquid appears in spin pumping through a dirty\njunction. Note that the approximate expression is inde-\npendent of the junction length Wfor a fixed thickness,\nsince the W-linear factor in NFI=WW′d/a′3cancels\nout the factor of Win Eq. (34).\nThe equation for the increase in the Gilbert damping\nfor the dirty interface has almost the same form as that\nfor 1/T1Tin NMR experiments where T1is the longitu-\ndinal relaxation time of nuclear spins[21–23]. Therefore,\nthe power law of the temperature dependence for the\ndirty interface is the same as in NMR experiments. This\nis because the spin transfer occurs at a spatially localized\npoint due to the impurity average at the dirty interface,\nleading to the same situation as the NMR experiment in\nwhich 1 /T1Tis related to the local dynamic spin suscep-\ntibility.\nV. EXPERIMENTAL RELEVANCE\nWe estimated the increase in the Gilbert damping δαG\nin two limiting situations, i.e., clean and dirty interfaces.\nIf we choose YIG as the ferromagnet, δαGshould be\nroughly in the range 10−5–10−2, because it should be\ncomparable to the Gilbert damping of bulk YIG, αG,\nwhich is of order of 10−5–10−3. For a clean interface,\nδαGis large enough to be measured in FMR experiments\n(see Figs. 2 and 3). Note that δαGcan be reduced by in-\ncreasing the thickness of YIG (denoted by d′). On the\nother hand, for a dirty interface, δαGis too small for it to\nbe observable by spin pumping (see Fig. 4). However, we\nwill moderate judgement on the possibility of observing\nδαGfor a dirty interface, because detailed information\non the interfacial exchange coupling is still lacking. We\nshould note that in the present modeling of randomness,\nthe increase in the Gilbert damping is given by a sum of\nthese two contributions, i.e., δαG=δαG,1+δαG,2, for an\narbitrary strength of interfacial randomness.\nOur calculation can be applied straightforwardly to\nother one-dimensional electron systems such as quasi-\none-dimensional magnets, whose low-energy states are\nalso described by the Tomonaga-Luttinger liquid model.\nIn particular, the low-energy states of spin systems with\nin-plane anisotropy are characterized by a Luttinger pa-\nrameter Kssmaller than 1. If Ksis sufficiently smaller\nthan 1, δαGshould show nontrivial power-law behavior\nwith respect to the temperature even for a clean inter-\nface.\nVI. SUMMARY\nWe theoretically studied spin pumping from a ferro-\nmagnetic insulator into carbon nanotubes. First, we for-\nmulated the increase in the Gilbert damping in terms\nof the spin susceptibility and described the interfacial7\nexchange coupling with a simple model, in which two\ntypes of spin-flip process, i.e., momentum-conserving and\nmomentum-nonconserving processes, coexist. Then, we\nanalytically calculated the increase in the Gilbert damp-\ning by treating electrons in carbon nanotubes in the\nframework of the Luttinger liquid. For a clean interface,\nthe increase in damping is proportional to the inverse of\nthe temperature at high temperatures while it is almost\nconstant at low temperatures. The crossover tempera-\nture includes information on the Fermi velocity in carbon\nnanotubes. We also found that the increase in damping is\nhighly sensitive to the deviation of the Luttinger param-\neter in the spin sector from unity. For a dirty interface,\nthe increase in damping shows a power-law dependence\non the temperature with a nontrivial exponent reflecting\nthe nature of the Tomonaga-Luttinger liquid. We also\nestimated the increase of the Gilbert damping using re-\nalistic parameters. Our results indicate a possible appli-\ncation of spin pumping for detecting power-law behavior\nof spin excitation in low-dimensional systems. Detection\nof other types of spin excitation in exotic many-body\nstates will be left as a future study.\nACKNOWLEDGMENTS\nThis French-Japanese collaboration is supported by\nthe CNRS International Research Project “Excitations\nin Correlated Electron Systems driven in the Giga-\nHertz range” (ESEC). This work received support from\nthe French government under the France 2030 invest-\nment plan, as part of the Initiative d’Excellence d’Aix-\nMarseille Universit´ e - A*MIDEX. We acknowledge sup-\nport from the institutes IPhU (AMX-19-IET-008) and\nAMUtech (AMX-19-IET-01X). T. K. acknowledges sup-\nport from the Japan Society for the Promotion of Sci-\nence (JSPS KAKENHI Grants No. 20K03831). M.\nM. acknowledges support by a Grant-in-Aid for Sci-\nentific Research B (23H01839 and 21H01800) and A\n(21H04565) from MEXT, Japan, and by the Priority Pro-\ngram of Chinese Academy of Sciences, under Grant No.\nXDB28000000.\nAppendix A: Correlation functions\nHere, we briefly summarize the calculation of the corre-\nlation function Crαr′α′(x, y, t ) defined in Eq. (23). Using\nthe bosonic fields, the correlation function is written as\nCrαr′α′(x, y, t )\n=1\n(2πa)2\u0002\n⟨eAeBeCeD⟩0− ⟨eCeDeAeB⟩0\u0003\n, (A1)\nA=−i(−rθα+(x, t) +ϕα+(x, t)), (A2)\nB=i(−r′θα′−(x, t) +ϕα′−(x, t)), (A3)\nC=−i(−r′θα′−(y,0) +ϕα′−(y,0)), (A4)\nD=i(−rθα+(y,0) +ϕα+(y,0)), (A5)where we set r= +1 ( r=−1) for the left-going (right-\ngoing) branch. Using the formula,\n⟨eA1eA2···eAN⟩= exp\n1\n2X\ni⟨A2\ni⟩+X\ni<j⟨AiAj⟩\n,\n(A6)\nwhich holds when [ Ai, Aj] is a c-number, we obtain\n⟨eAeBeCeD⟩0≡eFrαr ′α′(x−y,t)\n= exph1\n2⟨(A2+B2+C2+D2)⟩+⟨AB⟩+⟨CD⟩\n+⟨AC⟩+⟨AD⟩+⟨BC⟩+⟨BD⟩i\n, (A7)\n⟨eCeDeAeB⟩0=eFr′α′rα(y−x,−t). (A8)\nThe correlation functions of the bosonic fields, which are\ndefined as GXY\njδ=⟨X(x, t)Y(y,0)⟩, are calculated as [19]\nGθθ\njδ(x, t) =Kjδ\n4(I(x, t) +I(−x, t)), (A9)\nGϕϕ\njδ(x, t) =1\n4Kjδ(I(x, t) +I(−x, t)), (A10)\nGθϕ\njδ(x, t) =Gϕθ\njδ(x, t) =1\n4(I(x, t)−I(−x, t)),(A11)\nI(x, t) =−log\u00142iβℏvF\nLsinh\u0012π(ia−x−vFt)\nβℏvF\u0013\u0015\n.\n(A12)\nUsing these correlation functions, we obtain\nFrαr′α′(x, t) =F1δr,r′δα,α′+F2δr,−r′δα,α′\n+F3δr,r′δα,−α′+F4δr,−r′δα,−α′,(A13)\nF1(x, t) =˜Gθθ\ns++˜Gθθ\ns−+˜Gϕϕ\ns++˜Gϕϕ\ns−\n−r(˜Gθϕ\ns++˜Gθϕ\ns−+˜Gϕθ\ns++˜Gϕθ\ns−),(A14)\nF2(x, t) =˜Gθθ\nc++˜Gθθ\nc−+˜Gϕϕ\ns++˜Gϕϕ\ns−, (A15)\nF3(x, t) =˜Gθθ\ns++˜Gθθ\nc−+˜Gϕϕ\ns++˜Gϕϕ\nc−\n−r(˜Gθϕ\ns++˜Gθϕ\nc−+˜Gϕθ\ns++˜Gϕθ\nc−),(A16)\nF4(x, t) =˜Gθθ\nc++˜Gθθ\ns−+˜Gϕϕ\ns++˜Gϕϕ\nc−, (A17)\nwhere ˜GXY\njδ(x, t)≡GXY\njδ(x, t)−GXY\njδ(0,0). Com-\nbining these results enables the correlation function\nCrαr′α′(x, y, t ) to be obtained analytically.8\nAppendix B: Analytic expressions of integrals\nFor a clean interface, the increase in damping is given\nas\nδαG,1=−4S0J2\n1\nℏ2ω0(2πa)2NFIIγ, (B1)\nIγ=v2\nF\u0012βℏ\nπ\u00133Zw\n0dx′Zw\n0dy′Z∞\n0dusin(˜ω0u)\n×Im(\u0014sinh(iα)\nsinh(iα+x−y−u)\u0015γ+1\n×\u0014sinh(iα)\nsinh(iα−x+y−u)\u0015γ−1)\n, (B2)\nwhere ˜ ω0=βℏω0/π,u=πt/β ℏ,w=πW/β ℏvF,x′=\nπx/β ℏvF,y′=πy/β ℏvF, and α=πa/β ℏvF. Changing\nvariables from x′andy′with Z= (x+y)/2 and z=x−y,\nthe integral is modified as\nIγ=v2\nF\u0012βℏ\nπ\u00133Z∞\n0dusin(˜ω0u)\n×\"Zw/2\n0dZZ2Z\n−2Zdz+Zw\nw/2dZZ2(w−Z)\n−2(w−Z)dz#\n×Im(\u0014sinh(iα)\nsinh(iα+z−u)\u0015γ+1\n×\u0014sinh(iα)\nsinh(iα−z−u)\u0015γ−1)\n=−v2\nF\n4\u0012βℏ\nπ\u00133Zw/2\n0dZZ−2z\n2zdzZ∞\n−∞du\n×(ei˜ω0u−e−i˜ω0u)\n×\u0014sinh(iα)\nsinh(iα+z−u)\u0015γ+1\u0014sinh(iα)\nsinh(iα−z−u)\u0015γ−1\n.\n(B3)\nIn the last equation, we have used the relation\n\u0012sinh(iα)\nsinh(iα±z−u)\u0013∗\n=sinh(iα)\nsinh(iα∓z+u)(B4)\nand the symmetry of the integrand with respect to Z↔\nw−Zandz↔ −z. At this stage, it is useful to introduce\nBγ(ζ, z) =Z∞\n−∞du eiζu\u0014sinh(iα)\nsinh(iα+z−u)\u0015γ+1\n×\u0014sinh(iα)\nsinh(iα−z−u)\u0015γ−1\n, (B5)\nso that\nIγ=−v2\nF\n2\u0012βℏ\nπ\u00133Zw/2\n0dZZ2Z\n−2Zdz\n×[Bγ(˜ω0, z)− Bγ(−˜ω0, z)]. (B6)Setting v= 2uand rearranging the hyperbolic sine func-\ntion, we obtain\nBγ(ζ, z) =1\n2(1−e−2iα)2γe−2zZ∞\n−∞dv\n×e−v(γ−iζ/2)\n[e−v+e−2z+i(π−2α)]γ+1[e−v+e−2z+i(π−2α)]γ−1,\n(B7)\nwhich can be computed analytically, invoking the formula\n3.315.1 in Ref. [29] as\nBγ(ζ, z)\n=1\n2(1−e−2iα)2γe−i(π−2α)(γ+iζ/2)e−2z(γ−1−iζ/2)\n×|Γ(γ+iζ/2)|2\nΓ(2γ)F(γ−1, γ,2γ; 1−e−4z), (B8)\nwhere F(a, b, c ;x) is the Gauss hypergeometric function.\nTo leading order in the small parameter α(≪1), this\nreduces to\nBγ(ζ, z) =1\n2(2α)2γeπζ/2+iζze−2z(γ−1)\n×|Γ(γ+iζ/2)|2\nΓ(2γ)F(γ−1, γ,2γ; 1−e−4z).(B9)\nIn practice, we are mostly interested in the regime where\n˜ω0=βℏω0≪1 so we can focus on values of ζsuch\nthat|ζ| ≪1. This allows us to expand Bγ(ζ, z) for small\nvalues of ζ, which yields, after substituting back into the\nexpression for Iγ,\nIγ=−π\n2v2\nFω0\u0012βℏ\nπ\u00134\n(2α)2γΓ(γ)2\nΓ(2γ)Zw/2\n0dZ\n×Z2Z\n0dze−2z(γ−1)F(γ−1, γ,2γ; 1−e−4z),(B10)\nwhere we have used the symmetry of the integrand with\nrespect to z↔ −z. Combining this result with Eq. (B1),\nEqs. (32) and (33) can be derived, after rewriting the\nintegral variable as z′= 2Z.\nIn the limiting case, the integral I(w, γ) given in\nEq. (33) can be approximated into a simple form. For the\nshort-junction limit ( w/π=W/β ℏvF=W/L th≪g(γ)),\nwe obtain\nI(w, γ)≃1\nwZw\n0dz′Zz′\n0dz,e−2z(γ−1)=w\n2.(B11)\nFor the long-junction limit ( w/π≫g(γ)),\nI(w, γ)\n≃1\nwZw\n0dz′Z∞\n0dz,e−2z(γ−1)F(γ−1, γ,2γ; 1−e−4z)\n=π\n2g(γ), (B12)9\nwhere g(γ) is defined by Eq. (43). These analytical ex-\npressions lead to Eqs. (41) and (42) in the main text.\nFor a dirty interface, the increase in damping is ex-\npressed as\nδαG,2=−S0J2\n2aW\nℏ2ω0(πa)2NFIX\nr,r′,α,α′I′\nγrαr ′α′, (B13)\nI′\nγ=βℏ\nπZ∞\n0dusin(˜ω0u)Im(\u0014sinh(iα)\nsinh(iα−u)\u00152γ)\n,\n(B14)\nwith the same dimensionless variables as for a clean in-\nterface. By a similar way as the clean case, the integral\nI′\nγis modified as\nI′\nγ=−βℏ\n4π[Aγ(−˜ω0)−Aγ(−˜ω0)], (B15)\nAγ(ζ) =Z∞\n−∞du e−iζu\u0014sinh(iα)\nsinh(iα−u)\u00152γ\n. (B16)\nSetting v= 2uand rearranging the hyperbolic sine func-tion, we obtain\nAγ(ζ) =1\n2(1−e−2iα)2γZ��\n−∞dve−(γ+iζ/2)v\n(e−v+ei(π−2α))2γ.\n(B17)\nInvoking the formula 3.314 in Ref. [29], this can be com-\nputed as\nAγ(ζ) =1\n2(2 sin α)2γeαζe−πζ/2|Γ(γ+iζ/2)|2\nΓ(2γ),(B18)\nwhich then yields, to leading order in α(≪1),\nI′\nγ=−βℏ\n4π(2α)2γ|Γ(γ+iζ/2)|2\nΓ(2γ)sinh(π˜ω0/2).(B19)\nAssuming ˜ ω0=βℏω0/π≪1, this is further simplified as\nI′\nγ=−β2ℏ2\n8πω0\u00122πa\nβℏvF\u00132γΓ(γ)2\nΓ(2γ), (B20)\nwhich finally leads to Eq. (46) in the main text.\n[1] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, En-\nhanced gilbert damping in thin ferromagnetic films, Phys.\nRev. Lett. 88, 117601 (2002).\n[2] F. Hellman, A. Hoffmann, Y. Tserkovnyak, G. S. D.\nBeach, E. E. Fullerton, C. Leighton, A. H. MacDon-\nald, D. C. Ralph, D. A. Arena, H. A. D¨ urr, P. Fischer,\nJ. Grollier, J. P. Heremans, T. Jungwirth, A. V. Kimel,\nB. Koopmans, I. N. Krivorotov, S. J. May, A. K. Petford-\nLong, J. M. Rondinelli, N. Samarth, I. K. Schuller, A. N.\nSlavin, M. D. Stiles, O. Tchernyshyov, A. Thiaville, and\nB. L. Zink, Interface-induced phenomena in magnetism,\nRev. Mod. 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Ryzhik, Table of Integrals, Series,\nand Products (Elsevier Academic Press, 2007)."    },    {        "title": "2303.03852v1.Electrically_tunable_Gilbert_damping_in_van_der_Waals_heterostructures_of_two_dimensional_ferromagnetic_metals_and_ferroelectrics.pdf",        "content": "Page 1 of 15 \n Electrically tunable Gilbert damping in van der Waals heterostructures of two-\ndimensional ferromagnetic meta ls and ferroelectrics \nLiang Qiu,1 Zequan Wang,1 Xiao-Sheng Ni,1 Dao-Xin Yao1,2 and Yusheng Hou 1,* \nAFFILIATIONS \n1 Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, State \nKey Laboratory of Optoelectronic Materials and Technologies, Center for Neutron \nScience and Technology, School of Physics, Sun Yat-Sen University, Guangzhou, \n510275, China  \n2 International Quantum Academy, Shenzhen 518048, China \n \nABSTRACT  \nTuning the Gilbert damping of ferromagnetic (FM) metals via a nonvolatile way  is \nof importance to exploit and design next-generation novel spintronic devices. Through \nsystematical first-principles calculations, we study the magnetic properties of the van \nder Waals heterostructure of two-dimensional FM metal  CrTe 2 and ferroelectric (FE) \nIn2Te3 monolayers. The ferromagnetism of CrTe 2 is maintained in CrTe 2/In2Te3 and its \nmagnetic easy axis can be switched from in-plane to out- of-plane by reversing the FE \npolarization of In 2Te3. Excitingly, we find that the Gilbert damping of CrTe 2 is tunable \nwhen the FE polarization of In 2Te3 is reversed from upward to downward. By analyzing \nthe k-dependent contributions to the Gilbert damping, we unravel that such tunability \nresults from the changed intersections between the bands of CrTe 2 and Fermi level on \nthe reversal of the FE polarizations of In 2Te3 in CrTe 2/In2Te3. Our work provides a n \nappealing way to electrically tailor Gilbert dampings of two-dimensional FM metals by \ncontacting them with ferroelectrics.  \n \n*Authors to whom correspondence should be addressed:  \n[Yusheng Hou, houysh@mail.sysu.edu.cn]  \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: 10.1063/5.0145401Page 2 of 15 \n Since the atomically thin long-range ferromagnetic ( FM) orders at finite \ntemperatures are discovered in CrI 31 monolayer (ML) and Cr 2Ge2Te62 bilayer, two-\ndimensional (2D) van der Waals (vdW) FM materials have attracted intensive \nattention.3-5 Up to now, many novel vdW ferromagnets such as Fe 3GeTe 2,6 Fe5GeTe 2,7 \nVSe 28,9 and MnSe 210 have been synthesized in experiments. Due to the intrinsic \nferromagnetism in these vdW FM materials, it is highly fertile to engineer emergent \nphenomena through magnetic proximity effect in their heterostructures.11 For instance , \nan unprecedented control of the spin and valley pseudospins in WSe 2 ML is reported in \nCrI 3/WSe 2.12 By contacting the thin films of three-dimensional topological insulators \nand graphene with CrI 3, high-temperature quantum anomalous Hall effect and vdW spin \nvalves are proposed in CrI 3/Bi2Se3/CrI 313 and CrI 3/graphene/CrI 3,14 respectively. On the \nother hand, the magnetic properties of these vdW FM materials can also be controlled \nby means of external perturbations such as gating and moiré patterns.3 In CrI 3 bilayer, \nHuang et al. observed a voltage-controlled switching between antiferromagnetic (AFM) \nand FM states.15 Via an ionic gate, Deng et al. even increased the Curie temperature \n(TC) of the thin flake of vdW FM metal Fe 3GeTe 2 to room temperature, which is much \nhigher than its bulk TC.6 Very recently, Xu et al. demonstrated a coexisting FM and \nAFM state in a twisted bilayer CrI 3.16 These indicate that vdW FM materials are \npromising platforms to design and implement spintronic devices in the 2D limit.4,11  \nRecently, of great interest is the emergent vdW magnetic material CrTe 2 which is \na new platform for realizing room-temperature intrinsic ferromagnetism.17,18 Especially, \nCrTe 2 exhibits greatly tunable magneti sm. In the beginning, its ground state is believed \nto be the nonmagnetic 2 H phase,19 while several later researches suggest that either the \nFM or AFM 1 T phases should be the ground state of CrTe 2.17,18,20- 23 Currently, the \nconsensus is that the structural ground state of CrTe 2 is the 1 T phase. With respect to its \nmagnetic ground state, a first-principles study shows that the FM and AFM ground \nstates in CrTe 2 ML depend on its in-plane lattice constants.24 It is worth noting that the \nTC of FM CrTe 2 down to the few-layer limit can be higher than 300 K,18 making it have \nwide practical application prospects in spintronics.  \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: 10.1063/5.0145401Page 3 of 15 \n Building heterostructures of FM and ferroelectric (FE) materials offers an effective \nway to control nonvolatile magnetism via an electric field. Experimentally, Eerenstein \net al. presented an electric-field-controlled nonvolatile converse magnetoelectric effect \nin a multiferroic heterostructure La0.67Sr0.33MnO 3/BaTiO 3.25 Later, Zhang et al. reported \nan electric-field-driven control of nonvolatile magnetization in a heterostructure of FM \namorphous alloy Co40Fe40B20 and FE Pb(Mg 1/3Nb2/3)0.7Ti0.3O3.26 Theoretically, Chen et \nal. demonstrated based on first-principles calculations that the interlayer magnetism of \nCrI 3 bilayer in CrI 3/In2Se3 is switchable between FM and AFM couplings by the \nnonvolatile control of the FE polarization direction of In 2Se3.27 In spite of these \ninteresting findings, using FE substrates to electrically tune the Gilbert damping of \nferromagnets, an important factor determining the operation speed of spintronic devices, \nis rarely investigated in 2D FM/FE vdW heterostructures. Therefore, it is of great \nimportance to explore the possibility of tuning the Gilbert damping in such kind of \nheterostructures.  \nIn this work, we first demonstrate that the magnetic ground state of 1 T-phase CrTe 2 \nML will change from the zigzag AFM (denoted as z-AFM) to FM orders with increasing \nits in-plane lattice constants. By building a vdW heterostructure of CrTe 2 and FE In2Te3 \nMLs, we show that the magnetic easy axis of CrTe 2 can be tuned from in-plane to out-\nof-plane by reversing the FE polarization of In 2Te3, although its ferromagnetism is kept . \nImportantly, we find that the Gilbert damping of CrTe 2 is tunable with a wide range on \nreversing the FE polarization of In 2Te3 from upward to downward. Through looking \ninto the k-dependent contributions to the Gilbert damping, we reveal that such tunability \noriginates from the changed intersections between the bands of CrTe 2 and Fermi level \nwhen the FE polarizations of In 2Te3 is reversed in CrTe 2/In2Te3. Our work demonstrates \nthat putting 2D vdW FM metals on FE substrates is an attractive method to electrically \ntune their Gilbert dampings.  \nCrTe 2, a member of the 2D transition metal dichalcogenide family, can potentially \ncrystalize into several different layered structures such as 1 T, 1Td, 1H and 2 H phases.28 \nIt is believed that the 1 T phase is the most stable among all of the se possible phases 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: 10.1063/5.0145401Page 4 of 15 \n both bulk and ML. This phase has a hexagonal lattice and belongs to the P3_m1 space \ngroup, with each Cr atom surrounded by the octahedrons of Te atoms (Fig. 1a). In view \nof the hot debates on the magnetic ground state in CrTe 2 ML, we establish a 2× 2√3 \nsupercell and calculate the total energies of several different magnetic structures (Fig. \nS1 in Supplementary Materials) when its lattice constant varies from 3.65 to 4.00 Å. As \nshown in Fig. 1b, our calculations show that z-AFM order is the magnetic ground state \nwhen the lattice constant is from 3.65 to 3.80 Å. By contrast, the FM order is the \nmagnetic ground state when the lattice constant is in the range from 3.80 to 4.00 Å. \nNote that our results are consistent with the experimentally observed z- AFM23 and \nFM29 orders in CrTe 2 with a lattice constant of 3.70 and 3.95 Å, respectively. Since we \nare interested in the Gilbert damping of ferromagnets and the experimentally grow n \nCrTe 2 on ZrTe 2 has a lattice constant of 3.95 Å,29 we will focus on CrTe 2 ML with this \nlattice constant hereinafter.  \n \n \nFIG. 1. (a) Side (the top panel) and top (the bottom panel) views of CrTe 2 ML. The NN \nand second- NN exchange paths are shown by red arrows in the top view. (b) The phase \ndiagram of the magnetic ground state of CrTe 2 ML  with different lattice constants. Insets \nshow the schematic illustrations of the z-AFM and FM orders. The up and down spins \nare indicated by the blue and red balls, respectively. The stars highlight the experimental \nlattice constants of CrTe 2 in Ref.23 and Ref.29.  \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: 10.1063/5.0145401Page 5 of 15 \n To obtain an deeper understanding on the ferromagnetism of CrTe 2 ML, we adopt \na spin Hamiltonian consisting of Heisenberg exchange couplings and single-ion \nmagnetic anisotropy (SIA) as follows:30  \n𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2\n𝑖  (1) \nIn Eq. (1), J1 and J2 are the nearest neighbor (NN) and second- NN Heisenberg exchange \ncouplings. Note that a negative (positive) J means a FM (AFM) Heisenberg exchange \ncouples. Besides, A parameterizes the SIA term. First of all, our DFT calculations show \nthat the magnetic moment of CrTe 2 ML is 3.35 μB/Cr, consistent with previous DFT \ncalculations.31 As shown in Table I, the calculated J1 and J2 are both FM and J1 is much \nstronger than J2. Both FM  J1 and J2 undoubtedly indicate that CrTe 2 ML has a FM \nmagnetic ground state. Finally, the SIA parameter A is obtained by calculating the \nenergy difference between two FM states with out-of-plane and in-plane magnetizations. \nOur calculations obtain A=1.81 meV/Cr, indicating that CrTe 2 ML has an out-of-plane \nmagnetic easy axis. Hence, our calculations show that CrTe 2 ML exhibits an out-of-\nplane FM order, consistent with experimental observations.29  \n \nTABLE I. Listed are the in-plane lattice constant s a, Heisenberg exchange couplings J \n(in unit of meV) and SIA (in unit of meV/Cr) of CrTe 2 ML and CrTe 2/In2Te3. \nSystem  a (Å) J1 J2  A \nCrTe 2 3.95 -24.56  -0.88  1.81 \nCrTe 2/In2Te3(↑) 7.90 -20.90  -1.80  -1.44  \nCrTe 2/In2Te3(↓) 7.90 -19.33  -0.88   0.16  \n \nTo achieve an electrically tunable Gilbert damping in CrTe 2 ML, we establish its \nvdW heterostructure with F E In2Te3 ML. In building this heterostructure, w e stack a \n2×2 supercell of CrTe 2 and a √3 ×√3  supercell of In2Te3 along the (001) direction. \nBecause the magnetic properties of CrTe 2 ML are the primary topic and the electronic \nproperties of In2Te3 ML are basically not affected by a strain (Fig. S2), we stretch the \nlattice constant of the latter to match that of the former. Fig. 2a shows the most stable \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: 10.1063/5.0145401Page 6 of 15 \n stacking configuration in CrTe 2/In2Te3 with an upward FE polarization [denoted as \nCrTe 2/In2Te3(↑)]. At the interface in this configuration, one of four Cr atoms and one of \nfour Te atoms at the bottom of CrTe 2 sits on the top of the top-layer Te atom s of In2Te3. \nIn CrTe 2/In2Te3 with a downward FE polarization [denoted as CrTe 2/In2Te3(↓ )], the \nstacking configuration at its interface is same as that in CrTe 2/In2Te3(↑ ). The only \ndifference between CrTe 2/In2Te3(↑ ) and CrTe 2/In2Te3(↓ ) is that the middle-layer Te \natoms of In 2Te3 in the former is farther to CrTe 2 than that in the latter (Fig. 2a and 2c). \nIt is noteworthy that the bottom-layer Te atoms of CrTe 2 do not stay at a plane anymore \nin the relaxed CrTe 2/In2Te3 (see more details in Fig. S3), suggesting non-negligible \ninteractions between CrTe 2 and In 2Te3. \n \n  \nFIG. 2. (a) The schematic stacking configuration and (b) charge density difference 𝛥ρ \nof CrTe 2/In2Te3(↑). (c) and (d) same as (a) and (b) but for CrTe 2/In2Te3(↓). In (b) and \n(d), color bar indicates the weight of negative (blue) and positive (red) charge density \ndifferences. (e) The total DOS of CrTe 2/In2Te3. (f) and (g) show the PDOS of CrTe 2 and \nIn2Te3 in CrTe 2/In2Te3, respectively. In (e)-(g), upward and downward polarizations are \nindicated by black and red lines, respectively.  \n \nTo shed light on the effect of the FE polarization of In 2Te3 on the electronic \nproperty of CrTe 2/In2Te3, we first investigate the spatial distribution of charge density \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: 10.1063/5.0145401Page 7 of 15 \n difference \n2 2 3 2 2 3 CrTe In Te CrTe In Te     = − −  with different FE polarization directions. \nAs shown in Fig. 2b and 2d, we see that there is an obvious charge transfer at the \ninterfaces of both CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓), which is further confirmed by \nthe planar averaged 𝛥ρ (Fig. S4). Additionally, the charge transfer in CrTe 2/In2Te3(↑) \nis distinctly less than th at in CrTe 2/In2Te3(↓). Fig. 2e shows that the total density of \nstates (DOS) near Fermi level are highly different in CrTe 2/In2Te3(↑ ) and \nCrTe 2/In2Te3(↓). By projecting the DOS onto CrTe 2 and In 2Te3, Fig. 2f shows that the \nprojected DOS (PDOS) of CrTe 2 in CrTe 2/In2Te3(↑) is larger than that in CrTe 2/In2Te3(↓) \nat Fermi level. Interestingly, the PDOS of In 2Te3 in CrTe 2/In2Te3(↑) is larger than that \nin CrTe 2/In2Te3(↓) below Fermi level while the situation is inversed above Fermi level \n(Fig. 2g). By looking into the five Cr- d orbital projected DOS in CrTe 2/In2Te3(↑) and \nCrTe 2/In2Te3(↓) (Fig. S5), we see that there are obviously different occupations for xyd, \n22xyd− and 223zrd− orbitals near Fermi level. All of these imply that the reversal of the \nFE polarization of In 2Te3 may have an unignorable influence on the magnetic properites \nof CrTe 2/In2Te3.   \nDue to the presence of the FE In 2Te3, the inversion symmetry is inevitably broken \nand nonzero Dzyaloshinskii-Moriya interactions (DMIs) may exist in CrTe 2/In2Te3. In \nthis case, we add a DMI term into Eq. (1) to investigate the magnetism of CrTe 2/In2Te3 \nand the corresponding spin Hamiltonian is in the form of32  \n𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ +∑ 𝑫𝑖𝑗⋅ (𝑆 𝑖× 𝑆 𝑗) ⟨𝑖𝑗⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2\n𝑖  (2). \nIn Eq. (2), Dij is the DMI vector of the NN Cr-Cr pairs. As the C6-rotational symmetry \nwith respect to Cr atoms in CrTe 2 is reduced to the C3-rotational symmetry, the NN \nDMIs are split into four different DMIs (Fig. S6). For simplicity, the J1 and J2 are still \nregard ed to be six-fold. From Table I, we see that the NN J1 of both CrTe 2/In2Te3(↑) and \nCrTe 2/In2Te3(↓) are still FM but slightly smaller than that of free-standing CrTe 2 ML. \nMoreover, the second- NN FM J2 is obviously enhanced in CrTe 2/In2Te3(↑) compared \nwith CrTe 2/In2Te3(↓) and free-standing CrTe 2 ML. To calculated the NN DMIs, we build \na √3×√3 supercell of CrTe 2/In2Te3 and the four-state method33 is employed here. As \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: 10.1063/5.0145401Page 8 of 15 \n listed in Table S1, the FE polarization direction of In 2Te3 basically has no qualitative \neffect on the DMIs in CrTe 2/In2Te3 although it affects their magnitudes. More explicitly, \nthe magnitudes of the calculated DMIs range from 1.22 to 2.81 meV, which are about \none order smaller than the NN J1. Finally, we find that the SIA of CrTe 2/In2Te3 is \nstrongly dependent on the FE polarization of In 2Te3. When In 2Te3 has an upward FE \npolarization, the SIA of CrTe 2/In2Te3(↑) is negative, indicating an in-plane magnetic \neasy axis. However, when the FE polarization of In 2Te3 is downward, CrTe 2/In2Te3(↓) \nhas a positive SIA, indicating an out-of-plane magnetic easy axis. It is worth noting that \nCrTe 2/In2Te3(↓) has a much weak SIA than the free-standing CrTe 2 ML, although they \nboth have positive SIAs. The different Heisenberg exchange couplings, DMIs and SIAs \nin CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) clearly unveil that the magnetic properties of \nCrTe 2 are tuned by the FE polarization of In2Te3.  \nTo obtain the magnetic ground state of CrTe 2/In2Te3, MC simulations are carried \nout. As shown in Fig. S7, CrTe 2/In2Te3(↑) has an in-plane FM magnetic ground state \nwhereas CrTe 2/In2Te3(↓ ) has an out-of-plane one. Such magnetic ground states are \nunderstandable. Firstly, the ratios between DMIs and the NN Heisenberg exchange \ncouplings are small and most of them are out of the typical range of 0.1–0.2 for the \nappearance of magnetic skyrmions.34 Secondly, the SIAs of the CrTe 2/In2Te3(↑) and the \nCrTe 2/In2Te3(↓ ) prefer in-plane and out-of-plane magnetic easy axes, respectively . \nTaking them together, we obtain that the FM Heisenberg exchange couplings dominate \nover the DMIs and thus give rise to a FM magnetic ground state with its magnetization \ndetermined by the SIA,35 consistent with our MC simulated results.  \nFigure 3a shows the Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward \nand downward FE polarizations of In2Te3. Similar to previous studies,36,37 the Gilbert \ndampings of CrTe 2/In2Te3 decrease first and then increase as the scattering rate Γ \nincreases. Astonishingly, the Gilbert dampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) \nare distinctly different at the same scattering rate Γ ranging from 0.001 to 1.0 eV . To \nhave a more intuitive sense on the effect of the FE polarizations of In 2Te3 on the Gilbert \ndampings in CrTe 2/In2Te3, we calculate the ratio   =  at any given Γ, where \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: 10.1063/5.0145401Page 9 of 15 \n ( ) is the Gilbert damping of CrTe 2/In2Te3(↑) [CrTe 2/In2Te3(↓)]. As shown in Fig. \n3b, the ratio 𝜂 ranges from 6 to around 1.3 with increasing Γ. As the FE polarization \nof In 2Te3 can be switched from upward to downward by an external electric field, the \nGilbert damping of CrTe 2/In2Te3 is electrically tunable in practice.  \n \n \nFIG. 3. (a) The Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward (black line) \nand downward (red line) FE polarizations of In 2Te3. (b) The Gilbert damping ratio 𝜂 \nas a function of the scattering rate Γ.  \n \nTo gain a deep insight into how the FE polarization of In 2Te3 tunes the Gilbert \ndamping in CrTe 2/In2Te3, we investigate the k-dependent contributions to the Gilbert \ndampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). As shown in Fig. 4a and 4b, the bands \naround Fermi level have qiute different intermixing between CrTe 2 and In 2Te3 states \nwhen the FE polarizaiton of In 2Te3 is reversed. Explicitly, there are obvious intermixing \nbelow Fermi leve in CrTe 2/In2Te3(↑) while the intermixing mainly takes place above \nFermi level in CrTe 2/In2Te3(↓). Especially, the bands intersected by Fermi level are at \ndifferent k points in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). Through looking into the k-\ndependent contributions to the ir Gilbert dampings (Fig. 4c and 4d), we see that large \ncontributions are from the k points (highlighted by arrows in Fig. 4) at which the bands \nof CrTe 2 cross Fermi level. In addition, these large contributions are different. Such k-\ndipendent contribution to Gilbert dampings is understandable. Based on the scattering \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: 10.1063/5.0145401Page 10 of 15 \n theory of Gilbert damping 38, Gilbert damping parameter is calculated using the \nfollowing Eq. (3) 36  \n( ) ( ) , , , , , , (3)kk\nk i k j k j k i F k i F k j\nk ij SE E E EM u u\n      = − − −\nHH, \nwhere EF is Fermi level and Ek,i is the enery of band i at a given k point. Due to the delta \n( ) ( ) ,, F k i F k jE E E E −−  , only the valence and conduction bands near Fermi level \nmake dominant contribution to the Gilbert damping. Additionally, their contributions \nalso depend on factor , , , ,kk\nk i k j k j k iuu   \nHH. Overall , through changing the \nintersections between the bands of CrTe 2 and Fermi level, the reversal of the FE \npolarization of In 2Te3 can modulate the contributions to Gilbert damping. Consequently, \nthe total Gilbert dampings are different in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓).  \n \n \nFIG. 4. (a) Band structure calculated with spin-orbit coupling and (c) the k-dependent \ncontributions to the Gilbert damping in CrTe 2/In2Te3(↑). (b) and (d) same as (a) and (c) \nbut for CrTe 2/In2Te3(↓). In (a) and (b), Fermi levels are indicated by horizontal dash \nlines and the states from CrTe 2 and In 2Te3 are shown by red and blue, respectively.  \n \n    From experimental perspectives, the fabrication of CrTe 2/In2Te3 vdW \nheterostructure should be feasible. On the one hand, CrTe 2 with the lattice constant of \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: 10.1063/5.0145401Page 11 of 15 \n 3.95 Å has been successfully grown on ZrTe 2 substrate by the molecular beam epitaxy.29 \nOn the other hand, In 2Te3 is also synthesized.39 Taking these and the vdW nature of \nCrTe 2 and In 2Te3 together, a practical scheme of growing CrTe 2/In2Te3 is sketched in \nFig. S8 : first grow CrTe 2 ML on ZrTe 2 substrate29 and then put In2Te3 ML on CrTe 2 to \nform the desired CrTe 2/In2Te3 vdW heterostructure.     \nIn summary, by constructing a vdW heterostructure of 2D FM metal CrTe 2 and FE \nIn2Te3 MLs, we find that the magnetic properties of CrTe 2 are engineered by the reversal \nof the FE polariton of In 2Te3. Although the ferromagnetism of CrTe 2 is maintained in \nthe presence of the FE In2Te3, its magnetic easy axis can be tuned from in-plane to out-\nof-plane by reversing the FE polarization of In 2Te3. More importantly, the Gilbert \ndamping of CrTe 2 is tunable with a wide range when reversing the FE polarization of \nIn2Te3 from upward to downward. Such tunability of the Gilbert damping in \nCrTe 2/In2Te3 results from the changed intersections between the bands of CrTe 2 and \nFermi level on reversing the FE polarizations of In 2Te3. Our work introduces a \nremarkably useful method to electrically tune the Gilbert dampings of 2D vdW FM \nmetals by contacting them with ferroelectrics, and should stimulate more experimental \ninvestigations in this realm.  \n \nSee the supplementary material for the details of computational methods31,36,40- 50 \nand other results mentioned in the main text.  \n \nThis project is supported by National Nature Science Foundation of China (No. \n12104518, 92165204, 11974432), NKRDPC-2018YFA0306001, NKRDPC-\n2022YFA1402802, GBABRF-2022A1515012643 and GZABRF-202201011118 . \nDensity functional theory calculations are performed at Tianhe- II. \n \nAUTHOR DECLARATIONS  \nConflict of Interest  \nThe authors have no conflicts to disclose.  \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: 10.1063/5.0145401Page 12 of 15 \n Author Contributions  \nLiang Qiu : Investigation (equal); Methodology (equal); Writing –original draft (equal). \nZequan Wang : Methodology (equal). Xiao -sheng Ni : Investigation (equal); \nMethodology (equal).  Dao-Xin Yao : Supervision (equal); Funding acquisition (equal); \nInvestigation (equal); Writing – review &editing (equal). Yusheng Hou : \nConceptualization (equal); Funding acquisition (equal); Investigation (equal); Project \nadministration(equal); Resources (equal); Supervision (equal); Writing – review \n&editing (equal).  \n \nDATA A V AILABILITY  \nThe data that support the findings of this study are available from the \ncorresponding authors upon reasonable request.  \n \n \nREFERENCES  \n1 B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, \nD. Zhong, E. 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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: 10.1063/5.0145401This 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: 10.1063/5.0145401This 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: 10.1063/5.0145401This 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: 10.1063/5.0145401This 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: 10.1063/5.0145401"    },    {        "title": "2303.12380v2.Twisted_bilayer_graphene_reveals_its_flat_bands_under_spin_pumping.pdf",        "content": "arXiv:2303.12380v2  [cond-mat.mes-hall]  1 Sep 2023Twisted bilayer graphene reveals its flat bands under spin pu mping\nSonia Haddad1,2,3,∗Takeo Kato2, Jihang Zhu3, and Lassaad Mandhour1\n1Laboratoire de Physique de la Mati` ere Condens´ ee, Facult´ e des Sciences de Tunis,\nUniversit´ e Tunis El Manar, Campus Universitaire 1060 Tuni s, Tunisia\n2Institute for Solid State Physics, University of Tokyo, Kas hiwa, Chiba 277-8581, Japan\n3Max Planck Institute for the Physics of Complex Systems,\nN¨ othnitzer Strasse 38, Dresden 01187, Germany\n(Dated: September 4, 2023)\nThe salient property of the electronic band structure of twi sted bilayer graphene (TBG), at the\nso-called magic angle (MA), is the emergence of flat bands aro und the charge neutralitypoint. These\nbands are associated with the observed superconducting pha ses and the correlated insulating states.\nScanning tunneling microscopy combined with angle resolve d photoemission spectroscopy are usu-\nally used to visualize the flatness of the band structure of TB G at the MA. Here, we theoretically\nargue that spin pumping (SP) provides a direct probe of the fla t bands of TBG and an accurate\ndetermination of the MA. We consider a junction separating a ferromagnetic insulator and a het-\nerostructure of TBG adjacent to a monolayer of a transition m etal dichalcogenide. We show that\nthe Gilbert damping of the ferromagnetic resonance experim ent, through this junction, depends on\nthe twist angle of TBG, and exhibits a sharp drop at the MA. We d iscuss the experimental realiza-\ntion of our results which open the way to a twist switchable sp intronics in twisted van der Waals\nheterostructures.\nIntroduction. – Stacking two graphene layers with\na relative twist angle θresults in a moir´ e superstruc-\nture which is found to host, in the vicinity of the\nso-called magic angle (MA) θM∼1.1◦, unconven-\ntional superconductivity and strongly correlated insulat-\ning states1–3. There is a general consensus that such\nstrong electronic correlations originate from the moir´ e\nflat bands emerging at the MA around the charge neu-\ntrality point4–11. The tantalizing signature of the flat\nbands have been experimentally demonstrated by prob-\ning the corresponding peaks of the density of states us-\ning transport1–3,12,13, electroniccompressibilitymeasure-\nments14,15, scanning tunneling microscopy (STM) and\nspectroscopy (STS)16–23. The direct evidence of these\nflat bands has been reported by angle resolved photoe-\nmission spectroscopy (ARPES) measurements combined\nto different imaging techniques24–26. However, spectro-\nscopic measurements on magic-angle TBG raise many\ntechnical challenges related to the need of an accurate\ncontrol of the twist angle, and the necessity to have non-\nencapsulated samples which can degrade in air25.\nHere we propose a noninvasive method to probe the\nflat bands of TBG and accurately determine the MA.\nThis method is based on spin pumping (SP) induced by\nferromagnetic resonance (FMR)27–30, where the increase\nin the FMR linewidth, given by the Gilbert damping\n(GD) coefficient, provides insight into the spin excita-\ntions of the nonmagnetic (NM) material adjacent to the\nferromagnet31–33. SP is expected to be efficient if the\nNM has high spin-orbit coupling (SOC) strength34.\nIn our work, we consider spin injection from a ferromag-\nnetic insulator(FI) into aTBG alignedonamonolayerof\ntransition metal dichalcogenides (TMD) which are con-\nsidered as good substrate candidates to induce relatively\nstrong SOC in graphene and TBG35–58.\nWe theoretically study a planar junction of a FI and\na TBG adjacent to WSe 2(TBG/WSe 2) as depicted inFig.1. We consider the case where a microwave of a\nfrequency Ω is applied to this junction, and focus on the\ntwist angle dependence of the FMR linewidth59.\nFigure 1. Schematic representation of the junction between\na ferromagnetic insulator (FI) and a heterostructure of TBG\nadjacent to a monolayer of WSe 2. The labels (1) and (2) de-\nnote the graphene layers of TBG represented by the red and\nthe blue lines. The red arrow indicates the spin orientation\nof the FI characterized by an average spin /angbracketleftSFI/angbracketright= (S0,0,0),\nwritten in the coordinate frame of the FI magnetization. The\ngray lines represent the boron-nitride (hBN) layers encaps u-\nlating the TBG/WSe 2heterostructure.\nContinuum model. – In TBG with a twist angle θ, the\nHamiltonian hl(k) ofagraphenelayer l(l= 1,2), rotated\nat an angle θl, ishl(k) =eiθl\n2σzh(0)\nl(k)e−iθl\n2σz, where\nθ2=−θ1=θ\n2andh(0)\nl(k) is the unrotated monolayer\nHamiltonian. In the continuum limit, h(0)\n1(k) reduces to\nh(0)\n1(k) =−/planckover2pi1vFk·σ∗, wherevFis the Fermi velocity,\nσ∗= (ξσx,σy), andσi(i=x,y,z) are the sublattice-\nPauli matrices and ξis the valley index. We assume that\nthe SOC is only induced in the graphene layer adjacent\nto the TMD layer, since the SOC arises from overlaps\nbetween atomic orbitals50. This assumption is consistent\nwith recent studies on bilayer graphene and TBG aligned\non TMD layers50,56,60,61. Layer (2), in contact with the\nWSe2monolayer, is then descried by the Hamiltonian2\nh(0)\n2(k) =h(0)\n1(k) +hSOC+m\n2σz56, wherehSOCis given\nby\nhSOC=λI\n2ξsz+λR\n2(ξσxsy−σysx)+λKM\n2ξσzsz,(1)\nsi(i=x,y,z) are the spin-Pauli matrices, λI,λRand\nλKMcorrespond, respectively, to the Ising, Rashba and\nKane-Mele SOC parameters56. The variation ranges of\nthese parameters are λI∼1−5 meV,λR∼1−15 meV,\nwhileλKMis expected to be small37,44,50,51,62–64. The\nlast term in h(0)\n2(k) is due to the inversion symmetry\nbreaking induced by the TMD layer. Hereafter, we ne-\nglect this term regarding the small value of mcompared\nto the SOC parameters56.\nAs in the case of TBG65, the low-energy Hamiltonian of\nTBG/WSe 2reduces, at the valley ξ, to\nHξ,SOC(k) =\nh1(k)T1T2T3\nT†\n1h2,1(k) 0 0\nT†\n20h2,2(k) 0\nT†\n30 0 h2,3(k)\n.(2)\nHξ,SOC(k) is written in the basis Ψ =\n(ψ0(k),ψ1(k),ψ2(k),ψ3(k)) constructed on the four-component spin-sublattice spinor ψ0(k) andψj(k),\n(j= 1,2,3) corresponding, respectively, to layer (1)\nand layer (2) (see Secs. I and II of the Supplemental\nMaterial66and Refs.8,56,65,67–70). The momentum\nkis measured relatively to the Dirac point K1ξof\nlayer (1). In Eq. ( 2),Tjare the spin-independent\ninterlayer coupling matrices, h2,j(k) =h2(k+qjξ),\n(j= 1,2,3) where qjξare the vectors connecting\nK1ξto its three neighboring Dirac points K2ξof\nlayer (2) in the moir´ e Brillouin zone (mBZ)65, and\nare given by q1ξ=K1ξ−K2ξ,q2ξ=q1ξ+ξGM\n1,\nq3ξ=q1ξ+ξ/parenleftbig\nGM\n1+GM\n2/parenrightbig\n, where/parenleftbig\nGM\n1,GM\n2/parenrightbig\nis the\nmBZ basis (see Sec. I of the Supplemental Material66).\nIn the unrelaxed TBG, and choosing sublattice A as the\norigin of the unit cell in each layer, the Tjmatrices take\nthe form8T1=w(Iσ+σx),T2=w/parenleftig\nIσ−1\n2σx+ξ√\n3\n2σy/parenrightig\nandT3=w/parenleftig\nIσ−1\n2σx−ξ√\n3\n2σy/parenrightig\n66, where\nw∼110meV71is the interlayer tunneling ampli-\ntude and Iσis the identity matrix acting on the\nsublattice indices.\nUsing the perturbative approach of Ref.65, we de-\nrive, from Eq. ( 2), the effective low-energy Hamiltonian\nH(1)\nξ,SOC(k) of TBG/WSe 2(see Sec. II of the Supplemen-\ntal Material66). To the leading order in k,H(1)\nξ,SOC(k)\nreads as66\nH(1)\nξ,SOC(k) =/an}bracketle{tΨ|Hξ,SOC|Ψ/an}bracketri}ht\n/an}bracketle{tΨ|Ψ/an}bracketri}ht=ψ†\n0/bracketleftbig\nheff(k)+hSOC\neff/bracketrightbig\nψ0, (3)\nheff(k) =−/planckover2pi1vF\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/braceleftbigg\nkx/bracketleftbigg/parenleftbig\n1−3α2/parenrightbig\nξσxIs−3α2\n/planckover2pi1vFq0(ξλIσysz+λR(ξσysy−σxsx))/bracketrightbigg\n+ky/bracketleftbigg/parenleftbig\n1−3α2/parenrightbig\nσyIs−3α2\n/planckover2pi1vFq0(−λIσxsz+λR(σxsy+ξσysx))/bracketrightbigg/bracerightbigg\n, (4)\nhSOC\neff=3α2\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/bracketleftbigg\nξλIszIσ+λR\n2(sxσy−ξsyσx)/bracketrightbigg\n, (5)\nwhere/an}bracketle{tΨ|Ψ/an}bracketri}ht ∼1+6α2,α=w\n/planckover2pi1vFq0,q0=|qjξ|=4π\n3aθ,a\nis the graphene lattice constant and σi, (i=x,y,z) act\nnow on the band indices σ=±of the eigenenergies of\nH(1)\nξ,SOC, denotedEσ,±, and given to the leading orders in\nkandλI,R\n/planckover2pi1vFq0by\nE(k)σ,±=σ\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/radicalig\nf1(k)±6α2/radicalbig\nf2(k) (6)\nf1(k) = (/planckover2pi1vF)2/parenleftbig\n1−3α2/parenrightbig2||k||2+9\n2α4/parenleftbig\n2λ2\nI+λ2\nR/parenrightbig\nf2(k) = (/planckover2pi1vF)2/parenleftbig\n1−3α2/parenrightbig2||k||2/parenleftbigg\nλ2\nI+1\n4λ2\nR/parenrightbigg\n+9\n16α4λ4\nR.\nEquation 5shows that the SOC parameters λIandλRare renormalized by the moir´ e structure of TBG to\n˜λI∼6α2\n1+6α2λI,˜λR∼3α2\n1+6α2λR, (7)\nwhich increase by decreasing the twist angle.\nThe expression of H(1)\nξ,SOC[Eq. (3)] can be taken as a\nstarting point to unveil the role of SOC in the emer-\ngence of the stable superconducting phase observed, at\nθ∼0.8◦, in TBG adjacent to WSe 256.\nTo probe the validity of the effective Hamiltonian H(1)\nξ,SOC\n[Eq. (3)], we compared the corresponding eigenenergies\nwith the numerical band structure obtained within the\ncontinuum model and taking into account 148 bands per\nvalley and spin projection (see Sec. II of the Supplemen-\ntal Material66). The results show that H(1)\nξ,SOCdescribes3\ncorrectly the band structure of TBG/WSe 2down to a\ntwist angle θ∼0.7◦. At smaller angles, the effective\nFermi velocities of H(1)\nξ,SOCare overestimated. Such a\ndiscrepancy is expected since the lattice relaxation effect\nis important at small angles56. It is worth noting that,\nfor the sake of simplicity, we did not consider a relaxed\nTBG, since we are interested in the SP around the MA.\nGilbert damping. – In the absence of a junction,\nthe magnon Green function of the FI is defined as72–78\nG0(qm,iωn) =2S0//planckover2pi1\niωn−ωqm−αG|ωn|, whereωn= 2πn//planckover2pi1β\nare the Matsubara frequencies for bosons, S0is the am-\nplitude of the average spin per site, and αGis the GD\nstrength. The term −αG|ωn|describes the spin relax-\nation within the FI. In FMR experiments, the microwave\nexcitationinducesauniformspinprecession,whichlimits\nthe magnon self-energy to the processes with qm= 079.\nIn the presence of the interfacial coupling, a cor-\nrection,δαG(ω), to the GD term is induced by the\nadjacent heterostructure TBG/WSe 2.δαG(ω) can be\nexpressed in terms of the the self-energy ΣR\n0(ω)≡\nΣqm=0(iωn→ω+iδ), resulting from the interfacial ex-\nchange interactions, as79\nδαG(ω)≡ −2S0\n/planckover2pi1ωImΣR\n0(ω). (8)\nFor simplicity, we neglect the real part of ΣR\n0(ω) which\nsimply shifts the FMR line and did not affect the\nlinewidth, in which we are interested. The self-energy,\nin Eq. (8), includes the contributions of all the inter-\nfacial spin transfer processes and can be written as\nΣ0(iωn) =/summationtext\nqΣ0(q,iωn). Each process, described by\nthe self-energy Σ 0(q,iωn), is characterized by a momen-\ntum transfer qand a matrix element Tq,qm=0≡Tq,0.\nIn the second order perturbation, with respect to\nthe interfacial exchange interaction Tq,0, the self-energy\nΣ0(q,iωn), is written as78\nΣ0(q,iωn) =|Tq,0|2\n4β/summationdisplay\nk,iωmTr/bracketleftig\nσx′,−\nsˆg(k,ωm)\n×σx′,+\nsˆg(k+q,iωm+iωn)/bracketrightig\n.(9)\nσx′,±\nsare the electronic spin ladder operators written in\nthe coordinate system ( x′,y′,z′) of the FI magnetiza-\ntion characterized by an average spin /an}bracketle{tSFI/an}bracketri}ht= (S0,0,0).\nˆg(k,iωm) is the electronic Matsubara Green function\ngiven by ˆg(k,iωn) =/bracketleftig\niωnI−H(1)\nSOC(k)/bracketrightig−1\n, whereωn=\n(2n+ 1)π//planckover2pi1βare the fermionic Matsubara frequencies.\nIn the basis of the spin-band four-component spinor Ψ =\n(ψ+,↑,ψ+,↓,ψ−,↑,ψ−,↓)56, ˆg(k,iωn) reads as ˆg(k,iωn) =\nˆg0(k,iωn)Is+ˆ g(k,iωn)·s, where s= (sx,sy,sz) are\nthe spin-Pauli matrices; ˆ g= (ˆgx,ˆgy,ˆgz), ˆg0, and ˆgi\n(i=x,y,z) are expressed, to the leading order in the\nSOC, as a function of the band-Pauli matrices σi(see\nSec. III of the Supplemental Material66).Since the ferromagnetic peak, given by Im GR\n0, is sharp\nenough, namely αG+δαG≪1, one can replace the res-\nonance frequency ωqm=0by the FMR frequency Ω. The\nGD correction can then be expressed as66,78\nδαG(Ω) =−2S0\n/planckover2pi1ΩImΣR\n0(Ω). (10)\nIn general, the interfacialspin transferincludes cleanand\ndirty processes. The former (latter) take place with con-\nserved (non-conserved) electron momentum, which turns\nout to take q=0(q/ne}ationslash=0) in Eq. ( 979).\nWefirstconsideracleaninterface, forwhichananalytical\nexpression of the GD correction [Eq. 10] can be derived\n(see Sec. IV of the Supplemental Material66and refer-\nence78,79). The case of a dirty junction is discussed in\nthe next section.\nCarryingoutthe summationover ωminEq.(9), weob-\ntaintheanalyticalexpressionoftheinterfacialself-energy\n(see Sec. IV of the Supplemental Material66). The sum\nover the electronic states k= (k,ϕk) runs over the states\nincluded within a cutoff, kc∼q0/2, on the momentum\namplitudek, where the low-energy Hamiltonian [Eq. ( 3)]\nis expected to hold (see Sec. IV of the Supplemental\nMaterial66).\nIn the following, we discuss the behavior of the nor-\nmalized GD coefficient\nδαG/α0\nG=/parenleftbiggλ\n/planckover2pi1Ω/parenrightbigg2\n˜Σ(q=0,Ω), (11)\nwhere˜Σ is a dimensionless function depending on the\ntwist angle θ, temperature T, the chemical poten-\ntialµand the orientation of the FI magnetization,\nα0\nG= 2S0/parenleftig\n|T0|\nλ/parenrightig2\nandλ=λI+λR\n2is the average SOC\n(for details, see Sec. IV of the Supplemental Material66\nand reference80).\nDiscussion. – In Fig.2, we plotδαG/α0\nG[Eq. (11)],\nas a function of the twist angle θ, for the undoped\nTBG, at different temperatures and for a fixed FMR\nenergy/planckover2pi1Ω = 0.06meV which corresponds to the yttrium\niron garnet. The SOC parameters are λI= 3meV and\nλI= 4meV as in Ref. [ 56].\nFigure2shows that regardless of the temperature\nrange,δαGincreasesbydecreasing θbut dropssharplyat\nthe MA, where it exhibits a relatively small peak which\nis smeared out at low temperature.\nPutting aside its drop at the MA, the enhancement of\nδαG, by decreasing θ, can be, in a first step, ascribed to\nthe dependence of the self-energy [Eq. ( 9)] on the effec-\ntive SOC, given by Eq. ( 7), which increase by decreasing\nθ. However, to understand the behavior of δαGat the\nMA one needs to go back to the band structure, Eσ,±(k)\n[Eq. (6)], of the continuum Hamiltonian of TBG/WSe 2,\nwhich is depicted in Fig. 3at different twist angles.\nThe arrows indicate the out-of-plane electronic spin4\nFigure 2. Normalized GD, δαG/α0\nG[Eq. (11)], as a function of\nthe twist angle at different temperature ranges. Calculatio ns\nare done for λI= 3 meV, λR= 4 meV, µ= 0, and for a FMR\nenergy/planckover2pi1Ω = 0.06 meV.\nprojection /an}bracketle{tsz/an}bracketri}htwhich we have numerically calculated\nfor different twist angles in Sec. II of the Supplemental\nMaterial66.\nAway from the MA, the band dispersion gets larger\nasθdecreases and, in particular, the separation between\nbands with opposite /an}bracketle{tsz/an}bracketri}ht, involved in the SP process, in-\ncreases. This behavior is due to the angle dependence\nof the effective Fermi velocity v∗of TBG/WSe 2, which\nreduces, in the first order in the SOC, to that of TBG,\nnamely (see Secs. I and II of the supplemental Mate-\nrial66)\nv∗∼vF1−3α2\n1+6α2(12)\nThe expression of the GD [Eq. ( 11)] includes transitions\nbetweenbandswithopposite /an}bracketle{tsz/an}bracketri}ht(seeSec. IVoftheSup-\nplemental Material66). These transitions depend on the\nstatistical weight ∆ f(E) =f(E/angbracketleftsz/angbracketright)−f(E−/angbracketleftsz/angbracketright) where\nf(x) is the Fermi-Dirac function and E/angbracketleftsz/angbracketrightis the energy\nband with a spin orientation /an}bracketle{tsz/an}bracketri}ht.\nIn Fig.4, we plot a pictorial representationof the band\nstructureofthecontinuummodel[Eq.( 6)]andtheFermi-\nDirac distribution f(E) at a given temperature T. The\nband dispersiongets largeras θmovesawayfrom the MA\n(Fig.3) andthe separationbetween thebandswith oppo-\nsite/an}bracketle{tSz/an}bracketri}htincreases. As a consequence, the corresponding\nstatistical weight ∆ f(E) is enhanced compared to the\ncase around the MA. This behavior explains the drop of\nthe GD at the MA.\nAround the MA ( θ+\nMandθ−\nM), the statistical weight\n∆f(E) is reduced compared to that at the MA since the\nbandsE+,−andE−,−get closer (Fig. 3).\nThis behavior gives rise to the small peak at the MA\n(Fig.2), which disappearsat low temperature ( kBT <λ)\nwhere bands around the MA have the same statistical\nweight ∆f(E) = 1 (see Sec. IV of the Supplemental Ma-\nterial66). In this case, the GD is basically dependent on\nthe effective Fermi velocity v∗[Eq. (12)] which vanishesat exactly the MA. Such dependence is responsible for\nthe cancellation of several terms contributing to the self-\nenergy [Eq. ( 9)], as they are proportional to v∗[Eq. (12)]\n(see Sec. IV of the Supplemental Material66).\nLet us nowturn to thecaseofadirtyinterfacewherethe\nspin transfer should now also include the non-conserved\nmomentum processes. The corresponding self-energy\n[Eq. (9)] can also be expressed in terms of the thermal\nweight ∆f(E) governing the interband transitions (see\nSec. IV of the Supplemental Material66).\nRegarding the flatness of the bands, the dirty processes\nat the MA acquire, as in the clean limit, small thermal\nweights compared to the twist angles away from the MA,\nwhere the band are dispersive. In the dirty limit, the\nGilbert damping correction is, then, expected to drop at\nthe MA as found in the case of a clean interface.\nIt comes out that the twist angle dependence of δαG\nis a direct probe of the emergence of the flat bands in\nTBG. On the other hand, the temperature dependence\nof the fine structure around the MA provides an accurate\nmeasurement of the MA, with a precision below 0 .005◦\n(see Fig. S.4of the Supplemental Material66). It also\ngives an estimation of the SOC induced in TBG adjacent\nto a monolayer of TMD.\nIt is worth stressing that in our model we did not take\ninto account the electron-electron interactions which\nsignificantly distort the electronic band structure of\nTBG80–83. Near the MA, the dominant electron-electron\ninteraction is found to be the Coulomb interaction with\nan amplitude estimated to be 10-15 meV82, which is\nlarger than the width of the flat bands ∼2−5 meV\nand the SOC considered in the present work. How\nare the results of Fig.2 modified in the presence of\nCoulomb interaction? Treating this interaction within\nthe Hartree-Fock approximation revealed that the\nHartree term considerably widens the bands while the\nexchange term leads, basically, to broken-symmetry\nphases. At the charge neutrality, the Hartree term\nvanishes and the exchange potential, which concerns\nbands with identical spins, opens a gap of 4 meV80,82,83,\nwhich is of the order of the SOC amplitudes. As a\nconsequence, the statistical weight ∆ f(E) of the bands\nwith opposite /an}bracketle{tSz/an}bracketri}htis expected to increase, but keeping\nlarger values at small angles compared to the MA.\nMoreover, the bandwidth, around the MA, is found to\nrelatively increase under the exchange term80,82,83, but\nremains smaller than 3 meV, which preserve the flatness\nof the bands. It comes out, that our results hold in\nundoped TBG under Coulomb interaction, and can be\nused to extract the value of the MA at which the Gilbert\ndamping correction drops. Away from the neutrality,\nthe bands are substantially distorted by the Coulomb\ninteraction80,82,83and our results should be taken with a\ngrain of salt since they account for filling νfactors away\nfrom−0.5<ν <0.5, where the bandwidth, at the MA,\nis less than 4 meV.5\nFigure 3. Band structure of TBG/WSe 2in the continuum limit [Eq. ( 6)] atθ= 0.5◦(a),θ=θ−\nM= 1.043◦(b),θ=θ+\nM= 1.058◦\n(c) andθ= 1.2◦(d). The dashed lines represent the bands at the MA ( θM= 1.05◦). The red (blue) arrows correspond to the\nout-of-plane electronic spin projection /angbracketleftsz/angbracketright= +1 (/angbracketleftsz/angbracketright=−1)56. Calculations are done for λI= 3 meV and λR= 4 meV.\nFigure 4. Schematic representation of the band structure\nEσ,±(Eq.6) and the Fermi-Dirac distribution f(E). The\nbands in dashed and green lines correspond, respectively, t o\nthe MA and to a twist angle θfar from the MA. The red\n(blue) arrows represent the projection of the out-of-plane\nspin projection /angbracketleftSz/angbracketright= +1 (/angbracketleftSz/angbracketright=−1). Around the MA,\nthe bands are almost flat and the statistical weights ∆ f(E),\ncorresponding to the transitions between E−,+→E+,+and\nE−,−→E+,−, are small compared to the case of a twist angle\naway from the MA, where the band dispersion is larger.\nBesides interactions, strain is found to be a key pa-\nrameter in the emergence of flat bands in TBG68,70. The\neffect of strain can be included in our model by deriving\nthe strain induced correction to the Hamiltonian given\nby Eq. (3), taking into account the strain dependence of\nthe vectors qjconnecting the Dirac points70. The twist\nangle, at which δαGdrops, can then provide a way to\nmeasure the strain in TBG.\nExperimental realization. – Our proposed setup\nconsists of an interface between a FI and a fully hBN\nencapsulated TBG/WSe 2heterostructure (Fig. 1). The\nhBN layer acts as a tunnel barrier which prevents the\ndiffusion of the FI atoms into the graphene layer84.\nOn the other hand, the encapsulation provides a clean\ninterface and prevents the graphene degradation84\nwhich is a challenging issue in the STM and ARPES\nexperiments24–26, carried out on non-encapsulated TBG\nsamples.\nIt should be stressed that the hBN encapsulated\nTBG/WSe 2heterostructure has been already realizedin Refs. [ 56and57]. Furthermore, the spin transport\nthrough a clean interface between a FI and 2D material\nhas been experimentally achieved84,85. The 2D materials\nwere fully encapsulated by hBN84or covered by a thin\nlayer of an oxide insulator (as MgO)85to avoid the\ninterdiffusion with the FI.\nOurproposedtechniquetomeasurethe MAcan, then, be\nimplemented experimentally with a clean interface and\nat room temperature. Moreover, an insitumanipulation\nof the twist angle can be realized as in Refs. [ 86–89].\nConclusion. – To conclude, we have proposed an\nexperiment to probe the flat bands of TBG and to\nmeasure its MA accurately. The experiment is based\non a spin pumping measurement through a junction\nseparating a FI and a TBG adjacent to a monolayer of\nWSe2. We first derived the continuum model of TBG\nwith SOC, which constitutes a first step to develop an\nanalytical understanding of the emergence of a stable\nsuperconducting state at small twist angles observed\nin TBG in proximity to WSe 256. We then determined\nanalytically the Gilbert damping correction δαGinduced\nby the presence of the TBG/WSe 2heterostructure. Our\nresults show that the twist angle dependence of δαG\nexhibits a drop at the MA with a temperature-dependent\nfine structure. This feature provides an accurate de-\ntermination of the MA and an estimation of the SOC\ninduced in TBG by its proximity to the TMD layer. Our\nproposed set-up can be readily implemented regarding\nthe state-of-the art of the experimental realizations of\nSP in 2D materials and TBG-based heterostructure.\nOur work opens the gate to a twist tunable spintronics\nin twisted layered heterostructures.\nAcknowledgments. – We thank Mamoru Matsuo and\nShu Zhang for stimulating discussions. We are indebted\nto Jean-No¨ el Fuchs and Daniel Varjas for a critical read-\ning of the manuscript. S. H. acknowledges the kind hos-\npitality of the Institute for Solid State Physics (ISSP)\nwhere this work was carried out. S. H. also thanks the\nhospitality of the Max Planck Institute for the Physics of\nComplex Systems (MPI-PKS). S. H. acknowledges finan-\ncial support from the ISSP International visiting profes-6\nsors program and the MPI-PKS visitors program.\n∗sonia.haddad@fst.utm.tn\n1Y.Cao, V.Fatemi, S. Fang, K.Watanabe, T. Taniguchi, E.\nKaxiras, and P. 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Inbar, J. Birkbeck, J. Xiao, T. Taniguchi, K. Watan-\nabe, B. Yan, Y. Oreg, A. Stern, E. Berg and S. Ilani, The\nquantum twisting microscope, Nature 614, 682 (2023).9\nSUPPLEMENTAL MATERIAL\nTwisted bilayer graphene reveals its flat bands under spin pu mping\nSonia Haddad1,2,3, Takeo Kato2, Jihang Zhu3, and Lassaad Mandhour1\n1Laboratoire de Physique de la Mati` ere Condens´ ee, Facult´ e des Sciences de Tunis, Universit´ e Tunis El Manar,\nCampus Universitaire 1060 Tunis, Tunisia\n2Institute for Solid State Physics, University of Tokyo, Kashiwa, Ch iba 277-8581, Japan\n3Max Planck Institute for the Physics of Complex Systems, N¨ othnit zer Strasse 38, Dresden 01187, Germany\nS.1. I. DERIVATION OF THE LOW-ENERGY HAMILTONIAN OF TBG WITH OUT SOC\nWe start by a brief overview of the perturbative approach propos ed by Bistritzer and MacDonald65to derive the\ncontinuum model of TBG. We consider a TBG where the two layers l= 1,2 are rotated oppositely θ2=−θ1=θ\n2.\nThe Hamiltonian of a graphene layer lrotated at an angle θlis\nhl(k) =eiθl\n2σzh(0)\nl(k)e−iθl\n2σz(S.1)\nwhereh(0)\nl(k) is the Hamiltonian of the unrotated layer ( l) given, in the continuum limit, by\nh(0)\nl(k) =−/planckover2pi1vFk·σ∗, (S.2)\nwhere the momentum kis written relatively to the Dirac point Kl,ξ,vFis the Fermi velocity, ξis the valley index,\nσ∗= (ξσx,σy) andσi(i=x,y,z) are the sublattice-Pauli matrices.\nThe leading contributions of the interlayer tunneling can be limited to t hree nearest hopping processes in the\nmomentum space connecting states |k/an}bracketri}ht1, around the Dirac point K1,ξof layer (1), to the states |k+qjξ/an}bracketri}ht2around\nK2,ξ, the Dirac point of layer (2). The qjξvectors are given by65\nq1ξ=ξkθ(0,1),q2ξ=q1ξ+ξGM\n1=ξkθ/parenleftigg\n−√\n3\n2,−1\n2/parenrightigg\n,\nq3=q1ξ+ξ/parenleftbig\nGM\n1+GM\n2/parenrightbig\n=ξkθ/parenleftigg√\n3\n2,−1\n2/parenrightigg\n, (S.3)\nwherekθ= 2kDsinθ\n2∼θkDandkD=|K1,ξ|=|K2,ξ|=4π\n3a,abeing the graphene lattice parameter. The/parenleftbig\nGM\n1,GM\n2/parenrightbig\nis the moir´ e BZ basis given by GM\ni=RT\ntGi,Giare the lattice basis vectors of the monolayer reciprocal lattice\nG1=2π\na/parenleftbig\n1,−1/√\n3/parenrightbig\nandG2=2π\na/parenleftbig\n0,2/√\n3/parenrightbig\n.Rtis the rotation tensor written, in the sublattice basis, at a small\ntwist angle as\nR(θ) =/parenleftbigg\n0−θ\nθ0/parenrightbigg\n. (S.4)\nIn the basis {|k/an}bracketri}ht1,|k+qj,ξ/an}bracketri}ht2}, the Hamiltonian, at the valley ξ, reads as65\nH(k) =\nh1(k)T1T2T3\nT†\n1h2,1(k) 0 0\nT†\n20h2,2(k) 0\nT†\n30 0 h2,3(k)\n, (S.5)\nFor the relaxed TBG the Tjmatrices are given by69\nT1=/parenleftbigg\nw w′\nw′w′′/parenrightbigg\n,T2=eiξGM\n1·r/parenleftbigg\nw w′e−iξΦ\nw′eiξΦw′′/parenrightbigg\n,T3=eiξ(GM\n1+GM\n2)·r/parenleftbigg\nw w′eiξΦ\nw′e−iξΦw′′/parenrightbigg\n, (S.6)\nHereh1(k) is given by Eq. S.1andh2,j(k)≡h2(k+qjξ) =h1(k+qjξ), where the momentum kis written relatively\ntoK1,ξ. Φ =2π\n3andris the shortest inplane shifts between carbon atoms of the two laye rs8,67. Hereafter, we neglect\nthe relative sliding between the layers which is not relevant in the phys ics of TBG8,65. Choosing the A sublattice in10\nboth layers ( i= 1,2) as the origin of the unit cell, turns out to take r=0in Eq.S.68. The parameters w,w′andw′′\nare the tunneling amplitudes which take the same value w=w′=w′′∼110 meV in the rigid TBG68.\nIn the relaxed lattice, these amplitudes are no more equal w∼w′′∼90 meV and w′= 117 meV69. In the present\nwork, we do not consider the lattice relaxation effect, since the SOC parameters λI, λR∼4meV are small compared\nto the difference between the interlayer amplitudes ∆ w=w−w′∼20meV. In the unrelaxed lattice, the Tjmatrices\ncan be written as\nT1=w(Iσ+σx),T2=w/parenleftigg\nIσ−1\n2σx+ξ√\n3\n2σy/parenrightigg\n,T3=w/parenleftigg\nIσ−1\n2σx−ξ√\n3\n2σy/parenrightigg\n. (S.7)\nHere,Iσis the identity matrix acting on the sublattice indices.\nConsidering, in Eq. S.5, thekdependent term as a perturbation, the effective Hamiltonian can be written, to the\nleading order in k, as\nH(1)(k) =/an}bracketle{tΨ|H(k)|Ψ/an}bracketri}ht\n/an}bracketle{tΨ|Ψ/an}bracketri}ht, (S.8)\nwhere Ψ = ( ψ0(k),ψ1(k),ψ2(k),ψ3(k)) is the zero energy eigenstate of H(k=0). Ψ is constructed on the two-\ncomponent sublattice spinor ψ0(k) (ψj(k)) of layer 1 (layer 2) taken at the momentum k(k+qjξ) around the Dirac\npointK1,ξ(K2,ξ) at the valley ξ.ψ0is the zero energy eigenstate of h1. The Ψ components satisfy\nh1ψ0+/summationdisplay\njTjψj= 0,andT†\njψ0+hjψj= 0,withh1ψ0= 0, (S.9)\nwherehj≡h2(qj,ξ). Then\nψj=−h−1\njT†\njψ0,and/summationdisplay\njTjh−1\njT†\nj= 0. (S.10)\nTo the leading order in k,H(1)(k), takes the following form\nH(1)(k) =/an}bracketle{tΨ|H(k)|Ψ/an}bracketri}ht\n/an}bracketle{tΨ|Ψ/an}bracketri}ht=1\n/an}bracketle{tΨ|Ψ/an}bracketri}ht\nψ†\n0h1(k)ψ0+ψ†\n0/summationdisplay\njTjh−1\njhj(k)h−1\njT†\njψ0\n, (S.11)\nH(1)(k) =−/planckover2pi1v∗ψ†\n0k·σ∗ψ0, (S.12)\nEq.S.12is obtained by neglecting θinhj(k), which turns out to take qj,ξ=0inhj(k)65.\nv∗is the effective velocity of the energy band of TBG around the zero e nergy which vanishes at the MA θm, and is\ngiven by65\nv∗=vF1−3α2\n1+6α2(S.13)\nwhereα=w\n/planckover2pi1vFq0,q0=|qjξ| ∼4π\n3aθ.\nIn our numerical calculations (Fig. 3of the main text), we take w= 118 meV and /planckover2pi1vF/a∼2.68 eV which corresponds\ntoθm= 1.05◦for the first MA65.\nS.2. II. DERIVATION OF THE LOW-ENERGY HAMILTONIAN OF TBG WIT H SOC\nWe now consider the heterostructure consisting of TBG adjacent to a monolayer of WSe 2as shown in Fig.1 of the\nmain text, where we denote the graphene layer in contact with the T MD by layer (2). This layer is subject to a SOC\ninduced by proximity effect by the TMD, and the corresponding Hamilt onian can be written as56\nh2(k) =h1(k)+hSOC+m\n2σz (S.14)\nwherehSOCis given by Eq. 1of the main text.11\nTo derive the continuum model of TBG/WSe 2, we follow the perturbative approach of Ref.65presented in the\nprevioussection. Now, thebasisΨ = ( ψ0(k),ψ1(k),ψ2(k),ψ3(k)) isconstructedonthefour-componentspin-sublattice\nspinorψ0(k) andψj(k), (j= 1,2,3) corresponding, respectively, to layer (1) and layer (2). ψ0(k) is written as\nψ0(k)T= (ψ0,A↑,ψ0,A↓,ψ0,B↑,ψ0,B↓). In this basis, the Hamiltonian of TBG/WSe 2takes the form\nHξ,SOC(k) =\nh1(k)T1T2T3\nT†\n1h2,1(k) 0 0\nT†\n20h2,2(k) 0\nT†\n30 0 h2,3(k)\n. (S.15)\nThe momentum kis measured relatively to the Dirac point K1ξof layer (1), h2,j(k) =h2(k+qjξ), (j= 1,2,3) and\nh2(k) includes now the SOC terms (Eq. S.14).\nWe take the sublattice A as the origin of the unit cell in each layer. The Tjmatrices are written as the tensor product\nof those given by Eq. S.7, with the 2 ×2 identity spin-matrix Is.\nRegarding the small values of the SOC, we assume that Hξ,SOC(k) has a zero eigenenergy and the corresponding\neigenstate Ψ satisfies the condition given by Eq. S.10.\nFollowingthesameprocedureasintheprevioussection, wederivefr omEq.S.11theeffectivelowenergyHamiltonian\nH(1)\nξ,SOC(k) of TBG/WSe 2by substituting hj(k) by the Hamiltonian of layer (2), rotated at θ/2 and including SOC\nas\nh2,rot(k) =−/planckover2pi1k·σ∗Is+λI\n2ξsz+λR\n2(ξσxsy−σysx)+λKM\n2ξσzsz−λR\n2θ(ξσysy+σxsx), (S.16)\nHereafter, we neglect the Kane and Mele term whose contribution, to the leading order in k, is found to vanish. We\nalso disregard the last term in Eq. S.16, which results into a higher order correction in θ.\nTo the first orderin the SOC coupling, we obtain the continuum model of TBG/WSe 2described by the Hamiltonian\nH(1)\nξ,SOC(k) given by Eq. 3in the main text. This Hamiltonian contain a SOC term ( hSOC\neff) with renormalized Ising\nand Rashba interactions\n˜λI=6α2\n/an}bracketle{tΨ|Ψ/an}bracketri}htλI,˜λR=3α2\n/an}bracketle{tΨ|Ψ/an}bracketri}htλR,and/an}bracketle{tΨ|Ψ/an}bracketri}ht ∼1+6α2(S.17)\n˜λIand˜λRare enhanced by decreasing the twist angle from the MA.\nFigure S.1. Energy bands of TBG/WSe 2around zero energy as function of the dimensionless momentu m amplitude k/q0at\ndifferent twist angles. The bands are represented up to the cu toffkc=q0/2. Calculations are done for λI= 3 meV and\nλR= 4 meV56. The MA is θM= 1.05◦.12\nTo the leading order in k, the four eigenergies of the Hamiltonian H(1)\nξ,SOC(Eq.3of the main text), denoted E(k)σ,±,\nare given by\nE(k)σ,±=σ\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/radicalig\nf1(k)±6α2/radicalbig\nf2(k), (S.18)\nf1(k) = (/planckover2pi1vF)2/parenleftbig\n1−3α2/parenrightbig2||k||2+9\n2α4/parenleftbig\n2λ2\nI+λ2\nR/parenrightbig\n,\nf2(k) = (/planckover2pi1vF)2/parenleftbig\n1−3α2/parenrightbig2||k||2/parenleftbigg\nλ2\nI+1\n4λ2\nR/parenrightbigg\n+9\n16α4λ4\nR, (S.19)\nwhereσ=±is the band index. E(k)σ,±are depicted in Fig. S.1at different twist angles.\nAt the Dirac point, the eigenergies reduce to\nE±,+=±3α2\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/radicalig\nλ2\nI+λ2\nR,E±,−=±3α2\n/an}bracketle{tΨ|Ψ/an}bracketri}htλI. (S.20)\nIt is worth to note that in TBG, the flatness of the bands around th e charge neutrality point strongly depends on\nthe heterostrain which may emerge in the graphene layers during th e fabrication procedure70. The interplay between\nstrain and SOC in TBG/WSe 2goes beyond the scope of the present work.\nIn figures S.2andS.3, we plot the electronic band structure of TBG/WSe 2at different twist angles. The solid\nlines correspond to the numerical results obtained within the contin uum model taking into account 148 bands for\neach spin and valley. The calculations are done for the relaxed TBG wit h interlayer momentum hopping amplitudes\nw= 55meV and w= 105meV (Eq. S.6). We considered these values to reproduce the numerical band st ructure\nobtained in Ref.[ 56] using the continuum model. We have taken into account the lattice r elaxation, as in Ref. 56,\nsince it is expected to be important at small angles. The numerical re sults are compared to the eigenergies of the\neffective Hamiltonian H(1)\nξ,SOC, given by Eq. 3of the main text (dashed line), and to the approximated expression s\n(Eq.S.18) represented by dotted gray lines. It should be stressed that H(1)\nξ,SOCis derived for a rigid TBG, for which\nwe have taken an interlayer hopping amplitude w= 105meV, which gives rise to a MA θM∼1.1◦as in Ref [ 56].\nFor the sake of simplicity, we did not consider the relaxation effect, w hich is not significant around the MA56, where\nwe consider the spin pumping effect. The derivation of the effective H amiltonian of the relaxed TBG adjacent to\nWSe2is left to a future work.\nAs shownby Fig. S.2andS.3, the effective Hamiltonian H(1)\nξ,SOC(Eq.3ofthe main text) providesa good description\nof the band structure of TBG/WSe 2. It can be taken as a framework to unveil the origin of the observe d stable\nsuperconducting state in this heterostructure56. However, at relatively small angles ( ∼0.5◦), the Fermi velocities of\nH(1)\nξ,SOCare overestimated (Fig. S.2). This discrepancy is due to the assumption of a rigid TBG lattice which is not\njustified at small angles56.\nS.3. III. ELECTRONIC GREEN FUNCTION\nThe Matsubara Green function associated to the effective Hamilton ianH(1)\nξ,SOC(Eq.5) is\nˆg(k,iωn) = [iωnISIσ−Hξ,SOC(k)]−1, (S.21)\nwhereISandIσare the 2 ×2 spin and band identity matrices, respectively. ˆ g(k,iωn) can be expressed as\nˆg(k,iωn) = ˆg0(k,iωn)Is+ˆ g·s. (S.22)\ns= (sx,sy,sz) are spin-Pauli matrices, ˆ g0(k,iωn) and the components ˆ gi(i=x,y,z) ofˆ gare written as\nˆg0(k,iωn) =A0(k,iωn)+C0(k,iωn),ˆgx(k,iωn) =Bx(k,iωn)+Dx(k,iωn),\ngy(k,iωn) =By(k,iωn)+Dy(k,iωn),ˆgz(k,iωn) =Az(k,iωn)+Cz(k,iωn). (S.23)13\n(a) (b) (c)\nFigure S.2. Electronic band structure of TBG/WSe 2calculated, at a twist angle (a) θ= 0.5◦, (b)θ= 0.79◦, and (c) θ= 0.87◦.\nCalculations are based on the continuum model and including 148 bands for each moir´ e valley and spin. The line color deno tes\nthe value of the out-of-plane spin projection /angbracketleftSz/angbracketright. The dashed black lines represent the eigenergies of the fou r-band effective\nHamiltonian given by Eq. 3of the main text, and the gray dotted lines denote the approxi mated eigenergies given by Eq. S.18.\nCalculations are done for λR= 4meV and λI= 3meV. The band structure is represented in the moir´ e Brill ouin zone where κ\nandκ′correspond respectively to the Dirac point K1of layer (1) and K2layer (2) at the valley ξ= +. The bottom panel is a\nzoomed-in representation around the high symmetry point κ.\n(a) (b) (c)\nFigure S.3. Electronic band structure of TBG/WSe 2calculated around the MA θMat (a)θ= 1.05◦, (b)θM= 1.1◦, (c)\nθ= 1.2◦. The bottom panel is a zoomed-in representation around the h igh symmetry point κ. The data are the same as in\nFig.S.2.14\nTheA,B,CandDoperators are written in terms of the band-Pauli matrices σx,y,zand the corresponding identity\nmatrixIσ\nAi(k,iωn) =Ai1(k,iωn)Iσ+Aiz(k,iωn)σz\nCi(k,iωn) =Cix(k,iωn)σx+Ciy(k,iωn)σy\nBj(k,iωn) =Bj1(k,iωn)Iσ+Bjz(k,iωn)σz\nDj(k,iωn) =Djx(k,iωn)σx+Djy(k,iωn)σy (S.24)\nherej=x,yandi= 0,z.\nIn the limit of small SOC couplings λR,λI≪/planckover2pi1vFq0, A, B, C, and D become\nA01(k,iωm) =Y00−E2\n1\n2(E2\n3−E2\n1)/bracketleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2/bracketrightbigg\n+E2\n3−Y00\n2(E2\n3−E2\n1)/bracketleftbigg1\ni/planckover2pi1ωm−E3+1\ni/planckover2pi1ωm−E4/bracketrightbigg\n(S.25)\nY00=(/planckover2pi1vF)2(1−3α2)2\n/an}bracketle{tΨ|Ψ/an}bracketri}ht2||k||2+1\n2/parenleftbigg3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\n+/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\nA0z(k,iωm) =Y0\n2(E2\n1−E2\n3)/braceleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracerightbigg\n(S.26)\nY0=−3\n2/parenleftbigg3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht\nC0x(k,iωm) =1\n2(E2\n1−E2\n3)/braceleftbigg\nY2/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n−Y1/bracketleftbigg\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\nξkx (S.27)\nY1=/planckover2pi1vF(1−3α2)\n/an}bracketle{tΨ|Ψ/an}bracketri}ht, Y2=−Y1/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\nC0y(k,iωm) =1\n2(E2\n1−E2\n3)/braceleftbigg\nY2/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n−Y1/bracketleftbigg/parenleftbigg\nE11\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\nky (S.28)\nAz1(k,iωm) =ξ1\n2(E2\n1−E2\n3)/braceleftbigg\nY4/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n−Y3/bracketleftbigg\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\n(S.29)\nY3=−3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht, Y4=−Y3/bracketleftigg\n(/planckover2pi1vF)2\n/an}bracketle{tΨ|Ψ/an}bracketri}ht2(1−3α2)2||k||2−1\n2/parenleftbigg3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\n−/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2/bracketrightigg\nAzz(k,iωm) =ξY5\n2(E2\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\n(S.30)\nY5=1\n2/parenleftbiggλR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\nCzx(k,iωm) =Y6\n2(E2\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\nkx (S.31)\nY6=/planckover2pi1vF(1−3α2)6α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht215\nCzy(k,iωm) =Y6\n2(E2\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\nξky\n(S.32)\nBx1(k,iωm) =Y7\n2(E2\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\nky (S.33)\nY7=/planckover2pi1vF(1−3α2)3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht2\nBxz(k,iωm) =Y81\n2(E2\n1−E2\n3)ky×\n/braceleftbigg1\nE1/bracketleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/bracketrightbigg\n+1\nE3/bracketleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracketrightbigg/bracerightbigg\n(S.34)\nY8=−/planckover2pi1vF(1−3α2)3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht23α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht\nDxx(k,iωm) =Y91\n2(E2\n1−E2\n3)ξkxky×\n/braceleftbigg1\nE1/bracketleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/bracketrightbigg\n−1\nE3/bracketleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracketrightbigg/bracerightbigg\n(S.35)\nY9= (/planckover2pi1vF)2(1−3α2)23α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht\nDxy(k,iωm) =1\n2(E2\n1−E2\n3)/braceleftbigg\nY11/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n−Y10/bracketleftbigg\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\n(S.36)\nY10=−3α2λR\n2/an}bracketle{tΨ|Ψ/an}bracketri}ht, Y11=Y10/bracketleftigg\n(/planckover2pi1vF)2(1−3α2)2(k2\nx−k2\ny)+/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2/bracketrightigg\nBy1(k,iωm) =−Y71\n2(E3\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\nkx(S.37)\nByz(k,iωm) =−Y81\n2(E2\n1−E2\n3)kx/braceleftbigg1\nE1/bracketleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/bracketrightbigg\n−1\nE3/bracketleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracketrightbigg/bracerightbigg\n(S.38)\nDyx(k,iωm) =ξ1\n2(E2\n1−E2\n3)/braceleftbigg\nY12/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n+Y10/bracketleftbigg\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\n(S.39)\nY12=3α2λR\n3/an}bracketle{tΨ|Ψ/an}bracketri}ht/bracketleftigg\n−(/planckover2pi1vF)2(1−3α2)2(k2\nx−k2\ny)+/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2/bracketrightigg\nDyy(k,iωm) =−Y91\n(E2\n1−E2\n3)kxky/braceleftbigg1\nE1/bracketleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm+E1/bracketrightbigg\n−1\nE3/bracketleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracketrightbigg/bracerightbigg\n,(S.40)\nwhereE1=E+,+(k),E2=E−,+(k),E3=E+,−(k) andE4=E−,−(k) (Eq.S.18).16\nS.4. IV. MAGNON GREEN FUNCTION AND GILBERT DAMPING\nA. Interfacial exchange coupling between a ferro- magnetic insulator (FI) and a TBG\nWe consider the Hamiltonian of the ferromagnetic insulator (FI) in th e independent magnon approximation as\nHFI=/summationdisplay\nk/planckover2pi1ωkb†\nkbk, (S.41)\nwhere/planckover2pi1ωk≃ D|k|2+/planckover2pi1γhdcis a dispersion of magnons, Dis a spin stiffness, γis the gyromagnetic ratio, hdcis a\nstatic magnetic field. In the spin pumping setup, only the static part associated to k=0is relevant. Considering\nonly the uniform spin precession, the Hamiltonian of the FI can be simp ly written as\nHFI=/planckover2pi1ω0b†\n0b0, (S.42)\nwherebqis the Fourier transformation of the site representation bidefined as\nbi=1√NFI/summationdisplay\nqeiq·ribq≃1√NFIb0, (S.43)\nb†\ni=1√NFI/summationdisplay\nqe−iq·rib†\nq≃1√NFIb†\n0, (S.44)\nwhereNFIis the number of unit cells in the FI.\nWe consider the (retarded) magnon Green function as\nGR(q,ω) =/integraldisplay\ndtGR(q,t)eiωt, (S.45)\nGR(q,t) =−i\n/planckover2pi1θ(t)/an}bracketle{t[S+\nq(t),S−\nq(0)]/an}bracketri}ht=−2iS0\n/planckover2pi1θ(t)/an}bracketle{t[bq(t),b†\nq(0)]/an}bracketri}ht. (S.46)\nIn the absence of the junction, the magnon Green function is\nGR\n0(q,ω) =2S0//planckover2pi1\nω−ωq+iδ. (S.47)\nWe introduce spin relaxation of the bulk FI phenomenologically as\nGR\n0(q,ω) =2S0//planckover2pi1\nω−ωq+iαGω, (S.48)\nwhereαGis a dimensionless strength of the Gilbert damping, which is of order of 10−4–10−3. We note that a line\nshape of the ferromagnetic resonance is proportional to Im GR\n0(q=0,ω)79.\nIn the presence of the interfacial coupling and for a uniform spin pr ecession, the magnon Green function is given\nby the Dyson equation as\nGR\n0(q= 0,ω) =2S0//planckover2pi1\nω−ωq=0+iαGω−(2S0//planckover2pi1)ΣR(q= 0,ω), (S.49)\nwhere ΣR(ω) is the self-energy. Although the real part of ΣR(ω) is related to the shift of the ferromagnetic resonance,\nwe neglect it for simplicity. Then, the magnon Green function is rewrit ten as\nGR\n0(0,ω) =2S0//planckover2pi1\nω−ωq=0+i(αG+δαG)ω, (S.50)\nδαG(ω) =−2S0\n/planckover2pi1ωImΣR(q=0,ω). (S.51)\nWe note that δαG(ω) depends on ωin general. However, since the ferromagnetic resonance peak is sh arp enough\n(αG+δαG≪1), we can replace ωwithω0= Ω (the peak position of the ferromagnetic resonance):\nδαG≃ −2S0\n/planckover2pi1ΩΣR(q= 0,Ω). (S.52)17\nThe Hamiltonian of the interfacial coupling is given as\nHint=/summationdisplay\n/angbracketlefti,j/angbracketrightTij(S+\nis−\nj+h.c.). (S.53)\nHere,S±\niis a spin ladder operator of the FI and is described by magnon annihilat ion/creation operators ( biandb†\ni)\nas\nS+\ni=/radicalbig\n2S0bi, S−\ni=/radicalbig\n2S0b†\ni, (S.54)\nwhereS0is an amplitude of the localized spin in the FI. s±\njis a spin ladder operator of electrons in two-dimensional\nelectron systems (twisted bilayer graphene) and is described by th e electron annihilation/creation operators ( cjσand\nc†\njσ) as\ns+\nj=c†\nj↑cj↓, s−\nj=c†\nj↓cj↑. (S.55)\nWe define the Fourier transformation as\ncjσ=1√\nN/summationdisplay\nkeik·rjckσ, (S.56)\nc†\njσ=1√\nN/summationdisplay\nke−ik·rjc†\nkσ, (S.57)\nwhereNis the number of unit cells and rjis the position of the site jin TBG. Then, we obtain\ns+\nj=1\nN/summationdisplay\nk,k′e−ik·rj+ik′·rjc†\nk↑ck′↓, (S.58)\ns−\nj=1\nN/summationdisplay\nk,k′e−ik·rj+ik′·rjc†\nk↓ck′↑. (S.59)\nWe define the Fourier transformation of s±\njas\ns+\nj=1\nN/summationdisplay\nqeiq·rjs+\nq, (S.60)\ns−\nj=1\nN/summationdisplay\nqeiq·rjs−\nq. (S.61)\nFroms−\nj= (s+\nj)†, we obtain the relation s−\nq= (s+\n−q)†. The inverse Fourier transformation is given as\ns+\nq=/summationdisplay\nje−iq·rjs+\nj, (S.62)\ns−\nq= (s+\n−q) =/summationdisplay\nje−iq·rjs−\nj. (S.63)\nEspecially for q=0, we obtain\ns+\n0=/summationdisplay\njs+\nj, s−\n0=/summationdisplay\njs−\nj, (S.64)\nUsing Eqs. ( S.56) and (S.57), we obtain\ns+\nq=1\nN/summationdisplay\nje−iq·rj/summationdisplay\nk,k′e−ik·rj+ik′·rjc†\nk↑ck′↓=/summationdisplay\nkc†\nk↑ck+q↓, (S.65)\ns−\nq=1\nN/summationdisplay\nje−iq·rj/summationdisplay\nk,k′e−ik·rj+ik′·rjc†\nk↓ck′↑=/summationdisplay\nkc†\nk↓ck+q↑. (S.66)18\nFor a clean interface, we can set Tij=T. Then, using Eqs. ( S.43) and (S.44), the Hamiltonian of the interface is\nwritten as\nHint=T√2S0√NFI/summationdisplay\n/angbracketlefti,j/angbracketright(b0s−\nj+b†\n0s+\nj)≃T√2S0√NFI\nb0/parenleftig/summationdisplay\njs−\nj/parenrightig\n+b†\n0/parenleftig/summationdisplay\njs+\nj/parenrightig\n\n=/radicalbig\n2S0b0˜s−+/radicalbig\n2S0b0˜s+, (S.67)\nwhere ˜s±is defined as ˜ s±= (T/√NFI)s±\n0.\nBy the second-order perturbation, the self-energy of the magn on atq=0is calculated as\nΣR(ω) =/integraldisplay\ndtΣR(t)eiωt, (S.68)\nΣR(t) =−i\n/planckover2pi1θ(t)/an}bracketle{t[˜s+(t),˜s−(0)]/an}bracketri}ht. (S.69)\nThe self-energy can be related to a retarded component of the dy namic spin susceptibility per unit cell as\nΣR(ω) =−T2N\nNFIχ(0,ω), (S.70)\nχ(q,ω) =/integraldisplay\ndtχ(q,t)eiωt, (S.71)\nχ(q,t) =i\nN/planckover2pi1θ(t)/an}bracketle{t[s+\n−q(t),s−\nq(0)]/an}bracketri}ht. (S.72)\nχ(q,t) is calculated for one-band of TBG without spin-orbit interaction as\nχ(q,t) =1\nN/summationdisplay\nqf(ξk)−f(ξk+q)\n/planckover2pi1ω+iδ+ξk+q−ξk, (S.73)\nwhereξk=ǫk−µ,ǫkis a dispersion of electrons, µis a chemical potential. This is just a Lindhard function. We\nnote thatχ(q,t) is independent of the system size (area). For systems with spin-o rbit interaction, we have to extend\nthe Lindhard function into the spin-dependent one.\nThen, the enhancement of the Gilbert damping is written as\nδαG=−2S0\n/planckover2pi1ΩImΣR(q=0,Ω)\n=2S0T2N\nNFI/planckover2pi1ω0χ(q=0,Ω). (S.74)\nWe note that the number of the unit cell of twisted bilayer graphene is written as N=S/AwhereSis a area of the\njunction and Ais an area of a unit cell of twisted bilayer graphene. We also note that the number of the unit cell\nof the FI is written as NFI=Sd/a3wheredis a thickness of the FI, ais a lattice constant of the FI. Using these\nparameters, we obtain\nδαG=2S0T2a3\nAd/planckover2pi1ω0χ(q=0,Ω). (S.75)\nWe note that δαGis proportional to 1 /din consistent with experimental results. If YIG (Yttrium Iron Garn et) is\nchosen as the ferromagnet insulator, the parameter is given in the Table.\nB. Electronic spins in the FI magnetization frame\nRegarding the dependence on szof the electronic Hamiltonian (Eq. 3of the main text), one should consider a 3 D\nFI magnetization as in Ref.79. The average spin vector is along the orthoradial spherical vecto r/an}bracketle{tSFI/an}bracketri}ht=/an}bracketle{tSFI/an}bracketri}htux′.\nThe radial vector uz′forms an angle θmwith thezaxis perpendicular to the interface. The third axis y′is in the\n(xoy) plane and its unit vector is the orthoradial inplane vector uy′=−sinΦmux+cosΦ muyas shown in Fig. S.4.19\nTable I. Experimental parameters.\nMicrowave frequency ω01GHz\nAmplitude of spins of FI S010\nLattice constant of FI a12.376˚A\nThickness of FI d≥10nm\nInterfacial exchange coupling J∼1K (not known)\nFigure S.4. Magnetization-fixed coordinate frame ( x′,y′,z′) with respect to the Laboratory frame ( x,y,z).\nIn the FI spin frame ( x′,y′,z′), the components of the electronic spin operators are given by:\nsx′\nk=sk·ux′= cosθmcosΦmsx\nk+cosθmsinΦmsy\nk−sinθmsz\nk\nsy′\nk=sk·uy′=−sinΦmsx\nk+cosΦ msy\nk\nsz′\nk=sk·uz′= sinθmcosΦmsx\nk+sinθmsinΦmsy\nk+cosθmsz\nk\n(S.76)\nWe define the ladder electronic spin operators as\nsx′,±\nk=sy′\n±k±isz′\n±k=1\n2/summationdisplay\nσ,σ′,k′c†\nk′,σ/parenleftig\nσx′,±\ns/parenrightig\nσ,σ′c†\nk′±k,σ′ (S.77)\nwheresx′,±=sx(−sinΦm±isinθmcosΦm)+sy(cosΦm±isinθmsinΦm)±icosθmsz.\nC. Magnon self-energy\nIn the second order perturbation with respect to the interfacial exchange interaction Tq, the interfacial self-energy\nis given by78\nΣ(q,iωn) =|Tq|2\n4β/summationdisplay\nk,iωmTr/bracketleftig\nσx′,−\nsˆg(k,iωm)σx′,+\nsˆg(k+q,iωm+iωn)/bracketrightig\n(S.78)\nwhere ˆg(k,iωm) is the electronic Green function given by Eq. S.22.\nThe trace term is of the form:\nTr[a∗·σs(ˆg0+ˆg·σs)a·σs(ˆg′\n0+ˆg′·σs)] (S.79)\nwhere the vector a= (−sinΦm+isinθmcosΦm,cosΦm+isinθmsinΦm,icosθm) is written in the laboratory frame\n(x,y,z).\nWe set ˆg= ˆg(k,iωm) and ˆg′= ˆg(k,iωm+iωn). Taking into account the operator character of ˆ gone could use the\nidentity\n(a·σs)(b·σs) = (a·b)I+i(a×b)·σs (S.80)20\nGiven the expressions of ˆ gand ˆg′in Eq.S.23, the trace term (Eq. S.79) reduces to:\nTr[a∗·σs(ˆg0+ˆg·σs)a·σs(ˆg′\n0+ˆg′·σs)] =/summationdisplay\ni=0,1,2Fi(k,iωm,iωn) (S.81)\nwhere\nF0(k,iωm,iωn) = 4/parenleftbig\nA01A′\n01+A0zA′\n0z+C0xC′\n0x+C0yC′\n0y/parenrightbig\nF1(k,iωm,iωn) =−2{cosθmcosΦm/parenleftbig\nA01B′\nx1−Bx1A′\n01+A0zB′\nxz−A′\n0zBxz+C0xD′\nxx−C′\n0xDxx+C0yD′\nxy−C′\n0yDxy/parenrightbig\n+cosθmsinΦm/parenleftbig\nA01B′\ny1−By1A′\n01+A0zB′\nyz−A′\n0zByz+C0xD′\nyx−C′\n0xDyx+C0yD′\nyy−C′\n0yDyy/parenrightbig\n−sinθm/parenleftbig\nA01A′\nz1−A′\n01Az1+A0zA′\nzz−A′\n0zAzz+C0xC′\nzx−C′\n0xCzx+C0yC′\nzy−C′\nzyC0y/parenrightbig/bracerightbig\nF2(k,iωm,iωn) =−2cos2θmcos2Φm/parenleftbig\nBx1B′\nx1+BxzB′\nxz+DxxD′\nxx+DxyD′\nxy/parenrightbig\n−2cos2θmsin2Φm/parenleftbig\nBy1B′\ny1+ByzB′\nyz+DyxD′\nyx+DyyD′\nyy/parenrightbig\n−2sin2θm/parenleftbig\nAz1A′\nz1+AzzA′\nzz+CzxC′\nzx+CzyC′\nzy/parenrightbig\n−cos2θmsin2Φ m/parenleftbig\nBx1B′\ny1+BxzB′\nyz+DxxD′\nyx+DxyD′\nyy+B′\nx1By1+B′\nxzByz+D′\nxxDyx+D′\nxyDyy/parenrightbig\n+cosΦ msin2θm/parenleftbig\nBx1A′\nz1+BxzA′\nzz+DxxC′\nzx+DxyC′\nzyB′\nx1Az1+B′\nxzAzz+D′\nxxCzx+D′\nxyCzy/parenrightbig\n+sinΦ msin2θm/parenleftbig\nBy1A′\nz1+ByzA′\nzz+DyxC′\nzx+DyyC′\nzyB′\ny1Az1+B′\nyzAzz+D′\nyxCzx+D′\nyyCzy/parenrightbig\n(S.82)\nThe terms with a prime are expressed in terms of iω′\nn=iωn+iωm.\nRegarding the kdependence of the A,B,CandDoperators (Eqs. S.25-S.40), onlyF0, the last term in F1and the\nthree first terms in F2give non-vanishing contributions after summing over kin Eq.12.\nOn the other hand, the terms between parentheses in the first an d second line in F2expression give the same\ncontribution. As a result, the GD is found to be independent of the a zimuthal angle Φ m, which expresses isotropy of\nthe electronic band structure Eσ,±(Eq.S.18). However, the GD depends on the out-of-plane orientation of th e FI\nmagnetization via the angle θm.\nAccording to Eq. S.78, the terms to calculate are of the form\n/summationdisplay\nkF(k)/summationdisplay\nωm1\ni/planckover2pi1ωm−Ei1\ni/planckover2pi1ω′m−Ej, (S.83)\nwhereω′\nm=ωm+ωnandF(k) is a function of k= (k,ϕk).\nThe summation over ωmin Eq.S.83can be written as\n/summationdisplay\nωm1\ni/planckover2pi1ωm−Ei1\ni/planckover2pi1ω′m−Ej=1\ni/planckover2pi1ωn−(Ej−Ei)/summationdisplay\nωm/bracketleftbigg1\ni/planckover2pi1ωm−Ei−1\ni/planckover2pi1ω′m−Ej/bracketrightbigg\n=−1\ni/planckover2pi1ωn−(Ej−Ei)1\nkBT/integraldisplay\ncdz\n2πih(z)f(z) (S.84)\nwhereh(z) =1\nz−Ei−1\nz+i/planckover2pi1ωn−Ej,f(z) is the Fermi-Dirac function, Cis clockwise contour around the poles z=i/planckover2pi1ωm.\nEquation S.83reduces, then, to\n/summationdisplay\nωm1\ni/planckover2pi1ωm−Ei1\ni/planckover2pi1ω′m−Ej=f(Ej)−f(Ei)\ni/planckover2pi1ωn−(Ej−Ei)(S.85)\nTaking the analytic continuation i/planckover2pi1ωn=/planckover2pi1ω+iη, Eq.S.83becomes\nlim\nη→0/summationdisplay\nkF(k)ηf(Ej)−f(Ei)\n(/planckover2pi1ω−Ej+Ei)2+η2= lim\nη→0/summationdisplay\nkF(k)(f(Ej)−f(Ei))L(/planckover2pi1ω−(Ej−Ei)), (S.86)\nL(x) =η\nx2+η2being the Lorentzian function. The sum over k= (k,ϕk) in Eq.S.86reduces toA\n(2π)2/integraltextkc\n0kdk/integraltext2π\n0dϕk,\nwhereAis the moir´ e superlattice area, kcis a cutoff on the momentum amplitude k, below which the continuum\nmodel for the monolayer is justified. We take kc=q0/2, whereq0=4π\n3aθis the separation between the Dirac points\nK1,ξandK2,ξof, respectively, layer (1) and layer (2) at a given monolayer valley ξ.21\nD. Gilbert damping correction\nFor a uniform spin precession, the Gilbert damping correction δαG, at the FMR frequency Ω, can be expressed as78\nδαG=−2S0\n/planckover2pi1ΩImΣ(q=0,Ω) (S.87)\nThe imaginary part of the self-energy is of the form ImΣ( q=0,Ω) =T2\n0\n/planckover2pi1Ω˜Σ(q=0,Ω), where ˜Σ(q=0,Ω) is a dimen-\nsionless integral. Introducing the average SOC λ=1\n2(λI+λR),δαGcan be written as\nδαG=−α0\nG/parenleftbiggλ\n/planckover2pi1Ω/parenrightbigg2\n˜Σ(q=0,Ω) (S.88)\nwhereα0\nG= 2S0T2\n0\nλ2.\nIn Fig.S.5, we plot the normalized Gilbert damping correction δαG/α0\nGas a function of the twist angle θand\nthe FMR energy /planckover2pi1Ω at different temperatures. The bottom panels are a zoomed repr esentation around the MA.\nFig.S.5shows that, the GD increases by decreasing the twist angle and sha rply drops at the MA, regardless of the\ntemperature range and the FMR frequency .\nAt high temperature ( kBT >λ), the GD exhibits, around the MA, a fine structure characterized by a peak which\ndisappears at low temperature. The origin of this peak is, as discuss ed in the main text, due to the dispersion of the\nenergy bands of TBG/WSe 2and their corresponding thermal weights ∆ f(E) =f(E/angbracketleftSz/angbracketright)−f(E−/angbracketleftSz/angbracketright) (Eq.S.86).\nFigure S.5. Color plot of the normalized Gilbert damping cor rectionδαG/α0\nGas a function of the twist angle θand the FMR\nenergy/planckover2pi1Ω atkBT= 0.1 meV ((a) and (d)), kBT= 1 meV ((b) and (e)) and kBT= 25 meV ((c) and (f)). The bottom panels\nshow the behavior of the GD around the MA. Calculations are do ne forµ= 0,λI= 3 meV and λR= 4meV.\nIn Fig.S.6we plot ∆f(E) corresponding to the transitions between E−,+→E+,+andE−,−→E+,−in the case\nof the undoped system.\nFiguresS.6(a) and (b) show that, at high temperature ( kBT > λ I,λR), ∆f(E) increases as the band dispersion\ngets larger and reaches its minimal value at the MA. This behavior exp lains the drop of the GD at the MA and its\nenhancement at small twist angles.\nIn figure S.6(c), we plot ∆ f(E) around the MA, at relatively high thermal energy compared to the SOC, where the\nGD exhibits a peak at the MA (Fig. 2of the main text). In this case, ∆ f(E) is maximal at the MA and decreases at\nthe anglesθ+\nMandθ−\nMclose to the MA. This feature results from the decrease of the ene rgy separation between E−,−\nandE+,−atθ+\nMandθ−\nM, compared to that at θM(Fig.3of the main text). At low temperature, and around the\nMA, one gets ∆ f(E) = 1 for the transitions E−,−→E+,−andE−,+→E+,+. As a consequence, the GD behavior is\nnow only dependent on the effective Fermi velocity v∗which vanishes at the MA. As a consequence, the small peak22\nFigure S.6. Statistical weight ∆ f(E) corresponding to the transitions between E−,+→E+,+(a) andE−,−→E+,−((b) and\n(c)) at different temperatures. The dots represent the energ yE+,+(a) andE+,−((b) and (c)) at the MA and the arrows mark\nthe limit of the band E+,+(a) andE+,−((b) and (c)) at the indicated twist angle. In (c), ∆ f(E) is shown around the MA\nfor the transition between E−,−→E+,−. Calculations are done for the SOC λI= 3 meV, λR= 4 meV56and in the undoped\nTBG (µ= 0).\nof the GD, emerging at the MA at relatively high temperature, disapp ears.\nFigureS.7shows the behavior of the normalized GD correction δαG/α0\nGas function of the chemical potential µ\natkBT= 25meV and for the FMR energy /planckover2pi1Ω = 0.06meV. The decrease of δαGis a consequence of the thermal\nweight. The results shown in Fig. S.7are expected to hold in the presence of Coulomb interaction if the wid th of the\nbands at the MA remains less than 4 meV, which is the case of the filling f actorνsatisfying −0.5<ν <0.580.\nFigure S.7. Normalized GD correction δαG/α0\nGas function of the chemical potential µatkBT= 25meV and for different twist\nangles. The upper limit of µisµc=/planckover2pi1vFkccorresponding to the momentum cutoff kc=q0\n2. Calculations are done for the SOC\nλI= 3 meV, λR= 4 meV56,kBT= 25meV and for the FMR energy /planckover2pi1Ω = 0.06meV.\nIn Fig.S.8, we plot the normalized GD correction δαG/α0\nGas function of the SOC parameters, λIandλR, for\ndifferent twist angles, at kBT= 25meV, /planckover2pi1Ω = 0.06meV and in the case of the undoped system. The drop of δαG\nat the MA is a robust feature regardless of the amplitude of the SOC . However, there is a relative increase of δαG,\nat the MA, if the bands E−,+andE+,+(orE−,−andE+,−) are in resonance with the FMR energy, as shown in\nFig.S.8(c). This resonance can be only reached for relatively small values o fλR.\nAs shown in Fig. S.2, the energy spectrum of the effective model (dashed lines) are slig htly more dispersive, at\nsmall twist angles ( θ∼0.5◦), than those obtained by including higher bands (solid lines). This disc repancy should be\ntaken into account when fixing the value of the cutoff kcup to which the sum in Eq. S.86is evaluated. To determine\nthe role of the cutoff on the SP effect, we plot, in Fig. S.9, the GD correction δαGas a function of the twist angle at\ndifferent cutoffs kc≤q0\n2, whereq0=|K1,ξK2,ξ|is the momentum separation between the Dirac points K1,ξandK2,ξ\nof respectively layer (1) and layer (2) at a given valley ξ.\nFig.S.9shows that the GD correction drops at the MA regardless of the cu toff values. The larger the cutoff, the\nsharper the drop.23\nFigure S.8. Normalized GD correction δαG/α0\nGas function of the SOC λRandλIat a twist angle θ= 0.5◦(a),θ= 0.8◦(b)\nand at the MA θ= 1.05◦(c). Calculations are done for µ= 0,kBT= 25meV and for the FMR energy /planckover2pi1Ω = 0.06meV.\nFigure S.9. Normalized GD correction δαG/α0\nGas function of the twist angle for different values of the cuto ff parameter kc.\nCalculations are done for µ= 0,kBT= 25meV, λI= 3meV, λR= 4meV, and for the FMR energy /planckover2pi1Ω = 0.06meV."    },    {        "title": "2304.02248v2.Threshold_current_of_field_free_perpendicular_magnetization_switching_using_anomalous_spin_orbit_torque.pdf",        "content": "Threshold current of \feld-free perpendicular magnetization switching using\nanomalous spin-orbit torque\nTian Yi Zhang1, Cai Hua Wan1;2\u0003and Xiu Feng Han1;2;3\u0003\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, University of Chinese Academy of Sciences,\nChinese Academy of Sciences, Beijing 100190, China\n2Center of Materials Science and Optoelectronics Engineering,\nUniversity of Chinese Academy of Sciences, Beijing 100049, China\n3Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China\n(Dated: April 7, 2023)\nSpin-orbit torque (SOT) is a promising technique for next-generation magnetic random-access\nmemory (MRAM). Recent experiments have shown that materials with low-symmetry crystalline\nor magnetic structures can generate anomalous SOT with an out-of-plane component, which is\ncrucial for switching the perpendicular magnetization of adjacent ferromagnetic (FM) layers in\na \feld-free condition. In this study, we derive the threshold current for \feld-free perpendicular\nmagnetization switching using anomalous SOT and numerically calculate the magnetic moment\ntrajectory in an FM free layer for currents smaller and greater than the threshold current. We also\ninvestigate the dependence of switching time and energy consumption on applied current, \fnding\nthat the minimum energy consumption decreases with an increasing out-of-plane torque proportion.\nAdditionally, we explore the relationships between the threshold current and anisotropy strength,\nout-of-plane torque proportion, FM free layer thickness, and Gilbert damping constant. The results\nshow a negative correlation between the threshold current and out-of-plane torque proportion, and\npositive correlations with the other three parameters. Finally, we demonstrate that even when the\napplied current is smaller than the threshold current, it can still add an e\u000bective exchange bias \feld\nHbiason the FM free layer. The Hbiasis proportional to the applied current JSOT, facilitating\nthe determination of anomalous SOT e\u000eciency. Our \fndings provide insights into the design of\nspintronic devices that favor \feld-free switching of perpendicular magnetization using anomalous\nSOT and o\u000ber a means of adjusting the exchange bias \feld to control FM layer magnetization\ndepinning.\nI. INTRODUCTION\nThe spin-orbit torque (SOT) is a promising technique\nfor developing the next-generation magnetic random-\naccess memory (MRAM) [1{6]. Perpendicularly mag-\nnetized ferromagnetic (FM) layers have superior perfor-\nmance in thermostability, high density, and retention\ncompared to in-plane magnetized FM layers when used in\nMRAM [7]. However, ordinary SOT cannot easily switch\nthe perpendicular FM \flms in the \feld-free condition.\nTherefore, determining how to switch the perpendicular\nmagnetization of the FM free layer in the magnetic tun-\nnel junctions (MTJ) by SOTs has long been a frontier\nof SOT studies. According to the spin Hall e\u000bect (SHE)\n[4], when electron current jcis sourced along the x direc-\ntion, the spin current jstransports along the z direction\nand its polarization \u001b/js\u0002jcwill be along the y di-\nrection. The adjacent FM free layer is thus a\u000bected by\nthe spin current dominatingly through a damping-like\nSOT\u001cd/(m\u0002\u001b)\u0002m, which is also along the y di-\nrection. This torque is orthogonal to the perpendicular\neasy axis of the free layer; therefore, using pure SOT\nalone, we cannot deterministically switch the perpendic-\nular magnetization. Several attempts have been made\n\u0003xfhan@iphy.ac.cn; wancaihua@iphy.ac.to circumvent this problem, such as applying an in-plane\nmagnetic \feld [8, 9], using structural asymmetry [10, 11],\nmediating an in-plane exchange bias/coupling \feld [12{\n17], mediating the interlayer Dzyaloshinskii-Moriya in-\nteraction [18] or exploring materials with low-symmetric\ncrystalline or magnetic structures to generate an anoma-\nlous SOT [12, 19{24].\nEspecially, the groundbreaking studies on low-\nsymmetry materials have shown that the spin polariza-\ntion, denoted by \u001b, of an outofplanetransporting spin\ncurrent jscan have both in-plane and out-of-plane com-\nponents, despite being generated by an in-plane electron\ncurrent jc. These crystallized materials include Mn 3Ir\n[12], Mn 3Pt [20], Mn 3Sn [21], WTe 2[22], CuPt [25], and\nmore. This anomalous SOT is highly dependent on the\ncrystal or magnetic symmetry. By utilizing the out-of-\nplane component of \u001b, one can achieve a deterministic\nswitch of the magnetization of a FM free layer without\nan external magnetic \feld.\nIn order to gain a more thorough understanding of the\nanomalous SOT and its potential for switching a perpen-\ndicular magnetization, it is necessary to optimize and\nutilize relevant parameters that a\u000bect the switching dy-\nnamics. An analytical derivation of the threshold cur-\nrent in the coexistence of ordinary and anomalous SOTs\nwould be particularly bene\fcial for this purpose. Despite\nprevious work on formulating threshold currents for dif-\nferent SOT modes [26{30], there is still a lack of a speci\fcarXiv:2304.02248v2  [physics.app-ph]  6 Apr 20232\nthreshold current for the coexistence case of the anoma-\nlous and ordinary SOTs. This research aims to address\nthis gap.\nIn this paper, we analytically derive the threshold cur-\nrent required to generate an anomalous SOT for switch-\ning the perpendicular magnetization of an adjacent FM\nlayer. Additionally, we use macrospin simulations to in-\nvestigate the precessional trajectory of the FM layer's\nmagnetic moment both below and above the threshold,\n\fnding consistent results with our analytical model. We\nalso examine the dependence of switching time and en-\nergy consumption on the applied current, as well as the\nminimum energy consumption dependence on the pro-\nportion of out-of-plane torque. Furthermore, we calcu-\nlate the threshold current's dependence on anisotropy\nstrength, out-of-plane torque ratio, FM free layer thick-\nness, and Gilbert damping constant. Finally, we demon-\nstrate that an applied current below the threshold can\nstill produce an e\u000bective exchange bias \feld in the FM\nlayer, and provide the relationship between the e\u000bective\nexchange bias \feld and the applied current. This work\ncan be instructive to design SOT devices with the anoma-\nlous SOT materials.\nII. MODEL AND METHOD\nThe schematic diagram of a FM free layer magnetiza-\ntion switching driven by the anomalous SOT is shown\nin Fig. 1. The FM free layer with perpendicular mag-\nnetic anisotropy (PMA) is adjacent to a material with\nlow-symmetric crystalline structure. The applied elec-\ntron current JSOT is along the -y direction, spin current\nJSis along the z direction and the polarization \u001bof the\nspin current has components in both x and z directions\nas shown in the upper left pannel of Fig. 1. \fis the angle\nbetween\u001band the x axis. At the interface of the low-\nsymmetric material with the FM free layer, a pure spin\ncurrent with the \u001bpolarization di\u000buses into the FM free\nlayer and acts a damping-like SOT on the latter. Then\nthe magnetization of the FM layer will precess around an\ne\u000bective magnetic \feld or switch its magnetization under\nthe concerted interplay of the SOT with other torques\nfrom built-in \felds.\nThe spin dynamics of the FM layer can be described\nby the LLG formula [31]\n@m\n@t=\u0000\r\u00160(m\u0002HK) +\u000b\u0000\nm\u0002@m\n@t\u0001\n+\r\u00160HDL\nSOT((m\u0002\u001b)\u0002m)(1)\nwhere m=m\nMsis the unit vector along the direction of\nmagnetization, mis magnetic moment, Msis the sat-\nurated magnetization value, \r= 1:76\u00021011T\u00001s\u00001\nis the gyromagnetic ratio, \u00160= 4\u0019\u000210\u00007VsA\u00001m\u00001\nis the permeability of vacuum, HK=HKmzezis the\nanisotropic \feld, \u000bis Gilbert damping constant, \u001b=\n(\u001bx;0;\u001bz) = (cos\f; 0;sin\f ) is the unit vector along the\nelectron spin polarization direction, \fis the angle be-\ntween the spin polarization direction and the x axis,\nJSOT M\nJS'.\u0001GSFF\u0001MBZFS tθ\nφYZ[M\nβσFIG. 1. A schematic diagram of the FM free layer magnetiza-\ntion switch driven by an anomalous SOT with both in-plane\nand out-of-plane components. The applied electron current\nJSOT is along the -y direction and generates a spin current\nJSpropagating along the z direction. The spin current dif-\nfuses into the FM free layer with the perpendicular magnetic\nanisotropy to drive its magnetization dynamics.\nHDL\nSOT is the torque intensity generated by SOT, which\ncan be calculated by the following formula [32{36]\nHDL\nSOT=JSOT\u0012SH}\n2et\u00160Ms(2)\nwhereJSOTis the magnitude of the applied electron cur-\nrent density, \u0012SHis the spin Hall angle that represents\nconversion e\u000eciency from electron current to spin cur-\nrent, }= 1:05\u000210\u000034J\u0001sis the reduced Planck constant,\ne= 1:6\u000210\u000019Cis the elementary charge carried by an\nelectron, and tis the e\u000bective thickness of the free layer\nafter subtracting a dead layer.\nWhen a small current is applied, the anomalous SOT\nacting on the FM free layer is not large enough to switch\nthe magnetization of the FM layer. The magnetization\nwill precess under the SOT e\u000bect, and stabilize to a \fnal\ndirection due to the Gilbert damping. This direction is\nso-called the direction of the e\u000bective \feld Heff. Here\nwe constrain ourself in a \feld-free system which is exactly\nneeded in practice. When the applied current is above\na threshold Jc, the torque acting on the FM free layer\nbecomes large enough to make the magnetization pre-\ncession amplitude divergently increase and \fnally realize\nmagnetization reversal. In the following, we will give the\nanalytical derivation of the threshold current Jc. For\nthose readers who interest the dependence of Jcon var-\nious material parameters, they can directly skip to Eq.\n(15) where the \fnal results are directly gave out.\nBy crossing mleft at both sides of Eq. (1), we can\nreform the LLG formula Eq. (1) as in Eq. (3),\n@m\n@t=\u0000\r\u00160\n1+\u000b2[(m\u0002HK) +\u000bm\u0002(m\u0002HK)\n\u0000HDL\nSOT((m\u0002\u001b)\u0002m)\u0000\u000bHDL\nSOT(m\u0002\u001b)](3)3\nLet@m\n@t= 0 we can get\nm\u0002Heff= 0 (4)\nwhere the e\u000bective magnetic \feld Heffcan be written as\nHeff=HK\u0000HDL\nSOT(\u001b\u0002m)\n=HK(HDL\nSOT\nHKsin\fmy;\u0000HDL\nSOT\nHKsin\fmx+HDL\nSOT\nHKcos\fmz\n;\u0000HDL\nSOT\nHKcos\fmy+mz)\n(5)\nThe direction of Heffis also the \fnally stabilized direc-\ntion of the magnetization as JSOT< Jc. From Eqs. (4)\nand (5), we can then get\n8\n>><\n>>:HDL\nSOT\nHKsin\fmy=kmx\n\u0000HDL\nSOT\nHKsin\fmx+HDL\nSOT\nHKcos\fmz=kmy\n\u0000HDL\nSOT\nHKcos\fmy+mz=kmz(6)\nwhere the non-zero real number k satis\fes\nk3\u0000k2+\u0012HDL\nSOT\nHK\u00132\nk\u0000\u0012HDL\nSOT\nHK\u00132\nsin2\f= 0 (7)\nFrom Eq. (7), we get k=k\u0010\nHDL\nSOT\nHK;\f\u0011\n, Then the polar\nand azimuth angles ( \u0012H;'H) of the magnetization in the\nsteady state can be obtained, the schematic diagram of\npolar angle \u0012Hand azimuth angle 'Hin spherical coor-\ndinates is shown in the upper left pannel of Fig. 1.\n\u0012H=arctan0\nB@(1\u0000k)s\nsin2\f+\u0012\nkHK\nHDL\nSOT\u00132\nkcos\f1\nCA;\n'H=arctan\u0010\nkHK\nHDL\nSOTsin\f\u0011(8)\nAfter getting \u0012Hand'H, we can transform the coordi-\nnate system from the original system O to a new one O' inwhich Heffis directed at the z' axis, and the correspond-\ning transformation matrix between the two coordinates\nis\nR=0\n@cos\u0012H0\u0000sin\u0012H\n0 1 0\nsin\u0012H0cos\u0012H1\nA0\n@cos'Hsin'H0\n\u0000sin'Hcos'H0\n0 0 11\nA\n(9)\nAnd the relationship from the (x, y, z) coordinate to the\n(x', y', z') coordinate is simply\n0\n@x0\ny0\nz01\nA=R0\n@x\ny\nz1\nA (10)\nThe transformed coordination allows us to analyze dy-\nnamic stability of system straightforwardly. When\nJSOT<Jc, the components of the magnetization along\nthe x' and y' directions will converge to 0 after a long-\nenough damping, and as JSOT\u0015Jc, the precession am-\nplitude will go divergently and the magnetization will\nswitch to the opposite. At this time, the magnetization\nalong the x 'and y' components will gradually increase,\nwhich is our criterion to determine Jc. Speci\fcally, con-\nsidering the two magnetization components along the x'\nand y' directions, we rewrite the LLG formula Eq. (3) in\nthe following Eq. (11):\n\u00001 +\u000b2\n\r\u00160d\ndt\u0012mx0\nmy0\u0013\n=m\u0012mx0\nmy0\u0013\n+G (11)\nWhere mandGare 2\u00022 matrices, and their respective\ncomponents are explicitly shown below\n8\n>><\n>>:M11=HDL\nSOT\u0000\n\u0000sin\fcos2\u0012H\u0000cos\u0012Hcos'Hcos\fsin\u0012H\u0001\n+\u000b(cos4\u0012H\u0000cos2\u0012Hsin2\u0012H)HK\nM12=\u0000\u000bHDL\nSOT(sin\fcos\u0012H+cos'Hcos\fsin\u0012H) +cos3\u0012HHK\nM21=\u000bHDL\nSOT(sin\fcos\u0012H+cos'Hcos\fsin\u0012H) + (cos\u0012Hsin2\u0012H\u0000cos3\u0012H)HK\nM22= HDL\nSOT\u0000\n\u0000sin\fcos2\u0012H\u0000cos\u0012Hcos'Hcos\fsin\u0012H\u0001\n+\u000bcos4\u0012HHK(12)\n\u001aG1=\u0000\u000bHDL\nSOTcos\u0012Hcos\fsin' H\u0000HDL\nSOT\u0000\ncos3\u0012Hcos'Hcos\f\u0000sin\fcos2\u0012Hsin\u0012H\u0001\n+\u000bcos4\u0012Hsin\u0012HHk\nG2=\u0000\u000bHDL\nSOT\u0000\ncos2\u0012Hcos'Hcos\f\u0000sin\fcos\u0012Hsin\u0012H\u0001\n\u0000HDL\nSOTcos2\u0012Hcos\fSin' H\u0000cos3\u0012Hsin\u0012HHK(13)\nFrom Eq. (12), we can see that the eigenvalue of the 2 \u00022\nmatrix mis\u00151;2=M11+M22\u0006ip\n\u00004M12M21\u0000(M11\u0000M22)2\n2.\nWhenM11+M22<0, mx0and my0decay to 0 if any;\nin contrast, when M11+M22>0, they will diverge oncethe emergence of an even tiny jmx0jorjmy0jactivated\nby thermal \ructuations or other reasons. Therefore, the\nswitching criteria turns clear as\nM11+M22= 0 (14)4\nThe threshold current value Jccan be obtained from this\ncondition. Detailed derivation steps are shown in Ap-\npendix A, ifHDL\nSOT\nHK\u001c1 (widely applicable for most cases),\nwe can get that\nJc=e\u00160MsHKt\n}\u0012SH4\u000bp\nsin2\f+16\u000b2cos2\f+sin\f(15)\nWorth noting, according to recent experiment data [22],\ntwo typical values ofHDL\nSOT\nHKare 0:014and0:023 at Jc, so\nthe simpli\fcation conditionHDL\nSOT\nHK\u001c1 holds reasonable\nhere. When the out-of-plane torque is 0 or \f= 0, the re-\nsultJc=e\u00160MsHKt\n}\u0012SHbecomes simpli\fed in accordance with\nthe previously proposed J cfor the z-type SOT magnetiza-\ntion reversal at a small applied magnetic \feld [26, 30, 37].\nMore interesting, if\u000b\ntan\f\u001c1 (a small damping in the or-\nder of 10\u00002and a substantial anomalous SOT ratio not\nlower than\u00180.1 can qualify the condition), the above\nequation can be simpli\fed as\nJc=2e\u00160MsHK\u000bt\n}\u0012SHsin\f(16)\nThis threshold current density then shares a similar fash-\nion with the case of the spin-transfer torque switching\nmode for the perpendicular MTJ with polarization P of\nthe pinned layer replaced by the anomalous spin Hall an-\ngle\u0012SHsin\f.\nIII. RESULTS AND DISSCUSSIONS\nWe visualize the magnetization trajectory with di\u000ber-\nent out-of-plane torque ratio \u0011\u0011tan\f andJSOT, as\nshown in Fig. 2. Time step is set as dt= 1fs. The ini-\ntial direction of mis along the (0,0,1) in the O coordinate\nsystem, Simulation parameters are displayed in TABLE\nI [20, 38, 39]. For the situation without any out-of-plane\nSOT or\u0011= 0, when JSOT = 1:8\u00021013Am\u00002which is\nunable to destabilize the magnetization in the FM free\nlayer, mis \fnally stabilized at the direction of the equiv-\nalent e\u000bective \feld (0 :000;\u00000:451;0:893). AsJSOT =\n1:9\u00021013Am\u00002, the SOT is large enough to destabilize\nmto (\u00001;0;0) in the equatorial plane as shown in Fig.\n2(a) and (b). These scenarios produce the case of z-type\nmode without an external bias \feld. As for \u00116= 0, the\n\fnal state of JSOT> Jcbecomes di\u000berent. From Eq.\n(15), we can directly calculate that the threshold cur-\nrents for\u0011= 0:1 and 0.75 are Jc= 6:2\u00021012Am\u00002and\n1:1\u00021012Am\u00002respectively. As JSOT= 6\u00021012Am\u00002\nand 1\u00021012Am\u00002for\u0011= 0:1 and 0.75 respectively, the\nSOT acting on the FM free layer is not large enough,\nso the precession amplitude gets smaller and smaller,\nand \fnally mis stabilized at the direction of Heff\n(0:003;\u00000:135;0:991) and (\u00000:006;\u00000:0222;0:9997) for\n\u0011= 0:1 and 0.75 respectively, see Fig. 2(c) and (e).\nWhenJSOT = 7\u00021012Am\u00002and 2\u00021012Am\u00002for\u0011= 0:1 and 0.75 respectively, the precession ampli-\ntude is divergently increasing, and meventually turns to\nthe opposite direction ( \u00000:0035;\u00000:1575;\u00000:9875) and\n(\u00000:0003;0:0215;\u00000:9998) for\u0011= 0:1 and 0.75 respec-\ntively, as shown in Fig. 2(d) and (f), consistent with our\nprevious analysis. Worth mentioning, as JSOT> Jc,m\nwill not converge to the direction of Heffsince the pre-\nrequisite for the calculation is jm0\nxj;jm0\nyj\u001c1, which is\nviolated in this case. This notice does not undermine the\nstrictness of the criteria of deriving Jc.\nTABLE I. Parameters for numerical calculation (unless oth-\nerwise noted).\nParameters Quantity Value\nDamping constant \u000b 0.015 [39]\nAnisotropic \feld \u00160HK 0.85 T [38]\nSaturated Magnetization Ms 1:3\u0002106A/m [38]\nRatio of anomalous SOT tan\f 0.1 or 0.75 [20]\nthe FM thickness t 1 nm\nOverall spin Hall angle \u0012SH 0.075 [20]\nWe also calculate the relationship between the switch-\ning timets, switching energy consumption QSOT and\nJSOTwhenJSOT>Jc. Here\u0011= 0:75,tsis de\fned as the\ntime from sourcing current to the occurrence of a negative\nmzcomponent in the calculation, switching energy con-\nsumption is de\fned as QSOT\u0011J2\nSOTts, which scales with\nthe energy consumed in the switching process. We can\nsee from Fig. 3(a) that as JSOT increases, t sdecreases\nrapidly from 16.5 ns when JSOT = 1:2\u00021012Am\u00002to\n0.1 ns when JSOT = 1:11\u00021013Am\u00002, and the in\ru-\nence ofJSOT on tsis gradually reduced as JSOT in-\ncreases. From Fig. 3(a), we can see that QSOT mini-\nmizes when JSOTis near 2:4\u00021012Am\u00002, with minimum\nvalueQmin\nSOT = 7:7\u00021015A2sm\u00004. Then we study the\n\u0011-dependence of Qmin\nSOT, as shown in Fig. 3(b). As \u0011in-\ncreases,Qmin\nSOTgradually decreases, manifesting the larger\nout-of-plane torque ratio results in the less energy con-\nsumption. With the resistivity of conductive layer \u001a=\n200\u0016\ncm[20] and the SOT-channel width l= 100nm\nand length d= 300nm,QSOT = 1\u00021015A2sm\u00004cor-\nresponds to an energy consumption of 0.06 pJ.\nThen, we numerically calculate the dependence of Jc\non the anisotropy strength HK, out-of-plane torque ra-\ntio\u0011, FM free layer thickness tand Gilbert damping con-\nstant\u000b, as shown in Fig. 4. We can see that Jcincreases\nwith the increase in H Kas expected. In addition, we\ncan also see from Fig. 4(a) that Jcgradually decreases\nwith the increase in \u0011. This is because the decisive factor\nthat a\u000bects magnetization switching is the z component\nof anomalous SOT. When z component of e\u000bective \feld\ncaused by SOT is larger than e\u000bective anisotropy \feld,\nmagnetization switch happens. And as \u0011increases, the\nz component of SOT increases, then Jcbecomes lower\nif we still intend to switch the magnetization. We ex-\ntract three threshold currents corresponding to di\u000berent\nanisotropic properties, as shown in Fig. 4(b). Clearly5\n\tB\n \tC\n\tD\n \tE\n\tF\n \tG\n η = 0\nη = 0 \u000f\u0012 \nη = 0\u000f\u0018\u0016 η = 0\u000f\u0018\u0016 η = 0\u000f\u0012 η = 0\nFIG. 2. The magnetization trajectory with di\u000berent \u0011and\nJSOT. The two parameters are shown as follows, \u0011= 0, (a)\nJSOT = 1:8\u00021013Am\u00002and (b)JSOT = 1:9\u00021013Am\u00002;\n\u0011= 0:1, with threshold current value Jc= 1:1\u00021012Am\u00002,\n(c)JSOT = 1\u00021012Am\u00002and (d)JSOT = 2\u00021012Am\u00002;\n\u0011= 0:75, with threshold current value Jc= 6:2\u00021012Am\u00002,\n(e)JSOT= 6\u00021012Am\u00002and (f)JSOT= 7\u00021012Am\u00002.\nshown in the \fgure, the higher anisotropy results in the\ngreater in\ruence of \u0011onJc.\nWe also study the thickness tand Gilbert damping\n\u000bdependences of Jc, as shown in Fig. 4(c). As \u000bin-\ncreases,Jcincreases. This feature, similar to the classic\nspin-transfer torque (STT) switching scheme [40] , can\nbe explained as following. During the switching dynam-\nics with a larger \u000b,mwill be easier to converge to its\ninitialized direction due to an enhanced damping which\nneeds a larger Jcto battle against. And as the thickness\nof the magnetic layer is higher, Jcbecome larger too with\nno doubt in accordance with Eq. (15). We then extract\nthreeJccorresponding to di\u000berent thicknesses, as shown\nin Fig. 4(d). Jcscales linearly with \u000b, also in accordance\nwith Eq. (15) and the STT scheme. Worth noting, on\nthe other hand, when \u0011= 0, the threshold current Eq.\n(15) is equal to the threshold current for the z-type SOT\nmode with an applied magnetic \feld Hx0. The thresh-\nold current for the z-type SOT magnetization switching\n(a) (b)FIG. 3. The JSOT-dependence of the switching time t sand\nthe switching energy consumption QSOT. (b) The \u0011-\ndependence of minimum energy loss Qmin\nSOT. In the regime\nof\u0011\u00140.8, the increase in \u0011can signi\fcantly reduce the value\nofQmin\nSOT.\nis[30]\nJc=et\u00160MsHK\n}\u0012SH\n[s\n4\u000b(4\u000b+2\u000b(HFL\nSOT\nHDL\nSOT)2+HFL\nSOT\nHDL\nSOT)+(9\u000b2\u00004\u000b(HFL\nSOT\nHDL\nSOT)\u00008\u000b2(HFL\nSOT\nHDL\nSOT))(Hx=HK)2\n4\u000b+2\u000b(HFL\nSOT\nHDL\nSOT)2+(HFL\nSOT\nHDL\nSOT)\n\u00005\u000b(Hx=HK)\n4\u000b+2\u000b(HFL\nSOT\nHDL\nSOT)2+(HFL\nSOT\nHDL\nSOT)]\n(17)\nwith \feld-like torque intensity HFL\nSOT and applied mag-\nnetic \feldHx. When the HFL\nSOT= 0 andHx\u001cHK, the\nthreshold current is [26]\nJc=et\u00160Ms\n}\u0012SH(HK\u0000p\n2Hx) (18)\n(a) (b)\n(c) (d)\nFIG. 4. The dependence of Jcon the anisotropic \feld\nHKand anomalous ratio \u0011. (b) The\u0011dependence of Jcun-\nder\u000b= 0:015; t= 1nmextracted from Fig. 4(a). (c) The\ndependence of Jcon the thickness tand Gilbert damping \u000b.\n(d) The\u0011dependence of Jcunder\u00160HK= 0:85T; \u0011 = 0:75\nextracted from Fig. 4(c).\nThe experiment [25] and the above derivation have\nshown that the anomalous SOT can switch the magneti-6\nzation of FM free layer if the applied current is above the\nthresholdJc. However, even if JSOT< Jc, the anoma-\nlous SOT can still manifest itself by acting an e\u000bective\nexchange bias \feld Hbias, which facilitates us to deter-\nmine the anomalous SOT e\u000eciency. We calculate the\nhysteresis loops corresponding to di\u000berent JSOT<Jc, as\nshown in Fig. 5(a). Here the LLG equation has to take\nmagnetic \feld Hzinto account.\n@m\n@t=\u0000\r\u00160(m\u0002(HK+Hz)) +\u000b\u0000\nm\u0002@m\n@t\u0001\n+\r\u00160HDL\nSOT((m\u0002\u001b)\u0002m)(19)\nFig. 5(a) shows for an unbiased loop without JSOT, the\nforward and backward switching coercivity of the FM\nlayer is symmetric. However, when a positive (negative)\nJSOT; 0= 6:28\u00021011Am\u00002is applied, the hysteresis loop\nis biased leftward (rightward) or their H z-symmetrical\naxis is o\u000bset in the negative (positive) direction. There-\nfore, equivalently, a nonzero JSOT imposes an exchange\nbias \feldHbiasalong the z axis to the FM free layer,\nwhich determines the switching direction in the \feld-free\ncondition. Compared to z-type SOT [37] that switch\nmagnetization with the help of in-plane magnetic \feld,\nthis result give a novel method to control magnetization\nof FM layer, which is easier to integrate. We also cal-\nculate the dependence of Hbiason JSOT shown in Fig.\n5(b) which appears linear with each other and the slope\nis\u0012SH}sin\f\n2et\u00160Ms\u000b= 6:05\u000210\u00007m;ifHDL\nSOT\nHK\u001c1 andHbias\nHK\u001c1.\nThis result provides a direct way to determine the anoma-\nlous torque e\u000eciency. Compared with previous work that\ndetect anomalous torque with in-plane magnetization of\nCoFeB [13], our results shows a simpler linear relation-\nship between HbiasandJSOT.\n(a) (b)\nFIG. 5. (a) Hysteresis loop of FM free layer under di\u000berent\napplied currents JSOT, (b) Relationship between equivalent\nexchange bias \feld Hbiasand applied current JSOT. Here\n\u0011= 0:75.\nIV. CONCLUSIONS\nIn this paper, we present an analytical derivation of the\nthreshold current required to achieve \feld-free switch-\ning of perpendicular magnetization using the anomalous\nSOT in combination with an ordinary one. We also con-\nduct numerical simulations to investigate the magneti-\nzation trajectory of a FM free layer when the appliedcurrent is both below and above the threshold. Our ana-\nlytical and numerical results are in agreement. Further-\nmore, we explore the dependence of the switching time\nand energy consumption on the applied current and show\nthat the minimum energy consumption is negatively cor-\nrelated with the out-of-plane torque proportion. Addi-\ntionally, we investigate the e\u000bects of various parameters,\nincluding anisotropy strength, out-of-plane torque ratio,\nFM free layer thickness, and Gilbert damping constant,\non the threshold current. Our \fndings indicate a nega-\ntive correlation between the out-of-plane torque propor-\ntion and the threshold current, and a positive correlation\nbetween the other three parameters and the threshold\ncurrent. Finally, we demonstrate that when the applied\ncurrent is below the threshold, it can induce an exchange\nbias \feldHbiasin the FM free layer. Our numerical re-\nsults show that the exchange bias \feld Hbiasis propor-\ntional to the applied current JSOT. This study provides\ninsights into the design of spintronic devices that enable\n\feld-free switching of perpendicular magnetization us-\ning the anomalous spin-orbit torque and o\u000bers a direct\nmethod for adjusting the exchange bias \feld, which can\nbe useful in controlling FM layer magnetization pinning\nand depinning.\nACKNOWLEDGMENTS\nThis work is \fnancial supported by the National Key\nResearch and Development Program of China (MOST)\n(Grant No. 2017YFA0206200, 2021YFB3601300), the\nNational Natural Science Foundation of China (NSFC)\n(Grant No. 12134017 ,11974398 , 12061131012), and par-\ntially supported by the Strategic Priority Research Pro-\ngram (B) (Grant No. XDB33000000, Youth Innovation\nPromotion Association of CAS (2020008)).\nAppendix A: Derivation of Eq. (15)\nFrom Eq. (7), suppose k1=k\u00001\n3, we can get that\nk3\n1+ \u0012HDL\nSOT\nHK\u00132\n\u00001\n3!\nk1+ \n\u00002\n27+\u00121\n3\u0000sin2\f\u0013\u0012HDL\nSOT\nHK\u00132!\n= 0\n(S1)\nIfHDL\nSOT\nHK\u001c1, we can get that\nk1=3r\n1\n27\u00001\u00003sin2\f\n6\u0010\nHDL\nSOT\nHK\u00112\n+sin\f\n3p\n3HDL\nSOT\nHK\n+3r\n1\n27\u00001\u00003sin2\f\n6\u0010\nHDL\nSOT\nHK\u00112\n\u0000sin\f\n3p\n3HDL\nSOT\nHK\n=2\n3\u0000cos2\f\u0010\nHDL\nSOT\nHK\u00112(S2)\nSo that\nk= 1\u0000cos2\f\u0012HDL\nSOT\nHK\u00132\n(S3)7\nWe can get the pole angle and azimuth angle ( \u0012H;'H)\n8\n>>>>><\n>>>>>:tan\u0012H=(1\u0000k)p\nsin2\f+(kHK=HDL\nSOT)2\nkcos\f\n=cos\fHDL\nSOT\nHK\ncos'H=sin\fHDL\nSOT=HK p\n(1\u0000(cos\fH K=HDL\nSOT)2+(sin\fH K=HDL\nSOT)2\n=sin\fHDL\nSOT\nHK\n(S4)From Eq. (12), we can get threshold current using equa-\ntionM11+M22= 0:\nJc=et\u00160MsHK\n\u0012SH}4\u000bp\nsin2\f+16\u000b2cos2\f+sin\f(S5)\n[1] W. J. Kong, C. H. Wan, C. Y. Guo, C. Fang, B. S. Tao,\nX. Wang, and X. F. 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"1Thickness-dependentmagneticpropertiesinPt/[Co/Ni]n\nmultilayerswithperpendicularmagneticanisotropy*\nChunjieYan(晏春杰)1,LinaChen(陈丽娜)1,2,#,KaiyuanZhou(周恺元)1,Liupeng\nYang(杨留鹏)1,QingweiFu(付清为)1,WenqiangWang(王文强)1,Wen-Cheng\nYue(岳文诚)3,LikeLiang(梁立克)1,ZuiTao(陶醉)1,JunDu(杜军)1,Yong-Lei\nWang(王永磊)3andRonghuaLiu(刘荣华)1*\n1NationalLaboratoryofSolidStateMicrostructures,SchoolofPhysicsand\nCollaborativeInnovationCenterofAdvancedMicrostructures,NanjingUniversity,\nNanjing210093,China\n2SchoolofScience,NanjingUniversityofPostsandTelecommunications,Nanjing\n210023,China.\n3SchoolofElectronicsScienceandEngineering,NanjingUniversity,Nanjing210093,\nChina.\nWesystematicallyinvestigatedtheNiandCothickness-dependentperpendicular\nmagneticanisotropy(PMA)coefficient,magneticdomainstructures,and\nmagnetizationdynamicsofPt(5nm)/[Co(tConm)/Ni(tNinm)]5/Pt(1nm)multilayersby\ncombiningthefourstandardmagneticcharacterizationtechniques.The\nmagnetic-relatedhysteresisloopsobtainedfromthefield-dependentmagnetizationM\nandanomalousHallresistivity(AHR)xyfoundthatthetwoserialmultilayerswith\ntCo=0.2and0.3nmhavetheoptimumPMAcoefficientKUwellasthehighest\ncoercivityHCattheNithicknesstNi=0.6nm.Additionally,themagneticdomain\nstructuresobtainedbyMagneto-opticKerreffect(MOKE)microscopyalso\nsignificantlydependonthethicknessandKUofthefilms.Furthermore,the\nthickness-dependentlinewidthofferromagneticresonanceisinverselyproportionalto\nKUandHC,indicatingthatinhomogeneousmagneticpropertiesdominatethe\nlinewidth.However,theintrinsicGilbertdampingconstantdeterminedbyalinear\nfittingoffrequency-dependentlinewidthdoesnotdependonNithicknessandKU.OurresultscouldhelppromotethePMA[Co/Ni]multilayerapplicationsinvarious\nspintronicandspin-orbitronicdevices.\nKeywords:perpendicularmagneticanisotropy,magneticdomain,damping,\nmultiayers\nPACS：75.30.Gw,75.70.Kw,75.40.Gb,68.65.Ac1.INTRODUCTION\nMagneticmultilayerswithstrongperpendicularmagneticanisotropy(PMA)and\nlowmagneticdampinghaveattractedmuchattentionbecauseoftheirpotential\napplicationsinhigh-densitymagneticrandomaccessmemories(MRAM)[1-5]andspin\ntorquenano-oscillators[6-9].Comparedtothein-planemagnetizedferromagnets,\nferromagneticfilmswithPMAfacilitatetherealizationofnonvolatileMRAMwith\nlowerpowerandhigherdensitystoragebecausethelatterhaslowercriticalswitching\ncurrentandhigherthermalstabilitythantheformerasthecontinuousdownscalingof\nthecellsizeofdevices[10].Inaddition,PMAcanbeaneffectivemagneticfieldto\nachievezeroexternalmagneticfieldworkingspin-torquenano-oscillatorswith\nferromagnetswithstrongPMAandlowdampingasitsfreelayer.[11]Therefore,the\ncontrollabletailoringPMAofmagneticfilmsisanessentialprerequisitefor\ndevelopinghigh-performancespintronicdevices.Themagneticmultilayers,e.g.,\n[Co/Pd],[Co/Pt],and[Co/Ni],provideanopportunitytotunetheirmagnetic\npropertiesbychangingthethicknessratiocontrollablyandthenumberofbilayer\nrepeatsthankstotheinterface-inducedPMAduetointerfacialspin-orbitcouplingand\ninterfacialstrainrelevantmagnetoelasticeffects[12-17].AmongthesePMAmultilayers,\nthePMA[Co/Ni]multilayeralsoexhibitslowdampingconstant[14],whichgetsmuch\nattention,especiallyforthefieldsofcurrent-drivenauto-oscillationofmagnetization\nandexcitationandmanipulationofspin-waves[18,19].Furthermore,thePMA[Co/Ni]\nmultilayeralsousefulforspin-orbittorquedevices[20-22].Therefore,[Co/Ni]\nmultilayersareconsideredoneofthemostpromisingPMAferromagnetsinvarious\nspintronicdevices.Althoughthereareafewstudiesonthemagneticanisotropy,\nmagnetotransport,andmagneticdampingofPt/[Co/Ni]multilayers[6,14,23],the\nsystematicallystudiedevolutionofmagnetostaticproperties,includingthetopography\nofmagneticdomainsandmagneticdynamicswiththethicknessratioofCoandNi\nlayersforthismultilayerfilmstillneedstomakeathoroughinvestigationforfacilitatingitbetterusedinfurtherspintronics.\nHere,wesystematicallyinvestigatehowtocontrolthemagneticfilmPMAby\ntailoringtheinterfacialeffectbyvaryingthethicknessoftheNilayeranditsimpact\nonmagneticdomainstructureanddynamicaldampingintwoserialCo/Nimultilayers\nwithtCo=0.2and0.3nm.ThehighestPMAcoefficientKU~3×106ergcm-3and\ncoercivityHC~250OearefoundattheoptimumNithicknesstNi=0.6nmforthe\nstudiedtwoserials.Thenucleationofthemagneticdomainoccursatonlyafew\nnucleationsitesandgraduallyexpandswithmagneticfieldsforthemultilayerswith\ntheoptimumNithicknesses0.4nm~0.6nm.Finally,theintrinsicGilbertdamping\nconstantαisnotsensitivetothickness-dependentKUanddomainstructureseven\nthoughthelinewidthofferromagneticresonanceisinverselyproportionaltoKUand\nHC,whichisdominatedbyinhomogeneousmagneticproperties.\n2.EXPERIMENT\nTwoserialmultilayersofPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1)and\nPt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1),namedasPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)]\nrespectively,weredepositedonSi/SiO2substratesatroomtemperatureby\ndc-magnetronsputteringwithArpressure3×10-3torr.Theunitinparenthesesisthe\nthicknessinnm.Thebasepressureofthesputteringdepositionchamberisbelow2×\n10−8torr.Thedepositionratewasmonitoredbythequartzcrystalmonitorinsituand\ncalibratedbyspectroscopicellipsometry(SE).Thestaticmagneticpropertieswere\ncharacterizedbythevibratingsamplemagnetometer(VSM),theanomalousHall\nresistivity(AHR)measurement，andtheMagneto-opticKerreffect(MOKE)\nmicroscopy,respectively.Thefilms'ferromagneticresonance(FMR)spectra,\nobtainedbycombiningcoplanarwaveguide(CPW)andlock-intechniques,werealso\nadoptedtocharacterizetheirdynamicmagneticproperties.Allthesemagnetic\ncharacterizationswereperformedatroomtemperature.3.RESULTSANDDISCUSSION\n3.1Quasi-staticmagneticproperties\nTodirectlyobtainthethicknessdependenceofPMApropertiesintheCo/Nifilms,\nwefirstperformedthemagnetichysteresisloopsofsampleswithdifferentthicknesses\nusingVSM.Figure1showsthemagnetizationhysteresisloopswiththeout-of-plane\nandin-planefieldgeometriesforthetwoserialmultilayersofPt/[Co(0.2)/Ni(tNi)]and\nPt/[Co(0.3)/Ni(tNi)]samples.Thewell-definedsquareM-Hloopsunderout-of-plane\nfield[Figs.1(a)and(c)]indicatethattwostudiedserialPt/[Co/Ni]multilayersexhibit\naperpendicularmagneticanisotropy.Additionally,thesaturationmagnetizationMSof\nthemultilayersdecreaseswithincreasingthethicknessoftheNilayertNi,from673\nemucm-3to495emucm-3forPt/[Co(0.2)/Ni(tNi)]and723emucm-3to639emucm-3\nforPt/[Co(0.3)/Ni(tNi)],whichagreeswiththemuchlowerMS~484emucm-3ofthe\nmetalnickelcomparedtothatofthecobaltlayerMS~1422emucm-3.Basedonthe\nout-of-planeandin-planemagnetizationhysteresisloops,theperpendicular\nanisotropyfieldHKwasdeterminedbyusingthedefinedformulaforthePMA[12,24]:\nK=2\ns0S⊥−0S∥ \u0000 \u0000 +4πS.ThecalculatedHK,MSandthe\ncoercivityHC,obtainedfromtheM-Hloops,weresummarizedbelowinFig.5.\nFig.1.(a)-(b)MagnetizationloopsofthefilmsPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1)with\nout-of-plane(a)andin-plane(b)magneticfield.(c)-(d)Sameas(a)-(b),for\nPt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).\nThesestaticmagneticpropertiesofthemetalPt/[Co/Ni]multilayerfilmsalsocan\nbedeterminedbytheelectrictransportsinmagneticfield,e.g.,anomalousHall\nresistivity(AHR)andmagnetoresistiveeffect.Comparedtothestandard\nmagnetometerabove,theelectrictransportsinmagneticfieldmeasurementsprovide\nanalternativeapproachand,especially,moreusefulforspintronicnano-devices\nbecausetheycaneasilyaccessthemagneticpropertiesofthemicroscaleand\nnanoscalesamples[25,26].Therefore,wealsoperformtheout-of-planeandin-plane\nAHRloopsasafunctionoftheappliedmagneticfieldsforthestudiedtwoserial\nmultilayers,asshowninFig.2.ThecoercivityHCdeterminedfromtheout-of-plane\nAHRloopsarewellconsistentwiththevaluesobtainedbytheM-Hloops,andare\nalsosummarizedinthefollowingFig.5.Meanwhile,wecancalculateHKofthe\nstudiedfilmsfromthein-planeAHRloopsbyusingthefollowingrelation[27]:K=\n∥∙tanarcsinxy\nxy(0)+4πS,wherexy0istheAHRvalueatzeroin-planefield.\nTheevolutionofHKwiththethicknessoftheNilayerisoverallconsistentwiththe\nresultsdeterminedbytheVSMmeasurement.Furthermore,theAHRmeasurements\nalsoprovideustheadditionalinformation,whichcannotbeeasilyaccessedbyVSM,\naboutthestudiedtwoserialPt/[Co/Ni]multilayers.Forexample,wefindthatthe\nin-planeAHRnear-zeromagneticfieldismuchsmallerthantheout-of-planeAHR\nforthesampleswithcertainNithickness,indicatingthatthesesamplesformthe\nmulti-domainstructuresatthelowin-planemagneticfields.Therefore,thevalueof\nthedifferencebetweenout-of-planeandin-planeAHRatnear-zerofieldshintsthat\nthedifferentNithicknessfilmsmayexhibitdistinctmagneticdomainstructures[28].Fig.2.(a)-(b)AnomalousHallresistivityasafunctionofout-of-plane(a)and\nin-planemagneticfieldwith5otiltanglefromthefileplane(b)forthesamples\nPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).Theinsetshowsthegeometricrelationshipbetween\nmagneticfield,magnetizationandeffectiveanisotropicfieldKeff=K–4πS\n(c)-(d)Sameas(a)-(b),forPt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).AllAHRweremeasuredby\nusingthefilmspatternedintoa0.3×10mmHallcross.\n3.2Magneticdomainstructures\nTodirectlyrevealtheevolutionofmagneticdomainstructurewithNithicknessand\nthedetailofmagnetizationreversalprocessunderexternalfield,theMOKE\nmicroscopyisalsoperformedforPt/[Co(0.2)/Ni(tNi)]serialsamples.Figure3shows\nMOKEloopsandtherepresentativeMOKEimagesduringscanningout-of-plane\nmagneticfields.Brightanddarkregionsrepresentthedomainswithmagneticmoment\npoint-upandpoint-down,respectively.Exceptforthemultilayerwiththenickel\nthicknesstNi=0.2nm[Fig.3(a)],allfilmsexhibitawell-definedPMAcorresponding\nrectangularordumbbell-shapehysteresisloops[Figs.3(b)-(h)].TheMOKEimagesshowthatthesamplewithtNi=0.3nmbeginstonucleatewithnumerousnucleation\npointsapproximatelyuniformdispersiononthewholefilmatthefieldof138Oe,\nthengraduallyexpandwithincreasingfield,andsaturatetotheuniformstateat⊥˃\n300Oe[Fig.3(b)].Combiningwithfield-dependentmagneticsusceptibilityandAHR\ncharacterizations,HCandHKofPt/[Co(0.2)/Ni(tNi)]firstenhancewithincreasingtNi,\nreachthemaximumattNi=0.6nm,andthenreducewithcontinueincreasingtNito0.9\nnm.Forthesampleswith0.4nm≤Ni≤0.6nm,theMOKEloopsexhibita\nwell-definedrectangularshape.Meanwhile,incontrasttotNi=0.3nm,onlyafew\nnucleationpointsappearatthecriticalmagneticfield,andthenbegintoexpandthe\nmagneticdomainwithacontinuouslyincreasingfield.ForincreasingtNitoabove0.7\nnm,theMOKEloopsbegintotransferintothedumbbellshapefromtheprior\nwell-definedrectangularshape.ThecorrespondingMOKEimagesshowthatthefilms\nwithtNi≥0.7nmformtree-likedomainwallswithmoreirregularbranchingsas\nincreasingtNiduringthefieldrangenearbelowthesaturationfield[themiddlepicture\nofFig.3].Inotherwords,forthefilmswithtNi≥0.7nm,thelengthofthedomainwall\nincreaseswithincreasingtNi,whichisconsistentwiththetrendofdependenceofHK\nonNithickness.Asweknowthatthetotalenergydensityofthemagneticdomain\nwallperunitareawisproportionaltomagneticanisotropyenergy,exchangeenergy\nanddemagnetizationenergybasedonthewidelyrecognizedformula[29]:w=2\n+\nU\n2+S2\n4µ02\n+,whereAistheexchangeconstant,δisthethicknessofthedomain\nwall,KUisthePMAcoefficient,MSisthesaturationmagnetization,tisthethickness\noftheentirefilm,andμ0isthepermeabilityofvacuum.Tominimizethetotalenergy\noftheentirefilm,thevolume(orlength)ofthedomainwallneedstoreduceand\nincreasedemagnetizationenergywhenKUincreaseswithincreasingtNiintherangeof\n0.4nm~0.6nm.Fig.3.(a)-(h)Magneto-opticKerrhysteresisloopsandmagneticdomainimagesof\nthefilmsPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1)withlabeledthicknesstNi=0.2nm(a),0.3nm\n(b),0.4nm(c),0.5nm(d),0.6nm(e),0.7nm(f),0.8nm(g),0.9nm(h),respectively.\nThecorrespondingmagneticdomainimageswiththesizeof100µm×150µmwereobtainedatthelabeledout-of-planemagneticfields(alsomarkedasthereddots\nonloops).\n3.3Magnetizationdynamics\nTofurtherinvestigatetheNithickness-dependentmagnetizationdynamicsofCo/Ni\nmultilayers,weperformthebroadbandFMRmeasurementwiththeexternalfield\nperpendiculartothefilmplane.AllFMRmeasurementswerecarriedoutwitha\nhome-madedifferentialFMRmeasurementsystemcombiningLock-intechniqueat\nroomtemperature.Acontinuous-waveOerstedfieldwithaselectedradiofrequencyis\ngeneratedviaconnectingcoplanewaveguide(CPW)withanRFgenerator,which\nproducesamicrowavesignaltoexciteFMRofferromagneticfilm,whichwithfilm\nsurfacewasadheredontheCPW.TheRFpowerusedintheexperimentsis15dBm.\nToimprovethesignal-to-noiseratio(SNR),alock-indetectiontechniqueisemployed\nthroughthemodulationofsignals.Themodulationofadirectcurrent(DC)magnetic\nfieldHisprovidedbyapairofsecondaryHelmholtzcoilspoweredbyanalternating\ncurrent(AC)sourcewith129.9Hz[seeFig.4(a)][16,30].Thedifferentialabsorption\nsignalismeasuredbysweepingthemagneticfieldwithafixedmicrowavefrequency.\nTherepresentativeFMRspectrumofPt(5)/[Co(0.2)/Ni(0.3)]5/Pt(1)obtainedat9GHz\nisshownintheinsetofFig.4(b).ThedifferentialFMRspectrumcanbewellfittedby\nusingacombinationofsymmetricandantisymmetricLorentzianfunction,asfollows:\n\n=s4∆−res\n4−res2+∆22+∆2−4−res2\n4−res2+∆22, (1)\nwhereVSandVArepresentthesymmetricandantisymmetricfactors,Histhe\nexternalmagneticfield,Hresistheresonancefield,andΔHisthelinewidthofFMR\ncorresponding3timesofthepeak-to-dipwidthintheFMRspectrum.The\nrelationshipbetweenthefrequencyfandtheresonancefieldHresofthetwoseriesof\nPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)]samples[Figs.4(b)and(d)]canbewell\nfittedbytheKittelequation[31]=γ\n2πres+eff, (2)\nwhereγ\n2π=2.8MHzOe−1isthegyromagneticratio,Heffiseffective\ndemagnetization[32]eff=K−4S.Therefore,themagneticanisotropyfieldHK\nalsocanbedirectlydeterminedfromthedispersionrelationoffversusHresbyusinga\nparameterMSobtainedbyVSM.Inaddition,wecanobtaintheintrinsicGilbert\ndampingαbyfittingtheexperimentaldataoflinewidthΔHversusresonance\nfrequency[Figs.4(c)and(e)]withtheformula:∆=∆0+4πα\nγ,hereΔH0isan\ninhomogeneouslinewidthindependentofthefrequency,andthesecondtermisthe\nintrinsiclinewidthlinearlyproportionaltothefrequency.Theinhomogeneous\nlinewidthofsamplesisderivedfromroughness,defectsandinhomogeneousPMA\nandmagnetization[33].\nFig.4.(a)DifferentialFMRspectraexperimentalsetup.(b)Dependenceofthe\nresonancefieldHresonthefrequencyfwiththeout-of-planefieldforthefilms\nPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).SolidlinesindicatetheKittelfittingcurves.Theinsetis\ntherepresentativeFMRspectrumobtainedat9GHz,whichcanbewellfittedbyEq.(1)(solidredline).(c)Thelinewidthversusfrequency(symbols)forthesamples\nPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).Thesolidlineisalinearfitting,whichcanextractthe\ncorrespondingdampingconstantαbasedonEq.(2).(d)-(e)Sameas(b)-(c),forthe\nfilmsPt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).\nFigure5summarizesthedependenceofthedeterminedmaterialparameters:the\nsaturationmagnetizationMS,thecoercivityHC,theanisotropyfieldHK,the\ninhomogeneouslinewidthΔH0andthemagneticdampingconstantαonNithickness\ntNiforthestudiedtwoseriesofPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)]samples.\nThedeterminedHKbythreeindependentmethodsshowsanoverallconsistent\nbehavior.TheHKbeginstoincreasewithincreasingtNi,andreachesthemaximumat\ntNi~0.6nm,whereafterreducesagainwithcontinuingtoincreasetNi.Severalreasons\naccountforthisphenomenon.First,themagneticanisotropyofthestudiedmultilayer\nismainlycontributedfromtheinterfacialmagneticanisotropyoftheCo/NiandPt/Co\ninterfaces[34].Second,theCo/Nimultilayers'interfacequalitydependshighlyonthe\nNilayer'sthickness.Inotherwords,toothinnickellayermaynotgetagoodCo/Ni\ninterfaceduetoinevitableelementsdiffusionduringsputteringdeposition.However,\ntheHKwilldropduetoreducingtheratiooftheinterfacialanisotropytothevolume\nanisotropyenergyiftheNilayeristoothick.LikeHK,theHCshowsasimilartrend\nwithvaryingthicknessoftheNilayer.Aswewellknowthatthecoercivitydepends\nonPMA,aswellasdefects-inducedpinningeffects.But,inourcase,theresultsshow\nthatthecombinationofPMAandmagnetization-relevantdemagnetizationfield\ndominatethecoercivity,whichcanbewellexplainedbytheBrownformula[35]:C=\n2U\nS−S,whereKU=(MS*HK)/2andNarethemagneticanisotropyconstantand\nthedemagnetizationfactorofthefilm,respectively.\nFigures5(d)and5(i)showtheinhomogeneouslinewidth(H0)ofFMRspectraas\nafunctionoftNiforPt/[Co(0.2)/Ni(tNi)]andPt/[Co(0.3)/Ni(tNi)],respectively.Forthin\nthicknessNisamples,islandstructuresaremostlikelyformed.Thisresultsina\nbroadeningoftheresonancelinewidthduetoadistributionofeffectiveinternal\nanisotropyanddemagnetizationfields[37].OnecanseethattheminimumlinewidthoftwoserialsamplescorrespondstothemaximumPMAfieldHK,suggestingthe\ninhomogeneousmagneticanisotropy-inducedlinearbroadeningistheminimumatthe\noptimumPMAcondition[36].Althoughtheintrinsicdampingconstantisalmost\nindependentoftheNithicknessforthestudiedtwoserials,butthePt/[Co(0.2)/Ni(tNi)]\nfilmshavealowerdampingconstantα~0.04thanα~0.07ofPt/[Co(0.3)/Ni(tNi)].\nTheobviousdifferenceindampingconstantbetweenthetwoserialmultilayer\nsystemsindicatesthattheformerhasbettermagneticdynamicproperties.\nFig.5.(a)-(e)DependenceofthesaturationmagnetizationMS(a),thecoercivityHC(b),\ntheanisotropyfieldHK(c),theinhomogeneouslinewidthΔH0(d)andthemagnetic\ndampingconstantα(e)ontheNithicknesstNiinthefilms\nPt(5)/[Co(0.2)/Ni(tNi)]5/Pt(1).(f)-(j)Sameas(a)-(e)forthesamples\nPt(5)/[Co(0.3)/Ni(tNi)]5/Pt(1).MS,HC,andHKweredeterminedfromthepreviousmagnetizationloops,AHRloops,MOKEloops,andtheferromagnetresonance\nspectra.ThelinewidthwasdeterminedbyfittingtheexperimentalFMRspectrumwith\naLorentzianfunctionbasedonEq.(1).Themagneticdampingconstantwasobtained\nbyalinearfittingofΔHvs.fcurvesbasedonEq.(2).\n4.CONCLUSION\nNithicknesseffectonthestaticmagneticpropertiesandmagneticdynamicsofPt(5\nnm)/[Co(0.2nmand0.3nm)/Ni(tNinm)]5/Pt(1nm)multilayersdemonstratethatthe\ntwostudiedserialmultilayersystemsexhibittheoptimumPMAcoefficientKUwellas\nthehighestcoercivityHCattheNithicknesstNi=0.6nm.TheMOKEimagesfurther\nconfirmthatthemaximumKUcorrespondstothemagneticdomainstructurewiththe\nshortestlengthofdomainwallthroughminimizingthetotalenergy,whichconsistsof\nmagneticanisotropyenergy,exchangeenergy,anddemagnetizationenergy.\nFurthermore,thefrequency-dependentFMRspectrashowthatthedampingconstant\nremainsalmostconstantwiththedifferentNithicknessesforbothserials,butthe\nPt/[Co(0.2)/Ni(tNi)]multilayerserialhasalowerdampingconstantα~0.04than0.07\nofthePt/[Co(0.3)/Ni(tNi)]serial.Accordingtotheobtainedresults,wefindthatthe\noptimumPMAcoefficientKU=3.3×106ergcm-3,thehighestcoercivityHC=250\nOe,andaswellasthelowestdampingconstantα=0.04canbeachievedat\nPt(5)/[Co(0.2)/Ni(0.6)]5/Pt(1).Ourresultsofoptimizingmagneticpropertiesofthe\nPt/[Co/Ni]multilayerbytuningtheratioofCo/Nilayersishelpfultofacilitateits\napplicationsinvariousspintronicdevices.\nAcknowledgement：ProjectissupportedbytheNationalNaturalScienceFoundationof\nChina(GrantNos.11774150,12074178,12004171,12074189and51971109),theAppliedBasicResearchProgramsofScienceandTechnologyCommissionFoundationofJiangsuProvince,\nChina(GrantNo.BK20170627),theNationalKeyR&DProgramofChina(GrantNo.\n2018YFA0209002),theOpenResearchFundofJiangsuProvincialKeyLaboratoryfor\nNanotechnology,andtheScientificFoundationofNanjingUniversityofPostsand\nTelecommunications(NUPTSF)(GrantNo.NY220164).\nCorrespondingauthors,Email:chenlina@njupt.edu.cn,rhliu@nju.edu.cn\nReferences\n[1]LiQY,ZhangPH,LiHT,ChenLN,ZhouKY,YanCJ,LiLY,XuYB,ZhangWX,LiuB,MengH,\nLiuRHandDuYW2021Chin.Phys.B30047504\n[2]YangMS,FangLandChiYQ2018Chin.Phys.B27098504\n[3]ZhuT2014Chin.Phys.B23047504\n[4]SuYC,LeiHYandHuJG2015Chin.Phys.B24097506\n[5]FuQW,ZhouKY,ChenLN,XuYB,ZhouTJ,WangDH,ChiKQ,MengH,LiuB,LiuRHandDuY\nW2020Chin.Phys.Lett.37117501\n[6]LiuRH,LimWLandUrazhdinS2015Phys.Rev.Lett.114137201\n[7]ChenLN,GaoZY,ZhouKY,DuYWandLiuRH2021Phys.Rev.Appl.16034044\n[8]ChenLN,ChenY,ZhouKY,LiHT,PuY,XuYB,DuYWandLiuRH2021Nanoscale.137838\n[9]LiLY,ChenLN,LiuRHandDuYW2020Chin.Phys.B29117102\n[10]WangGZ,ZhangZZ,MaBandJinQY2013J.Appl.Phys.11317C111\n[11]GuoYY,ZhaoFF,XueHBandLiuZJ2016Chin.Phys.Lett.33037501\n[12]AndrieuS,HauetT,GottwaldM,RajanikanthA,CalmelsL,BatailleAM,MontaigneF,ManginS,OteroE,\nOhresserP,LeFèvreP,BertranF,RestaA,VladA,CoatiAandGarreauY2018Phys.Rev.Materials2064410\n[13]LiRZ,LiYC,ShengYandWangKY2021Chin.Phys.B30028506\n[14]SongHS,LeeKD,SohnJW,YangSH,ParkinSSP,YouCYandShinSC2013Appl.Phys.Lett.102\n102401\n[15]NakazawaS,ObinataA,ChibaDandUenoK2017Appl.Phys.Lett.110062406[16]LiQY,XiongSQ,ChenLN,ZhouKY,XiangRX,LiHT,GaoZY,LiuRHandDuYW2021\nChin.Phys.Lett.38047501\n[17]HuangLA,WangMY,WangP,YuanY,LiuRB,LiuTY,LuY,ChenJR,WeiLJ,ZhangW,YouB,Xu\nQYandDuJ2022Chin.Phys.B31027506\n[18]ChenLN,UrazhdinS,ZhouKY,DuYWandLiuRH2020Phys.Rev.Appl.13024034\n[19]ChenLN,GuYY,ZhouKY,LiZS,LiLY,GaoZY,DuYWandLiuRH2021Phys.Rev.B103\n144426\n[20]CaiKM,YangMY,JuHL,WangSM,JiY,LiBH,EdmondsKW,ShengY,ZhangB,ZhangN,LiuS,\nZhengHZandWangKY2017Nat.Mater.16712\n[21]YangMY,CaiKM,JuHL,EdmondsKW,YangG,LiuS,LiBH,ZhangB,ShengY,WangSG,JiY\nandWangKY2016Sci.Rep.620778\n[22]YangMY,DengYC,WuZH,CaiKM,EdmondsKW,LiYC,ShengY,WangSM,CuiY,LuoJ,JiY,\nZhengHZandWangKY2019IEEEElectronDeviceLett401554\n[23]CuiYW,FengXY,ZhangQH,ZhouHA,JiangWJ,CaoJW,XueDSandFanXL2021Phys.Rev.B\n103024415\n[24]BaiQW,GuoB,YinQandWangShY2022Chin.Phys.B31017501\n[25]ZhangP,XieKX,LinWW,WuDandSangH2014Appl.Phys.Lett.104082404\n[26]LohmannM,SuT,NiuB,HouYS,AlghamdiM,AldosaryM,XingWY,ZhongJN,JiaS,HanW,WuR\nQ,CuiYTandShiJ2019NanoLett.192397\n[27]ChibaD,KawaguchiM,FukamiS,IshiwataN,ShimamuraK,KobayashiKandOnoT2012NatCommu.3\n888\n[28]TangJX,XuGZ,YouYR,XuZ,ZhangZ,ChenX,GongYYandXuF2020Appl.Phys.Lett.117\n202402\n[29]HubertAandSchäferR1988magneticdomains(WorldPublishingCorporation)p.148\n[30]FuQW,LiY,ChenLN,MaFS,LiHT,XuYB,LiuB,LiuRHandDuYW2020Chin.Phys.Lett.37\n087503\n[31]NembachHT,SilvaTJ,ShawJM,SchneiderML,CareyMJ,MaatSandChildressJR2011Phys.Rev.B\n84054424[32]MaoSW,LuJ,YangL,RuanXZ,WangHL,WeiDH,XuYBandZJH2020Chin.Phys.Lett.37\n058501\n[33]XieHk,PanLN,ChengXH,ZhuZT,FengHM,WangJBandLiuQF2018J.Magn.Magn.Mater.461\n19\n[34]AroraM,HübnerR,SuessD,HeinrichBandGirtE2017Phys.Rev.B96024401\n[35]KronmüllerH,DurstK-D,SagawaM1988J.Magn.Magn.Mater.74291\n[36]ShawJM,NembachHTandSilvaTJ2010J.Appl.Phys.108093922\n[37]PlatowW,AnisimovAN,DuniferGL,FarleMandBaberschkeK1998Phys.Rev.B58561"    },    {        "title": "2304.09841v2.Inviscid_damping_of_monotone_shear_flows_for_2D_inhomogeneous_Euler_equation_with_non_constant_density_in_a_finite_channel.pdf",        "content": "arXiv:2304.09841v2  [math.AP]  1 Nov 2023INVISCID DAMPING OF MONOTONE SHEAR FLOWS FOR 2D\nINHOMOGENEOUS EULER EQUATION WITH NON-CONSTANT DENSITY\nIN A FINITE CHANNEL\nWEIREN ZHAO\nAbstract. We prove the nonlinear inviscid damping for a class of monoto ne shear flows with non-\nconstant background density for the two-dimensional ideal inhomogeneous fluids in T×[0,1] when\nthe initial perturbation is in Gevrey-1\ns(1\n2< s <1) class with compact support.\nContents\n1. Introduction 1\n2. Good unknown and coordinate system 9\n3. Linear inviscid damping 14\n4. Wave operator 28\n5. The good system 43\n6. Energy functional and bootstrap proposition 48\n7. Energy estimate 53\n8. Elliptic estimate for steam function 63\n9. Zero mode of the pressure 68\n10. Elliptic estimate for the pressure 71\n11. Transport terms 84\n12. The lower order terms in If 90\n13. Linear force term in Ia 93\n14. The nonlinear interactions with the pressure in If 95\nAppendix A. Gevrey spaces 98\nAppendix B. The Fourier transform of the integral operator 99\nAppendix C. Sturm-Liouville Equation 101\nAppendix D. Properties of multiplier A 114\nReferences 115\n1.Introduction\nWe consider the two-dimensional inhomogeneous incompress ible Euler system in a finite channel\nT×[0,1]:\n(1.1)\n\n∂tρ+v·∇ρ= 0,\nρ(∂tv+v·∇v)+∇P= 0,\ndivv= 0\n12 WEIREN ZHAO\nwhereρ >0 is the density, vis the velocity and Pis the pressure. By introducing m=ρ−1, we\nhave\n(1.2)\n\n∂tm+v·∇m= 0,\n∂tv+v·∇v+m∇P= 0,\ndivv= 0.\nThe system has a nontrivial steady state:\nvs= (u(y),0)ms=θ(y)Ps= cons..\nIn this paper, we study the global stability of this steady st ate. It is natural to introduce the\nperturbation: v= (u(y),0) +U,m=θ(y)+d(t,x,y) andP= cons. +P. We also introduce the\nperturbed vorticity ω=∇×Uand the associated stream function ψ. Then we obtain the equations\nof perturbation ( U,d,ω,ψ,P ):\n(1.3a)/braceleftBigg\n∂tUx+u(y)∂xUx+u′(y)∂xψ+U·∇Ux+θ∂xP+d∂xP= 0\n∂tUy+u(y)∂xUy+U·∇Uy+θ∂yP+d∂yP= 0,\nand\n(1.3b)/braceleftBigg\n∂tω+u(y)∂xω−u′′(y)∂xψ−θ′(y)∂xP+∂yP∂xd−∂xP∂yd+U·∇ω= 0,\n∂td+u(y)∂xd+θ′(y)∂xψ+U·∇d= 0,\nwhere\n(1.3c)\n\nU=∇⊥ψ,∆ψ=ω,div/parenleftbig\n(θ+d)∇P/parenrightbig\n=−div(u·∇u)−2u′(y)∂xxψ,\nψ(t,x,0) =ψ(t,x,1) = 0, ∂yP(t,x,0) =∂yP(t,x,1) = 0,\nω|t=0=ωin(x,y), d|t=0=din(x,y).\nIn the homogeneous density case θ= 1,d= 0, (1.3) is the Euler equation around a shear flow\n(u(y),0). In [30], Orr observed that for the Couette flow case u(y) =y, the velocity will tend to\na shear flow as t→ ∞. This phenomenon is called inviscid damping. Bedrossian an d Masmoudi\n[5] proved nonlinear inviscid damping around the Couette flo w in Gevrey- mclass (1 ≤m <2).\nDeng and Masmoudi [14] proved some instability for initial p erturbations in the Gevrey- mclass\n(m >2). We refer to [18, 19] and references therein for other rela ted interesting results. For the\nmonotone shear flow setting, Case [9] predicted the t−1decay for the velocity. It is later proved by\nWei, Zhang, and Zhao [33]. We also refer to [38, 21, 20]. The no nlinear inviscid damping for stable\nmonotone shear flow was proved by Ionescu-Jia [16], and Masmo udi-Zhao [28]. For non-monotone\nflows such as the Poiseuille flow and the Kolmogorov flow, anoth er dynamic phenomenon should be\ntaken into consideration, which is the so-called vorticity depletion, predicted by Bouchet and Morita\n[8] and later proved in [34, 35, 17]. See also [1, 26, 19] for si milar depletion in various systems. For\nthe inhomogeneous Euler equation, very recently, Chen, Wei , Zhang, and Zhang [11] proved the\nnonlinear inviscid damping for the Couette flow with the dens ity near a constant, namely, θ= 1.\nOur main result is\nTheorem 1.1. Supposeu(y),θ(y)are smooth functions defined on [0,1]which satisfy:\n(1)(Monotone) There exists c0>0such thatu′≥c0>0.\n(2)(Positive density) There exists c0>0such thatθ≥c0>0.\n(3)(Compact support) There exists κ0∈(0,1\n10]such that suppu′′⊂[4κ0,1−4κ0]andsuppθ′⊂\n[4κ0,1−4κ0].\n(4)(Linear stability) The distorted Rayleigh operator uId−/parenleftBig\nu′\nθ/parenrightBig′˜∆−1has no eigenvalue and no\nembedded eigenvalue.NONLINEAR INVISCID DAMPING 3\n(5)(Regularity) There exist K >1ands0∈(0,1)such that for all integers m≥0,\nsup\ny∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingledm(θ′(y))\ndym/vextendsingle/vextendsingle/vextendsingle/vextendsingle+sup\ny∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingledm(u′′(y))\ndym/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Km(m!)1\ns0(m+1)−2.\nFor all1> s0≥s >1/2andλin>0, there exist λin> λ∞=λ∞(λin,K,κ0,s)>0and\n0< ǫ0=ǫ0(λin,λ∞,κ0,s)≤1\n2such that for all ǫ≤ǫ0ifωin, dinhave compact support in\nT×[3κ0,1−3κ0]and satisfy\n/ba∇dblωin/ba∇dbl2\nGs,λin=/summationdisplay\nk/integraldisplay\n|/hatwideωin(k,η)|2e2λin/a\\}bracketle{tk,η/a\\}bracketri}htsdη≤ǫ2,/integraldisplay\nT×[0,1]ωin(x,y)dxdy= 0\nand\n/ba∇dbldin/ba∇dbl2\nGs,λin=/summationdisplay\nk/integraldisplay/vextendsingle/vextendsingle/vextendsingle/hatwidedin(k,η)/vextendsingle/vextendsingle/vextendsingle2\ne2λin/a\\}bracketle{tk,η/a\\}bracketri}htsdη≤ǫ2,/integraldisplay\nT×[0,1]din(x,y)dxdy= 0\nthen the smooth solution ω(t)andd(t)to(1.3)satisfy:\n1.(Compact support) For all t≥0,suppω(t)⊂T×[2κ0,1−2κ0]andsuppd(t)⊂T×[2κ0,1−\n2κ0].\n2.(Scattering) There exist f∞,d∞∈ Gs,λ∞withsuppf∞⊂T×[2κ0,1−2κ0]andsuppd∞⊂\nT×[2κ0,1−2κ0]such that for all t≥0,\n(1.4) /ba∇dblω(t,x+tu(y)+Φ(t,y),y)−f∞(x,y)/ba∇dblGs,λ∞/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht,\nand\n(1.5) /ba∇dbld(t,x+tu(y)+Φ(t,y),y)−d∞(x,y)/ba∇dblGs,λ∞/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht,\nwhereΦ(t,y)is given explicitly by\n(1.6) Φ( t,y) =1\n2π/integraldisplayt\n0/integraldisplay\nTUx(τ,x,y)dxdτ=u∞(y)t+O(ǫ2),\nwithu∞(y) =−1\n2π∂y/integraltext\nT∆−1f∞(x,y)dx.\n3.(Inviscid damping) The velocity field Usatisfies\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\n2π/integraldisplay\nUx(t,x,·)dx−u∞/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nGs,λ∞/lessorsimilarǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2, (1.7)\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleUx(t)−1\n2π/integraldisplay\nUx(t,x,·)dx/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht, (1.8)\n/vextenddouble/vextenddoubleUy(t)/vextenddouble/vextenddouble\nL2/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2. (1.9)\nWe list some remarks here.\n(1) By modifying the Fourier multiplier, one can prove the as ymptotic stability in the Gevrey-2\nclass.\n(2) The compact support assumptions on u′′,θ′andωin,dinseem necessary, which ensures the\nFourier analysis works in the estimate.\n(3) In the proof, we construct a wave operator to absorb the no nlocal term. Such kind of op-\nerators appears in the study of dispersive equations with po tential, (also called distorted\nFourier transform, see [13, 15, 25, 31, 32, 36] for more detai ls. Let us also mention that\nrecently, the wave operator was successfully used to solve i mportant problems in fluid me-\nchanics. In [24], the authors use the wave operator method to solve Gallay’s conjecture on4 WEIREN ZHAO\npseudospectral and spectral bounds of the Oseen vortices op erator. In [35], the wave oper-\nator method was used to solve Beck and Wayne’s conjecture. In [28], the authors use the\nwave operator method to prove the nonlinear inviscid dampin g for stable monotone shear\nflows.\n(4) In [11], the authors introduce a well-constructed weigh t A∗which captures the important\ndifference of regularities between the vorticity and density . There are other choices. The\nweight introduced in [11] is constructed in a clever way. We u se the same weight in this\npaper. It helps us to simplify the proof and reduce more than 2 0 pages of this paper.\n(5) It is quite interesting to study the stability threshold problem for the 2D inhomogeneous\nincompressible Navier-Stokes equations at a high Reynolds number. For the Navier-Stokes\nequations, we refer to [2, 3, 4, 6, 7, 10, 12, 22, 23, 27, 29] for the recent progress.\n1.1.Main idea. The paper contains mainly two parts. The first part, from sect ion 2 to section 5,\nis the more important part, where we do the reduction. In this part, we introduce a good unknown,\npressure decomposition, nonlinear change of coordinates, and the wave operator to deduce a good\nworking PDE system. The following map (1) shows the main idea of reaching a good working\nsystem. In each step, we ensure that the system is better for n onlinear energy estimates. In the\nsecond part, weclose the energy estimate by usingtime-depe ndent Fourier multipliers which capture\nthe growth of the solution at each frequency and each time.\nWe first introduce a good unknown to get rid of the linear press ure termθ′(y)∂xP, see section\n2 for more details. This offers us a better equation (see equati on (2.21)), at least at the linear\nlevel. One can easily compare the linearized equation (2.21 ) of the good unknown with the classical\nlinearized Euler equation and obtain linear (in)stability results parallelly. Due to the new good\nunknown, we have to study a distorted elliptic equation to re cover the stream function, even at\n(t,x,y) coordinate. Moreover, the equation for the pressure is als o distorted by the background\ndensity. In Appendix C, we study the Sturm-Liouville equati on and obtain the decay estimate of\nthe Fourier transform of the corresponding Green function, see Proposition 8.3 and Proposition\n10.1, which will be used in the elliptic estimates for the str eam function and pressure. This part is\nnew. Roughly speaking, after being distorted by the density and nonlinear change of coordinates,\nthe main part of the elliptic operator still behaves as ∂2\nz+(∂v−t∂z)2, namely, the Fourier transform\nof the Green function Gsatisfies\n|G(t,k,ξ,η)| ≤Cmin/braceleftBigg\ne−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(ξ−kt)2,e−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(η−kt)2/bracerightBigg\n.\nNote that the denominator k2+(ξ−kt)2is the Fourier transform of −∂2\nz−(∂v−t∂z)2.\nThe pressure term is one of the main difficulties: 1, the right- hand side of the pressure equation\n(2.2) is not compactly supported; 2, the pressure equation i s an elliptic equation with Neumann\nboundary conditions, which is not friendly to the zero mode, namely, the compatibility condition\nshould be kept in the decomposition. The second step is press ure decomposition, which is mainly\nbecause of the finite channel setting. Instead of solving the elliptic equation, we obtain the zero\nmode of the pressure from the vertical velocity equation. Se e (5.7i) for the equation and section 9\nfor detailed estimates. For the nonzero modes, we first divid e the pressure into the linear part Pl\nand the nonlinear part Pn. We divide the pressure into 6 terms,\nP=Pl,1+Pl,2+Pl,b+Pn,1+Pn,2+Pn,b.\nNow we introduce the main idea of decomposition. For the nonl inear part Pn, we consider the\nsteady case, namely\n−(θ+d)∂xPn=Ux∂xUx+Uy∂yUx=1\n2∂x(Ux)2+∂xψ(Ω+∂x∂xψ).NONLINEAR INVISCID DAMPING 5\nFigure 1. The map of main ideas\n∂tω+u(y)∂xω−u′′(y)∂xψ−θ′(y)∂xP+...= 0\nStep 1: good unknown ˜ ω=∂y/parenleftbig1\nθ∂yψ/parenrightbig\n+1\nθ∂xxψdef=˜∆ψ\ndiv/parenleftbig\n(θ+d)∇P/parenrightbig\n=−div(U·∇U)−2u′(y)∂xxψ\nStep 2: pressure decomposition\n∂tUy+u(y)∂xUy+U·∇Uy+θ∂yP+d∂yP= 0\nzero mode pressure\n∂t˜ω+u(y)∂x˜ω+P0(Ux)∂x˜ω−/parenleftBig\nu′\nθ/parenrightBig′\nUy+...= 0\nStep 3: change of coordinates ( t,x,y)→(t,z,v)\n∂tΩ−ϕ1∂zP/negationslash=Ψ+U·∇z,vΩ+NΩ[Ψ]+Na[Π] = 0\nStep 4: wave operatorinverse change of coordinates\n∂t˜ω+∂xR˜ω= 0\nThe working system (5.7)distorted elliptic equation\nSturm-Liouville Equation\ndiv/parenleftbig\n(θ+d)∇Pl,2/parenrightbig\n= (χ′′\n2u)(y)∂yψ/negationslash=\nelliptic commutator\nNoticing that near the boundary d+θis constant, Ω = 0, we then have\nPn=−((∂xψ)2+(∂yψ)2)+C\n2(θ+d)\nnear the boundary. We use this idea to construct two boundary correction terms Pl,bandPn,bto get\nrid of the boundary value on the right-hand side of the pressu re equation and make them compactly\nsupported. The same idea also works for the decomposition of the linear part, see section 2 for\nmore details. There is a derivative loss after the decomposi tion, which does not appear in the whole\nspaceT×Rsetting, discussed in [11]. For example, for the linear part , before the decomposition,\none can formally regard ∆ Pl≈ −∂xxψ. After the decomposition, there is χ′′\n2∂yψon the right-hand6 WEIREN ZHAO\nside of the equation of Pl,2, see (2.5). Note that the derivative loss in yis bad because it will\nlead to growth. Luckily, χ′′\n2vanishes on the support of θ′,d,χ1. So we first study ∆ Pl,2≈χ′′\n2∂yψ\nand control /a\\}b∇acketle{t∂x,∂y/a\\}b∇acket∇i}htPl,2byψ. Then we study ∆( Pl,2χ1)≈2χ′\n1∂yPl,2and control /a\\}b∇acketle{t∂x,∂y/a\\}b∇acket∇i}ht2Pl,2χ1by\n/a\\}b∇acketle{t∂x,∂y/a\\}b∇acket∇i}htPl,2, which can be formally regarded as the estimate of the commut ator\n[∆−1,χ1](χ′′\n2∂yψ).\nThe same problem appears in the pressure decomposition of th e nonlinear part. Roughly speaking,\nthe additional growth of Π l,2and Πn,2happens only near the boundary, which does not affect the\nvorticity equation due to the compact support of aandθ′. We use the same strategy to treat Π l,2\nand Πn,2, see section 10 for more details.\nThe third step is the nonlinear change of coordinates, which helps us to get rid of the non-\ndecaying zero mode in the transport term. The step will also d istort the elliptic equation for the\nstream function and pressure. But after separating the line ar equation with the time-independent\ncoefficients, the time-dependent (solution-dependent) par t is small, see (8.2) and (10.4) for the\ndecompositions, also see Proposition 8.3, Lemma 10.2, Lemm a 10.3, Lemma 10.4, and Lemma 10.5\nfor the estimates of the solution-dependent part.\nThe original nonlocal term −u′′∂xψ−θ′∂xP, which does not appear in the Couette flow with\na constant background density case, is hard to deal with. Aft er introducing the good unknown\n˜ω(see 2.1) and doing the nonlinear change of coordinates, we h ave a slightly better new nonlocal\ntermϕ1∂zP/\\e}atio\\slash=Ψ. The new nonlocal term may lead to instability. However, un der the linear stability\nassumption (4) in Theorem 1.1, we can construct a wave operat or to eliminate it. Similar to\n[28], we have the following observations: 1, the operator ca n be constructed for each frequency\ninzseparately; 2, the nonlocal term is small for high frequenci es in thezvariable, which also\ncorresponds to the linear stability of the distorted Raylei gh operator in large wave numbers. So the\nwave operator is not needed for higher frequencies; 3, the no nlocal term ( ϕ1◦u−1)(v)∂zP/\\e}atio\\slash=˚T−1\n1,D[Ω]\nwith time-independent coefficient ( ϕ1◦u−1)(v) is responsible for the linear instability. Thus, we\nonly construct a wave operator for lower frequencies in zto eliminate the time-independent part.\nDue to the change of coordinates, even the linearized nonloc al term˚T−1\n1,D[Ω] is time-dependent. We\nthen introduce theinverse linear change of coordinates to m ake the nonlocal term time-independent.\nWe arrive at (2.21), which is the same as the linearized equat ion in the original variables but has\na different meaning. The construction of the wave operator is h ighly non-trivial. In section 3,\nwe study (2.21). This part is new. We obtain the representati on formula of the solution, which\ngives the linear inviscid damping. The representation form ula leads us to the formula of the wave\noperator. In section 4, we study the basic properties of the w ave operator. The nice estimates of\nits Fourier kernel, make the wave operator well-adapted to n onlinear interactions. More precisely,\nthe Fourier kernel D(t,k,ξ1,ξ2) of the wave operator satisfies\n|D(t,k,ξ1,ξ2)|/lessorsimilare−λD|ξ1−ξ2|s0.\nIn terms of regularity, one can simply regard the effect of the w ave operator acting on the function\nfas multiplying fby a smooth function, whose Fourier transform is e−λD|ξ|s0. This is one reason\nwhy the wave operator is well-adapted to the nonlinear syste m. The second reason is related to\nthe famous Orr mechanism [30]. In physical terms, because th e mode of the vorticity in question\nis initially well-mixed, and then proceeds to un-mix under t he shear flow evolution, a transient\ngrowth happens near the Orr critical time. In the breakthrou gh paper [5], the authors estimate\nthis transient growth mathematically rigorously by introd ucing a multiplier A( t,∇). In terms of\nfrequency, the wave operator has two effects: 1, it changes the amplitude of the function at each\nfrequency; 2, it essentially mixes the information near eac h frequency, due to the fast decay of its\nFourier kernel. It ensures that the critical time does not mo ve too much. These good properties ofNONLINEAR INVISCID DAMPING 7\nthe wave operator allow us to use the same multiplier A of [5] a nd use the same strategy as in [28]\nto study the effect of the wave operator on the nonlinear terms.\nWith all the preparations done, we arrive at a good working sy stem (5.7). In the second part,\nwe study the nonlinear problem and close the energy estimate s. The main idea is to introduce\ntime-dependent Fourier multipliers which capture the differ ent growths of different frequencies at\ndifferent times. We use the classical multiplier A( t,∇) (see Appendix D for details) of [5] to capture\nthe nonlinear growth from the nonlinear interactions of the transport term U · ∇Ω. We use an\nadditional multiplier B( t,∇) introduced in [11] to gain the derivative for the density afor short\ntimes. It is useful in the estimate of the nonlinear interact ions between aand the pressure Π when\nais in higher frequencies. We also use the multipliers M3,M4,andM5by using the ideas from\n[11] to control the pressure. We refer to [11] for some explan ations about the new multipliers.\nSome ideas related to the energy estimates and applying the w eight are given in section 6 after we\nintroduce the working system and the Fourier multipliers, w hich are more technical.\nThe proof of the main theorem is based on a standard bootstrap argument. In section 6, we\nintroduce the bootstrap Proposition 6.7. In section 7, we gi ve the key propositions which will be\nused to prove Proposition 6.7 and finish the proof of Theorem 1 .1 by admitting those propositions.\nThe rest of the paper is to prove propositions in section 7. So me estimates are done in the previous\npapers [5, 28]. The key estimate is to control the pressure te rm, which is in section 9 and section\n10.\n1.2.Notations. Forf(z,v) smooth with compact support in T×(u(0),u(1)), we define the Fourier\ntransform in the first variable F1f(k,v), the Fourier transform in the second variable F2f(z,η) and\nthe Fourier transform in both variables ˆfk(η) where (k,η)∈Z×R,\nF1f(k,v) =1\n2π/integraldisplay\nTf(z,v)e−izkdz,\nF2f(z,η) =/integraldisplay\nRf(z,v)e−ivηdv,\nˆfk(η) =ˆf(k,η) =1\n2π/integraldisplay\nT×Rf(z,v)e−izk−ivηdzdv.\nWe use a convention is to denote M ,N as dyadic integers M ,N∈Dwhere\nD=/braceleftbigg1\n2,1,2,...,2j,.../bracerightbigg\n= 2N∪1\n2.\nThis will be useful when defining Littlewood-Paley projecti ons and paraproduct decompositions.\nWe will use the same A for A( t,∇)f= (A(η)ˆf(η))∨or Aˆf= A(η)ˆf(η), where A( t,∇) is a Fourier\nmultiplier. For function A( t,k,ξ), we denote the time derivative of A by ˙A(t,k,ξ) =∂tA(t,k,ξ).\nWe also denote the time derivative on the Fourier multiplier A(t,∇) by\n˙A(t,∇)f= (˙A(t,k,η)ˆf(t,k,η))∨.\nWe use the notation f/lessorsimilargwhen there exists a constant C >0 independent of the parameters\nof interest such that f≤Cg(we defineg/greaterorsimilarfanalogously). Similarly, we use the notation f≈g\nwhen there exists C >0 such that C−1g≤f≤Cg. We use the notation cto denote a constant\nsuch that 0 ≤c<1 which may be different from line to line.\nWe will denote the l1vector norm |k,η|=|k|+|η|, which by convention is the norm taken in our\nwork. Similarly, given a scalar or vector in Rnwe denote\n/a\\}b∇acketle{tv/a\\}b∇acket∇i}ht=/parenleftbig\n1+|v|2/parenrightbig1\n2.8 WEIREN ZHAO\nWe denote the projection to the kth mode of f(x,y) by\nPkf(x,y) =fk(x,y) =1\n2π/parenleftbigg/integraldisplay\nTf(x′,y)e−ikx′dx′/parenrightbigg\neikx,\nand denote the projection to the non-zero mode by\nP/\\e}atio\\slash=f(x,y) =f/\\e}atio\\slash==/summationdisplay\nk∈Z\\{0}fk(x,y).\nand the projection to zero mode by\n(P0f)(y) =1\n2π/parenleftbigg/integraldisplay\nTf(x′,y)dx′/parenrightbigg\n.\nWe also use ˆfk(ξ) to denote ˆf(k,ξ) to emphasize it is the Fourier transform of the kmode.\nIn some proofs, we use the Littlewood-Paley dyadic decompos ition. Let̺0be a smooth function\nsuch that̺0= 1 for |ξ| ≤1\n2and̺(ξ) = 0 for |ξ| ≥3\n4. We define ̺=̺0(ξ/2)−̺0(ξ) and\n̺M(ξ) =̺(M−1ξ) with dyadic numbers M = 1 ,2,4,....\nfM=̺M(|∇|)f,/hatwidefk(ξ)M=/hatwidefk(ξ)ρM(k,ξ)\nf<M=̺0(|∇|)f+/summationdisplay\nK∈2N:K<MfK.\nWe denote the standard L2norms by /ba∇dbl·/ba∇dbl2. We make common use of the Gevrey-1\nsnorm with\nSobolev correction defined by\n/ba∇dblf/ba∇dbl2\nGs,λ,σ=/summationdisplay\nk/integraldisplay/vextendsingle/vextendsingle/vextendsingleˆfk(η)/vextendsingle/vextendsingle/vextendsingle2\ne2λ|k,η|s/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht2σdη.\nIn the proof, we sometimes use Gs,σinstead where we omit λ=λ(t) for simplification and keep the\nimportant index σwhich represents the Sobolev corrections. We refer to this n orm as the Gs,λ,σ\nnorm and occasionally refer to the space of functions\nGs,λ,σ=/braceleftbig\nf∈L2:/ba∇dblf/ba∇dblGs,λ,σ<∞/bracerightbig\n.\nWe refer to section A.2 in [5] for a discussion of the basic properties of this norm an d some related\nuseful inequalities. For η≥0, we define E( η)∈Zto be the integer part. We define for η∈Rand\n1≤ |k| ≤E(/radicalbig\n|η|) withηk≥0,tk,η=/vextendsingle/vextendsingleη\nk/vextendsingle/vextendsingle−|η|\n2|k|(|k|+1)andt0,η= 2|η|and the critical intervals\nIk,η=/braceleftBigg\n[t|k|,η,t|k|−1,η] ifηk≥0 and 1≤ |k| ≤E(/radicalbig\n|η|),\n∅ otherwise.\nFor minor technical reasons, we define a slightly restricted subset as the resonant intervals\nIk,η=/braceleftBigg\nIk,η/radicalbig\n|η| ≤tk,η,\n∅ otherwise.\nWe use1Abe the characteristic function which means 1A(x) =/braceleftbigg1x∈A\n0x /∈A. We also use the\nsmooth cut-off functions χ1andχ2in Gevrey-2\ns0+1class which satisfies\nsuppχ1⊂[κ0,1−κ0], χ1(y)≡1 fory∈[1−2κ0,2κ0],\nsuppχ2⊂/bracketleftbigκ0\n2,1−κ0\n2/bracketrightbig\n, χ2(y)≡1 fory∈[κ0,1−κ0],NONLINEAR INVISCID DAMPING 9\nand there exist K1>1 such that for all integers m≥0\nsup\ny∈R/vextendsingle/vextendsingle/vextendsingle/vextendsingledmχi(y)\ndym/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Km\n1(m!)2\ns0+1(m+1)−2, i= 1,2. (1.10)\n2.Good unknown and coordinate system\nThere are two linear nonlocal terms −u′′∂xψand−θ′∂xPin the vorticity equation of (1.3). The\nfirstnonlocal termalsoappearsintheEulercase[28], where theauthorsintroducethewave operator\nto eliminate it. The second nonlocal term is new, which does n ot exist in the constant background\ndensity case θ= 1. The idea of eliminating this term is to use the horizontal velocity equation,\nwhich also contains ∂xPin the equation. We introduce the good unknown\n˜ω=ω\nθ−θ′\nθ2∂yψ=∂y/parenleftbigg1\nθ∂yψ/parenrightbigg\n+1\nθ∂xxψdef=˜∆ψ. (2.1)\n2.1.New system in good unknown. A direct calculation gives\n(2.2)\n\n∂t˜ω+u(y)∂x˜ω−/parenleftbiggu′\nθ/parenrightbigg′\nUy+U·∇˜ω+θ′\nθ∂xψ˜ω+/parenleftbiggθ′′\nθ2−(θ′)2\nθ3/parenrightbigg\n∂xψ∂yψ\n+∂xd∂yP−∂yd∂xP\nθ+θ′d∂xP\nθ2= 0,\n˜∆ψdef=∂y/parenleftbigg1\nθ∂yψ/parenrightbigg\n+1\nθ∂xxψ= ˜ω, U= (Ux,Uy) =∇⊥ψ,\nψ(t,x,0) =ψ(t,x,1) = 0\n∂td+u(y)∂xd+θ′(y)∂xψ+U·∇d= 0,\ndiv/parenleftbig\n(θ+d)∇P/parenrightbig\n=−div(U·∇U)−2u′(y)∂xxψ,\nLet us mention some observations of the new system here.\n•The new linear nonlocal term/parenleftBig\nu′\nθ/parenrightBig′\nUyhas compact support.\n•Ifd= 0,θ= 1, we are back to the Euler equation around shear flows.\n2.2.Pressure decomposition. The pressure equation is not good for applying the Fourier tr ans-\nform, because the right-hand side −div(U· ∇U)−2u′(y)∂xxψis not compactly supported. We\ndividePinto six terms: two linear terms Pl,1andPl,2, two nonlinear terms Pn,1andPn,2, and two\nboundary corrections Pl,bPn,b, namely,\nP=Pl,1+Pl,2+Pl,b+Pn,1+Pn,2+Pn,b.\nWe definePl,b=u∂yψ/negationslash=−u′ψ/negationslash=\nθ+d(1−χ2), whereψ/\\e}atio\\slash==ψ−1\n2π/integraltext\nTψ(t,x,y)dxis the projection of ψonto\nthe non-zero mode. Then by using the facts\n(1−χ2)ω= 0, u′′(1−χ2) = 0,(1−χ2)∇(1\nθ+d) = 0,\nwe have\n∂yPl,b=u(P/\\e}atio\\slash=[ω]−∂xxψ/\\e}atio\\slash=)\nθ+d(1−χ2)−u∂yψ/\\e}atio\\slash=−u′ψ/\\e}atio\\slash=\nθ+dχ′\n2\ndiv/parenleftbig\n(θ+d)∇Pl,b/parenrightbig\n= ∆/parenleftbig\n(u∂yψ−u′ψ)(1−χ2)/parenrightbig\n= (1−χ2)/parenleftbig\nu∂yxxψ/\\e}atio\\slash=−u′∂xxψ/\\e}atio\\slash=/parenrightbig\n−χ′′\n2(u∂yψ/\\e}atio\\slash=−u′ψ/\\e}atio\\slash=)\n−2χ′\n2(u∂yyψ/\\e}atio\\slash=−u′∂yψ/\\e}atio\\slash=)−2χ′\n2(u′∂yψ/\\e}atio\\slash=−u′′ψ/\\e}atio\\slash=)10 WEIREN ZHAO\n+(1−χ2)(u∂yyyψ/\\e}atio\\slash=+2u′∂yyψ/\\e}atio\\slash=−u′∂yyψ/\\e}atio\\slash=−u′′∂yψ/\\e}atio\\slash=−u′′′ψ/\\e}atio\\slash=)\n=−2(1−χ2)u′∂xxψ/\\e}atio\\slash=+2χ′\n2(u∂xxψ/\\e}atio\\slash=)−χ′′\n2(u∂yψ/\\e}atio\\slash=−u′ψ/\\e}atio\\slash=)\nwhich gives ∂yPl,b(t,x,0) =∂yPl,b(t,x,1) = 0 and\n∂xd∂yPl,b−∂yd∂xPl,b\nθ+θ′d∂xPl,b\nθ2≡0,andd∂xPl,b≡0. (2.3)\nThis means that the linear boundary correction does not affect the equation of ˜ ω. Then we define\nPlby solving the following equation:/braceleftBigg\ndiv/parenleftbig\n(θ+d)∇Pl,1/parenrightbig\n=−2(χ2u′)(y)∂xxψ−2(uχ′\n2)(y)∂xxψ−(u′χ′′\n2)(y)ψ/\\e}atio\\slash=,\n∂yPl,1(t,x,0) =∂yPl,1(t,x,1) = 0,(2.4)\nand/braceleftBigg\ndiv/parenleftbig\n(θ+d)∇Pl,2/parenrightbig\n= (χ′′\n2u)(y)∂yψ/\\e}atio\\slash=,\n∂yPl,2(t,x,0) =∂yPl,2(t,x,1) = 0.(2.5)\nIt is easy to check that Pl,1+Pl,2+Pl,bsolves\n/braceleftBigg\ndiv/parenleftbig\n(θ+d)∇(Pl,1+Pl,2+Pl,b)/parenrightbig\n=−2u′(y)∂xxψ,\n∂y(Pl,1+Pl,2+Pl,b)(t,x,0) =∂y(Pl,1+Pl,2+Pl,b)(t,x,1) = 0.(2.6)\nWe then define the nonlinear boundary correction:\nPn,b(t,x,y) =−1\n21\nθ+d/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig\n(1−χ2). (2.7)\nNotice that ∂yPn,b(t,x,0) =∂yPn,b(t,x,1) = 0 and\n∂xd∂yPn,b−∂yd∂xPn,b\nθ+θ′d∂xPn,b\nθ2≡0,andd∂xPn,b≡0\nwhich means that the nonlinear boundary correction does not affect the equation of ˜ ω.\nNow let us deduce the equation for Pn. A direct calculation gives\ndiv/parenleftbig\n(θ+d)∇Pn/parenrightbig\n=−div(U·∇U)+1\n2div/parenleftBig\n(θ+d)∇/parenleftBig1\nθ+d/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig\n(1−χ2)/parenrightBig/parenrightBig\n=−2(∂xyψ)2−2(∂xxψ)2+2ω∂xxψ\n+1\n2div/parenleftBig/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig\n(1−χ2)(θ+d)∇/parenleftBig1\nθ+d/parenrightBig/parenrightBig\n+1\n2div/parenleftBig\n(1−χ2)∇/parenleftBig/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig/parenrightBig/parenrightBig\n−1\n2∂y/parenleftBig/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig\nχ′\n2/parenrightBig\n.\nNoticing that (1 −χ2) = 0 on the compact support of ∇/parenleftBig\n1\nθ+d/parenrightBig\n, we obtain that\ndiv/parenleftbig\n(θ+d)∇Pn/parenrightbig\n=−div(U·∇U)+div/parenleftBig\n(θ+d)∇/parenleftBig1\nθ+d/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig\n(1−χ2)/parenrightBig/parenrightBig\n=−2(∂xyψ)2−2(∂xxψ)2+2ω∂xxψ\n+1\n2(1−χ2)div/parenleftBig\n∇/parenleftBig/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig/parenrightBig/parenrightBig\n−χ′\n2∂y/parenleftBig/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig/parenrightBig\n−1\n2χ′′\n2/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBigNONLINEAR INVISCID DAMPING 11\n=−2χ2(∂xyψ)2−2χ2(∂xxψ)2+2ω∂xxψ\n+(1−χ2)∂xω∂xψ+(1−χ2)∂yω∂yψ\n−χ′\n2/parenleftBig/bracketleftBig\n2(∂xψ)∂xyψ+2(∂yψ)∂xxψ/bracketrightBig/parenrightBig\n−1\n2χ′′\n2/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig\n.\nNoticing that (1 −χ2) = 0 on the compact support of ∇ω, we obtain the equation of Pn\n(2.8)\n\ndiv/parenleftbig\n(θ+d)∇Pn/parenrightbig\n=−2χ2(∂xyψ)2−2χ2(∂xxψ)2+2ω∂xxψ\n−χ′\n2/parenleftBig/bracketleftBig\n2(∂xψ)∂xyψ+2(∂yψ)∂xxψ/bracketrightBig/parenrightBig\n−1\n2χ′′\n2/bracketleftBig\n(∂xψ)2+(∂yψ)2/bracketrightBig\n.\n∂yPn(t,x,0) =∂yPn(t,x,1) = 0.\nNote that the last term ( ∂yψ)2in (2.8) requires an additional decomposition for Pn=Pn,1+Pn,2\nwith\n(2.9)\n\ndiv/parenleftbig\n(θ+d)∇Pn,1/parenrightbig\n=−2χ2(∂xyψ)2−2χ2(∂xxψ)2+2ω∂xxψ\n−χ′\n2/parenleftBig/bracketleftBig\n2(∂xψ)∂xyψ+2(∂yψ)∂xxψ/bracketrightBig/parenrightBig\n−1\n2χ′′\n2/bracketleftBig\n(∂xψ)2+(∂yP/\\e}atio\\slash=ψ)2/bracketrightBig\n.\n∂yPn,1(t,x,0) =∂yPn,1(t,x,1) = 0.\nand\n(2.10)\n\ndiv/parenleftbig\n(θ+d)∇Pn,2/parenrightbig\n=−χ′′\n2(∂yP/\\e}atio\\slash=ψ)(∂yP0ψ)−1\n2χ′′\n2(∂yP0ψ)2.\n∂yPn,2(t,x,0) =∂yPn,2(t,x,1) = 0.\nBy this decomposition, the functions of the right-hand side of equations (2.4), (2.5), (2.8), and\n(2.10) have compact support. Let us give a formal discussion about the behavior of each pressure.\nBy dropping the lower order terms, coefficients, and the disto rting effect in the elliptic equations,\nin terms of regularity, one can formally regard that these eq uations are\n∆Pl,1≈χ2∂xxψ,∆Pl,2≈χ′′\n2∂yψ/\\e}atio\\slash=,∆Pn,1≈ω∂xxψ,∆Pn,2≈χ′′\n2(∂yψ/\\e}atio\\slash=).\n2.3.Zero mode of the pressure. For the zero mode of the pressure, we use the equation\n∂tUy+u(y)∂xUy+U·∇Uy+θ∂yP+d∂yP= 0\nto estimate the zero mode of the pressure. We have\n(θ+P0(d))∂yP0(P)+P0[P/\\e}atio\\slash=(d)∂yP/\\e}atio\\slash=(P)] =P0(Ux∂xUy−Uy∂xUx) =P0(−∂yψ∂xxψ+∂xψ∂xyψ).\n2.4.Nonlinear coordinate transform. Let us now introduce the change of variables ( t,x,y)→\n(t,z,v) to eliminate the non-decaying zero mode:\n(2.11)v(t,y) =u(y)−χ1(y)\nt/integraldisplayt\n01\n2π/integraldisplay\nT∂yψ(t′,x,y)dxdt′,\nz(t,x,y) =x−tv(t,y).\nThus Ranu= Ranv= [u(0),u(1)]. Let us also define ∂tv(t,v) and∂yv(t,v) so that\n∂tv(t,v(t,y)) =∂tv(t,y), ∂yv(t,v(t,y)) =∂yv(t,y).12 WEIREN ZHAO\nIn order to simplify the notations, we introduce\n(2.12)ϕ1(y) =/parenleftbiggu′\nθ/parenrightbigg′\n(y), ϕ2(y) =θ′(y)\nθ(y), ϕ3(y) =/parenleftbiggθ′′\nθ2−(θ′)2\nθ3/parenrightbigg\n,\nϕ4(y) =1\nθ(y), ϕ5(y) =ϕ2(y)ϕ4(y), ϕ6(y) =−2(χ2u′)(y)−2(uχ′\n2)(y),\nϕ7(y) =−(u′χ′′\n2)(y), ϕ8(y) = (χ′′\n2u)(y), ϕ9(y) =u′(y), ϕ10(y) =θ′(y),\nϕ11(y) =θ(y).\nHere and below, for any function ϕ(y), we denote\nϕ(t,v(t,y)) =ϕ(y) and/tildewideϕ(u(y)) =ϕ(y).\nWe write two functions ϕ(t,v) and/tildewideϕ(v) both in (t,v) variable and regard /tildewideϕ(v) as the linear part of\nϕ(t,v). So we can expect\nϕδ(t,v)def=ϕ(t,v)−/tildewideϕ(v) (2.13)\nis small.\nRemark 2.1. Recall the notations ϕ(t,v(t,y)) =ϕ(y)and/tildewideϕ(u(y)) =ϕ(y), thenϕ(t,v)satisfies\nthe following transport equation\n∂tϕ(t,v)+∂tv(t,v)∂vϕ(t,v) = 0,\nandϕδ(t,v) =ϕ(t,v)−/tildewideϕ(v)satisfies\n∂tϕδ+∂tv(t,v)∂vϕδ=−∂tv(t,v)∂v/tildewideϕ(v).\nAlso note that ϕ4(t,v),ϕ9(t,v), andϕ11(t,v)do not have compact support, but ϕδ\nj(t,v)has compact\nsupport for j= 1,2,...,11\nDefine Ω(t,z,v),a(t,z,v), Π(t,z,v) and Ψ(t,z,v) so that\n(2.14)Ω(t,z(t,x,y),v(t,y)) = ˜ω(t,x,y), a(t,z(t,x,y),v(t,y)) =d(t,x,y),\nΠl,1(t,z(t,x,y),v(t,y)) =Pl,1(t,x,y),Πl,2(t,z(t,x,y),v(t,y)) =Pl,2(t,x,y),\nΠn,1(t,z(t,x,y),v(t,y)) =Pn,1(t,x,y),Πn,2(t,z(t,x,y),v(t,y)) =Pn,2(t,x,y),\nΨ(t,z(t,x,y),v(t,y)) =ψ(t,x,y), ω(t,z(t,x,y),v(t,y)) =ω(t,x,y)\nhence the original 2D inhomogeneous Euler system (1.3) is ex pressed in the ( t,z,v) variables as\n(2.15)\n\n∂tP0(ω)+∂yvP0(∂zΨ∂vω−∂vΨ∂zω)−∂yvP0(∂zΠl∂va−∂vΠl∂za)\n−∂yvP0(∂zΠn∂va−∂vΠn∂za) = 0,\n∂tΩ−ϕ1∂zP/\\e}atio\\slash=Ψ+U·∇z,vΩ+NΩ[Ψ]+Na[Π] = 0,\n∂ta+θ′∂zΨ+U·∇z,va= 0,\nϕ4∂zzΨ+∂yv(∂v−t∂z)/parenleftBig\nϕ4∂yv(∂v−t∂z)Ψ/parenrightBig\n= Ω,\nΨ(t,z,u(0)) = Ψ(t,z,u(1)) = 0,\nU = (0,∂tv)+∂yv∇⊥\nv,zP/\\e}atio\\slash=(Ψ),\nwhereNrepresents the nonlinear terms:\nNΩ[Ψ] =ϕ2∂zΨΩ+ϕ3∂zΨ∂yv(∂v−t∂z)Ψ (2.16)NONLINEAR INVISCID DAMPING 13\nandNa[Π] =Na[Πl]+Na[Πn] with Πl= Υ2Πl,1+Υ1Πl,2, Πn= Υ2Πn,1+Υ1Πn,2, and\n(2.17)Na[Πl] =ϕ4∂yv(∂vΠl∂za−∂zΠl∂va)+ϕ5a∂zΠl,\nNa[Πn] =ϕ4∂yv(∂vΠn∂za−∂zΠn∂va)+ϕ5a∂zΠn.\nHere (Pl,1,Pl,2,Pn,1,Pn,2) solve the following equations.\n(2.18a)\n\n∂z/parenleftBig\n(a+θ)∂zΠl,1/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Πl,1/parenrightBig\n=ϕ6∂zzΨ+ϕ7P/\\e}atio\\slash=(Ψ),\n(∂v−t∂z)Πl,1(t,z,u(0)) = (∂v−t∂z)Πl,1(t,z,u(1)) = 0,\nand\n(2.18b)\n\n∂z/parenleftBig\n(a+θ)∂zΠl,2/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Πl,2/parenrightBig\n=ϕ8∂yv(∂v−t∂z)P/\\e}atio\\slash=(Ψ),\n(∂v−t∂z)Πl,2(t,z,u(0)) = (∂v−t∂z)Πl,2(t,z,u(1)) = 0,\nand\n(2.18c)\n\n∂z/parenleftBig\n(a+θ)∂zΠn,1/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Πn,1/parenrightBig\n=−2χ2(∂yv(∂v−t∂z)∂zΨ)2−2∂zzΨ/parenleftBig\nΩ−χ2∂zzΨ/parenrightBig\n,\n−2χ′\n2∂yv/bracketleftBig\n(∂zΨ)(∂v−t∂z)∂zΨ+(∂v−t∂z)Ψ∂zzΨ/bracketrightBig\n−1\n2χ′′\n2/bracketleftBig\n(∂zΨ)2+(∂yv)2((∂v−t∂z)P/\\e}atio\\slash=Ψ)2/bracketrightBig\n,\n(∂v−t∂z)Πn,1(t,z,u(0)) = (∂v−t∂z)Πn,1(t,z,u(1)) = 0,\nand\n(2.18d)\n\n∂z/parenleftBig\n(a+θ)∂zΠn,2/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Πn,2/parenrightBig\n=−1\n2χ′′\n2/bracketleftBig\n2(∂yv)2((∂v−t∂z)P/\\e}atio\\slash=Ψ)∂vP0Ψ+(∂yv)2(∂vP0Ψ)∂vP0Ψ/bracketrightBig\n,\n(∂v−t∂z)Πn,2(t,z,u(0)) = (∂v−t∂z)Πn,2(t,z,u(1)) = 0.\nWe also have that the zero mode of the pressure solves\n(θ+P0(a))∂vP0(Π)+P0/bracketleftbig\nP/\\e}atio\\slash=(a)P/\\e}atio\\slash=(∂v−t∂z)P/\\e}atio\\slash=(Π)/bracketrightbig\n=P0/parenleftBig\n∂zΨ∂z(∂v−t∂z)Ψ−∂zzΨ∂vΨ/parenrightBig\n.\nNote that (2.15) is not our final working system. The main part of the nonlocal quasi-linear term\n−ϕ1∂zP/\\e}atio\\slash=(Ψ) will be eliminated by applying the wave operator.\n2.5.Linearization and the inverse linear change of coordinate. Let us also introduce the\nlinearization of (2.15) in ( t,z,v) coordinates\n(2.19)\n\n∂tΩ−/tildewiderϕ1∂zP/\\e}atio\\slash=Ψ = 0,\n˜∆l\ntΨdef=/tildewiderϕ4∂zzΨ+/tildewideu′(∂v−t∂z)/parenleftBig\n/tildewiderϕ4/tildewideu′(∂v−t∂z)Ψ/parenrightBig\n= Ω,\nHere we recall that /tildewiderϕ1(v) =/parenleftbigg/parenleftBig\nu′\nθ/parenrightBig′\n◦u−1/parenrightbigg\n(v),/tildewiderϕ4(v) =1\n(θ◦u−1)(v)and/tildewideu′(v) = (u′◦u−1)(v). Let us\nnow introduce the inverse linear change of coordinate, name ly, (t,z,v)→(t,x,y):\ny=u−1(v), x=z+tu(y) (2.20)14 WEIREN ZHAO\nDefine ˜ω(t,x,y) = Ω(t,x−tu(y),u(y)) andψ(t,x,y) = Ψ(t,x−tu(y),u(y)). Then we arrive at\n∂t˜ω+∂xR˜ω= 0, ψ=˜∆−1˜ω, (2.21)\nwhereR=uId−/parenleftBig\nu′\nθ/parenrightBig′˜∆−1is the distorted Rayleigh operator. Note that here we use the same\nnotation (t,x,y), ˜ω, andψ. They are different from the functions in (2.2) written in orig inal (t,x,y)\ncoordinates.\n3.Linear inviscid damping\nIn this section, we study the linearized equation (2.21). Th e main purpose is to introduce the\nwave operator associated with the distorted Rayleigh opera torR.\nThe wave operator related to a self-adjoint operator is well -known [13, 31, 15, 25, 32, 36]. Let\nA,Bbe two self-adjoint operators in the Hilbert space H, then the wave operator Drelated toA\nandBsatisfies\nAD=DB.\nIt can be defined by\nD= lim\nt→∞e−itAeitB.\nHowever, the wave operator related to non-self-adjoint ope rators is usually not easy to construct\nand estimate. Let us explain the intuition of the constructi on on the operator i∂xassociated with\nthe Fourier transform. More precisely, we know that λande−iλxare the generalized eigenvalue and\neigenfunction. For any function f∈L1∩L2we have the representation formula\nf(x) =/integraldisplay\nR/a\\}b∇acketle{tf,e−iλx/a\\}b∇acket∇i}hte−iλxdλ. (3.1)\nLet us define T(f)(λ) =/a\\}b∇acketle{tf,e−iλx/a\\}b∇acket∇i}ht=/integraltext\nRf(x)eixλdx, thenT◦(i∂x) =λT, and we can regard Tas a\nwave operator conjugating i∂xto the multiplication by λ.\n3.1.The distorted Rayleigh equation. By taking the Fourier transform in x, we have\n∂t˜∆kˆψ+iku(y)˜∆kˆψ−ik/parenleftbigu′\nθ/parenrightbig′ˆψ= 0.\nLet˜Lkˆψ=˜∆−1\nk/parenleftBig\nu(y)˜∆kˆψ−/parenleftbigu′\nθ/parenrightbig′ˆψ/parenrightBig\n. We then have\n(3.2) ˆψ(t,k,y) =1\n2πi/contintegraldisplay\n∂Se−ickt(c−˜Lk)−1ˆψindc\nwhereScontains the spectrum of ˜Lk. Next, we study the resolvent ΨI= (c−˜Lk)−1ˆψin. Then ΨI\nsolves the distorted inhomogeneous Rayleigh equation\n(u−c)∂y/parenleftbigg1\nθ∂yΨI/parenrightbigg\n−(u−c)k2\nθΨI−/parenleftbigu′\nθ/parenrightbig′ΨI=−˜∆kˆψin=−ˆ˜ωin (3.3)\nwithboundaryconditionΨI(0) = ΨI(1) = 0. WealsointroducethehomogeneousdistortedRayleig h\nequation\n(u−c)∂y/parenleftbigg1\nθ∂yΨH/parenrightbigg\n−(u−c)k2\nθΨH−/parenleftbigu′\nθ/parenrightbig′ΨH= 0. (3.4)\nRemark 3.1. Letu∈C4,u′/\\e}atio\\slash= 0andθ>0. Then we have the following facts:\n•Ranu⊂σ(˜Lk)is the continuous spectrum.NONLINEAR INVISCID DAMPING 15\n•Ifc /∈Ranuis an eigenvalue of ˜Lk, then(3.4)has a non-trivial solution in H2(0,1)∩\nH1\n0(0,1).\n•Ifc=u(yc)∈Ranuis an embedded eigenvalue, then/parenleftbigu′\nθ/parenrightbig′(yc) = 0. The boundary value\nu(0),u(1)are not the embedded eigenvalue.\n•If/parenleftbigu′\nθ/parenrightbig′does not change sign in [0,1], then˜Lkhas no eigenvalue or embedded eigenvalue.\n3.2.Homogeneous equation. In this section, we show the existence of a smooth solution to the\nhomogeneous equation (3.4).\nProposition 3.2. Lety,y′∈[0,1], and letφ1solve\n(3.5) φ1(y,y′,k) = 1+/integraldisplayy\ny′k2θ(z′)\n(u(z′)−u(y′))2/integraldisplayz′\ny′(u(z′′)−u(y′))2\nθ(z′′)φ1(z′′,y′,k)dz′′dz′\nor in the differential equation form\n(3.6)\n\n∂y/parenleftbigg(u(y)−u(y′))2\nθ(y)∂yφ1(y,y′,k)/parenrightbigg\n=k2(u(y)−u(y′))2\nθ(y)φ1(y,y′,k),\nφ1(y′,y′,k) = 1∂yφ1(y′,y′,k) = 0.\nThenφ(y,y′,k) = (u(y)−u(y′))φ1(y,y′,k)solves(3.4)withc=u(y′)∈Ranu.\nLetǫ0>0and for|ǫ|<ǫ0, letφ1(y,y′,ǫ,k)solve\n(3.7) φ1(y,y′,ǫ,k) = 1+/integraldisplayy\ny′/integraldisplayz′\ny′k2θ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2φ1(z′′,y′,ǫ,k)dz′′dz′\nor in the differential equation form\n(3.8)\n\n∂y/parenleftbigg(u(y)−u(y′)−iǫ)2\nθ(y)∂yφ1(y,y′,ǫ,k)/parenrightbigg\n=k2(u(y)−u(y′)−iǫ)2\nθ(y)φ1(y,y′,ǫ,k),\nφ1(y′,y′,ǫ,k) = 1∂yφ1(y′,y′,ǫ,k) = 0.\nThenφ(y,y′,ǫ,k) = (u(y)−u(y′)−iǫ)φ1(y,y′,ǫ,k)solves(3.4)withc=u(y′)+iǫ.\nMoreover, the following estimates hold for y,y′∈[0,1],k∈Z\\{0}, and|ǫ|<ǫ0.\nφ1(y,y′,k)≥1,(y−y′)∂yφ1(y,y′,k)≥0 (3.9)\nC−1eC−1|k|(|y−y′|)≤φ1(y,y′,k)≤CeC|k||y−y′|, (3.10)\nC−1|k|min{|k|(y−y′),1} ≤|∂yφ1(y,y′,k)|\nφ1(y,y′,k)≤C|k|min{|k|(y−y′),1} (3.11)\n|∂yyφ1(y,y′,k)| ≤Ck2φ1(y,y′,k), (3.12)\n0≤φ1(y,y′,k)−1≤Cmin{1,|k|2|y−y′|2}φ1(y,y′,k), (3.13)\nandφ1(y,y′,ǫ,k)→φ1(y,y′,k)asǫ→0, and forǫ0small enough (may depend on k)\n1\n2≤ |φ1(y,y′,ǫ,k)| ≤CeC|k||y−y′|. (3.14)\nHereCis a constant independent of k.\nProof.Existence part. We first show the existence of φ1(y,y′,ǫ,k) for any fixed k∈Z\\{0}and\n(y,y′,ǫ)∈[0,1]2×[−ǫ0,ǫ0]. We introduce the weighted norm\n/ba∇dblf/ba∇dblX0def= sup\n(y,y′)∈[0,1]2/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(y,y′)\ncoshA(y−y′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle,16 WEIREN ZHAO\n/ba∇dblf/ba∇dblXdef= sup\n(y,y′,ǫ)∈[0,1]2×[−ǫ0,ǫ0]/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(y,y′,ǫ)\ncoshA(y−y′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nand\n/ba∇dblf/ba∇dblY0def=2/summationdisplay\nk=0/summationdisplay\n|β|=kA−k/ba∇dbl∇β\ny,y′f/ba∇dblX0,\n/ba∇dblf/ba∇dblYdef=/ba∇dblf/ba∇dblX+1\nA/parenleftbig\n/ba∇dbl∂yf/ba∇dblX+/ba∇dbl∂y′f/ba∇dblX+/ba∇dbl∂ǫf/ba∇dblX/parenrightbig\n.\nConsider the linear operator for 0 ≤ǫ≤ |ǫ0|\nTǫ[φ1] =/integraldisplayy\ny′/integraldisplayz′\ny′θ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2φ1(z′′,y′,ǫ,k)dz′′dz′,\nit is easy to check that\n/ba∇dblT0[φ1]/ba∇dblX0≤C\nA2/ba∇dblφ1/ba∇dblX0,/ba∇dblTǫ[φ1]/ba∇dblX≤C\nA2/ba∇dblφ1/ba∇dblX.\nA direct calculation gives\n∂yTǫ[φ1] =/integraldisplayy\ny′θ(y)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(y)−u(y′)−iǫ)2φ1(z′′,y′,ǫ,k)dz′′,\nwhich gives\n/ba∇dbl∂yT0[φ1]/ba∇dblX0≤C\nA/ba∇dblφ1/ba∇dblX0,/ba∇dbl∂yTǫ[φ1]/ba∇dblX≤C\nA/ba∇dblφ1/ba∇dblX.\nWe also have\n∂yyTǫ[φ1] =φ1+/integraldisplayy\ny′θ′(y)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(y)−u(y′)−iǫ)2φ1(z′′,y′,ǫ,k)dz′′\n−2/integraldisplayy\ny′θ(y)u′(y)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(y)−u(y′)−iǫ)3φ1(z′′,y′,ǫ,k)dz′′,\nwhich gives\n/ba∇dbl∂yyT0[φ1]/ba∇dblX0≤C/ba∇dblφ1/ba∇dblX0,/ba∇dbl∂yyTǫ[φ1]/ba∇dblX≤C/ba∇dblφ1/ba∇dblX.\nA direct calculation gives for −ǫ0≤ǫ≤ǫ0\n(∂y+∂y′)Tǫ[φ1] =/integraldisplayy\ny′/integraldisplayz′\ny′(∂y′+∂z′+∂z′′)/parenleftbiggθ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2φ1(z′′,y′,ǫ,k)/parenrightbigg\ndz′′dz′\n=/integraldisplayy\ny′/integraldisplayz′\ny′(∂y′+∂z′+∂z′′)/parenleftbiggθ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2/parenrightbigg\nφ1(z′′,y′,ǫ,k)dz′′dz′\n+/integraldisplayy\ny′/integraldisplayz′\ny′θ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2(∂y′+∂z′′)φ1(z′′,y′,ǫ,k)dz′′dz′\nand\n(∂y+∂y′)2Tǫ[φ1]\n=/integraldisplayy\ny′/integraldisplayz′\ny′(∂y′+∂z′+∂z′′)2/parenleftbiggθ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2/parenrightbigg\nφ1(z′′,y′,ǫ,k)dz′′dz′NONLINEAR INVISCID DAMPING 17\n+/integraldisplayy\ny′/integraldisplayz′\ny′θ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2(∂y′+∂z′′)2φ1(z′′,y′,ǫ,k)dz′′dz′\n+/integraldisplayy\ny′/integraldisplayz′\ny′(∂y′+∂z′+∂z′′)/parenleftbiggθ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2/parenrightbigg\n(∂y′+∂z′′)φ1(z′′,y′,ǫ,k)dz′′dz′,\nwhich together with the fact that for 0 ≤y′≤z′′≤z′≤1 or 0≤z′≤z′′≤y′≤1\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂y′+∂z′+∂z′′)j/parenleftbiggθ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C,forj= 0,1,2,\ngives that\n/ba∇dbl(∂y+∂y′)T0[φ1]/ba∇dblX0≤C\nA2/parenleftbig\n/ba∇dbl(∂y+∂y′)φ1/ba∇dblX0+/ba∇dblφ1/ba∇dblX0/parenrightbig\n,\n/ba∇dbl(∂y+∂y′)Tǫ[φ1]/ba∇dblX≤C\nA2/parenleftbig\n/ba∇dbl(∂y+∂y′)φ1/ba∇dblX+/ba∇dblφ1/ba∇dblX/parenrightbig\n,\n/ba∇dbl(∂y+∂y′)2T0[φ1]/ba∇dblX0≤C\nA2/parenleftbig\n/ba∇dbl(∂y+∂y′)2φ1/ba∇dblX0+/ba∇dbl(∂y+∂y′)φ1/ba∇dblX0+/ba∇dblφ1/ba∇dblX0/parenrightbig\n,\n/ba∇dbl(∂y+∂y′)2Tǫ[φ1]/ba∇dblX≤C\nA2/parenleftbig\n/ba∇dbl(∂y+∂y′)2φ1/ba∇dblX+/ba∇dbl(∂y+∂y′)φ1/ba∇dblX+/ba∇dblφ1/ba∇dblX/parenrightbig\n.\nWe also have\n∂ǫTǫ[φ1] =−2i/integraldisplayy\ny′/integraldisplayz′\ny′θ(z′)(u(z′′)−u(y′)−iǫ)\nθ(z′′)(u(z′)−u(y′)−iǫ)2φ1(z′′,y′,ǫ,k)dz′′dz′\n+2i/integraldisplayy\ny′/integraldisplayz′\ny′θ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)3φ1(z′′,y′,ǫ,k)dz′′dz′\n+/integraldisplayy\ny′/integraldisplayz′\ny′θ(z′)(u(z′′)−u(y′)−iǫ)2\nθ(z′′)(u(z′)−u(y′)−iǫ)2∂ǫφ1(z′′,y′,ǫ,k)dz′′dz′,\nwhich gives\n/ba∇dbl∂ǫTǫ[φ1]/ba∇dblX≤C\nA/ba∇dblφ1/ba∇dblX+C\nA2/ba∇dbl∂ǫφ1/ba∇dblX.\nBy combining all the estimates, we arrive at\n/ba∇dblT0[φ1]/ba∇dblY0≤C\nA2/ba∇dblφ1/ba∇dblY0,/ba∇dblTǫ[φ1]/ba∇dblY≤C\nA2/ba∇dblφ1/ba∇dblY.\nBy using the equations\nφ1(y,y′,ǫ,k) = 1+k2Tǫ[φ1](y,y′,ǫ,k) andφ1(y,y′,k) = 1+k2Tǫ[φ1](y,y′,k)\nandtakingAlargeenough,weshowthat Tǫisthecontraction mapyieldstheexistenceof φ1(y,y′,ǫ,k)\nandφ1(y,y′,k).\nEstimate part. The estimate (3.9) follows directly from the fact that T0is a positive operator.\nWe also get that for 0 ≤y′≤z′′≤y≤1 or 0≤y≤z′′≤y′≤1,\n1≤φ1(z′′,y′,k)≤φ1(y,y′,k).\nThus\nφ1(y,y′,k)−1 =k2/integraldisplayy\ny′/integraldisplayz′\ny′θ(z′)(u(z′′)−u(y′))2\nθ(z′′)(u(z′)−u(y′))2φ1(z′′,y′,k)dz′′dz′\n≤Ck2|y−y′|2φ1(y,y′,k),18 WEIREN ZHAO\n|∂yφ1(y,y′,k)|=k2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayy\ny′θ(y)(u(z′′)−u(y′))2\nθ(z′′)(u(y)−u(y′))2φ1(z′′,y′,k)dz′′/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤k2|y−y′|φ1(y,y′,k),\nwhich gives (3.13) and part of (3.11).\nWe introduce f(y,y′,k) =∂yφ1(y,y′,k)\nθ(y)φ1(y,y′,k)thenf(y,y′,k)≥0 fory≥y′andf(y,y′,k)≤0 fory≤y′.\nWe also have\nf′+2u′\nu(y)−u(y′)f+f2θ=k2\nθ,\nandf′(y′,y′,k) =k2\n3θ(y′)>0. By the continuity of fand the fact that2u′\nu(y)−u(y′)f≥0, using a\nclassical contradiction argument, we have |f(y,y′,k)| ≤|k|\nθ(y), which gives (3.11). We also get that\n(3.15) |f′(y,y′,k)| ≤Ck2\nand\nφ1(z′′,y′,k)\nφ1(y,y′,k)= exp/parenleftBig/integraldisplayz′′\nyθ(z′)f(z′,y′,k)dz′/parenrightBig\n≥e−|k||y−z′′|.\nThus we have for |y−y′| ≤1\n|k|, 1≤φ1(z′′,y,k)≤φ1(y,y′,k)≤C\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yφ1(y,y′,k)\nφ1(y,y′,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=k2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayy\ny′θ(y)(u(z′′)−u(y′))2\nθ(z′′)(u(y)−u(y′))2φ1(z′′,y′,k)\nφ1(y,y′,k)dz′′/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≥C−1k2|y−y′|\nand for|y−y′| ≥1\n|k|and ify>y′\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yφ1(y,y′,k)\nφ1(y,y′,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=k2/integraldisplayy\ny′θ(y)(u(z′′)−u(y′))2\nθ(z′′)(u(y)−u(y′))2φ1(z′′,y′,k)\nφ1(y,y′,k)dz′′\n≥C−1k2/integraldisplayy\ny′(u(z′′)−u(y′))2\n(u(y)−u(y′))2e−|k||y−z′′|dz′′\n≥C−1k2/integraldisplayy\ny−1\n2|k|(u(z′′)−u(y′))2\n(u(y)−u(y′))2e−|k||y−z′′|dz′′≥C−1|k|,\nand ify<y′,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yφ1(y,y′,k)\nφ1(y,y′,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥C−1k2/integraldisplayy+1\n2|k|\ny(u(z′′)−u(y′))2\n(u(y)−u(y′))2e−|k||y−z′′|dz′′≥C−1|k|.\nWe now use\nφ1(y,y′,k) = exp/parenleftBig/integraldisplayy\ny′∂yφ1(z′,y′,k)\nφ1(z′,y′,k)dz′/parenrightBig\n≥eC−1|k||y−y′|,\nwhich give (3.10). The estimate (3.12) follows directly fro m (3.15) and (3.12). The estimate (3.14)\nfollows fromthefactthat φ1(y,y′,ǫ,k) iscontinuousin ǫand(3.10). Thusweprovedtheproposition.\n/squareNONLINEAR INVISCID DAMPING 19\n3.3.Inhomogeneous solution. In this section, we give the representation formula of the in ho-\nmogeneous solution to (3.3) and study the limit as ǫ→0. Let us first introduce\nρ(y′) = (u(y′)−u(0))(u(1)−u(y′)),\nand\n(3.16a)J1(y′,k)def=−(u(1)−u(y′))θ(0)\nu′(0)−(u(y′)−u(0))θ(1)\nu′(1)\n+ρ(y′)/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′\n−ρ(y′)H/bracketleftbig/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′χ[u(0),u(1)]/bracketrightbig\n(u(y′))\n=−(u(1)−u(y′))θ(0)\nu′(0)−(u(y′)−u(0))θ(1)\nu′(1)\n+ρ(y′)/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′\n+ρ(y′)P.V./integraldisplay1\n01\nu(z)−u(y′)/parenleftBigθ(z)\nu′(z)/parenrightBig′\ndz,\nand\n(3.16b) J2(y′) =ρ(y′)/parenleftbig\n(θ◦u−1)(u−1)/parenrightbig′(u(y′)) =−ρ(y′)θ(y′)2\nu′(y′)3/parenleftBigu′\nθ/parenrightBig′\n(y′).\nWe then have the following lemma.\nLemma 3.3. Lety′∈[0,1]andk∈Z\\{0}. Thenu(y′)∈Ranuis an embedded eigenvalue of ˜Lk,\nif and only if\nJ1(y′,k)2+J2(y′)2= 0.\nProof.SupposeJ1(y′,k)2+J2(y′)2= 0, then/parenleftBig\nu′\nθ/parenrightBig′\n(y′) = 0 andy′/\\e}atio\\slash= 0,1. We have for 0 ≤y<y′,\nϕ1(y,y′) =φ(0,y′,k)φ(y,y′,k)/integraldisplayy\n0θ(z)\nφ(z,y′,k)2dz\nsolves the homogeneous equation (3.4) with boundary condit ionϕ1(0,y′) = 0 and∂yϕ1(0,y′) = 1.\nNext, we show that ϕ1can naturally C2extend toy′≤y≤1 under the assumption/parenleftBig\nu′\nθ/parenrightBig′\n(y′) = 0.\nA direct calculation gives for 0 ≤y<y′,\nϕ1(y,y′) =φ(0,y′,k)φ(y,y′,k)/integraldisplayy\n0θ(z)\n(u(z)−u(y′))2dz\n+φ(0,y′,k)φ(y,y′,k)/integraldisplayy\n0θ(z)\n(u(z)−u(y′))2/parenleftBig1\nφ1(z,y′,k)2−1/parenrightBig\ndz\n=φ(0,y′,k)φ(y,y′,k)/integraldisplayu(y)\nu(0)(θ◦u−1)(˜u)(u−1)′(˜u)\n(˜u−u(y′))2d˜u\n+φ(0,y′,k)φ(y,y′,k)/integraldisplayy\n0θ(z)\n(u(z)−u(y′))2/parenleftBig1\nφ1(z,y′,k)2−1/parenrightBig\ndz\n=φ(0,y′,k)φ(y,y′,k)/integraldisplayu(y)\nu(0)(θ◦u−1)(˜u)(u−1)′(˜u)−(θ◦u−1)(u(y′))(u−1)′(u(y′))\n(˜u−u(y′))2d˜u20 WEIREN ZHAO\n−φ(0,y′,k)φ(y,y′,k)(θ◦u−1)(u(y′))(u−1)′(u(y′))\n˜u−u(y′)/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(y)\nu(0)\n+φ(0,y′,k)φ(y,y′,k)/integraldisplayy\n0θ(z)\n(u(z)−u(y′))2/parenleftBig1\nφ1(z,y′,k)2−1/parenrightBig\ndz=I1(y)+I2(y)+I3(y).\nNote that the assumption/parenleftBig\nu′\nθ/parenrightBig′\n(y′) = 0 gives/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′(u(y′)) = 0, which gives that the\nintegral inI1\n/integraldisplayu(y)\nu(0)[(θ◦u−1)(u−1)′](˜u)−[(θ◦u−1)(u−1)′](u(y′))\n(˜u−u(y′))2d˜u\n=/integraldisplayu(y)\nu(0)[(θ◦u−1)(u−1)′](˜u)−[(θ◦u−1)(u−1)′](u(y′))−/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′(u(y′))(˜u−u(y′))\n(˜u−u(y′))2d˜u\nis well defined for all y∈[0,1] andIj(y)∈C2, forj= 1,2,3. It is easy to check that I1+I2+I3is\nalso well defined for all y∈[0,1] which also solves homogeneous equation (3.4) with c=u(y′). We\nalso useϕ1(y,y′) to represent its extension. A direct calculation gives\nϕ1(1,y′) =I1(1)+I2(1)+I3(1) =−φ1(0,y′,k)φ1(1,y′,k)J1(y′,k) = 0.\nThusϕ1(y,y′) is an eigenfunction associated with the embedded eigenval ueu(y′).\nSupposeu(y′) is an embedded eigenvalue and ΦH∈H2is the corresponding eigenfunction. Then\n∂y/parenleftbig1\nθ∂yΦH/parenrightbig\n=k2\nθΦH+/parenleftbig\nu′\nθ/parenrightbig′\nΦH(y)\nu(y)−u(y′)∈L2. Thus either ΦH(y′) = 0 or/parenleftbigu′\nθ/parenrightbig′(y′) = 0. If ΦH(y′) = 0, we\nhave\n0 =/integraldisplay1\n01\nθ(|∂yΦH|2+k2|ΦH|2)dy+/integraldisplay1\n0/parenleftbigu′\nθ/parenrightbig′|ΦH(y)|2\nu(y)−u(y′)dy\n=/integraldisplay1\n01\nθ(|∂yΦH|2+k2|ΦH|2)dy−2Re/integraldisplay1\n0/parenleftbigu′\nθ/parenrightbig\n∂yΦH(y)ΦH(y)\nu(y)−u(y′)dy+/integraldisplay1\n0/parenleftbigu′\nθ/parenrightbig\nu′|ΦH(y)|2\n(u(y)−u(y′))2dy\n=/integraldisplay1\n01\nθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yΦH−u′ΦH(y)\nu(y)−u(y′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+k2\nθ|ΦH|2dy,\nwhich gives ΦH≡0. Thus/parenleftbigu′\nθ/parenrightbig′(y′) = 0 which means J2(y′) = 0. Then by the argument in the first\npart,ϕ1(y,y′) is aC2solution that solves the homogeneous equation (3.4) with c=u(y′). Then\nϕ1(y,y′) is an eigenfunction if and only if ϕ1(1,y′) = 0, which gives J1(y′,k) = 0. Thus we proved\nthe lemma. /square\nBy following a similar idea, we have the following lemma.\nLemma 3.4. Letk∈Z\\{0}andc=u(y′)+iǫwith0<|ǫ| ≤ǫ0. Thencis an eigenvalue of ˜Lkif\nand only if\n/integraldisplay1\n0θ(z′)\nφ(z′,y′,ǫ,k)2dz′= 0\nProof.The function\nϕ(y,y′,ǫ,k) =φ(0,y′,ǫ,k)φ(y,y′,ǫ,k)/integraldisplayy\n0θ(z′)\nφ(z′,y′,ǫ,k)2dz′∈H2\nisasolutiontothehomogeneousequation(3.4)with c=u(y′)+iǫwithboundaryvalue ϕ(0,y′,ǫ,k) =\n0 and∂yϕ(0,y′,ǫ,k) = 1. The integral is well-defined for all y∈[0,1] by taking ǫ0small enough.NONLINEAR INVISCID DAMPING 21\nThusϕ(y,y′,ǫ,k) is an eigenfunction associated with c=u(y′)+iǫif and only if ϕ(1,y′,ǫ,k) = 0.\nThe lemma follows directly from the fact that φ(0,y′,ǫ,k)φ(1,y′,ǫ,k)/\\e}atio\\slash= 0 forǫ/\\e}atio\\slash= 0. /square\nLemma 3.5. Under the assumption that ˜Lkhas no embedded eigenvalue, we have\nC−1(1+k2ρ(y′)2)≤J1(y′,k)2+π2J2(y′)2≤C(1+k2ρ(y′)2).\nProof.For|k| ≤M0, Lemma 3.3 implies\nJ1(y′,k)2+π2J2(y′)2≈M01.\nLet us now consider the case |k|>M0withM0large enough. By Proposition 3.2, we have\n/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′\n≤ −C−1/integraldisplay\n|z′−y′|≤1\n|k|1\n(z′−y′)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayy′\nz′∂yφ1(z′′,y′,k)dz′′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledz′\n≤ −C−1/integraldisplay\n|z′−y′|≤1\n|k|k2dz′≤ −C−1|k|.\nIt is easy to check that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C/integraldisplay1\n01\n(z′−y′)2min{k2|z′−y′|2,1}dz′≤C|k|,\nand/vextendsingle/vextendsingle/vextendsingleH/bracketleftbig/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′χ[u(0),u(1)]/bracketrightbig\n(u(y′))/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dbl/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′/ba∇dblH1.\nThus we have\nJ1(y′,k)2≥/parenleftBig\n−(u(1)−u(y′))θ(0)\nu′(0)−(u(y′)−u(0))θ(1)\nu′(1)/parenrightBig2\n+ρ(y′)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n−/parenleftBig\nρ(y′)H/bracketleftbig/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′χ[u(0),u(1)]/bracketrightbig\n(u(y′))/parenrightBig2\n≥C−1(1+k2ρ(y′)2)\nand\nJ1(y′,k)2≤/parenleftBig\n−(u(1)−u(y′))θ(0)\nu′(0)−(u(y′)−u(0))θ(1)\nu′(1)/parenrightBig2\n+ρ(y′)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/parenleftBig\nρ(y′)H/bracketleftbig/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′χ[u(0),u(1)]/bracketrightbig\n(u(y′))/parenrightBig2\n≤C(1+k2ρ(y′)2).\nThus we prove the lemma. /square22 WEIREN ZHAO\nNow let us study the inhomogeneous equation (3.3). By using t he homogeneous solution, we\nrewrite it as\n∂y/parenleftbiggφ2\nθ∂y/parenleftbiggΨI\nφ/parenrightbigg/parenrightbigg\n=−ˆ˜ωinφ1.\nThus, forc=u(y′)+iǫwithy′∈[0,1] and 0<|ǫ| ≤ǫ0, it holds that\n(3.17)ΨI(y,y′,ǫ,k) =−φ(y,y′,ǫ,k)/integraldisplayy\n0θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)φ1(z′′,y′,ǫ,k)dz′′\n(u(z′)−u(y′)−iǫ)2φ1(z′,y′,ǫ,k)2dz′\n+/integraltext1\n0θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)φ1(z′′,y′,ǫ,k)dz′′\n(u(z′)−u(y′)−iǫ)2φ1(z′,y′,ǫ,k)2dz′\n/integraltext1\n0θ(z′)\n(u(z′)−u(y′)−iǫ)2φ1(z′,y′,ǫ,k)2dz′φ(y,y′,ǫ,k)/integraldisplayy\n0θ(z′)\nφ(z′,y′,ǫ,k)2dz′\n=−φ(y,y′,ǫ,k)/integraldisplayy\n1θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)φ1(z′′,y′,ǫ,k)dz′′\n(u(z′)−u(y′)−iǫ)2φ1(z′,y′,ǫ,k)2dz′\n+/integraltext1\n0θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)φ1(z′′,y′,ǫ,k)dz′′\n(u(z′)−u(y′)−iǫ)2φ1(z′,y′,ǫ,k)2dz′\n/integraltext1\n0θ(z′)\n(u(z′)−u(y′)−iǫ)2φ1(z′,y′,ǫ,k)2dz′φ(y,y′,ǫ,k)/integraldisplayy\n1θ(z′)\nφ(z′,y′,ǫ,k)2dz′.\nNote that Lemma 3.4 ensures that the above representation fo rmula is well-defined by taking ǫ0\nsmall enough. By the continuity of φ(y,y′,ǫ,k) inǫ, it holds that\nlim\nǫ→0φ(y,y′,ǫ,k)/integraldisplayy\n1θ(z′)\nφ(z′,y′,ǫ,k)2dz′=φ(y,y′,k)/integraldisplayy\n1θ(z′)\nφ(z′,y′,k)2dz′fory′<y≤1\nlim\nǫ→0φ(y,y′,ǫ,k)/integraldisplayy\n0θ(z′)\nφ(z′,y′,ǫ,k)2dz′=φ(y,y′,k)/integraldisplayy\n0θ(z′)\nφ(z′,y′,k)2dz′for 0≤y<y′\nlim\nǫ→0φ(y,y′,ǫ,k)/integraldisplayy\njθ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)φ1(z′′,y′,ǫ,k)dz′′\n(u(z′)−u(y′)−iǫ)2φ1(z′,y′,ǫ,k)2dz′\n=φ(y,y′,k)/integraldisplayy\njθ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)φ1(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′forj= 0,1, y∈[0,1].\nWe have for y′∈(0,1)\n/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′)−iǫ)2φ1(z′,y′,ǫ,k)2dz′\n=/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′)−iǫ)2dz′+/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′)−iǫ)2/parenleftBig1\nφ1(z′,y′,ǫ,k)2−1/parenrightBig\ndz′\n=/integraldisplayu(1)\nu(0)(θ◦u−1)(u)(u−1)′(u)\n(u−u(y′)−iǫ)2du+/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′)−iǫ)2/parenleftBig1\nφ1(z′,y′,ǫ,k)2−1/parenrightBig\ndz′\n=−(θ◦u−1)(u)(u−1)′(u)\n(u−u(y′)−iǫ)/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(1)\nu(0)+/integraldisplayu(1)\nu(0)/parenleftbig\n(θ◦u−1)(u−1)/parenrightbig′(u)\nu−u(y′)−iǫdu\n+/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′)−iǫ)2/parenleftBig1\nφ1(z′,y′,ǫ,k)2−1/parenrightBig\ndz′\n→θ(0)(u−1)′(u(0))\n(u(0)−u(y′))−θ(1)(u−1)′(u(1))\n(u(1)−u(y′))NONLINEAR INVISCID DAMPING 23\n−H/bracketleftbig/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′χ[u(0),u(1)]/bracketrightbig\n(u(y′))±iπ/parenleftbig\n(θ◦u−1)(u−1)/parenrightbig′(u(y′))\n+/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′asǫ→0±,\nwhereH[f](u) =P.V./integraltextf(u′)\nu−u′du′is the Hilbert transform and χ[u(0),u(1)](u) =/braceleftbigg1, u∈[u(0),u(1)]\n0,else,is\nthe characteristic function. We also have for y′∈(0,1),\n/integraldisplay1\n0θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)φ1(z′′,y′,ǫ,k)dz′′\n(u(z′)−u(y′)−iǫ)2φ1(z′,y′,ǫ,k)2dz′\n=/integraldisplay1\n0θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)dz′′\n(u(z′)−u(y′)−iǫ)2dz′+/integraldisplay1\n0/integraldisplayz′\ny′θ(z′)ˆ˜ωin(z′′,k)\n(u(z′)−u(y′)−iǫ)2/parenleftBigφ1(z′′,y′,ǫ,k)\nφ1(z′,y′,ǫ,k)2−1/parenrightBig\ndz′′dz′\n=/integraldisplayu(1)\nu(0)/integraltext˜u\nu(y′)ˆ˜ωin(u−1(˜˜u),k)(u−1)′(˜˜u)d˜˜u\n(˜u−u(y′)−iǫ)2(θ◦u−1)(˜u)(u−1)′(˜u)d˜u\n+/integraldisplay1\n0/integraldisplayz′\ny′θ(z′)ˆ˜ωin(z′′,k)\n(u(z′)−u(y′)−iǫ)2/parenleftBigφ1(z′′,y′,ǫ,k)\nφ1(z′,y′,ǫ,k)2−1/parenrightBig\ndz′′dz′\n=−(θ◦u−1)(˜u)/integraltext˜u\nu(y′)ˆ˜ωin(u−1(˜˜u),k)(u−1)′(˜˜u)d˜˜u\n(˜u−u(y′)−iǫ)(u−1)′(˜u)/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(1)\nu(0)\n+/integraldisplayu(1)\nu(0)(θ◦u−1)(˜u)ˆ˜ωin(u−1(˜u),k)/parenleftbig\n(u−1)′(˜u)/parenrightbig2\n(˜u−u(y′)−iǫ)d˜u\n+/integraldisplayu(1)\nu(0)/integraltext˜u\nu(y′)ˆ˜ωin(u−1(˜˜u),k)(u−1)′(˜˜u)d˜˜u\n(˜u−u(y′)−iǫ)/parenleftBig\n(θ◦u−1)(˜u)(u−1)′/parenrightBig′\n(˜u)d˜u\n+/integraldisplay1\n0/integraldisplayz′\ny′θ(z′)ˆ˜ωin(z′′,k)\n(u(z′)−u(y′)−iǫ)2/parenleftBigφ1(z′′,y′,ǫ,k)\nφ1(z′,y′,ǫ,k)2−1/parenrightBig\ndz′′dz′\nǫ→0±− −−− → −(θ◦u−1)(˜u)/integraltext˜u\nu(y′)ˆ˜ωin(u−1(˜˜u),k)(u−1)′(˜˜u)d˜˜u\n(˜u−u(y′))(u−1)′(˜u)/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(1)\nu(0)\n−H/bracketleftbig\n(θ◦u−1)ˆ˜ωin(u−1(·),k)/parenleftbig\n(u−1)′/parenrightbig2χ[u(0),u(1)]/bracketrightbig\n(u(y′))±iπˆ˜ωin(y′,k)θ(y′)\n(u′(y′))2\n+/integraldisplayu(1)\nu(0)/integraltext˜u\nu(y′)ˆ˜ωin(u−1(˜˜u),k)(u−1)′(˜˜u)d˜˜u\n(˜u−u(y′))/parenleftBig\n(θ◦u−1)(˜u)(u−1)′/parenrightBig′\n(˜u)d˜u\n+/integraldisplay1\n0/integraldisplayz′\ny′θ(z′)ˆ˜ωin(z′′,k)\n(u(z′)−u(y′))2/parenleftBigφ1(z′′,y′,k)\nφ1(z′,y′,k)2−1/parenrightBig\ndz′′dz′\n=P.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)dz′′\n(u(z′)−u(y′))2dz′\n+/integraldisplay1\n0/integraldisplayz′\ny′θ(z′)ˆ˜ωin(z′′,k)\n(u(z′)−u(y′))2/parenleftBigφ1(z′′,y′,k)\nφ1(z′,y′,k)2−1/parenrightBig\ndz′′dz′±iπθ(y′)ˆ˜ωin(y′,k)\n(u′(y′))2\ndef= Π1[ˆ˜ωin]±iπΠ2[ˆ˜ωin].24 WEIREN ZHAO\nLemma 3.6. It holds that\n/ba∇dblΠ1[ˆ˜ωin]/ba∇dblL2≤C/ba∇dblˆ˜ωin/ba∇dblL2,\nwhereC >0is a constant independent of k.\nProof.By theL2estimate of the Hilbert transform and the maximum function, we have\n/vextenddouble/vextenddouble/vextenddouble−H/bracketleftbig\n(θ◦u−1)ˆ˜ωin(u−1(·),k)/parenleftbig\n(u−1)′/parenrightbig2χ[u(0),u(1)]/bracketrightbig\n(u(y′))/vextenddouble/vextenddouble/vextenddouble\nL2≤C/ba∇dblˆ˜ωin/ba∇dblL2,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble−(θ◦u−1)(˜u)/integraltext˜u\nu(y′)ˆ˜ωin(u−1(˜˜u),k)(u−1)′(˜˜u)d˜˜u\n(˜u−u(y′))(u−1)′(˜u)/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(1)\nu(0)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2≤C/ba∇dblˆ˜ωin/ba∇dblL2,\nand/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayu(1)\nu(0)/integraltext˜u\nu(y′)ˆ˜ωin(u−1(˜˜u),k)(u−1)′(˜˜u)d˜˜u\n(˜u−u(y′))/parenleftBig\n(θ◦u−1)(˜u)(u−1)′/parenrightBig′\n(˜u)d˜u/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2≤C/ba∇dblˆ˜ωin/ba∇dblL2.\nWe write\n/integraldisplay1\n0/integraldisplayz′\ny′θ(z′)ˆ˜ωin(z′′,k)\n(u(z′)−u(y′))2/parenleftBigφ1(z′′,y′,k)\nφ1(z′,y′,k)2−1/parenrightBig\ndz′′dz′\n=/integraldisplay\n|u(z′)−u(y′)|≤2\n|k|/integraldisplayz′\ny′θ(z′)ˆ˜ωin(z′′,k)\n(u(z′)−u(y′))2/parenleftBigφ1(z′′,y′,k)\nφ1(z′,y′,k)2−1)/parenrightBig\ndz′′dz′\n−/integraldisplay\n|u(z′)−u(y′)|≥2\n|k|θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)dz′′\n(u(z′)−u(y′))2dz′\n+/integraldisplay\n|u(z′)−u(y′)|≥2\n|k|/integraldisplayz′\ny′θ(z′)ˆ˜ωin(z′′,k)\n(u(z′)−u(y′))2φ1(z′′,y′,k)\nφ1(z′,y′,k)2dz′′dz′\n=I1+I2+I3.\nBy Proposition 3.2, we have\n/ba∇dblI1/ba∇dblL2≤C/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\n|u(z′)−u(y′)|≤2\n|k||k|\n|y′−z′|/integraldisplayz′\ny′|ˆ˜ωin(z′′,k)|dz′′dz′/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2≤C/ba∇dblˆ˜ωin/ba∇dblL2,\n/ba∇dblI3/ba∇dblL2≤C/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay\n|u(z′)−u(y′)|≥2\n|k||k|\n|y′−z′|/integraldisplayz′\ny′|ˆ˜ωin(z′′,k)|dz′′e−C−1|k||y′−z′|dz′/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2≤C/ba∇dblˆ˜ωin/ba∇dblL2.\nWe have\nI2=−χ{|u(y′)−u(0)|≥2\n|k|}(u(y′))/integraldisplayu(y′)−2\n|k|\nu(0)(θ◦u−1)(u)(u−1)′(u)/integraltextu\nu(y′)ˆ˜ωin(u−1(˜u),k)(u−1)′d˜u\n(u−u(y′))2du\n−χ{|u(y′)−u(1)|≥2\n|k|}(u(y′))/integraldisplayu(1)\nu(y′)+2\n|k|(θ◦u−1)(u)(u−1)′(u)/integraltextu\nu(y′)ˆ˜ωin(u−1(˜u),k)(u−1)′d˜u\n(u−u(y′))2du\n=χ{|u(y′)−u(0)|≥2\n|k|}(u(y′))(θ◦u−1)(u)(u−1)′(u)/integraltextu\nu(y′)ˆ˜ωin(u−1(˜u),k)(u−1)′d˜u\n(u−u(y′))/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(y′)−2\n|k|\nu(0)NONLINEAR INVISCID DAMPING 25\n+χ{|u(y′)−u(1)|≥2\n|k|}(u(y′))(θ◦u−1)(u)(u−1)′(u)/integraltextu\nu(y′)ˆ˜ωin(u−1(˜u),k)(u−1)′d˜u\n(u−u(y′))/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(1)\nu(y′)+2\n|k|\n−/integraldisplay\n|u−u(z′)|≥2\n|k|/parenleftBig\n(θ◦u−1)(u)(u−1)′(u)/parenrightBig′/integraltextu\nu(y′)ˆ˜ωin(u−1(˜u),k)(u−1)′d˜u\nu−u(y′)du\n−/integraldisplay\n|u−u(z′)|≥2\n|k|(θ◦u−1)(u)((u−1)′(u))2ˆ˜ωin(u−1(u),k)\nu−u(y′)du.\nBy theL2estimate of the maximal Hilbert transform, and the maximum f unction, we have\n/ba∇dblI2/ba∇dblL2(0,1)≤C/ba∇dblˆ˜ωin/ba∇dblL2.\nCombining the above estimates, we prove the lemma. /square\nWe conclude the above calculations by the following proposi tion.\nProposition 3.7. It holds that lim\nǫ→0±ΨI(y,y′,ǫ,k) = ΨI\n±(y,y′,k). where\nΨI\n±(y,y′,k) =\n\n−φ(y,y′,k)/integraldisplayy\n0θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)φ1(z′′,y���,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n+ρ(y′)(Π1[ˆ˜ωin]±iπΠ2[ˆ˜ωin])\nJ1(y′,k)±iπJ2(y′)φ(y,y′,k)/integraldisplayy\n0θ(z′)\nφ(z′,y′,k)2dz′,fory<y′\n−φ(y,y′,k)/integraldisplayy\n1θ(z′)/integraltextz′\ny′ˆ˜ωin(z′′,k)φ1(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n+ρ(y′)(Π1[ˆ˜ωin]±iπΠ2[ˆ˜ωin])\nJ1(y′,k)±iπJ2(y′)φ(y,y′,k)/integraldisplayy\n1θ(z′)\nφ(z′,y′,k)2dz′fory′<y.\nWe have the following uniform estimate.\nProposition 3.8. For any fixed k∈Z\\ {0}, suppose that ˜Lkhas no eigenvalue or embedded\neigenvalue. Then for all c /∈Ranu, letΨIsolve(3.3)with boundary condition ΨI(0) = ΨI(1) = 0.\nThen it holds that\n/ba∇dblΨI/ba∇dblH1≤C/ba∇dblˆ˜ωin/ba∇dblH1, (3.18)\nwhereCis a constant independent of c.\nProof.It is easy to check that for c∈ {c: dist(c,Ranu)>M}withMlarge enough, it holds that\n/ba∇dblΨI/ba∇dblH1≤C/ba∇dblˆ˜ωin/ba∇dblL2.\nSo we only focus on the case c∈ {c: dist(c,Ranu)≤M}. We prove the estimate (3.18) by a\ncontradiction argument. Suppose (3.18) does not hold, name ly, there is ( cn,ΨI\nn,ˆ˜ωin,n) such that\n{cn}n≥1⊂ {c: 0<dist(c,Ranu)≤M}and asn→ ∞,/ba∇dblΨI\nn/ba∇dblH1= 1,/ba∇dblˆ˜ωn/ba∇dblH1→0 and\n(u−cn)∂y/parenleftbigg1\nθ∂yΨI\nn/parenrightbigg\n−(u−cn)k2\nθΨI\nn−/parenleftbigu′\nθ/parenrightbig′ΨI\nn=−ˆ˜ωin,n.\nThen there exists a subsequence also denoted by ( cn,ΨI\nn,ˆ˜ωin,n), so that\ncn→c∞,ΨI\nn⇀ΨI\n∞ˆ˜ωin,n→0 inH1.26 WEIREN ZHAO\nIfc∞/∈Ranu, then/ba∇dblΨl\nn/ba∇dblH3≤C(/ba∇dblˆ˜ωin,n/ba∇dblH1+/ba∇dblΨl\nn/ba∇dblH1), thus ΨI\nn→ΨI\n∞and\n(u−c∞)∂y/parenleftbigg1\nθ∂yΨI\n∞/parenrightbigg\n−(u−c∞)k2\nθΨI\n∞−/parenleftbigu′\nθ/parenrightbig′ΨI\n∞= 0.\nIt implies that c∞is an eigenvalue that leads to a contradiction.\nIfc∞=u(y∞)∈Ranu(without loss of generality, we assume that Im cn>0), then we have\n∂y/parenleftbigg1\nθ∂yΨI\nn/parenrightbigg\n=k2\nθΨI\nn+/parenleftbigu′\nθ/parenrightbig′ΨI\nn−ˆ˜ωin,n\nu−cn\n=k2\nθΨI\nn+∂y/parenleftBigg/parenleftbigu′\nθ/parenrightbig′ΨI\nn−ˆ˜ωin,n\nu′ln(u−cn)/parenrightBigg\n−∂y/parenleftBigg/parenleftbigu′\nθ/parenrightbig′ΨI\nn−ˆ˜ωin,n\nu′/parenrightBigg\nln(u−cn)\nwhich gives that for 1 <p<2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nθ∂yΨI\nn−/parenleftbigu′\nθ/parenrightbig′ΨI\nn−ˆ˜ωin,n\nu′ln(u−cn)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2∩˙W1,p≤C. (3.19)\nTherefore, we have for any δ0>0\n/ba∇dbl∂yΨI\nn/ba∇dblL2((y∞−δ0,y∞+δ0)∩(0,1))≤Cδ1\n3\n0\nHere the constant Cis independent of n. We also have\nΨI\nn→ΨI\n∞inH1(0,1)\\(y∞−δ0,y∞+δ0).\nThus we have\nΨI\nn→ΨI\n∞inH1(0,1).\nFor anyϕ∈H1\n0, we also have\n0 =−/integraldisplay1\n0∂yΦI\nn(y)∂yϕ(y)\nθ(y)dy−/integraldisplay1\n0ΦI\nn(y)ϕ(y)\nθ(y)dy−/integraldisplay1\n0/parenleftbigu′\nθ/parenrightbig′(y)ΦI\nnϕ\nu−cndy+/integraldisplay1\n0ˆ˜ωin,nϕ\nu−cndy\n=−/integraldisplay1\n0∂yΦI\nn(y)∂yϕ(y)\nθ(y)dy−/integraldisplay1\n0ΦI\nn(y)ϕ(y)\nθ(y)dy−/integraldisplay1\n0/parenleftbigu′\nθ/parenrightbig′(y)ΦI\nnϕ\nu′∂y(ln(u−cn))dy\n+/integraldisplay1\n0ˆ˜ωin,nϕ\nu′∂y(ln(u−cn))dy\n=−/integraldisplay1\n0∂yΦI\nn(y)∂yϕ(y)\nθ(y)dy−/integraldisplay1\n0ΦI\nn(y)ϕ(y)\nθ(y)dy+/integraldisplay1\n0∂y/parenleftbigg/parenleftbigu′\nθ/parenrightbig′(y)ΦI\nnϕ\nu′/parenrightbigg\nln(u−cn)dy\n−/parenleftbigu′\nθ/parenrightbig′(y)ΦI\nnϕ\nu′ln(u(y)−cn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n0+ˆ˜ωin,nϕ\nu′ln(u(y)−cn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n0\n−/integraldisplay1\n0∂y/parenleftBiggˆ˜ωin,nϕ\nu′/parenrightBigg\nln(u−cn)dy\n→ −/integraldisplay1\n0∂yΦI\n∞(y)∂yϕ(y)\nθ(y)dy−/integraldisplay1\n0ΦI\n∞(y)ϕ(y)\nθ(y)dy+/integraldisplay1\n0∂y/parenleftbigg/parenleftbigu′\nθ/parenrightbig′(y)ΦI\n∞ϕ\nu′/parenrightbigg\nln(u−cn)dy\n−iπ/parenleftbigu′\nθ/parenrightbig′(y∞)ΦI\n∞(y∞)ϕ(y∞)\nu′(y∞).\nBy takingϕ(y) =ΦI∞(y), we have/parenleftbigu′\nθ/parenrightbig′(y∞)|ΦI\n∞(y∞)|2= 0. Thus we have ΦI\n∞∈H2∩H1\n0andc∞\nis an embedded eigenvalue, which leads to a contradiction. /squareNONLINEAR INVISCID DAMPING 27\n3.4.Linear damping, representation formula, and the wave operator. In this section, we\nprove the weak linear inviscid damping and introduce the rep resentation formula and the wave\noperator which will be used in the nonlinear problem.\nLemma 3.9. Letψbe the velocity solving the linearized equation (2.21). Then it holds that\n/ba∇dbl(∂yˆψ,ˆψ)/ba∇dblH1\ntL2y≤C/ba∇dblˆ˜ωin/ba∇dblH1y,\nwhich implies lim\nt→∞/ba∇dbl(∂yˆψ,ˆψ)/ba∇dblL2y= 0.\nProof.We have by (3.2), Proposition 3.8 , and Proposition 3.7 that\nˆψ(t,k,y) =1\n2πi/integraldisplay1\n0e−iu(y′)kt(ΨI\n−(y,y′,k)−ΨI\n+(y,y′,k))u′(y′)dy′\n∂tˆψ(t,k,y) =1\n2πi/integraldisplay1\n0(−iu(y′)k)e−iu(y′)kt(ΨI\n−(y,y′,k)−ΨI\n+(y,y′,k))u′(y′)dy′\nand\n/ba∇dblΨI\n±(y,y′,k)/ba∇dblL∞\ny′H1y≤C/ba∇dblˆ˜ωin/ba∇dblH1x,y.\nTherefore by Plancherel’s formula, we infer that\n/ba∇dbl(∂yˆψ,ˆψ)/ba∇dbl2\nH1\ntL2y=/integraldisplay\nR/parenleftbig\n/ba∇dbl(∂yˆψ,ˆψ)/ba∇dbl2\nL2y+/ba∇dbl∂t(∂yˆψ,ˆψ)/ba∇dbl2\nL2y/parenrightbig\ndt\n/lessorsimilar/integraldisplay1\n0/integraldisplay\nR/parenleftbig\n|ˆψ(t,k,y)|2+|∂yˆψ(t,k,·)|2+k2|∂tˆψ(t,k,y)|2+|∂t∂yˆψ(t,k,y)|2/parenrightbig\ndtdy\n/lessorsimilar/integraldisplay1\n0/integraldisplay\nRanu/parenleftbig\nk2|ΨI\n±(y,y′,k)|2+|∂yΨI\n±(y,y′,k)|2/parenrightbig\ndcdy\n≤C/integraldisplay\nRanu/ba∇dblˆ˜ωin(k,·)/ba∇dbl2\nH1ydc/lessorsimilarC/ba∇dblˆ˜ωin(k,·)/ba∇dbl2\nH1y.\nThe lemma follows directly from the Sobolev embedding. /square\nIn order to obtain the point-wise linear inviscid damping, o ne may need to study the regularity\nof ΨI\n±(y,y′,k) iny′. We may not discuss it here.\nWe now introduce the representation formula.\nProposition 3.10. Letψbe the velocity solving the linearized equation (2.21). Then it holds that\nˆψ(t,k,y) =−/integraldisplay1\n0e−iu(y′)ktJ1(y′,k)Π2[ˆ˜ωin](y′)−J2(y′)Π1[ˆ˜ωin](y′,k)\nJ1(y′,k)2+π2J2(y′)2ρ(y′)u′(y′)e(y,y′,k)dy′\nwhere\ne(y,y′,k) =\n\nφ(y,y′,k)/integraldisplayy\n0θ(z)\nφ(z,y′,k)2dz0≤y<y′,\nφ(y,y′,k)/integraldisplayy\n1θ(z)\nφ(z,y′,k)2dz y′<y≤1.(3.20)\nProof.The proposition follows directly from (3.2), Proposition 3 .8, and Proposition 3.7 with a\nsimple calculation. /square28 WEIREN ZHAO\nThe representation formula in Proposition 3.10 leads to the wave operator. By taking t= 0, we\nhave the following representation formula:\n(3.21) ˆψin(k,y) =−/integraldisplay1\n0Du,k/bracketleftBig\n˜∆kˆψin/bracketrightBig\n(y′,k)θ(y′)\nu′(y′)ρ(y′)e(y,y′,k)\n/radicalbig\nJ1(y′,k)2+π2J2(y′)2dy′,\nwhere\n(3.22)Du,k[˜ω](y′,k)def=u′(y′)\nθ(y′)u′(y′)J1(y′,k)Π2[˜ω](y′)−u′(y′)J2(y′)Π1[˜ω](y′,k)/radicalbig\nJ1(y′,k)2+π2J2(y′)2\n=J1(y′,k)˜ω(y′)+ρ(y′)θ(y′)\nu(y′)/parenleftBig\nu′\nθ/parenrightBig′\n(y′)Π1[˜ω](y′,k)\n/radicalbig\nJ1(y′,k)2+π2J2(y′)2.\n4.Wave operator\nIn this section, we give the basic properties of the wave oper ator and study its Fourier kernel.\nProposition 4.1. Fork/\\e}atio\\slash= 0, letRu,k=uId−/parenleftBig\nu′\nθ/parenrightBig′˜∆−1\nk. Suppose that for all k/\\e}atio\\slash= 0,Ru,khas no\neigenvalue and no embedded eigenvalue. Let Du,kbe defined in (3.22). Then there exists D1\nu,ksuch\nthat\nDu,kRu,k=uDu,k, (4.1)\nand forω,g∈L2(0,1)\n/integraldisplay1\n0Du,k(ω)(y)D1\nu,k(g)(y)dy=/integraldisplay1\n0ω(y)g(y)dy. (4.2)\nMoreover, there exists C >1independent of ksuch that\nC−1≤ /ba∇dblDu,k/ba∇dblL2→L2≤C, C−1≤ /ba∇dblD1\nu,k/ba∇dblL2→L2≤C. (4.3)\nNote that (4.3) implies that the wave operator Du,kis invertible and\nD1\nu,k= (D−1\nu,k)∗. (4.4)\nWe give the representation formula of D−1\nu,kin (4.12).\nProof.Proof of (4.1).Letψ=˜∆−1\nkω, namely∂y/parenleftbig1\nθ∂yψ/parenrightbig\n−k2\nθψ=ωandψ(0) =ψ(1) = 0. Then\nRu,k[ω](y′) =u(y′)ω(y′)−/parenleftBig\nu′\nθ/parenrightBig′\nψ(y′). A direct calculation gives\nDu,k[Ru,k[ω]](y′,k)\n=u′(y′)\nθ(y′)u′(y′)J1(y′,k)Π2[Ru,k[ω]](y′)−u′(y′)J2(y′)Π1[Ru,k[ω]](y′,k)/radicalbig\nJ1(y′,k)2+π2J2(y′)2\n=u(y′)u′(y′)J1(y′,k)Π2[ω]/radicalbig\nJ1(y′,k)2+π2J2(y′)2u′(y′)\nθ(y′)−J1(y′,k)θ(y′)\nu′(y′)/parenleftBig\nu′\nθ/parenrightBig′\nψ(y′)\n/radicalbig\nJ1(y′,k)2+π2J2(y′)2u′(y′)\nθ(y′)\n−u′(y′)\nθ(y′)u′(y′)J2(y′)P.V./integraltext1\n0θ(z′)/integraltextz′\ny′/parenleftbigg\nu(z′′)ω(z′′)−/parenleftBig\nu′\nθ/parenrightBig′\nψ(z′′)/parenrightbigg\nφ1(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n/radicalbig\nJ1(y′,k)2+π2J2(y′)2.NONLINEAR INVISCID DAMPING 29\nLet us now simplify the second term. Recall that φ(y,y′,k) = (u(y)−u(y′))φ1(y,y′,k) solves\n∂y/parenleftbigg1\nθ∂yφ/parenrightbigg\n−k2\nθφ=/parenleftbiggu′\nθ/parenrightbigg′\nφ1.\nThen by using the integration by part, we have\nP.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′/parenleftbigg\nu(z′′)ω(z′′)−/parenleftBig\nu′\nθ/parenrightBig′\nψ(z′′)/parenrightbigg\nφ1(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n=P.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′u(z′′)ω(z′′)φ1(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n−P.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′ψ(z′′)/parenleftBig\n∂y/parenleftbig1\nθ∂yφ(z′′,y′,k)/parenrightbig\n−k2\nθφ(z′′,y′,k)/parenrightBig\ndz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n=P.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′u(z′′)ω(z′′)φ1(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′+P.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′ψ(z′′)k2\nθφ(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n+P.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′∂yψ(z′′)/parenleftbig1\nθ∂yφ(z′′,y′,k)/parenrightbig\ndz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n−P.V./integraldisplay1\n0(u(z′)−u(y′))ψ(z′)∂yφ1(z′,y′,k)+u′(z′)ψ(z′)φ1(z′,y′,k)−θ(z′)ψ(y′)/parenleftBig\nu′(y′)\nθ(y′)/parenrightBig\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n=P.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′u(z′′)ω(z′′)φ1(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′−P.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′ω(z′′)φ(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n+P.V./integraldisplay1\n0∂yψ(z′)\n(u(z′)−u(y′))φ1(z′,y′,k)dz′−/integraldisplay1\n0ψ(z′)∂yφ1(z′,y′,k)\n(u(z′)−u(y′))φ1(z′,y′,k)2dz′\n−P.V./integraldisplay1\n0u′(z′)ψ(z′)φ1(z′,y′,k)−θ(z′)ψ(y′)/parenleftBig\nu′(y′)\nθ(y′)/parenrightBig\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n=u(y′)Π1[ω]+I1[ψ]+I2[ψ]+I3[ψ].\nWe write\nI1=/integraldisplay1\n0∂yψ(z′)\n(u(z′)−u(y′))/parenleftBig1\nφ1(z′,y′,k)−1/parenrightBig\ndz′+P.V./integraldisplay1\n0∂yψ(z′)\n(u(z′)−u(y′))dz′\nI2=/integraldisplay1\n0ψ(z′)\n(u(z′)−u(y′))∂y/parenleftBig1\nφ1(z′,y′,k)−1/parenrightBig\ndz′\nI3=/integraldisplay1\n0ψ(z′)∂y/parenleftBig1\n(u(z′)−u(y′))/parenrightBig/parenleftBig1\nφ1(z′,y′,k)−1/parenrightBig\ndz′\n−P.V./integraldisplay1\n0u′(z′)ψ(z′)−θ(z′)ψ(y′)/parenleftBig\nu′(y′)\nθ(y′)/parenrightBig\n(u(z′)−u(y′))2dz′\n+ψ(y′)u′(y′)\nθ(y′)/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′.30 WEIREN ZHAO\nTherefore, we obtain that\nI1+I2+I3=P.V./integraldisplay1\n0∂yψ(z′)\n(u(z′)−u(y′))dz′(4.5)\n+P.V./integraldisplay1\n0∂y/parenleftBig1\nu(z′)−u(y′)/parenrightBig/parenleftBig\nψ(z′)−θ(z′)\nu′(z′)ψ(y′)/parenleftbiggu′(y′)\nθ(y′)/parenrightbigg/parenrightBig\ndz′\n+ψ(y′)u′(y′)\nθ(y′)/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′\n=ψ(y′)u′(y′)\nθ(y′)P.V./integraldisplay1\n01\nu(z′)−u(y′)/parenleftBigθ(z′)\nu′(z′)/parenrightBig′\ndz′\n−θ(1)\nu′(1)ψ(y′)/parenleftBig\nu′(y′)\nθ(y′)/parenrightBig\nu(1)−u(y′)+θ(0)\nu′(0)ψ(y′)/parenleftBig\nu′(y′)\nθ(y′)/parenrightBig\nu(0)−u(y′)\n+ψ(y′)u′(y′)\nθ(y′)/integraldisplay1\n0θ(z′)\n(u(z′)−u(y′))2/parenleftBig1\nφ1(z′,y′,k)2−1/parenrightBig\ndz′\n=ψ(y′)u′(y′)\nθ(y′)ρ(y′)J1(y′,k),\nwhich gives that\nP.V./integraldisplay1\n0θ(z′)/integraltextz′\ny′/parenleftbigg\nu(z′′)ω(z′′)−/parenleftBig\nu′\nθ/parenrightBig′\nψ(z′′)/parenrightbigg\nφ1(z′′,y′,k)dz′′\n(u(z′)−u(y′))2φ1(z′,y′,k)2dz′\n=u(y′)Π1[ω]+ψ(y′)u′(y′)\nθ(y′)ρ(y′)J1(y′,k).\nThen we can conclude that\nDu,k[Ru,k[ω]](y′,k)\n=u(y′)u′(y′)\nθ(y′)u′(y′)J1(y′,k)Π2[ω]/radicalbig\nJ1(y′,k)2+π2J2(y′)2−u(y′)u′(y′)\nθ(y′)u′(y′)J2(y′)Π1[ω]/radicalbig\nJ1(y′,k)2+π2J2(y′)2=u(y′)Du,k[ω].\nProof of (4.2).Letg∈H2(0,1)∩H1\n0(0,1) andψ=˜∆−1\nkω, namely∂y/parenleftbig1\nθ∂yψ/parenrightbig\n−k2\nθψ=ωand\nψ(0) =ψ(1) = 0. Then\n/integraldisplay1\n0ω(y)g(y)dy=/integraldisplay1\n0ψ(y)˜∆kg(y)dy.\nBy the representation formula (3.21), we have\n/integraldisplay1\n0ψ(y)˜∆k[g]dy=−/integraldisplay1\n0/integraldisplay1\n0Du,k[ω](y′,k)θ(y′)\nu′(y′)ρ(y′)e(y,y′,k)\n/radicalbig\nJ1(y′,k)2+π2J2(y′)2dy′˜∆kg(y)dy\n=−/integraldisplay1\n0/integraldisplayy\n0Du,k[ω](y′,k)θ(y′)\nu′(y′)ρ(y′)φ(y,y′,k)/integraltexty\n1θ(z)\nφ(z,y′,k)2dz\n/radicalbig\nJ1(y′,k)2+π2J2(y′)2dy′˜∆kg(y)dy\n−/integraldisplay1\n0/integraldisplay1\nyDu,k[ω](y′,k)θ(y′)\nu′(y′)ρ(y′)φ(y,y′,k)/integraltexty\n0θ(z)\nφ(z,y′,k)2dz\n/radicalbig\nJ1(y′,k)2+π2J2(y′)2dy′˜∆kg(y)dyNONLINEAR INVISCID DAMPING 31\n=/integraldisplay1\n0Du,k[ω](y′,k)ρ(y′)/radicalbig\nJ1(y′,k)2+π2J2(y′)2θ(y′)\nu′(y′)/integraldisplay1\ny′θ(z)/integraltextz\ny′φ(y,y′,k)˜∆kg(y)dy\nφ(z,y′,k)2dzdy′\n−/integraldisplay1\n0Du,k[ω](y′,k)ρ(y′)/radicalbig\nJ1(y′,k)2+π2J2(y′)2θ(y′)\nu′(y′)/integraldisplayy′\n0θ(z)/integraltexty′\nzφ(y,y′,k)˜∆kg(y)dy\nφ(z,y′,k)2dzdy′\n=/integraldisplay1\n0Du,k[ω](y′,k)ρ(y′)/radicalbig\nJ1(y′,k)2+π2J2(y′)2θ(y′)\nu′(y��)/integraldisplay1\n0θ(z)/integraltextz\ny′φ(y,y′,k)˜∆kg(y)dy\nφ(z,y′,k)2dzdy′.\nWe now simplify the integral by using integration by parts\n/integraldisplay1\n0θ(z)/integraltextz\ny′φ(z′′,y′,k)˜∆kg(z′′)dy\nφ(z,y′,k)2dz\n=/integraldisplay1\n0θ(z)/integraltextz\ny′˜∆kφ(z′′,y′,k)g(z′′)dy\nφ(z,y′,k)2dz+P.V./integraldisplay1\n0∂yg(z)\n(u(z)−u(y′))φ1(z,y′,k)dz\n−/integraldisplay1\n0∂yφ1(z,y′,k)g(z)\n(u(z)−u(y′))φ1(z,y′,k)2dz−/integraldisplay1\n0u′(z)φ1(z,y′,k)g(z)−θ(z)1\nθ(y′)u′(y′)g(y′)\nφ(z,y′,k)2dz\n=P.V./integraldisplay1\n0θ(z)/integraltextz\ny′/parenleftBig\nu′\nθ/parenrightBig′\nφ1(z′′,y′,k)g(z′′)dy\nφ(z,y′,k)2dz+I1[g]+I2[g]+I3[g],\nwhereI1,I2,I3are defined in the previous step. Thus by (4.5), we get\n/integraldisplay1\n0θ(z)/integraltextz\ny′φ(z′′,y′,k)˜∆kg(z′′)dy\nφ(z,y′,k)2dz= Π1/bracketleftbigg/parenleftbiggu′\nθ/parenrightbigg′\ng/bracketrightbigg\n+u′(y′)\nθ(y′)ρ(y′)J1(y′,k)g(y′).\nWe define\nD1\nu,k[g] =θ(y′)\nu′(y′)ρ(y′)Π1/bracketleftbigg/parenleftBig\nu′\nθ/parenrightBig′\ng/bracketrightbigg\n+J1(y′,k)g(y′)\n/radicalbig\nJ1(y′,k)2+π2J2(y′)2, (4.6)\nand then we obtain (4.2) for g∈H2(0,1)∩H1\n0(0,1).\nBy Lemma 3.5 and Lemma 3.6, it is easy to obtain for any ω∈L2(0,1) andk/\\e}atio\\slash= 0,\n/ba∇dblDu,k[ω]/ba∇dblL2+/ba∇dblD1\nu,k[ω]/ba∇dblL2≤C/ba∇dblω/ba∇dblL2. (4.7)\nHereC >0 is a constant independent of k.\nDue to the fact that H2∩H1\n0is dense in L2, for anyg∈L2, there is a sequence {gn}n≥1such\nthatH2∩H1\n0∋gn→ginL2and\n/integraldisplay1\n0Du,k(ω)(y′)D1\nu,k(gn)(y′)dy′=/integraldisplay1\n0ω(y)gn(y)dy.\nThen\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0Du,k(ω)(yc)D1\nu,k(g)(y′)dy′−/integraldisplay1\n0ω(k,y)g(k,y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0Du,k(ω)(y′)D1\nu,k(gn)(y′)dy′−/integraldisplay1\n0ω(k,y)(g(y)−gn(y))dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C/ba∇dblDu,k(ω)/ba∇dblL2/ba∇dblD1\nu,k(g−gn)/ba∇dblL2+/ba∇dblω/ba∇dblL2/ba∇dblg−gn/ba∇dblL2→0.32 WEIREN ZHAO\nThus we proved (4.2). The lower bound of Du,kandD1\nu,kin (4.3) follow directly from a duality\nargument, and the upper bound (4.7). Thus we proved the propo sition. /square\nRemark 4.2. For any bounded function C−1< p(y′)< C, the operators p(y′)Du,kand1\np(y′)D1\nu,k\nalso satisfy the same properties in Proposition (4.1). The wave operator Du,kis normalized so that\nif/parenleftBig\nu′\nθ/parenrightBig′\n= 0, thenDu,k=D1\nu,k= Id.\nMoreover, it is easy to check that\nDu,k[ω](y′)−D1\nu,k[ω](y′) =θ(y′)\nu′(y′)ρ(y′)\n/radicalbig\nJ1(y′,k)2+π2J2(y′)2/bracketleftbigg/parenleftbiggu′\nθ/parenrightbigg′\n,Π1/bracketrightbigg\n[ω](y′). (4.8)\nwhich is a commutator.\n4.1.Wave operator in (t,z,v)coordinates. Recall that we construct the wave operator in\n(t,˜x,˜y) coordinates (we drop tildes and still use ( t,x,y) in the above calculations for convenience),\nwhich is from the nonlinear change of coordinates and then th e linear inverse change of coordinates.\nWe show the relationship between different coordinate system s in the following map.\n(t,x,y)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\noriginal systemnonlinear change of coordinates/ma√sto− −−−−−−−−−−−−−−−−−− → (t,z,v)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nworking objectinverse linear change of coordinate− −−−−−−−−−−−−−−−−−−−− ⇀ ↽−−−−−−−−−−−−−−−−−−−− −\nlinear change of coordinates(t,˜x,˜y)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nwave operator.\nIn this section, we give the representation formula of the wa ve operator in ( t,z,v) coordinates.\nLet us first rewrite Du,kandD1\nu,kin different forms. We first define\nb1(y′,k) =J1(y′,k)/radicalbig\nJ1(y′,k)2+π2J2(y′)2(4.9)\nb2(y′,k) =−ρ(y′)θ(y′)\nu′(y′)/radicalbig\nJ1(y′,k)2+π2J2(y′)2. (4.10)\nRecall that ϕ1(y) =/parenleftBig\nu′\nθ/parenrightBig′\n(y) defined in (2.12) and e(z,y′,k) defined in (3.20). A direct calculation\ngives that\nΠ1[ω](y′) =−P.V./integraldisplay1\n0e(z,y′,k)\nu(y′)−u(y)ω(z)dz\nwhich gives\n(4.11)Du,k(ω)(k,y) =b1(k,y)ω(y)+ϕ1(y)b2(k,y)p.v./integraldisplay1\n0e(k,y′,y)\nu(y′)−u(y)ω(y′)dy′,\nD1\nu,k(ω)(k,y) =b1(k,y)ω(y)+b2(k,y)/integraldisplay1\n0e(k,y′,y)ϕ1(y′)\nu(y′)−u(y)ω(y′)dy′.\nBy the (4.2), it is easy to obtain that the inverse of Du,kexists and has the following formula\nD−1\nu,k(ω)(k,y) =b1(k,y)ω(y)+ϕ1(y)/integraldisplay1\n0e(k,y,y′)\nu(y)−u(y′)b2(k,y′)ω(y′)dy′. (4.12)\nWe also define B1(v,k),B2(v,k) andE(v,v′,k) so that\nB1(u(y),k) =b1(y���,k), B2(u(y),k) =b2(y′,k), E(u(y),u(y′),k) =e(y,y′,k). (4.13)NONLINEAR INVISCID DAMPING 33\nLet us now introduce the following operators associated wit hDu,k,D−1\nu,k, andD1\nu,k. For any Schwartz\nfunctionf(t,z,v) defined on R×T×Ranu, we define\nDu,k/parenleftbig\nF1f(t,k,·)/parenrightbig\n(t,k,v) (4.14)\ndef=B1(v,k)F1f(t,k,v)\n+/tildewiderϕ1(v)B2(v,k)/integraldisplayu(1)\nu(0)E(k,v1,v)F1f(t,k,v1)e−i(v1−v)tk\nv1−v(u−1)′(v1)dv1,\nD−1\nu,k/parenleftbig\nF1f(t,k,·)/parenrightbig\n(t,k,v) (4.15)\ndef=B1(v,k)F1f(t,k,v)\n+/tildewiderϕ1(v)/integraldisplayu(1)\nu(0)E(k,v,v1)B2(k,v1)F1f(t,k,v1)ei(v1−v)tk\nv−v1(u−1)′(v1)dv1.\nand\nD1\nu,k/parenleftbig\nF1f(t,k,·)/parenrightbig\n(t,k,v) (4.16)\ndef=B1(v,k)F1f(t,k,v)\n+B2(v,k)/integraldisplayu(1)\nu(0)E(k,v1,v)/tildewiderϕ1(v1)F1f(t,k,v1)e−i(v1−v)tk\nv1−v(u−1)′(v)dv1.\nWith these definitions, we have the following proposition.\nProposition 4.3. Fork/\\e}atio\\slash= 0, let˜∆l\nt,k=−k2/tildewiderϕ4+/tildewideu′(∂v−itk)/parenleftBig\n/tildewiderϕ4/tildewideu′(∂v−itk)/parenrightBig\nand\nRu,k=vId−/tildewiderϕ1(v)/parenleftbig˜∆l\nt,k/parenrightbig−1\nbe a modified Rayleigh operator. Then it holds for any f(t,z,v)andg(t,z,v)that\nDu,kRu,k=vDu,k, (4.17)\nDu,kD−1\nu,k=D−1\nu,kDu,k= Id,D−1\nu,k= (D1\nu,k)∗, (4.18)\n[∂t,Du,k]/parenleftbig\nF1f(t,k,·)/parenrightbig\n(t,k,v) = [ikv,Du,k]/parenleftbig\nF1f(t,k,·)/parenrightbig\n(t,k,v), (4.19)/integraldisplay\nDu,k/parenleftbig\nF1f(t,k,·)/parenrightbig\n(t,k,v)D1\nu,k/parenleftbig\nF1g(t,k,·)/parenrightbig\n(t,k,v)dv=/integraldisplay\nF1f(k,u)F1g(k,v)dv, (4.20)\nC−1≤ /ba∇dblDu,k/ba∇dblL2→L2≤C, C−1≤ /ba∇dblD1\nu,k/ba∇dblL2→L2≤C, C−1≤ /ba∇dblD−1\nu,k/ba∇dblL2→L2≤C. (4.21)\nLetΩ(t,z,v)be defined in (2.14). Then applying the inverse linear change of coordinate (2.20), we\nget that\nDu,k/parenleftbig\nF1Ω(t,k,·)/parenrightbig\n(t,k,v)eikz/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=x−u(y)t, v=u(y)=Du,k/parenleftbig\nF1Ω(t,k,·)/parenrightbig\n(t,k,u(y))eik(x−tu(y))\n=Du,k/parenleftbig\nF1w(t,k,·)/parenrightbig\n(t,k,y)eikx,\nwherew(t,x,y) = Ω/parenleftbig\nt,x−u(y)t,u(y)/parenrightbig\nis different from ˜ω(t,x,y)in the original coordinate system.\nRemark 4.4. We have changed D1\nu,kin the(z,v)coordinate slightly to ensure that (4.20)holds.\nLetΩ(t,z,v)be defined in (2.14). It holds that\nD1\nu,k/parenleftbig\nF1Ω(t,k,·)/parenrightbig\n(t,k,v)eikz/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=x−u(y)t, v=u(y)=1\nu′(y)D1\nu,k/parenleftbig\nu′F1ω(t,k,·)/parenrightbig\n(t,k,y)eikx,34 WEIREN ZHAO\nwherew(t,x,y) = Ω/parenleftbig\nt,x−u(y)t,u(y)/parenrightbig\n.\nThe proposition follows directly from Proposition 4.1 and t he representation formulas (4.11) and\n(4.12) by a simple change of coordinates.\nRemark 4.5. By the compact support property of b2. It is easy to check Du,k(F1Ω)has the same\ncompact support as Ω.\n4.2.Gevrey regularity of the homogeneous solution. To obtain the Fourier kernels of Du,k,\nD−1\nu,k, andD1\nu,k, we need to study the regularity of the coefficients B1(v,k),B2(v,k) and the kernel\nE(v,v′,k). Indeed, all these functions are related to the homogeneou s solutionφ1(y,y′,k) obtained\nin Proposition 3.2.\nLet us define Φ( v,v′,k) and Φ 1(v,v′,k) to be such that\nΦ(u(y),u(y′),k) =φ(y,y′,k),Φ1(u(y),u(y′),k) =φ1(y,y′,k). (4.22)\nBy Proposition 3.2 and (3.5), we have\nΦ(v,v′,k) = (v−v′)Φ1(v,v′,k),\nand\n(4.23)Φ1(v,v′,k) = 1+/integraldisplayv\nv′k2˜θ(z′)(u−1)′(z′)\n(z′−v′)2/integraldisplayz′\nv′(z′′−v′)2\n˜θ(z′′)(u−1)′(z′′)Φ1(z′′,v′,k)dz′′dz′\ndef= 1+k2(T0◦T2,2)[Φ1],\nwhere˜θ(v) = (θ◦u−1)(v), and forj1≤j2+1,j2= 0,1,2,...\nT0[f](v,v′,k) =/integraldisplayv\nv′f(v1,v′,k)dv1\nTj1,j2[Φ1](z′,v′,k) =˜θ(z′)(u−1)′(z′)\n(z′−v′)2/integraldisplayz′\nv′(z′′−v′)2\n˜θ(z′′)(u−1)′(z′′)Φ1(z′′,v′,k)dz′′.\nFor anyk/\\e}atio\\slash= 0, letM≥1 be a large constant, then we define the following weighted Ge vrey spaces:\nGM|k|,s\nph,cosh([a,b]2)def={f∈C∞([a,b]2) :/ba∇dblf/ba∇dblGM|k|,s\nph,cosh([a,b]2)<∞}, (4.24)\nwith\n(4.25) /ba∇dblf/ba∇dblGM|k|,s\nph,cosh([a,b]2)def= sup\n(v,v′)∈[a,b]2,m≥m1≥0|∂m−m1v(∂v′+∂v)m1f(v,v′)|\nΓs(m)Mm|k|mcoshM|k|(v−v′),\nwhere Γs(m) = (m!)1\ns(m+1)−2.\nLemma 4.6. It holds that\nsup\n(v,v′)∈[a,b]2,m≥m1≥0|∂m−m1v∂m1\nv′f(v,v′)|\nΓs(m)2m1Mm|k|mcoshM|k|(v−v′)≤ /ba∇dblf/ba∇dblGM|k|,s\nph,cosh([a,b]2).\nProof.We have\nsup\n(v,v′)∈[a,b]2,m≥m1≥0|∂m−m1v∂m1\nv′f(v,v′)|\nΓs(m)2m1Mm|k|mcoshM|k|(v−v′)\n≤sup\nm1≥02−m1m1/summationdisplay\nm2=0m1!\nm2!(m1−m2)!sup\n(v,v′)∈[a,b]2,m≥m2≥0|∂m−m2v(∂v′+∂v)m2f(v,v′)|\nΓs(m)Mm|k|mcoshM|k|(v−v′)NONLINEAR INVISCID DAMPING 35\n≤sup\nm1≥02−m1m1/summationdisplay\nm2=0m1!\nm2!(m1−m2)!/ba∇dblf/ba∇dblGM|k|,s\nph,cosh([a,b]2)≤ /ba∇dblf/ba∇dblGM|k|,s\nph,cosh([a,b]2).\nThus we proved the remark. /square\nRemark 4.7. It holds that\n/ba∇dblfg/ba∇dblGM|k|,s\nph,cosh([a,b]2)≤C/ba∇dblf/ba∇dblGM|k|,s\nph,cosh([a,b]2)/ba∇dblg/ba∇dblGM|k|,s\nph,1([a,b]2).\nwhere the norm /ba∇dbl·/ba∇dblGM|k|,s\nph,1([a,b]2)is defined in (A.7).\nProposition 4.8. Suppose (u−1)′′,˜θ∈ GKu,s0\nph,1([u(0),u(1)]), then there exists M0=M0(Ku)≥1,\nsuch that for any M≥M0, it holds that\n/ba∇dblT0◦T2,2/ba∇dblGM|k|,s0\nph,cosh([u(0),u(1)]2)→GM|k|,s0\nph,cosh([u(0),u(1)]2)≤C\nM2k2.\nProof.In this proof, we use the following notations for convenienc e. LetL1(v) =˜θ(v)(u−1)′(v) and\nL2(v) =(u−1)′(v)\n˜θ(v), then\n/ba∇dblL1/ba∇dblGM|k|,s\nph,1([u(0),u(1)]2)+/ba∇dblL2/ba∇dblGM|k|,s\nph,1([u(0),u(1)]2)≤C.\nIt is easy to check that\n(∂v+∂v′)mT0◦T2,2f\n=/summationdisplay\nk1+k2≤mm!\n(m−k1−k2)!k1!k2!\n×/integraldisplayv\nv′L(k1+1)\n1(u1)\n(u1−v′)2/integraldisplayu1\nv′(u2−v′)2L(k2+1)\n2(u2)(∂u2+∂v′)m−k1−k2f(k,u2,v′)du2du1,\nwhich gives us that/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)mT0◦T2,2f\nΓs0(m)Mm|k|mcoshM|k|(v−v′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/summationdisplay\nk1+k2≤mm!\n(m−k1−k2)!k1!k2!Γs0(m−k1−k2)Γs0(k1)Γs0(k2)\n×/integraldisplayv\nv′1\n(u1−v′)2/integraldisplayu1\nv′(u2−v′)2coshM|k|(u2−v′)du2du1\n×sup\n0≤m1≤m/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)m1f\nΓs0(m1)Mm1|k|m1coshM|k|(v−v′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/ba∇dblL1/ba∇dblGM|k|,s0\nph,1([u(0),u(1)])/ba∇dblL2/ba∇dblGM|k|,s0\nph,1([u(0),u(1)])\n≤C\nM2k2sup\n0≤m1≤m/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)m1f\nΓs0(m1)Mm1|k|m1coshM|k|(v−v′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nwhich implies\nsup\nm≥0/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)mT0◦T2,2f\nΓs0(m)Mm|k|mcoshM|k|(v−v′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C\nM2k2sup\nm≥0/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)mf\nΓs0(m)Mm|k|mcoshM|k|(v−v′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nWe also have\n∂vT0◦T2,2f(v,v′) = T2,2f(v,v′)\n∂2\nvT0◦T2,2f(v,v′) = (v−v′)L′\n1(v)T3,2f−2T3,2f+L1(v)L2(v)f(v,v′)36 WEIREN ZHAO\nClaim: It holds for any s∈(0,1) andj≥0 that\n(4.26)sup\nm1,m≥0/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂m1v(∂v+∂v′)m−m1/parenleftBig\n1\n(v−v′)j+1/integraltextv\nv′(u′−v′)jf(u′,v′)du′/parenrightBig\nΓs(m)Mm|k|mcoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([u(0),u(1)]2)\n≤Csup\nm1,m≥0/vextenddouble/vextenddouble/vextenddouble/vextenddouble(∂v′+∂v)m−m1∂m1vf(v,v′)\nΓs(m)Mm|k|mcoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([u(0),u(1)]2).\nLet us admit the claim and finish the proof of the proposition fi rst. Indeed, it is easy to check that\n2/summationdisplay\nm=0m/summationdisplay\nm1=0/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂m1v∂m−m1\nv′T0◦T2,2f\n(M|k|)mcoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([u(0),u(1)]2)≤C\nM2k22/summationdisplay\nm=0/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂m\nv′f\ncoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([u(0),u(1)]2).\nForm1≥2, by using the claim, we have that/vextenddouble/vextenddouble/vextenddouble/vextenddouble(∂v′+∂v)m−m1∂m1vT0◦T2,2f\nΓs0(m)(M|k|)mcoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([u(0),u(1)]2)\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble(∂v′+∂v)m−m1∂m1−2\nv/parenleftBig\n(v−v′)L′\n1(v)T3,2f/parenrightBig\nΓs0(m)(M|k|)mcoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([u(0),u(1)]2)\n+2/vextenddouble/vextenddouble/vextenddouble/vextenddouble(∂v′+∂v)m−m1∂m1−2\nvT3,2f\nΓs0(m)(M|k|)mcoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([u(0),u(1)]2)\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble(∂v′+∂v)m−m1∂m1−2\nvL1(v)L2(v)f(v,v′))\nΓs0(m)(M|k|)mcoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([u(0),u(1)]2)\n≤C\nM2k2sup\nm2≤m1−2,m3≤m−m1/vextenddouble/vextenddouble/vextenddouble/vextenddouble(∂v′+∂v)m3∂m2vf(v,v′)\nΓs0(m2+m3)(M|k|)m2+m3coshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞([u(0),u(1)]2).\nFinally, let us prove the claim. Indeed we have\n∂m1v(∂v+∂v′)m−m1/parenleftbigg1\n(v−v′)j+1/integraldisplayu\nv′(u′−v′)jf(u′,v′)du′/parenrightbigg\n=∂m1v(∂v+∂v′)m−m1/parenleftbigg/integraldisplay1\n0tjf(v′+(v−v′)t,v′)dt/parenrightbigg\n=/integraldisplay1\n0tjtm1/parenleftbig\n∂m1v(∂v+∂v′)m−m1f/parenrightbig\n(v′+(v−v′)t,v′)dt,\nwhich gives that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂m1v(∂v+∂v′)m−m1/parenleftBig\n1\n(v−v′)j+1/integraltextu\nv′(u′−v′)jf(u′,v′)du′/parenrightBig\ncoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞[u(0),u(1)]2\n≤/integraltext1\n0tjtm1coshtM|k|(v−v′)dt\ncoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂m1v(∂v+∂v′)m−m1f\ncoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞[u(0),u(1)]2\n≤C/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂m1v(∂v+∂v′)m−m1f\ncoshM|k|(v−v′)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞[u(0),u(1)]2.\nThus we proved the proposition. /square\nBy (4.23), Proposition 4.8 and Lemma 4.6, we have the followi ng corollary:NONLINEAR INVISCID DAMPING 37\nCorollary 4.9. Suppose that u,θare the same as in Proposition 4.8 and Φ1satisfies (4.23), then\nthere existM0andC(k)>0such that for any |k| /\\e}atio\\slash= 0, for all integers m≥m1≥0andM≥M0,\n/vextendsingle/vextendsingle∂m1v∂m−m1\nv′Φ1(v,v′,k)/vextendsingle/vextendsingle≤C(k)Γs0(m)(2M)m|k|m.\nMoreover there exists C≥1such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m1v∂m−m1\nv′/parenleftBigΦ1(v,v′,k)−1\n(v−v′)2/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(k)Γs0(m)(CM)m|k|m.\nProof.Thesecondinequalityfollowsdirectlyfromthefirstinequa lityandthefactthatΦ 1(v,v′,k)/vextendsingle/vextendsingle\nv=v′=\n1 and∂v′Φ1(v,v′,k)/vextendsingle/vextendsingle\nv=v′=∂vΦ1(v,v′,k) = 0/vextendsingle/vextendsingle\nv=v′. So we only need to prove the first inequality.\nBy takingM0large enough, it follows from (4.23) and Proposition 4.8 tha t\n/ba∇dblΦ1/ba∇dblGM|k|,s0\nph,cosh([u(0),u(1)]2)≤ /ba∇dblk2T0◦T2,2Φ1/ba∇dblGM|k|,s0\nph,cosh([u(0),u(1)]2)+C\n≤C\nM2/ba∇dblΦ1/ba∇dblGM|k|,s0\nph,cosh([u(0),u(1)]2)+C,\nwhich gives us that\n/ba∇dblΦ1/ba∇dblGM|k|,s0\nph,cosh([u(0),u(1)]2)≤C.\nThen the corollary follows from Lemma 4.6. /square\n4.3.Gevrey regularity of the coefficients and Fourier kernels. In order to estimate the\nFourier kernels of Du,k,D−1\nu,k, andD1\nu,k, we need write the kernel E(v,v′,k) in a good form. Recall\nthat in (y,y′) coordinates we have\ne(y,y′,k) =\n\nφ(y,y′,k)/integraldisplayy\n0θ(z)\nφ(z,y′,k)2dz0≤y<y′,\nφ(y,y′,k)/integraldisplayy\n1θ(z)\nφ(z,y′,k)2dz y′<y≤1.\nA direct calculation gives us that for j= 0,1\nφ(y,y′,k)/integraldisplayy\njθ(w)\nφ(y,w,k)2dw\n=φ(y,y′,k)/integraldisplayy\njθ(w)\n(u(w)−u(y′))2/parenleftBig1\nφ1(w,y′,k)2−1/parenrightBig\ndy′\n+φ(y,y′,k)/integraldisplayy\njθ(w)\n(u(w)−u(y′))2dw\n=φ(y,y′,k)/integraldisplayy\njθ(z)\n(u(w)−u(y′))2/parenleftBig1\nφ1(w,y′,k)2−1/parenrightBig\ndw\n+φ(y,y′,k)/integraldisplayu(y)\nu(j)/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u1)−/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u(y′))−/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(u(y′))(u1−u(y′))\n(u1−u(y′))2du1\n−/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u(y′))φ(y,y′,k)\nu(w)−u(y′)/vextendsingle/vextendsingle/vextendsingle/vextendsinglew=y\nw=j+φ(y,y′,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(u(y′))ln|u(w)−u(y′)|/vextendsingle/vextendsingle/vextendsingle/vextendsinglew=y\nw=j.38 WEIREN ZHAO\nThus we obtain\n(4.27)φ(y,y′,k)/integraldisplayy\njθ(w)\nφ(y,w,k)2dw\n=−/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u(y′))+φ(y,y′,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(u(y′))ln|u(y)−u(y′)|+φre\nj(y,y′,k),\nwhereφre\njis the regular part\nφre\nj(y,y′,k)\n=φ(y,y′,k)/integraldisplayy\njθ(w)\n(u(w)−u(y′))2/parenleftBig1\nφ1(w,y′,k)2−1/parenrightBig\ndw\n+φ(y,y′,k)/integraldisplayu(y)\nu(j)/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u1)−/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u(y′))−/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(u(y′))(u1−u(y′))\n(u1−u(y′))2du1\n+/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u(y′))(1−φ1(y,y′,k))+/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u(y′))u(y)−u(y′)\nu(j)−u(y′)φ1(y,y′,k)\n−φ(y,y′,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(u(y′))ln|u(j)−u(y′)|.\nNote that the integral φ(y,y′,k)/integraltexty\njθ(w)\nφ(w,y′,k)2dwis well defined for y <y′ifj= 0 and for y >y′if\nj= 1. However, the right-hand side of (4.27) can be defined for y/\\e}atio\\slash=y′.\nTherefore, we have\n(4.28)e(y,y′,k) =−/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u(y′))+φ(y,y′,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(u(y′))ln|u(y)−u(y′)|\n+φre\n0(y,y′,k)(1R−(y−y′))+φre\n1(y,y′,k)(1R+(y−y′))\nand thus\n(4.29)E(v,v′,k) =−/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v′)+Φ(v,v′,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v′)ln|v−v′|\n+Φre\n0(v,v′,k)(1R−(v−v′))+Φre\n1(v,v′,k)(1R+(v−v′))\nwhere forj= 0,1, Φre\nj(v,v′,k) satisfies Φre\nj(u(y),u(y′),k) =φre\nj(y,y′,k), namely,\nΦre\nj(v,v′,k)\n= Φ(v,v′,k)/integraldisplayv\nu(j)˜θ(w)u−1(w)\n(w−v′)2/parenleftBig1\nΦ1(w,v′,k)2−1/parenrightBig\ndw\n+Φ(v,v′,k)/integraldisplayv\nu(j)/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u1)−/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v′)−/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v′)(u1−v′)\n(u1−v′)2du1\n+/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v′)(1−Φ1(v,v′,k))+/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v′)v−v′\nu(j)−v′Φ1(v,v′,k)\n−Φ(v,v′,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v′)ln|u(j)−v′|.NONLINEAR INVISCID DAMPING 39\nWe define\n(4.30)Φre\nj,1(v,v′,k) =Φre\nj(v,v′,k)\nv−v′\n= Φ1(v,v′,k)/integraldisplayv\nu(j)˜θ(w)u−1(w)\n(w−v′)2/parenleftBig1\nΦ1(w,v′,k)2−1/parenrightBig\ndw\n+Φ1(v,v′,k)/integraldisplayv\nu(j)/parenleftBig\n˜θ(u−1)′/parenrightBig\n(u1)−/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v′)−/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v′)(u1−v′)\n(u1−v′)2du1\n+/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v′)(1−Φ1(v,v′,k))\nv−v′+/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v′)1\nu(j)−v′Φ1(v,v′,k)\n−Φ1(v,v′,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v′)ln|u(j)−v′|.\nWe have the following proposition:\nProposition 4.10. It holds that\n|∂m1v∂m−m1\nv′B1(v,k)|+|∂m1v∂m−m1\nv′B2(v,k)| ≤C(k)Γs(m)(M)m.\nRecall that ˜χ2=χ2◦u−1is a smooth function with compact support such that suppχ2⊂[κ0\n2,1−κ0\n2]\nand satisfies (1.10).\n|∂m1v∂m−m1\nv′˜χ2(v)˜χ2(v′)Φre\nj(v,v′k)| ≤C(k)Γs(m)(M)m,forj= 0,1,\nand\n|∂m1v∂m−m1\nv′˜χ2(v)˜χ2(v′)Φre\nj,1(v,v′k)| ≤C(k)Γs(m)(M)m,forj= 0,1.\nProof.First, we write the numerator J1(y′,k) ofb1(y′,k) inv′coordinate\nJ1(v′,k)def=−(u(1)−v′)θ(0)\nu′(0)−(v′−u(0))θ(1)\nu′(1)\n+ ˜ρ(v′)/integraldisplay1\n0˜θ(w)(u−1)′(w)\n(w−v′)2/parenleftBig1\nΦ1(w,v′,k)2−1/parenrightBig\ndw\n−˜ρ(v′)H/bracketleftbig/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′χ[u(0),u(1)]/bracketrightbig\n(v′)\nwhere ˜ρ(v′) = (v′−u(0))(u(1)−v′). Notice that/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′has compact support. Thus by\nRemark A.2, we have\n/vextendsingle/vextendsingleFv→ξ[(θ◦u−1)(u−1)′](ξ)/vextendsingle/vextendsingle≤Ce−λu,θ/a\\}bracketle{tξ/a\\}bracketri}hts\nwhich implies/vextendsingle/vextendsingle/vextendsingleFv′→ξ/bracketleftbig\nH/bracketleftbig/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′χ[u(0),u(1)]/bracketrightbig\n(v′)/bracketrightbig/vextendsingle/vextendsingle/vextendsingle≤Ce−λu,θ/a\\}bracketle{tξ/a\\}bracketri}hts.\nThus by Lemma A.1, we have/vextendsingle/vextendsingle/vextendsingle∂m\nv′/bracketleftbig\nH/bracketleftbig/parenleftbig\n(θ◦u−1)(u−1)′/parenrightbig′χ[u(0),u(1)]/bracketrightbig\n(v′)/vextendsingle/vextendsingle/vextendsingle≤CΓs(m)(M)m.\nThe estimates of B1,B2follow directly from (3.9), Corollary 4.9, Lemma 3.5, and (A .9).\nNow we prove the regularity of Φre\njand Φre\nj,1. Notice that with the cut-off function, the ‘bad’\nterms ˜χ2(v)˜χ2(v′)/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v′)v−v′\nu(j)−v′Φ1(v,v′,k) is regular. For the term that contains logarithmic\nfunction Φ( v,v′,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v′)ln|u(j)−v′|, we notice that/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v′) has compact support.\nThus by using (3.9), Corollary 4.9, Lemma 3.5, and (A.9), we p rove the proposition. /square40 WEIREN ZHAO\nAlthough the wave operator is nonlocal both in physical spac e and frequency space, the next\nproposition shows that the wave operator does not move frequ encies a lot. Now we introduce the\nFourier kernels and the estimates.\nProposition 4.11. Recall that ˜χ2=χ2◦u−1is a smooth function with compact support such that\nsuppχ2⊂[κ0\n2,1−κ0\n2]and satisfies (1.10). Then for any 0<|k| ≤kM, there exist D(t,k,ξ1,ξ2),\nD1(t,k,ξ1,ξ2)andD−1(t,k,ξ1,ξ2)such that\nF2/parenleftBig\nDu,k/parenleftbig\n˜χ2F1f(t,k,·)/parenrightbig/parenrightBig\n(t,k,ξ1) =/integraldisplay\nD(t,k,ξ1,ξ2)ˆfk(t,ξ2)dξ2,\nand\nF2/parenleftBig\n˜χ2D1\nu,k/parenleftbig\n˜χ2F1f(t,k,·)/parenrightbig/parenrightBig\n(t,k,ξ1) =/integraldisplay\nD1(t,k,ξ1,ξ2)ˆfk(t,ξ2)dξ2,\nand\nF2/parenleftBig\nD−1\nu,k/parenleftbig\n˜χ2F1f(t,k,·)/parenrightbig/parenrightBig\n(t,k,ξ1) =/integraldisplay\nD−1(t,k,ξ1,ξ2)ˆfk(t,ξ2)dξ2.\nMoreover, there exists λD=λD(λ0,κ0,s0,s1,kM)independent of tsuch that\n|D(t,k,ξ1,ξ2)|+/vextendsingle/vextendsingleD1(t,k,ξ1,ξ2)/vextendsingle/vextendsingle+/vextendsingle/vextendsingleD−1(t,k,ξ1,ξ2)/vextendsingle/vextendsingle/lessorsimilare−λD|ξ1−ξ2|s0. (4.31)\nProof.Note that by (4.14), (4.15) and (4.16), the wave operator Du,kis similar to the operator D1\nu,k\nand the inverse wave operator D−1\nu,kin structure. We only present the proof for the wave operator\nDu,k. The estimates of D1\nu,kandD−1\nu,kare similar and we omit them.\nWe divide the nonlocal part of the wave operator into four diffe rent types of integral operators\nwhose kernels have different singularities:\nDu,k/parenleftbig\nF1˜f(t,k,·)/parenrightbig\n(t,k,v′) =B1(v,k)F1˜f(t,k,v)+4/summationdisplay\nj=1Πj(˜f), (4.32)\nwhere\nΠ1(˜f) =/tildewiderϕ1(v′)B2(v′,k)/integraldisplayu(1)\nu(0)F1˜f(u1)e−i(u1−v′)tk(u−1)′(u1)Φre\n1,1(u1,v′,k)du1,\nΠ2(˜f) =/tildewiderϕ1(v′)B2(v′,k)/integraldisplayu(1)\nu(0)F1˜f(u1)e−i(u1−v′)tk(u−1)′(u1)Φre\n0,1(u1,v′,k)(1R−(u1−v′))du1\n−/tildewiderϕ1(v′)B2(v′,k)/integraldisplayu(1)\nu(0)F1˜f(u1)e−i(u1−v′)tk(u−1)′(u1)Φre\n1,1(u1,v′,k)1R−(u1−v′)du1,\nΠ3(˜f) =/tildewiderϕ1(v′)B2(v′,k)/integraldisplayu(1)\nu(0)F1˜f(u1)e−i(u1−v′)tk(u−1)′(u1)\n×/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v′)Φ1(u1,v′,k)ln|u1−v′|du1,\nΠ4(˜f) =−/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v′)/tildewiderϕ1(v′)B2(v′,k)/integraldisplayu(1)\nu(0)F1˜f(u1)e−i(u1−v′)tk\nu1−v′(u−1)′(u1)du1.\nBy Lemma B.1, we get that there exists D(t,k,ξ1,ξ2) such that\nF2/parenleftBig\nDu,k/parenleftbig\n˜χ2F1f(t,k,·)/parenrightbig/parenrightBig\n(t,k,ξ1) =/integraldisplay\nD(t,k,ξ1,ξ2)ˆfk(t,ξ2)dξ2.NONLINEAR INVISCID DAMPING 41\nThe behavior of D(t,k,ξ1,ξ2) dependsonly on the regularity of D1,D2andE. Therefore by Lemma\nB.1, Remark A.2, Corollary 4.9, Proposition 4.10, and Lemma B.2, we obtain Proposition 4.11. /square\n4.4.Commutator. In this section, let us study the difference between D1\nu,kandDu,kand prove\nthe following proposition.\nProposition 4.12. There exists Dcom(t,k,ξ,ξ 1)such that\nF2/parenleftBig\n˜χ2Du,k(˜χ2F1f)−˜χ2D1\nu,k(˜χ2F1f)/parenrightBig\n(ξ) =/integraldisplay\nDcom(t,k,ξ,ξ 1)ˆfk(t,ξ1)dξ1.\nMoreover, there exists λD=λD(λ0,θ0,s0,s1,kM)independent of tsuch that\n/vextendsingle/vextendsingleDcom(t,k,ξ,ξ 1)/vextendsingle/vextendsingle/lessorsimilarmin/parenleftBigg\ne−λD|ξ−ξ1|s0\n1+|ξ−kt|2,e−λD|ξ−ξ1|s0\n1+|ξ1−kt|2/parenrightBigg\n.\nProof.We have by (4.29) that\n˜χ2Du,k(F1f)−˜χ2D1\nu,k(F1f)\n= ˜χ2B2(v,k)/integraldisplayu(1)\nu(0)E(k,v1,v)F1f(v1)e−i(v1−v)tk\nv1−v/bracketleftBig\n/tildewiderϕ1(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′(v)/bracketrightBig\ndv1\n=−˜χ2B2(k,v)/integraldisplayu(1)\nu(0)/parenleftBig\n˜θ(u−1)′/parenrightBig\n(v)F1f(v1)e−i(v1−v)tk\nv1−v/bracketleftBig\n/tildewiderϕ1(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′(v)/bracketrightBig\ndv1\n+ ˜χ2B2(k,v)/integraldisplayu(1)\nu(0)Φ1(v1,v,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v)ln|v1−v|F1f(v1)e−i(v1−v)tk\n×/bracketleftBig\n/tildewiderϕ1(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′(v)/bracketrightBig\ndv1\n+ ˜χ2B2(v,k)/integraldisplayu(1)\nu(0)Φre\n0,1(v1,v,k)(1R−(v1−v))F1f(v1)e−i(v1−v)tk\n×/bracketleftBig\n/tildewiderϕ1(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′(v)/bracketrightBig\ndv1\n+ ˜χ2B2(v,k)/integraldisplayu(1)\nu(0)Φre\n1,1(v1,v,k)(1R+(v1−v))F1f(v1)e−i(v1−v)tk\n×/bracketleftBig\n/tildewiderϕ1(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′(v)/bracketrightBig\ndv1.\nLetDcom(t,k,ξ,ξ 1) be the Fourier kernel of the operator ˜ χ2Du,k(˜χ2F1f)−˜χ2D1\nu,k(˜χ2F1f), which\nmeans that\nF2/parenleftBig\n˜χ2Du,k(˜χ2F1f)−˜χ2D1\nu,k(˜χ2F1f)/parenrightBig\n(ξ) =/integraldisplay\nDcom(t,k,ξ,ξ 1)ˆfk(t,ξ1)dξ1.\nThen by the same argument as in the proof of Proposition 4.11, we get that there exists λDsuch\nthat/vextendsingle/vextendsingleDcom(t,k,ξ,ξ 1)/vextendsingle/vextendsingle/lessorsimilare−λD|ξ−ξ1|s0.\nLet us also study the derivate ( ∂v−itk) acting on ˜ χ2Du,k(F1f)−˜χ2D1\nu,k(F1f):\n(∂v−itk)/parenleftBig\n˜χ2Du,k(F1f)−˜χ2D1\nu,k(F1f)/parenrightBig\n= ˜χ2B2(v,k)/integraldisplayu(1)\nu(0)Φ1(v1,v,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v)ln|v1−v|F1f(v1)e−i(v1−v)tk42 WEIREN ZHAO\n×∂v/bracketleftBig\n/tildewiderϕ1(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′(v)/bracketrightBig\ndv1\n+ ˜χ2B2(v,k)/integraldisplayu(1)\nu(0)Φre\n0,1(v1,v,k)(1R−(v1−v))F1f(v1)e−i(v1−v)tk\n×∂v/bracketleftBig\n/tildewiderϕ1(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′(v)/bracketrightBig\ndv1\n+ ˜χ2B2(v,k)/integraldisplayu(1)\nu(0)Φre\n1,1(v1,v,k)(1R+(v1−v))F1f(v1)e−i(v1−v)tk\n×∂v/bracketleftBig\n/tildewiderϕ1(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′(v)/bracketrightBig\ndv1\n+good terms .\nHere note that the derivative ∂vacting on ln |v1−v|gives a good term:\n/tildewiderϕ1(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′(v)\nv1−v∈ GKu,s\nph,1([u(0),u(1)]2),\nfor someKu>0.\nTherefore, there exists Dcom,1(t,k,ξ,ξ 1) such that\n(iξ−ikt)F2/parenleftBig\n˜χ2Du,k(˜χ2F1f)−˜χ2D1\nu,k(˜χ2F1f)/parenrightBig\n(ξ) =/integraldisplay\nDcom,1(t,k,ξ,ξ 1)ˆfk(t,ξ1)dξ1.\nMoreover, Lemma B.2 gives us that there exists λ′\nDsuch that\n|Dcom,1(t,k,ξ,ξ 1)|/lessorsimilare−λ′\nD|ξ−ξ1|s0,\nwhich together with the fact that\ni(ξ−kt)Dcom(t,k,ξ,ξ 1) =Dcom,1(t,k,ξ,ξ 1),\ngives us that\n/vextendsingle/vextendsingleDcom(t,k,ξ,ξ 1)/vextendsingle/vextendsingle/lessorsimilare−λ′\nD|ξ−ξ1|s0\n1+|ξ−kt|.\nWe can repeat the above argument once more and get that\n(∂v−itk)2/parenleftBig\n˜χ2Du,k(F1f)−˜χ2D1\nu,k(F1f)/parenrightBig\n= ˜χ2B2(v,k)/integraldisplayu(1)\nu(0)Φ1(v1,v,k)/parenleftBig\n˜θ(u−1)′/parenrightBig′\n(v)F1f(v1)\nv1−ve−i(v1−v)tk\n×/bracketleftBig\n/tildewiderϕ1′(v)(u−1)′(v1)−/tildewiderϕ1(v1)(u−1)′′(v)/bracketrightBig\ndv1\n+ ˜χ2B2(v,k)Φre\n0,1(v,v,k)F1f(v)/bracketleftBig\n/tildewiderϕ1′(v)(u−1)′(v)−/tildewiderϕ1(v)(u−1)′′(v)/bracketrightBig\n+ ˜χ2B2(v,k)Φre\n1,1(v,v,k)F1f(v)/bracketleftBig\n/tildewiderϕ1′(v)(u−1)′(v)−/tildewiderϕ1(v)(u−1)′′(v)/bracketrightBig\n+good terms .\nTherefore, there exists Dcom,2(t,k,ξ,ξ 1) such that\n(iξ−ikt)2F2/parenleftBig\n˜χ2Du,k(˜χ2F1f)−˜χ2D1\nu,k(˜χ2F1f)/parenrightBig\n(ξ) =/integraldisplay\nDcom,2(t,k,ξ,ξ 1)ˆfk(t,ξ1)dξ1.\nMoreover, there exists λ′\nDsuch that\n|Dcom,2(t,k,ξ,ξ 1)|/lessorsimilare−λ′\nD|ξ−ξ1|s0,NONLINEAR INVISCID DAMPING 43\nwhich together with the fact that\n−(ξ−kt)2Dcom(t,k,ξ,ξ 1) =Dcom,2(t,k,ξ,ξ 1),\ngives us that\n/vextendsingle/vextendsingleDcom(t,k,ξ,ξ 1)/vextendsingle/vextendsingle/lessorsimilare−λ′\nD|ξ−ξ1|s0\n1+|ξ−kt|2.\nBy using the fact that\n1+|ξ1−kt|2\n1+|ξ−kt|2/lessorsimilar1+|ξ−kt|2+|ξ−ξ1|2\n1+|ξ−kt|2/lessorsimilar/a\\}b∇acketle{tξ−ξ1/a\\}b∇acket∇i}ht2,\nwe obtain Proposition 4.12 for some λD<λ′\nD. /square\n5.The good system\nNow we apply the wave operator on Ω in (2.15). Let us first intro duce the new good unknown\n(5.1)f(t,z,v)def=P0(Ω)(t,v)+P|k|≥kMΩ(t,z,v)\n+/summationdisplay\n0<|k|<kMDu,k(F1Ω(t,k,·))(t,k,v)eizk,\nwhere\nP|k|≥kMΩ(t,z,v) =/summationdisplay\n|k|≥kMF1Ω(t,k,v)eikz.\nLet us give the following remark and determine kM.\nRemark 5.1 (Determination of kM).To make the choice of kMmore precise, let us first introduce\nthree constants:\n(1)The constant Cell≥1only depends on the background flow. It is the constant appeari ng in\nthe linear elliptic estimate.\n(2)The constant Cϕ′′\n1= 1+/ba∇dbl/tildewiderϕ1/ba∇dblGs,σ+2only depends on the Gevrey norm of /tildewiderϕ1.\n(3)The constant C0≥1is a universal constant that depends on sandσ.\nWe choose kMsuch that\n(5.2) kM≥100C0Cu′′Cell.\nSinceCu′′andCellonly depend on the background flow, the constant kMis fixed once the background\nflow is given.\nBy Remark 4.5, fhas the same compact support as Ω.\nLet us now deduce the equation of the new good unknown f. An easy calculation shows that\n∂tf=∂tP0(Ω)+∂tPk≥kMΩ\n+∂t/summationdisplay\n0<|k|<kMDu,k(F1Ω(t,k,·))(t,k,v)eizk\n=−P0(U·∇z,vΩ)−P0(NΩ[Ψ])−P0(Na[Π])+ϕ1∂zPk≥kMΨ\n−Pk≥kM/parenleftBig\nU·∇z,vΩ/parenrightBig\n−Pk≥kM(NΩ[Ψ])−Pk≥kM(Na[Π])\n+/summationdisplay\n0<|k|<kM[∂t,Du,k](F1Ω(t,k,·))(t,k,v)eizk44 WEIREN ZHAO\n+/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nϕ1∂z/parenleftbig\nΨ/parenrightbig/parenrightBig/parenrightBig\n(t,k,v)eizk\n−/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig\n(t,k,v)eizk\n−/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nNΩ[Ψ]/parenrightBig/parenrightBig\n(t,k,v)eizk−/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nNa[Π]/parenrightBig/parenrightBig\n(t,k,v)eizk.\nBy the fact that,\nDu,k/parenleftBig\nF1/parenleftBig\nϕ1∂zΨ/parenrightBig/parenrightBig\n(t,k,v)\n=Du,k/parenleftBig\nF1/parenleftBig\n/tildewiderϕ1∂z/parenleftbig/parenleftbig˜∆l\nt/parenrightbig−1Ω/parenrightbig/parenrightBig/parenrightBig\n(t,k,v)+Du,k/parenleftBig\nF1/parenleftBig\nϕ1∂zΨ−/tildewiderϕ1∂z/parenleftbig/parenleftbig˜∆l\nt/parenrightbig−1Ω/parenrightbig/parenrightBig/parenrightBig\n(t,k,v)\n=Du,k/parenleftBig\nF1/parenleftBig\n/tildewiderϕ1∂z/parenleftbig/parenleftbig˜∆l\nt/parenrightbig−1Ω/parenrightbig/parenrightBig/parenrightBig\n(t,k,v)+Du,k/parenleftBig\nF1/parenleftBig\nϕδ\n1∂zΨ−/tildewiderϕ1∂z/parenleftbig/parenleftbig˜∆l\nt/parenrightbig−1Ω−Ψ/parenrightbig/parenrightBig/parenrightBig\n(t,k,v)\n=−ikvDu,k(F1Ω(t,k,·))(t,k,v)+Du,k(ikF1vΩ(t,k,·))(t,k,v)\n+Du,k/parenleftBig\nF1/parenleftBig\nϕδ∂zΨ−/tildewiderϕ1∂z/parenleftbig/parenleftbig˜∆l\nt/parenrightbig−1Ω−Ψ/parenrightbig/parenrightBig/parenrightBig\n(t,k,v),\nwe obtain by (4.19) that\n∂tf=−P0(U·∇z,vΩ)−P0(NΩ[Ψ])−P0(Na[Π])+ϕ1∂zP|k|≥kMΨ (5.3)\n−P|k|≥kM/parenleftBig\nU·∇z,vΩ/parenrightBig\n−Pk≥kM(NΩ[Ψ])−Pk≥kM(Na[Π])\n+/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nϕδ\n1∂zΨ−/tildewiderϕ1∂z/parenleftbig/parenleftbig˜∆l\nt/parenrightbig−1Ω−Ψ/parenrightbig/parenrightBig/parenrightBig\n(t,k,v)eizk\n−/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig\n(t,k,v)eizk\n−/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nNΩ[Ψ]/parenrightBig/parenrightBig\n(t,k,v)eizk\n−/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nNa[Π]/parenrightBig/parenrightBig\n(t,k,v)eizk.\nHere we recall the notations used in the previous calculatio ns\nϕδ\n1(t,v) =ϕ1(t,v)−/tildewiderϕ1(v)\n˜∆l\nt=/tildewiderϕ4∂zz+/tildewideu′(∂v−t∂z)/parenleftBig\n/tildewiderϕ4/tildewideu′(∂v−t∂z)/parenrightBig\n.\nWe also have\n∂tv(t,y) =−χ1(y)\nt2/integraldisplayt\n0P0(Ux)(t′,y)dt′+χ1(y)\ntP0(Ux)(t,y),\nand\n∂ttv(t,y) =∂t/parenleftbig\n∂tv(t,v(t,y))/parenrightbig\n=∂t/parenleftbig\n∂tv/parenrightbig\n(t,v(t,y))+∂tv(t,y)∂v/parenleftbig\n∂tv/parenrightbig\n(t,v(t,y))\nwhich gives that\n∂ttv(t,y) =2χ1(y)\nt3/integraldisplayt\n0P0(Ux)(t′,y)dt′−2χ1(y)\nt2P0(Ux)(t,y)+χ1(y)\ntP0(∂tUx)(t,y)NONLINEAR INVISCID DAMPING 45\n=−2\nt∂tv(t,y)+χ1(y)\ntP0(∂tUx)(t,y).\nBy using the fact that\n∂tUx+u(y)∂xUx+u′(y)Uy+θ(y)∂xP+d∂xP+Ux∂xUx+Uy∂yUx= 0,\nwe obtain that\n(5.4)∂t(∂tv)+∂tv∂v(∂tv)+2\nt(∂tv) =−χ1\nt∂yvP0/parenleftBig\n∇⊥\nz,vP/\\e}atio\\slash=(Ψ)·∇z,vUx/parenrightBig\n−χ1\ntP0/parenleftBig\nP/\\e}atio\\slash=(a)∂zP/\\e}atio\\slash=(Πl+Πn)/parenrightBig\n,\nwhereUx(t,z(t,x,y),v(t,y)) =Ux(t,x,y).\nWe also have\n∂yv(t,y) =u′(y)−1\nt/integraldisplayt\n0P0(ω)(t′,y)dt′,\nthen it holds that\n∂t/parenleftBig\nt/parenleftbig\n∂yv(t,y)−u′(y)/parenrightbig/parenrightBig\n=−P0(ω)(t,y).\nLeth(t,v(t,y)) =∂yv(t,y)−u′(y), thenh(t,v) satisfies\n(5.5) ∂th+∂tv∂vh=1\nt/parenleftbig\n−P0(ω)−h/parenrightbigdef=¯h=∂yv∂v∂tv.\nWe then get\n(5.6)∂t¯h=−¯h\nt−1\nt(∂tP0(ω)+∂th)\n=−¯2h\nt−∂tv∂v¯h+∂yv\nt/parenleftBig\nP0(∂zΨ∂vω−∂vΨ∂zω)−P0(∂zΠl∂va−∂vΠl∂za)\n−P0(∂zΠn∂va−∂vΠn∂za)/parenrightBig\n.\nWe now introduce the cut-off function Υ 1(v),Υ2(v)∈ GM(κ0),s+1\n2\nph,1which satisfies\nsuppΥ 1(v)⊂[u(κ0),u(1−κ0)]\nsuppΥ 2(v)⊂/bracketleftbig\nu(κ0\n2),u(1−κ0\n2)/bracketrightbig\nand Υ(v)≡1,forv∈[u(κ0),u(1−κ0)].46 WEIREN ZHAO\n5.1.The working system. We then rewrite the system by considering the compact suppor t.\n(5.7a)\n\n∂tf=−P0(Υ2U·∇z,vΩ)−P0(NΩ[Υ2Ψ])\n−P0(Na[Υ2Πl])−P0(Na[Υ2Πn])\n−P|k|≥kM/parenleftBig\nΥ2U·∇z,vΩ/parenrightBig\n+ϕ1∂zP|k|≥kM(Υ2Ψ)−P|k|≥kM(NΩ[Υ2Ψ])\n−P|k|≥kM(Na[Πl])−P|k|≥kM(Na[Πn])\n+/summationdisplay\n0<|k|<kMΥ2Du,k/parenleftBig\nF1/parenleftBig\nϕδ\n1∂z(Υ2Ψ)−/tildewiderϕ1∂z/parenleftbig˚T−1\n1,D[Ω]−Υ2Ψ/parenrightbig/parenrightBig/parenrightBig\n(t,k,v)eizk\n−/summationdisplay\n0<|k|<kMΥ2Du,k/parenleftBig\nF1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig\n(t,k,v)eizk\n−/summationdisplay\n0<|k|<kMΥ2Du,k/parenleftBig\nF1/parenleftBig\nNΩ[Υ2Ψ]/parenrightBig/parenrightBig\n(t,k,v)eizk−Υ2D(Na[Π])\n∂ta=−θ′∂z(Υ2Ψ)−Υ2U·∇z,va,\nU = (0,∂tv)+∂yv∇⊥\nv,zP/\\e}atio\\slash=(Υ2Ψ),\nΩ =ϕ4∂zzΨ+∂yv(∂v−t∂z)/parenleftBig\nϕ4∂yv(∂v−t∂z)Ψ/parenrightBig\n.\nwhere the operator ˚T−1\n1,Dis defined in (8.4) and D(Na[Π]) =D(Na[Πl])+D(Na[Πn]) with\n(5.7b)D(Na[Πl]) =/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nΥ2Na[Πl]/parenrightBig/parenrightBig\n(t,k,v)eizk\nD(Na[Πn]) =/summationdisplay\n0<|k|<kMDu,k/parenleftBig\nF1/parenleftBig\nΥ2Na[Πn]/parenrightBig/parenrightBig\n(t,k,v)eizk.\nwithNa(Π) being defined in (2.17) and\n(5.7c) Π l= Υ2Πl,1+Υ1Πl,2,Πn= Υ2Πn,1+Υ1Πn,2.\nWe can recover Ω from f.\nΩ(t,z,v) =P0(f)+/summationdisplay\n0<|k|<kMD−1\nu,k/bracketleftbig\nF1f/bracketrightbig\n(t,k,v)eizk+P|k|≥kM[f](t,z,v). (5.7d)\nThe pressure terms Π are solved from the elliptic systems:\n(5.7e)\n\n∂z/parenleftBig\n(a+θ)∂zΠl,1/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Πl,1/parenrightBig\n=ϕ6∂zz(Υ2Ψ)+ϕ7P/\\e}atio\\slash=(Υ2Ψ),\n(∂v−t∂z)Πl,1(t,z,u(0)) = (∂v−t∂z)Πl,1(t,z,u(1)) = 0,\nand\n(5.7f)\n\n∂z/parenleftBig\n(a+θ)∂zΠl,2/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Πl,2/parenrightBig\n=ϕ8∂yv(∂v−t∂z)P/\\e}atio\\slash=(Υ2Ψ),\n(∂v−t∂z)Πl,2(t,z,u(0)) = (∂v−t∂z)Πl,2(t,z,u(1)) = 0,NONLINEAR INVISCID DAMPING 47\nand\n(5.7g)\n\n∂z/parenleftBig\n(a+θ)∂zΠn/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Πn/parenrightBig\n=−2χ2(∂yv(∂v−t∂z)∂z(Υ2Ψ))2−2∂zz(Υ2Ψ)/parenleftBig\nΩ−χ2∂zz(Υ2Ψ)/parenrightBig\n,\n���2χ′\n2∂yv/bracketleftBig\n(∂z(Υ2Ψ))(∂v−t∂z)∂z(Υ2Ψ)+(∂v−t∂z)(Υ2Ψ)∂zz(Υ2Ψ)/bracketrightBig\n−1\n2χ′′\n2/bracketleftBig\n(∂z(Υ2Ψ))2+(∂yv)2((∂v−t∂z)(Υ2Ψ))2/bracketrightBig\n,\n(∂v−t∂z)Πn(t,z,u(0)) = (∂v−t∂z)Πn(t,z,u(1)) = 0.\nand\n(5.7h)\n\n∂z/parenleftBig\n(a+θ)∂zΠn,2/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Πn,2/parenrightBig\n=−1\n2χ′′\n2/bracketleftBig\n2(∂yv)2((∂v−t∂z)P/\\e}atio\\slash=(Υ2Ψ))∂vP0(Υ2Ψ)+(∂yv)2(∂vP0(Υ2Ψ))2/bracketrightBig\n,\n(∂v−t∂z)Πn,2(t,z,u(0)) = (∂v−t∂z)Πn,2(t,z,u(1)) = 0.\nWe have that the zero mode of the pressure solves\n(5.7i)∂vP0(ΠΥ1)+(ϕ4Υ2)P0(a)∂vP0(ΠΥ1)+(ϕ4Υ2)P0/bracketleftbig\nP/\\e}atio\\slash=(a)P/\\e}atio\\slash=(∂v−t∂z)P/\\e}atio\\slash=(ΠΥ1)/bracketrightbig\n= (ϕ4Υ2)P0/parenleftBig\n∂z(ΨΥ2)∂z(∂v−t∂z)(ΨΥ2)−∂zz(ΨΥ2)(∂v−t∂z)(ΨΥ2)/parenrightBig\n.\nWe also list the equations related to the change of coordinat es.\n(5.7j)∂t(∂tv)+∂tv∂v(∂tv)+2\nt(∂tv) =−χ1\nt∂yvP0/parenleftBig\n∇⊥\nz,vP/\\e}atio\\slash=(Υ2Ψ)·∇z,v(Υ2Ux)/parenrightBig\n−χ1\ntP0/parenleftBig\nP/\\e}atio\\slash=(a)∂zP/\\e}atio\\slash=/parenleftbig\n(Υ2Πl,1)+(Υ 2Πl,2)+(Υ 2Πn)/parenrightbig/parenrightBig\n,\nwhere\nUx(t,z,v) =−∂yv(∂v−t∂z)(Υ2Ψ).\nForj= 1,2,...,11, we have\n(5.7k) ∂tϕδ\nj+∂tv(t,v)∂vϕδ\nj=−∂tv(t,v)∂v/tildewiderϕj(v)\nWe also have that h=∂yv−u′solves\n(5.7l) ∂th+∂tv∂vh=¯h.\nwhere¯hsolves\n(5.7m)∂t¯h=−2¯h\nt−∂tv∂v¯h\n+∂yv\nt/parenleftBig\nP0(∂z(Υ2Ψ)∂vω−∂v(Υ2Ψ)∂zω)−P0(∂z(Πl)∂va−∂v(Πl)∂za)\n−P0(∂z(Υ2Πn)∂va−∂v(Υ2Πn)∂za)/parenrightBig\n.48 WEIREN ZHAO\n6.Energy functional and bootstrap proposition\nWe will use the same multiplier A( t,∇) introduced in [5].\nAk(t,η) =eλ(t)/a\\}bracketle{tk,η/a\\}bracketri}hts/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}htσJk(t,η).\nThe indexλ(t) is the bulk Gevrey-1\nsregularity and will be chosen to satisfy\nλ(t) =3\n4λ0+1\n4λ′, t≤1\n˙λ(t) =−δλ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2˜q(1+λ(t)), t∈(1,∞)\nwhereλ0,λ′are parameters that depend on the regularity of the backgrou nd flowu(y), densityθ(y)\nand chosen by the proof, δλ≈λ0−λ′is a small parameter that ensures λ(t)≥λ0\n2+λ′\n2fort≥0\nand1\n2<˜q≤s\n8+7\n16is a parameter chosen by the proof.\nThe main multiplier for dealing with the Orr mechanism and th e associated nonlinear growth is\n(6.1) J k(t,η) =eµ|η|1\n2\nwk(t,η)+eµ|k|1\n2,\nwhere\n(6.2) wk(t,η) =\n\nwk(tE(√η),η,η)t<tE(√η),η\nwNR(t,η)t∈[tE(√η),η,2η]\\Ik,η\nwR(t,η)t∈Ik,η\n1t≥2η.\nHere (wR(t,η),wNR(t,η)) is defined in the following way:\nLetwNRbe a non-decreasing function of time with wNR(t,η) = 1 fort≥2η. For definiteness,\nwe remark here that for |η| ≤1,wNR(t,η) = 1, which will be a consequence of the definition. For\nk= 1,2,3,...,E(√η), we define\n(6.3)wNR(t,η) =/parenleftbiggk2\nη/parenleftBig\n1+bk,η/vextendsingle/vextendsingle/vextendsinglet−η\nk/vextendsingle/vextendsingle/vextendsingle/parenrightBig/parenrightbiggCκ1\nwNR(tk−1,η,η),\n∀t∈IR\nk,η=/bracketleftBigη\nk,tk−1,η/bracketrightBig\n,\nand\n(6.4)wNR(t,η) =/parenleftBig\n1+ak,η/vextendsingle/vextendsingle/vextendsinglet−η\nk/vextendsingle/vextendsingle/vextendsingle/parenrightBig−1−Cκ1wNR/parenleftBigη\nk,η/parenrightBig\n,\n∀t∈IL\nk,η=/bracketleftBig\ntk,η,η\nk/bracketrightBig\n.\nThe constant bk,ηis chosen to ensure thatk2\nη/parenleftBig\n1+bk,η/vextendsingle/vextendsingle/vextendsingletk−1,η−η\nk/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n= 1, hence for k≥2\n(6.5) bk,η=2(k−1)\nk/parenleftbigg\n1−k2\nη/parenrightbigg\nandb1,η= 1−1\nηand similarly,\n(6.6) ak,η=2(k+1)\nk/parenleftbigg\n1−k2\nη/parenrightbigg\n.NONLINEAR INVISCID DAMPING 49\nOn each interval I k,η, we define wR(t,η) by\nwR(t,η) =k2\nη/parenleftBig\n1+bk,η/vextendsingle/vextendsingle/vextendsinglet−η\nk/vextendsingle/vextendsingle/vextendsingle/parenrightBig\nwNR(t,η),∀t∈IR\nk,η=/bracketleftBigη\nk,tk−1,η/bracketrightBig\n, (6.7)\nwR(t,η) =k2\nη/parenleftBig\n1+ak,η/vextendsingle/vextendsingle/vextendsinglet−η\nk/vextendsingle/vextendsingle/vextendsingle/parenrightBig\nwNR(t,η),∀t∈IL\nk,η=/bracketleftBig\ntk,η,η\nk/bracketrightBig\n. (6.8)\nWe also define JR(t,η) and AR(t,η) to assign resonant regularity at every critical time:\n(6.9)JR(t,η) =\n\neµ|η|1\n2w−1\nR(tE(√η),η,η)t<tE(√η),η\neµ|η|1\n2w−1\nR(t,η)t∈[tE(√η),η,2η]\neµ|η|1\n2t≥2η,\nAR(t,η) =eλ(t)/a\\}bracketle{tη/a\\}bracketri}hts/a\\}b∇acketle{tη/a\\}b∇acket∇i}htσJR(t,η).\nOne may refer to [5] and Appendix D for more basic properties o f the multiplier A( t,∇).\nWe also introduce the Fourier multiplier A∗(t,∇) =F−1(A∗\nk(t,η)/hatwidefk(t,η)) from [11],\nA∗\nk(t,η) = Ak(t,η)Bk(t,η),Bk(t,η) =/parenleftBig\n1+k2+|η|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/parenrightBig1\n2.\nThe weight is well-constructed to capture the important reg ularity difference between the vorticity\nfand the density a.\nRemark 6.1. It holds that\n∂tA∗\nk(t,η)\nA∗\nk(t,η)=∂tAk(t,η)\nAk(t,η)−t(k2+|η|)\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4+(k2+|η|)/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2≈∂tAk(t,η)\nAk(t,η).\nProof.The remark follows directly from the fact that\nt(k2+|η|)\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4+(k2+|η|)/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/lessorsimilar/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3\n2+/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/lessorsimilar/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}hts\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2¯q.\n/square\nLet us mention some ideas for choosing the multiplier B. It is used to solve the problem from\nthe nonlinear interactions between the pressure and the den sity, which can be formally written as\n∂zΠ∂vaand∂vΠ∂za. On one hand, if the pressure is in higher frequencies, namel y,∂zΠhigh∂valow\nand∂vΠhigh∂zalow, both terms behavior as the Reaction terms ∂zΨhigh∂vΩlowand∂vΨhigh∂zΩlow.\nThey can be controlled easily. We only need to control\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΠΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\nby theCKterms. However, it is not enough to close the elliptic estima te for the pressure Π. Indeed,\nunlike the elliptic equation of the stream function, the ell iptic error terms in the pressure equations\nare not just zero mode, see Sa[Π] and its estimates in Lemma 10.3. Notice that Π behaves bet ter\nthan the stream function Ψ, which allows us to replace/angbracketleftBig\n∂v\nt∂z/angbracketrightBig−1\nA(t,∇) by a stronger weight M3,\nsee Propositions 7.8, 7.9, and 7.10, and section 9 and sectio n 10 for more details. On the other\nhand, when the density ais in higher frequencies, namely, ∂zΠlow∂vahighand∂vΠlow∂zahigh, we\nhave derivative loss. Note that in similar terms that involv e the stream function and vorticity, we\nuse the commutator to gain a 1 /2 derivative. Here we put half more derivative in the multipl ier of50 WEIREN ZHAO\nafor short time, which allows us to control these terms. Then w e should be careful in estimating\nthe linear term −θ′∂z(Υ2Ψ) in the equation of a, because the added weight B will act on Ψ, see\nProposition 7.14 and section 13 for more details. The other p lace where we take advantage of the\noperator B is the estimate of elliptic error term Sa(Π), see the proof of Lemma 10.3 for more details.\nLet us now remark here that by the nice Fourier kernel estimat es, the operator A commutates\nwell with the wave operator. The following two remarks can be found in [28].\nRemark 6.2. There exists C=C(kM)≥1independent of tsuch that for any λ<λD,\nC−1/ba∇dblΩ/ba∇dblGs,λ,σ(T×[u(0),u(1)])≤ /ba∇dblf/ba∇dblGs,λ,σ(T×[u(0),u(1)])≤C/ba∇dblΩ/ba∇dblGs,λ,σ(T×[u(0),u(1)]),\nand that there exists C=C(kM)≥1independent of tsuch that for any λ(t)<λD,\nC−1/ba∇dblAΩ/ba∇dbl2\n2≤ /ba∇dblAf/ba∇dbl2\n2≤C/ba∇dblAΩ/ba∇dbl2\n2.\nRemark 6.3. Proposition 4.11 gives us that for λ0andλ′sufficiently small,\n−˙λ(t)/vextenddouble/vextenddouble/vextenddouble|∇|s/2AΩ/vextenddouble/vextenddouble/vextenddouble2\n2+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n∂tw\nw˜AΩ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarCKf,\nwhere\nCKf=/summationdisplay\nk∈Z/integraldisplay\nR|˙Ak(t,η)|Ak(t,η)|/hatwidefk(t,η)|2dη.\nRemark 6.4. The following fact will be frequently used: for any function ϕof zero mode, we write\nϕ=/tildewideϕ+ϕδ\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n∂tw\nw˜A(ϕP/\\e}atio\\slash=f)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilar/ba∇dbl/tildewideϕ/ba∇dblGs,σ+2CKf+/ba∇dblϕδ/ba∇dblGs,σ−6CKf+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n∂tw\nwARϕδ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2/ba∇dblf/ba∇dblGs,σ−6\n/lessorsimilarCKf+ǫCKϕδ,\nwhere\nCKϕδ=/integraldisplay\nR˙AR(t,η)AR(t,η)|/hatwiderϕδ(t,η)|2dη.\nWe give the following remark, related to the operator A.\nRemark 6.5. Letℓ=k−landζ=ξ−η, then\n(1)ifℓ/\\e}atio\\slash= 0,k/\\e}atio\\slash= 0, andl/\\e}atio\\slash= 0, then\nAk(ξ)/lessorsimilarAl(η)ecλ|ℓ,ζ|s/parenleftBig\n1+|ξ|\nk2+(ξ−kt)21t∈Ik,ξ∩Ik,η1c|ξ|≤|η|≤1\nc|ξ|/parenrightBig\n1LH\n+Aℓ(ζ)ecλ|l,η|s/parenleftBig\n1+|ξ|\nk2+(ξ−kt)21t∈Ik,ξ∩Ik,ζ1c|ξ|≤|ζ|≤1\nc|ξ|/parenrightBig\n1HL\n+ecλ|l,η|secλ|ℓ,ξ|s/a\\}b∇acketle{tl,η/a\\}b∇acket∇i}htσ/2+1/a\\}b∇acketle{tℓ,ξ/a\\}b∇acket∇i}htσ/2+11HH,\n(2)ifℓ= 0andk/\\e}atio\\slash= 0, then\nAk(ξ)/lessorsimilarAk(η)ecλ|ζ|s1LH+AR(ζ)ecλ|k,η|s1HL\n+ecλ|k,η|secλ|ξ|s/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}htσ/2+1/a\\}b∇acketle{tξ/a\\}b∇acket∇i}htσ/2+11HH,\nwhereLH ={4|ℓ,ζ| ≤ |l,η|},HL ={|ℓ,ζ| ≥4|l,η|}, andHH ={1\n4|ℓ,ζ| ≤ |l,η| ≤4|ℓ,ζ|}.NONLINEAR INVISCID DAMPING 51\n6.1.Energy functionals. Now we define the energy functionals:\nEf=/summationdisplay\nk∈Z/integraldisplay\nR|Ak(t,η)/hatwidefk(t,η)|2dη,CKf=/summationdisplay\nk∈Z/integraldisplay\nR|˙Ak(t,η)|Ak(t,η)|/hatwidefk(t,η)|2dη,\nEa=/summationdisplay\nk∈Z/integraldisplay\nR|A∗\nk(t,η)/hatwidefk(t,η)|2dη,CKa=/summationdisplay\nk∈Z/integraldisplay\nR|˙A∗\nk(t,η)|A∗\nk(t,η)|/hatwidefk(t,η)|2dη,\nEh=/integraldisplay\nR|AR(t,η)/hatwideh(t,η)|2dη,CKh=/integraldisplay\nR|˙AR(t,η)|AR(t,η)|/hatwideh(t,η)|2dη\nE¯h=/integraldisplay\nR/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+2s\n/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht2s|A0(t,η)/hatwider(∂yv∂v∂tv)(t,η)|2dη,\nCK¯h=/integraldisplay\nR/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+2s\n/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht2s|˙A0(t,η)|A0(t,η)|/hatwider(∂yv∂v∂tv)(t,η)|2dη,\nEϕδ\nj=/integraldisplay\nR|AR(t,η)/hatwiderϕδ\nj(t,η)|2dη,CKϕδ\nj=/integraldisplay\nR˙AR(t,η)AR(t,η)|/hatwiderϕδ\nj(t,η)|2dη, j= 1,2,...,11.\nWe also introduce an assistant functional:\nE∂tv=/integraldisplay\nR/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4−KDǫe2λ(t)/a\\}bracketle{tη/a\\}bracketri}hts/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht2σ−12|/hatwider∂tv(t,η)|2dη.\nWe define the total energy functional\nE(t) =Ef+1\nKaEa+E¯h+E∂tv+1\nKv/parenleftBig11/summationdisplay\nj=1Eϕδ\nj/parenrightBig\nwhere K D≥1, Ka≥1 and Kv≥1 are large constants determined in the proof. Indeed, K D\ndepends on s,λ,λ′, Kais mainly determined by the constant Cε0in Proposition 7.14 and K vis\nmainly determined by the Gevrey norms of /tildewiderϕjand/tildewideθ.\n6.2.Determine kM.In this section, we explain why the constant kMis determined as it is in\nRemark 5.1. We define\nIL\nf,5def=1\n2π/summationdisplay\n|k|≥kM/integraldisplay\nRAk(t,η)2/hatwider/parenleftBig\n/tildewiderϕ1∂z˚T−1\n1,D(Ω)/parenrightBig\nk(t,η)/hatwideΩk(t,η)dη. (6.10)\nThis is the linear part of If,5which is defined in (7.1). We have the following Lemma.\nLemma 6.6. It holds for all kM≥1that\n|IL\nf,5| ≤Cϕ1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblAP|k|≥kMf/ba∇dbl2\nL2+C0CellCϕ1\nkMCKf. (6.11)\nHere bothCell>1andCϕ1>1are constants only depending on the background flow and densi ty.\nMore precisely, the constant Cellis from Proposition C.9, and the constant Cϕ1is given by the\nGevrey norm of /tildewiderϕ1.\nProof.We have\nIL\nf,5=1\n2π/summationdisplay\n|k|≥kM/integraldisplay\n|η−ξ|≥8|k,ξ|Ak(t,η)2/parenleftBig/hatwider/tildewiderϕ1(η−ξ)ik/hatwider˚T−1\n1,D(Ω)k(t,ξ)/parenrightBig\n/hatwideΩk(t,η)dξdη\n+1\n2π/summationdisplay\n|k|≥kM/integraldisplay\n|η−ξ|≤1\n8|k,ξ|Ak(t,η)2/parenleftBig/hatwider/tildewiderϕ1(η−ξ)ik/hatwider˚T−1\n1,D(Ω)k(t,ξ)/parenrightBig\n/hatwideΩk(t,η)dξdη52 WEIREN ZHAO\n+1\n2π/summationdisplay\n|k|≥kM/integraldisplay\n1\n8|k,ξ|≤|η−ξ|≤8|k,ξ|Ak(t,η)2/parenleftBig/hatwider/tildewiderϕ1(η−ξ)ik/hatwider˚T−1\n1,D(Ω)k(t,ξ)/parenrightBig\n/hatwideΩk(t,η)dξdη\n=IL,HL\nf,5+IL,LH\nf,5+IL,HH\nf,5.\nOn the support of the integrand of IL,HL\nf,5, we have\nAk(t,η)≤AR(t,η−ξ)ecλ/a\\}bracketle{tk,ξ/a\\}bracketri}hts,\nwhich implies that\n|IL,HL\nf,5| ≤C/ba∇dblAR/tildewiderϕ1/ba∇dblL2/ba∇dblP|k|≥kM˚T−1\n1,D(Ω)/ba∇dblGs,λ,4/ba∇dblAP|k|≥kMΩ/ba∇dblL2\n≤Cell/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblAR/tildewiderϕ1/ba∇dblL2/ba∇dblP|k|≥kMf/ba∇dblGs,λ,6/ba∇dblAP|k|≥kMf/ba∇dblL2.\nHere we use Proposition C.9\n/summationdisplay\n|k|≥kM/ba∇dbl/a\\}b∇acketle{tk,ξ/a\\}b∇acket∇i}ht4ecλ/a\\}bracketle{tk,ξ/a\\}bracketri}hts/hatwider˚T−1\n1,D(Ω)k(t,ξ)/ba∇dbl2\nL2\n=/summationdisplay\n|k|≥kM/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{tk,ξ/a\\}b∇acket∇i}ht4ecλ/a\\}bracketle{tk,ξ/a\\}bracketri}hts/integraldisplay\nGD,1(t,k,ξ,η)/hatwideΩk(t,η)dη/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2\n≤/summationdisplay\n|k|≥kM/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayeλ/a\\}bracketle{tk,ξ−η/a\\}bracketri}htse−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(η−kt)2/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht4ecλ/a\\}bracketle{tk,η/a\\}bracketri}hts/hatwideΩk(t,η)dη/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2\n≤/summationdisplay\n|k|≥kM/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplaye−1\n2λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht6ecλ/a\\}bracketle{tk,η/a\\}bracketri}hts/hatwideΩk(t,η)dη/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2≤ /a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−4Cell/ba∇dblP|k|≥kMf/ba∇dbl2\nGs,λ,6.\nA similar argument gives that\n|IL,HH\nf,5| ≤Cell/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblAR/tildewiderϕ1/ba∇dblL2/ba∇dblAP|k|≥kMf/ba∇dbl2\nL2.\nTo estimate |IL,LH\nf,5|, we have by Proposition C.9 that,\n|IL,LH\nf,5|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2π/summationdisplay\n|k|≥kM/integraldisplay\n|η−ξ|≤1\n8|k,ξ|Ak(t,η)2/parenleftBig/hatwider/tildewiderϕ1(η−ξ)ikGD,1(t,ξ,ζ)/hatwideΩk(ζ)/parenrightBig\n/hatwideΩk(t,η)dζdξdη/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤Cell/summationdisplay\n|k|≥kM/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n|η−ξ|≤1\n8|k,ξ|Ak(t,η)2/parenleftBig\ne−λu,θ/a\\}bracketle{tη−ξ/a\\}bracketri}hts|k|e−λ∆/a\\}bracketle{tζ−ξ/a\\}bracketri}hts\nk2+(ζ−kt)2/hatwideΩk(ζ)/parenrightBig\n/hatwideΩk(t,η)dζdξdη/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤CellCϕ1/summationdisplay\n|k|≥kM/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n|η−ξ|≤1\n8|k,ξ|/parenleftBig\ne−3\n4λu,θ/a\\}bracketle{tη−ξ/a\\}bracketri}hts|k|e−3\n4λ∆/a\\}bracketle{tζ−ξ/a\\}bracketri}hts\nk2+(ζ−kt)2/hatwiderAfk(ζ)/parenrightBig\n/hatwiderAfk(t,η)dζdξdη/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nIt is easy to check that for k≥kM\n1\nk2+(ζ−kt)2/lessorsimilar1\nkM/radicalBigg\n˙Ak(t,η)\nAk(t,η)/radicalBigg\n˙Ak(t,ζ)\nAk(t,ζ)/a\\}b∇acketle{tζ−ξ,η−ξ/a\\}b∇acket∇i}ht3\nwhich implies that there is a universal constant C0such that\n|IL,LH\nf,5| ≤C0CellCϕ1\nkMCKf.\nThus we obtain (6.11) and finish the proof. /squareNONLINEAR INVISCID DAMPING 53\nOne can regard kMas the threshold after which we no longer need to apply the wav e operator.\n6.3.The bootstrap argument. The local wellposedness of the inhomogeneous incompressib le\nEuler equation in the Gevrey class is classical. In this way, we may safely ignore the time interval\n[0,1] by further restricting the size of the initial data.\nThe goal is next to prove by a continuity argument that this en ergyE(t) (together with some\nrelated quantities) is uniformly bounded for all time if ǫis sufficiently small.\nWe define the following controls referred to in the sequel as t he bootstrap hypotheses for t≥1\nand some constant C(kM),\n(B1)E(t)≤10C(kM)ǫ2;\n(B2) Compact support of Ω and a:\nsuppΩ⊂/bracketleftBig\n2.5κ0−10C(kM)ǫ,1−2.5κ0+10C(kM)ǫ/bracketrightBig\n,\nsuppa⊂/bracketleftBig\n2.5κ0−10C(kM)ǫ,1−2.5κ0+10C(kM)ǫ/bracketrightBig\n;\n(B3) ‘CK’ integral estimates\n/integraldisplayt\n1/bracketleftbigg\nCKf+1\nKaCKa+CK¯h+CK∂tv+1\nKv/parenleftBig11/summationdisplay\nj=1CKϕδ\nj/parenrightBig/bracketrightbigg\ndτ≤20C(kM)ǫ2,\nProposition 6.7 (Bootstrap) .LetkMsatisfy(5.2). There exists an ǫ0∈(0,1\n2)depending only on\nkM,λ0,λ′,sandσsuch that if ǫ<ǫ0and on[1,T∗]the bootstrap hypotheses (B1)-(B3) hold, then\nfor∀t∈[1,T∗],\n1.E(t)≤8C(kM)ǫ2,\n2.Compact support of Ωanda:\nsuppΩ⊂/bracketleftBig\n2.5κ0−8C(kM)ǫ,1−2.5κ0+8C(kM)ǫ/bracketrightBig\n,\nsuppa⊂/bracketleftBig\n2.5κ0−8C(kM)ǫ,1−2.5κ0+8C(kM)ǫ/bracketrightBig\n;\n3.and theCKcontrols satisfy:\n/integraldisplayt\n1/bracketleftbigg\nCKf+1\nKaCKa+CK¯h+CK∂tv+1\nKv/parenleftBig11/summationdisplay\nj=1CKϕδ\nj/parenrightBig/bracketrightbigg\ndτ≤18C(kM)ǫ2,\nfrom which it follows that T∗= +∞.\nThe remainder of this paper is devoted to the proof of Proposi tion 6.7.\n7.Energy estimate\n7.1.Energy estimate for Ef.A direct calculation gives\n1\n2d\ndtEf+CKf\n=−1\n2π/summationdisplay\nk=0,|k|≥kM/integraldisplay\nRAk(t,η)2/hatwider(Υ2U·∇z,vΩ)k(t,η)/hatwideΩk(t,η)dη\n−1\n2π/summationdisplay\n0<|k|<kM/integraldisplay\nRAk(t,η)2F2/parenleftBig\nDu,k/parenleftBig\nF1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig/parenrightBig\nk(t,η)/hatwidefk(t,η)dη54 WEIREN ZHAO\n−1\n2π/summationdisplay\nk=0,|k|≥kM/integraldisplay\nRAk(t,η)2/hatwider(NΩ[Υ2Ψ])k(t,η)/hatwideΩk(t,η)dη\n−1\n2π/summationdisplay\nk=0,|k|≥kM/integraldisplay\nRAk(t,η)2/hatwider(Na[Π])k(t,η)/hatwideΩk(t,η)dη\n+1\n2π/summationdisplay\n|k|≥kM/integraldisplay\nRAk(t,η)2/hatwider/parenleftBig\nϕ1∂z(Υ2Ψ)/parenrightBig\nk(t,η)/hatwideΩk(t,η)dη\n+1\n2π/summationdisplay\n0<|k|<kM/integraldisplay\nRAk(t,η)2F2/parenleftBig\nDu,k/parenleftBig\nF1/parenleftBig\nϕδ\n1∂z(Υ2Ψ)−/tildewiderϕ1∂z/parenleftbig˚T−1\n1,D[Ω]−Υ2Ψ/parenrightbig/parenrightBig/parenrightBig/parenrightBig\nk(t,η)/hatwidefk(t,η)dη\n−1\n2π/summationdisplay\n0<|k|<kM/integraldisplay\nRAk(t,η)2F2/parenleftBig\nDu,k/parenleftBig\nF1/parenleftBig\nNΩ[Ψ]/parenrightBig/parenrightBig/parenrightBig\nk(t,η)/hatwidefk(t,η)dη\n−1\n2π/summationdisplay\n0<|k|<kM/integraldisplay\nRAk(t,η)2/hatwider/parenleftbig\nD(Na[Π])/parenrightbig\nk(t,η)/hatwidefk(t,η)dη=8/summationdisplay\nj=1If,j.(7.1)\nWe first treat If,2. The main idea is to recover the transport structure togethe r withIf,1. A usual\nway to recover the transport structure U ·∇z,vDΩ is to study the commutator\n/bracketleftbig\nD,U·∇z,v/bracketrightbig\nΩ.\nHowever, it is not good here, because the commutator will mix the information of different frequen-\ncies inz. The new idea is to use an important identity (4.20), which fo rmally is that\n/integraldisplay\nD(f)D1(g)dzdv=/integraldisplay\nfgdzdv, (7.2)\nwhereD1= (D−1)∗(see (4.4) for more details). More precisely, we need to cont rol the following\nterm/integraldisplay\nAD(U·∇z,vΩ)AD(Ω)dzdv.\nThe idea is to remove the wave operator Dusing (7.2) and recover the transport structure:\n/integraldisplay\nAD(U·∇z,vΩ)AD(Ω)dzdv\n=/integraldisplay\nA(U·∇z,vΩ)A(Ω)dzdv+/integraldisplay\n[A,D1](U·∇z,vΩ)DA(Ω)dzdv\n+/integraldisplay\nAD1(U·∇z,vΩ)[A,D](Ω)dzdv+/integraldisplay\nA(D−D1)(U·∇z,vΩ)AD(Ω)dzdv.\nWith the previous brief idea in mind, now we present a precise calculation.\nFor 0<|k|<kM, let\nIk\nf,2=/integraldisplay\nAk(η)F2Du,k(F1Ω)k(η)Ak(η)F2/parenleftBig\nDu,k/parenleftBig\nF1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig\n(t,k,·)/parenrightBig\n(η)dη\nthenIf,2=/summationtext\n0<|k|<kMIk\nf,2and\nIk\nf,2=/integraldisplay\nAk(η)F2Du,k(F1Ω)k(η)Ak(η)F2/parenleftBig\nDu,k/parenleftBig\nF1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig\n(t,k,·)/parenrightBig\n(η)dηNONLINEAR INVISCID DAMPING 55\n=/integraldisplay\nAk(η)F2Du,k(F1Ω)k(η)Ak(η)F2/parenleftBig\n˜χ2D1\nu,k/parenleftBig\n˜χ2F1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig\n(t,k,·)/parenrightBig\n(η)dη\n+/integraldisplay\nAk(η)F2Du,k(F1Ω)k(η)Ak(η)F2/bracketleftBig/parenleftbig\n˜χ2Du,k−˜χ2D1\nu,k/parenrightbig/parenleftBig\n˜χ2F1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig\n(t,k,·)/bracketrightBig\n(η)dη\n=/integraldisplay\nAk(η)/hatwideΩk(η)Ak(η)/parenleftbig\nU·∇z,vΩ/parenrightbig/logicalandtext\nk(η)dη\n+/integraldisplay\n(Ak(η)−Ak(ξ))D(t,k,η,ξ)ˆΩk(ξ)Ak(η)F2/parenleftBig\n˜χ2D1\nu,k/parenleftBig\n˜χ2F1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig\n(t,k,·)/parenrightBig\n(η)dξdη\n+/summationdisplay\nl/integraldisplay\nF2Du,k/parenleftbig\nF−1\n2(Ak(η)/hatwideΩk)/parenrightbig\n(η)\n×(Ak(η)−Ak(ξ))D1(t,k,η,ξ)/parenleftBig\n/hatwideUk−l(ξ−ξ1)·(il,iξ1)/hatwideΩl(ξ1)/parenrightBig\ndξ1dξdη\n+/integraldisplay\nAk(η)¯ˆfk(η)Ak(η)F2/bracketleftBig/parenleftbig\n˜χ2Du,k−˜χ2D1\nu,k/parenrightbig/parenleftBig\n˜χ2F1/parenleftBig\nU·∇z,vΩ/parenrightBig/parenrightBig\n(t,k,·)/bracketrightBig\n(η)dη\n=/integraldisplay\nAk(η)/hatwideΩk(η)Ak(η)/parenleftbig\nU·∇z,vΩ/parenrightbig/logicalandtext\nk(η)dη+Icom1\nf,2,k+Icom2\nf,2,k+Icom3\nf,2,k\nand then\nIf,1+If,2=−1\n2π/summationdisplay\nk∈Z/integraldisplay\nRAk(t,η)2/hatwider(ΥU·∇z,vΩ)k(t,η)/hatwideΩk(t,η)dη\n+/summationdisplay\n0<|k|<kM/parenleftbig\nIcom1\nf,2,k+Icom2\nf,2,k+Icom3\nf,2,k/parenrightbigdef=Itr\nf+Icom\nf,2.(7.3)\nWe have the following proposition for Itr\nf.\nProposition 7.1. Under the bootstrap hypotheses,\n|Itr\nf|/lessorsimilarǫCKf+ǫ2CKh+ǫ2CKϕδ\n9+ǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.(7.4)\nThe proposition is proved in section 11.\nNote that although the wave operator in this paper is new, we s till have the same estimates of\nthe Fourier kernels of Du,k,D1\nu,k, and ˜χ2Du,k−˜χ2D1\nu,kas that in [28], see Proposition 4.11 and\nProposition 4.12. Thus, the commutator term Icom\nf,2which involves the wave operator can be treated\nin the same way as [28].\nProposition 7.2. Under the bootstrap hypotheses,\n|Icom\nf,2|/lessorsimilarǫCKf+ǫCKh+ǫCKϕδ\n9+ǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nWe refer to the estimate of Πcomterm in [28] for the proof and more details.\nThe last term in the previous two propositions will be contro lled by the elliptic estimate.56 WEIREN ZHAO\nProposition 7.3 (Precision elliptic control) .Under the bootstrap hypotheses,\n(7.5)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP|k|≥kM(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n≤2Cell/bracketleftBig\nCKf+C(kM)ǫ2/parenleftBig\nCKh+CKϕδ\n9+CKϕδ\n4/parenrightBig/bracketrightBig\n,\nThe constant Cellis the same constant in Lemma 6.6.\nGenerally, it holds that\n(7.6)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n/lessorsimilarkMCKf+ǫ2/parenleftBig\nCKh+CKϕδ\n9+CKϕδ\n4/parenrightBig\n.\nThe basic idea of proving this proposition is to control the e lliptic error terms, which is similar\nto previous papers [5, 28]. However, a new Fourier analysis o f the Green function is required, which\nis given in Appendix C. The proof of this proposition is given in section 8.\nThe termIf,3andIf,7are arose from introducing the good unknown ˜ ω. They are lower-order\nterms since there is no derivative loss. They are also differen t from the nonlocal error term I1\nf,6or\nnonlocal small term If,7that will be discussed later. In If,3andIf,7, the function ∂z(Υ2Ψ) interacts\nwith the nonzero modes.\nProposition 7.4. Under the bootstrap hypotheses,\n|If,3|+|If,7|/lessorsimilarǫCKf+ǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nThe termIf,6contains two terms\nIf,6=1\n2π/summationdisplay\n0<|k|<kM/integraldisplay\nRAk(t,η)2F2/parenleftBig\nDu,k/parenleftBig\nF1/parenleftBig\nϕδ\n1∂z(Υ2Ψ)/parenrightBig/parenrightBig/parenrightBig\nk(t,η)/hatwidefk(t,η)dη\n+1\n2π/summationdisplay\n0<|k|<kM/integraldisplay\nRAk(t,η)2F2/parenleftBig\nDu,k/parenleftBig\nF1/parenleftBig\n/tildewiderϕ1∂z/parenleftbig˚T−1\n1,D[Ω]−Υ2Ψ/parenrightbig/parenrightBig/parenrightBig/parenrightBig\nk(t,η)/hatwidefk(t,η)dη\ndef=I1\nf,6+I2\nf,6.\nThe first term I1\nf,6can be treated as the reaction term, which is similar to Πu′′,ǫ1term in [28]. The\nsecond term I2\nf,6is relate to the elliptic error term ˚T−1\n1,D[S1[Ψ]] (see (8.5)), which is similar to Πu′′,ǫ2\nterm in [28].\nProposition 7.5. Under the bootstrap hypotheses,\n|I1\nf,6|+|I2\nf,6|/lessorsimilarǫCKf+ǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2+ǫCKϕδ\n1+ǫCKh+ǫCKϕδ\n9+ǫCKϕδ\n4\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.NONLINEAR INVISCID DAMPING 57\nThe termsIf,5have the same structure and can betreated in the same way. Ind eed,If,5is similar\nto the term Πu′′in [28]. The key observation for this term is as follows:\n1. we can treat it as the reaction term;\n2. we can get an extra smallness if |k| ≥kMwithkMsufficiently large.\nProposition 7.6. Under the bootstrap hypotheses\n|If,5| ≤2C0Cϕ1Ef(t)\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+1\n50CKf+2C0CellCϕ1\nkM/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1|∇|s/2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts∆LP|k|≥kMA(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n+2C0CellCϕ1\nkM/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1R/radicalbigg\n∂tw\nw∆L˜AP|k|≥kM(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2,\nwhere1R= 1t∈Ik,ηfor somekη>0.\nRemark 7.7. By applying (7.5)and using (5.2), we get that\n|If,5| ≤2C0Cϕ1Ef(t)\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+1\n10CKf+C(kM)ǫ2/parenleftBig\nCKh+CKϕδ\n9+CKϕδ\n4/parenrightBig\n.\nProof.Let us outline the main idea of the proof of Proposition 7.6. A s in the proof of Lemma 6.6,\nwe divideI5\nfinto high-low, low-high, and high-high interactions based on the frequencies. For the\nhigh-low and high-high interactions, we use the facts that\n/ba∇dblPk≥kMΥ2Ψ/ba∇dblGs,σ−6≤C\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblPk≥kMΩ/ba∇dblL2≤C\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2E1\n2\nf\n/ba∇dblARϕ1/ba∇dblL2≤ /ba∇dblAR/tildewiderϕ1/ba∇dblL2+/ba∇dblARϕδ\n1/ba∇dblL2≤9\n8Cϕ1,\nand obtain that\n|I5\nf,HL|+|I5\nf,HH| ≤2C0Cϕ1Ef(t)\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2.\nFor the low-high interactions, we get that\n|I5\nf,LH|/lessorsimilar/summationdisplay\nk/\\e}atio\\slash=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n|ξ−η|≤1\n8|k,ξ|A/hatwidefk(η)Ak(t,η)/parenleftBig\n/hatwiderϕ1(η−ξ)1|k|≥kM|k|/hatwider(ΨΥ)k(ξ)/parenrightBig\ndξdη/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilar/summationdisplay\nk/\\e}atio\\slash=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n|ξ−η|≤1\n8|k,ξ|[1NR,NR+1NR,R+1R,NR+1R,R]1|k|≥kM\n×A/hatwidefk(η)Ak(t,η)/parenleftBig\n/hatwiderϕ1(η−ξ)|k|/hatwider(ΨΥ)k(ξ)/parenrightBig\ndξdη/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n=I5;NR,NR\nf,LH+I5;NR,R\nf,LH+I5;R,NR\nf,LH+I5;R,R\nf,LH.\nForI5;NR,NR\nf,LHandI5;R,NR\nf,LH, we use the fact that there is Csuch that for all kM\n(7.7)/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts|k|\n/a\\}b∇acketle{tk,ξ/a\\}b∇acket∇i}hts/21t/∈Ik,ξ1|k|≥kM≤C\nkM/angbracketleftbiggξ\ntk/angbracketrightbigg−1|k,ξ|s/2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts(k2+(ξ−kt)2),\nand obtain that\n|I5;NR,NR\nf,LH|+|I5;R,NR\nf,LH| ≤C/ba∇dblϕ1/ba∇dblGs,λ,σ−6/ba∇dbl|∇|s/2Af/ba∇dbl2/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∂z|\n/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts/21NR1|k|≥kMP/\\e}atio\\slash=A(ΨΥ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n258 WEIREN ZHAO\n≤C0/10\nkM/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2s/ba∇dblϕ1/ba∇dblGs,λ,σ−6/ba∇dbl|∇|s/2Af/ba∇dbl2\n2\n+C0/10\nkM/ba∇dblϕ1/ba∇dblGs,λ,σ−6/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1|∇|s/2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts∆LP|k|≥kMA(ΨΥ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nForI5;NR,R\nf,LH, we use the fact that\n/radicalBigg\nw(t,ξ)\n∂tw(t,ξ)|k|wR(t,ξ)\nwNR(t,ξ)1|k|≥kM1t∈Ik,ξ\n≈/parenleftBig\n1+/vextendsingle/vextendsinglet−ξ\nk/vextendsingle/vextendsingle/parenrightBig1\n2|k|3/parenleftBig\n1+/vextendsingle/vextendsinglet−ξ\nk/vextendsingle/vextendsingle/parenrightBig\nξ1t∈Ik,ξ1|k|≥kM≈1\nkM/parenleftbig\nk2+(ξ−kt)2/parenrightbig/radicalBigg\n∂tw(t,ξ)\nw(t,ξ).\nand obtain that\n|I5;NR,R\nf,LH| ≤C0/10\nkM/ba∇dblϕ1/ba∇dblGs,λ,σ−6/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n∂tw\nw˜Af/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2+C0/10\nkM/ba∇dblϕ1/ba∇dblGs,λ,σ−6/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s/2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsAf/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n+C0/10\nkM/ba∇dblϕ1/ba∇dblGs,λ,σ−6/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n∂tw\nw∆L˜AP|k|≥kM(ΨΥ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nForI5;R,R\nf,LH, we use the fact that\n/radicalBigg\nwk(t,ξ)\n∂twk(t,ξ)|k|1t∈Ik,ξ1|k|≥kM/lessorsimilar|k|/radicalbigg\n1+/vextendsingle/vextendsingle/vextendsinglet−ξ\nk/vextendsingle/vextendsingle/vextendsingle1t∈Ik,ξ1|k|≥kM\n/lessorsimilar|k|/parenleftbigg\n1+/vextendsingle/vextendsingle/vextendsinglet−ξ\nk/vextendsingle/vextendsingle/vextendsingle/parenrightbigg/radicalBigg\n∂twk(t,ξ)\nwk(t,ξ)1t∈Ik,ξ1|k|≥kM/lessorsimilar/parenleftbig\nk2+(kt−ξ)2/parenrightbig\nkM/radicalBigg\n∂twk(t,ξ)\nwk(t,ξ)1t∈Ik,ξ1|k|≥kM,\nand obtain that\n|I5;R,R\nf,LH| ≤C0/10\nkM/ba∇dblϕ1/ba∇dblGs,λ,σ−6/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n∂tw\nw˜Af/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2+C0/10\nkM/ba∇dblϕ1/ba∇dblGs,λ,σ−6/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/radicalbigg\n∂tw\nw∆L˜AP|k|≥kM1R(ΨΥ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nHereC0>1 is a universal constant. By combining all the above estimat es and using the bootstrap\nassumptions, we obtain the proposition.\nOne may refer to the proof of Πu′′in [28] for more details. Indeed, the linear part of If,5is treated\nin Lemma 6.6, which is used to determine the kMbefore the bootstrap argument. /square\nNow we control the pressure. We first estimate the zero mode.\nProposition 7.8. Under the bootstrap hypotheses,\n/vextenddouble/vextenddoublet(t2+|∂v|2)(2−∂2\nv)−1A0P0/parenleftbig\n∂v(ΠΥ1)/parenrightbig/vextenddouble/vextenddouble\n2/lessorsimilarǫNONLINEAR INVISCID DAMPING 59\nand/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublet(t2+|∂v|2)(2−∂2\nv)−1/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\nA0P0/parenleftbig\n∂v(ΠΥ1)/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n/lessorsimilarǫ4\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2s+ǫ2CKa+ǫ2CKϕδ\n4\n+ǫ2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nProposition 7.8 is proved in section 9.\nThe estimates of the non-zero modes of pressure are highly no n-trivial.\nProposition 7.9. Under the bootstrap hypotheses,/vextenddouble/vextenddouble/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}ht)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig\nAP/\\e}atio\\slash=Π⋆/vextenddouble/vextenddouble\n2/lessorsimilarǫ\nand/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}ht)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig/parenleftBigg\n/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=Π⋆/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ2/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nϕδ\n6+CK1\n2\nϕδ\n7+CK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n8+CK1\n2\nϕδ\n9+CK1\n2a/parenrightBig\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublet(t2+|∂v|2)(2−∂2\nv)−1/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\nA0P0/parenleftbig\n∂v(ΠΥ1)/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2,\nwhereΠ⋆∈ {Υ2Πl,1,Υ1Πl,2}.\nProposition 7.10. Under the bootstrap hypotheses,/vextenddouble/vextenddouble/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}ht)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig\nAP/\\e}atio\\slash=Π⋆/vextenddouble/vextenddouble\n2/lessorsimilarǫ2\nand/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}ht)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig/parenleftBigg\n/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=Π⋆/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫCK1\n2\nf+ǫ2/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nϕδ\n4+CK1\n2\nϕδ\n6+CK1\n2\nϕδ\n7+CK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n8+CK1\n2\nϕδ\n9+CK1\n2a/parenrightBig\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublet(t2+|∂v|2)(2−∂2\nv)−1/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\nA0P0/parenleftbig\n∂v(ΠΥ1)/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2,\nwhereΠ⋆∈ {Υ2Πn,1,Υ1Πn,2}.\nThere are four key ideas in estimates: 1, the pressure decomp osition in section 2 ensures the\nFourier analysis works in the problem; 2, the new Fourier ana lysis of the Green function associated\nwith the linearized equation is given in Appendix C; 3, the we ll-designed weights B, M3,M4, and60 WEIREN ZHAO\nM5from [11] help control the elliptic error terms easily; 4, th e iteration method helps us gain back\nthe derivative loss in the equation of Π l,2and Πn,2, see section 10 for more details. These two\npropositions are proved in section 10.\nAs a direct corollary, we have the point-wise decay estimate for the pressure.\nCorollary 7.11. Under the bootstrap hypotheses,\n/ba∇dbl∂vP0(Υ2Π)/ba∇dblGs,σ−6/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3,\nand\n/ba∇dblP/\\e}atio\\slash=(Π⋆)/ba∇dblGs,σ−6/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4,\nwhereΠ⋆∈ {Υ2Πl,1,Υ2Πn,1,Υ1Πl,2,Υ1Πn,2}.\nThetermsIf,4andIf,8are new, which are the interactions between the density aand the pressure\nΠ. If the pressure is in higher frequencies, these terms beha ve as the Reaction term in [5]. If the\ndensityais in higher frequencies, there is a derivative loss. So we ne ed the multiplier B to gain half\nderivative. We have the following proposition for the estim ates.\nProposition 7.12. Under the bootstrap hypotheses,\n|If,4|+|If,8|/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3+ǫCKf+ǫCKa\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublet(t2+|∂v|2)(2−∂2\nv)−1/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\nA0P0/parenleftbig\n∂v(ΠΥ1)/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}ht)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig/parenleftBigg\n/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=Π⋆/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nwhereΠ⋆= Υ2Πl,1+Υ2Πn,1+Υ1Πl,2+Υ1Πn,2.\nThe proposition is proved in section 14.\n7.2.Energy estimate of Ea.A direct calculation gives\n1\n2d\ndtEa+CKa=−/integraldisplay\nA∗aA∗(Υ2U·∇z,va)dzdv−/integraldisplay\nA∗aA∗/parenleftbig\nθ′∂z(Υ2Ψ)/parenrightbig\ndzdv\ndef=Itr\na+Il\na.\nWe have the following proposition for the estimate of Itr\na.\nProposition 7.13. Under the bootstrap hypotheses,\n|Itr\na|/lessorsimilarǫCKa+ǫ2CKh+ǫ2CKϕδ\n9+ǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.(7.8)\nThe proposition is proved in section 11.\nThe linear term force Il\nais new because of the non-constant background density θ. However, it\nisUy∂yθwhich can be regarded as the Reaction term Ra\nNbut without smallness.NONLINEAR INVISCID DAMPING 61\nProposition 7.14. Under the bootstrap hypotheses,\n|Il\na| ≤Cǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+CǫCKϕδ\n10+ε0CKa\n+Cε0/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nThe proposition is proved in section 13.\n7.3.Energy estimates of the coordinate system and coefficients. The coordinate system is\neasy to control, which is similar to Proposition 2.5 in [5].\nProposition 7.15 (Coordinate system controls) .Under the bootstrap hypotheses, for ǫsufficiently\nsmall and Kvsufficiently large there is a K>0such that\nEh+1\n4/integraldisplayt\n1CKh(τ)dτ≤1\n2Kvǫ2, (7.9)\nE¯h+/integraldisplayt\n1CK¯h(τ)+dτ≤C(kM)ǫ2+Kǫ3, (7.10)\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4−KDǫ/ba∇dbl∂tv/ba∇dbl2\nGs,λ(t),σ−6≤C(kM)ǫ2+Kǫ3. (7.11)\nProof.Weusetotheestimateof ωinCorollary8.4tocontroltheterm P0(∂z(Υ2Ψ)∂vω−∂v(Υ2Ψ)∂zω).\nThe term which involves P0(∂zΠl∂va−∂vΠl∂za) andP0(∂zΠn∂va−∂vΠn∂za) can be controlled by\nusing the same idea of the proof of Proposition 7.12 in sectio n 14. /square\nProposition 7.16. Under the bootstrap hypotheses, for ǫsufficiently small and Kvsufficiently large\nthere is a K>0such that for j= 1,2,...,11,\nEϕδ\nj+1\n4/integraldisplayt\n1CKϕδ\nj(τ)dτ≤1\n2Kvǫ2. (7.12)\nThe energy estimates for the coefficients ϕδ\nj(j=1,...,11) is easier than h. They have the same\ntransport structure as the equation of h:\n(7.13) ( ∂t+∂tv∂v)U=F,\nwhereFrepresents one of the forcing terms ∂tv∂v/tildewiderϕj. Due to the fact that /tildewiderϕjare smoother given\nfunctions compared to the solutions, Fhas the same behavior as ∂tvwhich is better than the force\nterm¯h=∂yv∂v∂tvin the equation of h(5.5). By following the estimate of hinsection 8.2 of [5],\none can obtain the estimate of Eϕδ\nj(t) easily.\n7.4.Proof of the main theorem. Now, we prove the compact support result in Proposition 6.7\nand conclude the proof. Under the bootstrap assumption, we h ave\n∆ψ(t,x,y) =ω(t,x,y) =ω(t,x−tv(t,y),v(t,y)), ψ(x,0) =ψ(x,1) = 0.\nTaking the Fourier transform in x, we get\nF1ψ(t,k,y) =/integraldisplay1\n0Gk(y,z)ω(t,k,v(t,z))e−iktv(t,z)dz, (7.14)\nand\n∂yF1ψ(t,k,y) =/integraldisplay1\n0∂yGk(y,z)ω(t,k,v(t,z))e−iktv(t,z)dz, (7.15)62 WEIREN ZHAO\nwhere\nGk(y1,y2) =1\nksinhk×/braceleftBigg\nsinh(k(1−y2))sinhky1, y1≤y2,\nsinh(ky2)sinh(k(1−y1)), y1≥y2,\nand\nG0(y1,y2) =/braceleftBigg\n(1−y2)y1, y1≤y2,\ny2(1−y1), y1≥y2,\nIntegrating by parts in zin the identities (7.14)(twice) and (7.15) (once), we get th at\n|Uy(t,x,y)|/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2,|Ux−<Ux>|/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht. (7.16)\nThen let us define\n\nd(X1,X2)\ndt(t,x,y) = (u(y)+Ux(t,X1,X2),Uy(t,X1,X2)),\n(X1,X2)/vextendsingle/vextendsingle\nt=0= (x,y).\nThend\ndtω(t,X1,X2) = (u′′∂xψ)(t,X1,X2) and|X2(t,x,y)−y| ≤Cǫ, thus the for y /∈[4κ0−Cǫ,1−\n4κ0+Cǫ],d\ndtω(t,X1,X2) = 0. Note that weonly need that u′′tobeequal to0 close to the boundary.\nThus the support of ω(t) will always be away from the boundary.\nThis ends the proof of the bootstrap Proposition 6.7. /square\nLet us now go back to the original coordinate system and prove Theorem 1.1. Applying the same\nmethod in section 2.4 of [5], we get for ω1(t,z,y) =ω(t,z,v) =ω(t,x,y),P1(t,z,y) =P(t,x,y) =\nΠ(t,z,v),d1(t,z,y) =d(t,x,y) =a(t,z,v) andψ1(t,z,y) = Ψ(t,z,v) =ψ(t,x,y),\n∂tω1+∇⊥\nz,yP/\\e}atio\\slash=ψ1·∇z,yω1−u′′∂zψ1−θ′(y)∂zP1+∂yP1∂zd1−∂zP1∂ya= 0,\nand\n∂td1+θ′∂zψ1+∇⊥\nz,yP/\\e}atio\\slash=ψ1·∇z,yd1= 0\nThen we can define\nf∞=ω1(1)−/integraldisplay∞\n1∇⊥\nz,yP/\\e}atio\\slash=ψ1(s)·∇z,yω1(s)ds+/integraldisplay∞\n1u′′∂zψ1(s)ds\n+/integraldisplay∞\n1θ′(y)∂zP1(s)ds−/integraldisplay∞\n1∂yP1∂zd1(s)ds+/integraldisplay∞\n1∂zP1∂ya(s)ds,\nand\nd∞=d1(1)−/integraldisplay∞\n1θ′∂zψ1(s)ds−/integraldisplay∞\n1∇⊥\nz,yP/\\e}atio\\slash=ψ1·∇z,yd1(s)ds.\nTherefore there exists f∞,d∞such that,\n/ba∇dblω(t,x+tu(y)+Φ(t,y)χ1(y),y)−f∞(x,y)/ba∇dblGs,λ∞/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht.\n/ba∇dbld(t,x+tu(y)+Φ(t,y)χ1(y),y)−d∞(x,y)/ba∇dblGs,λ∞/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht.\nNote thatω(t,x,y) = 0 fory /∈[2κ0,1−2κ0] so isf∞, thus we can replace Φ( t,y)χ1(y) by Φ(t,y)\nwhich gives (1.4) and (1.5).\nNow, we prove (1.7). We have\n−∂yP0(Ux) =P0(ω), ∂tP0(Ux)+P0(Uyω) = 0,NONLINEAR INVISCID DAMPING 63\nwhich implies that P0(Ux)(t,y) =CUfor anyt≥1 andy∈[0,2κ0]∪[1−2κ0,1]. Let˜Ux(t,z,y) =\nUx(t,x,y), then/parenleftBig\nP0(˜Ux)(t,y)−CU/parenrightBig\nhas the same compact support as ω1and satisfies\n∂t/parenleftBig\nP0(˜Ux)(t,y)−CU/parenrightBig\n=−P0(∇⊥\nz,yP/\\e}atio\\slash=ψ1·∇z,y˜Ux)χ1(y)\n=−P0(−∂yP/\\e}atio\\slash=(Υ2ψ1)∂z(∂y−t∂yv∂z)(Υ2ψ1)+∂zP/\\e}atio\\slash=(Υ2ψ1)∂y∂z(Υ2ψ1)).\nThen (1.7) follows from the fact that\n/ba∇dblΥ2ψ1/ba∇dblGs,λ′∞,2/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2. (7.17)\nThis ends the proof of Theorem 1.1. /square\n8.Elliptic estimate for steam function\nIn this section, we study the following elliptic equation\n(8.1)\n\nϕ4∂zzΨ+∂yv(∂v−t∂z)/parenleftBig\nϕ4∂yv(∂v−t∂z)Ψ/parenrightBig\n= Ω,\nΨ(t,z,u(0)) = Ψ(t,z,u(1)) = 0,\nWe rewrite the equation by separating its linear part and the nonlinear part.\n(8.2)Ω =ϕ4∂zzΨ+∂yv(∂v−t∂z)/parenleftBig\nϕ4∂yv(∂v−t∂z)Ψ/parenrightBig\n=T1[Ψ]+ϕδ\n4∂zzΨ+∂yv(∂v−t∂z)/parenleftBig\nϕ4(∂yv−/tildewideu′)(∂v−t∂z)Ψ/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\nϕδ\n4/tildewideu′(∂v−t∂z)Ψ/parenrightBig\n+(∂yv−/tildewideu′)(∂v−t∂z)/parenleftBig\n/tildewiderϕ4/tildewideu′(∂v−t∂z)Ψ/parenrightBig\n.\nwhere\nT1[Ψ] =/tildewiderϕ2∂zzΨ+/tildewideu′(∂v−t∂z)/parenleftBig\n/tildewiderϕ2/tildewideu′(∂v−t∂z)Ψ/parenrightBig\n. (8.3)\nIt is easy to see that T1is an elliptic second-order differential operator. We define T−1\n1,Dto be the\ninverseoperatorwithDirichlet boundaryconditions, name lyforanyfunction f(t,z,v),T1/bracketleftbig\nT−1\n1,D[f]/bracketrightbig\n=\nfwith\nT−1\n1,D[f](t,z,0) =T−1\n1,D[f](t,z,1)\nWe also define\n˚T−1\n1,D= Υ2T−1\n1,D. (8.4)\nProposition 8.1. LetΩbe compactly supported. Then there is GD,1(t,k,ξ,η)such that\n/hatwider(˚T−1\n1,D[Ω])(t,k,ξ) =/integraldisplay\nRGD,1(t,k,ξ,η)/hatwideΩ(t,k,η)dη,\nwith estimate\n|GD,1(t,k,ξ,η)| ≤Cmin/braceleftBigg\ne−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(ξ−kt)2,e−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(η−kt)2/bracerightBigg\n.\nThe proposition is proved in Appendix C.1.\nBy applying T−1\n1,Don equation (8.2) and multiplying the cut-off function Υ 2, we get that\n˚T−1\n1,D[Ω]−Υ2Ψ =˚T−1\n1,D[S1[Ψ]] (8.5)64 WEIREN ZHAO\nwhere\n(8.6)S1[Ψ] =ϕδ\n4∂zzΨ+∂yv(∂v−t∂z)/parenleftBig\nϕ4(∂yv−/tildewideu′)(∂v−t∂z)Ψ/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\nϕδ\n4/tildewideu′(∂v−t∂z)Ψ/parenrightBig\n+(∂yv−/tildewideu′)(∂v−t∂z)/parenleftBig\n/tildewiderϕ4/tildewideu′(∂v−t∂z)Ψ/parenrightBig\n=ϕδ\n4∂zzΨ+h1(∂v−t∂z)2(Υ2Ψ)+h2(∂v−t∂z)(Υ2Ψ)\nhas compact support with supp S1[Ψ]⊂[κ0,1−κ0] with\nh1=∂yvϕ4(∂yv−/tildewideu′)+∂yvϕδ\n4/tildewideu′+(∂yv−/tildewideu′)/tildewiderϕ4/tildewideu′\nh2=∂yv∂v/parenleftbig\nϕ4(∂yv−/tildewideu′)/parenrightbig\n+∂yv∂v/parenleftbig\nϕδ\n4/tildewideu′/parenrightbig\n+(∂yv−/tildewideu′)∂v(/tildewiderϕ4/tildewideu′).\n8.1.Lossy estimate. We have the following lemmas.\nLemma 8.2. It holds for all 0≤s1≤σ−1that\n/ba∇dbl∂2\nzΥ2P/\\e}atio\\slash=Ψ/ba∇dblGs,λ(t),s1−2/lessorsimilar/ba∇dblΩ/ba∇dblGs,λ(t),s1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2. (8.7)\nWith the estimate of the Fourier kernel of ˚T−1\n1,D, the lossy estimate is simple. We refer to the\nproof of Lemma 3.4 in [28].\n8.2.Precision elliptic control. The precision elliptic control is slightly more complicate d than\nthat in [28]. We have more terms here. The idea is to show that S1[Ψ] is small. We have the\nfollowing proposition.\nProposition 8.3. Under the bootstrap hypotheses,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP⋆/parenleftbig˚T−1\n1,D[S1[Ψ]]/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n/lessorsimilarǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP⋆(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n+ǫCKh+ǫCKϕδ\n9+ǫCKϕδ\n4,(8.8)\nand\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)P⋆/parenleftbig˚T−1\n1,D[S1[Ψ]]/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n/lessorsimilarǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)P⋆(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2+ǫ2,(8.9)\nwhereP⋆∈/braceleftbig\nP0<|k|<kM,P|k|≥kM/bracerightbig\nis the projecction.\nProof.By applying Proposition 8.1, we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP⋆/parenleftbig˚T−1\n1,D[S1[Ψ]]/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilar/vextenddouble/vextenddoubleM1/parenleftbig\n[S1[Ψ]]/parenrightbig/vextenddouble/vextenddouble\n2+/vextenddouble/vextenddoubleM2/parenleftbig\n[S1[Ψ]]/parenrightbig/vextenddouble/vextenddouble\n2NONLINEAR INVISCID DAMPING 65\nand/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)P⋆/parenleftbig˚T−1\n1,D[S1[Ψ]]/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilar/vextenddouble/vextenddoubleM0/parenleftbig\n[S1[Ψ]]/parenrightbig/vextenddouble/vextenddouble\n2\nwhere\nM0(t,∇) =/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\nAP⋆,\nM1(t,∇) =/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1|∇|s/2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsAP⋆,\nM2(t,∇) =/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1/radicalbigg\n∂tw\nw˜AP⋆.\nLet\nT1=h1(∂v−t∂z)2(ΨΥ2),T2=h2(∂v−t∂z)(ΨΥ2),T3=ϕδ\n4∂zz(Υ2Ψ). (8.10)\nDivide each via a paraproduct decomposition in the vvariable only\nT1=/summationdisplay\nM≥8(h1)M(∂v−t∂z)2(ΨΥ)<M/8+/summationdisplay\nM≥8(h1)<M/8(∂v−t∂z)2(ΨΥ)M (8.11)\n+/summationdisplay\nM∈D/summationdisplay\n1\n8M≤M′≤8M(h1)M′(∂v−t∂z)2(ΨΥ)M\n= T1\nHL+T1\nLH+T1\nR,\nT2=/summationdisplay\nM≥8(h2)M(∂v−t∂z)(ΨΥ)<M/8+/summationdisplay\nM≥8(h2)<M/8(∂v−t∂z)(ΨΥ)M (8.12)\n+/summationdisplay\nM∈D/summationdisplay\n1\n8M≤M′≤8M(h2)M′(∂v−t∂z)(ΨΥ)M\n= T2\nHL+T2\nLH+T2\nR,\nT3=/summationdisplay\nM≥8(ϕδ\n4)M(∂z)2(ΨΥ)<M/8+/summationdisplay\nM≥8(ϕδ\n4)<M/8(∂z)2(ΨΥ)M (8.13)\n+/summationdisplay\nM∈D/summationdisplay\n1\n8M≤M′≤8M(ϕδ\n4)M′(∂z)2(ΨΥ)M\n= T3\nHL+T3\nLH+T3\nR.\nNote that T1=h1(∂v−t∂z)2(ΨΥ) has a similar form as the term T1= (1−(v′)2)(∂v−t∂z)2φin [5].\nThe support of the functions is away from the boundary, which guarantees that the Fourier analysis\nworks well. Similarly, the term T2has the same structure. The new term T3here is even better\nthan T1. The key point in proof of Proposition 2.4, section 4.2.1 — 4.2.3 in [5] is that the Fourier\nmultipliers M0,M1andM2have some algebraic type properties. It is the same here. Thu s we\nhave\n3/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleM0Tj\nLH/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarǫ2/ba∇dblM0∆LP⋆(ΨΥ)/ba∇dbl2\n2\n3/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleM1Tj\nLH/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarǫ2/ba∇dblM1∆LP⋆(ΨΥ)/ba∇dbl2\n266 WEIREN ZHAO\n3/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleM2Tj\nLH/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarǫ2/ba∇dblM1∆LP⋆(ΨΥ)/ba∇dbl2\n2+ǫ2/ba∇dblM2∆LP⋆(ΨΥ)/ba∇dbl2\n2,\nand\n3/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleM0Tj\nHL/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarǫ2/ba∇dblM0∆LP⋆(ΨΥ)/ba∇dbl2\n2+ǫ4,\n3/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleM1Tj\nHL/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarǫ2/ba∇dblM1∆LP⋆(ΨΥ)/ba∇dbl2\n2+ǫ2CK2\nh+ǫ2CK2\nϕδ\n4+ǫ2CK2\nϕδ\n9,\n3/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleM2Tj\nHL/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarǫ2/ba∇dblM2∆LP⋆(ΨΥ)/ba∇dbl2\n2+ǫ2/ba∇dblM1∆LP⋆(ΨΥ)/ba∇dbl2\n2\n+ǫ2CK1\nh+ǫ2CK1\nϕδ\n4+ǫ2CK2\nϕδ\n9,\nand\n3/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleM0Tj\nR/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarǫ2/ba∇dblM0∆LP⋆(ΨΥ)/ba∇dbl2\n2,\n3/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleM1Tj\nR/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarǫ2/ba∇dblM1∆LP⋆(ΨΥ)/ba∇dbl2\n2,\n3/summationdisplay\nj=1/vextenddouble/vextenddouble/vextenddoubleM2Tj\nR/vextenddouble/vextenddouble/vextenddouble2\n2/lessorsimilarǫ2/ba∇dblM1∆LP⋆(ΨΥ)/ba∇dbl2\n2+ǫ2/ba∇dblM2∆LP⋆(ΨΥ)/ba∇dbl2\n2,\nwhere ∆L=∂2\nz+(∂v−t∂z)2. Thus we prove the proposition. /square\nNow we prove Proposition 7.3.\nProof.Weconsider thecase |k| ≥kMfirst. Inthiscase, Pkf=PkΩ. Byusing(8.5), andProposition\n8.3, we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP|k|≥kM(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n≤CellCKf+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP|k|≥kM/parenleftbig˚T−1\n1,D[S1[Ψ]]/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n≤CellCKf+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP|k|≥kM(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n+ǫCKh+ǫCKϕδ\n9+ǫCKϕδ\n4\nwhich gives (7.5).\nFor the case 0 <|k|<kM, we havePkf(t,z,v) =Du,k[F1Ω(t,k,·)](t,k,v)eizkand thus\nPkΩ(t,z,v) =D−1\nu,k[F1f(t,k,·)](t,k,v)eizk.NONLINEAR INVISCID DAMPING 67\nBy following the same argument, we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP|k|≥kM(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2\n≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP|k|≤kM(Ω)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2+ǫCKh+ǫCKϕδ\n9+ǫCKϕδ\n4.\nTo control the first term by CKf, we use Remark 6.3. Thus we prove the proposition. /square\nWeconcludethissectionbyintroducingthefollowingcorol lary, whichwillbeusedintheestimates\nof pressure terms.\nCorollary 8.4. Under the bootstrap hypotheses,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)AP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarǫ,\nand\n/ba∇dbl/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}htA0P0(Υ2Ψ)/ba∇dbl2/lessorsimilarǫ,/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}htA0P0(Υ2Ψ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarCK1\n2\nf+ǫ/parenleftBig\nCK1\n2\nh+CK1\n2\nϕδ\n4+CK1\n2\nϕδ\n9/parenrightBig\n.\nMoreover, we have estimate for ω(t,z,v) =θΩ+ϕ2∂yv(∂v−t∂z)Ψ. It holds that\n/ba∇dblAω/ba∇dblL2/lessorsimilarǫ,\nand/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig/radicalbigg\n∂tw\nwA+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts˜A/parenrightBig\nω/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarCK1\n2\nf+ǫ/parenleftBig\nCK1\n2\nh+CK1\n2\nϕδ\n2+CK1\n2\nϕδ\n4+CK1\n2\nϕδ\n9/parenrightBig\n.\nProof.The estimates for nonzero modes and the estimates for ωin the corollary follow directly from\n(8.5), (8.9) and Remark 6.2 and the estimate of zero mode of Υ 2Ψ.\nFor the zero mode, similarly, we have the decomposition\n(8.14) P0(Ω) =P0(T1[Ψ])+h1∂vvP0(Υ2Ψ)+h2∂vP0(Υ2Ψ),\nwhere\nP0(T1[Ψ]) =/tildewideu′∂v/parenleftBig\n/tildewiderϕ2/tildewideu′∂vP0(Ψ)/parenrightBig\n. (8.15)\nWe then have\n/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}htA0P0(Υ2Ψ) =/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}ht˚T−1\n1,D/bracketleftBig\nP0(Ω)−h1∂vvP0(Υ2Ψ)−h2∂vP0(Υ2Ψ)/bracketrightBig\n.\nNotice that Proposition 8.1 holds also for k= 0. Note that for |η| ≥2|ξ−η|\nA0(ξ)\nAR(η)/lessorsimilarecλ/a\\}bracketle{tξ−η/a\\}bracketri}hts,for some 0 <c<1.\nThus we conclude that\n/ba∇dbl/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}htA0P0(Υ2Ψ)/ba∇dbl2/lessorsimilar/vextenddouble/vextenddouble/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}ht−1A0P0Ω/vextenddouble/vextenddouble\n2+/vextenddouble/vextenddouble/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}ht−1A0(h1∂vvP0(Υ2Ψ))/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddouble/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}ht−1A0(h2∂vP0(Υ2Ψ))/vextenddouble/vextenddouble\nL2\n/lessorsimilar/ba∇dblA0P0Ω/ba∇dbl2+ǫ/ba∇dbl/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}htA0P0(Υ2Ψ)/ba∇dbl268 WEIREN ZHAO\n+/parenleftBig\n/ba∇dblARh1/ba∇dblL2+/ba∇dblAR/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}ht−1h2/ba∇dblL2/parenrightBig\n/ba∇dblP0(Υ2Ψ)/ba∇dblGs,σ−6.\nSimilarly, we have/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}htA0P0(Υ2Ψ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarCK1\n2\nf+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}ht−1A0/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n(h1∂vvP0(Υ2Ψ))/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}ht−1A0/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n(h2∂vP0(Υ2Ψ))/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilarCK1\n2\nf+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}htA0P0(Υ2Ψ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+/parenleftBig\nCK1\n2\nh+CK1\n2\nϕδ\n4+CK1\n2\nϕδ\n9/parenrightBig\n/ba∇dblP0(Υ2Ψ)/ba∇dblGs,σ−6,\nwhich gives the corollary. /square\nTheCorollary8.4gives theestimates of ω(t,z,v) whichisthesameasΩ. Theestimates ofthezero\nmode of the stream function P0(Υ2Ψ) are used in the estimate of A4=χ′\n2∂yv∂vP0(Υ2Ψ)∂zz(Υ2Ψ)\nin the estimates of the pressure, see Lemma 10.7.\n9.Zero mode of the pressure\nIn this section, we estimate the zero mode of the pressure. Un like the whole space case, we will\nnot use the elliptic equation to study the zero mode of the pre ssure. Instead, we use (5.7i). Let us\ndefine\nM00(t,ξ) =t(t2+|ξ|2)\n2+ξ2A0(ξ),\nM01(t,ξ) =t(t2+|ξ|2)\n2+ξ2/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{tξ/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\nA0(ξ)\nBy applying M00andM01on (5.7i), we have for j= 0,1\n(9.1)M0j∂vP0(ΠΥ1)\n=−M0j/parenleftBig\n(ϕ4Υ2)P0(a)∂vP0(ΠΥ1)/parenrightBig\n−M0j/parenleftBig\n(ϕ4Υ2)P0/bracketleftbig\nP/\\e}atio\\slash=(a)P/\\e}atio\\slash=(∂v−t∂z)P/\\e}atio\\slash=(ΠΥ1)/bracketrightbig/parenrightBig\n+M0j/bracketleftBig\n(ϕ4Υ2)P0/parenleftBig\n∂z(ΨΥ2)∂z(∂v−t∂z)(ΨΥ2)−∂zz(ΨΥ2)(∂v−t∂z)(ΨΥ2)/parenrightBig/parenrightBig/bracketrightBig\n.\nWe first show that the first two terms in (9.1) are small.\nProposition 9.1. Under the bootstrap hypotheses,/vextenddouble/vextenddouble/vextenddoubleM00/parenleftBig\n(ϕ4Υ2)P0(a)∂vP0(ΠΥ1)/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarǫ/ba∇dblM00∂vP0(ΠΥ1)/ba∇dblL2,\n/vextenddouble/vextenddouble/vextenddoubleM00/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}ht/parenleftBig\n(ϕ4Υ2)P0/bracketleftbig\nP/\\e}atio\\slash=(a)P/\\e}atio\\slash=(∂v−t∂z)P/\\e}atio\\slash=(ΠΥ1)/bracketrightbig/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarǫ2,\n/vextenddouble/vextenddouble/vextenddoubleM01/parenleftBig\n(ϕ4Υ2)P0(a)∂vP0(ΠΥ1)/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarǫ/ba∇dblM01∂vP0(ΠΥ1)/ba∇dblL2+ǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+ǫCK1\n2a\n/vextenddouble/vextenddouble/vextenddoubleM01/parenleftBig\n(ϕ4Υ2)P0/bracketleftbig\nP/\\e}atio\\slash=(a)P/\\e}atio\\slash=(∂v−t∂z)P/\\e}atio\\slash=(ΠΥ1)/bracketrightbig/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2NONLINEAR INVISCID DAMPING 69\n/lessorsimilarǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+ǫ/parenleftbig\nCK1\n2a+CK1\n2\nϕδ\n4/parenrightbig\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(1+/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}ht+|∂v|)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig\nAP/\\e}atio\\slash=(ΠΥ1)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2.\nProof.It is easy to check that\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3/ba∇dbl∂vP0(ΠΥ1)/ba∇dblGs,σ−6/lessorsimilar/ba∇dblM00∂vP0(ΠΥ1)/ba∇dblL2.\nWe have/vextenddouble/vextenddouble/vextenddoubleM00/parenleftBig\n(ϕ4Υ2)P0(a)∂vP0(ΠΥ1)/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilar/ba∇dblAR(ϕ4Υ2)/ba∇dblL2/ba∇dblA∗a/ba∇dblL2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3/ba∇dbl∂vP0(ΠΥ1)/ba∇dblGs,σ−6+ǫ/ba∇dblM00∂vP0(ΠΥ1)/ba∇dblL2\n/lessorsimilarǫ/ba∇dblM00∂vP0(ΠΥ1)/ba∇dblL2.\nand/vextenddouble/vextenddouble/vextenddoubleM01/parenleftBig\n(ϕ4Υ2)P0(a)∂vP0(ΠΥ1)/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilar/ba∇dbl(ϕ4Υ2)/ba∇dblGs,σ−6CK1\n2a/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3/ba∇dbl∂vP0(ΠΥ1)/ba∇dblGs,σ−6+ǫ/ba∇dblM01∂vP0(ΠΥ1)/ba∇dblL2\n+ǫ/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nϕδ\n4/parenrightBig\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dbl∂vP0(ΠΥ1)/ba∇dblGs,σ−6\n/lessorsimilarǫ/ba∇dblM01∂vP0(ΠΥ1)/ba∇dblL2+ǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+ǫCK1\n2a.\nWe also have for |ξ−η| ≤1\n2|η|\nt(t2+|ξ|2)\n2+ξ22+k2+η2\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)/lessorsimilar(t+|ξ|)(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)2+k2+η2\n2+ξ2\n/lessorsimilar/a\\}b∇acketle{tk,ξ−η/a\\}b∇acket∇i}ht3/parenleftBig\n|η−kt|+|k|/parenrightBig\n1|η−kt|≥1\n2|kt|+|η|\n|k|1|η−kt|≤1\n2|kt|,t∈Ik,η+|η−kt|1|η−kt|≤1\n2|kt|,t/∈Ik,η\nand by (D.1), we get\nA0(ξ)\nAk(η)/lessorsimilar/parenleftbigg|k|(|k|+|η−kt|)\n|ξ|1t∈Ik,η+1/parenrightbigg\necλ|η−ξ|se−cλ|k|s. (9.2)\nThe case |ξ−η| ≥1\n2|η|is simple, in which case, we switch A 0(ξ) to A∗\n−k(ξ−η). Thus we have\n/vextenddouble/vextenddouble/vextenddoubleM00/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}ht/parenleftBig\n(ϕ4Υ2)P0/bracketleftbig\nP/\\e}atio\\slash=(a)P/\\e}atio\\slash=(∂v−t∂z)P/\\e}atio\\slash=(ΠΥ1)/bracketrightbig/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilar/ba∇dblAR(ϕ4Υ2)/ba∇dblL2/ba∇dblA∗a/ba∇dblL2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4/ba∇dblP/\\e}atio\\slash=(ΠΥ1)/ba∇dblGs,σ−6\n+ǫ/vextenddouble/vextenddouble/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(1+/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}ht+|∂v|)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig\nAP/\\e}atio\\slash=(ΠΥ1)/vextenddouble/vextenddouble\nL2/lessorsimilarǫ2\nand by Lemma D.2\n/vextenddouble/vextenddouble/vextenddoubleM01/parenleftBig\n(ϕ4Υ2)P0/bracketleftbig\nP/\\e}atio\\slash=(a)P/\\e}atio\\slash=(∂v−t∂z)P/\\e}atio\\slash=(ΠΥ1)/bracketrightbig/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilar/ba∇dbl(ϕ4Υ2)/ba∇dblGsCK1\n2a/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4/ba∇dblP/\\e}atio\\slash=(ΠΥ1)/ba∇dblGs,σ−6\nǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(1+/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}ht+|∂v|)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig\nAP/\\e}atio\\slash=(ΠΥ1)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL270 WEIREN ZHAO\n+ǫ/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nϕδ\n4/parenrightBig\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4/ba∇dblP/\\e}atio\\slash=(ΠΥ1)/ba∇dblGs,σ−6\n/lessorsimilarǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+ǫ/parenleftbig\nCK1\n2a+CK1\n2\nϕδ\n4/parenrightbig\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(1+/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}ht+|∂v|)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig\nAP/\\e}atio\\slash=(ΠΥ1)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2.\nThus, we prove the proposition. /square\nWe now prove Proposition 7.8.\nProof.By Proposition 9.1, we only need to control the last term in (9 .1). Indeed, we only need to\nconsider\nIj=M0j/bracketleftBig\n(ϕ4Υ2)P0/parenleftBig\n∂zz(ΨΥ2)(∂v−t∂z)(ΨΥ2)/parenrightBig/parenrightBig/bracketrightBig\ndue to the projection P0. By using the same idea in the proof of Proposition 7.8, we hav e for\n|ξ−η| ≤1\n2|η|\nt(t2+|ξ|2)\n2+ξ2/angbracketleftBigη\nkt/angbracketrightBig\n≈t(t2+|ξ|2)\n2+ξ2|η|+|kt|\n|kt|\n/lessorsimilar/parenleftBig\n|η−kt|+|k|/parenrightBig\n1|η−kt|≥1\n2|kt|+|η|\n|k|1|η−kt|≤1\n2|kt|,t∈Ik,η+|η−kt|1|η−kt|≤1\n2|kt|,t/∈Ik,η.\nThen by (9.2),\n/ba∇dblI0/ba∇dblL2/lessorsimilar/ba∇dblAR(ϕ4Υ2)/ba∇dblL2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4/ba∇dblP/\\e}atio\\slash=(ΨΥ2)/ba∇dbl2\nGs,σ−6\n+/ba∇dbl(ϕ4Υ2)/ba∇dblGs,σ−6/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblP/\\e}atio\\slash=(ΨΥ2)/ba∇dblGs,σ−6/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)AP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ2,\nand by Lemma D.2\n/ba∇dblI1/ba∇dblL2/lessorsimilar/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2a+CK1\n2\nϕδ\n4/parenrightBig\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4/ba∇dblP/\\e}atio\\slash=(ΨΥ2)/ba∇dbl2\nGs,σ−6\n+/ba∇dbl(ϕ4Υ2)/ba∇dblGs,σ−6/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblP/\\e}atio\\slash=(ΨΥ2)/ba∇dblGs,σ−6\n×/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+ǫ2CK1\n2a+ǫ2CK1\n2\nϕδ\n4\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2,\nwhich gives Proposition 7.8. /squareNONLINEAR INVISCID DAMPING 71\n10.Elliptic estimate for the pressure\nIn this section, we study the following elliptic equation\n(10.1)\n\n∂z/parenleftBig\n(a+θ)∂zΠ/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Π/parenrightBig\n= Ξ,\n(∂v−t∂z)Π(t,z,u(0)) = (∂v−t∂z)Π(t,z,u(1)) = 0,\nwhere Π ∈ {Πl,1,Πl,2,Πn,1,Πn,2}and Ξ represents the right-hand side of equations (2.18) res pec-\ntively.\nWe rewrite the equation by separating its linear part and the nonlinear part.\nΞ =T2[Π]+∂z/parenleftBig\n(a+θδ)∂zΠ/parenrightBig\n+(∂yv−/tildewideu′)(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)Π/parenrightBig\n+/tildewideu′(∂v−t∂z)/parenleftBig\n(a+θδ)∂yv(∂v−t∂z)Π/parenrightBig\n+/tildewideu′(∂v−t∂z)/parenleftBig\n/tildewideθ(∂yv−/tildewideu′)(∂v−t∂z)Π/parenrightBig\n,\nwhere\n(10.2) T2[Π] =/tildewideθ∂zzΠ+/tildewideu′(∂v−t∂z)/parenleftBig\n/tildewideθ/tildewideu′(∂v−t∂z)Π/parenrightBig\nSimilar to T1, we define T−1\nNto be the inverse operator of T2with Neumann boundary conditions,\nnamely for any function f(t,z,v),T2/bracketleftbig\nT−1\nN[f]/bracketrightbig\n=fwith\n∂vT−1\nN[f](t,z,0) =∂vT−1\nN[f](t,z,1)\nWe also define\n˚T−1\nj,N= ΥjT−1\nN, j= 1,2. (10.3)\nProposition 10.1. LetΞbe compactly supported. Then for k/\\e}atio\\slash= 0there isGN,j(t,k,ξ,η)such that\nforj= 1,2,\n/hatwider(˚T−1\nj,N[Ξ])(t,k,ξ) =/integraldisplay\nRGN,j(t,k,ξ,η)/hatwideΞ(t,k,η)dη,\nwith estimate\n|GN,j(t,k,ξ,η)| ≤Cmin/braceleftBigg\ne−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(ξ−kt)2,e−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(η−kt)2/bracerightBigg\n.\nThe proposition is proved in Appendix C.3.\nNext, werewritethepressureequation byseparatingintera ctions with zero-modeandinteractions\nwitha.\n(10.4) Ξ =T2[Π]+S2[Π]+Sa[Π],\nwhere the ‘pressure error’ terms are\nS2[Π] =θδ∂zz(Υ2Π)+g1(∂v−t∂z)2(Υ2Π)+g2(∂v−t∂z)(Υ2Π),\nSa[Π] =∂z/parenleftbig\na∂z(Υ2Π)/parenrightbig\n+g3(∂v−t∂z)/parenleftbig\na(∂v−t∂z)(Υ2Π)/parenrightbig\n+g4a(∂v−t∂z)(Υ2Π).\nwith\ng1= (∂yv−/tildewideu′)θ∂yv+˜u′θδ∂yv+/tildewideu′˜θ(∂yv−/tildewideu′)\ng2= (∂yv−/tildewideu′)∂v(θ∂yv)+˜u′∂v(θδ∂yv)+/tildewideu′∂v(˜θ(∂yv−/tildewideu′))\ng3= (∂yv−/tildewideu′)∂yv+˜u′∂yv+/tildewideu′(∂yv−/tildewideu′)72 WEIREN ZHAO\ng4= (∂yv−/tildewideu′)∂v(∂yv)+˜u′∂v(∂yv)+/tildewideu′∂v((∂yv−/tildewideu′)).\nBy acting T−1\nNon (10.4) and multiplying the cut-off function, we get that\n˚T−1\nj,N[P/\\e}atio\\slash=Ξ]−ΥjP/\\e}atio\\slash=Π =˚T−1\nj,N/bracketleftbig\nP/\\e}atio\\slash=S2[Π]+P/\\e}atio\\slash=Sa[Π]/bracketrightbig\n. (10.5)\nWe introduce three multipliers, which are constructed in [1 1]: fork∈Z\\{0}, ξ∈R, let\nM3(t,k,ξ) =/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2Ak(ξ),\nM4(t,k,ξ) =/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2|k,ξ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsAk(ξ),\nM5(t,k,ξ) =/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2/radicalbigg\n∂tw\nw˜Ak(ξ).\nIt is easy to check that\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2=|kt|\n|ξ|+|kt|(|ξ|+|kt|)(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\nk2+ξ2+1≥|kt|\n|ξ|+|kt|≈/angbracketleftbiggξ\nkt/angbracketrightbigg−1\n, (10.6)\nwhich implies M4(t,k,ξ)≥ M1(t,k,ξ) andM5(t,k,ξ)≥ M2(t,k,ξ).\nWe first control the two error terms ˚T−1\nj,N/bracketleftbig\nP/\\e}atio\\slash=S2[Π]/bracketrightbig\nand˚T−1\nj,N/bracketleftbig\nP/\\e}atio\\slash=Sa[Π]/bracketrightbig\nLemma 10.2. Under the bootstrap hypotheses,/vextenddouble/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nS2[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarǫ/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble\n2\n/vextenddouble/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nS2[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarǫ/parenleftbig\nCK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightbig\n+ǫ/ba∇dbl(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/ba∇dbl2.\nLemma 10.3. Under the bootstrap hypotheses,/vextenddouble/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nSa[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarǫ/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble\n2+ǫ2,\n/vextenddouble/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nSa[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarǫ/parenleftbig\nCK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9+CK1\n2a+ǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightbig\n+ǫ/ba∇dbl(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/ba∇dbl2\n+ǫ/ba∇dblM01∂vP0(Υ2Π)/ba∇dblL2.\nBy applying Proposition 10.1, we have/vextenddouble/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nS2[Π]+Sa[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilar/ba∇dblM3P/\\e}atio\\slash=S2[Π]/ba∇dbl2+/ba∇dblM3P/\\e}atio\\slash=Sa[Π]/ba∇dbl2,/vextenddouble/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nS2[Π]+Sa[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilar/ba∇dbl(M4+M5)P/\\e}atio\\slash=S2[Π]/ba∇dbl2+/ba∇dbl(M4+M5)P/\\e}atio\\slash=Sa[Π]/ba∇dbl2.\nThe idea of the proof of these two lemmas is the same as the proo fs ofLemma 3.2, Lemma 3.3, and\nLemma 3.4 of [11].\nProof.Estimate for S2[Π]:By fact that\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2/lessorsimilarc/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)\n1+k2+η2ecmin{/a\\}bracketle{tξ−η/a\\}bracketri}ht1\n2,/a\\}bracketle{tk,η/a\\}bracketri}ht1\n2},NONLINEAR INVISCID DAMPING 73\n|k,ξ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/lessorsimilar|k,η|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+|η−ξ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts,\nand Remark 6.5, we get\nM3(t,k,ξ)/lessorsimilarAR(ξ−η)/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)\n1+k2+η2ecλ|k,η|s+ecλ|ξ−η|sM3(t,k,η)\nM4(t,k,ξ)/lessorsimilar|η−ξ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsAR(ξ−η)/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)\n1+k2+η2ecλ|k,η|s+ecλ|ξ−η|sM4(t,k,η).\nBy Lemma D.2, we have\nM5(t,k,ξ)/lessorsimilar/parenleftBig|η−ξ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+/radicalbigg\n∂tw\nw/parenrightBig\nAR(ξ−η)M3(t,k,η)ecλ|k,η|s\n+ecλ|ξ−η|s/parenleftBig\nM4(t,k,η)+M5(t,k,η)/parenrightBig\n.\nThus we have/vextenddouble/vextenddouble/vextenddoubleM3P/\\e}atio\\slash=[θδ∂zz(Υ2Π)+g1(∂v−t∂z)2(Υ2Π)]/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilarǫ/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble\n2,\n/vextenddouble/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\nP/\\e}atio\\slash=[θδ∂zz(Υ2Π)+g1(∂v−t∂z)2(Υ2Π)]/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar/parenleftbig\nCK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightbig/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble\n2\n+ǫ/ba∇dbl(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/ba∇dbl2.\nFor theg2(∂v−t∂z)Π term, since there is one derivative loss in g2, we need to use the decay of\nΠ and the multiplier|k|t(t+|k,η|)\nk2+η2to gain the regularity, which plays a similar role to the mult iplier\n/angbracketleftbigη\nkt/angbracketrightbig−1inM1andM2. We have for |ξ−η| ≥2|k,η|(the caseg2is in higher frequencies)\n(10.7)/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ21+k2+η2\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)/lessorsimilar/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht2\n1+k2+ξ2/parenleftBig\n1+|η−ξ|\n(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)/parenrightBig\n/lessorsimilar/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht2\n/a\\}b∇acketle{tη−ξ/a\\}b∇acket∇i}ht,\nand for|ξ−η| ≤2|k,η|\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ21+k2+η2\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)/lessorsimilar/parenleftBig\n1+|ξ−η||ξ+η|\n1+k2+ξ2/parenrightBig/parenleftBig\n1+|ξ−η|\nt+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|/parenrightBig\n/lessorsimilar/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht3.\nThen we have by Remark 6.5 and Lemma D.2,\n/ba∇dblM3[g2(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π)]/ba∇dblL2/lessorsimilarǫ/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble\n2,/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\nP/\\e}atio\\slash=[g2(∂v−t∂z)(Υ2Π)]/vextenddouble/vextenddouble\nL2\n/lessorsimilar/parenleftbig\nCK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightbig/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble\n2\n+ǫ/ba∇dbl(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/ba∇dbl2.\nThis gives the lemma. /square\nProof.Estimate for Sa[Π]:For the estimate of Sa[Π], we take advantage of the divergence form.\nNote that switching from A k(ξ) to Al(η), we will loseξ\nk2+(ξ−kt)2in the case t∈Ik,ξ∩Ik,η, k/\\e}atio\\slash=l. By\ntaking out one derivative ( ∂z,∂v−t∂z), in the worst case t∈Ik,ξ∩Ik,η, k/\\e}atio\\slash=l, we multiply a small\nfactξ−kt\nη−lt.74 WEIREN ZHAO\nWe firstconsider the firsttwo terms in Sa[Π] when Π is non-zero modes, namely, ∂z(a∂zP/\\e}atio\\slash=(Υ2Π))\nandg3(∂v−t∂z)(a(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π)). Since g3is the zero-mode, using the idea of the proof of\nS2[Π], we have\n/vextenddouble/vextenddoubleM3P/\\e}atio\\slash=/bracketleftbig\ng3(∂v−t∂z)(a(∂v−t∂z)(Υ2Π))/bracketrightbig/vextenddouble/vextenddouble\nL2/lessorsimilar/ba∇dblM3P/\\e}atio\\slash=[(∂v−t∂z)(a(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π))]/ba∇dblL2/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\nP/\\e}atio\\slash=[g3(∂v−t∂z)(a(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π))]/vextenddouble/vextenddouble\nL2\n/lessorsimilar/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\nP/\\e}atio\\slash=[(∂v−t∂z)(a(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π))]/vextenddouble/vextenddouble\nL2\n+/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightBig\n/ba∇dblM3P/\\e}atio\\slash=[(∂v��t∂z)(a(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π))]/ba∇dblL2\nThen we treat the last term in Sa[Π], namely, g4a(∂v−t∂z)(Υ2Π). Ifg4is in lower frequencies, the\ntreatment is easier than g3(∂v−t∂z)(a(∂v−t∂z)(Υ2Π)). The difficulty happens when g4is in higher\nfrequencies. In such a case, we follow the idea of the treatme nt ofg2(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π) term, then\nby (10.7) we have\n/vextenddouble/vextenddouble/vextenddoubleM3/bracketleftBig\ng4P/\\e}atio\\slash=/parenleftBig\na(∂v−t∂z)(Υ2Π)/parenrightBig/bracketrightBig/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleM3(|∂z|+|∂v−t∂z|)P/\\e}atio\\slash=/parenleftBig\na(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π)/parenrightBig/bracketrightBig/vextenddouble/vextenddouble/vextenddouble\nL2\n+/vextenddouble/vextenddouble/vextenddoubleM3(|∂z|+|∂v−t∂z|)P/\\e}atio\\slash=/parenleftBig\na∂vP0(Υ2Π)/parenrightBig/bracketrightBig/vextenddouble/vextenddouble/vextenddouble\nL2/vextenddouble/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\nP/\\e}atio\\slash=/bracketleftBig\ng4/parenleftBig\na(∂v−t∂z)(Υ2Π)/parenrightBig/bracketrightBig/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightBig/vextenddouble/vextenddouble/vextenddoubleM3(|∂z|+|∂v−t∂z|)P/\\e}atio\\slash=/parenleftBig\na(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π)/parenrightBig/bracketrightBig/vextenddouble/vextenddouble/vextenddouble\nL2\n+/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightBig/vextenddouble/vextenddouble/vextenddoubleM3(|∂z|+|∂v−t∂z|)P/\\e}atio\\slash=/parenleftBig\na∂vP0(Υ2Π)/parenrightBig/bracketrightBig/vextenddouble/vextenddouble/vextenddouble\nL2\n+ǫ/ba∇dbl(M4+M5)(|∂z|+|∂v−t∂z|)P/\\e}atio\\slash=(a(∂v−t∂z)(Υ2Π))/ba∇dbl2.\nWe have by Lemma D.4 and Remark 6.5 that for |k−l,η−ξ| ≤ |l,η|,k/\\e}atio\\slash= 0,l/\\e}atio\\slash= 0\n|k|M3(t,k,ξ)/angbracketleftbigg\nt−ξ\nk/angbracketrightbigg\n/lessorsimilarM3(t,l,η)(|l|+|η−lt|)|k|+|ξ−kt|\n|l|+|η−lt|/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n/a\\}b∇acketle{tl/a\\}b∇acket∇i}ht(t+/a\\}b∇acketle{tl/a\\}b∇acket∇i}ht+|η|)1+l2+η2\n1+k2+ξ2ecλ|k−l,η−ξ|s\n+Al(η)/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2|ξ|(|k|+|ξ−kt|)\nk2+(ξ−kt)21t∈Ik,ξ∩Ik,η,k/\\e}atio\\slash=l\n/lessorsimilarM3(t,l,η)(|l|+|η−lt|)ecλ|k−l,η−ξ|/a\\}b∇acketle{tk−l,η−ξ/a\\}b∇acket∇i}ht5+Al(η)ξ/k\nη/l|l|/angbracketleftBig\nt−η\nl/angbracketrightBig\necλ|k−l,η−ξ|\n/lessorsimilarecλ|k−l,η−ξ|M3(t,l,η)(|l|+|η−lt|)\nFor|k−l,η−ξ| ≥ |l,η|,k/\\e}atio\\slash= 0,l/\\e}atio\\slash= 0,k/\\e}atio\\slash=l, we have by Remark 6.5\n|k|M3(t,k,ξ)/angbracketleftbigg\nt−ξ\nk/angbracketrightbigg\n/lessorsimilar/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2(|k|+|ξ−kt|)Ak−l(ξ−η)ecλ|l,η|s\n+/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2Ak−l(ξ−η)|ξ|(|k|+|ξ−kt|)\nk2+|ξ−kt|2ecλ|l,η|s1t∈Ik,ξ∩Ik,ξ−η,k/\\e}atio\\slash=k−lNONLINEAR INVISCID DAMPING 75\n/lessorsimilarA∗\nk−l(ξ−η)/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3ecλ|l,η|s.\nSimilarly, by Lemma D.2, we have\n|k|(M4(t,k,ξ)+M5(t,k,ξ))/angbracketleftbigg\nt−ξ\nk/angbracketrightbigg\n/lessorsimilarecλ|k−l,η−ξ|(M4(t,l,η)+M5(t,l,η))(|l|+|η−lt|)\n+/parenleftBig/radicalbigg\n∂tw\nw+|k−l,ξ−η|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\nA∗\nk−l(ξ−η)/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2ecλ|l,η|s.\nThus, we get\n/ba∇dblM3P/\\e}atio\\slash=[(∂v−t∂z)(a(∂v−t∂z)P/\\e}atio\\slash=((Υ2Π)))]/ba∇dblL2\n/lessorsimilarǫ/vextenddouble/vextenddoubleM3/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig\nP/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble\n2+ǫ/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4/ba∇dblP/\\e}atio\\slash=(Υ2Π)/ba∇dblGs,σ−6\n/ba∇dbl(M4+M5)P/\\e}atio\\slash=[(∂v−t∂z)(a(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π))]/ba∇dblL2\n/lessorsimilarǫ/vextenddouble/vextenddouble(M4+M5)/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig\nP/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble\nL2+/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4/ba∇dblP/\\e}atio\\slash=(Υ2Π)/ba∇dblGs,σ−6CK1\n2a.\nNext, weconsiderthesecondtermin Sa[Π]whenΠisthezeromode,namely, g3(∂v−t∂z)(P/\\e}atio\\slash=(a)∂vP0(Υ2Π)).\nAgain, we first get rid of g3:\n/vextenddouble/vextenddoubleM3/bracketleftbig\ng3(∂v−t∂z)(P/\\e}atio\\slash=a∂vP0(Υ2Π))/bracketrightbig/vextenddouble/vextenddouble\nL2/lessorsimilar/ba∇dblM3[(∂v−t∂z)(P/\\e}atio\\slash=a∂vP0(Υ2Π))]/ba∇dblL2/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\n[g3(∂v−t∂z)(P/\\e}atio\\slash=a∂vP0(Υ2Π))]/vextenddouble/vextenddouble\nL2\n/lessorsimilar/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\nP/\\e}atio\\slash=[(∂v−t∂z)(a∂vP0(Υ2Π))]/vextenddouble/vextenddouble\nL2\n+/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightBig\n/ba∇dblM3[(∂v−t∂z)(P/\\e}atio\\slash=a∂vP0(Υ2Π))]/ba∇dblL2.\nWe have for |ξ−η| ≤2|k,η|, (which is associated with the case ais in higher frequencies,)\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2|ξ−kt|\nBk(η)/lessorsimilar/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2t|ξ−kt|\nt+|ξ|1\n2+|k|/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht1\n2\n/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht1\n2,\nand for|ξ−η| ≥2|k,η|,\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2|ξ−kt|/parenleftBigt(t2+(ξ−η)2)\n2+(ξ−η)2/parenrightBig−1\n/lessorsimilar/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht6|ξ−kt|\nt+|ξ|.\nThus we obtain by Remark 6.5 that\n/ba∇dblM3[(∂v−t∂z)(P/\\e}atio\\slash=a∂vP0(Υ2Π))]/ba∇dblL2/lessorsimilar/ba∇dblA∗a/ba∇dblL2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3/ba∇dbl∂vP0(Υ2Π)/ba∇dblGs,σ−6\n+/ba∇dbla/ba∇dblGs,σ−6/ba∇dblM00∂vP0(Υ2Π)/ba∇dblL2/lessorsimilarǫ2\nand by Lemma D.2\n/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\nP/\\e}atio\\slash=[(∂v−t∂z)(a∂vP0(Υ2Π))]/vextenddouble/vextenddouble\nL2\n/lessorsimilarǫCK1\n2a+ǫ/ba∇dblM01∂vP0(Υ2Π)/ba∇dblL2.\nThus we proved the lemma. /square\nNow we introduce two lemmas with weaker weight.76 WEIREN ZHAO\nLemma 10.4. Under the bootstrap hypotheses,\n/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nS2[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble/vextenddouble\n2/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nS2[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ/parenleftbig\nCK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightbig\n+ǫ/ba∇dbl(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Π)/ba∇dbl2.\nLemma 10.5. Under the bootstrap hypotheses,\n/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nSa[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ1Π)/vextenddouble/vextenddouble\n2+ǫ2,/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nSa[Π]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ/parenleftbig\nCK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9+CK1\n2a+ǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightbig\n+ǫ/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ1Π)/vextenddouble/vextenddouble\n2\n+ǫ/ba∇dblM01∂vP0(Υ2Π)/ba∇dblL2.\nNoticing that, by applying Proposition 10.1, we only need to control\n/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)−1\n2P/\\e}atio\\slash=/parenleftBig\nS2[Π]+Sa[Π]/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2\nand/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)−1\n2P/\\e}atio\\slash=/parenleftBig\nS2[Π]+Sa[Π]/parenrightBig/vextenddouble/vextenddouble/vextenddouble\n2.\nNotice that whenthegaining ( k2+(ξ−kt)2)−1\n2is weak, namely, |ξ| ≈ |kt|, theweight/a\\}bracketle{tk/a\\}bracketri}htt(t+/a\\}bracketle{tk/a\\}bracketri}ht+|ξ|)\n1+k2+ξ2≈\n1. The above two lemmas can be proved by using the same idea in t he proof of Lemma 10.2 and\nLemma 10.3, and following the proofs step by step.\nProof.We have for |ξ−η| ≤2|k,η|andk/\\e}atio\\slash= 0 that\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2Ak(ξ)\n(k2+(ξ−kt)2)1\n2/lessorsimilar/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)\n1+k2+ξ2/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht4\n(k2+(η−kt)2)1\n2Ak(η)ecλ/a\\}bracketle{tξ−η/a\\}bracketri}hts\nand for|ξ−η| ≥2|k,η|andk/\\e}atio\\slash= 0, we have (10.7). Then we have by Remark 6.5 and Lemma D.2,\n/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)−1\n2[g2(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π)]/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilarǫ/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble/vextenddouble\n2,\n/vextenddouble/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\n(−∂2\nz−(∂v−t∂z)2)−1\n2P/\\e}atio\\slash=[g2(∂v−t∂z)(Υ2Π)]/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar/parenleftbig\nCK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightbig/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Π)/vextenddouble/vextenddouble\n2\n+ǫ/ba∇dbl(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Π)/ba∇dbl2.\nThis gives Lemma 10.4. /squareNONLINEAR INVISCID DAMPING 77\nProof.By using the same idea of the proof of Lemma 10.3, we only need t o estimate\n/ba∇dblM3P/\\e}atio\\slash=[(a(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π))]/ba∇dblL2,/ba∇dblM3P/\\e}atio\\slash=[(a∂zP/\\e}atio\\slash=(Υ2Π))]/ba∇dblL2\nand\n/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\nP/\\e}atio\\slash=[(a(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π))]/vextenddouble/vextenddouble\nL2.\nConsidering the support of a, we havea(∂v−t∂z)P/\\e}atio\\slash=(Υ2Π) =a(∂v−t∂z)P/\\e}atio\\slash=(Υ1Π). Thus we use\nΥ1Π and estimate\n/ba∇dblM3P/\\e}atio\\slash=[(a(∂v−t∂z)P/\\e}atio\\slash=(Υ1Π))]/ba∇dblL2,/ba∇dblM3P/\\e}atio\\slash=[(a∂zP/\\e}atio\\slash=(Υ1Π))]/ba∇dblL2\nand\n/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\nP/\\e}atio\\slash=[(a(∂v−t∂z)P/\\e}atio\\slash=(Υ1Π))]/vextenddouble/vextenddouble\nL2.\nWe have by Lemma D.4 and Remark 6.5 that for |k−l,η−ξ| ≤ |l,η|,k/\\e}atio\\slash= 0,l/\\e}atio\\slash= 0\nM3(t,k,ξ)(|l|+|η−lt|)\n/lessorsimilarM3(t,l,η)(|l|+|η−lt|)eλ|k−l,η−ξ|s\n+Al(η)/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2|ξ|(|l|+|η−lt|)\nk2+(ξ−kt)21t∈Ik,ξ∩Ik,η,k/\\e}atio\\slash=l\n/lessorsimilarM3(t,l,η)ecλ|k−l,η−ξ|/a\\}b∇acketle{tk−l,η−ξ/a\\}b∇acket∇i}ht5+Al(η)ξ/k\nη/l|l|2/angbracketleftBig\nt−η\nl/angbracketrightBig2\necλ|k−l,η−ξ|\n/lessorsimilarecλ|k−l,η−ξ|M3(t,l,η)(|l|+|η−lt|)2.\nThe lemma follows from Remark 6.5 and Lemma D.2. /square\n10.1.Estimate of Πl,1.In this section, we obtain the estimate for P/\\e}atio\\slash=(Πl,1). We only need to\nestimate M3(∂2\nz+ (∂v−t∂z)2)˚T−1\nj,N[P/\\e}atio\\slash=Ξ1],M4(∂2\nz+ (∂v−t∂z)2)˚T−1\nj,N[P/\\e}atio\\slash=Ξ1], andM5(∂2\nz+ (∂v−\nt∂z)2)˚T−1\nj,N[P/\\e}atio\\slash=Ξ1] where\nΞ1=ϕ6∂zz(Υ2Ψ)+ϕ7P/\\e}atio\\slash=(Υ2Ψ)\nBy Proposition 10.1, we have\n(10.8)/vextenddouble/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)˚T−1\nj,N[P/\\e}atio\\slash=Ξ1]/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilar/ba∇dblM3[P/\\e}atio\\slash=Ξ1]/ba∇dbl2,\n/vextenddouble/vextenddouble/vextenddoubleM4(∂2\nz+(∂v−t∂z)2)˚T−1\nj,N[P/\\e}atio\\slash=Ξ1]/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilar/ba∇dblM4[P/\\e}atio\\slash=Ξ1]/ba∇dbl2,\n/vextenddouble/vextenddouble/vextenddoubleM5(∂2\nz+(∂v−t∂z)2)˚T−1\nj,N[P/\\e}atio\\slash=Ξ1]/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilar/ba∇dblM5[P/\\e}atio\\slash=Ξ1]/ba∇dbl2.\nBy using\n(|t|+|k|+|ξ|)|k|/lessorsimilar/a\\}b∇acketle{tk,ξ/a\\}b∇acket∇i}ht(|ξ−kt|+|k|),fork/\\e}atio\\slash= 0, (10.9)\nand Remark 6.5, we then have for 2 |k,η| ≥ |ξ−η|,k/\\e}atio\\slash= 0,\n|k|2/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2Ak(ξ)/lessorsimilar|k2|Ak(η)/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht3/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)\n1+k2+η2ecλ|ξ−η|s\n/lessorsimilart|k|2|k,η|(|k|+|tk−η|)\n1+k2+η2|η|+|kt|\n|kt|/angbracketleftBigη\nkt/angbracketrightBig−1\nAk(η)ecλ|ξ−η|s\n/lessorsimilar(k2+(η−kt)2)/angbracketleftBigη\nkt/angbracketrightBig−1\nAk(η)ecλ|ξ−η|s.78 WEIREN ZHAO\nand for 2 |k,η| ≤ |ξ−η|,k/\\e}atio\\slash= 0,\n|k|/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2Ak(ξ)/lessorsimilarAR(ξ−η)eλ|k,η|s/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht2\nThus we get that\n/ba∇dblM3[P/\\e}atio\\slash=Ξ1]/ba∇dbl2/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblP/\\e}atio\\slash=(Υ2Ψ)/ba∇dblGs,σ−6+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)AP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2/lessorsimilarǫ. (10.10)\nBy Lemma D.2, we have\n/vextenddouble/vextenddouble/parenleftbig\nM4+M5/parenrightbig\n[P/\\e}atio\\slash=Ξ1]/vextenddouble/vextenddouble\n2/lessorsimilar/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nϕδ\n6+CK1\n2\nϕδ\n7/parenrightBig\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblP/\\e}atio\\slash=(Υ2Ψ)/ba∇dblGs,σ−6\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.\nNow we prove Proposition 7.9 for Π ⋆= Υ2Πl,1.\nProof.By (10.5), Lemma 10.2, Lemma 10.3, (10.8), and (10.10), we ge t that\n/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Πl,1)/vextenddouble/vextenddouble\n2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)˚T−1\n2,N[P/\\e}atio\\slash=Ξ1]/vextenddouble/vextenddouble/vextenddouble\n2+/vextenddouble/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)˚T−1\nj,N/bracketleftbig\nP/\\e}atio\\slash=S2[Πl,1]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)˚T−1\n2,N/bracketleftbig\nP/\\e}atio\\slash=Sa[Πl,1]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ+ǫ/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Πl,1)/vextenddouble/vextenddouble\n2+ǫ2\nand/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Πl,1)/vextenddouble/vextenddouble\n2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)˚T−1\n2,N[P/\\e}atio\\slash=Ξ1]/vextenddouble/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)˚T−1\nj,N/bracketleftbig\nP/\\e}atio\\slash=S2[Πl,1]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)˚T−1\n2,N/bracketleftbig\nP/\\e}atio\\slash=Sa[Πl,1]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nϕδ\n6+CK1\n2\nϕδ\n7+CK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9+CK1\n2a/parenrightBig\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+ǫ/ba∇dbl(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Πl,1)/ba∇dbl2\n+ǫ/ba∇dblM01∂vP0(Υ2Π)/ba∇dblL2,\nwhich gives/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Πl,1)/vextenddouble/vextenddouble\n2/lessorsimilarǫ,\nand/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ2Πl,1)/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nϕδ\n6+CK1\n2\nϕδ\n7+CK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9+CK1\n2a/parenrightBigNONLINEAR INVISCID DAMPING 79\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+ǫ/ba∇dblM01∂vP0(Υ2Π)/ba∇dblL2.\nThus we prove the estimates for P/\\e}atio\\slash=(Υ2Πl,1). /square\n10.2.Estimate of Πl,2.The Πl,2term arose from the boundary correction. It seems worse than\nΠl,1. However, on the support of ϕ8, Υ1= 0. Thus to estimate Π l,2, we use two steps. First, we\nget the estimate of Π l,2using equation (5.7f). Second, we obtain an improved estima te for Υ 1Πl,2,\nnamely, we study the following equation:\n(10.11)\n\n∂z/parenleftBig\n(a+θ)∂z(Υ1Πl,2)/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)(Υ1Πl,2)/parenrightBig\n= 2(∂yv)2(∂vΥ1)/parenleftBig\nθ(∂v−t∂z)(Υ2Πl,2)/parenrightBig\n+(∂yv)2/parenleftBig\nθ(∂2\nvΥ1)(Υ2Πl,2)/parenrightBig\n,\n(∂v−t∂z)Πl,2(t,z,u(0)) = (∂v−t∂z)Πl,2(t,z,u(1)) = 0.\nNote that on the support of ∂vΥ1or∂2\nvΥ1, bothθand∂yvare constant value.\nWe show the idea in the following inequalities:\nStep 1. /ba∇dblΥ2Πl,2/ba∇dblY/lessorsimilarǫ/ba∇dblΥ2Πl,2/ba∇dblY+ǫ/ba∇dblΥ1Πl,2/ba∇dblX+good terms ,\nStep 2. /ba∇dblΥ1Πl,2/ba∇dblX/lessorsimilarǫ/ba∇dblΥ1Πl,2/ba∇dblX+/ba∇dblΥ2Πl,2/ba∇dblY+good terms .\nLet us first estimate P/\\e}atio\\slash=(Υ2Πl,2).\nLemma 10.6. Under the bootstrap hypotheses,\n/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/vextenddouble/vextenddouble/vextenddouble\nL2/lessorsimilarǫ+ǫ/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ1Πl,2)/vextenddouble/vextenddouble\n2,\nand/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar+ǫ/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ1Πl,2)/vextenddouble/vextenddouble\n2\n+ǫ/parenleftBig\nCK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n8+CK1\n2\nϕδ\n9+CK1\n2a+1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n+ǫ/ba∇dblM01∂vP0(Υ2Πl,2)/ba∇dblL2\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP|k|≥kM(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.\nMoreover, it holds that\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3/ba∇dblP/\\e}atio\\slash=(Υ2Πl,2)/ba∇dblGs,σ−6+/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/vextenddouble/vextenddouble/vextenddouble(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/vextenddouble/vextenddouble/vextenddouble\nGs,σ−6\n/lessorsimilarǫ+ǫ/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ1Πl,2)/vextenddouble/vextenddouble\n2.\nProof.By (10.5) Lemma 10.4, and Lemma 10.5, we have\n/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\n2,N[P/\\e}atio\\slash=Ξ2]/vextenddouble/vextenddouble/vextenddouble\nL2\n+/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nS2[Πl,2]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n280 WEIREN ZHAO\n+/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nSa[Πl,2]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ2+/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)−1\n2[P/\\e}atio\\slash=Ξ2]/vextenddouble/vextenddouble/vextenddouble\nL2+ǫ/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/vextenddouble/vextenddouble/vextenddouble\nL2\n+ǫ/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ1Πl,2)/vextenddouble/vextenddouble\n2,\nand/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\n2,N[P/\\e}atio\\slash=Ξ2]/vextenddouble/vextenddouble/vextenddouble\nL2\n+/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nS2[Πl,2]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2˚T−1\nj,NP/\\e}atio\\slash=/bracketleftbig\nSa[Πl,2]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ/ba∇dbl(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/ba∇dbl2\n+ǫ/ba∇dbl(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ1Πl,2)/ba∇dbl2\n+ǫ/parenleftBig\nCK1\n2\nθδ+CK1\n2\nh+CK1\n2\nϕδ\n9+CK1\n2a+ǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\n+/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)−1\n2[P/\\e}atio\\slash=Ξ2]/vextenddouble/vextenddouble/vextenddouble\nL2\n+ǫ/ba∇dblM01∂vP0(Υ2Πl,2)/ba∇dblL2,\nNowwecontrol/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)−1\n2[P/\\e}atio\\slash=Ξ2]/vextenddouble/vextenddouble/vextenddouble\nL2and/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)−1\n2[P/\\e}atio\\slash=Ξ2]/vextenddouble/vextenddouble/vextenddouble\nL2.\nwhere\nΞ2=ϕ8∂yv(∂v−t∂z)P/\\e}atio\\slash=(Υ2Ψ)\nBy (10.9), we have for |ξ−η| ≤2|k,η|\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2(η−kt)\n|k|+|ξ−kt|\n/lessorsimilar(η−kt)\n1+|k|+|ξ||η|+|kt|\nk/angbracketleftBigη\nkt/angbracketrightBig−1\n/lessorsimilar/a\\}b∇acketle{tη−ξ/a\\}b∇acket∇i}ht(k2+(η−kt))2/angbracketleftBigη\nkt/angbracketrightBig−1\nand for|ξ−η| ≥2|k,η|\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2(η−kt)\n|k|+|ξ−kt|/lessorsimilar(η−kt)t\n1+|k|+|ξ|/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2|k,η|.\nThus we get by Remark 6.5 and Lemma D.2 that\n/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)−1\n2[P/\\e}atio\\slash=Ξ2]/vextenddouble/vextenddouble/vextenddouble\nL2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)AP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+(1+/ba∇dblAR(ϕδ\n8)/ba∇dbl2)(1+/ba∇dblARh/ba∇dbl2+/ba∇dblARϕδ\n9/ba∇dbl2)/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblP/\\e}atio\\slash=(ΨΥ2)/ba∇dbl/lessorsimilarǫ,\nand/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)−1\n2[P/\\e}atio\\slash=Ξ2]/vextenddouble/vextenddouble/vextenddouble\nL2NONLINEAR INVISCID DAMPING 81\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP|k|≥kM(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+ǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+ǫCK1\n2\nϕδ\n8+ǫCK1\n2\nϕδ\n9+ǫCK1\n2\nh.\nThus we proved the lemma. /square\nNow we use (10.11) to prove Proposition 7.9 for Π ⋆= Υ1Πl,2.\nProof.By (10.5), Lemma 10.2, Lemma 10.3, (10.8), and (10.10), we ge t that\n/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ1Πl,2)/vextenddouble/vextenddouble\n2\n/lessorsimilar/ba∇dblM3[P/\\e}atio\\slash=Ξ3]/ba∇dbl2+/vextenddouble/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)˚T−1\n1,N/bracketleftbig\nP/\\e}atio\\slash=S2[Πl,2]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddouble/vextenddoubleM3(∂2\nz+(∂v−t∂z)2)˚T−1\n1,N/bracketleftbig\nP/\\e}atio\\slash=Sa[Πl,2]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2.\nand\n/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)P/\\e}atio\\slash=(Υ1Πl,2)/vextenddouble/vextenddouble\n2\n/lessorsimilar/ba∇dbl(M4+M5)[P/\\e}atio\\slash=Ξ3]/ba∇dbl2\n+/vextenddouble/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)˚T−1\n1,N/bracketleftbig\nP/\\e}atio\\slash=S2[Πl,2]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddouble/vextenddouble(M4+M5)(∂2\nz+(∂v−t∂z)2)˚T−1\n1,N/bracketleftbig\nP/\\e}atio\\slash=Sa[Πl,2]/bracketrightbig/vextenddouble/vextenddouble/vextenddouble\n2,\nwith\nΞ3= 2(∂yv)2(∂vΥ1)/parenleftBig\nθ(∂v−t∂z)(Υ2Πl,2)/parenrightBig\n+(∂yv)2/parenleftBig\nθ(∂2\nvΥ1)(Υ2Πl,2)/parenrightBig\nWe have for |ξ−η| ≤2|k,η|\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2/lessorsimilar/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η|)\n1+k2+η2/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht3,\nand for|ξ−η| ≥2|k,η|\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2.\nThus we have by Remark 6.5 that\n/ba∇dblM3[P/\\e}atio\\slash=Ξ3]/ba∇dbl2/lessorsimilar/vextenddouble/vextenddouble/vextenddoubleM3(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/vextenddouble/vextenddouble/vextenddouble\nL2\n+/ba∇dblAR(∂yvΥ1)/ba∇dbl2\n2/ba∇dblARθ/ba∇dblL2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dbl(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/ba∇dblGs,σ−6/lessorsimilarǫ.\nand by Lemma D.2\n/ba∇dbl(M4+M5)[P/\\e}atio\\slash=Ξ3]/ba∇dbl2\n/lessorsimilar/vextenddouble/vextenddouble/vextenddouble(M4+M5)(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/vextenddouble/vextenddouble/vextenddouble\nL2\n+/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nh+CK1\n2\nθ/parenrightBig\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dbl(−∂2\nz−(∂v−t∂z)2)1\n2P/\\e}atio\\slash=(Υ2Πl,2)/ba∇dblGs,σ−6\nBy using Lemma 10.6, we proved the estimates for P/\\e}atio\\slash=(Υ1Πl,2). /square82 WEIREN ZHAO\n10.3.Estimate of Πn,1.Inthis section, wetreat thenonlinearpressure. Similarly , wefirstestimate\n/ba∇dblM3[P/\\e}atio\\slash=Ξ4]/ba∇dbl2,and/ba∇dbl(M4+M5)[P/\\e}atio\\slash=Ξ4]/ba∇dbl2\nwhere\nΞ4=−2χ2(∂yv(∂v−t∂z)∂z(Υ2Ψ))2−2∂zz(Υ2Ψ)/parenleftBig\nΩ−χ2∂zz(Υ2Ψ)/parenrightBig\n,\n−2χ′\n2∂yv/bracketleftBig\n(∂z(Υ2Ψ))(∂v−t∂z)∂z(Υ2Ψ)+(∂v−t∂z)(Υ2Ψ)∂zz(Υ2Ψ)/bracketrightBig\n−1\n2χ′′\n2/bracketleftBig\n(∂z(Υ2Ψ))2+(∂yv)2((∂v−t∂z)P/\\e}atio\\slash=(Υ2Ψ))2/bracketrightBig\n.\nWe have the following lemma:\nLemma 10.7. Under the bootstrap hypotheses,\n/ba∇dblM3[P/\\e}atio\\slash=Ξ4]/ba∇dbl2/lessorsimilarǫ2,\nand\n/ba∇dbl(M4+M5)[P/\\e}atio\\slash=Ξ4]/ba∇dbl2/lessorsimilarǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+ǫCK1\n2\nf+ǫ2CK1\n2\nh+ǫ2CK1\n2\nϕδ\n4+ǫ2CK1\n2\nϕδ\n9\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.\nProof.We separate terms in Ξ 4into four classes:\nA1=/braceleftBig\nP/\\e}atio\\slash=/parenleftbig\nχ2(∂yv(∂v−t∂z)∂z(Υ2Ψ))2/parenrightbig\n, P/\\e}atio\\slash=/parenleftbig\nχ′′\n2(∂yv)2((∂v−t∂z)P/\\e}atio\\slash=(Υ2Ψ))2/parenrightbig/bracerightBig\nA2=/braceleftBig\nP/\\e}atio\\slash=/parenleftbig\n∂zz(Υ2Ψ)P/\\e}atio\\slash=Ω/parenrightbig\n, P/\\e}atio\\slash=/parenleftbig\nχ2(∂zz(Υ2Ψ))2/parenrightbig\n, P/\\e}atio\\slash=/parenleftbig\nχ′′\n2(∂z(Υ2Ψ))2/parenrightbig\n,\nP/\\e}atio\\slash=/parenleftbig\nχ′\n2∂yv(∂z(Υ2Ψ))(∂v−t∂z)∂z(Υ2Ψ)/parenrightbig\n, P/\\e}atio\\slash=/parenleftbig\nχ′\n2∂yv(∂v−t∂z)P/\\e}atio\\slash=(Υ2Ψ)∂zz(Υ2Ψ)/parenrightbig/bracerightBig\nA3=/braceleftBig\n∂zz(Υ2Ψ)P0Ω/bracerightBig\nA4=/braceleftBig\nχ′\n2∂yv∂vP0(Υ2Ψ)∂zz(Υ2Ψ)/bracerightBig\nThe first term in each class is the typical term. By using (10.9 ), we have for |k−l,ξ−η| ≤ |l,η|\nthat\n/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2(|η−lt|)|l|/lessorsimilar/a\\}b∇acketle{tk−l,ξ−η/a\\}b∇acket∇i}ht4/a\\}b∇acketle{tl/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tl/a\\}b∇acket∇i}ht+|η|)\n1+l2+η2(|η−lt|)|l|\n/lessorsimilar/a\\}b∇acketle{tk−l,ξ−η/a\\}b∇acket∇i}ht4/angbracketleftBigη\nlt/angbracketrightBig−1\n(l2+(η−lt)2)\nwhich together with Remark 6.5 and Lemma D.2, gives that for f∈A1,/ba∇dblM3f/ba∇dbl2/lessorsimilarǫ2, and\n/ba∇dbl(M4+M5)f/ba∇dbl2/lessorsimilarǫ2CK1\n2\nh+ǫ2CK1\n2\nϕδ\n9+ǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.\nWe havek,l/\\e}atio\\slash= 0 andk/\\e}atio\\slash=l,\n(k−l)2/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2Ak(ξ)NONLINEAR INVISCID DAMPING 83\n/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2ecλ|k−l,ξ−η|sAl(η)\n+/a\\}b∇acketle{tl,η/a\\}b∇acket∇i}ht4/angbracketleftbiggξ−η\n(k−l)t/angbracketrightbigg−1\n((k−l)2+(ξ−η−(k−l)t)2)Ak−l(ξ−η)ecλ|l,η|s,\nwhich together with Remark 6.5 and Lemma D.2 gives for f∈A2,/ba∇dblM3f/ba∇dbl2/lessorsimilarǫ2, and\n/ba∇dbl(M4+M5)f/ba∇dbl2/lessorsimilarǫ2CK1\n2\nh+ǫ2CK1\n2\nϕδ\n9+ǫCK1\n2\nf+ǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.\nBy Remark 6.5 and (10.9), we have for k/\\e}atio\\slash= 0\nk2/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2Ak(ξ)\n/lessorsimilar/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht3(|k|+|η−kt|)tk2\n|k|+|η|η+kt\nkt/angbracketleftBigη\nkt/angbracketrightBig−1\nAk(η)ecλ|ξ−η|s\n+/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht4(|k|+|η−kt|)tk2\n|k|+|η|η\n(k2+(η−kt)2)A0(ξ−η)ecλ|k,η|s\n+/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht4/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2A0(ξ−η)ecλ|k,η|s\n/lessorsimilar/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht5/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2A0(ξ−η)ecλ|k,η|s+/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht3(|k|2+|η−kt|2)/angbracketleftBigη\nkt/angbracketrightBig−1\nAk(η)ecλ|ξ−η|s\nwhich gives that for f∈A3,/ba∇dblM3f/ba∇dbl2/lessorsimilarǫ2. By Lemma D.2, it holds that\n/ba∇dbl(M4+M5)f/ba∇dbl2/lessorsimilarǫCK1\n2\nf+ǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.\nBy Remark 6.5 and (10.9), we have for\n|ξ−η|/a\\}b∇acketle{tk/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|ξ|)\n1+k2+ξ2Ak(ξ)k2\n/lessorsimilar/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht4t(/a\\}b∇acketle{tk/a\\}b∇acket∇i}ht+|η−kt|)\n1+|k|+|η|η+kt\nkt/angbracketleftBigη\nkt/angbracketrightBig−1\nAk(ξ)k2ecλ|ξ−η|s\n+/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht4t+|ξ−η|\nt/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht(|k|+|ξ|)|ξ−η|A0(ξ−η)ecλ|k,η|sξ\nk2+(ξ−kt)2\n/lessorsimilar/a\\}b∇acketle{tξ−η/a\\}b∇acket∇i}ht4(|k|2+|η−kt|2)/angbracketleftBigη\nkt/angbracketrightBig−1\nAk(η)ecλ|ξ−η|s\n+/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht4|ξ−η|A0(ξ−η)ecλ|k,η|s,\nwhich together with Corollary 8.4 and Lemma D.2 gives for f∈A4,/ba∇dblM3f/ba∇dbl2/lessorsimilarǫ2, and\n/ba∇dbl(M4+M5)f/ba∇dbl2/lessorsimilarǫCK1\n2\nf+ǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+ǫ2/parenleftBig\nCK1\n2\nh+CK1\n2\nϕδ\n4+CK1\n2\nϕδ\n9/parenrightBig\n+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.84 WEIREN ZHAO\nThus we prove the lemma. /square\n10.4.Estimate of Πn,2.The strategy of estimating Π n,2is similar to the estimate of Π l,2. We first\nget estimate for P/\\e}atio\\slash=(Υ2Πn,2), where we use Corollary 8.4. Then we study the elliptic equa tion for\nP/\\e}atio\\slash=(Υ1Πn,2):\n(10.12)\n\n∂z/parenleftBig\n(a+θ)∂z(Υ1Πn,2)/parenrightBig\n+∂yv(∂v−t∂z)/parenleftBig\n(a+θ)∂yv(∂v−t∂z)(Υ1Πn,2)/parenrightBig\n= 2(∂yv)2(∂vΥ1)/parenleftBig\nθ(∂v−t∂z)(Υ2Πn,2)/parenrightBig\n+(∂yv)2/parenleftBig\nθ(∂2\nvΥ1)(Υ2Πn,2)/parenrightBig\n,\n(∂v−t∂z)Πn,2(t,z,u(0)) = (∂v−t∂z)Πn,2(t,z,u(1)) = 0.\nThe proof is similar. We omit the details.\n11.Transport terms\nThe transport structure in the equation of aandfare similar. Here we only present the proof\nfora. We rewrite Itr\naas\nItr\na=−1\n2/integraldisplay\n∇·(ΥU)|A∗a|2dzdv+/integraldisplay\nA∗a[A∗(ΥU·∇a)−ΥU·∇A∗a]dzdv\n=−1\n2/integraldisplay\n∇·(ΥU)|A∗a|2dzdv+1\n2π/summationdisplay\nN∈D(Ta\nN+Ra\nN)+1\n2πRa,\nwhere T, R, and Rare given by using the paraproduct decomposition:\n(11.1)Ta\nN= 2π/integraldisplay\nA∗a/bracketleftbig\nA∗/parenleftbig\n(ΥU)<N/8·∇z,vaN/parenrightbig\n−U<N/8·∇z,vA∗aN/bracketrightbig\ndx,\nRa\nN= 2π/integraldisplay\nA∗a/bracketleftbig\nA∗/parenleftbig\n(ΥU)N·∇z,va<N/8/parenrightbig\n−UN·∇z,vA∗a<N/8/bracketrightbig\ndx,\nRa= 2π/summationdisplay\nN∈D/summationdisplay\n1\n8N≤N′≤8N/integraldisplay\nA∗a[A∗((ΥU)N·∇z,vaN′)−(ΥU)N·∇z,vA∗aN′]dx.\nSimilarly, we have the decomposition for Itr\nfas in (11.1) by replacing A∗by A and replacing aby\nf. We denote them by Tf\nN, Rf\nN, andRfrespectively, also see (2.26)of [5].\nBy Sobolev embedding, σ>5 and the bootstrap hypotheses, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n∇·(ΥU)|AΩ|2dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ /ba∇dbl∇(ΥU)/ba∇dbl∞/ba∇dblAΩ/ba∇dbl2\n2/lessorsimilarǫ\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2/ba∇dblAΩ/ba∇dbl2\n2/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2. (11.2)NONLINEAR INVISCID DAMPING 85\n11.1.Treatment of Ta\nN.We rewrite\n(11.3)Ta\nN=/summationdisplay\nk=0/integraldisplay\nA∗\nk(η)/hatwideak(η)i/hatwider∂tv(η−ξ)<N\n8ξA∗\nk(ξ)/hatwideak(ξ)N/parenleftBigg/radicalBigg\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|η|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|ξ|−1/parenrightBigg\nAk(η)\nAk(ξ)dξdη\n+/summationdisplay\nk/\\e}atio\\slash=0/summationdisplay\nl/\\e}atio\\slash=k/integraldisplay\nA∗\nk(η)/hatwideak(η)/hatwider∂yvΥ2\n2Ψk−l(η−ξ)<N\n8(kξ−lη)A∗\nl(ξ)/hatwideal(ξ)N\n×/parenleftBigg/radicalBigg\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|η|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+l2+|ξ|−1/parenrightBigg\nAk(η)\nAl(ξ)dξdη\n+/summationdisplay\nk=0/integraldisplay\nA∗\nk(η)/hatwideak(η)i/hatwider∂tv(η−ξ)<N\n8ξA∗\nk(ξ)/hatwideak(ξ)N/parenleftBigAk(η)\nAl(ξ)−1/parenrightBig\ndξdη\n+/summationdisplay\nk/\\e}atio\\slash=0/summationdisplay\nl/\\e}atio\\slash=k/integraldisplay\nA∗\nk(η)/hatwideak(η)/hatwider∂yvΥ2\n2Ψk−l(η−ξ)<N\n8(kξ−lη)A∗\nl(ξ)/hatwideal(ξ)N/parenleftBigAk(η)\nAl(ξ)−1/parenrightBig\ndξdη\ndef= Ta\nN,1+Ta\nN,2+Ta\nN,3+Ta\nN,4.\nFor Tf\nNterm, we have an even simpler decomposition and there are no Tf\nN,1and Tf\nN,2.\nBy fact that |l,ξ| ≈ |k,η|,|kξ−lη| ≤ |k−l,η−ξ||l,ξ|, and\n|l,ξ|/parenleftBigg/radicalBigg\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|η|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+l2+|ξ|−1/parenrightBigg\nAk(η)\nAl(ξ)\n/lessorsimilar(|k−l|+|η−ξ|)|l,ξ|/radicalbig\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+l2+|ξ|Ak(η)\nAl(ξ)/lessorsimilare11µ|k−l,η−ξ||l,ξ|s/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−2s,\nwe obtain that\n/summationdisplay\nNTa\nN,1/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−2s+2˜q/ba∇dbl∂tv/ba∇dblGσ−6CKa/lessorsimilarE1\n2\n∂tvCKa/lessorsimilarǫCKa,\n/summationdisplay\nNTa\nN,2/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−2s+2˜q/ba∇dbl∂yvΥ2/ba∇dblGσ−6/ba∇dblΥ2Ψ/ba∇dblGσ−6CKa/lessorsimilarǫCKa.\nAs in [5], we write\nAk(η)\nAl(ξ)−1 = [eλ|k,η|s−λ|l,ξ|s−1]+eλ|k,η|s−λ|l,ξ|s/bracketleftBigJk(η)\nJl(ξ)−1/bracketrightBig/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}htσ\n/a\\}b∇acketle{tl,ξ/a\\}b∇acket∇i}htσ\n+eλ|k,η|s−λ|l,ξ|s/bracketleftBig/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}htσ\n/a\\}b∇acketle{tl,ξ/a\\}b∇acket∇i}htσ−1/bracketrightBig\n=CA\n1+CA\n2+CA\n3.\nIt is easy to check that\n||l,ξ|CA\n1|/lessorsimilar|l,ξ|secλ|k−l,η−ξ|s,||l,ξ|CA\n3|/lessorsimilarecλ|k−l,η−ξ|s.\nBy Lemma D.4, Lemma D.5, we have by following the steps in section 5 of [5].\n|CA\n2|l,ξ||/lessorsimilarecλ|k−l,η−ξ|se20µ|k−l,η−ξ|1\n2/parenleftBig\n|l,ξ|s+t2/radicalBigg\n∂twk(t,η)\nwk(t,η)/radicalBigg\n∂twl(t,ξ)\nwl(t,ξ)1k/\\e}atio\\slash=l/parenrightBig\n,86 WEIREN ZHAO\nwhich gives that\n/summationdisplay\nNTa\nN,3/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2˜q/ba∇dbl∂tv/ba∇dblGσ−6CKa/lessorsimilarE1\n2\n∂tvCKa/lessorsimilarǫCKa,\n/summationdisplay\nNTa\nN,4/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dbl∂yvΥ2/ba∇dblGσ−6/ba∇dblΥ2Ψ/ba∇dblGσ−6CKa/lessorsimilarǫCKa.\n11.2.Treatment of Ra\nN.We rewrite\nRa\nN=/summationdisplay\nk/integraldisplay\nA∗\nk(η)2/hatwideak(η)i(η−ξ)/hatwider∂tv(ξ)N/hatwideak(η−ξ)<N/8dξdη\n+/summationdisplay\nM≥8/summationdisplay\nk/\\e}atio\\slash=0,l/\\e}atio\\slash=0/integraldisplay\nA∗\nk(η)2/hatwideak(η)i(kζ−lη)/parenleftBig\n/hatwider∂yvΥ2(ξ−ζ)<M/8/hatwidestΥ2Ψl(ζ)M/parenrightBig\nN/hatwideak−l(η−ξ)<N/8dξdηdζ\n+/summationdisplay\nM≥8/summationdisplay\nk/\\e}atio\\slash=0,l/\\e}atio\\slash=0/integraldisplay\nA∗\nk(η)2/hatwideak(η)i(kζ−lη)/parenleftBig\n/hatwider∂yvΥ2(ξ−ζ)M/hatwidestΥ2Ψl(ζ)<M/8/parenrightBig\nN/hatwideak−l(η−ξ)<N/8dξdηdζ\n+/summationdisplay\nM/summationdisplay\nM/8≤M′≤8M/summationdisplay\nk/\\e}atio\\slash=0,l/\\e}atio\\slash=0/integraldisplay\nA∗\nk(η)2/hatwideak(η)i(kζ−lη)/parenleftBig\n/hatwider∂yvΥ2(ξ−ζ)M/hatwidestΥ2Ψl(ζ)M′/parenrightBig\nN\n×/hatwideak−l(η−ξ)<N/8dξdηdζ\n−/summationdisplay\nk,l/integraldisplay\nA∗\nk(η)/hatwideak(η)/parenleftBig/hatwider(Υ2\n2U)l(ξ)/parenrightBig\nN·A∗\nk−l(η−ξ)/hatwider∇ak−l(η−ξ)<N/8dξdη\n= Ra\nN,1+Ra\nN,2+Ra\nN,3+Ra\nN,4+Ra\nN,5.\n11.2.1.Treatment of Ra\nN,1.For Ra\nN,1, we have\n/radicalBigg\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|η|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|η−ξ|/lessorsimilar/radicalbig\n|η|1t+|k|≤√η+1t+|k|≥√η\nwhich together with Lemma D.4 gives that\n|Ra\nN,1|/lessorsimilar/summationdisplay\nk/integraldisplay\nA∗\nk(η)|/hatwideak(η)|Ak(η)\nA0(ξ)/parenleftBig/radicalbig\n|η|1t+|k|≤√η+1t+|k|≥√η/parenrightBig\nA0(ξ)|/hatwider∂tv(ξ)N||/hatwider∇ak(η−ξ)<N/8|dξdη\n/lessorsimilar/ba∇dbla/ba∇dblGσ−6/ba∇dblA0|∂v|1\n2(∂tv)N/ba∇dblL2/ba∇dblA∗a∼N/ba∇dblL2+/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/ba∇dbla/ba∇dblGσ−6/ba∇dblA0(∂tv)N/ba∇dblL2/ba∇dblA∗a∼N/ba∇dblL2,\nwhich together with the\n/summationdisplay\nN|Ra\nN,1|/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2s. (11.4)\nWe have for |k−l,η−ξ| ≤6\n32|l,ξ|that\n(11.5)Bk(t,η)\nBk−l(t,η−ξ)=/radicalBigg\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|η|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+(k−l)2+|η−ξ|\n/lessorsimilar/parenleftBig\n1+|ζ|1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|ζ|1\n2,|l|2≤1\n8|ζ|+|l|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|l|,|l|2≥8|ζ|/parenrightBig\n/a\\}b∇acketle{tk−l,η−ξ/a\\}b∇acket∇i}ht/a\\}b∇acketle{tζ−ξ/a\\}b∇acket∇i}htNONLINEAR INVISCID DAMPING 87\n11.2.2.Treatment of Ra\nN,2.For Ra\nN,2, we have\n|ξ−ζ| ≤6\n32|l,ζ|,|k−l,η−ξ| ≤6\n32|l,ξ|\nwhich gives |l,ζ| ≈ |l,ξ| ≈ |k,η|. We then have\nA∗\nk(η)\nAl(ζ)/radicalbig\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+(k−l)2+|η−ξ|\n/lessorsimilarAk(η)\nAl(ζ)/radicalBigg\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|η|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+(k−l)2+|η−ξ|\n/lessorsimilarAk(η)\nAl(ζ)/a\\}b∇acketle{tk−l,η−ξ/a\\}b∇acket∇i}ht/a\\}b∇acketle{tζ−ξ/a\\}b∇acket∇i}ht/parenleftBig\n1+|ζ|1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|ζ|1\n2,|l|2≤1\n8|ζ|+|l|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|l|,|l|2≥8|ζ|/parenrightBig\ndef=CA\n4+CA\n5+CA\n6.\nWe then have by following the argument in section 6 or [5] that\n(11.6)|l,ζ|CA\n4/lessorsimilarecλ|k−l,η−ξ|secλ|ξ−ζ|se20µ|k−l,η−ξ|1\n2e20µ|ζ−ξ|1\n2\n×/parenleftbigg\n|k,η|s/2|l,ζ|1−s/21t/∈Il,ζ\n+/radicalBigg\n∂twk(η)\nwk(η)/radicalBigg\n∂twl(ζ)\nwl(ζ)|ζ|2\nl2/a\\}b∇acketle{tζ/t−l/a\\}b∇acket∇i}ht212|η|1\n2≤t≤2|η|1t/∈Il,ζ\n+/bracketleftBig/radicalBigg\n∂twk(η)\nwk(η)+|k,η|s/2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/bracketrightBigwR(ζ)\nwNR(ζ)/radicalBigg\nwl(ζ)\n∂twl(ζ)|l,ζ|1t∈Il,ζ\n+/radicalBigg\n∂twk(η)\nwk(η)/radicalBigg\nwl(ζ)\n∂twl(ζ)|l|1t∈Il,ζ/parenrightbigg\n/lessorsimilarecλ|k−l,η−ξ|secλ|ξ−ζ|s/bracketleftBig/radicalBigg\n∂twk(η)\nwk(η)+|k,η|s/2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/bracketrightBig\n×/angbracketleftbiggζ\nlt/angbracketrightbigg−1/bracketleftBig|l,ζ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+/radicalBigg\n∂twl(ζ)\nwl(ζ)/bracketrightBig\n(l2+(ζ−lt)2)1l/\\e}atio\\slash=0.\nThe last inequality follows directly from Lemma 6.1 in [5]. We also have by Lemma D.4 that\n|l,ζ|CA\n5/lessorsimilarecλ|k−l,η−ξ|secλ|ξ−ζ|s|ζ|3\n2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht−11/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|ζ|1\n2,|lt|≤1\n2|ζ|\n/lessorsimilarecλ|k−l,η−ξ|secλ|ξ−ζ|s1\n/a\\}b∇acketle{tζ/a\\}b∇acket∇i}ht1\n2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht(l2+(ζ−lt)2)1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|ζ|1\n2\n/lessorsimilarecλ|k−l,η−ξ|secλ|ξ−ζ|s|ζ|1\n2\n|l|/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2(l2+(ζ−lt)2)/angbracketleftbiggζ\nlt/angbracketrightbigg−1\n/lessorsimilarecλ|k−l,η−ξ|secλ|ξ−ζ|s|η|s\n2|ζ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2(l2+(ζ−lt)2)/angbracketleftbiggζ\nlt/angbracketrightbigg−188 WEIREN ZHAO\nand\n|l,ζ|CA\n6/lessorsimilarecλ|k−l,η−ξ|secλ|ξ−ζ|s|l|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/parenleftBig\n1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|l|,|l|2≥8|ζ|,|lt|≥2|ζ|+1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|l|,|l|2≥8|ζ|,|lt|≤2|ζ|/parenrightBig\n/lessorsimilarecλ|k−l,η−ξ|secλ|ξ−ζ|s1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2(l2+(ζ−lt)2)/angbracketleftbiggζ\nlt/angbracketrightbigg−1\n.\nTherefore, we have\n/summationdisplay\nN|Ra\nN,2|/lessorsimilarCK1\n2a/ba∇dbl∂yvΥ2/ba∇dblGσ−6/ba∇dblBa/ba∇dblGσ−6\n×/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2.\nFor the term Rf\nN,2, the estimate is simpler where the estimates of CA\n5andCA\n6are not needed.\n11.2.3.Treatment of Ra\nN,3.The treatment of Ra\nN,3is similar to the treatment of Rǫ,1\nN;HLin section\n6.2.1of [5]. Here we present the key steps and main estimates. In su pport of the integrand function,\nwe have\n|ξ−ζ| ≈ |l,ξ| ≈ |k,η|.\nWe have\nBk(t,η)\nBk−l(t,η−ξ)=/radicalBigg\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|η|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+(k−l)2+|η−ξ|\n/lessorsimilar/parenleftBig\n1+|ζ−ξ|1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|ζ−ξ|1\n2,|l|2≤1\n8|ζ−ξ|+|l|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|l|,|l|2≥8|ζ−ξ|/parenrightBig\n.(11.7)\nWe will divide the integral based on the relative size of landξ. For the case 16 |l| ≥ |ξ|, we have on\nthe support of the integrand,\n|k−l,η−ξ| ≤3\n16|l,ξ|,|ξ−ζ|/lessorsimilar|l|,\nthen by (11.7), we have\n|l,ζ|A∗\nk(η)\nAl(ξ)Bk−l(t,η−ξ)=|l,ζ|Bk(t,η)\nBk−l(t,η−ξ)Ak(η)\nAl(ξ)\n/lessorsimilar|l|/parenleftBig\n1+|l|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|l|,|l|2≥8|ζ−ξ|/parenrightBigAk(η)\nAl(ξ)def=CA\n7+CA\n8.\nWe then get\nCA\n7/lessorsimilarecλ|k−l,η−ξ|secλ|ζ−ξ|s|k,η|s\n2|l|1−s\n21t/∈Il,ξ,l/\\e}atio\\slash=0 (11.8)\n/lessorsimilarecλ|k−l,η−ξ|secλ|ζ−ξ|s|k,η|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/angbracketleftbiggζ\nlt/angbracketrightbigg−1|l,ζ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts(l2+(ζ−lt)2)1l/\\e}atio\\slash=0,\nCA\n8/lessorsimilarecλ|k−l,η−ξ|secλ|ζ−ξ|s|l|2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht21/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|l|,|l|2≥8|ζ−ξ|(11.9)\n/lessorsimilarecλ|k−l,η−ξ|secλ|ζ−ξ|s1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2(l2+(ζ−lt)2)/angbracketleftbiggζ\nlt/angbracketrightbigg−1\n1l/\\e}atio\\slash=0.NONLINEAR INVISCID DAMPING 89\nFor the case 16 |l|<|ξ|, we have on the support of the integrand,\n|k−l,η−ξ| ≤3\n16|l,ξ|,|l,ζ| ≤67\n100|ξ−ζ|,\nand then\n|l,ζ|A∗\nk(η)\nAR(ξ−ζ)Bk−l(t,η−ξ)=|l,ζ|Bk(t,η)\nBk−l(t,η−ξ)Ak(η)\nAR(ξ−ζ)\n/lessorsimilar|l,ζ|/parenleftBig\n1+|ζ−ξ|1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|ζ−ξ|1\n2,|l|2≤1\n8|ζ−ξ|+|l|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|l|,|l|2≥8|ζ−ξ|/parenrightBigAk(η)\nAR(ξ−ζ)\n/lessorsimilar/parenleftBig\n1+|ζ−ξ|1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1/a\\}bracketle{tt/a\\}bracketri}ht≤1\n8|ζ−ξ|1\n2,|l|2≤1\n8|ζ−ξ|/parenrightBig\necλ|k−l,η−ξ|ecλ|l,ζ|def=CA\n9+CA\n10.\nWe get that\nCA\n9/lessorsimilarecλ|k−l,η−ξ|ecλ|l,ζ|(11.10)\nCA\n10/lessorsimilarecλ|k−l,η−ξ|ecλ|l,ζ||k,η|s\n2|ζ−ξ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht. (11.11)\nCombining all the estimates, we arrive at\n/summationdisplay\nN|Ra\nN,3|/lessorsimilarCK1\n2a/ba∇dbl∂yvΥ2/ba∇dblGσ−6/ba∇dblBa/ba∇dblGσ−6\n×/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA∗a/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsAR(∂yvΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2s−1/ba∇dblΥ2Ψ/ba∇dblGσ−6/ba∇dblBa/ba∇dblGσ−6\n+1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblA∗a/ba∇dblL2/vextenddouble/vextenddouble/vextenddoubleAR(∂yvΥ2)/vextenddouble/vextenddouble/vextenddouble\nL2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblΥ2Ψ/ba∇dblGσ−6/ba∇dblBa/ba∇dblGσ−6\n/lessorsimilarǫCK1\n2a/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n+ǫ2CK1\n2a/parenleftbigg1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+CK1\n2\nh+CK1\n2\nϕδ\n9/parenrightbigg\n+ǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2.\n11.2.4.Treatment of Ra\nN,4.The treatment of Ra\nN,4is similar to that of Rǫ,1\nN,HLin section 6.2.1 of [5].\nWe divide it into two cases |l|>100|ζ|and|l| ≤100|ζ|.\nFor the first case |l|>100|ζ|, on the support of the integrand, we have\n|k−l,η−ξ| ≤6\n32|l,ξ|,|ξ−ζ| ≤24|ζ| ≤24\n100|l,ζ|,\nwhich gives\n|ξ|/lessorsimilar|ζ|/lessorsimilar|l|,|l,ξ| ≈ |l| ≈ |k,η|90 WEIREN ZHAO\n|l,ζ|A∗\nk(η)\nAl(ζ)Bk−l(t,η−ξ)/lessorsimilar|l|Ak(η)\nAl(ζ)Bk(t,η)\nBk−l(t,η−ξ)=Ak(η)\nAl(ζ)/radicalBigg\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+k2+|η|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+(k−l)2+|η−ξ|\n/lessorsimilar|l|1−s\n2|k,η|s\n2/parenleftBig\n1+|l|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1|l|≥100t/parenrightBig\n1t/∈Il,ζeλ|k−l,η−ξ|seλ|ξ−ζ|s\n/lessorsimilar|k,η|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/angbracketleftbiggζ\nlt/angbracketrightbigg−1|l,ζ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts(l2+(ζ−lt)2)1l/\\e}atio\\slash=0eλ|k−l,η−ξ|seλ|ξ−ζ|s.\nFor the second case |l| ≤100|ζ|, on the support of the integrand, we have\n|k−l,η−ξ| ≤3\n16|l,ξ|,|ξ−ζ| ≤24|ζ| ≤24|l,ζ|,|l,ζ| ≤101|ζ| ≤2424|ξ−ζ|\nWe have\neλ|k,η|s≤eλ|l,ξ|s+cλ|k−l,η−ξ|s≤ecλ|l,ζ|s+cλ|ξ−ζ|s+cλ|k−l,η−ξ|s\nand\nJk(η)/lessorsimilare2µ|k,η|1\n2≤e2µ|l,ζ|1\n2+2µ|ξ−ζ|1\n2+2µ|k−l,η−ξ|1\n2\nand\nBk(t,η)\nBk−l(t,η−ξ)/lessorsimilar/a\\}b∇acketle{tk,η/a\\}b∇acket∇i}ht/lessorsimilar1+|l,ζ|+|ξ−ζ|+|k−l,η−ξ|\nThus we have\n/summationdisplay\nN≥8|Ra\nN,4|/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+ǫCKa+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsAP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2. (11.12)\n11.2.5.Treatment of Ra\nN,5.The treatment of Ra\nN,5is simple. We have\n|Ra\nN,5|/lessorsimilar/ba∇dblA∗a∼N/ba∇dblL2/ba∇dbl(Υ2U)N/ba∇dblHσ−6/ba∇dblA∗a<N/8/ba∇dblL2,\nwhich gives that\n/summationdisplay\nN≥8|Ra\nN,5|/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2.\n11.3.Treatment of Ra.The treatment of Rais similar to that of Ra\nN,4, one can follow section 7\nof [5] and obtain that\n|Ra|/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2.\n12.The lower order terms in If\nIn this section, we estimate If,3andIf,9. Recall that\nIf,3=−1\n2π/summationdisplay\n|k|≥kM/integraldisplay\nRAk(t,η)2/hatwider/parenleftBig\nϕ2∂zΨΩ/parenrightBig\nk(t,η)/hatwidefk(t,η)dη\n−1\n2π/summationdisplay\n|k|≥kM/integraldisplay\nRAk(t,η)2F/parenleftBig\nϕ3∂zΨ∂yv(∂v−t∂z)Ψ/parenrightBig\nk(t,η)/hatwidefk(t,η)dη\n=I1\nf,3+I2\nf,3,NONLINEAR INVISCID DAMPING 91\nand\nIf,9=−1\n2π/summationdisplay\n0<|k|<kM/integraldisplay\nRAk(t,η)2F2/parenleftBig\nDu,k/parenleftBig\nF1/parenleftBig\nϕ2∂zΨΩ/parenrightBig/parenrightBig/parenrightBig\nk(t,η)/hatwidefk(t,η)dη\n−1\n2π/summationdisplay\n0<|k|<kM/integraldisplay\nRAk(t,η)2F2/parenleftBig\nDu,k/parenleftBig\nF1/parenleftBig\nϕ3∂zΨ∂yv(∂v−t∂z)Ψ/parenrightBig/parenrightBig/parenrightBig\nk(t,η)/hatwidefk(t,η)dη\n=I1\nf,9+I2\nf,9.\nThe estimates are similar to each other. Here we present the t reatment of the most technical term\nI1\nf,9which involves the wave operator. By Proposition 4.11, we ha ve\n|I1\nf,9|/lessorsimilar/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\n0<|k|<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4Ak(η)2D(t,k,η,ξ 1)/hatwiderϕ2(ξ1−ξ2)il/hatwidestΥ2Ψl(ξ3)/hatwideΩk−l(ξ2−ξ3)/hatwidefk(η)dηdξ1dξ2dξ3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilar/summationdisplay\n0<|k|<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4Ak(η)2e−λD|η−ξ1|s|/hatwiderϕ2(ξ1−ξ2)||l/hatwidestΥ2Ψl(ξ3)||/hatwideΩk−l(ξ2−ξ3)||/hatwidefk(η)|dηdξ1dξ2dξ3\n/lessorsimilar/summationdisplay\n0<|k|<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4e−cλD|η−ξ1|sAk(ξ1)|/hatwiderϕ2(ξ1−ξ2)||l/hatwidestΥ2Ψl(ξ3)||/hatwideΩk−l(ξ2−ξ3)||/hatwiderAfk(η)|dηdξ1dξ2dξ3\n=/summationdisplay\nN≥8/summationdisplay\n0<|k|<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4e−cλD|η−ξ1|sAk(ξ1)/parenleftBig\n|/hatwiderϕ2(ξ1−ξ2)||l/hatwidestΥ2Ψl(ξ3)|/parenrightBig\n<N/8\n×|/hatwideΩk−l(ξ2−ξ3)N||/hatwiderAfk(η)|dηdξ1dξ2dξ3\n+/summationdisplay\nN≥8/summationdisplay\nM≥8/summationdisplay\n0<|k|<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4e−cλD|η−ξ1|sAk(ξ1)/parenleftBig\n|/hatwiderϕ2(ξ1−ξ2)M||l/hatwidestΥ2Ψl(ξ3)<M/8|/parenrightBig\nN\n×|/hatwideΩk−l(ξ2−ξ3)<N/8||/hatwiderAfk(η)|dηdξ1dξ2dξ3\n+/summationdisplay\nN≥8/summationdisplay\nM≥8/summationdisplay\n0<|k|<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4e−cλD|η−ξ1|sAk(ξ1)/parenleftBig\n|/hatwiderϕ2(ξ1−ξ2)<M/8||l/hatwidestΥ2Ψl(ξ3)M|/parenrightBig\nN\n×|/hatwideΩk−l(ξ2−ξ3)<N/8||/hatwiderAfk(η)|dηdξ1dξ2dξ3\n+/summationdisplay\nN≥8/summationdisplay\nM/summationdisplay\n1\n8M≤M′≤8M/summationdisplay\n0<|k|<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4e−cλD|η−ξ1|sAk(ξ1)/parenleftBig\n|/hatwiderϕ2(ξ1−ξ2)M′||l/hatwidestΥ2Ψl(ξ3)M|/parenrightBig\nN\n×|/hatwideΩk−l(ξ2−ξ3)<N/8||/hatwiderAfk(η)|dηdξ1dξ2dξ3\n+/summationdisplay\nN/summationdisplay\n1\n8N≤N′≤8N/summationdisplay\n0<|k|<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4e−cλD|η−ξ1|sAk(ξ1)/parenleftBig\n|/hatwiderϕ2(ξ1−ξ2)||l/hatwidestΥ2Ψl(ξ3)|/parenrightBig\nN′\n×|/hatwideΩk−l(ξ2−ξ3)N||/hatwiderAfk(η)|dηdξ1dξ2dξ3\ndef=I1\nf,9,LH+I1\nf,9,HL,HL+I1\nf,9,HL,LH+I1\nf,9,HL,HH+I1\nf,9,HH.\n12.1.The low-high interactions. For the low-high interaction, the estimate is easy. We have b y\nLemma D.4\nAk(ξ1)\nAk−l(ξ2−ξ3)/lessorsimilarecλ|ξ1−ξ2|+cλ|l,ξ3|/parenleftBig\n1+ξ1\nk21t∈Ik,ξ1/parenrightBig\n/lessorsimilarecλ|ξ1−ξ2|+cλ|l,ξ3|/a\\}b∇acketle{tk−l,ξ2−ξ3/a\\}b∇acket∇i}ht1\n2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1\n2,92 WEIREN ZHAO\nwhich gives that\nI1\nf,9,LH/lessorsimilarkM/ba∇dblϕ2/ba∇dblGσ−6/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/ba∇dblΥ2Ψ/ba∇dblGσ−6/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3\n4AΩ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2CK1\n2\nf/lessorsimilarkMǫCKf. (12.1)\n12.2.The high-low interactions. The treatment of I1\nf,9,HL,HLis similar to the Reaction term\nRa\nN,3, but simpler. We will divide the integral based on the relati ve size oflandξ3. For the case\n16|l| ≥ |ξ3|, we have on the support of the integrand,\n|l,ξ3| ≤3\n16|ξ1−ξ2|,|k−l,ξ2−ξ3| ≤3\n16|l,ξ1−ξ2+ξ3|/lessorsimilar|ξ1−ξ2|.\nThen by (11.8), we have\n|l|Ak(ξ1)\nAl(ξ3)/lessorsimilarecλ|k−l,ξ2−ξ3|s+cλ|ξ1−ξ2|s|k,ξ1|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/angbracketleftbiggξ3\nlt/angbracketrightbigg−1|l,ξ3|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts(l2+(ξ3−lt)2)1l/\\e}atio\\slash=0\n/lessorsimilar/a\\}b∇acketle{tξ1−η/a\\}b∇acket∇i}htecλ|k−l,ξ2−ξ3|s+cλ|ξ1−ξ2|s|k,η|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/angbracketleftbiggξ3\nlt/angbracketrightbigg−1|l,ξ3|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts(l2+(ξ3−lt)2)1l/\\e}atio\\slash=0.\nHere/a\\}b∇acketle{tξ1−η/a\\}b∇acket∇i}htwill be absorbed by e−cλD|ξ1−η|s.\nFor the case 16 |l|<|ξ3|, we have on the support of the integrand,\n|ξ3| ≈ |l,ξ3| ≤3\n16|ξ1−ξ2|,|k−l,ξ2−ξ3| ≤3\n16|l,ξ1−ξ2+ξ3|/lessorsimilar|ξ1−ξ2|.\nThen by (11.10), we have\n|l|Ak(ξ1)\nAR(ξ1−ξ2)/lessorsimilarecλ|k−l,ξ2−ξ3|s+cλ|ξ3|s.\nWe then get that\nI1\nf,9,HL,HL/lessorsimilarǫCK1\n2\nf/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2+ǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2.\nWe then treat I1\nf,9,HL,LH, which is similar to the Reaction term Ra\nN,2, but simpler, because the\nderivative loss is only in z. On the support of the integrand, we have that\n|ξ1−ξ2| ≤6\n32|l,ξ3|,|k−l,ξ2−ξ3| ≤6\n32|l,ξ1−ξ2+ξ3|\nwhich gives |l,ξ1−ξ2+ξ3| ≈ |l,ξ3| ≈ |k,ξ1|. We then have by (11.6),\n|l|Ak(ξ1)\nAl(ξ3)/lessorsimilarecλ|k−l,ξ2−ξ3|s+cλ|ξ1−ξ2|s/bracketleftBig/radicalBigg\n∂twk(ξ1)\nwk(ξ1)+|k,ξ1|s/2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/bracketrightBig\n×/angbracketleftbiggξ3\nlt/angbracketrightbigg−1/bracketleftBig|l,ξ3|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+/radicalBigg\n∂twl(ξ3)\nwl(ξ3)/bracketrightBig\n(l2+(ξ3−lt)2)1l/\\e}atio\\slash=0\n/lessorsimilar/a\\}b∇acketle{tξ1−η/a\\}b∇acket∇i}htecλ|k−l,ξ2−ξ3|s+cλ|ξ1−ξ2|s/bracketleftBig/radicalBigg\n∂twk(η)\nwk(η)+|k,η|s/2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/bracketrightBig\n×/angbracketleftbiggξ3\nlt/angbracketrightbigg−1/bracketleftBig|l,ξ3|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts+/radicalBigg\n∂twl(ξ3)\nwl(ξ3)/bracketrightBig\n(l2+(ξ3−lt)2)1l/\\e}atio\\slash=0,NONLINEAR INVISCID DAMPING 93\nwhich gives that\nI1\nf,9,HL,LH/lessorsimilarCK1\n2\nf/ba∇dblϕ2/ba∇dblGσ−6/ba∇dblΩ/ba∇dblGσ−6\n×/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫCKf+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nThe treatment of I1\nf,9,HL,HHis similar to that of Ra\nN,4. We have\nI1\nf,9,HL,HH/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+ǫCKf+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsAP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\n12.3.The high-high interactions. The treatment of I1\nf,9,HHis even simpler than that of Ra. We\nhave\nI1\nf,9,HH/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2−KDǫ/2.\n13.Linear force term in Ia\nIn this section, we estimate Il\naand prove Proposition 7.14. We write\nIl\na=−1\n2π/summationdisplay\nk/\\e}atio\\slash=0/integraldisplay\nA∗\nk(η)2/hatwideak(η)/hatwideθ′(η−ξ)ik/hatwidestΥ2Ψk(ξ)dξdη\n=−1\n2π/summationdisplay\nN≥8/summationdisplay\nk/\\e}atio\\slash=0/integraldisplay\nA∗\nk(η)2/hatwideak(η)/hatwideθ′(η−ξ)<N/8ik/hatwidestΥ2Ψk(ξ)Ndξdη\n−1\n2π/summationdisplay\nN≥8/summationdisplay\nk/\\e}atio\\slash=0/integraldisplay\nA∗\nk(η)2/hatwideak(η)/hatwideθ′(η−ξ)Nik/hatwidestΥ2Ψk(ξ)<N/8dξdη\n−1\n2π/summationdisplay\nN/summationdisplay\n1\n8N≤N′≤8N/summationdisplay\nk/\\e}atio\\slash=0/integraldisplay\nA∗\nk(η)2/hatwideak(η)/hatwideθ′(η−ξ)N′ik/hatwidestΥ2Ψk(ξ)Ndξdη\ndef=Il\na,LH+Il\na,HL+Il\na,HH.\nForIl\na,LH, on the support of the integrand, we have\n|η−ξ| ≤3\n16|k,ξ|\nand then by Lemma D.4, it holds that\n|k|A∗\nk(η)\nAk(ξ)/lessorsimilar|k|ecλ|η−ξ|s/radicalBigg\n1+k2+|ξ|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2=CA\n11.\nWe consider the following four cases:\n1. ift≤2/radicalbig\n|ξ|,|k| ≤1\n4/radicalbig\n|ξ|, then\nCA\n11≤ |k|ecλ|η−ξ|s/radicalbig\n|ξ|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/lessorsimilarecλ|η−ξ|s/angbracketleftbiggξ\nkt/angbracketrightbigg−1ξ2\nk/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/lessorsimilarecλ|η−ξ|s/angbracketleftbiggξ\nkt/angbracketrightbigg−1(ξ−kt)2+k2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2.94 WEIREN ZHAO\n2. ift≤2/radicalbig\n|ξ|,|k| ≥1\n4/radicalbig\n|ξ| ≥1\n8t, then\nCA\n11≤ |k|ecλ|η−ξ|s|k|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/lessorsimilarecλ|η−ξ|s/angbracketleftbiggξ\nkt/angbracketrightbigg−1|k||ξ|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht21|kt|≤2|ξ|+ecλ|η−ξ|s/angbracketleftbiggξ\nkt/angbracketrightbigg−1|k|2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht1|kt|≥2|ξ|\n/lessorsimilarecλ|η−ξ|s/parenleftBig\n(ξ−kt)2+k2/parenrightBig/parenleftBig/angbracketleftbiggξ\nkt/angbracketrightbigg−1|ξ|1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht21|k|≈√\n|ξ|+/angbracketleftbiggξ\nkt/angbracketrightbigg−11\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3/parenrightBig\n/lessorsimilarecλ|η−ξ|s/parenleftBig\n(ξ−kt)2+k2/parenrightBig/angbracketleftbiggξ\nkt/angbracketrightbigg−1|ξ|s\n2|k,η|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2.\n3. if 2|ξ|>t≥2/radicalbig\n|ξ|, thent∈Il,ξ∩Ij,ηfor somel,j, and\nCA\n11/lessorsimilar|k|2ecλ|η−ξ|s/radicalBigg\n∂twl(ξ)\nwl(ξ)/radicalBigg\n∂twj(η)\nwj(η)/parenleftBig\n1+|ξ\nl−t|/parenrightBig1\n2/parenleftBig\n1+|η\nj−t|/parenrightBig1\n2\n/lessorsimilar|k|2ecλ|η−ξ|s/radicalBigg\n∂twl(ξ)\nwl(ξ)/radicalBigg\n∂twj(η)\nwj(η)/parenleftBig\n1+|ξ\nk−t|/parenrightBig/angbracketleftbiggξ\nkt/angbracketrightbigg−1\n1|k|/greaterorsimilar|l|≈|j|\n+|k|2ecλ|η−ξ|s/radicalBigg\n∂twl(ξ)\nwl(ξ)/radicalBigg\n∂twj(η)\nwj(η)/parenleftBig\n1+|ξ\nk−t|/parenrightBig\n1|kt|≤1\n2|ξ|\n/lessorsimilarecλ|η−ξ|s/radicalBigg\n∂twl(ξ)\nwl(ξ)/radicalBigg\n∂twj(η)\nwj(η)/parenleftbig\nk2+|ξ−tk|2/parenrightbig/angbracketleftbiggξ\nkt/angbracketrightbigg−1\n.\n4. ift≥2|ξ|, then\nCA\n11/lessorsimilark2ecλ|η−ξ|sk2+|ξ−tk|2\nk2t2/angbracketleftbiggξ\nkt/angbracketrightbigg−1\n.\nThus, we have\nIl\na,LH/lessorsimilarCK1\n2a/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n≤ε0CKa+Cε0/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.(13.1)\nForIl\na,HL, we consider two cases 16 |k| ≤ |ξ|and 16|k|>|ξ|. For the first case 16 |k| ≤ |ξ|, on the\nsupport of the integrand, we have\n|ξ| ≈ |k,ξ| ≤3\n16|η−ξ| ≈ |k,η| ≈ |η|.\nThen we have\nA∗\nk(η)\nAR(η−ξ)/lessorsimilarecλ|k,ξ|s/parenleftBig\n1+/a\\}b∇acketle{tη/a\\}b∇acket∇i}ht1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/parenrightBig\n/lessorsimilarecλ|k,ξ|s/parenleftBig\n1+|k,η|s\n2|η−ξ|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/parenrightBig\n.\nFor the second case 16 |k|>|ξ|, on the support of the integrand, we have\n|k| ≈ |k,ξ| ≤3\n16|η−ξ| ≈ |k,η|.\nThen we have\nA∗\nk(η)\nAk(ξ)/lessorsimilarkecλ|η−ξ|/parenleftBig\n1+|ξ|+k2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2/parenrightBig1\n2NONLINEAR INVISCID DAMPING 95\n/lessorsimilar|k|ecλ|η−ξ|/parenleftBig\n1+|k|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/parenrightBig\n1|k|\n16≤|ξ|<16|k|+|k|ecλ|η−ξ|/parenleftBig\n1+|k|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht+/radicalbig\n|ξ|\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/parenrightBig\n1|ξ|<1\n16|k|\n/lessorsimilarecλ|η−ξ|/angbracketleftbiggξ\ntk/angbracketrightbigg−1\n(k2+(ξ−kt)2)/parenleftBig|k|1\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3\n2+|ξ|s\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2s/parenrightBig\n.\nTherefore, we have\n|Il\na,HL|/lessorsimilarǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+ǫCK1\n2a/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htARθ′/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2\n+CK1\n2a/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+ǫCK1\n2a/parenleftBig1\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht+CK1\n2\nϕδ\n10/parenrightBig\n+CK1\n2a/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n2\n/lessorsimilarǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+ǫCKϕδ\n10+ε0CKa\n+Cε0/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(ΨΥ2)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nThe treatment of Il\na,HHis simple. We have\n|Il\na,HH|/lessorsimilarǫ2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2.\n14.The nonlinear interactions with the pressure in If\nIn this section, we treat the If,4andIf,8. We have by (2.17) and Proposition 4.11 that\nIf,4=−1\n2π/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR3A0(η)/bracketleftBig\nh1(ξ′)(l(ξ+ζ))/hatwidest(Π⋆)l(ξ)/hatwidea−l(ζ)/bracketrightBig\nA0(η)/hatwidef0(η)dξdζdη\n−1\n2π/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR3A0(η)/bracketleftBig\nh2(ξ′)(il)/hatwidest(Π⋆)l(ξ)/hatwidea−l(ζ)/bracketrightBig\nA0(η)/hatwidef0(η)dξdζdη\n−1\n2π/summationdisplay\nk≥kM/summationdisplay\nl/integraldisplay\nR3Ak(η)/bracketleftBig\nh1(ξ′)(−kξ+l(ξ+ζ))/hatwidest(Π⋆)l(ξ)/hatwideak−l(ζ)/bracketrightBig\nAk(η)/hatwidefk(η)dξdζdη\n−1\n2π/summationdisplay\nk≥kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR3Ak(η)/bracketleftBig\nh2(ξ′)(il)/hatwidest(Π⋆)l(ξ)/hatwideak−l(ζ)/bracketrightBig\nAk(η)/hatwidefk(η)dξdζdη\nIf,8=−1\n2π/summationdisplay\n0<k<kM/summationdisplay\nl/integraldisplay\nR4Ak(η′)D(t,k,η′,η)\n×/bracketleftBig\nh1(ξ′)(−kξ+l(ξ+ζ))/hatwidest(Π⋆)l(ξ)/hatwideak−l(ζ)/bracketrightBig\n/hatwiderAfk(η′)dξdζdηdη′\n−1\n2π/summationdisplay\n0<k<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4Ak(η′)D(t,k,η′,η)96 WEIREN ZHAO\n×/bracketleftBig\nh2(ξ′)(il)/hatwidest(Π⋆)l(ξ)/hatwideak−l(ζ)/bracketrightBig\n/hatwiderAfk(η′)dξdζdηdη′\n=I1\nf,8+I2\nf,8,\nwhereξ′=η−ξ−ζ,\nh1(ξ′) =/integraldisplay\nR/hatwider(ϕ4Υ2)(ξ′′)/hatwider(Υ2∂yv)(ξ′−ξ′′)dξ′′,h2(ξ′) =/hatwiderϕ5(ξ′).\nand Π⋆∈/braceleftbig\nP/\\e}atio\\slash=(Υ2Πl,1), P/\\e}atio\\slash=(Υ1Πl,2), P/\\e}atio\\slash=(Υ2Πn,1), P/\\e}atio\\slash=(Υ1Πn,2), P0(Υ2Π)/bracerightbig\n.\nWe only present the estimates for I1\nf,8which involves the wave operator, other terms are similar\nor even easier. We have by Proposition 4.11 that\n|I1\nf,8|/lessorsimilar/summationdisplay\n0<k<kM/summationdisplay\nl/integraldisplay\nR4e−cλD|η−η′|s0Ak(η)/parenleftBig\n1|ξ′|≥4|k,η|+14|ξ′|≤|k,η|+11\n4|k,η|≤|ξ′|≤4|k,η|/parenrightBig\n×/bracketleftBig\n|h1(ξ′)||−kξ+l(ξ+ζ)||/hatwidest(Π⋆)l(ξ)||/hatwideak−l(ζ)|/bracketrightBig\n|/hatwiderAfk(η′)|dξdζdηdη′\n=IHL\nf,8+ILH\nf,8+IHH\nf,8,\nThen we have by Remark 6.5 and Corollary 7.11 and fact that\n/ba∇dblARh1/ba∇dbl2/lessorsimilar(1+/ba∇dblARϕδ\n4/ba∇dbl2)(1+/ba∇dblARh/ba∇dbl2+/ba∇dblARϕδ\n9/ba∇dbl2)/lessorsimilar1\nthat\nIHL\nf,8+IHH\nf,8/lessorsimilar/ba∇dblARh1/ba∇dbl2/ba∇dblP/\\e}atio\\slash=Π⋆/ba∇dblGs,σ−6/ba∇dbla/ba∇dblGs,σ−6/ba∇dblAf/ba∇dbl2\n+/ba∇dblARh1/ba∇dbl2/ba∇dblP0∂vΠ/ba∇dblGs,σ−6/ba∇dbla/ba∇dblGs,σ−6/ba∇dblAf/ba∇dbl2/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3.\nForILH\nf,8, by Remark 6.5, we have\nILH\nf,8/lessorsimilar/summationdisplay\n0<k<kM/summationdisplay\nl/integraldisplay\nR4e−cλD|η−η′|s0ecλ|ξ′|s|h1(ξ′)|/parenleftBig\n14|ξ′|≤|k,η|/parenrightBig\nAk(ξ+ζ)\n×/bracketleftBig\n|−kξ+l(ξ+ζ)||/hatwidest(Π⋆)l(ξ)||/hatwideak−l(ζ)|/bracketrightBig\n|/hatwiderAfk(η′)|dξdζdηdη′\n=/summationdisplay\n0<k<kM/integraldisplay\nR4e−cλD|η−η′|s0ecλ|ξ′|s|h1(ξ′)|14|ξ′|≤|k,η|Ak(ξ+ζ)\n×/parenleftBig\n1|ξ|≥4|k,ζ|+14|ξ|≤|k,ζ|+11\n4|k,ζ|≤|ξ|≤4|k,ζ|/parenrightBig\n×/bracketleftBig\n|−kξ||/hatwiderP0(Π)(ξ)||/hatwideak(ζ)|/bracketrightBig\n|/hatwiderAfk(η′)|dξdζdηdη′\n+/summationdisplay\n0<k<kM/summationdisplay\nl/\\e}atio\\slash=0/integraldisplay\nR4e−cλD|η−η′|s0ecλ|ξ′|s|h1(ξ′)|14|ξ′|≤|k,η|Ak(ξ+ζ)\n×/parenleftBig\n1|l,ξ|≥4|k,ζ|+14|l,ξ|≤|k,ζ|+11\n4|k,ζ|≤|l,ξ|≤4|k,ζ|/parenrightBig\n×/bracketleftBig\n|−kξ+l(ξ+ζ)||/hatwidest(Π⋆)l(ξ)||/hatwideak−l(ζ)|/bracketrightBig\n|/hatwiderAfk(η′)|dξdζdηdη′\n=IHL\nf,8,0+ILH\nf,8,0+IHH\nf,8,0+IHL\nf,8,/\\e}atio\\slash=+ILH\nf,8,/\\e}atio\\slash=+IHH\nf,8,/\\e}atio\\slash=.\nThe termsIHH\nf,8,0andIHH\nf,8,/\\e}atio\\slash=are easy. We have\nIHH\nf,8,0/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht3, IHH\nf,8,/\\e}atio\\slash=/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht4.NONLINEAR INVISCID DAMPING 97\nFor the high-low interactions in IHL\nf,8,0, we have\nAk(ξ+ζ)/lessorsimilarA0(ξ)/parenleftBig\n1+ξ+ζ\nk2+(ξ+ζ−kt)21t∈Ik,ξ+ζ/parenrightBig\necλ|k,ζ|s\n/lessorsimilarA0(ξ)/a\\}b∇acketle{tξ/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htst(t2+ξ2)\n2+ξ2/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht2−s\n2\nt1−s(t2+ξ2)ecλ|k,ζ|s\n+A0(ξ)t(t2+ξ2)\n2+ξ2/a\\}b∇acketle{tξ/a\\}b∇acket∇i}ht3/k2\nt(t2+ξ2)/parenleftBig\n/a\\}b∇acketle{tk,ζ/a\\}b∇acket∇i}ht2+/a\\}b∇acketle{tη−η′/a\\}b∇acket∇i}ht2+/a\\}b∇acketle{tξ′/a\\}b∇acket∇i}ht2/parenrightBig\n(1+(η′/k−t)2)1\n2(1+(ξ/k−t)2)1\n21t∈Ik,ξ+ζecλ|k,ζ|s\n/lessorsimilarA0(ξ)/a\\}b∇acketle{tξ/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htst(t2+ξ2)\n2+ξ2/a\\}b∇acketle{tk,η′/a\\}b∇acket∇i}hts\n2\nt/parenleftBig\n/a\\}b∇acketle{tk,ζ/a\\}b∇acket∇i}hts\n2+/a\\}b∇acketle{tη−η′/a\\}b∇acket∇i}hts\n2+/a\\}b∇acketle{tξ′/a\\}b∇acket∇i}hts\n2/parenrightBig\necλ|k,ζ|s\n+A0(ξ)t(t2+ξ2)\n2+ξ2/radicalBigg\n∂tw(ξ)\nw(ξ)/radicalBigg\n∂twk(η′)\nwk(η′)/parenleftBig\n/a\\}b∇acketle{tk,ζ/a\\}b∇acket∇i}ht2+/a\\}b∇acketle{tη−η′/a\\}b∇acket∇i}ht2+/a\\}b∇acketle{tξ′/a\\}b∇acket∇i}ht2/parenrightBig\necλ|k,ζ|s,\nwhich gives that\nIHL\nf,8,0/lessorsimilarǫCKf+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublet(t2+|∂v|2)(2−∂2\nv)−1/parenleftBig/radicalbigg\n∂tw\nw+/a\\}b∇acketle{t∂v/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/parenrightBig\nA0P0/parenleftbig\n∂v(ΠΥ1)/parenrightbig/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nThe high-low interactions in IHL\nf,8,/\\e}atio\\slash=is similar to the Reaction term Ra\nN, by using (11.6), we have\nIHL\nf,8,/\\e}atio\\slash=/lessorsimilarǫCKf+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/angbracketleftbigg∂v\nt∂z/angbracketrightbigg−1\n(∂2\nz+(∂v−t∂z)2)/parenleftBigg\n|∇|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=(Π⋆)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nThen by (10.6), we have\nIHL\nf,8,/\\e}atio\\slash=/lessorsimilarǫCKf+ǫ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/a\\}b∇acketle{t∂z/a\\}b∇acket∇i}htt(t+/a\\}b∇acketle{t∇/a\\}b∇acket∇i}ht)(1−∆)−1/parenleftbig\n(∂v−t∂z)2+∂2\nz/parenrightbig/parenleftBigg\n/a\\}b∇acketle{t∇/a\\}b∇acket∇i}hts\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}htsA+/radicalbigg\n∂tw\nw˜A/parenrightBigg\nP/\\e}atio\\slash=Π⋆/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n2.\nTo treat the low-high interactions in ILH\nf,8,0andILH\nf,8,/\\e}atio\\slash=, we take advantage of the multiplier B. We\nhave\n|k|Ak(ξ+ζ)/lessorsimilar|k|A∗\nk(ζ)ecλ|ξ|st\nt+|k|+|ζ|1\n2/lessorsimilarA∗\nk(ζ)ecλ|ξ|s/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht,\nwhich gives that\nILH\nf,8,0/lessorsimilar/ba∇dblA∗a/ba∇dbl2/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht/ba∇dbl∂vP0(Υ2Π)/ba∇dblGs,σ−6/ba∇dblAf/ba∇dblL2/lessorsimilarǫ3\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2.\nWe also have\n|−kξ+l(ξ+ζ)|Ak(ξ+ζ)\n/lessorsimilar/a\\}b∇acketle{tl,ξ/a\\}b∇acket∇i}ht/a\\}b∇acketle{tk−l,ζ/a\\}b∇acket∇i}htA∗\nk−l(ζ)ecλ|l,ξ|st\nt+|k−l|+|ζ|1\n2/parenleftBig\n1+ξ+ζ\nk2+(ξ+ζ−kt)21t∈Ik,ξ+ζ∩Ik,ζl/\\e}atio\\slash=0/parenrightBig\n/lessorsimilar/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2+2s|ζ|s\n2|k,η′|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}ht2sA∗\nk−l(ζ)ecλ|l,ξ|s,\nwhich gives that\nILH\nf,8,/\\e}atio\\slash=/lessorsimilarǫCKa+ǫCKf.98 WEIREN ZHAO\nAppendix A.Gevrey spaces\nThe physical space characterization of Gevrey functions is also used in this paper. We start\nwith a characterization of the Gevrey spaces on the physical side. See Lemma A.1 in [18] for the\nelementary proof.\nLemma A.1 ([18]).Suppose that d= 1,2,0<s<1,K >1andg∈C∞(Rd)withsuppg⊂[a,b]d\nsatisfies the bounds\n|Dαg(x)| ≤Km(m+1)m/s, x∈Rd(A.1)\nfor all integers m≥0and multi-indices αwith|α|=m. Then\n|ˆg(ξ)|/lessorsimilarK,sLe−µ|ξ|s, (A.2)\nfor allξ∈Rdand someµ=µ(K,s)>0.\nConversely, assume that, for some µ>0ands∈(0,1),\n|ˆg(ξ)| ≤Le−µ|ξ|s, (A.3)\nfor allξ∈Rd. Then there is K >1depending on sandµsuch that\n|Dαg(x)|/lessorsimilarµ,sKm(m+1)m/s, x∈Rd(A.4)\nfor all integers m≥0and multi-indices αwith|α|=m.\nForx∈[a,b]dand parameters s∈(0,1) andM≥1, we define the spaces\n(A.5) GM,s\nph([a,b]d)def=/braceleftbigg\ng: [a,b]d→C:/ba∇dblg/ba∇dblGM,s\nph([a,b]d)<∞/bracerightbigg\n.\nwhere\n(A.6) /ba∇dblg/ba∇dblGM,s\nph([a,b]d)def= sup\nx∈[a,b]d,m≥0,|α|≤m|Dαg(x)|\n(m+1)m/sMm.\nHere ‘ph’ represents the physical side.\nWe define the spaces\n(A.7) GM,s\nph,1([a,b]d)def=/braceleftbigg\ng: [a,b]d→C:/ba∇dblg/ba∇dblGM,s\nph,1([a,b]d)<∞/bracerightbigg\n.\nwhere\n(A.8) /ba∇dblg/ba∇dblGM,s\nph,1([a,b]d)def= sup\nx∈[a,b]d,m≥0,|α|≤m|Dαg(x)|\nΓs(m)Mm,\nwith Γs(m) = (m!)1\ns(m+1)−2, also see [37] for more details.\nBy Stirling’s formula N!∼√\n2πN(N/e)N, it is easy to check that there exist K1<K2such that\nKm\n1(m+1)m/s/lessorsimilarΓs(m)/lessorsimilarKm\n2(m+1)m/s,\nwhich implies\nGK1M,s\nph([a,b]d)⊂ GM,s\nph,1([a,b]d)⊂ GK2M,s\nph([a,b]d). (A.9)\nWe also have\n(A.10)k/summationdisplay\nj=0k!\nj!(k−j)!Γs(j)Γs(k−j)<Γs(k),NONLINEAR INVISCID DAMPING 99\nwhich implies\n/ba∇dblfg/ba∇dblGM,s\nph,1([a,b]d)≤ /ba∇dblf/ba∇dblGM,s\nph,1([a,b]d)/ba∇dblg/ba∇dblGM,s\nph,1([a,b]d). (A.11)\nRemark A.2. Supposeg∈ GM,s\nph,1(Rd)withsuppg⊂[a,b]d, then for all integers m≥0and multi-\nindicesαwith|α|=m. Then\n|ˆg(ξ)|/lessorsimilarK,sLe−µ|ξ|s, (A.12)\nfor allξ∈Rdand someµ=µ(K,s)>0.\nWe also introduce the composition lemma in [37].\nLemma A.3 ([37]).LetI,Jbe real open intervals and let f:J→Rbe aC∞-function such that\nf′∈ GL,s\nph,1(J)andg:I→Jbe aC∞-function such that g′∈ GM,s\nph,1(I). LetNbe a real constant such\nthat\nN≥max(M,L/ba∇dblg′/ba∇dblGM,s\nph,1(I)).\nThen the derivative (f◦g)′of the composite function f◦gbelongs to GN,s\nph,1(I)and satisfies\n/ba∇dbl(f◦g)′/ba∇dblGN,s\nph,1(I)≤N\nL/ba∇dblf′/ba∇dblGL,s\nph,1(J).\nWe also introduce the estimate of the inverse function in the Gevrey class.\nLemma A.4 ([37]).LetIandJbe real open intervals. Let f:I→Jbe aC∞-surjection such that\n|f′(x)| ≥1\nA, x∈I\nfor someA >0and such that f′′∈ GL,s\nph,1(I). Thenfhas aC∞-inversef−1:J→Isuch that\n(f−1)′∈ GM,s\nph,1(J)for someM >0.\nAppendix B.The Fourier transform of the integral operator\nIn this section, we make some preparations to study the Gevre y regularity of the nonlocal part of\nthe wave operator. Indeed, we will write the nonlocal part in to the following four types of integral\noperators:\nΠm(F)(v′) =/integraldisplay\nRK(v,v′)µm(v−v′)e−ikt(v−v′)F(v)dv, m= 1,2,3,4,\nΠ∗\nm(F)(v) =/integraldisplay\nRK(v,v′)µm(v−v′)e−ikt(v−v′)F(v′)dv′, m= 1,2,3,4,\nwhereK(v,v′) represents a smooth kernel with compact support which may v ary from one line to\nthe other, and µ1∈C∞(R) with compact support such that\nµ1(u) =/braceleftBigg\n1,|u| ≤u(1)−u(0),\n0,|u| ≥2(u(1)−u(0)),\nµ2(u) =µ1(u)1R−(u) andµ3(u) =µ1(u)ln|u|,µ4(u) =p.v.1\nu. Thus it holds that\n|/hatwiderµ1(ξ)|+|/hatwiderµ2(ξ)|+|/hatwiderµ3(ξ)|+|/hatwiderµ4(ξ)|/lessorsimilar1 (B.1)\nIt is easy to check that Π∗\nm, m= 1,2,3,4 are the dual operators of Π m, m= 1,2,3,4.\nFor a smooth kernel with compact support defined on R2, let/hatwideKbe the Fourier transform in the\nfirst variable and/hatwide/hatwideKbe the Fourier transform in both variables.100 WEIREN ZHAO\nLemma B.1. SupposeK(v,v′)∈C∞(R2)with compact support, then it holds for m= 1,2,3,4that\n/hatwiderΠm(F)(η) =1\n2π/integraldisplay\nR2ˆF(ξ)/hatwide/hatwideK(−ξ−ξ′,η+ξ′)/hatwiderµm(ξ′+kt)dξ′dξ,\nand\n/hatwiderΠ∗m(F)(ξ) =1\n2π/integraldisplay\nR2ˆF(η)/hatwide/hatwideK(−ξ−ξ′,η+ξ′)/hatwiderµm(ξ′+kt)dξ′dη.\nProof.LetKL(v−v′,v′) =K(v,v′), then we have that\nΠm(F)(v′) =/integraldisplay\nRKL(v−v′,v′)µm(v−v′)e−ikt(v−v′)F(u)dv\n=1\n2π/integraldisplay\nRKL(v−v′,v′)µm(v−v′)e−ikt(v−v′)/integraldisplay\nRˆF(ξ)eiξvdξdv\n=1\n2π/integraldisplay\nRˆF(ξ)eiv′ξ/integraldisplay\nRKL(v−v′,v′)µm(v−v′)e−ikt(v−v′)eiξ(v−v′)dvdξ\n=1\n2π/integraldisplay\nRˆF(ξ)eiv′ξF1/parenleftbig\nKL(·,v′)µm/parenrightbig\n(−ξ+kt)dξ\n=1\n2π/integraldisplay\nR/integraldisplay\nRˆF(ξ)eiv′ξ/hatwiderKL(−ξ−ξ′,v′)/hatwiderµm(ξ′+kt)dξ′dξ.\nThus we get that\n/hatwiderΠm(F)(η) =1\n2π/integraldisplay\nR2ˆF(ξ)/hatwider/hatwiderKL(−ξ−ξ′,η−ξ)/hatwiderµm(ξ′+kt)dξ′dξ.\nWe also have\n/hatwider/hatwiderKL(ξ,η) =/integraldisplay\nR2KL(w,v′)e−iwξ−icηdwdv′\n=/integraldisplay\nR2KL(v−v′,v′)e−i(v−v′)ξ−iv′ηdvdv′\n=/integraldisplay\nR2K(v,v′)e−i(v−v′)ξ−iv′ηdvdv′=/hatwide/hatwideK(ξ,η−ξ),\nwhich gives the first identity and the second identity can be o btained by the same argument. /square\nLemma B.2. SupposeK(v,v′)∈ GM,s0\nph,1(R2)with compact support in [u(0),u(1)]2, then there exists\nλ=λ(M,s0), such that for m= 1,2,3,4,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n/hatwide/hatwideK(−ξ−ξ′,η+ξ′)/hatwiderµm(ξ′+kt)dξ′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilare−λ|η−ξ|s0. (B.2)\nProof.This lemma follows directly from (B.1), Remark A.2 and the fa ct that\n/integraldisplay\nRe−λ′(|ξ+ξ′|2+|η+ξ′|2)s02dξ′/lessorsimilar/integraldisplay\nRe−λ|ξ−η|s0\n1+|ξ+ξ′|2+|η+ξ′|2dξ′/lessorsimilare−λ|ξ−η|s0,\nholds for 0 <λ<λ′. /square\nRemark B.3. We also point out that if the regularity assumption on K(v,v′)is replaced by\nsup\nm≥0/ba∇dbl(∂v+∂v′)m(1−∂vv−∂v′v′)K(v,v′)/ba∇dblL1\nΓs(m)Mm≤C, (B.3)NONLINEAR INVISCID DAMPING 101\nthen(B.2)also holds. Indeed, the weaker regularity assumption (B.3)implies that there is λsuch\nthat\n|/hatwide/hatwideK(ξ,η)|/lessorsimilare−λ|ξ+η|s\n1+|ξ|2+|η|2. (B.4)\nThen(B.2)follows directly from (B.1).\nAppendix C.Sturm-Liouville Equation\nIn this section, we consider two elliptic equations:\n(C.1)\n\n/tildewiderϕ2∂zzΨ+/tildewideu′(∂v−t∂z)/parenleftBig\n/tildewiderϕ2/tildewideu′(∂v−t∂z)Ψ/parenrightBig\n= Ω,\nΨ(t,z,u(0)) = Ψ(t,z,u(1)) = 0,\nand\n(C.2)\n\n/tildewideθ∂zzΠ+/tildewideu′(∂v−t∂z)/parenleftBig\n/tildewideθ/tildewideu′(∂v−t∂z)Π/parenrightBig\n= ΞΥ,\n(∂v−t∂z)Π(t,z,u(0)) = (∂v−t∂z)Π(t,z,u(1)) = 0.\nFor the first equation (C.1), if we let\n(C.3) Q=F1[Ψ](t,k,v)e−itkv\nthenQsolves\n/braceleftBigg\n∂v(h1(v)∂vQ)−k2h2(v)Q=F(v,k)\nQ(α,k) =Q(β,k) = 0\nwhere\n(C.4)α=u(0), β=u(1),F(v,k) =/tildewideu′(v)−1F1[Ω]e−iktv,\nh1(v) =/tildewiderϕ2/tildewideu′(v), h2(v) =/tildewiderϕ2(v)/tildewideu′(v)−1.\nFor the second equation (C.2), if we let\nR=F1[Π](t,k,v)e−itkv(C.5)\nthenRsolves\n/braceleftBigg\n∂v(h1(v)∂vR)−k2h2(v)R=G(v,k)\n∂vR(α,k) =∂vR(β,k) = 0\nwhere\n(C.6)α=u(0), β=u(1),G(v,k) =/tildewideu′(v)−1F1[Ξ]e−iktv,\nh1(v) =/tildewideθ/tildewideu′(v), h2(v) =/tildewideθ(v)/tildewideu′(v)−1.\nBoth equations are Sturm-Liouville Equations.102 WEIREN ZHAO\nC.1.Dirichlet boundary conditions. Inthissection, westudythegeneralSturm-LiouvilleEqua-\ntion/braceleftBigg\n∂v(h1(v)∂vQ)−k2h2(v)Q=F(v,k)\nQ(α,k) =Q(β,k) = 0(C.7)\nwhereh1,h2are positive functions and supp h′\nj⊂[α1,β1] forj= 1,2 withα < λ1< β1< β. We\nalso assume that h′\n1,h′\n2∈ GM,s\nph([α,β]). Let us first study the homogeneous equation.\nLemma C.1. There exist positive functions qα(v,k),qβ(v,k)≥1such that qα(α,k) =qβ(β,k) = 1,\nq′\nα(α,k) =q′\nβ(α,k) = 0, andqα(v,k)solves\nqα(v,k) = 1+k2/integraldisplayv\nα1\nh1(v′)/integraldisplayv′\nαh2(v′′)qα(v′′,k)dv′′dv′,\nandqβ(v,k)solves\nqβ(v,k) = 1+k2/integraldisplayv\nβ1\nh1(v′)/integraldisplayv′\nβh2(v′′)qβ(v′′,k)dv′′dv′.\nMoreover, it holds for v∈[α,β]that\nC−1min{|k|2(v−α),|k|} ≤q′\nα(v,k)\nqα(v,k)≤Cmin{|k|2(v−α),|k|}\nC−1min{|k|2(β−v),|k|} ≤ −q′\nβ(v,k)\nqβ(v,k)≤Cmin{|k|2(β−v),|k|}(C.8)\neC−1|k||v−α|≤qα(v,k)≤eC|k||v−α|, eC−1|k||v−β|≤qβ(v,k)≤eC|k||v−β|. (C.9)\nThe Wronskian is\nW[qα,qβ](v) =q′\nβ(α)h1(α)\nh1(v)=−q′\nα(β)h1(β)\nh1(v)<0.\nwhich also implies\n−q′\nβ(α)\nq′α(β)≈qβ(α)\nqα(β)≈1.\nProof.We only show the proof for qα.\nExistence. It is easy to show that T[qα] =/integraltextv\nα1\nh1(v′)/integraltextv\nαh2(v′′)qα(v′′,k)dv′′dv′is a contraction map\nin the weighted Gevrey space GM|k|,s\nph,cosh([α,β])def={f∈C∞([α,β]) :/ba∇dblf/ba∇dblGM|k|,s\nph,cosh([α,β])<∞}by taking\nMlarge enough, where\n/ba∇dblf/ba∇dblGM|k|,s\nph,cosh([α,β])def= sup\nv∈[α,β],m≥0|∂m\nvf|\nΓs(m)Mm|k|mcoshM|k|(v−α)\nand Γs(m) = (m!)1\ns(m+1)−2. The proof is simpler than the proof of Proposition 4.8. We om it the\ndetails. Note that Tis a positive operator, then qα(v,k)≥1 andq′\nα(v,k)≥0.\nEstimates. Letfα=h1q′\nα/qα, thenf′\nα+f2\nα/h1=k2h2which gives that 0 ≤fα≤ |k|√h2h1and\nthus 0≤q′\nα/qα≤ |k|/radicalbig\nh2/h1≤C|k|. Therefore, we have for v′′≤v\n1≥qα(v′′,k)\nqα(v,k)≥e−C|k||v′′−v|.NONLINEAR INVISCID DAMPING 103\nWe have for |v−α| ≤1\n|k|,\nq′\nα(v,k)\nqα(v,k)=k2\nh1(v)/integraldisplayv\nαh2(v′′)qα(v′′,k)\nqα(v,k)dv′′≈k2(v−α).\nAs for|v−α| ≥1\n|k|, we have\nq′\nα(v,k)\nqα(v,k)≥k2\nh1(v)/integraldisplayv\nv−1\n2|k|h2(v′′)qα(v′′,k)\nqα(v,k)dv′′≥C−1|k|.\nCombining the above estimates, we get (C.8), (C.9) follows d irectly.\nLet us now computer the Wronskian. A direct calculation give s\n∂v/parenleftBig\nh1W[qα,qβ]/parenrightBig\n= 0\nwhich gives the formula of the Wronskian. /square\nLet us now construct two linearly independent solutions qlandqr:\nql(v,k) =qα(v,k)−qβ(v,k)/qβ(α,k) (C.10)\nqr(v,k) =qβ(v,k)−qα(v,k)/qα(β,k) (C.11)\nWith the homogeneous solution qrandql, we get that\nQ(v,k) =1\nh1(v)qr(v,k)\nW[qr,ql]/integraldisplayv\n0ql(v′,k)F(v′,k)h1(v′)dv′+1\nh1(v)ql(v,k)\nW[qr,ql]/integraldisplay1\nvqr(v′,k)F(v′,k)h1(v′)dv′,\nwhere\nW[qr,ql] =/parenleftBig\n1−1\nqα(β)qβ(α)/parenrightBig\nW[qα,qβ](v) =/parenleftBig\n1−1\nqα(β)qβ(α)/parenrightBigq′\nβ(α)h1(α)\nh1(v)\nWe also express Q(v,k) by using qαandqβ.\n(C.12)Q(v,k) =1\nh1(v)1\nW[qr,ql]/integraldisplayv\nα/parenleftBig\n1+1\nqβ(α)qα(β)/parenrightBig\nqβ(v,k)qα(v′,k)F(v′,k)h1(v′)dv′\n+1\nh1(v)1\nW[qr,ql]/integraldisplayβ\nv/parenleftBig\n1+1\nqβ(α)qα(β)/parenrightBig\nqα(v,k)qβ(v′,k)F(v′,k)h1(v′)dv′\n−1\nh1(v)1\nW[qr,ql]/integraldisplayβ\nαqα(v,k)qα(v′,k)\nqα(β)F(v′,k)h1(v′)dv′\n−1\nh1(v)1\nW[qr,ql]/integraldisplayβ\nαqβ(v,k)qβ(v′,k)\nqβ(α)F(v′,k)h1(v′)dv′\nWe now introduce the Fourier kernel for the Sturm-Liouville type elliptic problem (C.7). For\nany given fixed k, and positive Gevrey functions h1,h2∈ GM,s\nph([α,β]) with constant values near the\nboundary and any Gevrey cut-off function Υ with compact suppo rt suppΥ ⊂(α,β), we define a\nlinear operator △−1\nD,kto be such that for a function F,\n△−1\nD,kF= ΥQ (C.13)\nwhereQsolves/braceleftBigg\n∂v(h1(v)∂vQ)−k2h2(v)Q=F(v,k)Υ(v)\nQ(α,k) =Q(β,k) = 0.104 WEIREN ZHAO\nFor the case k= 0, we have\n△−1\nD,0F=/integraldisplayβ\nαGD(v,v′,0)Υ(v)Υ(v′)F(v′,0)dv′\nwhere\nGD(v,v′,0) =\n\n/parenleftBig/integraldisplayβ\nα1\nh1(v)dv/parenrightBig−1/parenleftBig/integraldisplayv\nα1\nh1(w)dw/parenrightBig/parenleftBig/integraldisplayv′\nβ1\nh1(w)dw/parenrightBig\nh1(v′)α≤v≤v′≤β\n/parenleftBig/integraldisplayβ\nα1\nh1(v)dv/parenrightBig−1/parenleftBig/integraldisplayv′\nα1\nh1(w)dw/parenrightBig/parenleftBig/integraldisplayv\nβ1\nh1(w)dw/parenrightBig\nh1(v′)α≤v′≤v≤β.\nBy (C.12), we have for k/\\e}atio\\slash= 0\n△−1\nD,kF=/integraldisplayβ\nαGD(v,v′,k)Υ(v)Υ(v′)F(v′,k)dv′\nwhere\n(C.14)GD(v,v′,k) =C1\nα,β1\n|k|qα(v′,k)qβ(v,k)\nqβ(α,k)h1(v′)\nh1(v)χR+(v−v′)\n+C1\nα,β1\n|k|qα(v,k)qβ(v′,k)\nqβ(α,k)h1(v′)\nh1(v)χR+(v′−v)\n+C2\nα,β1\n|k|qα(v,k)qα(v′,k)\nqα(β,k)qα(β,k)h1(v′)\nh1(v)\n+C3\nα,β1\n|k|qβ(v,k)qβ(v′,k)\nqβ(α,k)qβ(α,k)h1(v′)\nh1(v)\nwith constants\n(C.15)C1\nα,β=|k|qβ(α)\nq′\nβ(α)1\nh1(α)/parenleftBig\n1+1\nqβ(α)qα(β)/parenrightBig/parenleftBig\n1−1\nqα(β)qβ(α)/parenrightBig−1\nC2\nα,β=−/parenleftBig\n1−1\nqα(β)qβ(α)/parenrightBig−1|k|qα(β)\nq′\nβ(α)h1(α)\nC3\nα,β=−/parenleftBig\n1−1\nqα(β)qβ(α)/parenrightBig−1|k|qβ(α)\nq′\nβ(α)h1(α),\nwhereqαandqβare obtained in Lemma C.1.\nA direct calculation gives that\n∂v△−1\nD,kF=/integraldisplayβ\nαGD(v,v′,k)Υ′(v)Υ(v′)F(v′,k)dv′\n−h′\n1(v)\nh1(v)/integraldisplayβ\nαGD(v,v′,k)Υ(v)Υ(v′)F(v′,k)dv′\n+/integraldisplayβ\nαG1\nD(v,v′,k)Υ(v)Υ(v′)F(v′,k)dv′,(C.16)NONLINEAR INVISCID DAMPING 105\nand\n∂vv△−1\nD,kF=/integraldisplayβ\nαGD(v,v′,k)Υ′′(v)Υ(v′)F(v′,k)dv′\n−2h′\n1(v)\nh1(v)/integraldisplayβ\nαGD(v,v′,k)Υ′(v)Υ(v′)F(v′,k)dv′\n+2/integraldisplayβ\nαG1\nD(v,v′,k)Υ′(v)Υ(v′)F(v′,k)dv′\n−/parenleftBigh′\n1(v)\nh1(v)/parenrightBig′/integraldisplayβ\nαGD(v,v′,k)Υ(v)Υ(v′)F(v′,k)dv′\n+/integraldisplayβ\nαG2\nD(v,v′,k)Υ(v)Υ(v′)F(v′,k)dv′\n−2h′\n1(v)\nh1(v)/integraldisplayβ\nαG1\nD(v,v′,k)Υ(v)Υ(v′)F(v′,k)dv′\n+C1\nα,βh1(v)\n|k|/parenleftbiggqα(v,k)\nqβ(α,k)/parenleftBigqβ(v,k)\nh1(v)/parenrightBig′\n−qβ(v,k)\nqβ(α,k)/parenleftBigqα(v,k)\nh1(v)/parenrightBig′/parenrightbigg\nΥ(v)F(v,k)(C.17)\nwhere\nG1\nD(v,v′,k) =C1\nα,β1\n|k|q′\nβ(v,k)\nqβ(v,k)qα(v′,k)qβ(v,k)\nqβ(α,k)h1(v′)\nh1(v)χR+(v−v′)\n+C1\nα,β1\n|k|q′\nα(v,k)\nqα(v,k)qα(v,k)qβ(v′,k)\nqβ(α,k)h1(v′)\nh1(v)χR+(v′−v)\n+C2\nα,β1\n|k|q′\nα(v,k)\nqα(v,k)qα(v,k)qα(v′,k)\nqα(β,k)qα(β,k)h1(v′)\nh1(v)\n+C3\nα,β1\n|k|q′\nβ(v,k)\nqβ(v,k)qβ(v,k)qβ(v′,k)\nqβ(α,k)qβ(α,k)h1(v′)\nh1(v),\nG2\nD(v,v′,k) =C1\nα,β1\n|k|q′′\nβ(v,k)\nqβ(v,k)qα(v′,k)qβ(v,k)\nqβ(α,k)h1(v′)\nh1(v)χR+(v−v′)\n+C1\nα,β1\n|k|q′′\nα(v,k)\nqα(v,k)qα(v,k)qβ(v′,k)\nqβ(α,k)h1(v′)\nh1(v)χR+(v′−v)\n+C2\nα,β1\n|k|q′′\nα(v,k)\nqα(v,k)qα(v,k)qα(v′,k)\nqα(β,k)qα(β,k)h1(v′)\nh1(v)\n+C3\nα,β1\n|k|q′′\nβ(v,k)\nqβ(v,k)qβ(v,k)qβ(v′,k)\nqβ(α,k)qβ(α,k)h1(v′)\nh1(v).\nC.2.Regularity of the Green’s function. In this section, we study the Gevrey regularity of the\nGreen’s function. According to the above decompositions, w e only need to estimateqα(v′,k)qβ(v,k)\nqβ(α,k)\nwithv′≤v,qα(v,k)qβ(v′,k)\nqβ(α,k)withv′≥vandqα(v,k)\nqα(β,k)withv < β−δ,qβ(v,k)\nqβ(α,k)withv≥α+δfor some\nδ >0. By Lemma C.1, it is easy to check all of the four terms are in G evrey. So we only focus on\nthe uniformity in kof the Gevrey norm. Now we assume |k| ≥k0>0 withk0large enough which\nis determined by h1,h2andδ. Without loss of generality, we assume k≥k0>0.\nNow let us first treatqα(v,k)\nqα(β,k)withv <β−δ,qβ(v,k)\nqβ(α,k)withv≥α+δfor someδ >0 and give the\nfollowing lemma:106 WEIREN ZHAO\nLemma C.2. It holds for α+δ<v≤β−δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m\nv/parenleftBigqα(v,k)\nqα(β,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CΓs(m)Mm, (C.18)\nand forα+δ<v<β −δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m\nv/parenleftBigqβ(v,k)\nqβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CΓs(m)Mm. (C.19)\nProof.Let us prove the first estimate and the second one can be obtain ed by the same argument.\nNotice that q={qα,qβ}solves\n∂v(h1∂vq) =k2h2q.\nThus form1≥2 orm1≥2, a direct calculation gives\n���m1vq=∂m1−2\nv/parenleftBig−h′\n1\nh1∂vq+k2h2\nh1q/parenrightBig\n=m1−2/summationdisplay\nj=0(m1−2)!\nj!(m1−2−j)!∂j\nv/parenleftBig−h′\n1\nh1/parenrightBig\n∂m1−2−j+1\nv q+k2m1−2/summationdisplay\nl=0(m1−2)!\nl!(m1−2−l)!∂l\nv/parenleftBigh2\nh1/parenrightBig\n∂m1−2−l\nvq\nWe now use mathematical induction to prove the following res ult:\nFor allN≥0, it holds that\n(C.20)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂N\nv/parenleftBigqα(v,k)\nqα(β,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftBigN/summationdisplay\nj=0N!\nj!(N−j)!Γs(N−j)CN−j(C|k|)je−C−1δ|k|/parenrightBig\n.\nIt follows directly by the monotonicity of qα(v,k) that\nqα(v,k)\nqα(β,k)/lessorsimilarqα(β−δ,k)\nqα(β,k)/lessorsimilare−/integraltextβ\nβ−δfα(v)/h1(v)dv/lessorsimilareC−1kδ,\nand by (C.8), it holds that\nq′\nα(v,k)\nqα(β,k)/lessorsimilarkqα(v,k)\nqα(β,k)/lessorsimilarke−/integraltextβ\nβ−δfα(v)/h1(v)dv/lessorsimilarkeC−1kδ.\nLet us now assume that (C.21) holds for any N≤m−1 withm≥2, then for N=m, and by\n(A.10), we get that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m\nvqα\nqα(β,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m−2\nv/parenleftBig−h′\n1\nh1∂vq+k2h2\nh1q/parenrightBig\n/qα(β,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤m−2/summationdisplay\nj=0(m−2)!\nj!(m−2−j)!/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j\nv/parenleftBig−h′\n1\nh1/parenrightBig∂m−2−j+1\nvqα(v)\nqα(β,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+k2m−2/summationdisplay\nl=0(m−2)!\nl!(m−2−l)!/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂l\nv/parenleftBigh2\nh1/parenrightBig∂m−2−l\nvqα\nqα(β,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤m−2/summationdisplay\nj=0(m−2)!\nj!(m−2−j)!Mj\n1Γs(j)/parenleftBigmj/summationdisplay\nj1=0mj!\nj1!(mj−j1)!Γs(mj−j1)Cmj−j1(C|k|)j1e−C−1δ|k|/parenrightBig\n+k2m−2/summationdisplay\nl=0(m−2)!\nl!(m−2−l)!Ml\n1Γs(l)/parenleftBigml−1/summationdisplay\nl1=0(ml−1)!\nl1!((ml−1)−l1)!Γs(ml−1−l1)Cml−1−l1(C|k|)l1e−C−1δ|k|/parenrightBig\n≤m−2/summationdisplay\nj=0mj/summationdisplay\nj1=0(m−2)!\nj!(m−2−j)!Γs(j)mj!\nj1!(mj−j1)!Γs(m−1−j−j1)Cm−1−j1(C|k|)j1e−C−1δ|k|NONLINEAR INVISCID DAMPING 107\n+m−2/summationdisplay\nl=0ml−1/summationdisplay\nl1=0(m−2)!\nl!(m−2−l)!Γs(l)(ml−1)!\nl1!((ml−1)−l1)!Γs(m−l−2−l1)Cm−2−l1(C|k|)l1+2e−C−1δ|k|\n≤m−1/summationdisplay\nj1=0(m−1)!\nj1!(m−1−j1)!Γs(m−1−j1)Cm−1−j1(C|k|)j1e−C−1δ|k|\n+m−2/summationdisplay\nl1=0(m−l1)!\n(l1+2)!(m−2−l1)!Γs(m−2−l1)Cm−2−l1(C|k|)l1+2e−C−1δ|k|,\nwheremj=m−1−j. Thus we proved (C.20). The lemma follows directly by applyi ng\n(C|k|)je−C−1δ|k|/lessorsimilar(Cδ)−j(jj/ej)<(Cδ)−jΓs(j),\nand (A.10). /square\nLet us rewrite qβ. Note that qα(v) and\nh1(α)qα(v)/integraldisplayv\nα1\nh1(w)qα(w)2dw\nare two linearly independent solutions. It is easy to get tha t\nqβ(v) =qβ(α)qα(v)+q′\nβ(α)h1(α)qα(v)/integraldisplayv\nα1\nh1(w)qα(w)2dw\n=qα(v)\nqα(β)−h1(β)q′\nα(β)qα(v)/integraldisplayv\nβ1\nh1(w)qα(w)2dw.\nLemma C.3. It holds for v′<v−δandv,v′∈[α+δ,β−δ], that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m−m1v∂m1\nv′/parenleftBigqβ(v,k)qα(v′,k)\nqβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CΓs(m)Mm,\nand forv′>v+δ\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m−m1v∂m1\nv′/parenleftBigqα(v,k)qβ(v′,k)\nqβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CΓs(m)Mm.\nProof.The two estimates can be proved by the same argument. Here we o nly prove the first\ninequality. We write\n/parenleftBigqβ(v,k)qα(v′,k)\nqβ(α,k)/parenrightBig\n=/parenleftBigqβ(v,k)qα(v,k)\nqβ(α,k)/parenrightBig/parenleftBigqα(v′,k)\nqα(v,k)/parenrightBig\n=qα(v)qα(v′,k)\nqα(β)qβ(α,k)−h1(β)q′\nα(β)\nqβ(α,k)qα(v)qα(v′)/integraldisplayv\nβ1\nh1(w)qα(w)2dw\nFor the first term, we have by Lemma C.2 that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m−m1v∂m1\nv′/parenleftBigqα(v)qα(v′,k)\nqα(β)qβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CΓs(m)Mm,\nThen by the monotonicity of qα, we have 0 <qα(v′,k)\nqα(v,k)/lessorsimilare−C−1δkandqα(v)\nqα(w)/lessorsimilare−C−1k|v−w|forv≤w,\nwhich gives that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleqα(v)qα(v′)/integraldisplayv\nβ1\nh1(w)qα(w)2dw/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilark/integraldisplayv\nβe−2C−1k|v−w|dwe−C−1k|v−v′|/lessorsimilare−C−1δk,108 WEIREN ZHAO\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v,∂v′)/parenleftBig\nqα(v)qα(v′)/integraldisplayv\nβ1\nh1(w)qα(w)2dw/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilark2/integraldisplayv\nβe−2C−1k|v−w|dwe−C−1k|v−v′|/lessorsimilarke−C−1δk,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v,∂v′)2/parenleftBig\nqα(v)qα(v′)/integraldisplayv\nβ1\nh1(w)qα(w)2dw/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilark3/integraldisplayv\nβe−2C−1k|v−w|dwe−C−1k|v−v′|/lessorsimilark2e−C−1δk.\nWe now use mathematical induction to prove the following res ult:\nFor allN=N1+N2, it holds that\n(C.21)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂N1v∂N2v/parenleftBigqβ(v,k)qα(v′,k)\nqβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftBigN/summationdisplay\nj=0N!\nj!(N−j)!Γs(N−j)CN−j(C|k|)je−C−1δ|k|/parenrightBig\n.\nSuppose that (C.21) holds for N1+N2= 0,1,2,...,m1+m2+1, we get for N1=m1+m2+2 Thus\nwe have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m1+2\nvqβ(v)∂m2\nv′qα(v′)\nqβ(α,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤m1/summationdisplay\nj=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle(m1)!\nj!(m1−j)!∂j\nv/parenleftBig−h′\n1\nh1/parenrightBig\n∂m1+1−j\nvqβ(v)∂m2\nv′qα(v′)/qβ(α,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+k2m1/summationdisplay\nl=0/vextendsingle/vextendsingle/vextendsingle/vextendsingle(m1)!\nl!(m1−l)!∂l\nv/parenleftBigh2\nh1/parenrightBig\n∂m1−l\nvqβ(v)∂m2\nv′qα(v′)/qβ(α,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤m1/summationdisplay\nj=0(m1)!\nj!(m1−j)!CjΓs(j)/parenleftBigNj/summationdisplay\nj1=0Nj!\nj1!(Nj−j1)!Γs(Nj−j1)CNj−j1(C|k|)j1e−C−1δ|k|/parenrightBig\n+C−2m1/summationdisplay\nl=0(m1)!\nl!(m1−l)!ClΓs(l)/parenleftBigNl/summationdisplay\nj2=0Nl!\nj2!(Nl−j2)!Γs(Nl−j2)CNl−j2(C|k|)j2+2e−C−1δ|k|/parenrightBig\n=I1+I2,\nwhereNj=m1+m2+1−jandNl=m1+m2−l.\nFor the first term, we have\nI1=m1/summationdisplay\nj=0Nj/summationdisplay\nj1=0(m1)!\nj!(m1−j)!Γs(j)Nj!\nj1!(Nj−j1)!Γs(Nj−j1)Cm1+m2+1−j1(C|k|)j1e−C−1δ|k|\n=m1/summationdisplay\nj=0Nj/summationdisplay\nj1=0/parenleftbiggm1!(m1+m2+1−j)!\n(m1−j)!(m1+m2+1−j1)!/parenrightbigg(m1+m2+1−j1)!\nj1!(m1+m2+1−j1)!\n×/parenleftbigg(Nj−j1+j)!\nj!(Nj−j1)!Γs(Nj−j1)Γs(j)/parenrightbigg\nCm1+m2+1−j1(C|k|)j1e−C−1δ|k|\n≤m1+m2+1/summationdisplay\nj1=0(m1+m2+1−j1)!\nj1!(m1+m2+1−j1)!Γs(m1+m2+1−j1)Cm1+m2+1−j1(C|k|)j1e−C−1δ|k|.\nSimilarly, for the second term, we have\nI2=C−2m1/summationdisplay\nl=0Nl/summationdisplay\nj2=0(m1)!\nl!(m1−l)!Γs(l)Nl!\nj2!(Nl−j2)!Γs(Nl−j2)Cm1+m2−j2(C|k|)j2+2e−C−1δ|k|NONLINEAR INVISCID DAMPING 109\n≤C−2m1+m2/summationdisplay\nj2=0(m1+m2+2−j2)!\n(j2+2)!(m1+m2−j2)!Γs(m1+m2−j2)Cm1+m2−j2(C|k|)j2+2e−C−1δ|k|.\nCombining the above estimate,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m1+2\nvqβ(v)∂m2\nv′qα(v′)\nqβ(α,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftBigN/summationdisplay\nj=0N!\nj!(N−j)!Γs(N−j)CN−j(C|k|)je−C−1δ|k|/parenrightBig\nwithN=m1+m2+2. Using a similar argument, we can also get for any m1,m2,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m1vqβ(v)∂m2+2\nv′qα(v′)\nqβ(α,k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftBigN/summationdisplay\nj=0N!\nj!(N−j)!Γs(N−j)CN−j(C|k|)je−C−1δ|k|/parenrightBig\nholds forN=m1+m2+2. Thus we proved (C.21). The lemma follows directly by appl ying\n(C|k|)je−C−1δ|k|/lessorsimilar(Cδ)−j(jj/ej)<(Cδ)−jΓs(j),\nand (A.10). /square\nBy using the fact that h1,h2are constant-valued functions near the boundary, and the bo undary\ncondition of qα(v), we obtain that for α≤v≤α+δ,\nqα(v) = cosh/parenleftBig\nk/integraldisplayv\nα/radicalbigg\nh2\nh1(w)dw/parenrightBig\nand\nk/radicalbig\nh1(α)h2(α)qα(v)/integraldisplayv\nα1\nh1(w)qα(w)2dw= sinh/parenleftBig\nk/integraldisplayv\nα/radicalbigg\nh2\nh1(w)dw/parenrightBig\n.\nLet\neα,+(v) =qα(v)+h1(α)qα(v)/integraldisplayv\nα1\nh1(w)qα(w)2dw=e/integraltextv\nαsα,+(w)/h1(w)dw, (C.22)\nthen a direct calculation gives that eα,+(v) solves\n∂v(h1∂veα,+(v)) =k2h2eα,+(v)\nand that sα,+(v) =h1(v)e′\nα,+(v)\neα,+(v)solves\ns′\nα,+(v)+sα,+(v)2\nh1(v)=k2h2(v),\nwith boundary condition sα,+(α) =k/radicalbig\nh1(α)h2(α).\nNote that the above calculation for sα,+is similar to fα.\nLemma C.4. There isk0>0and there are CandMindependent of k, so that for any k>k0, it\nholds that\nsup\nm≥0/vextendsingle/vextendsingle/vextendsingle∂m\nv/parenleftbig\nsα,+(v)−k/radicalbig\nh1(v)h2(v)/parenrightbig/vextendsingle/vextendsingle/vextendsingle≤CΓs(m)Mm.\nProof.We use the idea of WKB approximation and rewrite sα,+(v) as\nsα,+(v) =k/radicalbig\nh1(v)h2(v)−(/radicalbig\nh1(v)h2(v))′\n2/radicalbig\nh2(v)/h1(v)+r(v),110 WEIREN ZHAO\nthenrsolves\nr′+2k/radicalbig\nh2/h1r−(/radicalbig\nh1(v)h2(v))′\n/radicalbig\nh2(v)/h1(v)r+r2\nh1−/parenleftBig(/radicalbig\nh1(v)h2(v))′\n2/radicalbig\nh2(v)/h1(v)/parenrightBig′\n+/parenleftBig(/radicalbig\nh1(v)h2(v))′\n2/radicalbig\nh2(v)/parenrightBig2\n= 0.\nLetR(w) =r(v) withw(v) =/integraltextv\nα/radicalbig\nh2/h1(z)dz, then\nR′+2kR+A(w)R+B(w)R2+C(w) = 0,\nwith initial data R(0) = 0, where\nA(w) =−/radicalbig\nh1/h2(/radicalbig\nh1(v)h2(v))′\n/radicalbig\nh2(v)/h1(v)\nB(w) =/radicalBigg\n1\nh1(v)h2(v)\nC(w) =−/radicalbig\nh1/h2/parenleftBig(/radicalbig\nh1(v)h2(v))′\n2/radicalbig\nh2(v)/h1(v)/parenrightBig′\n+/radicalbig\nh1/h2/parenleftBig(/radicalbig\nh1(v)h2(v))′\n2/radicalbig\nh2(v)/parenrightBig2\n.\nBy the Gevrey regularity of h1,h2and composition Lemma A.3, it is easy to check that there are\nCandM1such that\nsup\nm≥0|∂m\nw(A,B,C)(w)| ≤CΓs(m)Mm\n1.\nLetDm=∂m\nw\nΓs(m)Mm, then we get that\n1\n2d\ndw|DmR|2+2k|DmR|2≤C( sup\n0≤m1≤m|Dm1R|+1)2|DmR|.\nBy the Gronwall’s inequality, we get that for k0>0 large enough and k>k0,\nsup\nm|DmR(w)|2/lessorsimilar1\nk,\nwhich together with Lemma A.3 gives\nsup\nm≥0|∂m\nvr(v)| ≤CΓs(m)Mm\n1√\nk.\nThus we proved the lemma. /square\nThus we have the following remark:\nRemark C.5. It holds that for v≥α+δ\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m+1\nv/parenleftBig1\neα,+(v)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓs(m)Mme−C−1k|v−α|\nProof.We rewrite1\neα,+(v)=ge(v)e−/integraltextv\nαk√\nh1h2(w)dw. Then Lemma C.4 gives that\n/vextendsingle/vextendsingle∂m+1\nvge(v)/vextendsingle/vextendsingle/lessorsimilarΓs(m)Mm.\nWe also have for v≥α+δ/vextendsingle/vextendsingle/vextendsingle∂m+1\nve−/integraltextv\nαk√\nh1h2(w)dw/vextendsingle/vextendsingle/vextendsingle≤Γs(m)Mme−C−1k|v−α|\nwhich gives the remark. /squareNONLINEAR INVISCID DAMPING 111\nLemma C.6. For anyβ−α\n100>δ >0, there areMandCindependent of k, such that for |v′−v| ≤δ\nsup\nm≥0/integraldisplayβ−δ\nα+δ/integraldisplay\n{0<v−v′≤δ}∩[α,β]/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)m\nΓs(m)Mm|k|−1(1−∂vv−∂v′v′)/parenleftBigqα(v′)\nqα(v)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingledvdv′≤C.\nand/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)m\nΓs(m)Mm/parenleftBigqα(v′)\nqα(v)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C\nProof.Let us first use eα,+to represent qα(v), by matching the boundary condition, we have\nqα(v) =eα,+(v)−k/radicalbig\nh1(α)h2(α)eα,+(v)/integraldisplayv\nα1\nh1(w)eα,+(w)2dw.\nThus we have\nqα(v)\nqα(w)=eα,+(v)\neα,+(w)/parenleftBigg\n1+k/radicalbig\nh1(α)h2(α)/integraltextv\nw1\nh1(z)eα,+(z)2dz\n1−k/radicalbig\nh1(α)h2(α)/integraltextv\nα1\nh1(w)eα,+(w)2dw/parenrightBigg\n(C.23)\nBy the fact that qα(v)≥1 and (C.22), it is easy to\n1−k/radicalbig\nh1(α)h2(α)/integraldisplayv\nα1\nh1(w)eα,+(w)2dw≥c0>0\nfor somec0independent of k. By Remark C.5, we have that for m≥1 andw≥α+δ\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m\nv/parenleftBig\n1−k/radicalbig\nh1(α)h2(α)/integraldisplayv\nα1\nh1(w)eα,+(w)2dw/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilark/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m−1\nv/parenleftBig1\nh1(w)eα,+(w)2/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓs(m)Mm\nand/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂w)m/parenleftbigg\nk/radicalbig\nh1(α)h2(α)/integraldisplayv\nw1\nh1(z)eα,+(z)2dz/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilark/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayv\nw∂m\nz/parenleftbigg1\nh1(z)eα,+(z)2/parenrightbigg\ndz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarΓs(m)Mm,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble(∂v+∂w)m(1−∂vv−∂ww)/parenleftbigg\nk/radicalbig\nh1(α)h2(α)/integraldisplayv\nw1\nh1(z)eα,+(z)2dz/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL1v,w({0<v<w−δ}∩[α+δ,β−δ]2)\n/lessorsimilar/integraldisplayβ−δ\nα+δk/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m+1\nv/parenleftbigg1\nh1(v)eα,+(v)2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledv+/integraldisplayβ−δ\nα+δk/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m+1\nw/parenleftbigg1\nh1(w)eα,+(w)2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingledw\n/lessorsimilarΓs(m)Mm\nhere we use the fact that\neα,+(w)≥e−C−1k|w−α|.\nNote that\neα,+(v)\neα,+(w)=ge(v)\nge(w)e/integraltextv\nwk√\nh1h2(z)dz,\nwherege(v) is defined in Remark C.5. Thus we obtain that for 0 ≤w−v≤δ\n(∂v+∂w)me/integraltextv\nwk√\nh1h2(z)dz112 WEIREN ZHAO\n= (∂v+∂w)m−1e/integraltextv\nwk√\nh1h2(z)dzk/parenleftBig/integraldisplayv\nw(/radicalbig\nh1h2(z))′dz/parenrightBig\n+e/integraltextv\nwk√\nh1h2(z)dzk/parenleftBig/integraldisplayv\nwdm\ndzm(/radicalbig\nh1h2(z))dz/parenrightBig\n,\nwhich gives that\n|(∂v+∂w)me/integraltextv\nwk√\nh1h2(z)dz| ≤e−C−1k|v−w|m/summationdisplay\nj=0(k|v−w|)jCm−jΓs(m−j)\n/lessorsimilarΓs(m)Mm,\nand/vextendsingle/vextendsingle/vextendsingle(1−∂vv−∂ww)(∂v+∂w)me/integraltextv\nwk√\nh1h2(z)dz/vextendsingle/vextendsingle/vextendsingle/lessorsimilark2Γs(m)Mme−C−1k|w−v|.\nCombining all the estimates, together with (C.23), we get th e lemma. /square\nLemma C.7. It holds for |v′−v| ≤δ,v,v′∈[α+δ,β−δ]that\nsup\nm≥0/integraldisplayβ−δ\nα+δ/integraldisplay\n{0<v−v′≤δ}∩[α,β]/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)m\nΓs(m)Mm|k|−1(1−∂vv−∂v′v′)/parenleftBigqβ(v,k)qα(v′,k)\nqβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingledvdv′≤C.\nand forv−δ<v′<v/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)m\nΓs(m)Mm/parenleftBigqβ(v,k)qα(v′,k)\nqβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C\nand\nsup\nm≥0/integraldisplayβ−δ\nα+δ/integraldisplay\n{0<v′−v≤δ}∩[α,β]/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)m\nΓs(m)Mm|k|−1(1−∂vv−∂v′v′)/parenleftBigqα(v,k)qβ(v′,k)\nqβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingledvdv′≤C.\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂m−m1v∂m1\nv′/parenleftBigqβ(v,k)qα(v′,k)\nqβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤CΓs(m)Mm,\nand forv′−δ <v<v′\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle(∂v+∂v′)m\nΓs(m)Mm/parenleftBigqα(v,k)qβ(v′,k)\nqβ(α,k)/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C.\nProposition C.8. Let/hatwideGDbe the Fourier transform of the kernel GDdefined in (C.14), namely,\n/hatwideGD(k,ξ1,ξ2) =1\n2π/integraldisplay\nR2/hatwideGD(v,v′,k)Υ(v)Υ(v′)e−ivξ1−iv′ξ2dvdv′.\nThen there is λ∆such that for all k∈Z,\n|/hatwideGD(k,ξ1,ξ2)| ≤Cmin/braceleftBigg\ne−λ∆/a\\}bracketle{tξ1+ξ2/a\\}bracketri}hts\n1+k2+ξ2\n1,e−λ∆/a\\}bracketle{tξ1+ξ2/a\\}bracketri}hts\n1+k2+ξ2\n2/bracerightBigg\n. (C.24)\nProof.Notice that/integraldisplay\nR2/hatwideGD(v,v′,k)Υ(v)Υ(v′)e−ivξ1−iv′ξ2dvdv′\n=/integraldisplay\nR2/hatwideGD(v,v+w,k)Υ(v)Υ(v+w)e−iv(ξ1+ξ2)−iv′ξ2dvdw(C.25)\nThe upper bound (C.24) follows from (C.14), (C.16), (C.17) L emma C.2, Lemma C.3, Lemma C.7,\nLemma A.1, Lemma B.1, Lemma B.2 and Remark B.3. /squareNONLINEAR INVISCID DAMPING 113\nBy using (C.3) and (C.4), it is easy for us to obtain the follow ing proposition which is equivalent\nto Proposition 8.1.\nProposition C.9. LetΨsolve(C.1)withΩcompactly supported. Then there is GD,1(t,k,ξ,η)such\nthat fork/\\e}atio\\slash= 0\n/hatwider(ΨΥ)(t,k,ξ) =/integraldisplay\nRGD,1(t,k,ξ,η)/hatwideΩ(t,k,η)dη,\nwith estimate\n|GD,1(t,k,ξ,η)| ≤Cellmin/braceleftBigg\ne−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(ξ−kt)2,e−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(η−kt)2/bracerightBigg\n.\nNotice that from ∆ ψ=ω, we have\n(C.26)\n\n∂zzΨ+/tildewideu′(∂v−t∂z)/parenleftBig\n/tildewideu′(∂v−t∂z)Ψ/parenrightBig\n=ω\nΨ(t,z,u(0)) = Ψ(t,z,u(1)) = 0.\nIf we set Q=F1(Ψ)(t,k,v)e−itkvandh1(v) =/tildewideu′(v),h2(v) =/tildewideu′(v)−1andF=F1[ω]e−itkv, then we\nalso obtain the Sturm-Liouville Equation (C.7). We then hav e the following corollary:\nCorollary C.10. LetΨsolve(C.26)withωcompactly supported. Then there is G2(t,k,ξ,η)such\nthat\n/hatwider(ΨΥ)(t,k,ξ) =/integraldisplay\nRG2(t,k,ξ,η)/hatwideω(t,k,η)dη,\nwith estimate\n|G2(t,k,ξ,η)| ≤Cellmin/braceleftBigg\ne−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(ξ−kt)2,e−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(η−kt)2/bracerightBigg\n.\nC.3.Neumann boundary conditions. We also study the general Sturm-Liouville Equation with\nNeumann boundary conditions\n/braceleftBigg\n∂v(h1(v)∂vR)−k2h2(v)R=G(v,k)\n∂vR(α,k) =∂vR(β,k) = 0(C.27)\nwhereh1,h2are positive functions and supp h′\nj⊂[α1,β1] forj= 1,2. We also assume that\nh′\n1,h′\n2∈ GM,s\nph([α,β]).\nBy Lemma C.1, it is easy to get that\nR(v,k) =1\nh1(v)qβ(v,k)\nW[qβ,qα]/integraldisplayv\n0qα(v′,k)G(v′,k)h1(v′)dv′\n+1\nh1(v)qα(v,k)\nW[qβ,qα]/integraldisplay1\nvqβ(v′,k)G(v′,k)h1(v′)dv′,(C.28)\nwith\nW[qβ,qα](v) =−W[qα,qβ](v) =−q′\nβ(α)h1(α)\nh1(v)=q′\nα(β)h1(β)\nh1(v)>0.\nWe now introduce the Fourier kernel for the Sturm-Liouville type elliptic problem (C.27). For any\ngiven fixed k, and positive Gevrey functions h1,h2∈ GM,s\nph([α,β]) with constant values near the114 WEIREN ZHAO\nboundary and any Gevrey cut-off function Υ with compact suppo rt suppΥ ⊂(α,β), we define a\nlinear operator △−1\nN,kto be such that for a function G,\n△−1\nN,kG= ΥR (C.29)\nwhereRsolves/braceleftBigg\n∂v(h1(v)∂vR)−k2h2(v)R=G(v,k)Υ(v)\n∂vR(α,k) =∂vR(β,k) = 0.\nBy (C.28), we have for k/\\e}atio\\slash= 0\n△−1\nN,kG=/integraldisplay1\n0GN(v,v′,k)Υ(v)Υ(v′)G(v′,k)dv′\nwith\n(C.30)GN(v,v′,k) =1\nh1(v)qβ(v,k)\nW[qβ,qα]qα(v′,k)h1(v′)χR+(v−v′)\n+1\nh1(v)qα(v,k)\nW[qβ,qα]qβ(v′,k)h1(v′)χR+(v′−v).\nSimilarly to Proposition C.9, by using the representation f ormula of the Green function (C.30) and\napplying Lemma C.3 and Lemma A.1, we have the following propo sition.\nProposition C.11. LetΠsolve(C.2)withΞcompactly supported. Then there is GN(t,k,ξ,η)such\nthat\n/hatwide(ΠΥ)(t,k,ξ) =/integraldisplay\nRGN(t,k,ξ,η)/hatwideΞ(t,k,η)dη,\nwith estimate\n|GN(t,k,ξ,η)| ≤Cmin/braceleftBigg\ne−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(ξ−kt)2,e−λ∆/a\\}bracketle{tξ−η/a\\}bracketri}hts\n1+k2+(η−kt)2/bracerightBigg\n.\nAppendix D.Properties of multiplier A\nIn this section, we list some key properties of the multiplie r A from [5].\nLemma D.1 (Lemma 3.3 in [5]) .Fort∈Ik,ηandt>2/radicalbig\n|η|, we have the following with τ=t−η\nk\n∂twNR(t,η)\nwNR(t,η)≈1\n1+|τ|≈∂twR(t,η)\nwR(t,η).\nLemma D.2 (Lemma 3.4 in [5]) .The following holds.\n(1)Fort≥1, andk,l,η,ξsuch that max{2/radicalbig\n|ξ|,2/radicalbig\n|η|}<t<2min{|ξ|,|η|},\n∂twk(t,η)\nwk(t,η)wl(t,ξ)\n∂twl(t,ξ)/lessorsimilar/a\\}b∇acketle{tη−ξ/a\\}b∇acket∇i}ht.\n(2)For allt≥1, andk,l,η,ξ, such that for some α≥1,1\nα|ξ| ≤ |η| ≤α|ξ|,\n/radicalBigg\n∂twl(t,ξ)\nwl(t,ξ)/lessorsimilarα/bracketleftBigg/radicalBigg\n∂twk(t,η)\nwk(t,η)+|η|s\n2\n/a\\}b∇acketle{tt/a\\}b∇acket∇i}hts/bracketrightBigg\n/a\\}b∇acketle{tη−ξ/a\\}b∇acket∇i}ht.\nLemma D.3 (Lemma 3.5 in [5]) .For allt,η,ξ, we have\nwNR(t,ξ)\nwNR(t,η)/lessorsimilareµ|η−ξ|1\n2.NONLINEAR INVISCID DAMPING 115\nLemma D.4 (Lemma 3.6 in [5]) .We have for t∈Ik,η∩Ik,ξwithk/\\e}atio\\slash=lthat\nJk(η)\nJl(ξ)/lessorsimilar|η|\nk2+(η−kt)2e9µ|k−l,η−ξ|1\n2.\nwe also rewrite it as\nJk(η)\nJl(ξ)/lessorsimilar|η|\nk2/radicalBigg\n∂twk(t,η)\nwk(t,η)/radicalBigg\n∂twl(t,ξ)\nwl(t,ξ)e20µ|k−l,η−ξ|1\n2.\nIn other cases, we have improved estimates\nJk(η)\nJl(ξ)/lessorsimilare10µ|k−l,η−ξ|1\n2.\nMoreover, if t∈Il,ξ,t /∈Ik,ηand1\nα|ξ| ≤ |η| ≤α|ξ|for someα>0, then\nJk(η)\nJl(ξ)/lessorsimilar|l|(l+|ξ−lt|)\n|ξ|e11µ|k−l,η−ξ|1\n2. 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Amer. Math. Soc., 369 (2017), pp. 8799–\n8855.\n(W. Zhao) Department of Mathematics, New York University Abu Dhabi, S aadiyat Island, P.O. Box\n129188, Abu Dhabi, United Arab Emirates.\nEmail address :zjzjzwr@126.com, wz19@nyu.edu"    },    {        "title": "2305.00814v2.Coherent_and_incoherent_magnons_induced_by_strong_ultrafast_demagnetization_in_thin_permalloy_films.pdf",        "content": "Coherent and incoherent magnons induced by strong ultrafast demagnetization in\nthin permalloy films\nAnulekha De,1,∗Akira Lentfert,1Laura Scheuer,1Benjamin Stadtmüller,1, 2\nGeorg von Freymann,1, 3Martin Aeschlimann,1and Philipp Pirro1,†\n1Department of Physics and Research Center OPTIMAS,\nRheinland-Pfälzische Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany\n2Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany\n3Fraunhofer Institute for Industrial Mathematics, 67663 Kaiserslautern, Germany\n(Dated: August 14, 2023)\nUnderstanding spin dynamics on femto- and picosecond timescales offers new opportunities for\nfaster and more efficient spintronic devices. Here, we experimentally investigate the coherent spin\ndynamics after ultrashort laser excitation by time-resolved magneto optical Kerr effect (TR-MOKE)\nin thin Ni80Fe20films. We provide a detailed study of the magnetic field and pump fluence depen-\ndenceofthecoherentprecessionaldynamics. Weshowthatthecoherentprecessionlifetimeincreases\nwith the applied external magnetic field which cannot be understood by viscous Gilbert damping\nof the coherent magnons. Instead, it can be explained by nonlinear magnon interactions and by the\nchange in the fraction of incoherent magnons. This interpretation is in agreement with the observed\ntrends of the coherent magnon amplitude and lifetime as a function of the exciting laser fluence.\nOur results provide a new insight into the magnetization relaxation processes in ferromagnetic thin\nfilms, which is of great importance for further spintronic applications.\nI. INTRODUCTION\nThe microscopic mechanism of laser-induced magneti-\nzation dynamics in femto-, pico- and nanosecond time-\nscales remains still a challenge in condensed matter\nphysics. Using the time-resolved magneto-optical Kerr\neffect (TR-MOKE), one can directly address the pro-\ncesses responsible for the excitation and relaxation of\na magnetic system on their characteristic timescales [1–\n4]. The pioneering work of Beaurepaire et al. in 1996\non femtosecond laser-induced ultrafast demagnetization\nopened up a new avenue for ultrafast manipulation of\nthe magnetization in magnetic materials [5]. Only a few\nyears later, experiments could show that the ultrafast\nlaser pulses also generate coherent magnons, manifested\nas precessional dynamics on the nanosecond time scale.\nThe precessional region in magnetic materials is a key\nparameter for encoding and transferring information in\nspintronic devices. This regime allows the study of the\nmagnetic anisotropy, damping, and precession frequency\nof different dynamic modes in continuous thin films and\npatterned nanostructures [2–4, 6–10]. However, the tran-\nsition between the two regimes of the magnetization dy-\nnamics, i.e., the ultrafast demagnetization and the coher-\nent precessional motion on the other side, is very crucial\nand raises intriguing questions. Many efforts have been\nmade to understand the leading mechanisms as well as\nto explore the characteristic time scales. For example,\nvan Kampen, et al. [2] demonstrated using TR-MOKE\nthat an optical pump pulse can induce coherent uniform\nspin precession in a ferromagnet. The excitation of these\n∗ade@rptu.de\n†ppirro@rptu.demagnons was explained by a transient change in mag-\nnetic anisotropy. Later, several works were concerned\nwith exploring the precessional relaxation mechanisms in\nferromagnets after ultrafast excitation [11–15]. A recent\nwork showed the excitation of the precession dynamics in\nferromagnets with two non-collinear optical pulses, that\ncan affect precessional relaxation mechanisms and damp-\ning [16].\nIn this article, we take a closer look at the relaxation\nof the coherent precession induced by ultrafast demag-\nnetization and study in detail the decay time of the\nmeasured coherent oscillations as a function of magnetic\nfield and pump fluence. We use femtosecond amplified\nlaser pulses to excite and detect magnetization dynamics\nin thin permalloy films, including ultrafast demagneti-\nzation, fast and slow remagnetization and precessional\ndynamics. We observe an unusual magnetic field depen-\ndence of the precessional relaxation time which is not\nin accordance with the expectations from the Gilbert\nmodel. Extrinsic contributions such as two-magnon scat-\ntering, magnetic anisotropy and spin pumping [11, 17–\n22] could also affect the magnetization relaxation process\nand the damping, but we show that these effects can be\nneglected in out case. Instead, we can relate our observa-\ntions to nonlinear magnon interactions and the variable\ncontribution of the incoherent magnon background after\nexcitation by a femtosecond optical pulse. Our findings\nare of general importance for the interpretation of the\ncoherent dynamics measured in similar experiments af-\nter ultrafast stimuli.\nII. MATERIALS AND METHODS\nPermalloy ( Ni80Fe20, Py) films of 5 nm and 2.8 nm\nthickness were deposited on MgO substrates by molecu-2\nlar beam epitaxy (MBE) technique in an ultrahigh vac-\nuum chamber. The samples were capped with a 3 nm\nthick layer of Al2O3to protect them from environmen-\ntal degradation, oxidation and laser ablation during the\npump-probe experiment using femtosecond laser pulses.\nFIG. 1. (a) Schematic diagram of the measurement geometry.\n(b) Typical experimental TR-MOKE data showing different\ntemporal regimes of the magnetization dynamics for 5 nm Py\nsample measured at µ0H= 113 mT and F= 4.6 mJ cm−2.\nThe magnetization dynamics were measured using a\nTR-MOKE setup based on a two-color, non-collinear op-\ntical pump-probe technique. A schematic diagram of the\nexperimental geometry is shown in Fig. 1(a). In this\nexperiment, we use the fundamental output of an ampli-\nfied femtosecond laser system with wavelength λ= 800\nnm, repetition rate of 1 kHz, and pulse width of ∼35\nfs (Libra, Coherent Inc.) as the pump pulse, while its\nsecond harmonic with λ= 400 nm is used to probe the\ndynamics. The probe is normally incident on the sam-\nple, while the pump is incident obliquely ( ∼30°) with\nrespect to the surface normal. During the measurements,\nwe have applied a magnetic field inclined at a small an-\ngle of ∼15°to the sample plane. The inclination of the\nmagnetization provides a finite out-of-plane (OOP) de-\nmagnetization field which is transiently modified by the\npump pulse, inducing a coherent precession of the mag-\nnetization[1,2]. Thissymmetrybreakingisimportantto\nobtainthesamestartingphaseoftheprecessionalmotion\nforeachlaserpumppulsetoavoidthatthelossofthepre-\ncession signal in a TR-MOKE experiment averaged over\nthousands of pump-probe cycles. For a completely in-\nplane configuration of the external magnetic field, we ob-\nserve a complete reduction of the precessional signal [see\nSupplemental Materials]. To obtain the intrinsic mag-\nnetic response, we performed the measurements for two\nopposite magnetization directions of the sample and ex-\ntracted the pure magnetic response from the difference\nof the two resulting Kerr signals. This is done to elim-\ninate any nonmagnetic signal, i.e., any signal that does\nnot depend of the direction of the sample magnetization\n[23]. We have used a specially designed photodetector\nconnected to a lock-in amplifier to measure the dynamic\nKerr rotation signal. All measurements have been per-\nformed under ambient condition and room temperature.III. RESULTS AND DISCUSSION\nSeveral processes occur when a femtosecond laser pulse\ninteracts with a ferromagnetic thin film in its saturation\ncondition. First of all, the magnetization of the sys-\ntem is partially or completely lost within hundreds of\nfemtoseconds, which is known as ultrafast demagnetiza-\ntion [5]. This is generally followed by a fast recovery of\nthe magnetization within sub-picoseconds to a few pi-\ncoseconds and a slower recovery within hundreds of pi-\ncoseconds, known as the fast and slow remagnetization.\nThe slower recovery is accompanied by a precession of\nthe magnetization. On a much longer time scale of a\nfew nanoseconds, the magnetization returns to its ini-\ntial equilibrium, which can be described phenomenolog-\nically by the Gilbert damping [24]. Figure 1(b) shows\nthe representative Kerr rotation data of the 5 nm Py\nsample for pump fluence F= 4.6 mJ cm−2andµ0H=\n113mT consisting of three temporal regions of the mag-\nnetization dynamics, i.e., the ultrafast demagnetization,\nthe fast remagnetization followed by the slow remagne-\ntization superposed with the damped precession within\nthe time window of 900 ps. The slow remagnetization is\nmainly due to heat diffusion from the lattice to the sub-\nstrate and the surroundings. Region I is characterized by\nthe demagnetization time τM, region II is characterized\nby the fast remagnetization time τE. For region III, we\ncharacterize the dynamics by the precession frequency f\nand the precessional relaxation time τd.\nFIG. 2. Ultrafast demagnetization traces at different pump\nfluences for 5 nm Py sample in (a) polar (b) longitudinal\nMOKE geometry. (c) Demagnetization times ( τM) vs. pump\nfluence (d) Fast remagnetization times ( τE) vs. pump fluence\nand (e) quenching vs. pump fluence measured in both po-\nlar and longitudinal geometry. Solid circles and open circles\nrepresent the data corresponding to polar and longitudinal\nMOKE respectively.\nWe start our discussion with the ultrafast demagne-\ntization and fast remagnetization processes of our sys-\ntem to study their dependence on the pump laser fluence3\nand the external magnetic field. In general, the ultrafast\ndemagnetization studies are performed in the longitudi-\nnal MOKE geometry. However, to measure the coherent\nprecessionofthemagnetization, weneedtoperformmea-\nsurementsinapolarMOKEgeometrywithaslightlyout-\nof-planeinclinedexternalmagneticfield[1,2]. Therefore,\nwe first check whether there is a significant difference in\nthe laser-induced ultrafast demagnetization in the two\nMOKE geometries. Figure 2(a) and (b) show the ultra-\nfast demagnetization traces obtained for 5 nm Py in the\npolar and longitudinal MOKE geometries, respectively.\nThe pump fluence is varied between 4.6 - 8.4 mJ cm−2\nby varying the power of the pump pulse. We restrict\nourselves to the low pump fluence regime to avoid sam-\nple damage, and the probe fluence is kept constant at\na very low value ( ∼fewµW) to avoid any additional\ncontribution to the spin dynamics by probe excitation.\nThe ultrafast demagnetization traces for both geometries\nshow qualitatively similar trends. However, minor dis-\ncrepancies between thequantitative values obtained from\ntwo different geometries arise due to slight differences in\nthe pump spot sizes. The amplitude of the maximum\nquenching of the Kerr rotation signal increases almost\nlinearly with the laser fluence. Closer inspection of the\ntraces also reveals an increase in the τMandτMwith\nincreasing fluence. To quantify this increase, we fit our\ndemagnetization traces with a phenomenological thermo-\ndynamic model, the so-called three-temperature model\n(3TM) [25], which is obtained by solving the energy rate\nequation between three different degrees of freedom, e.g.\nelectron, spin and lattice, under low pump fluence con-\nditions [see Supplemental Materials].\nFIG. 3. (a) Ultrafast demagnetization traces for 5 nm Py\nsample measured at different values of magnetic field and for\nfixedF= 4.6 mJ cm−2. (b) upper panel: Demagnetization\ntimes (τM) vs. magnetic field and lower panel: Fast remagne-\ntization times ( τE) vs. magnetic field.\nThe fluence-dependent behavior of τM,τEand quench-\ning in both MOKE geometries, as shown in Fig. 2(c),\n(d) and (e), respectively, is an indication of the spin-\nflip process-dominated ultrafast demagnetization in our\nsystems [26–28]. The values of τMextracted from our\nexperiments are on the same time scale as previous re-\nports [29], and are too large to represent superdiffusive\ntransport driven demagnetization [30]. These values areslightly larger in the longitudinal geometry compared to\nthe polar geometry. However, they do not vary signifi-\ncantly with the applied external magnetic field (as shown\nin Fig. 3), indicating that, as expected, the compara-\ntively small variations in Zeeman energy as well as the\nsmall change in magnetization direction associated with\nthe change in magnetic field strength do not affect the\nultrafast magnetization dynamics.\nAfter quantifying the ultrafast demagnetization and\nfast remagnetization dynamics (region I and II), we turn\nto the region III, which is characterized by the coher-\nent precessional magnetization dynamics induced by the\npump laser pulse. These dynamics in the GHz- range\naregenerallydescribedbythephenomenologicalLandau-\nLifshitz-Gilbert (LLG) equation [24],\nd⃗M\ndt=−γ⃗M×/bracketleftigg\nµ0⃗Heff−α\nγMSd⃗M\ndt/bracketrightigg\n(1)\nwhereγisthegyromagneticratio, MSisthesaturation\nmagnetization, αis the Gilbert damping constant, and\n⃗Heffis the effective magnetic field consisting of several\nfield components. The first term on the right side of\nEq. (1) accounts for the precession of the magnetization\nvector (⃗M) around⃗Heff. The second term with the first-\norder time derivative of ⃗M, is the Gilbert damping term\n[24], which models the transfer of energy and angular\nmomentum of ⃗Mto the surrounding degrees of freedom\n(relaxation of ⃗Mtowards⃗Heff). Figure 4(a) shows the\nbackground subtracted time-resolved Kerr rotation data\n(precessional part) for two different values of the applied\nmagnetic field, fitted with a damped sinusoidal function,\nM(t) =M(0)e−t/τdsin(2πft) (2)\nHere,M(0)istheinitialamplitudeoftheprecession, τd\nis the relaxation time of the coherent precession obtained\nas a fitting parameter, and fis the precession frequency,\nwhichcanalsobeextracteddirectlyfromtheFastFourier\nTransform (FFT) of the precessional oscillation. Due to\nthe size of the laser spot (D ∼500µm), our measure-\nment basically detects only magnon wavevectors up to\napproximately k ∼π/500 rad/µm, thus essentially only\nthe ferromagnetic resonance (FMR). The effective mag-\nnetization ( Meff), which includes the saturation magneti-\nzation and potential additional out-of-plane anisotropies,\nis calculated from the magnetic field dependence of the\nprecession frequencies (Fig. 4(b)) and fitting the data\npoints with the Kittel formula [31],\nf=1\n2π/radicalbig\nωH(ωH+ωM) (3)\nwhereωH=γµ0H,ωM=γµ0MeffandHis the ex-\nternally applied magnetic field and γ= 1.83×1011rad\ns−1T−1for Py. Strictly speaking, Eq. 3 is only valid4\nfor a completely in-plane magnetized film, but we have\nverified using micromagnetic simulations that it approxi-\nmates our experimental situation very well. From the fit,\nMeffis obtained to be ∼700±25kA m−1for 5 nm Py\n[and∼670±20kA m−1for 2.8 nm Py, see Supplemental\nMaterials] measured at F= 4.6 mJ cm−2.\nFIG. 4. Magnetic field dependent dynamics: (a) Background-\nsubtracted time-resolved Kerr rotation data for 5 nm Py sam-\nple measured at two different magnetic fields and at F= 4.6\nmJ cm−2. Solid lines are fitting lines. (b) Magnetic field\ndependence of precession frequency. Solid line represent the\nKittel fit to the data points. (c) Magnetic field dependence\nof quenching (green) and of precessional amplitude (red). (d)\nMagnetic field dependence of the precessional relaxation time\n(τd) measured after ultrafast demagnetization in TR-MOKE\n(red), usingmicrowave spectroscopy (blue) andanalytical cal-\nculation (black).\nWhenstudyingtheexcitationofthecoherentmagnons,\npoints of interest are the dependence of the precession\nfrequency, amplitude and lifetime on the external con-\nditions. Concerning the amplitude, Fig. 4(c) shows the\ndependence of quenching and precessional amplitude on\nthe magnetic field for a fixed pump fluence ( F= 4.6\nmJ cm−2). As expected, the quenching, which is a mea-\nsure of the energy initially introduced into the magnetic\nsystem, is independent of the applied magnetic field.\nSurprisingly at first glance, the precessional amplitude,\nwhich is usually considered as a measure for the energy of\nthe coherent oscillations, is strongly dependent on mag-\nnetic field. We interpret the observed increase of the\nprecession amplitude with increasing magnetic field as an\nincrease of the part of the coherent precession that has\na well-defined and constant phase relation to the pump\npulse. This can be explained by the fact that the static\nout-of-plane component of the magnetization, which is\nrequired for TR-MOKE to measure the coherent preces-\nsion [1, 2], increases with the strength of the applied bias\nmagnetic field.Interestingly, we also observe that the precessional re-\nlaxation time ( τd) as obtained from TR-MOKE mea-\nsurements, increases with increasing magnetic field (red\ndots in Fig. 4(d)). This is unexpected if one assumes\nthat Gilbert damping, described by a material parameter\n(Gilbert damping constant α) is responsible for the de-\ncayoftheprecession. SinceGilbertdampingisviscous, it\npredictsadecreaseofthemagnonlifetimewithincreasing\nfrequency, which is equivalent to an increase of the mag-\nnetic field strength in the presented geometry. Analyt-\nical calculations of the magnon characteristics based on\nthe Gilbert model [32–35] show that the Gilbert-induced\nlifetime decreases with increasing magnetic field (black\nsolid line in Fig. 4(d)). For the analytical calculations,\nwe have assumed an in-plane magnetic field, but we have\nverified again with micromagnetic simulations that this\napproximation is well justified. The lifetime ( τd) of the\nhomogeneousFMRmodeforanin-planemagnetized(the\napplied in-plane magnetic field is assumed to be H) thin\nfilm is calculated using the following expression [35]:\n1\nτd=α/parenleftig\nωH+ωM\n2/parenrightig\n(4)\nA possible interpretation for this intriguing discrep-\nancy between the lifetimes measured with different ex-\ncitation mechanisms could be the contribution of non-\nGilbert damping mechanisms. One of these is the\nso-called two-magnon scattering mechanism, where the\nmagnon energy is redistributed from the FMR ( k= 0)\ntoothershort-wavelengthmagnons( k>0)duetodefect-\ninduced scattering. Arias and Mills [17] developed a the-\nory describing the contributions of two-magnon scatter-\ning to the FMR linewidth. The two-magnon process is\nlinear in magnon amplitude, thus the broadening of the\nFMR linewidth is independent of the magnon amplitude.\nWoltersdorf et al., showed by TR-MOKE experiments\nthat different capping layers can affect spin relaxation\nand damping of Fe films in different ways [19]. They\nalso showed the increase of relaxation time or decrease of\ndamping with increasing magnetic field for Cu capped Fe\nfilms and interpreted it by two-magnon scattering. Liu\net al. [21] showed that the effective damping constant\ndecreases with the increasing magnetic field, suggesting\na contribution of magnetic anisotropy to the enhanced\ndamping. Some other reports have also discussed the en-\nhancement of damping with decreasing magnetic fields\ndue to two-magnon scattering, magnetic anisotropy or\nspin pumping effects [11, 20, 22].\nTo clarify a possible contribution of two-magnon scat-\ntering from defects at surfaces and interfaces [17, 18]\nto the measured decay, we performed additional inde-\npendent measurements of the FMR lifetime using induc-\ntive microwave spectroscopy measured with a vector net-\nwork analyzer (VNA), as shown by the blue dots in Fig.\n4(d). Thistechniqueisknowntobesensitivetolinewidth\nbroadeninginducedbytwo-magnonscattering[36]. How-\never, the measurements show a decrease of lifetime in ac-\ncordance with the Gilbert model. In addition, a theoreti-5\ncal work [18] predicts that a two-magnon contribution to\nthe linewidth should increase with resonance frequency\nand magnetic field in our experimental case (small angles\nof less than 15 °to the films plane). Thus, we can con-\nclude that the defect induced two-magnon scattering is\nnot responsible for the increase in lifetime ( τd) observed\nin our TR-MOKE measurements. Instead, we interpret\nthe change in lifetime due to coherent precession as fol-\nlows. Both incoherent and coherent magnons are excited\nwhenthesampleishitwithafslaserpulseinTR-MOKE.\nThe ratioof coherentto incoherent magnonsis influenced\nby the out-of-plane component of the static magnetiza-\ntion, which breaks the symmetry of the system. As in the\ncase of the precession amplitude, a higher external field\nstrength increases the static out-of-plane component and\nthus the relative proportion of coherent magnons with\na defined and constant phase relationship to the laser\npulses. Due to the reduced excitation of the incoher-\nent magnons at higher fields, the dephasing of the pre-\ncession signal is weaker, leading to a longer lifetime of\nthe measured coherent precession signal. As the external\nmagnetic field decreases, the relative proportion of inco-\nherent magnons to coherent magnons increases, leading\nto a lower lifetime. This is in contrast to microwave spec-\ntroscopymeasurementswhereonlycoherentmagnonsare\nexcited and only a negligible amount of thermally ex-\ncited incoherent magnons are present. This explains the\ndifferent trends in the lifetime of magnons excited and\nmeasured by microwave spectroscopy and TR-MOKE.\nFIG. 5. Pump fluence dependent dynamics: (a) Background-\nsubtracted time-resolved Kerr rotation data for 5 nm Py sam-\nple measured at µ0H= 113 mT and different pump fluences.\nSolid lines are fitting lines. (b) Pump fluence dependence of\nprecessional frequencies at µ0H= 113 mT. (c) Pump fluence\ndependence of quenching (olive) and precessional amplitude\n(brown). Pump fluence dependent quenching plotted here\n(olive) is the same as plotted by solid circles in Fig. 2(e). (d)\nPump fluence dependence of precessional relaxation time ( τd)\nmeasured after ultrafast demagnetization in TR-MOKE.\nAnother interesting parameter to study is the influenceof the excitation intensity, which in our case is given by\nthe pump fluence, on the coherent dynamics. The back-\nground subtracted time-resolved Kerr rotation data for\nthe 5 nm Py sample measured at different pump fluences\n(F) are shown in Fig. 5(a). The energy deposited by\nthe pump pulse, in the form of heat within the probed\nvolume, plays a very crucial role in the modification of\nthe local magnetic properties, i.e. the magnetic moment,\nanisotropy, coercivity, magnetic susceptibility, etc., as\nwell as the precession frequency can experience a vari-\nation with the pump fluence[12, 37]. However, we do not\nobserve any significant frequency shift within our exper-\nimental fluence range as shown in Fig. 5(b). This also\nindicates that the temperature of the sample is not sig-\nnificantly increased on longer timescales after the initial,\nfastremagnetization. Consequently, Meffcalculatedfrom\nEqn. (3) shows no significant dependence on Fwithin\nour experimental fluence range. Thus, we conclude that\nas the pump fluence increases, there is no further change\nin the anisotropy field that can modify the effective mag-\nnetization of the system to this extent [21]. Figure 5(c)\nshows that the quenching increases with pump fluence as\nexpected. However, although the initial quenching is a\nmeasure of the absorbed energy and thus the source of\ncoherent precession, the precession amplitude decreases\nwith fluence. The precessional relaxation time ( τd) also\nshows a decrease with pump fluence (Fig. 5(d)). The\ndecrease of both the precession amplitude and the pre-\ncessionrelaxationtimecanbeexplainedbythedephasing\nof the magnons due to nonlinear magnon-magnon inter-\nactions. These interactions generally increase with an\nincrease in the total magnon population (incoherent and\ncoherent),whichwebelieveisproportionaltothequench-\ning. In addition, the probability of excitation of incoher-\nent magnons relative to coherent magnons increases with\nthe increase of disorder in the system and thus with the\nquenching / pump fluence.\nIV. SUMMARY\nIn summary, the spin dynamics on different time scales\nin thin Ni80Fe20films have been studied using an all-\noptical TR-MOKE technique. We study the preces-\nsion dynamics in the GHz range after femtosecond laser-\ninduced ultrafast demagnetization. The demagnetiza-\ntion time, fast remagnetization time, and magnetization\nquenching studied in both longitudinal and polar geome-\ntry show an increasing trend with excitation fluence, con-\nsistent with a spin-flip scattering-dominated demagneti-\nzation process. On a longer timescale of several hundreds\nof picoseconds, we observe an increase of the coherent\nprecessional relaxation time with magnetic field / reso-\nnance frequency, which cannot be explained by viscous\nGilbert damping. Using standard FMR techniques, we\nconclude that two-magnon scattering is not responsible\nfor this behavior. Instead, we can consistently explain all\nobserved trends by considering the different relative con-6\ntributions of coherent and incoherent magnons produced\nin the ultrafast demagnetization process and the nonlin-\near interaction between them. This interpretation also\nexplains the dependence of the coherent magnon ampli-\ntude and relaxation time on the excitation fluence. We\nexpectthatourresultswillpavethewayforfutureexper-\nimental and theoretical investigations towards a deeper\nunderstanding of the photon-to-magnon conversion in ul-trafast demagnetization processes.\nV. 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Exp. 1, 121301\n(2008).Supplemental Materials  \nCoherent and incoherent magnons induced by strong ultrafast demagnetization in thin \npermalloy films  \nAnulekha De,1,∗ Akira Lentfert,1 Laura Scheuer,1 Benjamin Stadtmüller,1,2 Georg von Freymann,1,3 Martin \nAeschlimann,1 and Philipp Pirro1,† \n1Department of Physics and Research Center OPTIMAS, Rheinland -Pfälzische Technische Universität \nKaiserslautern -Landau, 67663 Kaiserslautern, Germany  \n2Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany  \n3Fraunhofer Institute for Industrial Mathematics, 67663 Kaiserslautern, Germany  \n \nS1. Three Temperature Model : \nThe dynamics of the spin fluctuations after excitation by  ultrafast laser pulses can be described by a \nphenomenological  thermodynamic model, the so -called three -temperature model  (3TM), which pictures \nhow energy is redistributed among  electrons, spins, and the lattice after the absorption of the laser  power by \nthe electronic system. The energy flow ultimately  leads to an increase in the spin temperat ure, thereby \nreducing  the magnetization. The expression is given by,  \n−∆𝑀\n𝑀= {[𝐴1\n(1+𝑡𝜏0⁄)12⁄−𝐴2𝜏𝐸−𝐴1𝜏𝑀\n𝜏𝐸−𝜏𝑀𝑒−𝑡𝜏𝑀⁄−𝜏𝐸(𝐴1−𝐴2)\n𝜏𝐸−𝜏𝑀𝑒−𝑡𝜏𝐸⁄]𝐻(𝑡)+𝐴3𝛿(𝑡)}𝐺(𝑡)   \nHere, τ M and τ E are ultrafast demagnetization and first remagnetization  times respectively. A 1 represents the \namplitude  of magnetization after fast relaxation, A 2 is proportional to  the maximum electron temperature \nrise, and A 3 represents  the state filling effects during pump -probe temporal overlap.  HS(t) is the Heaviside \nstep function, δ(t) is the Dirac delta  distribution, and G(t) is a Gaussian function corresponding to  the laser \npulse. τ 0 represents the cooling time through heat  diffusion. This model is very useful in analysing \nexperimental  data and extracting quantitative information on the timescales  of the different processes taking \nplace during the laser induced  ultrafast demagnetization.  \nS2. Precession dynamics for 2.8 nm Py sample : \nFigure S1(a) shows the background subtracted time -resolved  Kerr rotation data ( precessional part) measured \nat different  values of magnetic fields, fitted with a damped sinusoidal  function (Eqn. 2 of the main article) \nfor 2.8 nm Py sample.  All measurements are done at particular pump fluence of F =  4.6 mJ cm−2. Figure \nS1(b) shows magn etic field dependence  of the precession frequencies obtained from the fast Fourier  \ntransform (FFT) of the precessional oscillation, from which  we calculate the effective magnetization (M eff) \nusing the Kittel  formula (Eqn. (3) of the main article). The valu e of M eff obtained from fit is 670 ± 20 kA \nm−1, which is slightly  less than the thicker (5 nm Py ) film. Figure S1(c) shows the  variation of precessional \nrelaxation time (τ d) with magnetic  as obtained after ultrafast demagnetization in TR -MOKE  measurements \n(red), microwave spectroscopy measurements  (blue) and analytical calculations (black). We observe that  τd \nincreases with magnetic field for TR -MOKE measurements which is in contrast to  both the microwave \nspectroscopy measurements and analytical  calculation s. Similar results are obtained from 5 nm Py sample  \nand are thoroughly discussed in the main article.  \n \n \n  \nS3. Effect of in -plane magnetic field : \nFor the in -plane configuration of the magnetic field, we  observe a reduction in precessional amplitude \nleading to a  poor signal -to-noise ratio (as shown in S2(a)). As the magnetic  field is tilted slightly ( ∼ 15°) in \nthe out -of-plane direction,  the precessional amplitude increases and we observe a clear  time-resolve d Kerr \nrotation trace resulting clear fast Fourier  transformed magnon modes. In the in -plane configuration the  \ndominance of nonmagnetic noise due to two -magnon scattering has suppressed the features of magnetic \npeaks and only a  few spurious peaks are prese nt in the spectra. A comparison  between the FFT powers for \nin-plane and tilted magnetic fields  are shown in Fig. S2(c), where the power is negligibly small in  in-plane \nconfiguration as compared to the tilted configuration  of magnetic field.    \nFig. S1:  Magnetic field dependent dynamics of 2.8 nm Py sample: (a) Background -subtracted time -resolved Kerr rotation data \nmeasured at different values of magnetic field and at F = 4.6 mJ cm−2. Solid lines are fitting lines. (b) Magnetic field dependence \nof precession frequency. Solid line represent the Kittel fit to the data points. (c) Magnetic field dependence of precessiona l \nrelaxation time (τ d) measured after ultrafast demagnetization in TR -MOKE (red), microwave spectroscopy (blue) and analytical \ncalculation (black).  \n \nFig. S 2: (a) Precessional dynamics at two different values of in -plane magnetic field. (b) The corresponding FFT power spectra \n(c) The comparison between FFT spectra at in -plane and tilted magnetic fields. The values of magnetic fields are mentioned in \nthe respective graphs.  \n"    },    {        "title": "2305.09231v1.Non_Hermitian_Casimir_Effect_of_Magnons.pdf",        "content": "Non-Hermitian Casimir Effect of Magnons\nKouki Nakata1,\u0003and Kei Suzuki1,y\n1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan\n(Dated: May 17, 2023)\nThere has been a growing interest in non-Hermitian quantum mechanics. The key concepts of\nquantum mechanics are quantum fluctuations. Quantum fluctuations of quantum fields confined in a\nfinite-size system induce the zero-point energy shift. This quantum phenomenon, the Casimir effect,\nis one of the most striking phenomena of quantum mechanics in the sense that there are no classical\nanalogs and has been attracting much attention beyond the hierarchy of energy scales, ranging from\nelementary particle physics to condensed matter physics, together with photonics. However, the\nnon-Hermitian extension of the Casimir effect and the application to spintronics have not yet been\ninvestigated enough, although exploring energy sources and developing energy-efficient nanodevices\nare its central issues. Here we fill this gap. By developing a magnonic analog of the Casimir effect\ninto non-Hermitian systems, we show that this non-Hermitian Casimir effect of magnons is enhanced\nas the Gilbert damping constant (i.e., the energy dissipation rate) increases. When the damping\nconstant exceeds a critical value, the non-Hermitian Casimir effect of magnons exhibits an oscillating\nbehavior, including a beating one, as a function of the film thickness and is characterized by the\nexceptional point. Our result suggests that energy dissipation serves as a key ingredient of Casimir\nengineering.\nIntroduction. —Recently, non-Hermitian physics has\nbeen drawing considerable attention not only for funda-\nmental science but also for applications such as energy-\nefficient nanodevices [1]. Thanks to the complete absence\nof any metallic elements from conduction electrons, insu-\nlating magnets are free from drawbacks of conventional\nelectronics,e.g.,substantialenergylossduetoJouleheat-\ning. However, the effect of energy dissipation on spins is\nunavoidable even in insulating magnets. Hence, to ex-\nplore energy sources and develop the potential for appli-\ncations is a crucial issue.\nWe shed light on this problem by using a quantum ef-\nfect, the Casimir effect [2–4], which arises from the zero-\npoint energy. Quantum fluctuations of quantum fields\nrealize a zero-point energy shift under spatial boundary\nconditions. This Casimir effect is one of the most strik-\ning phenomena of quantum mechanics in the sense that\nthere areno classical analogs. Although the original plat-\nform for the Casimir effect [2–4] is the photon field [5],\nthe concept can be extended to various fields such as\nscalar, tensor, and spinor fields. Thanks to this universal\nproperty, the Casimir effects have been investigated in\nvarious research areas [6] beyond the hierarchy of energy\nscales [7–12], ranging from elementary particle physics\nto condensed matter physics, together with photonics.\nHowever, the non-Hermitian extension of the Casimir ef-\nfect and the application to spintronics remain missing in-\ngredients, although exploring energy sources and devel-\noping the potential for energy-efficient nanodevices are\nthe central issues of spintronics [13–16].\nHere we fill this gap. The Casimir effects are charac-\nterized by the energy dispersion relation. We therefore\n\u0003(Equal contribution) nakata.koki@jaea.go.jp\ny(Equal contribution) k.suzuki.2010@th.phys.titech.ac.jp\nFIG. 1. Schematic of the thin film of the AFMs.\nincorporate the effect of energy dissipation into the en-\nergy dispersion relation of magnons through the Gilbert\ndamping constant [17] and thus develop a magnonic ana-\nlog of the Casimir effect, called the magnonic Casimir ef-\nfect[18], intonon-Hermitiansystems. Wethenshowthat\nthisnon-HermitianextensionofthemagnonicCasimiref-\nfect, which we call the magnonic non-Hermitian Casimir\neffect, is enhanced as the Gilbert damping constant (i.e.,\nthe energy dissipation rate) increases. When the damp-\ning constant exceeds a critical value, the magnonic non-\nHermitian Casimir effect exhibits an oscillating behavior\nas a function of the film thickness and is characterized by\nthe exceptional point [19] (EP). We refer to this behav-\nior as the magnonic EP-induced Casimir oscillation. We\nemphasize that this magnonic EP-induced Casimir oscil-\nlation is absent in the dissipationless system of magnons.\nThe magnonic EP-induced Casimir oscillation exhibits a\nbeating behavior in the antiferromagnets (AFMs) where\nthe degeneracy between two kinds of magnons is lifted.\nOur result suggests that energy dissipation serves as a\nnew handle on Casimir engineering [20] to control and\nmanipulate the Casimir effect of magnons. Thus, we\npave a way for magnonic Casimir engineering through\nthe utilization of energy dissipation.\nMagnonic non-Hermitian system. —We consider the\ninsulating AFMs of two-sublattice systems in three di-\nmensions described by the Heisenberg model (see Fig. 1),arXiv:2305.09231v1  [quant-ph]  16 May 20232\nwhere the AFMs have the Néel magnetic order and there\nexists the zero-point energy [21, 22]. Throughout this\nstudy, we work under the assumption that the Néel phase\nremains stable in the presence of energy dissipation. El-\nementary magnetic excitations are two kinds of magnons\nhaving the spin angular momentum \u001b~with the index\n\u001b=\u0006and the reduced Planck constant ~. By incor-\nporating the effect of energy dissipation into the energy\ndispersion relation of magnons through the two-coupled\nLandau-Lifshitz-Gilbert equation where the value of the\nGilbert damping constant \u000b > 0for each sublattice is\nidentical to each other, we study the low-energy magnon\ndynamics [23] described by the energy dispersion relation\n\u000f\u001b;k;\u000b2Cand the wavenumber k= (kx;ky;kz)2R[24]\nin the long wavelength limit as [25]\n\u000f\u001b;k;\u000b=2S\n1 +\u000b2\u0010\n\u0000i\u000bC+q\n(E\u001b;k;\u000b)2\u0011\n(1)\nand\n(E\u001b;k;\u000b)2:=A\u001b;\u000b2(ak)2+\u000e\u001b2\u0000D\u001b2\u000b2;(2)\nwherek:=jkj, the length of a magnetic unit cell is a, the\nspin moment in a magnetic unit cell is S, and the others\nare material-dependent parameters which are indepen-\ndent of the wavenumber, 0< A\u001b;\u000b2R,0< \u000e\u001b2R,\n0<D\u001b2R, and 0< C2R: In the presence of spin\nanisotropies, the parameters are given as [25]\nA\u001b;\u000b=r\n(1 +\u000b2)\u0010\nJ2+\u001bKh\n2J\u0011\n; (3a)\n\u000e\u001b=p\nKe(2J+Ke) +Kh(J\u0000\u001bJ+Ke);(3b)\nD\u001b=s\nJ2+\u001bKhJ+Kh2\n4; (3c)\nC=J+Ke+Kh\n2; (3d)\nwhereJ > 0parametrizes the antiferromagnetic ex-\nchange interaction between the nearest-neighbor spins,\nKh>0is the hard-axis anisotropy, and Ke>0is the\neasy-axis anisotropy along the ydirection (see Fig. 1).\nThese are generally Kh=J\u001c1andKe=J\u001c1. In the\nabsence of the hard-axis anisotropy Kh= 0, two kinds\nof magnons \u001b=\u0006are in degenerate states, whereas the\ndegeneracy is lifted by Kh>0. Note that, in general, the\neffect of dipolar interactions is negligibly small in AFMs,\nand we neglect it throughout this study.\nThe Gilbert damping constant \u000bis a dimensionless\nconstant, and the energy dissipation rate increases as the\nGilbert damping constant grows. In the dissipationless\nsystem [18], the Gilbert damping constant (i.e., the en-\nergy dissipation rate) is zero \u000b= 0. The dissipative sys-\ntem of\u000b>0described by Eq. (1) can be regarded as a\nnon-Hermitian system for magnons in the sense that the\nenergy dispersion takes a complex value. Note that the\nconstant term in Eq. (1), \u0000i\u000bC, is independent of thewavenumber and just shifts the purely imaginary part of\nthe magnon energy dispersion \u000f\u001b;k;\u000b. For this reason [see\nalso Eq. (9a)], the constant term, \u0000i\u000bC, is not relevant\nto the magnonic Casimir effect [26]. We then define the\nmagnon energy gap of Eq. (1) as \u0001\u001b;\u000b:=Re(\u000f\u001b;k=0;\u000b),\ni.e.,\n\u0001\u001b;\u000b=2S\n1 +\u000b2Re\u0010q\n(E\u001b;k=0;\u000b)2\u0011\n; (4)\nand investigate the \u000b-dependence of the magnon energy\ndispersion.\nMagnonic EP. —Whenthedampingconstant \u000bissmall\nand (E\u001b;k=0;\u000b)2>0,E\u001b;k=0;\u000btakes a real value and\ndecreases as \u000bincreases. This results in\nd\u0001\u001b;\u000b\nd\u000b<0: (5)\nThus, the magnon energy gap decreases as the damping\nconstant increases [27] [compare the solid line with the\ndashed one in the left panel of Fig. 2 (i)]. When the\ndamping constant is large enough, the magnon energy\ngap vanishes \u0001\u001b;\u000b= 0at\u000b=\u000bcri\n\u001b,\n\u000bcri\n\u001b:=\u000e\u001b\nD\u001b; (6)\nwhere there exists the gapless magnon mode which be-\nhaves like a relativistic particle with the linear energy\ndispersion. From the property of Eq. (5), we call (i)\n\u000b\u0014\u000bcri\n\u001bthe gap-melting regime.\nWhen the damping constant exceeds the critical value\n\u000bcri\n\u001b, i.e.,\u000b > \u000bcri\n\u001b,E\u001b;k=0;\u000btakes a purely imaginary\nvalue as (E\u001b;k=0;\u000b)2<0. In this regime, the real part of\nthe magnon energy dispersion remains zero Re (\u000f\u001b;k;\u000b) =\n0for the region 0\u0014k\u0014kcri\n\u001b;\u000b,\nkcri\n\u001b;\u000b:=1\nas\nD\u001b2\u000b2\u0000\u000e\u001b2\nA\u001b;\u000b2; (7)\nwhereas Re (\u000f\u001b;k;\u000b)>0fork >kcri\n\u001b;\u000b[see the highlighted\nin yellow in the left panel of Figs. 2 (ii) and (iii)]. The\ncritical point kcri\n\u001b;\u000bcan be regarded as the EP [16, 27] for\nthe wavenumber k, and we refer to it as the magnonic\nEP.Asthevalueofthedampingconstantbecomeslarger,\nthat of the EP increases\ndkcri\n\u001b;\u000b\nd\u000b>0: (8)\nAt the EP k=kcri\n\u001b;\u000b, the group velocity v\u001b;k;\u000b:=\nRe[@\u000f\u001b;k;\u000b=(@~k)]becomes discontinuous [see the solid\nlines in the left panel of Figs. 2 (ii) and (iii)]. We em-\nphasize that this behavior is absent in the dissipationless\nsystem\u000b= 0[18]. In the presence of the EP, the group\nvelocity becomes much larger than the usual such as in\nthe gap-melting regime (i) [compare the solid lines in the\nleftpanelofFigs.2(ii)and(iii)withtheoneofFig.2(i)].3\nAssuming\u000bcri\n\u001b=+< \u000bcri\n\u001b=\u0000, the non-Hermitian system\nfor magnons described by Eq. (1) with \u000b>0can be di-\nvided into three regimes (i)-(iii) in terms of the magnonic\nEPs as follows [see the left panel of Figs. 2 (i), (ii),\nand (iii)]:\n(i)\u000b\u0014\u000bcri\n\u001b=+<\u000bcri\n\u001b=\u0000. No magnonic EPs.\n(ii)\u000bcri\n\u001b=+<\u000b<\u000bcri\n\u001b=\u0000. One EP,kcri\n\u001b=+;\u000b.\n(iii)\u000bcri\n\u001b=+<\u000bcri\n\u001b=\u0000\u0014\u000b. Two EPs, kcri\n\u001b=+;\u000bandkcri\n\u001b=\u0000;\u000b.\nMagnonic Casimir energy. —The magnonic analog of\nthe Casimir energy, called the magnonic Casimir en-\nergy [18], is characterized by the energy dispersion re-\nlation of magnons (see Ref. [18] for dissipationless sys-\ntems). Therefore, by incorporating the effect of energy\ndissipationintotheenergydispersionrelationofmagnons\nthrough the Gilbert damping constant [see Eq. (1)], a\nnon-Hermitian extension of the magnonic Casimir effect\ncan be developed. We remark that the Casimir energy\ninduced by quantum fields on the lattice, such as the\nmagnonic Casimir energy [18], can be defined by using\nthe lattice regularization [28–34]. In this study, we focus\non thin films confined in the zdirection (see Fig. 1). In\nthe two-sublattice systems, the wavenumber on the lat-\ntice is replaced as (akj)2!2[1\u0000cos(akj)]along thej\naxis forj=x;y;z. Here by taking into account the Bril-\nlouin zone (BZ), we set the boundary condition for the\nzdirection in wavenumber space so that it is discretized\naskz!\u0019n=Lz, i.e.,akz!\u0019n=Nz, whereLz:=aNz\nis the film thickness, Nj2Nis the number of magnetic\nunit cells along the jaxis, andn= 1;2;:::;2Nz. Thus,\nthe magnonic Casimir energy ECas[18] per the number\nof magnetic unit cells on the surface for Nzis defined as\nthe difference between the zero-point energy Esum\n0for the\ndiscrete energy \u000f\u001b;k;\u000b;ndue to discrete kz[see Eq. (9b)]\nand the one Eint\n0for the continuous energy \u000f\u001b;k;\u000b[see\nEqs. (9c) and (1)] as follows [28–34]:\nECas(Nz) :=Esum\n0(Nz)\u0000Eint\n0(Nz); (9a)\nEsum\n0(Nz) :=X\n\u001b=\u0006Z\nBZd2(ak?)\n(2\u0019)2\"\n1\n2\u00101\n22NzX\nn=1\u000f\u001b;k;\u000b;n\u0011#\n;\n(9b)\nEint\n0(Nz) :=X\n\u001b=\u0006Z\nBZd2(ak?)\n(2\u0019)2\"\n1\n2NzZ\nBZd(akz)\n2\u0019\u000f\u001b;k;\u000b#\n;\n(9c)\nwherek?:=q\nkx2+ky2,d2(ak?) =d(akx)d(aky),\nthe integral is over the first BZ, and the factor 1=2in\nEqs. (9b) and (9c) arises from the zero-point energy.\nWe remark that [23] assuming thin films of Nz\u001c\nNx;Ny(see Fig. 1), the zero-point energy in the thin film\nof the thickness NzisEsum\n0(Nz)NxNyand consists of two\nparts asEsum\n0(Nz) =ECas(Nz)+Eint\n0(Nz)[see Eq. (9a)],\nwhereEint\n0(Nz)exhibits the behavior of Eint\n0(Nz)/Nz\n[see Eq. (9c)]. Then, to see the film thickness dependence\nofECas(Nz), we introduce the rescaled Casimir energyC[b]\nCasin terms of Nzbforb2Ras\nC[b]\nCas(Nz) :=ECas\u0002Nzb(10)\nand callC[b]\nCasthe magnonic Casimir coefficient in the\nsense thatECas=C[b]\nCasNz\u0000b.\nNote that the zero-point energy arises from quan-\ntum fluctuations and does exist even at zero tempera-\nture. The zero-point energy defined at zero temperature\ndoes not depend on the Bose-distribution function [see\nEqs. (9b) and (9c)]. Throughout this work, we focus on\nzero temperature [23].\nMagnonic non-Hermitian Casimir effect. —Finally, we\ninvestigate the magnonic Casimir effect in the non-\nHermitian system \u000b > 0, which we call the magnonic\nnon-HermitianCasimireffect, foreachregime(i)-(iii). As\nan example, we consider NiO, an insulating AFM. From\nRefs. [25, 35, 36], we roughly estimate the model param-\neter values for NiO as follows [see Eq. (1)]: J= 47:0859\nmeV,Kh= 0:039 5212 meV,Ke= 0:001 71829 meV,\nS= 1:206 83, anda= 0:417nm. NiO is a biaxial\nAFM ofKh>0andKe>0. Due to the hard-axis\nanisotropy Kh>0, the degeneracy between two kinds\nof magnons \u001b=\u0006is lifted in NiO. These parameters\nprovide\u000bcri\n\u001b=+\u00180:008 5414< \u000bcri\n\u001b=\u0000\u00180:041 8709. Fig-\nure 2 shows the magnon energy dispersion [Eq. (1)] and\nthe magnonic Casimir energy [Eq. (9a)] with its Casimir\ncoefficient [Eq. (10)] for each regime (i)-(iii).\n(i) Gap-melting regime \u000b\u0014\u000bcri\n\u001b=+< \u000bcri\n\u001b=\u0000. The\nmagnonic Casimir energy takes a real value as shown\nin the middle and right panels of Fig. 2 (i), see also\nEq. (10), and there are no magnonic EPs [see the left\npanel of Fig. 2 (i)]. This property is the same as its\nCasimir effect in the dissipationless system \u000b= 0[18].\nWhen\u000b < \u000bcri\n\u001b=+, the magnon energy gap for both\n\u001b=\u0006is nonzero \u0001\u001b=\u0006;\u000b>0and both magnons \u001b=\u0006\nare the gapped modes. For each gapped mode, the abso-\nlute value of the magnonic Casimir coefficient C[3]\nCasde-\ncreases and approaches asymptotically to zero as the film\nthickness increases. We emphasize that the magnon en-\nergy gap decreases as the damping constant \u000bincreases\n[see Eq. (5)]. Then, the magnitude of the magnonic\nCasimir energy and its coefficient increase as the value\nof the damping constant becomes larger and approaches\nto the critical value \u000b!\u000bcri\n\u001b=+[see the middle panel of\nFig. 2 (i)].\nWhen\u000b=\u000bcri\n\u001b=+, the magnon \u001b=\u0000remains the\ngapped mode, whereas the magnon energy gap for \u001b= +\nvanishes \u0001\u001b=+;\u000b= 0and the magnon \u001b= +becomes\nthe gapless mode which behaves like a relativistic par-\nticle with the linear energy dispersion. In the gapless\nmode, the magnonic Casimir coefficient C[3]\nCasapproaches\nasymptotically to a nonzero constant as the film thick-\nness increases. This means that although the magnonic\nCasimir effect is realized on the lattice, the behavior of\nthe gapless magnon mode is analogous to the conven-\ntional Casimir effect of a massless scalar field in con-4\nNiOMagnon energy dispersion \u000f\u001b;k;\u000b Re(ECas) and Re(C[b]\nCas) Im(ECas) and Im(C[b]\nCas)\n(i)\n−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\nα=0.00854Re(εσ=−)\nRe(εσ=+)\nIm(εσ=±)Energy εσ,k,α [meV]\nWavenumber akεσ=± at α=0\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20Nzα=0.00854Casimir energy Re( ECas) [meV]\nNz: Thickness of magnet in units of aα=0\n         \n−1.5��1−0.5\n 0  20  40  60  80Re(CCas[3]) [meV]\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20Nzα=0.00854Casimir energy Im( ECas) [meV]\nNz: Thickness of magnet in units of aα=0\n         \n−0.1 0 0.1\n 0  20  40  60  80Im(CCas[3]) [meV]\n(ii)\n−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\nα=0.04Re(εσ=−)\nRe(εσ=+)\nIm(εσ=+)\nIm(εσ=−)EPEnergy εσ,k,α [meV]\nWavenumber ak\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20α=0.04\nNzCasimir energy Re( ECas) [meV]\nNz: Thickness of magnet in units of a−20−15−10−5 0 5 10\n 0 50 100  150  200  250  300Re(CCas[1.5]) [meV]\n 0 0.05 0.1 0.15 0.2\n 0  50  100  150  200  250  300α=0.04NzCasimir energy Im( ECas) [µeV]\nNz: Thickness of magnet in units of a−10−5 0 5 10\n 0 50 100  150  200  250  300Im(CCas[1.5]) [µeV]\n(iii)\n−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\nα=0.05Re(εσ=−)\nRe(εσ=+)\nIm(εσ=+)\nIm(εσ=−)EP EPEnergy εσ,k,α [meV]\nWavenumber ak−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20α=0.05\nNzCasimir energy Re( ECas) [meV]\nNz: Thickness of magnet in units of a−20−10 0 10 20\n 0 50 100  150  200  250  300Re(CCas[1.5]) [meV]\n 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35\n 0  50  100  150  200  250  300α=0.05NzCasimir energy Im( ECas) [µeV]\nNz: Thickness of magnet in units of aα=0.05\n−20−10 0 10 20\n 0 50 100  150  200  250  300Im(CCas[1.5]) [µeV]\nFIG. 2. Plots of the magnon energy dispersion \u000f\u001b;k;\u000b, the real part of the magnonic Casimir energy Re (ECas), and the\nimaginary part Im (ECas)for NiO in (i) the gap-melting regime, (ii) the oscillating regime, and (iii) the beating regime. Inset:\nEach magnonic Casimir coefficient C[b]\nCas[see Eq. (10)].\ntinuous space [37] except for a-dependent lattice effects,\nwhereas that of the gapped magnon modes is similar\nto the Casimir effect known for massive degrees of free-\ndom [37, 38].\n(ii) Oscillating regime \u000bcri\n\u001b=+< \u000b < \u000bcri\n\u001b=\u0000. The\nmagnonic Casimir energy takes a complex value as\nshown in the middle and right panels of Fig. 2 (ii), see\nalso Eq. (10). There is one EP, e.g., akcri\n\u001b=+;\u000b=0:04\u0018\n0:039 05437773 for\u000b= 0:04[see the left panel of\nFig. 2 (ii)]. Then, the magnonic non-Hermitian Casimir\neffect exhibits an oscillating behavior as a function of Nz\nfor the film thickness Lz:=aNz.\nAn intuitive explanation for the oscillation of the\nmagnonic non-Hermitian Casimir effect and its relation\nto the EP is given as follows: Through the lattice regu-\nlarization, the magnonic Casimir energy is defined as the\ndifference [see Eq. (9a)] between the zero-point energy\nwith the discrete wavenumber kz[see Eq. (9b)] and the\none with the continuous wavenumber [see Eq. (9c)]. On\nthe lattice, the wavenumber kzunder the boundary con-\ndition is discretized in units of \u0019=aNzaskz!(\u0019=aNz)n.\nAs the film thickness Nzincreases, the unit becomessmaller, and finally, it matches the EP as \u0019=aNz=kcri\n\u001b;\u000b,\ni.e.,Nz=\u0019=akcri\n\u001b;\u000b, where the magnonic non-Hermitian\nCasimir effect is enhanced because the group velocity be-\ncomes much larger than the usual such as in the gap-\nmelting regime (i) due to the EP [compare the solid lines\nin the left panel of Figs. 2 (ii) and (iii) with the one of\nFig. 2 (i)]. Then, the magnonic non-Hermitian Casimir\neffect is enhanced periodically where the film thickness\nNzis multiples of \u0019=akcri\n\u001b;\u000b. Thus, the oscillating behavior\nofthemagnonicnon-HermitianCasimireffectstemsfrom\nthe EP,kcri\n\u001b;\u000b, and the oscillation is characterized in units\nof\u0019=akcri\n\u001b;\u000b. We refer to this oscillating behavior as the\nmagnonic EP-induced Casimir oscillation. The period of\nthis Casimir oscillation is\n\u0003Cas\n\u001b;\u000b:=\u0019\nakcri\u001b;\u000b: (11)\nAs an example, the period is \u0003Cas\n\u001b=+;\u000b=0:04\u001880:441 49815\nfor\u000b= 0:04. This agrees with the numerical result in the\nmiddle and right panels of Fig. 2 (ii), see the highlighted\nin red. We call (ii) \u000bcri\n\u001b=+< \u000b < \u000bcri\n\u001b=\u0000the oscillat-\ning regime. The middle and right panels of Fig. 2 (ii)5\nshow that the magnonic EP-induced Casimir oscillation\nis characterized by its Casimir coefficient C[b]\nCasofb= 1:5.\n(iii) Beating regime \u000bcri\n\u001b=+< \u000bcri\n\u001b=\u0000\u0014\u000b. The\nmagnonic Casimir energy takes a complex value as shown\nin the middle and right panels of Fig. 2 (iii), see also\nEq. (10). There are two EPs, kcri\n\u001b=+;\u000bandkcri\n\u001b=\u0000;\u000b, which\ninduce two types of the Casimir oscillations character-\nized by \u0003Cas\n\u001b=+;\u000band\u0003Cas\n\u001b=\u0000;\u000b, respectively. As an exam-\nple,akcri\n\u001b=+;\u000b=0:05\u00180:049 21389535 andakcri\n\u001b=\u0000;\u000b=0:05\u0018\n0:027 28830018 provide \u0003Cas\n\u001b=+;\u000b=0:05\u001863:835 48044 and\n\u0003Cas\n\u001b=\u0000;\u000b=0:05\u0018115:125 9929, respectively, for \u000b= 0:05\n[see the left panel of Fig. 2 (iii)]. Due to the interference\nbetween the two Casimir oscillations, the magnonic non-\nHermitian Casimir effect exhibits a beating behavior as\na function of Nzfor the film thickness Lz:=aNzwith a\nperiod of\n1\nj1=\u0003Cas\n\u001b=+;\u000b\u00001=\u0003Cas\n\u001b=\u0000;\u000bj: (12)\nAs an example, the period is j1=\u0003Cas\n\u001b=+;\u000b=0:05\u0000\n1=\u0003Cas\n\u001b=\u0000;\u000b=0:05j\u00001\u0018143:284 2588 for\u000b= 0:05. This\nagrees with the numerical result in the middle and right\npanels of Fig. 2 (iii), see the highlighted in blue. We call\n(iii)\u000bcri\n\u001b=+<\u000bcri\n\u001b=\u0000\u0014\u000bthe beating regime. The middle\nand right panels of Fig. 2 (iii) show that the beating be-\nhavior of the magnonic EP-induced Casimir oscillation is\ncharacterized by its Casimir coefficient C[b]\nCasofb= 1:5.\nWe remark that in the absence of the hard-axis\nanisotropy Kh= 0, two kinds of magnons \u001b=\u0006\nare in degenerate states [see Eq. (1)]. This results in\n\u000bcri\n\u001b=+=\u000bcri\n\u001b=\u0000[see Eq. (6)] and \u0003Cas\n\u001b=+;\u000b= \u0003Cas\n\u001b=\u0000;\u000b[see\nEq. (11)]. Hence, the Casimir oscillation is one type with\naperiodof \u0003Cas\n\u001b;\u000b. Thismeansthatthebeatingbehavioris\nabsent in the uniaxial AFMs of Kh= 0andKe>0[23].\nMagnonic Casimir engineering. —In each regime (i)-\n(iii), the Gilbert damping (i.e., energy dissipation) serves\nas a key ingredient of Casimir engineering [20] to con-trol and manipulate the Casimir effect of magnons. The\nGilbert damping can be enhanced and controlled by the\nestablished experimental techniques of spintronics such\nas spin pumping [23]. In addition, microfabrication tech-\nnology can control the film thickness and manipulate the\nmagnonic non-Hermitian Casimir effect. The Casimir\npressure of magnons, which stems from the real part\nof its Casimir energy, contributes to the internal pres-\nsure of thin films. We find from the middle panel of\nFigs. 2 (ii) and (iii) that depending on the film thick-\nness, the sign of the real part of the magnonic Casimir\ncoefficient changes. This means that by tuning the film\nthickness, we can control and manipulate the direction\nof the magnonic Casimir pressure as well as the magni-\ntudethankstotheEP-inducedCasimiroscillation. Thus,\nour study utilizing energy dissipation, the magnonic non-\nHermitian Casimir effect, provides the new principles of\nnanoscale devices, such as highly sensitive pressure sen-\nsors and magnon transistors [39], and paves a way for\nmagnonic Casimir engineering.\nConclusion. —We have shown that as the Gilbert\ndamping constant (i.e., the energy dissipation rate) in-\ncreases, the non-Hermitian Casimir effect of magnons in\nantiferromagnets is enhanced and exhibits the oscillat-\ning behavior which stems from the exceptional point.\nThis exceptional point-induced Casimir oscillation also\nexhibits the beating behavior when the degeneracy be-\ntween two kinds of magnons is lifted. These magnonic\nCasimir oscillations are absent in the dissipationless sys-\ntem of magnons. 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Princep, R. A. Ewings, S. Ward, S. Tóth, C. Dubs,\nD. Prabhakaran, and A. T. Boothroyd, npj Quantum\nMater. 2, 1 (2017).\n[67] Y. Nambu and S.-i. Shamoto, J. Phys. Soc. Jpn. 90,\n081002 (2021), arXiv:2106.15752.\n[68] O. I. Gorbatov, G. Johansson, A. Jakobsson,\nS. Mankovsky, H. Ebert, I. Di Marco, J. Minár,\nand C. Etz, Phys. Rev. B 104, 174401 (2021).7\nSupplemental Material\nIn this Supplemental Material, we add an explanation\nabout the Casimir energy induced by quantum fields on\nthe lattice and provide some details about the magnonic\nCasimir effect in the absence of the hard-axis anisotropy.\nWe also add remarks on, in order, observation of the\nmagnonic Casimir effect in the AFMs, Casimir effects\nfrom other origins, thermal effects, higher energy bands,\nedge or surface magnon modes, and the effect of the edge\ncondition.\nI. THE CASIMIR ENERGY ON THE LATTICE\nIn the main text, following the Casimir energy\nfor photon fields (i.e., quantum fields in continuous\nspace) [2], the magnonic Casimir energy is defined as in\nEqs. (9a), (9b), and (9c) through the lattice regulariza-\ntion. In contrast to the Casimir effect for photon fields\n(i.e., quantum fields in continuous space), the magnonic\nCasimir energy is induced byits quantum field on the lat-\ntice, and there is no ultraviolet divergence in each com-\nponent [see Eqs. (9b) and (9c)]. Here we remark that the\nCasimir energy induced by quantum fields on the lat-\ntice, such as the magnonic Casimir energy ECas(Nz)[see\nEq. (9a)], plays a key role in finding the film thickness\ndependence of the zero-point energy in the thin film (see\nFig. 1). The zero-point energy in the thin film of the\nthicknessNzisEsum\n0(Nz)NxNy[see Eq. (9b)] and con-\nsists of two parts as Esum\n0(Nz) =ECas(Nz) +Eint\n0(Nz)\n[see Eq. (9a)], where Eint\n0(Nz)exhibits the behavior of\nEint\n0(Nz)/Nz[see Eq. (9c)].\nII. THE HARD-AXIS ANISOTROPY\nA. In the absence of the hard-axis anisotropy\nIn the main text, we have considered NiO. NiO is a bi-\naxial AFM of Kh>0andKe>0: There exist not only\nthe easy-axis anisotropy Ke= 0:001 71829 meV but also\nthe hard-axis anisotropy Kh= 0:039 5212 meV, see the\nmain text for other parameter values. Here, by chang-\ning only the value of KhtoKh= 0with leaving other\nparameter values unchanged, we estimate the magnonic\nCasimir effect and provide some details about its behav-\nior in the absence of the hard-axis anisotropy.\nFigure S1 shows the magnon energy dispersion \u000f\u001b;k;\u000b\nfor the gap-melting regime (i) in the absence of the\nhard-axis anisotropy Kh= 0. Figure S2 shows the real\npart of the magnonic Casimir energy Re (ECas)for the\ngap-melting regime (i) in the absence of the hard-axis\nanisotropy Kh= 0and that in the presence of hard-axis\nanisotropy Kh= 0:039 5212 meV. The latter is the same\nas the middle panel of Fig. 2 (i).\nIn the absence of the hard-axis anisotropy Kh= 0, two\nkinds of magnons \u001b=\u0006are in degenerate states [see\n−6−4−2 0 2 4 6 8 10 12 0  0.02  0.04  0.06  0.08  0.1\nα=0.00854, Kh=0\nRe(εσ=±)\nIm(εσ=±)Energy εσ,k,α [meV]\nWavenumber akεσ=± at α=0, Kh=0FIG. S1. Plots of the magnon energy dispersion \u000f\u001b;k;\u000bfor\nthe gap-melting regime (i) in the absence of the hard-axis\nanisotropy Kh= 0, where\u000bcri\n\u001b=+=\u000bcri\n\u001b=\u0000= 0:008 54.\n−3−2.5−2−1.5−1−0.5 0\n 0  2  4  6  8 10  12  14  16  18  20Nzα=0.00854 α=0.00854, Kh=0Casimir energy Re( ECas) [meV]\nNz: Thickness of magnet in units of aα=0\n             α=0,Kh=0\n            \n−1.5−1−0.5\n 0  20  40  60  80Re(CCas[3]) [meV]\nFIG.S2. PlotsoftherealpartofthemagnonicCasimirenergy\nRe(ECas)for the gap-melting regime (i) in the absence of the\nhard-axis anisotropy Kh= 0and those in the presence of\nhard-axis anisotropy Kh= 0:039 5212 meV. The latter is the\nsame as the middle panel of Fig. 2 (i). Inset: Its Casimir\ncoefficientC[b]\nCas=ECas\u0002Nzb.\nEq. (1)]. This results in \u000bcri\n\u001b=+=\u000bcri\n\u001b=\u0000= 0:008 54[see\nEq. (6)]. When the damping constant reaches the critical\nvalue\u000b=\u000bcri\n\u001b=+=\u000bcri\n\u001b=\u0000= 0:008 54, the magnon energy\ngaps for both \u001b=\u0006vanish, \u0001\u001b=\u0006;\u000b= 0, and both\nmagnons\u001b=\u0006become the gapless modes which behave\nlike relativistic particles with the linear energy disper-\nsion (see the solid line in Fig. S1). Then, the magnonic\nCasimir coefficient C[3]\nCasasymptotically approaches to\na nonzero constant as the film thickness increases (see\nFig. S2), which means that its Casimir energy exhibits\nthe behavior of ECas/1=Nz3. Figure S2 also shows\nthat the magnitude of the magnonic Casimir energy and\nits coefficient for Kh= 0become larger than that for\nKh= 0:039 5212 meV.8\nTABLE I. Magnonic non-Hermitian Casimir effects in the\nAFMs ofKe>0.\nKh>0Kh= 0\nThe degeneracy of magnons ( \u001b=\u0006) No Yes\nThe number of the magnonic EPs 2 1\nThe EP-induced Casimir oscillation Yes Yes\nThe beating behavior of the oscillation Yes No\nB. An example of the uniaxial AFM\nAs an example, Cr 2O3can be regarded as a uniaxial\nAFMofKh= 0andKe>0, wheretwokindsofmagnons\n\u001b=\u0006areindegeneratestates. Hence, themagnonicEP-\ninduced Casimir oscillation is one type, and its beating\nbehavior is absent in Cr 2O3. Magnonic non-Hermitian\nCasimir effects in the AFMs of Ke>0are summarized\nin Table I. Note that Ref. [56] reported the experimental\nrealization of sub-terahertz spin pumping in Cr 2O3, and\nRef. [57] reported that in NiO.\nIII. REMARKS ON OBSERVATION\nIn the main text, we have explained that the Gilbert\ndampingcanbeenhancedandcontrolledbytheuseofthe\nestablished experimental techniques of spintronics such\nas spin pumping. Here we add remarks on observation of\nour theoretical prediction. We expect that the magnonic\nCasimir effect in the AFMs can be experimentally ob-\nserved in principle through measurement of magnetiza-\ntion. The reason is as follows.\nExternal magnetic fields induce magnetostriction,\nwhich can be regarded as a kind of lattice deformation,\nand its correction for the length of a magnetic unit cell\nais characterized by the magnetostriction constant [58–\n65]. The magnonic Casimir energy of the AFMs does not\ndepend on external magnetic fields usually, whereas the\nmagnonic Casimir effect is influenced by magnetostric-\ntion, and its correction for the magnonic Casimir energy\ndepends on magnetic fields and contributes to magne-\ntization. Thus, although the correction is small, the\nmagnonic Casimir effect in the AFMs can be experimen-\ntally observed in principle through measurement of mag-\nnetization and its film thickness dependence by using ex-\nternal magnetic fields (i.e., magnetostriction).\nWe remark that the magnetic-field derivative of the\nreal part of the Helmholtz free energy is magnetiza-\ntion. At zero temperature, assuming thin films of Nz\u001c\nNx;Ny(see Fig. 1), the Helmholtz free energy of quan-\ntum fields for magnons in the thin film of the thickness\nNzisEsum\n0(Nz)NxNy[see Eq. (9b)] and consists of two\nparts asEsum\n0(Nz) =ECas(Nz)+Eint\n0(Nz)[see Eq. (9a)],\nwhereEint\n0(Nz)exhibits the linear-in- Nzbehavior as\nEint\n0(Nz)/Nz[see Eq. (9c)]. Since ECas(Nz)exhibits an\noscillatingandabeatingbehaviorasafunctionofthefilmthickness in the regimes (ii) and (iii), respectively [see\nEq. (10) and the middle panels of Figs. 2 (ii) and (iii)],\nthe Helmholtz free energy of the thin film shows a dif-\nferentNz-dependence from the linear-in- Nzbehavior. In\nother words, magnetization of the thin film exhibits an\noscillating or a beating behavior as a function of the film\nthickness due to the magnonic non-Hermitian Casimir ef-\nfect. Hence, our prediction, the non-Hermitian Casimir\neffect of magnons, can be observed in principle through\nmeasurement of magnetization, its oscillating or beating\nbehavior as a function of the film thickness.\nIV. CASIMIR EFFECTS FROM OTHER\nORIGINS\nIn the main text, we have focused on the magnonic\nCasimir effect. Here we add a remark on Casimir ef-\nfects from other origins such as phonons and photons.\nEven excluding magnetostriction, the energy dispersion\nof magnons depends strongly on magnetic fields through\nZeeman coupling, whereas those of phonons and photons\ndo not. Therefore, we expect that the magnonic Casimir\neffect can be distinguished experimentally from the oth-\ners by manipulating external magnetic fields.\nV. THERMAL EFFECTS\nIn the main text, we have focused on zero tempera-\nture. Here we remark on thermal effects. At nonzero\ntemperature, a thermal contribution to the Helmholtz\nfree energy, called the thermal Casimir energy, arises ad-\nditionally and is characterized by the Boltzmann factor.\nIt should be emphasized that, although it is called the\nthermal Casimir energy, there is a significant distinction\nbetween the thermal Casimir effect and the Casimir ef-\nfect: The thermal Casimir effect is independent of the\nzero-pointenergy. ThethermalCasimireffectarisesfrom\nthermal fluctuations and is affected by temperatures,\nwhereas the Casimir effect arises from the zero-point en-\nergy due to quantum fluctuations and is not affected by\ntemperatures. Hence, we expect that the Casimir effect\nof magnons can be distinguished experimentally from its\nthermalCasimireffectbymanipulatingtemperature. For\ndetailsofamagnonicanalogofthethermalCasimireffect\nin a Hermitian system, see Ref. [48] as an example.\nVI. HIGHER ENERGY BANDS\nIn the main text, we have assumed that the magnonic\nCasimir energy of the AFM, NiO, is dominated by the\ntwo bands of Eq. (1). Here we remark on the contri-\nbution from higher energy bands than those of Eq. (1).\nThe magnonic Casimir energy or the zero-point energy\n[see Eq. (9a)] arises from quantum fluctuations and does9\nexistevenatzerotemperature. Thezero-pointenergyde-\nfined at zero temperature does not depend on the Bose-\ndistribution function [see Eqs. (9b) and (9c)]. Hence,\nhigher energy bands than those of Eq. (1) also can con-\ntribute to the magnonic Casimir energy. However, the\ncontribution becomes smaller as the shape of the bands\nis flatter. Numerical calculations of Refs. [66–68] show\nthat higher energy bands of a ferrimagnet tend to be flat.\nThe ferrimagnet has an alternating structure of up and\ndown spins like the Néel magnetic order of the AFM, and\nin this sense, the ferrimagnet is similar to the AFM. We\ntherefore assume that higher energy bands of the AFM\nalso tend to be flat. Thus, throughout this study, we\nwork under the assumption that the magnonic Casimir\nenergy of the AFM, NiO, is dominated by the two bands\nof Eq. (1). For a more accurate estimation, inelastic neu-\ntron scattering measurement of its higher energy bands\nis essential.\nVII. EDGE OR SURFACE MAGNON MODES\nWe add an explanation about the effect of edge or\nsurface magnon modes on the magnonic Casimir energy.\nThe magnonic Casimir effect in our setup (see the thin\nfilm of Fig. 1) is induced by quantum fields for magnons\nof wavenumbers kzdiscretized by small Nz: Its necessary\ncondition is a kz-dependent dispersion relation through\nthe discretization of kz. In this study, we consider thin\nfilms ofNz\u001cNx;Ny. Even if there are edge or surface\nmagnon modes, they are confined only on the xyplane,\nand their wavenumber in the zdirection is always zero,\ni.e.,kz= 0, where its energy dispersion relation is in-\ndependent of kz. Therefore, such edge or surface modes\ncannot contribute to the magnonic Casimir effect. In\nthis sense, the magnonic Casimir effect in our setup (see\nFig. 1) is not affected by the presence or absence of edge\nor surface magnon modes.\nVIII. THE EFFECT OF EDGE CONDITIONS\nWe add a remark on the edge condition. Details of\nthe edge condition, such as the presence or absence of\ndisorder, may affect the boundary condition for the wave\nfunction of magnons, but the magnonic Casimir effect\nis little influenced as long as one does not assume an\nultrathin film such as Nz= 1;2;3. Even if there is a\nchange in the magnon band structure near the edge due\nto some reasons, such as changed spin anisotropies, the\nexistence of the magnonic Casimir effect remains valid\nas long as its necessary condition (see Sec. VII of this\nSupplemental Material) is satisfied."    },    {        "title": "2305.10111v1.Material_Parameters_for_Faster_Ballistic_Switching_of_an_In_plane_Magnetized_Nanomagnet.pdf",        "content": "arXiv:2305.10111v1  [cond-mat.mes-hall]  17 May 2023Journal of the Physical Society of Japan FULL PAPERS\nMaterial Parameters for Faster Ballistic Switching of an In -plane Magnetized\nNanomagnet\nToshiki Yamaji1*and Hiroshi Imamura1 †\n1National Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan\nHigh-speed magnetization switching of a nanomagnet is nece ssary for faster information processing. The ballistic\nswitching by a pulsed magnetic filed is a promising candidate for the high-speed switching. It is known that the switch-\ning speed of the ballistic switching can be increased by incr easing the magnitude of the pulsed magnetic field. However\nit is difficult to generate a strong and short magnetic field pulse in a sm all device. Here we explore another direction\nto achieve the high-speed ballistic switching by designing material parameters such as anisotropy constant, saturati on\nmagnetization, and the Gilbert damping constant. We perfor m the macrospin simulations for the ballistic switching of\nin-plane magnetized nano magnets with varying material par ameters. The results are analyzed based on the switching\ndynamics on the energy density contour. We show that the puls e width required for the ballistic switching can be re-\nduced by increasing the magnetic anisotropy constant or by d ecreasing the saturation magnetization. We also show that\nthere exists an optimal value of the Gilbert damping constan t that minimizes the pulse width required for the ballistic\nswitching.\n1. Introduction\nIn modern information technologies huge amount of data\nare represented as the direction of the magnetization in a sm all\nmagnet such as magnetic grains in magnetic tapes or hard\ndisk drives. To write information on the conventional mag-\nnetic recording media an external magnetic field is applied i n\nthe opposite direction of the magnetization to switch the di -\nrection of the magnetization. During the switching the mag-\nnetization undergoes multiple precessions around the loca l ef-\nfective field consisting of the external field, anisotropy fie ld,\nand demagnetizing field. The typical switching time or write\ntime is of the order of nanoseconds.\nTo meet the growing demand for fast information process-\ning it is important to develop a faster switching scheme. The\nballistic switching is a promising candidate for high-spee d\nswitching, and much e ffort has been devoted to developing\nthe ballistic switching both theoretically1–8)and experimen-\ntally.9–16)In ballistic switching a pulsed magnetic field is ap-\nplied perpendicular to the easy axis to induce the large-ang le\nprecession around the external magnetic field axis. The dura -\ntion of the pulse is set to a half of the precession period. Aft er\nthe pulse the magnetization relaxes to the equilibrium dire c-\ntion opposite to the initial direction. The switching speed of\nthe ballistic switching can be increased by increasing the m ag-\nnitude of the pulsed field. However, it is di fficult to generate a\nstrong and short field pulse in a small device. It is desired to\nfind a way to speed up the ballistic switching without increas -\ning magnetic field.\nThe magnetization dynamics of the ballistic switching is\ndetermined by the torques due to the external magnetic field,\nthe uniaxial anisotropy field, the demagnetizing field, and t he\nGilbert damping. The torques other than the external mag-\nnetic field torque are determined by the material parameters\nsuch as the anisotropy constant, the saturation magnetizat ion,\nand the Gilbert damping constant. There is room to speed up\n*toshiki-yamaji@aist.go.jp\n†h-imamura@aist.go.jpthe ballistic switching by designing the appropriate mater ial\nparameters.\nIn the early 2000s the several groups each independently\nreported the optical microscope measurements of the ballis -\ntic switching by picosecond pulse magnetic field.9–13)Then\nthe mechanism of a ballistic switching was analyzed in terms\nof the nonlinear dynamics concepts such as a fixed point, at-\ntractors, and saddle point.2, 3, 6)Especially the minimal field\nrequired for a ballistic switching was investigated by comp ar-\ning the so-called Stoner-Wohlfarth (SW) type.2, 3)The damp-\ning constant dependence of the minimal switching field was\nalso studied.2)The characteristics of the parameters of a pulse\nmagnetic field, i.e., magnitude, direction, and rise /fall time on\nthe mechanism of a ballistic switching had been also studied\nby the simulations and experiments.6, 7, 14, 15)\nAs described above, in 2000s and 2010s a ballistic switch-\ning technique had received much attention for the fast magne -\ntization reversal with ringing suppression by fine-tuning t he\nmagnetic pulse parameters. Due to the recent advance of an\nultra-fast measurement17)the studies of a ballistic switching\nhave attracted much attention again. Last year the in-plane\nmagnetization switching dynamics as functions of the pulse\nmagnetic field duration and amplitude was calculated and\nanalyzed by using the conventional Landau-Lifshitz-Gilbe rt\n(LLG) equation and its inertial form, the so-called iLLG\nequation.16)The solutions of both equations were compared\nin terms of the switching characteristics, speed and energy\ndensity analysis. Both equations return qualitatively sim ilar\nswitching dynamics. However the extensive material param-\neter dependences of a ballistic switching region have not\nyet been sufficiently explored. Therefore it is worth clearing\nthe extensive material parameter dependences of the ballis tic\nswitching of an in-plane magnetized nanomagnet.\nIn this paper, we study the ballistic switching of an in-\nplane magnetized nanomagnet with systematically varying\nthe material parameters by using the macrospin simulations .\nThe results show that the pulse width required for the bal-\nlistic switching can be reduced by increasing the magnetic\n1J. Phys. Soc. Jpn. FULL PAPERS\nHp\nmz\nyx(a)\n(c) my at t = 10 ns (b) \n(d) 0 200 400-1 01\nt [ps]my\ntp [ps]0 1 2 3 4 5tSW  [ps] \n110 10 210 3\ntl tutSW \n0 1 -1 \n0 1 2 3 4 502.55.010.0\n7.5\ntp [ps]Hp [T] \nFig. 1. (a) Schematic illustration of the in-plane magnetized nano magnet.\nThe pulse field, Hp, is applied along the x-direction. The initial direction of\nthe magnetization is in the positive y-direction. (b) Gray scale map of myat\nt=10 ns as a function of the pulse field width, tp, and Hp. The black and\nwhite regions represent the success and failure of switchin g. The parameters\nareµ0Ms=0.92 T,µ0HK=0.1 T, andα=0.023. (c) Typical example of\nthe time evolution of mywhen the magnetization switches ( Hp=5 T and tp\n=0.4 ps). The switching time, tSW, is defined as the time when mychanges\nthe sign. (d) tpdependence of tSWalong the dashed horizontal line at Hp=5\nT shown in Fig. 1(b). tlandtuare 3.15 ps and 3.93 ps, respectively. tSWat\ntl≤tp≤tuis 1.7 ps.\nanisotropy constant or by decreasing the saturation magnet i-\nzation. There exists an optimal value of the Gilbert damping\nconstant that minimizes the pulse width required for ballis -\ntic switching. The simulation results are intuitively expl ained\nby analyzing the switching trajectory on the energy density\ncontour.\n2. Model and Method\nIn this section we show the theoretical model, the numer-\nical simulation method, and the analysis using the trajecto ry\nin the limit ofα→0. The macrospin model of the in-plane\nmagnetized noanomagnet and the equations we solve to simu-\nlate the magnetization dynamics are given in Sec. 2.1. In Sec .\n2.2 we show that the switching conditions can be analyzed by\nusing the trajectory on the energy density contour in the lim it\nofα→0 if theα≪1.\n2.1 Macrospin Model Simulation\nFigure 1(a) shows the schematic illustration of the in-\nplane magnetized nanomagnet. The pulsed magnetic field,\nHp, is applied along the x-direction. The unit vector m=\n(mx,my,mz) indicates the direction of the magnetization. The\nsize of the nanomagnet is assumed to be so small that the dy-\nnamics of mcan be described by the macrospin LLG equation\ndm\ndt=−γm×/parenleftBigg\nHeff−α\nγdm\ndt/parenrightBigg\n, (1)\nwhere tis time,γis the gyromagnetic ratio, αis the Gilbert\ndamping constant. The e ffective field, Heff=Hp+HK+Hd,\ncomprises the pulse field, Hp, the anisotropy field, HK, andthe demagnetizing field, Hd. The anisotropy field and the de-\nmagnetizing field are defined as\nHK=/bracketleftbig2K/(µ0Ms)/bracketrightbigmyey, (2)\nand\nHd=µ0Msmzez, (3)\nrespectively, where Kis the uniaxial anisotropy constant, µ0\nis the magnetic permeability of vacuum, Msis the saturation\nmagnetization, and ejis the unit vector along the j-axis ( j=\nx,y,z).\nThe switching dynamics are calculated by numerically\nsolving the LLG equation. The initial ( t=0) direction is set\nasmy=1. The rectangular shaped pulse magnetic field with\nduration of tpis applied at t=0. The time evolution of magne-\ntization dynamics are calculated for 10 ns. Success or failu re\nof switching is determined by whether my<−0.5 att=10\nns.\nFigure 1(b) shows the gray scale plot of myatt=10 ns\non the tp-Hpplane. Following Ref. 16 the parameters are as-\nsumed to beµ0Ms=0.92 T, K=2.3 kJ/m3, i.e.µ0HK=\n0.1 T, andα=0.023. The black and white regions represent\nthe success and failure of switching, respectively. The wid e\nblack region at upper right of Fig. 1(b) represents the balli stic\nswitching region (BSR). A typical example of the time evolu-\ntion of mywhen the magnetization switches is shown in Fig.\n1(c). The switching time, tSW, is defined as the time when my\nchanges the sign. Figure 1(d) shows the tpdependence of tSW\nalong the horizontal line shown in Fig. 1(b), i.e. at Hp=5\nT. The BSR indicated by shade appears between tl=3.15\nps and tu=3.93 ps, where tSW=1.7 ps independent of tp.\nThe lower and upper boundary of the BSR are represented by\ntlandtu, respectively. We investigate the material parameter\ndependence of tlandtuwith keeping Hp=5 T.\n2.2 Analysis of the Switching Conditions for α≪1\nIf the Gilbert damping constant is much smaller than unity\nthe approximate value of tlandtucan be obtained without\nperforming macrospin simulations. In the limit of α→0, the\ntrajectory is represented by the energy contour because the en-\nergy is conserved during the motion of m. The energy density,\nE, of the nanomagnet is defined as18)\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ), (4)\nwhereθandφare the polar and azimuthal angles of the mag-\nnetization, respectively. The color plot of the energy dens ity\ncontour is shown in Fig. 2. The separatrix representing the\nenergy contour with E=Kis indicated by the white curve,\nwhich is expresses as\n1\n2µ0M2\nscos2θ−Ksin2θsin2φ=0. (5)\nThe green dot indicates the initial direction of matt=0. The\nblack curve represents the trajectory of munder the pulse field\nofHpin the limit ofα→0. Under the pulse field the energy\ndensity is given by\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ. (6)\n2J. Phys. Soc. Jpn. FULL PAPERS\n01 5 4 3 26E/K\ntltu\nθ\nφ\nFig. 2. (Color online) Color plot of the energy density contour give n by\nEq. (4).θandφare the polar and azimuthal angles of the magnetization, re-\nspectively. The material parameters, MsandKare same as in Fig. 1. The\nseparatrix given by Eq. (5) is indicated by the white curve. T he initial direc-\ntion of mis indicated by the green dot at ( θ,φ)=(π/2,π/2). The black curve\nrepresents the trajectory of the magnetization under the fie ld of Hp=5 T in\nthe limit ofα→0, which is given by Eq. (7). The yellow stars indicate the\nintersection points of the separatrix and the trajectory, w hich correspond to tp\n=tlandtu. If the pulse is turned o ffattl≤t≤tu, the magnetization switches\nballistically. The yellow triangle indicates the turning p oint of the trajectory\nof the magnetization near mz=1, at whichφ=0.\nSince the energy density of the initial direction, θ=φ=π/2,\nisE=0, the trajectory under the pulse field is expressed as\n1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ=0. (7)\nThe yellow stars indicate the points where the trajectory\ncrosses the separatrix surrounding the equilibrium point a t\nφ=−π/2. The upper and lower points indicates the direc-\ntion of mat the end of the pulse with tp=tuandtl, re-\nspectively. The corresponding angles ( θl,φl) and (θu,φu) can\nbe obtained by solving Eqs. (5) and (7) simultaneously. If\ntl≤tp≤tu, the magnetization relaxes to the equilibrium di-\nrection at (θ,φ)=(π/2,−π/2) after the pulse to complete the\nswitching. We can obtain the approximate expressions of tl\nandtuas follows. Assuming that the pulse field is much larger\nthan the other fields, the angular velocity of the precession ,ω,\nis approximated as γHp/(1+α2), and tlandtuare analytically\nobtained as\ntl=π−2θturn\nω−1\n2∆θ\nω, (8)\nand\ntu=π−2θturn\nω+1\n2∆θ\nω, (9)\nwhere∆θ=θu−θl, andθturnis the polar angle at the turning\npoint (φ=0) indicated by the yellow triangle.3. Results and Discussion\nIn this section we discuss the dependence of the BSR on\nthe material parameters by analyzing the numerical simula-\ntion results and Eqs. (8) and (9). The results for the variati on\nof the magnetic anisotropy constant, K, saturation magnetiza-\ntion, Ms, and the Gilbert damping constant, α, will be given\nin Secs. 3.1, 3.2, and 3.3, respectively.\n3.1 Anisotropy Constant Dependence of the BSR\nFigure 3(a) shows the anisotropy constant, K, dependence\nof the BSR. The parameters are Hp=5 T,µ0Ms=0.92 T, and\nα=0.023. The simulation results of tlandtuare indicated\nby the orange and blue dots, respectively. The analytical ap -\nproximations of tlandtuobtained by solving Eqs. (5),(7),(8),\nand (9) are represented by the orange and blue curves, respec -\ntively. The simulation and analytical results agree well wi th\neach other because the Gilbert damping constant is as small a s\n0.023. As shown in Fig. 3(a), tlis a monotonically decreasing\nfunction of Kwhile tuis a monotonically increasing function\nofK. As a result the width of the BSR, tu-tl, is a monoton-\nically increasing function of Kas shown in the inset of Fig.\n3(a).\nIn the left panel of Fig. 3(b) the separatrix and the trajecto ry\nwithα=0 for K=2.3 kJ/m3are shown by the blue and\nblack curves, respectively. The same plot for K=9.3 kJ/m3\nis shown in the right panel. As shown in these panels, the\nincrease of Kdoes not change the trajectory much. However,\nthe increase of Kchanges the separatrix significantly through\nthe second term of Eq. (5). Assuming that the angular velocit y\nof the precession is almost constant, the spread of the area\nsurrounded by the separatrix results in the spread of the tim e\ndifference between tlandtu. As a result the BSR is spread by\nthe increase of Kas shown in Fig. 3(a)\n3.2 Saturation Magnetization Dependence of the BSR\nFigure 4(a) shows the saturation magnetization dependence\nof the BSR obtained by the numerical simulation and the ana-\nlytical approximations. The horizontal axis represents th e sat-\nuration magnetization in unit of T, i.e µ0Ms. The parameters\nareHp=5 T,K=2.3 kJ/m3, andα=0.023. The symbols are\nthe same as in Fig. 3(a). The lower boundary of the BSR, tl,\nincreases as theµ0Msincreases while the upper boundary of\nthe BSR, tu, decreases with increase of µ0Ms. Therefore, the\nfaster switching is available for smaller Ms. Theµ0Msdepen-\ndence of the BSR ( tu-tl) is also shown in the inset of Fig. 4(a).\nThe BSR decreases with increase of µ0Ms. In other words, the\nwider BSR is obtained for smaller Ms.\nIn the right panel of Fig. 4(b) the separatrix and the trajec-\ntory withα=0 forµ0Ms=0.35 T are shown by the blue and\nblack curves, respectively. The same plot for µ0Ms=0.92 T is\nshown in the left panel. As shown in these panels, the increas e\nofMsdoes not change the trajectory much but decrease the\nseparatrix significantly through the first term of Eq. (5). As -\nsuming that the angular velocity of the precession is almost\nconstant, the reduction of the area surrounded by the separa -\ntrix results in the reduction of the time di fference between tl\nandtu. As a result the BSR decreases with increase of Msas\nshown in Fig. 4(a)\n3J. Phys. Soc. Jpn. FULL PAPERStl, t u [ps] \n0 10 20 30 40 2.03.04.05.0\nK [kJ/m3]ballistic switching region (a)\ntltu\ntu - t l [ps] \nK [kJ/m 3]0 10 20 30 40 0.0 0.5 1.0 1.5 2.0 2.5 \n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ(b) K = 2.3 kJ/m3K = 9.3 kJ/m3\ntltltutu\nFig. 3. (Color online) (a) Anisotropy constant, K, dependence of the BSR\n(orange shade). Simulation results of tlandtuare plotted by the orange and\nblue dots, respectively. The analytical results are indica ted by the solid curves\nwith the same color. The parameters are Hp=5 T,µ0Ms=0.92 T, andα=\n0.023. In the inset the simulation and analytical results of the width of the\nBSR, tu-tl, are plotted by the dots and the solid curve, respectively. ( b)\nTypical examples of the trajectory of the magnetization (bl ack curve) and the\nseparatrix (blue curve). The left and right panels show the r esults for K=2.3\nkJ/m3andK=9.3 kJ/m3, respectively. The orange and blue stars indicate\nthe direction at t=tlandtu, respectively. The green dots indicate the initial\ndirection of m.\n3.3 Gilbert Damping Constant Dependence of the BSR\nFigure 5(a) shows the simulation results of the Gilbert\ndamping constant, α, dependence of the BSR. The width of\nthe BSR is shown in the inset. The symbols are the same as\nin Fig. 3(a). The approximate values obtained by Eqs. (8) and\n(9) are not shown because the αis not limited toα≪1. The\nparameters are Hp=5 T, K=2.3 kJ/m3, andµ0Ms=0.92\nT. There exists an optimal value of αthat minimizes tl. The\noptimum value in Fig. 5 (a) is αopt=0.35.\nTo understand the mechanism for minimization of tlat a\ncertain value ofαone need to consider two di fferent effects of\nαon the magnetization dynamics. One e ffect is the decrease\nof the precession angular velocity with increase of α. The pre-\ncession angular velocity around the e ffective field of Heffis\ngiven by (γHeff)/(1+α2), which decreases with increase of α.\nThis effect causes the increase of tlandtu.\nThe other effect is the increase of the energy dissipation rate\nwith increase ofα. Let us consider the trajectory in the cases\nof small damping ( α=0.023) and large damping ( α=αopt).\nIn Fig. 5 (b) the typical examples of the trajectory for the\nsmall damping are shown by the yellow and green curves\nand dots on the energy density contour. The pulse widths are\ntp=tl(=3.15 ps) and 3.14 ps. The trajectories during the\npulse are represented by the solid curves and the trajectori es\nafter the pulse are represented by the dots. The white curve\nshows the separatrix and the black dot indicates the initial di-\ntl, t u [ps] \n2.03.04.05.0\n0.0 0.3 0.6 0.9 1.2\nμ0Ms [T]ballistic switching region \ntltu(a)\n(b) μ0Ms = 0.92 T μ0Ms = 0.35 T\n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ\ntl tltu tu1.5tu - t l [ps] \n0123\n0.0 0.3 0.6 0.9 1.2 \nμ0Ms [T] 1.5 \nFig. 4. (Color online) (a) Saturation magnetization dependence of the\nBSR. The horizontal axis represents the saturation magneti zation in unit of T,\ni.eµ0Ms. The parameters are Hp=5 T, K=2.3 kJ/m3, andα=0.023. The\nsymbols are the same as in Fig. 3 (a). (b) Typical examples of t he trajectory\nof the magnetization (black curve) and the separatrix (blue curve). The right\nand left panels show the results for µ0Ms=0.35 T and 0.92 T, respectively.\nThe symbols are the same as in Fig. 3 (b).\nrection. The yellow and green stars indicate the points wher e\nthe trajectories cross the separatrix surrounding the targ et and\ninitial states, respectively. The arrows indicate the dire ction\nof the movement of the magnetization. For the small damp-\ning, even very close to the separatrix around the target stat e at\nthe end of the pulse, the magnetization flows to the sepatrari x\naround the initial state and relax to the initial state after many\nprecessions with the slow energy dissipation.\nFigure 5 (c) shows the tpdependence of tSWat the large\ndamping (α=αopt). All parameters except αare the same\nas in Fig. 1 (d). t′\nl,tl, and tuare 0.82 ps, 1.98 ps, and 4.54\nps, respectively. t′\nlis the time when for the large damping the\nmagnetization goes across the e ffective separatrix around the\ninitial state during the pulse duration. In Fig. 5 (d) the typ ical\nexamples of the trajectory for the large damping are shown\nby the yellow ( tp=0.9 ps), green ( tp=tl=1.98 ps), and\npurple ( tp=4.55 ps) curves and dots on the energy density\ncontour. The symbols are the same as in Fig. 5 (b). In the\nregion 1 ( tp<t′\nl) of Fig. 5 (c) the magnetization is located on\nthe inside of the effective separatrix at the end of the pulse and\nreturn to the initial state. The trajectory is not shown in Fi g. 5\n(d). In the region 2 ( t′\nl≤tp<tl) of Fig. 5 (c) after the pulse is\nremoved the magnetization move toward the target state unde r\nthe effective field of Heffand goes across the separatrix. Then\nthe magnetization finishes the switching. The typical examp le\nof the trajectory is shown by the yellow curve and dots in Fig.\n5 (d). In the region 3 ( tl≤tp≤tu) after the pulse is removed\nthe magnetization ballistically switches. The typical exa mple\n4J. Phys. Soc. Jpn. FULL PAPERS\ntl, t u [ps] \n2.04.06.08.0\n0.0 0.2 0.4 0.6 0.8\nαballistic switching region \ntltu(a)\n0\ntu - t l [ps] \n0.00123\n0.2 0.4 0.6 0.8 \nα45\nαopt \nαopt\n01 5 4 3 26E/K \nα = 0.023 \ntp = t l = 3.15 ps \ntp = 3.14 ps (b)\n0 -π/2 π/2 π -π0π/2 π\nφθ\n< <<\n< <\nˑ\nˑ\n(c)\ntp [ps] 0 1 2 3 4 5tSW  [ps] \n110 10 210 3\ntl tu1 2 4 3\ntl’\n01 5 4 3 26E/K \nα = 0.35 \ntp = 0.9 ps \ntp = t l = 1.98 ps \ntp = 4.55 ps (d)\n0 -π/2 π/2 π -π0π/2 π\nφθ\n<\n<\n<ˑˑ <\n< <\nˑ\nFig. 5. (Color online) (a) Simulation results of the tlandtuas a function\nofα. The parameters are Hp=5.0 T, K=2.3 kJ/m3, andµ0Ms=0.92\nT. The symbols are the same as in Fig. 3 (a). (b) The trajectori es atα=\n0.023 with tp=3.15 ps (yellow) and 3.14 ps (green) on the energy density\ncontour. The trajectory during the pulse is represented by t he solid curve. The\ntrajectory after the pulse is represented by the dots. The wh ite curve shows\nthe separatrix.The direction of the trajectory is indicate d by the arrow. The\nstar indicates the intersection point of the trajectory and the separatrix. The\ninitial direction is indicated by the black dot. (c) tpdependence of tSWat\nα=αopt. All parameters except αare the same as in Fig. 1 (d). t′\nl,tl, and tu\nare 0.82 ps, 1.98 ps, and 4.54 ps, respectively. tSWattl≤tp≤tuis 1.98 ps.\n(d) The trajectories at α=αoptwith tp=0.9 ps (yellow), 1.98 ps (green), and\n4.55 ps (purple) on the energy density contour. The symbols a re the same as\nin Fig. 5 (b).\nof the trajectory is shown by the green curve and dots in Fig.5 (d). As explained in Sec. 2.1, in this study the tSWis defined\nas the time when mychanges the sign, in other words, the\nmagnetization reaches the turning point ( φ=0) on the energy\ndensity contour. Therefore the tlfor the large damping is equal\nto the ballistic tSW, i.e. the time when under the pulse field the\nmagnetization reaches the turning point. The ballistic tSWat\nthe region 3 is 1.98 ps.\nAs described above, for the large damping the magnetiza-\ntion can relax to the target state even at greater distances f rom\nthe separatrix by moving under Heffwith the fast energy dis-\nsipation. The effect can be regarded as the e ffective spread\nof the separatrix and results in the decrease of tland the in-\ncrease of tuwith increase ofα. Therefore, there exists the\noptimal value ofαthat minimizes tlwhile tumonotonically\nincreases with increase of αas shown in Fig. 5(a). In the re-\ngion 4 ( tp>tu) after the pulse is removed the magnetization\nmoves toward the separatrix around the initial state under Heff\nand relaxes to the initial state. We find that the BSR for the\nlarge damping can be explained by the anisotropic spread of\nthe effective separatrix with increasing α, which is fundamen-\ntally due to the breaking of the spatial inversion symmetry o f\nthe spin dynamics. The broken symmetry of the spatial inver-\nsion of the spin dynamics for the large damping can be easily\nconfirmed by comparing Fig. 5 (c) with Fig. 1 (d).\n4. Summary\nIn summary, we study the material parameter dependence\nof the ballistic switching region of the in-plane magnetize d\nnanomagnets based on the macrospin model. The results show\nthat the pulse width required for the ballistic switching ca n be\nreduced by increasing the magnetic anisotropy constant or b y\ndecreasing the saturation magnetization. The results also re-\nvealed that there exists an optimal value of the Gilbert damp -\ning constant that minimizes the pulse width required for the\nballistic switching. The simulation results are explained by\nanalyzing the trajectories on the energy contour. The resul ts\nare useful for further development of the high-speed inform a-\ntion processing using the ballistic switching of magnetiza tion.\nThis work is partially supported by JSPS KAKENHI Grant\nNumber JP20K05313.\n1) L. He and W. D. Doyle: J. Appl. Phys. 79(1996) 6489.\n2) Z. Z. Sun and X. R. Wang: Phys. Rev. B 71(2005) 174430.\n3) D. Xiao, M. Tsoi, and Q. Niu: Journal of Applied Physics 99(2006)\n013903.\n4) Y . Nozaki and K. Matsuyama: Jpn. J. Appl. Phys. 45(2006) L758.\n5) Y . Nozaki and K. Matsuyama: Journal of Applied Physics 100(2006)\n053911.\n6) Q. F. Xiao, B. C. Choi, J. Rudge, Y . K. Hong, and G. Donohoe: J ournal\nof Applied Physics 101(2007) 024306.\n7) P. P. Horley, V . R. Vieira, P. M. Gorley, V . K. Dugaev, J. Ber akdar, and\nJ. Barna´ s: Journal of Magnetism and Magnetic Materials 322(2010)\n1373.\n8) Y . B. Bazaliy: Journal of Applied Physics 110(2011) 063920.\n9) T. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ ar, and T . Rasing:\nNature 418(2002) 509.\n10) I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann , J. St¨ ohr,\nG. Ju, B. Lu, and D. Weller: Nature 428(2004) 831.\n11) H. W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas, and J. Miltat:\nPhys. Rev. Lett. 90(2003) 017204.\n12) W. K. Hiebert, L. Lagae, J. Das, J. Bekaert, R. Wirix-Spee tjens, and\nJ. De Boeck: Journal of Applied Physics 93(2003) 6906.\n13) W. K. Hiebert, L. Lagae, and J. De Boeck: Phys. Rev. B 68(2003)\n5J. Phys. Soc. Jpn. FULL PAPERS\n020402.\n14) H. W. Schumacher: Appl. Phys. Lett. 87(2005) 042504.\n15) N. Kikuchi, Y . Suyama, S. Okamoto, O. Kitakami, and T. Shi matsu:\nAppl. Phys. Lett. 104(2014) 112409.\n16) K. Neeraj, M. Pancaldi, V . Scalera, S. Perna, M. d’Aquino , C. Serpico,\nand S. Bonetti: Phys. Rev. B 105(2022) 054415.17) K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Zhou Hagstr ¨ om, S. S.\nP. K. Arekapudi, A. Semisalova, K. Lenz, B. Green, J.-C. Dein ert,\nI. Ilyakov, M. Chen, M. Bawatna, V . Scalera, M. d’Aquino, C. S erpico,\nO. Hellwig, J.-E. Wegrowe, M. Gensch, and S. Bonetti: Nat. Ph ys.17\n(2021) 245.\n18) W. F. Brown: Phys. Rev. 130(1963) 1677.\n6"    },    {        "title": "2305.13564v1.Current_driven_motion_of_magnetic_topological_defects_in_ferromagnetic_superconductors.pdf",        "content": "Current-driven motion of magnetic topological defects in ferromagnetic\nsuperconductors\nSe Kwon Kim1,∗and Suk Bum Chung2, 3,†\n1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea\n2Department of Physics and Natural Science Research Institute,\nUniversity of Seoul, Seoul 02504, Republic of Korea\n3School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea\n(Dated: May 24, 2023)\nRecent years have seen a number of instances where magnetism and superconductivity intrinsically\ncoexist. Our focus is on the case where spin-triplet superconductivity arises out of ferromagnetism,\nand we make a hydrodynamic analysis of the effect of a charge supercurrent on magnetic topological\ndefects like domain walls and merons. We find that the emergent electromagnetic field that arises out\nof the superconducting order parameter provides a description for not only the physical quantities\nsuch as the local energy flux density and the interaction between current and defects but also\nthe energy dissipation through magnetic dynamics of the Gilbert damping, which becomes more\nprominent compared to the normal state as superconductivity attenuates the energy dissipation\nthrough the charge sector. In particular, we reveal that the current-induced dynamics of domain\nwalls and merons in the presence of the Gilbert damping give rise to the nonsingular 4 πand 2 π\nphase slips, respectively, revealing the intertwined dynamics of spin and charge degrees of freedom\nin ferromagnetic superconductors.\nI. INTRODUCTION\nWhile magnetism has traditionally been regarded as\ninimical to superconductivity, recent years have seen ob-\nservation of ferromagnetism and superconductivity co-\nexisting or cooperating in varieties of materials which\nincludes uranium heavy-fermion compounds [1–3] and\ntwo-dimensional moir´ e materials such as twisted bilayer\ngraphene [4–6]. It has been known that such coexistence\ncan be naturally accommodated by the Cooper pairing\nof spin-polarized electrons [7]. In such cases, it is natural\nto question what effect, if any, ferromagnetism may have\non superconductivity and vice versa.\nIt is well established in magnetism and spintronics\nthat the current-induced motions of spin textures such\nas domain walls in magnetic metals give rise to the spin\nand energy dissipation into the baths of quasiparticles or\nphonons, commonly known as the Gilbert damping [8, 9].\nThe conservation of energy dictates that the dissipated\nenergy should be externally supplied by the input power.\nIn the case of normal metals, however, resistivity-induced\nenergy dissipation is present regardless of the presence\nor the absence of any spin textures. Hence the Gilbert\ndamping gives rise to only an additional term in the en-\nergy dissipation and, in this sense, its presence can be\ndifficult to confirm solely through charge transport.\nCharge transport detection of the Gilbert damping in\nferromagnetic superconductors may be more straightfor-\nward despite involving a feature unconventional for su-\nperconductors. To maintain a steady-state motion of spin\ntextures in the presence of the Gilbert damping, a fer-\nromagnetic superconductor needs the finite input power\n∗sekwonkim@kaist.ac.kr\n†sbchung0@uos.ac.kr\n(a)(b)\nFIG. 1. (a) The illustration of the mutually orthogonal unit\nvectors ˆs,ˆu, and ˆvthat describe the directional degrees of\nfreedom of the order parameter of a ferromagnetic supercon-\nductor. (b) The configuration of the triad {ˆs,ˆu,ˆv}for a do-\nmain wall in a ferromagnetic superconductor with easy-axis\nspin anisotropy along the zdirection.\nthat goes out of the superconductor solely in the form of\nthe Gilbert damping. This indicates voltage arising in-\nside the superconductor in the direction of the current by\nthe dynamics of spin textures. The mechanisms by which\na superconductor acquires a finite voltage difference be-\ntween two points is referred to as phase slips [10, 11].\nIn conventional superconductors, these phase slips gen-\nerally accompany the singularities, i.e.the vanishing of\nthe order parameter at a certain time during the phase\nslips.\nIn this paper, we show that this is not necessar-\nily the case for ferromagnetic superconductors by us-\ning the concrete example of the current-induced mo-\ntions of two types of magnetic defects, domain walls and\nmerons, which are schematically illustrated in Fig. 1(b)\nand Fig. 3, respectively. To this end, we begin by ex-\namining the order parameter of the spin-polarized super-\nconductor and show how the Cooper pair spin rotation\naround the spin polarization direction is actually equiv-\nalent to the twisting of the overall phase. This gives rise\nto a channel for the interaction between ferromagnetism\nand superconductivity, namely the coupling of CooperarXiv:2305.13564v1  [cond-mat.supr-con]  23 May 20232\npairs to the effective gauge field arising from spin tex-\nture [7, 12, 13].\nWe then proceed to show how such formalism can be\nused to obtain the current-induced motion of topologi-\ncal spin defects such as a domain wall and a meron in\npresence of a background superflow. First, for a domain\nwall, we show that the current-induced motion of do-\nmain walls in the presence of the Gilbert damping ac-\ncompanies the precessional dynamics of the local spin\npolarization and this in turn gives rise to the nonsingu-\nlar 4πphase slips through the generation of an emergent\nelectric field. The induced phase slip opens a channel\nthrough which the ferromagnetic superconductor can ac-\nquire input power, which is shown to be dissipated by the\nspin dynamics entirely via the Gilbert damping. Also,\na current-induced motion of a meron is shown to give\nrise to the nonsingular 2 πphase slips perpendicular to\nits motion, engendering a channel for the input power\nthat is dissipated via the Gilbert damping. The gener-\nation of the 2 πphase slips can be understood from the\nemergent electromagnetic field associated with the meron\ndynamics. For ferromagnetic metals, the emergent elec-\ntromagnetic fields associated with spin textures and their\ndynamics have been discussed theoretically [14–17] and\nconfirmed experimentally [18–21]. However, their man-\nifestations in the dynamics of magnetic defects in fer-\nromagnetic superconductors and the resultant nonsingu-\nlar phase slips have not been discussed yet. Our work\nreveals that the current-induced dynamics of magnetic\ndefects exemplify the intertwined dynamics of spin and\ncharge degrees of freedom in ferromagnetic superconduc-\ntors, where the emergent electromagnetic fields play cru-\ncial roles.\nThe paper is organized as follows. The general formal-\nism for the order parameter and its dynamics of ferro-\nmagnetic superconductors is developed phenomenologi-\ncally in Sec. II. The current-induced dynamics of a do-\nmain wall and its relation to the nonsingular 4 πphase\nslips are discussed in Sec. III. Section IV concerns the\ncurrent-induced dynamics of a meron and its relation to\nthe nonsingular 2 πphase slips. We conclude the paper\nin Sec. V with discussions.\nII. GENERAL FORMALISM\nA. Order parameter\nThe order parameter of a fully spin-polarized triplet su-\nperconductor provides a starting point for understanding\nhow superconductivity and magnetism are intertwined\nthrough the emergent gauge field. In the d-vector for-\nmalism defined by i(d·σσy)s,s′≡∆s,s′, it is given\nby [7, 12, 13]\nd=√ρ\n2eiϕ(ˆu+iˆv) =rρ\n2eiϕ(ˆu+iˆv)√\n2≡rρ\n2ˆd,(1)where ρ= 2d∗·dis the number density of the Cooper\npairs, ˆuandˆvare perpendicular unit vectors, and ˆd∗·ˆd=\n1; the simplest example would be ˆu=ˆx,ˆv=ˆywhich\ngives ∆ s,s′= 0 except for ∆ ↑↑(see Appendix A for the\ndetails). There is an ambiguity here in defining ϕas\nthe above order parameter remains invariant under the\nfollowing simultaneous change of ϕandˆuandˆv:\neiϕ(ˆu+iˆv) =ei(ϕ+δϕ)\u0002\ne−iδϕ(ˆu+iˆv)\u0003\n≡ei(ϕ+δϕ)(ˆu′+iˆv′),\n(2)\nwhere ˆu′,ˆv′are obtained by rotating ˆu,ˆvby +δϕaround\nˆu׈v. As the spin density in units of ℏcan be written\nas\ns= 2id×d∗=ρˆu׈v≡ρˆs, (3)\nEq. (2) denotes the U(1) ϕ+sorder parameter redundancy\n[7, 12], i.e.the invariance of the order parameter when\nthe angle of the spin rotation around ˆsequals the change\ninϕ. Such redundancy implies the existence of an effec-\ntive gauge field arising from the spin degrees of freedom.\nSee Fig. 1(a) for the illustration of the three mutually\northogonal unit vectors ˆs,ˆu, and ˆv, which are depicted\nby red, green, and blue arrows, respectively.\nFor deriving the vector potential and magnetostatics\nof this effective gauge field, the above order parameter\nsuffices. From the spin rotation angle around ˆsdefined\nasα, the effective vector gauge can be written as [14]\nai≡ℏ\nq∂iα=ℏ\nqˆs·(ˆu×∂iˆu) ; (4)\nhence the emergent gauge is a direct consequence of the\nU(1) ϕ+sorder parameter redundancy of Eq. (2). Indeed,\nthe emergent gauge field of spatial curvature in the chi-\nral superconductor has been attributed to the analogous\norder parameter redundancy there [22–25]. Here, while\nwe have kept the charge, q, generic, q=−2e <0 holds in\nsuperconductors. From this emergent vector potential, it\nis straightforward to obtain the emergent magnetic field,\nbi=ϵijk∂jak=−ℏϵijk\n2qˆs·(∂jˆs×∂kˆs) ; (5)\nnote that this is in the same form as the well-known\nMermin-Ho relation between the orbital angular momen-\ntum texture and superfluid velocity in the 3He-A super-\nfluid [26, 27]. Yet this discussion does not include any\ndynamics, for which we shall adopt a two-step approach\nof first formulating the simplest free energy for the order\nparameter [Eq. (2)] and then use its Lagrangian to obtain\nthe equations of motion.\nB. Free energy\nGiven that we seek results relevant to wide-ranging su-\nperconductors whose common attributes may not extend\nbeyond the spin-polarized Cooper pairing [1–6] we will\nconsider for our free energy the simplest minimal model3\nthat includes the spin anisotropy and the Zeeman cou-\npling:\nF[d] =Z\ndVF′\n0[d] +Z\ndVU\n2\u0000\n2|d|2−ρ0\u00012,(6)\nwhere\nF′\n0=A′\nd\n2|(∇−iq\nℏA)d|2+ρ\u0014A′\ns\n2|∇ˆs|2−D\n2(ˆs·ˆz)2−Hsz+qV\u0015\n,\nwhere A′\nsrepresents the excess spin stiffness; note that,\nin contrast to previous analysis [12], our treatment will\nencompass both the easy-axis anisotropy D > 0 and the\neasy-plane anisotropy D < 0. As we will focus on the\ncases where the fluctuation of the condensate density ρ≡\n2|d|2is strongly suppressed, it is convenient to separatelygroup together terms dependent on ρfluctuations [12, 13]\nF=Z\ndV ρF0+Z\ndV\u0014A′\nd\n16ρ(∇ρ)2+U\n2(ρ−ρ0)2\u0015\n,\nwhere\nF0=A′\nd\n2|(∇−iq\nℏA)ˆd|2+A′\ns\n2|∇ˆs|2−D\n2(ˆs·ˆz)2−Hsz+qV\nis the free energy density per unit density. The gauge\ntransformation is implemented as\nA7→A+∇Λ,d7→eiqΛ/ℏd.\nThe free energy can be recast into the following form:\nF=Z\ndV ρ\u001aAc\n2h\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)i2\n+As\n2(∂iˆs)2−D\n2(ˆs·ˆz)2−Hsz+qV\u001b\n+Z\ndV\u0014Ac\n8ρ(∇ρ)2+U\n2(ρ−ρ0)2\u0015\n,\n(7)\nwhere Ac=A′\ndandAs=A′\ns+A′\nd/2. Here, Acand\nAsrepresent the charge stiffness and the spin stiffness,\nrespectively. The similar expression without the sec-\nond and the third terms can be found in Eq. (2.3) and\nEq. (2.5) of Ref. [28].\nAccordingly, the charge supercurrent density is modi-\nfied to\nJi=−δF\nδAi=q\nℏρAc(∂iϕ−q\nℏAi−q\nℏai), (8)\nwith the velocity field is given by\nvi=Ji\nqρ=Ac\nℏ(∂iϕ−q\nℏAi−q\nℏai). (9)\nIt satisfies the following equation (by assuming a nonsin-\ngular θ):\n∇×v=−qAc\nℏ2(B+b). (10)\nThe free energy formalism provides a convenient\nspringboard for extending our analysis to dynamics as\nwell. In particular, such analysis helps us understand\nhow emergent electric field would arise, in analogy with\nthe standard electrodynamics. This is accomplished by\nconsidering the Langrangian for this minimal model.\nC. Equations of motion\nFor the dynamics analysis, we now obtain the classical\nequation of motion for both the charge and the spin com-ponent of the order parameter through considering the\nLagrangian of the spin-polarized superconductor. This\ncan be written as\nL=Z\ndV2iℏd∗·∂td−F\n=−Z\ndVℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]−F . (11)\nThe first term of the above Lagrangian arises from 2 iℏd∗\nbeing the conjugate variable to d; the detailed derivation\nof its relation to ρ, ϕ,ˆscan be found in Appendix B.\nThe low-energy dynamics of the order parameter can be\ndescribed by the three Euler-Lagrange equations for ϕ, ρ,\nandˆs.\nThe equations for ρandϕare basically analogous to\nthose of the conventional superconductors. The equation\nof motion for the density ρ,\n˙ρ=−1\nℏ∂in\nρAch\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)io\n=−1\nq∂iJi,\n=−∇·(ρv) (12)\nis obtained from δL/δϕ = 0 and is none other than the\ncontinuity equation for the Cooper pair density. Simi-\nlarly, the equation of motion for the phase ϕ\n−ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =F0+U(ρ−ρ0)−Ac\n4ρ∇2ρ(13)\nobtained from δL/δρ = 0 (where only terms constant or\nlinear in ρare retained) comes out to be the Josephson\nrelation. We, however, want to obtain a hydrodynamic\nequation of motion for Cooper pairs, for which purpose\nwe take the spatial derivative of the Josephson relation:4\n−ℏh\n∂t∂iϕ−q\nℏ(Ei+ei)−q\nℏ∂t(Ai+ai)i\n=ℏ2\nAcv·∂iv+∂i\u0014\nρ′\ne+U(ρ−ρ0)−Ac\n4ρ∇2ρ\u0015\n,\nwhere\nei=−ℏ\nqˆs·(∂iˆs×∂tˆs) (14)\nis the emergent electric field and\nρ′\ne=As(∂iˆs)2\n2−Ds2\nz\n2−Hsz (15)\nis the magnetic energy density (per unit density). By using\nv·∂iv=1\n2∂i(v2) = (v·∇)vi+ϵijkvj(∇×v)k= (v·∇)vi−qAc\nℏ2ϵijkvj(Bk+bk)\nand defining the material derivative Dt≡∂t+v·∇and the effective mass of a Cooper pair, m≡ℏ2/Ac, we obtain\nmDtv=q(E+e) +qv×(B+b)−∂i\u0014\nρ′\ne+U(ρ−ρ0)−Ac\n4ρ∇2ρ\u0015\n. (16)\nThe novelty in the ferromagnetic superconductor is the\nequation of motion for the spin direction ˆsthat is derived\nfrom δL/δˆs= 0 (see Appendix C for details):\nℏρ∂tˆs=−ℏJi\nq∂iˆs+∂i[ρAs(ˆs×∂iˆs)]+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z,\n(17)\nwhich is identical to the Landau-Lifshitz equation [29]\naugmented by the adiabatic spin-transfer torque [9, 30,\n31]. This can be also written as\nℏρDtˆs=∂i[ρAs(ˆs×∂iˆs)] +ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z.\nBy using ˙ ρ=−∂iJi/q, we can obtain the spin continuity\nequation:\n∂t(ℏρˆs) =−∂iJs\ni+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z,(18)\nwhere\nJs\ni=ℏJi\nqˆs−ρAs(ˆs×∂iˆs),\nis the spin current density. The first term and the second\nterm on the right-hand side are longitudinal spin currents\nproportional to the charge current and the transverse\nspin current that is carried by a spin texture, respec-\ntively.\nA complete set of equations describing the hydrody-\nnamics of a ferromagnetic superconductor in the absence\nof external fields ( E= 0 and B= 0) can now be given;\nthe analogous equations have been written down for a\nspinor BEC [32]. It is convenient to measure energy in\nthe unit of the anisotropy energy absolute value |D|andlength in the unit derived from the combination of |D|\nwith the spin stiffness As,i.e.\nl=s\nAs\n|D|, ϵ=|D|.\nAlso, we will use ˜ ρ≡ρ/ρ0, Uρ/D ≡η. This gives us the\ndimensionless equations\n−Dt˜ρ= ˜ρ(∇·v),\n˜m(∇×v) =−b,\n˜mDtv=e+v×b\n−∂i[ρe+η(˜ρ−1)− ∇2˜ρ/4],\n˜ρDtˆs=∂i[˜ρ(ˆs×∂iˆs)] + ˜ρν(ˆs·ˆz)ˆs׈z+ ˜ρhˆs׈z,\n(19)\nwhere ν≡sgn(D), ˜m≡As/Acthe dimensionless mass\nwhich is on the order of unity, h≡H/D the dimension-\nless external field, and\nρe=1\n2(∂iˆs)2−ν1\n2(ˆs·ˆz)2−hsz\nthe dimensionless magnetic energy density; for zero ex-\ncess spin stiffness ˜ m= 1/2. The emergent electromag-\nnetic fields are now re-defined as\nei=−ˆs·(∂iˆs×∂tˆs), b i=−ϵijk\n2ˆs·(∂jˆs×∂kˆs),\nwhere the charge qis absorbed into the fields.5\nD. Gilbert damping\nDue to the inevitable nonconservation of spin angu-\nlar momentum in solids, it is reasonable to expect the\ndamping of spin dynamics and the associated energy dis-\nsipation, which are not included in the hydrodynamics\nequations [Eq. (19)], to play an important role in spin\ndynamics of the ferromagnetic superconductors as in any\nother solid-state systems. The spin sinks can be quasi-\nparticles, phonons, and any other excitations that can\npossess angular momentum [33–36]. The spin dissipation\ncan be treated phenomenologically with the addition of\nthe Gilbert damping term α˜ρˆs×∂tˆs[8] to the spin equa-\ntion of motion in Eq. (19),\n˜ρDtˆs+α˜ρˆs×∂tˆs=∂i[˜ρ(ˆs×∂iˆs)]+ ˜ρν(ˆs·ˆz)ˆs׈z+ ˜ρhˆs׈z.\n(20)\nIn the incompressible limit η→ ∞ where ˜ ρis uniform and\nconstant, this gives us the energy continuity equation,\n∂tρe+∇·je=−v·e−α(∂tˆs)2, (21)\nwhere\nje=−∂tˆs·∇ˆs (22)\nis the magnetic energy flux density (per unit density).\nThis continuity equation, which has been previously\nnoted in literature [37, 38], can be derived by taking the\nproduct of both sides of Eq. (20) with ˆs×∂tˆs. One can\nnote that the first term on the right-hand side of Eq. (21)\n(−v·e) is the power dissipated (supplied) by the su-\nperflow vflowing parallel (antiparallel) to the direction\nof the emergent electric field e, while the second term\n(−α(∂tˆs)2) is the energy dissipation through the Gilbert\ndamping.\nEquation (21) implies that, in the incompressible limit,\nthe energy dissipated by the Gilbert damping is equal to\nthe work done by the emergent electric field when spin\ntexture is transported without any distortion. This is\nbecause the total magnetic energy should be unchanged\nin this process and hence the left-hand side of Eq. (21)\nintegrated over the whole system should be zero.\nIII. DOMAIN WALL\nFor the supercurrent-driven motion of topological de-\nfects, we first consider an easy-axis ferromagnetic super-\nconductor with spin-anisotropy sign ν= sgn( D) = +1.\nA domain wall is a generically stable topological defect\nbetween two different ground states for easy-axis spin\nsystems and intrinsically has no skyrmion density, i.e.\nno emergent magnetic field. Therefore for the rest of this\nsection, we will take the Cooper pair velocity to be ir-\nrotational, i.e.∇×v= 0 [Eq. (19)]. In addition, we\nalso set the applied magnetic field to be zero and hence\nh= 0.A. Deriving dynamics from a static solution\nThe solution for a domain-wall motion in the back-\nground of the constant and uniform background super-\nflow can be straightforwardly constructed from the static\ndomain wall solution in absence of any background su-\nperflow for the incompressible limit η→ ∞ . For this\ncase, the absence of the emergent magnetic field allows\nus to consider only the spin equation of motion [40]\n(∂t+v·∇+αˆs×∂t)ˆs=∂i[(ˆs×∂iˆs)] + (ˆs·ˆz)ˆs׈z; (23)\nfrom the hydrodynamic equations Eq. (19); others are\neither irrelevant due to b= 0 or merely provides the\nconstraint ˜ ρ= 1 in the incompressible limit. Given the\nintrinsically quasi-one-dimensional nature of the domain\nwall, we can set the boundary condition\nˆs(x→ ±∞ ) =±ˆz,\nfor any domain-wall configuration. For the static domain\nwall at x= 0 in absence of background superflow, the\nsolution is given by the following Walker ansatz [41]:\nˆs0= (ˆxcosφ0+ˆysinφ0)sech x+Qˆztanhx, (24)\nwhere Q=±1 represents the domain wall type, satisfies\nthe static domain-wall equation\n0 =∂x[(ˆs0×∂xˆs0)] + ( ˆs0·ˆz)ˆs0׈z, (25)\nderived from Eq. (23), for an arbitrary domain-wall angle\nφ0; it is important to note here that Eq. (25) is sufficient\nas Eq. (24) gives us v= 0 and b= 0 everywhere.\nFrom Eq. (25), it can be shown that the general solu-\ntion can be obtained by applying to the static solution\nboth giving boost in the spatial direction and precession\naround the easy-axis:\nˆs= (ˆxcos Ω t+ˆysin Ωt)sech( x−V t) +Qˆztanh( x−V t).\n(26)\nIn deriving the domain-wall velocity ˆxVand the preces-\nsion rate Ω, it is convenient to note that Eq. (26) also\nsatisfies Eq. (25) as the latter equation involves no time\nderivatives. Also, as Eq. (26) is obtained from boost and\nprecession,\n∂tˆs= (−V ∂x+ˆzΩ×)ˆs.\nThe velocity Vand the precession rate Ω therefore can\nbe obtained from\n[v∂x+ (1 + αˆs×)(−V ∂x+ˆzΩ×)]ˆs= 0, (27)\nwhere we set the background superflow to be perpendic-\nular to the domain wall without any loss of generality,\nv=vˆx. By taking the scalar product of the above equa-\ntion with ˆzandˆz׈swe obtain\nv−V=QαΩ,Ω =−QαV6\n(a)<latexit sha1_base64=\"mvV1IC9/vAkApJWzfWyUx3QadJw=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVpiZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBSnJC1</latexit>\n(b)<latexit sha1_base64=\"Lyp4rKMmNObzzWYxIHNaXwltWlk=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVoKzibdQtEtuzPQZeJlpEgy1LqFr04v4kkICrlkxrQ9N0Y/ZRoFlzDJdxIDMeMjNoC2pYqFYPx0dvGEnlqlR/uRtqWQztTfEykLjRmHge0MGQ7NojcV//PaCfYv/VSoOEFQfL6on0iKEZ2+T3tCA0c5toRxLeytlA+ZZhxtSHkbgrf48jJpnJe9SvnqrlKsXmdx5MgxOSEl4pELUiW3pEbqhBNFnskreXOM8+K8Ox/z1hUnmzkif+B8/gBUIpC2</latexit>\n(c)<latexit sha1_base64=\"eT4ibbwfOOWkpDCTAUUKdbMXNmE=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVriZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBVqJC3</latexit>\nFIG. 2. (a) A series of snapshots of a precessing domain wall moving to the right, where xandtare the spatial and the\ntemporal coordinates, respectively. The red arrows, blue arrows, and green arrows represent ˆs,ˆu, and ˆv, respectively. The\ndomain-wall position is denoted by the gray dot. The domain-wall angle, which is the azimuthal angle of ˆsat the center of the\ndomain wall, changes from φ0= 0 to φ0=−2πgradually with increasing time from bottom to top. On the left end, ˆu(green\narrow) rotates by −2πabout ˆs(red arrow), whereas on the right end, ˆurotates by 2 πabout ˆs. This process of a domain-wall\nprecession can be considered as a nonsingular 4 πphase slip since these opposite 2 πrotations of ˆuaround ˆsat the left and\nthe right ends induces a finite voltage across the wire. (b) Mapping of the instantaneous configuration of the two vectors ˆs\n(red arrows) and ˆu(green arrows) onto the unit sphere with the normal vector identified with ˆs. The yellow line represents\na spatial dimension of the system. (c) Collection of the mapping of the configuration of ˆsandˆuonto the unit sphere with ˆs\nidentified with the normal vector for all the snapshots shown in (a). Note that the unit tangent vector field ˆuis not uniquely\ndetermined at the north and the south poles as dictated by the Poincar´ e-Hopf theorem [39]. Rather, ˆurotates once around\nˆscounterclockwise (counterclockwise) at the north (south) pole as the domain wall completes one cycle of rotation, which is\nconsistent with the Euler number 2 of the sphere. See the main text for further detailed discussions.\nrespectively, giving us\nV=1\n1 +α2v ,Ω =−Qα\n1 +α2v; (28)\nNote that in absence of the Gilbert damping, α= 0, there\nwould have been no precession and the domain wall would\nhave remained static with respect to the background su-\nperflow. See Fig. 2 for the illustration of the domain-wall\ndynamics with precession.\nFrom the above solution, it is straightforward to con-\nfirm that all work done by the emergent electric field is\ndissipated through the Gilbert damping. The work done\nby the emergent electric field is\n−v·e=vjˆs·(∂jˆs×∂tˆs) =QvΩ[1−(ˆs·ˆz)2],(29)\nwhich gives the total energy input of\nW=Z\ndx(−v·e) = 2 QvΩ. (30)\nThe energy dissipation rate per unit density is given by\nα(∂tˆs)2=α(V2+ Ω2)[1−(ˆs·ˆz)2]. (31)\nThis energy dissipation through the spin dynamics and\nthe work rate done by the emergent electric field on the\nsuperflow are the same since\nQvΩ =α\n1 +α2v2(32)\nand\nα(V2+ Ω2) =α\n1 +α2v2. (33)B. 4πphase slips from a domain-wall dynamics\nDue to the U(1) ϕ+sorder parameter redundancy, the\nenergy dissipation from the damping-induced precession\nin the domain-wall motion can be regarded as an equiva-\nlent of 4 πphase slips. As shown in Fig. 2, ˆuandˆvrotate\naround ˆsby±2πby adiabatically following the dynamics\nof the local spin direction ˆsin one cycle of precession. To\nsee this, note that if we adopt the condition ˆv·ˆz= 0 in\ndefining ˆv, Eq. (26) will give us ˆv=−ˆxsin Ωt+ˆycos Ω t.\nBut given the U(1) ϕ+sredundancy, this is equivalent to\nthe±2πphase twist on the left and the right end, re-\nspectively.\nThe voltage arising from this precession can be under-\nstood either as arising from the emergent electric field, or\nequivalently, arising from the constant rate of 4 πphase\nslips. When φincreases at the rate ˙ φ= Ω, the emergent\nscalar potential at the two ends of the wire, x=±∞, is\ngiven by\n˜Ve=−ˆs·(ˆu×∂tˆu) =(\n−QΩ at x=−∞,\nQΩ at x=∞,(34)\nwhich is exactly the Josephson voltage for the 4 Qπphase\nslip occurring at the rate of Ω /2π. Then, to maintain the\nfinite superflow, there must be work given by\nW=v[˜Ve(x=∞)−˜Ve(x=−∞)] = 2 QvΩ,(35)\nby external reservoirs on the system, matching the work7\n[Eq. (30)]. Figure 2(a) shows the time evolution of the\ntriad{ˆs,ˆu,ˆv}associated with the domain-wall that both\nmoves and precesses. Note that ˆu(green arrow) at the\nleft end ( x→ −∞ ) and the right end ( x→ ∞ ) ro-\ntates clockwise and counterclockwise, respectively, about\nˆs(red arrow), engendering the nonsingular phase slips\nacross the xdirection. In a nutshell, the domain-wall an-\ngular dynamics produces the nonsingular 4 πphase slips\nthat give rise to a finite voltage difference between the\ntwo ends of the superconducting wire, which constitutes\nour first main result.\nThere is a topological reason why one precession of a\nmagnetic domain wall induces a 4 πphase slip, which can\nbe derived from the Poincar´ e-Hopf theorem or Poincar´ e-\nBrower theorem [39]. For concrete discussion on this, we\nwill consider in the following the case with Q= 1 as\nshown in Fig. 2(b) and (c). At a given time, the instan-\ntaneous configuration of ˆsandˆucan be mapped onto a\nline connecting the north pole [ ˆs(x→ ∞ )] and the south\npole [ ˆs(x→ −∞ )] on the unit sphere by identifying ˆs\nwith the surface normal as shown in Fig. 2(b). When we\nconsider the collection of the configuration of {ˆs,ˆu}onto\nthe unit sphere during one complete precession of the do-\nmain wall, i.e., φ0→φ0+2π,ˆucan be regarded as a unit\ntangent vector field on the sphere since it is perpendic-\nular to ˆs, i.e., the surface normal as shown in Fig. 2(c).\nHere, note that ˆuis not uniquely determined at the north\npole and the south pole, which is consistent with the well-\nknown topological property of the sphere that the unit\ntangent vector field cannot be defined without a singu-\nlarity on it. Instead of being uniquely determined, ˆu\nrotates by 2 πaround the north pole and rotates by −2π\naround the south pole as the domain wall completes one\ncycle of precession, which gives rise to a 4 πphase slip\nacross the wire as discussed above [Eq. (34)]. This can\nbe understood by applying the Poincar´ e-Hopf theorem to\nthe unit tangent vector field ˆuon the sphere. The Euler\nnumber of the unit sphere is 2, meaning that the sum of\nthe indices of the isolated singularities of the unit tan-\ngent vector field on the sphere must be 2. In our case,\nthe indices of the north pole and the south pole associ-\nated with ˆuare both 1, adding up to 2, agreeing with the\nEuler number of the unit sphere.\nIV. MERON\nWe now consider ferromagnetic superconductors with\neasy-plane spin anisotropy ( ν= sgn( D) =−1). For easy-\nplane spin systems, a meron, with its one-half skyrmion\ncharge, is a generically stable topological defect [42–\n45]. Therefore, we consider the rotational Cooper pair\nvelocity in absence of the applied magnetic field i.e.\n∇×v=−b/˜mwith h= 0. See Fig. 3 for the schematic\nillustration of a meron.\n(a)<latexit sha1_base64=\"mvV1IC9/vAkApJWzfWyUx3QadJw=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVpiZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBSnJC1</latexit>\n(b)<latexit sha1_base64=\"Lyp4rKMmNObzzWYxIHNaXwltWlk=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVoKzibdQtEtuzPQZeJlpEgy1LqFr04v4kkICrlkxrQ9N0Y/ZRoFlzDJdxIDMeMjNoC2pYqFYPx0dvGEnlqlR/uRtqWQztTfEykLjRmHge0MGQ7NojcV//PaCfYv/VSoOEFQfL6on0iKEZ2+T3tCA0c5toRxLeytlA+ZZhxtSHkbgrf48jJpnJe9SvnqrlKsXmdx5MgxOSEl4pELUiW3pEbqhBNFnskreXOM8+K8Ox/z1hUnmzkif+B8/gBUIpC2</latexit>\n(c)<latexit sha1_base64=\"eT4ibbwfOOWkpDCTAUUKdbMXNmE=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVriZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBVqJC3</latexit>\nFIG. 3. (a) An illustration of the meron with polarity p= 1\nand vorticity n= 1, which is a nonsingular topological de-\nfect of ferromagnetic superconductors with easy-plane spin\nanisotropy. The red, the green, and the blue arrows repre-\nsentˆs,ˆu, and ˆv, respectively. The meron core is denoted by\nthe gray dot. The local spin direction ˆsrotates by 2 πcoun-\nterclockwise about the zaxis when we follow the infinitely\ndistant trajectory encircling the meron core counterclockwise\n(and thus the vorticity n= 1). Note that ˆuandˆvalso change\nspatially to keep their orthonormality to ˆs; they rotate by 2 π\nclockwise about the local spin direction ˆswhen we enclose\nthe meron center counterclockwise. (b) Mapping of the con-\nfiguration of ˆs(red arrows) and ˆu(green arrows) along the\ninfinitely distant circle in (a) onto the equator with the surface\nnormal identified with ˆs. (c) Mapping of the configuration of\nˆsand ˆuof the entire system onto the northern hemisphere\nwith ˆsidentified with the surface normal. Note that ˆuis a\nwell-defined unit tangent vector field on the northern hemi-\nsphere without any singularity. Since the Euler number of\nthe hemisphere is 1, if the unit tangent vector field defined on\nthe northern hemisphere has no singularity, it should rotate\naround the surface normal by 2 πalong the equator according\nto the Poincar´ e-Hopf theorem [39], which is exactly what ˆu\n(unit tangent vector) does around ˆs(surface normal).\nA. Static solution\nAnalogous to the case of the domain wall in the previ-\nous section, a straightforward construction of the meron\nmotion solution here in the background of the constant\nand uniform background superflow is possible from the\nstatic meron solution in absence of any background su-\nperflow for the incompressible limit, η→ ∞ [32, 46].\nThe static solution can be obtained from the following8\ntwo equations;\n˜m(∇×v) =−b,\n(v·∇)ˆs=∂i[(ˆs×∂iˆs)]−(ˆs·ˆz)ˆs׈z.(36)\nIt is important to note here that while we are dealing\nwith a static configuration we still have v̸= 0 due to\nthe intrinsic emergent magnetic flux of the meron. In\naddition, this solution would possess the axial symmetry,\ni.e.\nˆs0= (sin θcosφ,sinθsinφ,cosθ), (37)\nwith θ=θ(r) and φ=nχ+ Φ where ( r, χ) are polar\ncoordinates for the two-dimensional system, and follow\nthe universal boundary conditions for merons are given\nby\nθ(r= 0) = (1 −p)π\n2, θ(r→ ∞ ) =π\n2;\np=±1 here is the polarity, which is the z-component of\nthe local spin direction ˆsat the meron center, and n∈Z\nthe vorticity, which counts how many times ˆswinds in\nthe easy plane along the closed trajectory encircling the\nmeron center. For our purpose, obtaining the differential\nequation for θ(r) is sufficient for showing Eq. (37) to be\nthe solution of Eq. (36). We start by noting that the\nemergent magnetic field is aligned entirely along the z-\naxis and\nbz=−ˆs0·(∂xˆs0×∂yˆs0) =−nsinθ\nrdθ\ndr(38)\nis function only for θ, which gives us the well-known re-\nsult\nZ\ndxdyb z=Z\nrdχdr\u0012\n−nsinθ\nrdθ\ndr\u0013\n=−2πpn (39)\nfor the total emergent magnetic flux. With this emergent\nmagnetic field, the first equation of Eq. (36) requires the\ncirculating velocity v=ˆφv(r) around the meron with\n1\nrd(rv)\ndr=nsinθ\n˜mrdθ\ndr.\nInserting this relation into the second equation of\nEq. (36) gives us\nsinθ(1−cosθ)1\n˜mr2=1\nrd\ndr\u0012\nrdθ\ndr\u0013\n+cos θsinθ\u0012\n1−1\nr2\u0013\n,\nfrom which θ(r) can be obtained numerically.\nTo find explicit expressions for ˆuandˆv, note that the\nfollowing three unit vectors form an orthonormal triad:\nˆs0=ˆer≡(sinθcosφ,sinθsinφ,cosθ),\nˆeθ≡(cosθcosφ,cosθsinφ,−sinθ),\nˆeφ≡(−sinφ,cosφ,0),which gives us\nˆu0(r, χ) = cos φ(χ)ˆeθ(r, χ)−sinφ(χ)ˆeφ(r, χ),(40)\nˆv0(r, χ) = sin φ(χ)ˆeθ(r, χ) + cos φ(χ)ˆeφ(r, χ).(41)\nThe local configuration of the triad ( ˆs0,ˆu0,ˆv0) for a\nmeron with p= 1 and n= 1 is shown in Fig. 3. Note\nthat it is nonsingular, differing from a conventional vor-\ntex of a s-wave superconductor [10]. An analogous non-\nsingular topological defects that give rise to 4 πnonsin-\ngular phase slips has been discussed by Anderson and\nToulouse [47] and has been termed the skyrmion solu-\ntion in the more recent literature [28, 32]. By contrast,\nour solution {ˆs0,ˆu0,ˆv0}[Eqs. (37,40,41)] represents an\nexplicit solution for the nonsingular topological defect\nin the easy-plane case that gives rise to 2 πnonsingular\nphase slip, as can be seen from Eqs (36) and (39): it\nharbors the emergent magnetic flux −2πpnand thus its\nmotion gives rise to the emergent electric field, i.e., phase\nslips perpendicular to its motion.\nThe non-trivial emergent gauge field ai, and thus the\nquantized non-zero emergent magnetic flux, of our non-\nsingular topological defect can be understood from the\nrotation of ˆuaround ˆsas stated in Eq. (4). There exists\na topological constraint dictating that a meron texture\nofˆsshould trap a quantized non-zero emergent magnetic\nflux, which can be understood by invoking the Poincar´ e-\nHopf theorem [27, 39] as follows. For the given meron\nconfiguration with p= 1, let us consider the mapping of\ntwo unit vectors ˆsandˆuonto the northern hemisphere\nsuch that ˆsis identified with the surface normal. Then,\nˆubecomes a unit tangent vector field on the hemisphere\nsince it is perpendicular to ˆs, i.e., the surface normal as\nshown in Fig. 3(c). The Euler number of the hemisphere\nis 1, and thus the Poincar´ e-Hopf theorem dictates that, if\nthe unit tangent vector field is nonsingular on the north-\nern hemisphere, it should rotate exactly one time about\nthe surface normal while traversing the equator, which is\nexactly what ˆudoes around ˆsin Fig. 3(b). Therefore, the\none-time rotation of ˆuaround ˆsalong the closed loop that\ncontains, but is also infinitely far from, the meron core\nas shown in Fig. 3(a), which gives rise to the quantized\nemergent magnetic flux, can be regarded as the physical\nmanifestation of the topological constraint that should\nbe satisfied by the nonsingular unit tangent vector field\ndefined on the hemisphere.\nB. Dynamics with a background superflow\nAnalogous to the domain wall motion, it is straight-\nforward to work out the spin equation of motion for the\nmeron motion driven by a uniform constant background\nsuperflow v0if we assume the meron motion to be rigid,\ni.e.ˆs(r, t) =ˆs0(r−Vt) (the same holds for ˆuandˆv). The\nCooper pair velocity would then be given by v=vm+v0,\nwhere vmis the Cooper pair velocity around a static\nmeron. We can therefore employ the collective coordi-\nnate approach to describe the dynamics of a meron, and9\nFIG. 4. Snapshots of a meron moving in the ydirection in the increasing time from (a) to (d), where the meron core is depicted\nby a gray dot. At x→ −∞ ,ˆu(green arrow) rotates about ˆs(red arrow) counterclockwise, whereas at x→ ∞ ,ˆurotates about\nˆsclockwise, inducing phase slips and thereby generating a finite voltage in the xdirection.\nuse the fact that ˆs0(r−Vt) should satisfy the original\nspin equation of motion of Eq. (20) with, as we are in\nthe incompressible limit, the constant ˜ ρwhile ˆs0(r) also\nsatisfies the equation for the static meron of Eq. (36).\nSubtraction between the two equations, together with\n∂tˆs0(r−Vt) =−V·∇ˆs0(r−Vt) gives us\n−(1 +αˆs0×)V·∇ˆs0=−v0·∇ˆs0.\nTaking the scalar product with s0×∂is0on both sides\ngives us\n−Vj[s0·(∂iˆs0×∂jˆs0)]−αVj(∂iˆs0·∂jˆs0)\n=−v0,j[s0·(∂iˆs0×∂jˆs0)].(42)\nStrictly speaking, this result shows that whereas we have\nan exact rigid motion solution with V=v0in absence\nof damping, no rigid motion solution can be exact in\npresence of damping. Yet to the zeroth order in v0and\nalso in the spirit of collective coordinate, we can average\nout the effect of the spin texture, i.e.defining\nGij=Z\ndxdys0·(∂iˆs0×∂jˆs0), D ij=Z\ndxdy∂ iˆs0·∂jˆs0,\nknown respectively as the gyrotropic coefficients and the\ndissipation coefficients [48–50] ( Dij=DδijandGxy=\n−Gyx≡G= 2πpnfor a meron). By integrating Eq. (42),\nwe have\n(αD+Gˆz×)V=Gˆz×v0, (43)\nwhich yields the following solution for the velocity V:\nV=G\nG2+α2D2(G+αDˆz×)v0. (44)\nTo consider a concrete example, we will hereafter restrict\nthe discussion to the case where the background super-\nflow flows in the xdirection: v0= (v0,0). In this case,we have\n\u0012\nVx\nVy\u0013\n=G\nG2+α2D2\u0012\nGv0\n−αDv 0\u0013\n. (45)\nNote that the presence of damping gives rise to the com-\nponent of the meron velocity transverse to the uniform\nsuperflow in proportion to the skyrmion number G/2πof\nthe meron, Vy=−α[GD/(G2+α2D2)]v0, exhibiting the\nso-called skyrmion Hall effect [16, 51–53]. This transverse\nmotion of the meron with respect to the superflow gives\nrise to the finite voltage in the direction of the superflow\nvia the 2 πphase slips, which we turn our attention now.\nC. 2πphase slips from a meron motion\nAgain analogous to the domain wall motion, the\nU(1) ϕ+sorder parameter redundancy allows the energy\ndissipation due to the meron motion as equivalent to the\n2πphase slips. This can be seen from the emergent elec-\ntric field\ne=−ˆs·(∇ˆs×∂tˆs)\n=ˆs0·(∇ˆs0×V·∇ˆs0) =−V×b, (46)\nwhich is in the same form as the Josephson electric field\narising from the vortex motion [10]. The same can natu-\nrally be said about the input power density required for\ndriving the uniform constant background superflow\n−v0·e=v0,xˆs·(∂xˆs×∂tˆs),\n=−v0,xVyˆs·(∂xˆs×∂yˆs).\nIt can be checked explicitly that the total energy rate for\ndriving the superflow\n−Z\ndxdyv0·ˆe=−v0,xVyG=αG2D\nG2+α2D2v2\n0,10\nis equal to the energy dissipation rate\nZ\ndxdyα (∂tˆs)2=Z\ndxdyαV iVj(∂iˆs0·∂jˆs0)\n=αDV2=αG2D\nG2+α2D2v2\n0.\nThis energy must come from external reservoirs\nthrough the boundary of the system, meaning that there\nshould be a development of a finite voltage across the\nsystem in the xdirection. This can be explicitly seen\nfrom\n˜Ve=−ˆs·(ˆu×∂tˆu) = (1 −cosθ)∂tφ . (47)\nTo see how a finite voltage is generated by the motion\nof a vortex, let us assume that a vortex moves in the y\ndirection, V=Vyˆy. Then for a given point at large x,\nZ∞\n−∞dtˆs·(ˆu×∂tˆu) =−Z∞\n−∞dt∂tφ=nπ , forx→+∞.\n(48)\nThe vector ˆu(x→ ∞ ) rotates by nπaround the ˆs. Also,\nZ∞\n−∞dtˆs·(ˆu×∂tˆu) =−Z∞\n−∞dt∂tφ=−nπ , forx→ −∞ .\n(49)\nThe vector ˆu(x→ −∞ ) rotates by −nπaround the\nˆs. This indicates that the motion of a meron in the\nydirection induces a nonsingular 2 πphase slips across\nthexdirection. To keep the superflow in the xdirec-\ntion constant, we need to counteract the effect of these\nphase slips. The corresponding work on the system is\ndissipated to external baths such as quasiparticles or\nphonons through the Gilbert damping. Figure 4 shows\nthe schematic illustration of the process. As the vor-\ntex moves in the positive ydirection from Fig. 4(a) to\nFig. 4(d), ˆuatx→ ∞ andx→ −∞ rotates about the\nlocal spin direction ˆscounterclockwise and clockwise, re-\nspectively, producing phase slips in the xdirection. This\nis our second main result: The dynamics of a meron en-\ngenders the nonsingular 2 πphase slips perpendicular to\nits motion through the generation of the emergent elec-\ntric field, showcasing the intertwined dynamics of spin\nand charge degrees of freedom of ferromagnetic super-\nconductors.\nV. DISCUSSION\nWithin the phenomenological framework for the dy-\nnamics of the order parameter of ferromagnetic super-\nconductors, we have shown that the current-induced dy-\nnamics of magnetic defects in the presence of the spin\ndissipation, i.e., the Gilbert damping, give rise to nonsin-\ngular phase slips via the emergent electromagnetic fields.\nThe input power, which is the product of the applied\ncurrent that drives the magnetic defects and the voltage\ngenerated by the phase slip, is shown to be equivalent tothe dissipated energy in the form the Gilbert damping\ninto the baths of quasiparticles or phonons. Our work on\nthe dynamics of magnetic defects showcases the intrinsic\ninterplay of spin and charge dynamics of ferromagnetic\nsuperconductors.\nA few remarks are on order about the limitations of\nour work. First, we did not include the effects of the\nnon-adiabatic spin-transfer torque by a supercurrent on\nthe dynamics of magnetic defects, which is expected to be\npresent on general grounds whenever the Gilbert damp-\ning is present [9, 40, 54, 55]. While we believe that the\ninclusion of the non-adiabatic spin-transfer torque in our\nmodel would not qualitatively change the relations that\nwe have found between the dynamics of magnetic defects\nand nonsingular phase slips, it will certainly enrich the\nphysics of the interplay of spin and charge dynamics in\nferromagnetic superconductors. Secondly, in this work,\na ferromagnetic superconductor is assumed to be a fully\nspin-polarized triplet superconductor as in Ref. [12], by\nleaving the generalization to a partially spin-polarized\ncase as future work. Thirdly, our analysis shows that\nthe assumption of rigid motion is not exact for merons\nin presence of damping. The coupling between current\nand magnons bound to meron cores may be a relevant\ntopic for future study. Lastly, the dynamics of magnetic\ndefects has been discussed in the incompressible limit,\nwhere the dynamics of the order-parameter amplitude\nis frozen. Releasing this assumption would allow us to\nstudy the interplay of spin dynamics and longitudinal\norder-parameter dynamics, which is beyond the scope of\nthe current work.\nACKNOWLEDGMENTS\nWe thank Mike Stone, Grigori Volovik, Daniel\nAgterberg and Jim Sauls for useful discussions.\nS.K.K. was supported by Brain Pool Plus Program\nthrough the National Research Foundation of Korea\nfunded by the Ministry of Science and ICT (NRF-\n2020H1D3A2A03099291) and by the National Research\nFoundation of Korea funded by the Korea Government\nvia the SRC Center for Quantum Coherence in Con-\ndensed Matter (NRF-RS-2023-00207732). S.B.C. was\nsupported by the National Research Foundation of Korea\n(NRF) grants funded by the Korea government (MSIT)\n(NRF-2023R1A2C1006144, NRF-2020R1A2C1007554,\nand NRF-2018R1A6A1A06024977).11\nAppendix A: Details of the order parameter\nThe multicomponent superconducting gap is given by\nˆ∆ =\u0012\n∆↑↑∆↑↓\n∆↓↑∆↓↓\u0013\n≡\u0012\n−˜dx+i˜dy˜dz\n˜dz˜dx+i˜dy\u0013\n=i(d·σ)σy. (A1)\nThen,\nˆ∆ˆ∆†=\u0012\n∆↑↑∆↑↓\n∆↓↑∆↓↓\u0013\u0012∆∗\n↑↑∆∗\n↓↑\n∆∗\n↑↓��∗\n↓↓\u0013\n=\u0012|∆↑↑|2+|∆↑↓|2∆↑↑∆∗\n↓↑+ ∆↑↓∆∗\n↓↓\n∆↓↑∆∗\n↑↑+ ∆↓↓∆∗\n↑↓|∆↓↓|2+|∆↓↑|2\u0013\n=\u0012\n−˜dx+i˜dy˜dz\n˜dz˜dx+i˜dy\u0013\u0012−˜d∗\nx−i˜d∗\ny˜d∗\nz\n˜d∗\nz˜d∗\nx−i˜d∗\ny\u0013\n=\u0012|˜dx|2+|˜dy|2+|˜dz|2+i(˜dx˜d∗\ny−˜d∗\nx˜dy) ( −˜dx˜d∗\nz+i˜dy˜d∗\nz) + c.c.\n(−˜d∗\nx−i˜d∗\ny)˜dz) + c.c. |˜dx|2+|˜dy|2+|˜dz|2−i(˜dx˜d∗\ny−˜d∗\nx˜dy)\u0013\n=|d|2σ0+i(d×d∗)·σ, (A2)\nwhere d= (˜dx,˜dy,˜dz). The total number density of the Cooper pairs is given by\n|∆↑↑|2+|∆↑↓|2+|∆↓↑|2+|∆↓↓|2= Tr[ ˆ∆ˆ∆†] = 2d·d∗= 2|d|2. (A3)\nThe expected spin angular momentum is given in units of ℏby\nTr[ˆ∆ˆ∆†σ] = Tr[ i(d×d∗)·σσ] = 2id×d∗. (A4)\nFor a fully spin-polarized triplet superconductor, we can use\nd=deiϕ(ˆu+iˆv), (A5)\nwith ˆu⊥ˆv. Then, we have d·d∗= 2d2. Therefore, d=√ρ/2. The spin polarization is then given by\ns= 2id×d∗= 4d2(ˆu׈v)\n≡4d2ˆs=ρˆs. (A6)\nTo see if the result makes sense, let us consider ˆs=ˆzwith ˆu=ˆxandˆv=ˆy. Then, d=deiϕ(ˆx+iˆy). Then,\n˜dx=deiϕand˜dy=deiϕi. Then, ∆ ↑↑=−2d,∆↓↓= ∆↑↓= ∆↓↑= 0. The condensate density (of the Cooper pairs) is\ngiven by ρ=|∆↑↑|2= 4d2and the spin density is given by s= 4d2ˆs=ρˆs.\nAppendix B: Kinetic term of the Lagrangian\nThe kinetic term of the Lagrangian density is given by\nLK= 2iℏd∗·∂td\n=iℏ\n2√ρe−iϕ(ˆu−iˆv)·∂t\u0002√ρeiϕ(ˆu+iˆv)\u0003\n=iℏ\n2√ρe−iϕ(ˆu−iˆv)·\u0014∂tρ\n2√ρeiϕ(ˆu+iˆv) + (i∂tϕ)√ρeiϕ(ˆu+iˆv) +√ρeiϕ(∂tˆu+i∂tˆv)\u0015\n=iℏ\n2√ρ(ˆu−iˆv)·\u0014∂tρ\n2√ρ(ˆu+iˆv) + (i∂tϕ)√ρ(ˆu+iˆv) +√ρ(∂tˆu+i∂tˆv)\u0015\n=iℏ\n2\u0014∂tρ\n22 + (i∂tϕ)ρ2 +ρ(iˆu·∂tˆv−iˆv·∂tˆu)\u0015\n=−ℏρ(∂tϕ+ˆu·∂tˆv)\n=−ℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]. (B1)12\nThe factor of 2 in front is to ensure the commutation relation, [ d∗\ni(r), dj(r′)] = 2 iℏδ(r−r′). Then, the total Lagrangian\ndensity is given by\nL=−ℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]\n−ρ\u001aAc\n2h\n∂iϕ+q\nℏAi+ˆs·(ˆu×∂iˆu)i2\n+As\n2(∂iˆs)2−D\n2(ˆs·ˆz)2−Hsz+qV\u001b\n−\u001aAc\n16(∇ρ)2+U\n2(ρ−ρ0)2\u001b\n.(B2)\nFrom this, we can read the emergent scalar potential\nVe=−ℏ\nqˆs·(ˆu×∂tˆu), (B3)\nand the emergent vector potential\nai=ℏ\nqˆs·(ˆu×∂iˆu). (B4)\nAppendix C: Equations of motion\nThe dynamics of the order parameter can be uniquely characterized by the dynamics of three variables, the con-\ndensate number density of the Cooper pairs ρ, the phase of the order parameter ϕ, and the spin direction ˆs.\nFirst, the equation of motion for ϕcan be obtained by\nδL\nδρ= 0,\n⇒ −ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =δF\nδρ,\n⇒ −ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =\u001aAc\n2h\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)i2\n+As\n2(∂iˆs)2−D\n2(ˆs·ˆz)2−Hsz+qV\u001b\n−Ac\n8∇2ρ+U(ρ−ρ0).\n(C1)\nSecond, the equation of motion for ρcan be obtained by\nd\ndt\u0012δL\nδ˙ϕ\u0013\n−δL\nδϕ= 0,\n⇒ −ℏ˙ρ=−δF\nδϕ,\n⇒ −ℏ˙ρ=∂in\nρAch\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)io\n,\n⇒˙ρ=−1\nq∂iJi, (C2)\nwhere Jiis the charge current density. This is nothing but the continuity equation.\nThird, to obtain the equation of motion for ˆs, by considering infinitesimal variations of the three vectors that\nmaintain the orthonormality conditions,\nˆs=ˆs0+δˆs=ˆs0+aˆu0+bˆv0,\nˆu=ˆu0−aˆs0,\nˆv=ˆv0−bˆs0, (C3)\nwe observe that, to zeroth order in aandb,\nδ\nδˆs(ˆs·(ˆu×∂tˆu)) = ˆu0∂\n∂a(ˆs·(ˆu×∂tˆu)) +ˆv0∂\n∂b(ˆs·(ˆu×∂tˆu))\n=−ˆu0(ˆs0·(ˆu0×∂tˆs0)) +ˆv0(ˆv0·(ˆu0×∂tˆu0))\n=ˆs0×∂tˆs0. (C4)13\nAlso, by the analogous steps,\nδ\nδˆs(ˆs·(ˆu×∂iˆu)) =ˆs×∂iˆs. (C5)\nThen, from the Lagrangian, we obtain\nδL\nδˆs= 0,\n⇒ℏρˆs×∂tˆs=δF\nδˆs,\n⇒ℏρ∂tˆs=−ℏJi\nq∂iˆs+∂i[ρAs(ˆs×∂iˆs)] +ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z. (C6)\nBy using ˙ ρ=−(∂iJi)/q, the last equation can be recast into\n∂t(ℏρˆs) =−∂i\u0014ℏJi\nqˆs−[ρAs(ˆs×∂iˆs)]\u0015\n+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z. (C7)\nThe left-hand side is the spin density, s=ℏρˆs. 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Recall th at the decoding time\nfor classical Goppa codes of codeword length nisO(n2logνn) in operations of a field\nwhose size is about the size of the set of locations. By comparison, r educed Goppa\ncodes can be decoded much quicker that classical Goppa codes.\nKeywords : Reed-Solomoncodes,BCHcodes,Goppacodes,Gilbert-Varsham ovBound,\nBerlekamp-Massey algorithm.\n1INTRODUCTION\nReed-Solomon codes [RS], BCH codes [BC], [Ho] and Goppa codes [Go] a re widely\nusederrorcorrectingcodesduetotheirfastdecodingalgorithms basedontheBerlekamp-\nMassey algorithm [Be68,Ma]. We shall construct new codes from Gop pa codes. These\nnew codes will have a faster decoding algorithm and have performan ces competent to\nthe classical Goppa codes.\nWe recall the notion of Goppa codes in a sense as narrow as possible. As any\nlinear code, a Goppa code has a Galois field GF( q) as its symbol field. To ensure quick\ndecoding, we require qto be a power of a small prime number. Its location field is an\nextension of GF( q), say GF( qm). Correspondingly, its set of locations is Z/(qm−1),\nwhich is canonically isomorphic to the multiplicative group GF( qm)×. To ensure a\nlower bound for its minimal distance, it has a designed distance δ. To get a code of\ngood performance, it has a structure polynomial gof degree δ−1 having no roots in\nGF(qm)×. For simplicity, we require gto have coefficients in GF( q). The Goppa code\n∗Shanghai Jiao Tong Univ., Shanghai 200240, clliu@sjtu.edu .cn.with these parameters is denoted as Γ( g,δ,m,q). Its codewords satisfy the congruence\nqm−2/summationdisplay\ni=0ci\nx−βi≡0(modg(x)),\nwhereβis a fixed primitive element of GF( qm).\nThe Fr¨ obenius of the Galois group of GF( qm) over GF( q) acts on Γ( g,δ,m,q). Its\nfixed codewords form a subcode ΓF(g,δ,m,q). Deleting the redundance caused by\nrepetition, we obtain a new code RΓ( g,δ,m,q). We call this code a reduced Goppa\ncode. The precise construction of reduced Goppa codes is given in D efinition 2.7. It is\nproved that the reduced Goppa codes approach to the Gilbert-Va rshamov bound (see\nTheorem 4.5). It is also proved that the decoding time for a reduced Goppa code of\ncodeword length nisO(n2logνn) inq-ary operations (see Theorem 5.15). Here the\nconstant νis determined by the performance of polynomial factorisation algor ithms.\nRecall that decoding the classical Goppa codes of codeword length n=qm−1 requires\nO(n2logνn)inqm-aryoperations. Bycomparison,reducedGoppacodescanbedec oded\nmuch quicker than classical Goppa codes.\nAcknowledgement. The author thanks Jiyou Li for pointing out some mistakes.\n2 Construction\nIn this section we introduce the action of the Fr¨ obenius on Goppa c odes, and give the\nconstruction of reduced Goppa codes in Definition 2.7.\nDefinition 2.1 The Fr¨ obenius Frobon the group Z/(qm−1)is a permutation on\nZ/(qm−1)defined by the formula:\nFrob(i+(qm−1)Z) =qi+(qm−1)Z.\nDefinition 2.2 The Fr¨ obenius Frobon the space GF(q)Z/(qm−1)is a permutation on\nGF(q)Z/(qm−1)defined by the formula:\nFrob(r)i=rqm−1i.\nLemma 2.3 The Goppa code Γ(g,δ,m,q)is stable under the Fr¨ obenius.\nProof.Letc∈Γ(g,δ,m,q). That is\nqm−2/summationdisplay\ni=0ci\nx−βi≡0(modg(x)).\nAsghas coefficients in GF( q), we infer\nqm−2/summationdisplay\ni=0ci\nx−βqi≡0(modg(x)).\n2Then\nqm−2/summationdisplay\ni=0Frob(c)i\nx−βi=qm−2/summationdisplay\ni=0cqm−1i\nx−βi(modg(x))\n=qm−2/summationdisplay\ni=0ci\nx−βqi(modg(x))\n≡0(modg(x)).\nThat is, Frob( c)∈Γ(g,δ,m,q). The lemma is proved.\nDefinition 2.4 The subcode of Γ(g,δ,m,q)formed of Fr¨ obenius fixed codewords is the\ncode\nΓF(g,δ,m,q) ={c∈Γ(g,δ,m,q)|Frob(c) =c}.\nDefinition 2.5 The set of orbits of Z/(qm−1)under the Fr¨ obenius is denoted as\nO(m,q).\nDefinition 2.6 Letc∈ΓF(g,δ,m,q)}. Foro∈O(m,q), we define\nco=ci,∀i∈o.\nDefinition 2.7 The code\nRΓ(g,δ,m,q) ={(co)o∈O(m,q)|c∈ΓF(g,δ,m,q)}\nis called a reduced Goppa code.\nWe see that RΓ( g,δ,m,q) is obtained from ΓF(g,δ,m,q) by deleting redundance caused\nby repetition, and, as a space over GF( q), is canonically isomorphic to ΓF(g,δ,m,q).\n3 Lower bounds on minimal distance and dimension\nIn thissectionwegivelowerbounds forthe minimal distanceanddimen sionofreduced\nGoppa codes. The main results are Theorems 3.1 and 3.6.\nTheorem 3.1 (distance bound)\ndistRΓ(g,δ,m,q)≥1\nmdistΓF(g,δ,m,q)≥δ\nm.\nProof.Letc∈RΓ(g,δ,m,q) be of minimal distance d. Fori∈Z/(qm−1), define\nci=co, i∈o.\nThen (ci)i∈Z/(qm−1)∈ΓF(g,δ,m,q), and\nw(ci)i∈Z/(qm−1)≤md.\n3It follows that\ndistΓF(g,δ,m,q)≤md≤m·distRΓ(g,δ,m,q).\nThe theorem now follows.\nLemma 3.2 Ifg(x) =/summationtextδ−1\ni=0gixi, then the polynomials\nδ−1−j/summationdisplay\ni=1gi+jxi, j= 0,1,···,δ−2\nare linearly independent in GF(q)[x]/(g(x)).\nProof.This follows from the fact that (/summationtextδ−1−j\ni=1gi+jxi)j=0,1,···,δ−2is equal to\n(xδ−1,xδ−2,···,x)\ngδ−10 0 0\ngδ−2gδ−10 0\n.........0\ng1g2···gδ−1\n\nLemma 3.3 (codeword equations) c∈Γ(g,δ,m,q)if and only if\nqm−2/summationdisplay\ni=0ci\ng(βi)βij= 0, j= 0,1,···,δ−2.\nProof.We have\nqm−2/summationdisplay\ni=0g(x)−g(βi)\ng(βi)ci\nx−βi=qm−2/summationdisplay\ni=0ci\ng(βi)δ−1/summationdisplay\nu=1guxu−βiu\nx−βi\n=qm−2/summationdisplay\ni=0ci\ng(βi)δ−1/summationdisplay\nu=1guu−1/summationdisplay\nj=0xu−jβij\n=qm−2/summationdisplay\ni=0ci\ng(βi)δ−2/summationdisplay\nj=0βijδ−1/summationdisplay\nu=j+1guxu−j\n=δ−2/summationdisplay\nj=0/parenleftiggqm−2/summationdisplay\ni=0ciβij\ng(βi)/parenrightiggδ−1−j/summationdisplay\ni=1gi+jxi.\nBy the last lemma, the polynomials\nδ−1−j/summationdisplay\ni=1gi+jxi, j= 0,1,···,δ−2\n4are linearly independent. It follows that\nqm−2/summationdisplay\ni=0ci\ng(βi)βij= 0, j= 0,1,···,δ−2\n⇔qm−2/summationdisplay\ni=0g(x)−g(βi)\ng(βi)ci\nx−βi≡0(modg(x))\n⇔qm−2/summationdisplay\ni=0ci\nx−βi≡0(modg(x))\n⇔c∈Γ(g,δ,m,q).\nThe lemma is proved.\nLemma 3.4 (codeword congruence) c∈Γ(g,δ,m,q)if and only if\nqm−2/summationdisplay\ni=0ci\ng(βi)1\n1−βix≡0(modxδ−1).\nProof.We have\nqm−2/summationdisplay\ni=0ci\ng(βi)1\n1−βix=qm−2/summationdisplay\ni=0ci\ng(βi)+∞/summationdisplay\nj=0βijxj\n≡δ−2/summationdisplay\nj=0xjqm−2/summationdisplay\ni=0ci\ng(βi)βij(modxδ−1).\nThus\nqm−2/summationdisplay\ni=0ci\ng(βi)1\n1−βix≡0(modxδ−1)\n⇔δ−2/summationdisplay\nj=0xjqm−2/summationdisplay\ni=0ci\ng(βi)βij≡0(modxδ−1)\n⇔qm−2/summationdisplay\ni=0ci\ng(βi)βij= 0, j= 0,···,δ−2\n⇔c∈Γ(g,δ,m,q),\nwhere the last equivalence follows from the last lemma. The lemma is pro ved.\nLemma 3.5 (codeword equations) (co)o∈O(m,q)∈RΓ(g,δ,m,q)if and only if\n/summationdisplay\no∈O(m,q)co/summationdisplay\ni∈oβij\ng(βi)= 0, j= 0,1,···,δ−2,\nif and only if/summationdisplay\no∈O(m,q)co/summationdisplay\ni∈o1\ng(βi)(1−βix)≡0(modxδ−1).\n5Proof.Let (co)o∈O(m,q)∈GF(q)O(m,q). Fori∈Z/(qm−1), define\nci=co, i∈o.\nThen\nqm−2/summationdisplay\ni=0ci\ng(βi)βij=/summationdisplay\no∈O(m,q)co/summationdisplay\ni∈oβij\ng(βi), j= 0,1,···,δ−2.\nIt follows that\n(co)o∈O(m,q)∈RΓ(g,δ,m,q)\n⇔(ci)i∈Z/(qm−1)∈ΓF(g,δ,m,q)\n⇔qm−2/summationdisplay\ni=0ci\ng(βi)βij= 0, j= 0,1,···,δ−2\n⇔/summationdisplay\no∈O(m,q)co/summationdisplay\ni∈oβij\ng(βi)= 0, j= 0,1,···,δ−2.\nSimilarly, c∈Γ(g,δ,m,q) if and only if\nqm−2/summationdisplay\ni=0ci\ng(βi)1\n1−βix≡0(modxδ−1).\nThe lemma is proved.\nTheorem 3.6 (dimension bound)\ndimRΓ(g,δ,m,q)≥ |O(m,q)|−δ+1.\nProof.This follows from the last lemma as well as the fact that\n/summationdisplay\ni∈oβij\ng(βi)∈GF(q).\n4 The Gilbert-Varshamov bound\nIn this section we prove Theorem 4.5, which says that the reduced G oppa codes\napproach the Gilbert-Varshamov bound as classical Goppa codes [B e73].\nDefinition 4.1 (Entropy) Theq-ary entropy function on (0,1)is defined by the for-\nmula\nHq(x) =xlogq(q−1)−xlogqx−(1−x)logq(1−x).\nLemma 4.2 (Entropy lemma) Ifd\nn≤q−1\nq, then\nHq(d\nn)≥1\nnlogqd/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(q−1)i.\n6Proof.By assumption, we haved\nn−d≤q−1.Thus, if 0 ≤i≤d, we have\nq−nHq(d\nn)= (q−1)−d(d\nn)d(n−d\nn)n−d\n≤(q−1)−i(d\nn)i(n−d\nn)n−i\nIt follows that\nq−nHq(d\nn)d/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(q−1)i≤d/summationdisplay\ni=0/parenleftbiggn\ni/parenrightbigg\n(d\nn)i(n−d\nn)n−i\n≤1.\nThe lemma now follows.\nDefinition 4.3 The set of monic irreducible polynomials over GF(q)is denoted as\nIrr(δ−1,q).\nLemma 4.4\n#{g∈Irr(δ−1,q)|dist(RΓ( g,δ,m,q))≤d}\n≤md\nδ−1d/summationdisplay\ni=0/parenleftbigg|O(m,q)|\ni/parenrightbigg\n(q−1)i.\nProof.We have\n#{g∈Irr(δ−1,q)|c∈RΓ(g,δ,m,q)}\n≤#{g∈Irr(δ−1,q)|qm−2/summationdisplay\ni=0ci\nx−βi≡0(modg(x))}\n≤m·wt(c)\nδ−1.\nIt follows that\n#{g∈Irr(δ−1,q)|dist(RΓ(g,δ,m,q))≤d}\n≤d/summationdisplay\ni=1/summationdisplay\nwt(c)=i#{g∈Irr(δ−1,q)|c∈RΓ(g,δ,m,q)}\n≤md\nδ−1d/summationdisplay\ni=0/parenleftbigg|O(m,q)|\ni/parenrightbigg\n(q−1)i.\nThe lemma is proved.\nTheorem 4.5 (Asymptotic Gibert-Varshamov bound) Letqbe any prime power.\nLetε >0. Then there is a tuple (g,δ,m)such that RΓ(g,δ,m,q)makes sense, such that\ndistRΓ(g,δ,m,q)\n|O(m,q)|≤q−1\nq,\n7and such that\nHq(distRΓ(g,δ,m,q)\n|O(m,q)|)+dimRΓ(g,δ,m,q)\n|O(m,q)|≥1−ε.\nProof.First, we choose N >0 such that, for all m > N, and for all δwith\nqm\nm2≤δ≤qm\nm+qm\n2+1,\nwe have\nlogq(1−q−δ−1\n2−q−δ+1)+logq(δ−1)−m−logqm\n|O(m,q)|≥ −ε.\nSecond, we choose mandδsuch that m > N, such that δ−1 is prime to m, and such\nthatqm\nm2≤δ≤q−1\nq·|O(m,q)|.\nThird, we choose dto be the largest integer such that\nmd\nδ−1d/summationdisplay\ni=0/parenleftbigg|O(m,q)|\ni/parenrightbigg\n(q−1)i< qδ−1−qδ−1\n2−1,\nwhich implies that\n#{g∈Irr(δ−1,q)|dist(RΓ(g,δ,m,q))≤d}< qδ−1−qδ−1\n2−1<#Irr(δ−1,q).\nFourth, we choose g∈Irr(δ−1,q) such that\ng/ne}ationslash∈ {g∈Irr(δ−1,q)|dist(RΓ(g,δ,m,q))≤d}.\nAsδ−1 is prime to m,ghas no roots in GF( qm)×. Thus RΓ( g,δ,m,q) makes sense,\nand\nD= dist(RΓ( g,δ,m,q))> d.\nWith these chosen parameters, we have\ndist(RΓ(g,δ,m,q))\n|O(m,q)|≤|O(m,q)|−dimRΓ(g,δ,m,q)+1\n|O(m,q)|\n≤δ\n|O(m,q)|\n≤q−1\nq,\nwhere the first inequality follows from the singleton bound, and the s econd inequality\n8follows from our bound for the dimensions of reduced Goppa codes. Moreover,\nHq(distRΓ(g,δ,m,q)\n|O(m,q)|)≥1\n|O(m,q)|logqD/summationdisplay\ni=0/parenleftbigg|O(m,q)|\ni/parenrightbigg\n(q−1)i\n≥logq(qδ−1−qδ−1\n2−1)+logq(δ−1)−logq(mD)\n|O(m,q)|\n≥logq(qδ−1−qδ−1\n2−1)+logq(δ−1)−m−logq(m)\n|O(m,q)|\n≥δ−1\n|O(m,q)|−ε\n≥|O(m,q)|−dimRΓ(g,δ,m,q)\n|O(m,q)|−ε.\nThe theorem is proved.\n5 Decoding\nIn this section we describe the decoding algorithm for reduced Gopp a codes. The\nmain result is Theorem 5.15.\nDefinition 5.1 Letr∈GF(q)O(m,q). If there is a codeword c∈RΓ(g,δ,m,q)such\nthat/summationdisplay\no∈O(m,q):ro/negationslash=co|o| ≤δ−1\n2,\nthen it is called a B-M correctable word for RΓ(g,δ,m,q).\nDefinition 5.2 Ifr∈GF(q)O(m,q)is a B-M correctable word for RΓ(g,δ,m,q), then\nthe error location set of ris\nE={o∈O(m,q)|ro/ne}ationslash=co},\nand the error of rise=r−c.\nDefinition 5.3 Ifr∈GF(q)O(m,q)is a B-M correctable word for RΓ(g,δ,m,q), then\nSj=/summationdisplay\no∈O(m,q)ro/summationdisplay\ni∈oβij\ng(βi), j= 0,1,···,δ−2.\nare called syndromes of r.\nLemma 5.4 Ifr∈GF(q)O(m,q)is a B-M correctable word for RΓ(g,δ,m,q), then its\nsyndromes can be calculated out within O(nδ)time inq-ary field operations, where nis\nthe length of r.\n9Proof.This follows from the fact that\n/summationdisplay\ni∈oβij\ng(βi)∈GF(q), j= 0,1,···,δ−2,\nwhich can be calculated out in pre-computation.\nLemma 5.5 (syndrome congruence) Ifr∈GF(q)O(m,q)is a B-M correctable word\nforRΓ(g,δ,m,q), then\nδ−2/summationdisplay\nj=0Sjxj≡/summationdisplay\no∈O(m,q)eo/summationdisplay\ni∈o1\ng(βi)(1−βix)(modxδ−1).\nProof.We have\nδ−2/summationdisplay\nj=0Sjxj=δ−2/summationdisplay\nj=0xj/summationdisplay\no∈O(m,q)ro/summationdisplay\ni∈oβij\ng(βi)\n=/summationdisplay\no∈O(m,q)ro/summationdisplay\ni∈o/summationtextδ−2\nj=0xjβij\ng(βi)\n≡/summationdisplay\no∈O(m,q)ro/summationdisplay\ni∈o1\ng(βi)(1−βix)(modxδ−1).\nThe lemma now follows from the equality r=c+eand the codeword equation\n/summationdisplay\no∈O(m,q)co/summationdisplay\ni∈o1\ng(βi)(1−βix)≡0(modxδ−1).\nDefinition 5.6 Ifr∈GF(q)O(m,q)is a B-M correctable word for RΓ(g,δ,m,q), then\nthe polynomial pair (ω(x),σ(x))such that degω(x)<degσ(x)≤δ−1\n2, and\nω(x)\nσ(x)≡δ−2/summationdisplay\nj=0Sjxj(modxδ−1)\nis called a B-M pair of r.\nLemma 5.7 Ifr∈GF(q)O(m,q)is a B-M correctable word for RΓ(g,δ,m,q), then,\nafter its syndromes have been calculated out, its B-M pair (ω(x),σ(x))can be calculated\nout within O(δ2)time inq-ary field operations.\nProof. This follows from the time estimation for the well-known Berlekamp-Ma ssey\nalgorithm, as well as the fact that\nSj∈GF(q), j= 0,···,δ−2.\n10Definition 5.8 The location polynomial of o∈O(m,q)is the irreducible polynomial\npo(x) =/productdisplay\ni∈o(x−β−i).\nLemma 5.9 (syndrome equation) Ifr∈GF(q)O(m,q)is a B-M correctable word for\nRΓ(g,δ,m,q), then\nω(x)\nσ(x)=/summationdisplay\no∈O(m,q)eo/summationdisplay\ni∈o1\ng(βi)(1−βix).\nProof.By the syndrome congruence, as well as the definition of B-M pair, w e have\nω(x)\nσ(x)≡δ−2/summationdisplay\nj=0Sjxj(modxδ−1)\n≡/summationdisplay\no∈O(m,q)eo/summationdisplay\ni∈o1\ng(βi)(1−βix)(modxδ−1).\nComparing degrees of related polynomials, we see that\nω(x)\nσ(x)=/summationdisplay\no∈O(m,q)eo/summationdisplay\ni∈o1\ng(βi)(1−βix).\nThe lemma is proved.\nLemma 5.10 (error locating formula) Ifr∈GF(q)O(m,q)is a B-M correctable\nword for RΓ(g,δ,m,q), then\nE={o∈O(m,q) :po|σ(x)}.\nProof.This follows from the last lemma.\nConsulting historical results on the factorisation of polynomials ove r finite fields\n[Be67,Be68,Be70], [Mc69], [Ca83], [MOV], [Ni93a,Ni93b], [RZ], [GS], we intro duce the\nfollowing definition.\nDefinition 5.11 Letqis a power of a small prime number. Let νbe the least non-\nnegative integer such that every polynomial of degree noverGF(q)can be factored into\nirreducible polynomials GF(q)withinO(n2(log(nq))ν)time inq-ary field operations.\nLemma 5.12 (error locating time) Letqis a power of a small prime number. If r\nis a B-M correctable word for RΓ(g,δ,m,q), then, after its B-M pair has been calculated\nout, its error location set Ecan be calculated out within O(δ2(logδ)ν)time inq-ary\nfield operations.\nProof.This follows from the last lemma, and the fact that σ(x) is a polynomial over\nGF(q).\n11Lemma 5.13 (error correction formula) Ifr∈GF(q)O(m,q)is a B-M correctable\nword for RΓ(g,δ,m,q), ando∈E, then\neo≡ −ω(x)g(1/x)\nxσ′(x)(modpo(x)).\nProof.In fact, for i∈o, we have\nω(x)≡ −eoβ−iσ(x)\ng(βi)(x−β−i)(mod(x−β−i))\n≡ −eoxσ′(x)\ng(1/x)(mod(x−β−i)).\nThus, by Chinese remainder theorem,\nω(x)≡ −eoxσ′(x)\ng(1/x)(modpo(x)).\nThe lemma now follows.\nLemma 5.14 (error correction time) Ifr∈GF(q)O(m,q)is a B-M correctable word\nforRΓ(g,δ,m,q), then, after its error location set Ehas calculated out, its error can be\ncorrected within O(δ2)time inq-ary field operations.\nProof.This follows from the last lemma.\nTheorem 5.15 (decoding time) Letqis a power of a small prime. If ris a B-M\ncorrectable word for RΓ(g,δ,m,q), andn=|O(m,q)|, thenrcan be decoded within\nO(n2(logn)ν)time inq-ary field operations.\nProof.Theorem 5.15 follows form Lemmas 5.4, 5.7, 5.12, and 5.14.\nRecall that, if ris a received word of Γ( g,δ,m,q) and there is a codeword csuch that\ndist(r,c)≤δ−1\n2,\nthenrcan be decoded within O(n2(logn)ν) time in qm-ary field operations with n=\nqm−1. By comparison, reduced Goppa codes can be decoded much quick er that the\nclassical Goppa codes.\n6 Comparison with Goppa codes\nIn this section we compare the m-fold direct product RΓ( g,δ,m,q)mwith the clas-\nsical Goppa code Γ( g,δ,m,q). We illustrate their performances in the following table.\n12RΓ(g,δ,m,q)mΓ(g,δ,m,q)\ncodeword length qm−1≤n≤qm+mqm\n2+1n=qm−1\ndimension k≥n−m(δ−1) k≥n−m(δ−1)\nminimal distance d≥δ d≥δ\nAsymptotic meets meets\nGilbert-Varshamov bound\ndecoding time O(n2(logn)ν−1) O(n2(logn)ν)\ninq-ary field operations inqm-ary field operations\nReferences\n[Be67] E. R. Berlekamp, ”Factoring polynomials over finite fields,” Bell Syst. Tech. J.,\nvol. 46, pp. 1853-1859, 1967.\n[Be68] E. R. Berlekamp, ”Algebraic coding theory,” New York, McGra w-Hill, 1968.\n[Be70] E. R. Berlekamp, ”Factoring polynomials over large finite fields ,” Math. Com-\nput., vol. 24, pp. 713-735, 1970.\n[Be73] E. R. Berlekamp, ”Goppa codes,” IEEE Trans. Inf. Theory, vol. 19, pp. 590-592,\n1973.\n[BC] R. C. Boseand D. K. Ray-Chaudhuri, ”Ona class oferrorcorre ctionbinary group\ncodes,” Inform. Control, vol. 3, no. 1, pp. 68-79, 1960.\n[Ca83] P. Camion, ”A deterministic algorithm for factoring polynomials overFq[X],”\nAnn. Discrete Math., vol. 17, pp. 149-157, 1983.\n[Ch] C. L. Chen, ”High speed decoding of BCH codes,” IEEE Trans. In f. Theory, vol.\n27, pp. 254-256, 1981.\n[GS] J. von zur Gathen and V. Shoup, ”Computing Frobenius maps an d factoriing\npolynomials,” Comput. Complexity, vol. 2, pp. 187-224, 1992.\n[Go] V. D. Goppa, ”Anew class oflinear error-correctingcodes,”Pr obl. Inform. Trans-\nmission., vol.6, pp. 207-212, 1970.\n[Ho] A. Hocquenghem, ”Codes correcteurs d’erreurs,” Chiffres, v ol. 2, pp. 147-156,\n1959.\n[Ma] J. L. Massey, ”Shift register synthesis and BCH decoding,” IEE E Trans. Inform.\nTheory, vol. 15, pp. 122-127, 1969.\n[Mc69] R. J. McEliece, ”Factoring polynomials over finite fields,” Math. Comput., vol.\n23, pp. 861-867, 1970.\n[MOV] A. J. Menezes, P. C. van Oorschot and A. Vanstone, ”Some c omputational\naspects of root finding GF( qm),” LNCS 358, pp. 259-270, 1988, Springer-Verlag.\n[Ni93a] H. Niederreiter, ”Factorization of polynomials and some linear algebra prob-\nlems,” Liner. Algbr. Appl., vol. 192, pp. 301-328, 1993.\n13[Ni93b] H. Niederreiter, ”A new efficient factorization algorithm for p olynomials over\nsmall finite fileds,” Appl. Algbr. Eng. Commun. Comput., vol. 4, pp. 81- 87, 1993.\n[RS] I. S. Reed and G. Solomon, ”Polynomialcodes overcertain finite fields,” J. SIAM,\nvol. 8, pp. 300-304, 1960.\n[RZ] M. Rothstein and H. Zassenhaus, ”Deterministic analysis of alea toric methods of\npolynomial factorization over finite fields,” J. Number Theory, vol. 4 7, pp. 20-42,\n1994.\n14"    },    {        "title": "2306.02770v1.Damping_of_coronal_oscillations_in_self_consistent_3D_radiative_MHD_simulations_of_the_solar_atmosphere.pdf",        "content": "Astronomy &Astrophysics manuscript no. waves ©ESO 2023\nJune 6, 2023\nDamping of coronal oscillations in self-consistent 3D radiative\nMHD simulations of the solar atmosphere\nP. Kohutova1,2, P. Antolin3, M. Szydlarski1,2, and M. Carlsson1,2\n1Rosseland Centre for Solar Physics, University of Oslo, P.O. Box 1029, Blindern, NO-0315 Oslo, Norway\ne-mail: petra.kohutova@astro.uio.no\n2Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029, Blindern, NO-0315 Oslo, Norway\n3Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle Upon Tyne NE1 8ST, UK\nReceived; accepted\nABSTRACT\nContext. Oscillations are abundant in the solar corona. Coronal loop oscillations are typically studied using highly idealised models\nof magnetic flux tubes. In order to improve our understanding of coronal oscillations, it is necessary to consider the e ffect of realistic\nmagnetic field topology and density structuring.\nAims. We analyse the damping of coronal oscillations using a self-consistent 3D radiation-MHD simulation of the solar atmosphere\nspanning from the convection zone into the corona, the associated oscillation dissipation and heating, and finally the physical processes\nresponsible for the damping and dissipation. The simulated corona formed in such a model does not depend on any prior assumptions\nabout the shape of the coronal loops.\nMethods. We analyse the evolution of a bundle of magnetic loops by magnetic field tracing.\nResults. We find that the bundle of magnetic loops shows damped transverse oscillations in response to perturbations in two separate\ninstances with oscillation periods of 177 s and 191 s, velocity amplitudes of 10 km s−1and 16 km s−1and damping times of 176 s\nand 198 s, respectively. The coronal oscillations lead to the development of velocity shear in the simulated corona resulting in the\nformation of vortices seen in the velocity field caused by the Kelvin-Helmholtz instability, contributing to the damping and dissipation\nof the transverse oscillations.\nConclusions. The oscillation parameters and evolution observed are in line with the values typically seen in observations of coronal\nloop oscillations. The dynamic evolution of the coronal loop bundle suggests the models of monolithic and static coronal loops with\nconstant lengths might need to be re-evaluated by relaxing the assumption of highly idealised waveguides.\nKey words. Magnetohydrodynamics (MHD) – Sun: corona – Sun: magnetic fields – Sun: oscillations\n1. Introduction\nCoronal structures act as waveguides for a variety of MHD os-\ncillation modes (Nakariakov & Verwichte 2005; Nakariakov &\nKolotkov 2020). There is extensive observational evidence that\ntransverse oscillations are ubiquitous in the solar corona, in both\nclosed coronal loops and open coronal structures (Tomczyk et al.\n2007; McIntosh et al. 2011; Anfinogentov et al. 2015). The trans-\nverse oscillations commonly observed in coronal loops are iden-\ntified as standing kink modes (Aschwanden et al. 1999; Nakari-\nakov et al. 1999). Coronal oscillations carry magnetic energy,\nwhich is deposited through the oscillation damping and dissipa-\ntion. Commonly proposed physical mechanisms responsible for\nthe oscillation damping and dissipation include resonant absorp-\ntion (Hollweg & Yang 1988; Goossens et al. 2002), phase mix-\ning (Heyvaerts & Priest 1983) and Kelvin-Helmholtz instability\n(Terradas et al. 2008; Antolin et al. 2014). Such mechanisms\nhave been mostly analysed using simplified models of coronal\nloops as straight magnetic flux-tubes clearly distinct from the\nsurroundings. There are several advantages to this approach: It is\nstraightforward to isolate individual e ffects and processes linked\nto wave evolution as well as to have full control over the param-\neters of the modelled loop. They also allow for very high spatial\nresolution, necessary for modelling of certain physical processes\nsuch as resonant absorption (Van Doorsselaere et al. 2004). Such\nmodels, however, also require making several assumptions aboutthe shape, density structure and morphology of coronal loops\nwhich might be too idealised.\nThe solar corona is in fact a dynamic environment with\ncomplex density structuring and the coronal magnetic field is\ncontinuously changing and evolving. Models of static and ide-\nalised coronal loops therefore neglect this evolution. In order to\naccount for realistic magnetic field configurations and density\nstructuring in the solar corona, a more self-consistent approach\nto modelling the evolution of coronal structures is necessary.\nThis can be achieved by taking advantage of realistic convection-\nzone-to-corona models (e.g. Carlsson et al. 2016; Cheung et al.\n2019; Kohutova & Popovas 2021; Breu et al. 2022). The evolu-\ntion of the corona in such models is self-consistently driven by\nthe dynamics of the lower solar atmosphere. This type of simu-\nlations therefore reflects the dynamic and continuously evolving\nnature of the solar corona.\nOne common feature of such simulations is the lack of\nclearly-identifiable coronal loops with well-defined boundaries.\nIn the realistic solar simulations the actual 3D structure of coro-\nnal features with increased emissivity in the optically thin coro-\nnal emission lines is much more complex, despite appearing as\nthin, well-defined loops in forward-modelled emission due to\nline-of-sight e ffects. Such structure of the corona is referred to as\n’the coronal veil’, and has been described using a self-consistent\nMURaM simulation spanning from the convection zone into the\nArticle number, page 1 of 9arXiv:2306.02770v1  [astro-ph.SR]  5 Jun 2023A&A proofs: manuscript no. waves\ncorona (Malanushenko et al. 2022). Di fferent codes capable of\nproducing self-consistent convection-zone-to-corona models, in-\ncluding MURaM (Rempel 2016) and Bifrost (Gudiksen et al.\n2011), seem to reproduce the ’coronal veil’ structure of the so-\nlar corona. The question remains how well does the simulated\ncorona, which forms in these models, represent the real solar\ncorona. On one hand, the coronal veil model can explain cer-\ntain properties of the coronal loops seen in Extreme Ultraviolet\n(EUV) observations such as loop cross-sections that appear to be\nconstant with height (Aschwanden & Nightingale 2005), which\nare di fficult to explain otherwise; we note that alternative expla-\nnations include a specific density and temperature distribution\nacross the magnetic field (Peter & Bingert 2012) and the pres-\nence of magnetic twist (Li et al. 2020). It would however also\nbring the applicability of coronal loop oscillation models into\nquestion, as most rely on the assumption of coronal loops being\nwell represented by idealised magnetic cylinders. On the other\nhand, the existence and omnipresence of transverse MHD waves\nsuch as kink waves directly points to the existence of struc-\ntures acting as waveguides that lead to collective behaviour in\nthe coronal volume. Also, the occurrence of coronal rain show-\ners (¸ Sahin & Antolin 2022) and long period intensity-pulsations\n(Froment et al. 2015), ubiquitous over active regions, indicate\nthe existence of coronal entities (that we refer as loops or loop\nbundles) with similar thermodynamic behaviour.\nMost models for wave dissipation rely on modelling coronal\nloops as straight magnetic flux-tubes. The mechanism of reso-\nnant absorption depends on conversion of a global kink mode\ninto local azimuthal Alfvén modes in an inhomogeneous layer\nwith an Alfvén speed gradient at the boundary of a cylindrical\ncoronal loop (Goossens et al. 2002; Pascoe et al. 2010; How-\nson et al. 2019). Similarly, the development of Kelvin-Helmholtz\nInstability vortices at the boundary of a transversely oscillating\nloop has been mostly studied using models of cylindrical loops\n(e.g. Antolin et al. 2014; Karampelas et al. 2017; Howson et al.\n2017, 2019) with a distinct density or magnetic interface sepa-\nrating them from the background plasma.\nThe main shortcoming of these models is that the observa-\ntional evidence for the proposed damping and dissipation mech-\nanisms is largely inconclusive. Signatures reminiscent of reso-\nnant absorption have been observed in oscillating prominence\nthreads (Antolin et al. 2015; Okamoto et al. 2015) and in trans-\nversely oscillating spicules (Antolin et al. 2018). To the best of\nour knowledge, these are the only observational evidence of such\nmechanisms to date. Models for damping of coronal oscillations\nwould therefore benefit from extending into more realistic se-\ntups. Numerical studies using setups which do not correspond to\nmagnetic cylinders were done for both oscillations in the chro-\nmosphere (Leenaarts et al. 2015; Khomenko & Cally 2012) and\nin the corona (Matsumoto 2018), the latter two however lack the\nself-consistent treatment of the lower solar atmosphere driving\nthe corona.\nIn this work we therefore focus on analysing coronal oscilla-\ntions in a more advanced numerical setup. We exploit the poten-\ntial of convection-zone-to-corona simulations with the radiation-\nMHD code Bifrost for coronal studies. We have previously\nshown that coronal oscillations are abundant in self-consistent\nconvection-zone-to-corona simulations and that the detected os-\ncillation modes and regimes match those seen in solar observa-\ntions (Kohutova & Popovas 2021). Here we analyse the evolu-\ntion of an oscillating bundle of magnetic loops which forms in\nthe simulation and focus on the oscillation damping. Despite the\ncomplex evolution of the bundle, variable length etc. the indi-\nvidual magnetic field lines contained in the bundle are found toexhibit collective evolution during extended periods of time, in-\ncluding collective oscillations triggered by impulsive events and\nsubsequent oscillation damping.\nThe manuscript is structured as follows. Section 2 describes\nthe numerical model. Section 3 describes the methods used for\nthe analysis of the evolution of a bundle of magnetic loops and of\nthe corresponding oscillatory behaviour seen in the simulation.\nIn Section 4 we focus on the damping of the oscillating loops\nand on the physical mechanism responsible for the damping. In\nSection 5 we discuss the results and implications for the coronal\nloop models. We summarize our conclusions in Section 6.\n2. Numerical model\nWe analyse the evolution, oscillatory behaviour and damping of\na bundle of magnetic loops in the numerical simulation of a mag-\nnetically enhanced network spanning from the upper convection\nzone to the corona using the Bifrost code (Gudiksen et al. 2011).\nThis simulation corresponds to the extended run of the pub-\nlic Bifrost simulation of the enhanced network (Carlsson et al.\n2016). The simulation subset with 2000 s duration analysed in\nthis work covers the time range from t =100 s to t =2100 s of\nthe extended run of the enhanced network simulation run from\nthe last snapshot of the public simulation, while having the same\nphysical setup.\nBifrost is a 3D radiation-MHD code which solves the resis-\ntive MHD equations and includes radiative transfer with scatter-\ning in the photosphere and low chromosphere, and parametrised\nradiative losses and heating in the upper chromosphere, transi-\ntion region and corona. The e ffects of field-aligned thermal con-\nduction and the non-equilibrium ionisation of hydrogen in the\nequation of state are included in the simulation.\nThe physical size of the simulation grid is 24 ×24×16.8\nMm and the grid resolution is 504 ×504×496. The grid spans\nfrom 2.4 Mm below the photosphere to 14.4 Mm in the corona.\nThe photosphere is located at z=0 surface and corresponds to\nthe (approximate) height where the optical depth τ500is equal\nto unity. The grid spacing is 48 km and uniform in the xandy\ndirection, while in the zdirection it varies from 19 km to 98 km\nin order to resolve steep density and temperature gradients in the\nlower solar atmosphere.\nThe simulation domain uses periodic boundaries in the x\nandydirections and open boundaries in the zdirection. The top\nboundary uses characteristic boundaries which transmit distur-\nbances with minimal reflection (Gudiksen et al. 2011). At the\nbottom boundary the flows are let through and the magnetic field\nis passively advected without introducing any additional mag-\nnetic field into the domain. The average horizontal pressure is\ndriven towards a constant value with a characteristic timescale\nof 100 s, creating a pressure node at the bottom boundary. This\nleads to acoustic wave reflection resembling the refraction of\nwaves in the deeper solar atmosphere, resulting in global radial\nbox oscillations with a period of 450 s, which are a simulation\ncounterpart of solar p-modes (Stein & Nordlund 2001; Carlsson\net al. 2016).\nThe photospheric magnetic field is concentrated in two\npatches of opposite polarity and has an average unsigned value\nof about 50 G (Fig. 1). The dipolar structure of the magnetic field\ncreates several magnetic loops in the simulated corona. The con-\nvective motions in the lower solar atmosphere lead to magnetic\nfield braiding. The Ohmic and viscous heating associated with\nthe braiding together maintain high temperatures in the chromo-\nsphere and corona. An artificial heating term is employed for\nplasma with temperatures below 2500 K, in order to prevent the\nArticle number, page 2 of 9Kohutova et al.: Damping of coronal oscillations in 3D MHD simulations\nFig. 1. Left: Bundle of magnetic loops and the photospheric magnetic field at t=880 s.Top right: The cut across the density structure at x=13\nMm intersecting the apex of the magnetic bundle. Bottom right: The cut across the temperature structure at x=13 Mm. The white points mark\nthe positions at which the individual loops in the bundle intersect the cuts. An animation of this figure is available online.\n0 5 10 15 20\nx[Mm]0510z[Mm]\n0 5 10 15 20\ny[Mm]0510z[Mm]\n0 5 10 15 20\nx[Mm]5101520y[Mm]\nFig. 2. A projected view of the magnetic bundle in the xz-plane (left), yz-plane (centre) and xy-plane (right) at t=880 s. An animation of this\nfigure is available online.\ntemperature from dropping too low in regions that are rapidly\nexpanding. Only few isolated regions in the simulation domain\nare affected by this, and the heating in the vast majority of the\ndomain is self-consistent. Contributing to the heating are small-\nscale reconnection events which heat the plasma through a com-\nbination of direct Ohmic dissipation and by inducing shear flows\nwhich are then converted into heat by viscous dissipation as well\nas dissipating oscillations. In order to ensure numerical stability\nthe code employs a di ffusive operator; this consists of a small\nglobal di ffusion term as well as of a directionally-dependent hy-\nper di ffusion component which enhances the di ffusion locally.\nFurther details of the numerical setup can be found in e.g. Carls-\nson et al. (2016); Kohutova et al. (2020).\n3. Evolution of a magnetic loop bundle\nThe corona in the simulation is filled with closed magnetic loops\nwhich can extend up to heights of 10 −14 Mm. The density\nstructure of the simulated corona is complex and there are sev-\neral structures with enhanced densities compared to the sur-\nroundings. Most of the overdense structures are filled by chro-\nmospheric evaporation in response to heating (Kohutova et al.\n2020). A cut across the simulation domain shows a lack of loopswith clearly defined cross-sections in the density or temperature\nstructure (Fig. 1).\nThe magnetic field configuration in the simulation domain is\ndriven by the dynamics of the lower solar atmosphere and the\nfootpoints of coronal structures are shu ffled and dragged around\nby the convective motions. The magnetic loops in the corona are\ntherefore continuously evolving and undergoing complex mo-\ntions including sideways displacement, oscillatory and torsional\nmotion and vertical expansion /contraction.\nWe focus on the evolution of a bundle of magnetic loops lo-\ncated in the centre of the simulation domain shown in Fig. 1\nreaching a height of around 10 Mm. Due to the evolving nature\nof the magnetic field in the simulated corona, in order to obtain\nthe evolution of such a bundle in three dimensions it is necessary\nto trace the evolution of the corresponding magnetic field lines\nin the bundle through both time and space. To do this, we use\na field-tracing method described in Leenaarts et al. (2015) and\nKohutova & Popovas (2021).\nA magnetic field line is defined as a curve in a 3D space\nr(s) parametrised by the arc length along the curve s, for which\ndr/ds=B/|B|and is therefore a representation of magnetic\nconnectivity of the coronal plasma. We trace the evolution of\nmagnetic field lines by inserting seed points into the simulation\nArticle number, page 3 of 9A&A proofs: manuscript no. waves\n0 250 500 750 1000 1250 1500 1750 2000\nt[s]7891011121314z[Mm]\nFig. 3. The evolution of the height of the individual loops in the bundle\natx=13 Mm as a function of time. Blue regions indicate the two\ninstances of damped oscillatory motion in the vertical direction.\ndomain at the apex of the magnetic bundle. Using the velocity\nat the seed point position the seed points are then passively ad-\nvected forward and backward in time. The magnetic field is then\ntraced through the instantaneous seed point position in order to\ndetermine the spatial coordinates of the traced field line at every\ntime-step. The accuracy of this method is given by the size of\nthe time-step between two successive simulation snapshots (i.e.,\n1 second in this case). We find that the method works well for 10\ns step-size and there are no major di fferences in field-line evo-\nlution between 1 s and 10 s time-step size. The 10 s time-step\nsize is therefore used for the field line tracing in the following\nanalysis. We note that this approach requires that the evolution\nis smooth and there are no large-amplitude velocity variations\noccurring on timescales shorter than the size of the time-step.\nSimilarly, in the instances where magnetic reconnection occurs,\nthis approach fails and the tracing leads to a jump in the field-line\nevolution. This is, however, not the case in the coronal part of the\nmagnetic bundle during the analysed time-period. We note there\nare 2 instances of rapid transverse displacement occuring close\nto the foootpoints of the traced loops at t =830 s and t =1100 s\ncaused either by an external perturbation or change of magnetic\nconnectivity in the lower solar atmosphere. These however do\nnot lead to discontinuities in the evolution of physical quantities\nin the coronal part of the analysed loops.\nThe magnetic bundle is shown in di fferent projections in Fig.\n2 with the bundle evolution shown in the online animation. The\nindividual magnetic loops display a large degree of collective be-\nhaviour and the magnetic bundle behaves as a coherent structure\nduring most of the duration of the simulation. The footpoints\nof the magnetic bundle are not static, and the bundle is continu-\nously changing and evolving. The lengths of the individual loops\nin the magnetic bundle change significantly over the duration of\nthe simulation, sometimes on a timescale of minutes.\n4. Oscillations and damping\nFull spatial coordinates of the magnetic loops in the bundle at\nany point in time enable us to obtain the evolution of the physical\nquantities along the loops, including both scalar and vector quan-\ntities. To aid the oscillation analysis, we decompose the velocity\nvector into three velocity components relative to the direction\n0 250 500 750 1000 1250 1500 1750−50050v[km s−1]vT vR vN\n0 250 500 750 1000 1250 1500 1750−6−4−2log (Q[W m−3]) QJoule Qvisc\n0 250 500 750 1000 1250 1500 1750\nt[s]102030L[Mm]5.505.756.006.25\nlog (T[K])\n200300400\nvA[km s−1]Fig. 4. Top: The evolution of the longitudinal (green), normal (orange)\nand the binormal velocity component (blue) at the apex of the bundle.\nMiddle: The evolution of the Joule volumetric heating rate (red dashed),\nthe viscous volumetric heating rate (red dotted) and the temperature\n(black) at the apex of the bundle. Bottom: The evolution of the average\nlength (blue) and the Alfvén velocity at the apex of the loops in the\nbundle. Blue regions indicate the time-range corresponding to the two\noscillations.\nof the magnetic field, vT,vNandvR. The longitudinal velocity\nvT=v·Tcorresponds to the velocity component aligned with the\ntangent vector of the magnetic field line given by T=B/|B|. The\nnormal velocity vN=v·Ncorresponds to the velocity component\nalong the normal vector of the field line given by N=dT\nds/|dT\nds|.\nIn the case of a closed magnetic loop, vNrepresents the motion\nin the plane of the loop and perpendicular to the loop tangent.\nFinally the third velocity component aligned with the binormal\nvector is given by vR=v·Rwhere R=T×N. In the case of a\nclosed magnetic loop, vRcorresponds to transverse motion per-\npendicular to the plane of the loop and to the loop tangent. Unit\nvectors T,N, and Rtogether form an orthogonal coordinate sys-\ntem known as the Frenet frame of reference. Such coordinate\nsystem is well-suited to analysing oscillations in complex 3D\nmagnetic field geometries (e.g., Carlsson & Bogdan 2006; Fe-\nlipe 2012; Leenaarts et al. 2015; González-Morales et al. 2019;\nKohutova & Popovas 2021).\nThe collective behaviour of the loop bundle is apparent from\nFig. 3, which shows the evolution of the height of the individ-\nual magnetic loops in the bundle at x=13 Mm. The evolution\nshows two clear instances of oscillatory behaviour, starting at\nt=400 s and t=1100 s and lasting about 400 s in both cases,\nas indicated marked by blue regions in Figs. 3 and 4. The os-\ncillations are damped and occur in a plane perpendicular to the\nbundle axis. From the evolution shown in animations of Figs.\n1 and 2 it is clear that these oscillations correspond to a trans-\nverse standing mode of the bundle. The point of maximum os-\ncillation displacement and hence maximum oscillation velocity\namplitude lies close to the apex of the bundle with the bundle\nfootpoints acting as nodes of the standing oscillation. The oscil-\nlations are triggered by impulsive events in the corona associated\nArticle number, page 4 of 9Kohutova et al.: Damping of coronal oscillations in 3D MHD simulations\n400 450 500 550 600 650 700 750 800−10−50510vN[km s−1]A = -10 km/s\nτ= 176 s\nP = 177 s\nk = -0.04\n1100 1150 1200 1250 1300 1350 1400−10010vN[km s−1]A = -16 km/s\nτ= 198 s\nP = 191 s\nk = -0.28\n400 450 500 550 600 650 700 750 800−6−5−4−3log (Q[W m−3])QJoule Qvisc\n1100 1150 1200 1250 1300 1350 1400−5.0−4.5−4.0−3.5−3.0log (Q[W m−3])QJoule Qvisc\n400 450 500 550 600 650 700 750 800\nt[s]5.75.85.96.0log (T[K])\n1100 1150 1200 1250 1300 1350 1400\nt[s]6.056.106.15log (T[K])\n0.030.040.05\nω[m s−1]\n0.030.040.050.060.07\nω[m s−1]|ωx||ωz|\nFig. 5. Top: The evolution of the detrended normal velocity component at the bundle apex averaged over the oscillating fieldlines in the bundle.\nBest-fit is shown in red. Middle: The evolution of the Joule (dashed line) and viscous volumetric heating rate (dotted line) at the bundle apex.\nBottom: Temperature (black), the x-component of the vorticity averaged in the vicinity of the bundle apex (solid blue line) and the z-component\nof the vorticity averaged in the vicinity of the right bundle footpoint (dotted blue line). The time intervals shown correspond to the time ranges\nhighlighted in blue in Figs. 3 and 4.\nwith a peak in the Joule heating and subsequent large-scale dis-\nplacement of the analysed bundle.\nWe analyse the evolution of the vRandvNcomponents by\naveraging the velocity components over the oscillating loops in\nthe bundle at the loop apex, which we define as the point halfway\nbetween the two footpoints of the loop (Fig 4). The oscillations\nare most clearly detectable in the vNcomponent. In both cases\nthe oscillations follow large scale perturbations of the magnetic\nbundle.\nTo remove the large-scale trends in the vNevolution corre-\nsponding to a bulk motion of the bundle, the oscillation time-\nseries are detrended by subtracting a best-fit second-degree poly-\nnomial. The time-series are then fitted with the function\nv(t)=v0exp\u0012\n−t\nτ\u0013\nsin\u00122πt\nP+kt−Φ\u0013\n(1)\nwhich includes both the oscillation damping and linear change\nin the oscillation period (Fig. 5). Here v0corresponds to the ve-\nlocity amplitude, τis the decay time, Pis the oscillation period,\nkis a parameter controlling the linear change in period and Φis\nthe phase. We find that the initial velocity amplitudes are 10 km /s\nand 16 km /s, these are damped with decay times of 176 s and 198\ns respectively, corresponding to roughly 3 oscillation periods be-\ning detectable before the oscillation decays. The periods for the\nfirst and second oscillation are 177 s and 191 s respectively. The\noscillation periods in the both cases decrease over the duration of\nthe oscillation; by approximately 15 s for oscillation 1 and 100 s\nfor oscillation 2. The Joule and viscous volumetric heating ratesshown in Fig. 5 are also averaged over the oscillating loops at the\napex bundle. Over the duration of oscillations the heating rates at\nthe apex of the bundle are variable, but overall have an increasing\ntrend during the later stages of both oscillations. Similarly, the\naverage temperature at the bundle apex increases by around 0.5\nMK over the duration of the oscillation in both cases. Finally, we\nshow the evolution of the absolute value of the x-component of\nthe vorticity ωxin the y−zplane at x=13 Mm, averaged over the\narea surrounding the bundle apex. The ωx-component is chosen\nbecause the axis of the loop at the apex is roughly aligned with\nthex-axis and the oscillatory motion mostly occurs in the plane\nperpendicular to the loop axis. During the first oscillation, the\nvorticity in the surroundings of the oscillating bundle increases\nto reach the maximum value around 170 s after the oscillation\nonset, followed by a gradual decrease to the pre-oscillation val-\nues. In the later case, the vorticity in the vicinity of the bundle\napex decreases during the entire oscillation duration. For the sec-\nond oscillation we also calculate the evolution of the ωzvorticity\ncomponent at the right footpoint at the height of 3 Mm, as this\nundergoes strong transverse displacement. Similar vorticity peak\nas at the apex position of the first oscillation can be seen in the\nevolution here.\nTo further understand the evolution of the vorticity close\nto the oscillating bundle, we show the velocity field given by\nvyey+vzezin the y−zplane at x=13 Mm which is nearly per-\npendicular to the bundle axis and in the plane of the bundle os-\ncillations (Fig 6). We calculate the velocity field before the onset\nof the oscillation, during the oscillation and at the end of the os-\nArticle number, page 5 of 9A&A proofs: manuscript no. waves\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 240 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 500 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 700 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 900 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 1100 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 1390 s\n−40−2002040\nvz[km s−1]\n−40−2002040\nvz[km s−1]\nFig. 6. Top: The vzvelocity component and the corresponding velocity field streamlines in cuts parallel to the y−zplane at x=13 Mm shown before\nthe oscillation onset (left), during the oscillation (middle) and at the end of the oscillation (right). Bottom: Same as above but for oscillation 2. The\nred cross markers correspond to the positions at which the individual loops in the bundle intersect the cut. The locations of the enhanced shear\nin the vzvelocity component correspond to strong counter-directional flows. In both cases several vortices develop in the vicinity of oscillating\nstructures.\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 240 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 500 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 700 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 900 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 1100 s\n10 12 14 16 18\ny[Mm]8101214z[Mm]t = 1390 s\n0200400600800\nvA[km s−1]\n0200400600800\nvA[km s−1]\nFig. 7. Top: The Alfvén speed and the corresponding velocity field streamlines in cuts parallel to the y−zplane at x=13 Mm shown before the\noscillation onset (left), during the oscillation (middle) and at the end of the oscillation (right). Bottom: Same as above but for oscillation 2. The\nred cross markers correspond to the positions at which the individual loops in the bundle intersect the cut.\ncillation. The shear flows are abundant in the vicinity of the bun-\ndle, this shows as counter-directional large magnitude flows in\nthevzcomponent of the velocity. We find that the velocity shear\nis strongest before and during the oscillation. Associated with\nthis is the development of several vortices visible in the velocity\nstreamlines in the close proximity of the oscillating structures.\nThe velocity shear in this region weakens and the vortices mostly\ndisappear once the oscillations have decayed. The vortices visi-\nble in the velocity streamlines are always located in the regions\nwith strong velocity shear, suggesting they originate due to the\ndevelopment of the Kelvin-Helmholtz instability. The size of the\nvortices is of the order of few Mm, which matches the transverse\nlength scale in the Alfvén speed variation (Fig. 7), as predicted\nby the straight flux-tube models (Antolin & Van Doorsselaere2019). The Kelvin Helmholtz instability in magnetised plasmas\nis inhibited by the magnetic tension, in this case however, the di-\nrection of the magnetic field is perpendicular to the flow velocity\nvector due to the oscillatory motion. The vortices can therefore\ndevelop without distorting the magnetic field, which would lead\nto stabilizing magnetic tension (Hillier et al. 2019; Barbulescu\net al. 2019). We note that both presence of vortices and shear\nflows will contribute to the increased values of ωxin the anal-\nysed region.\nArticle number, page 6 of 9Kohutova et al.: Damping of coronal oscillations in 3D MHD simulations\n5. Discussion\n5.1. Oscillation parameters\nDisplacement and velocity amplitudes in both instances of bun-\ndle oscillations are in agreement with the values typically ob-\nserved in active region coronal loops (e.g. White & Verwichte\n2012), although this also depends on the type and magnitude of\nthe perturbation responsible for the excitation of the oscillation.\nThe detected oscillation periods are also within the range com-\nmonly seen in the observations (Nechaeva et al. 2019). We note\nthat the oscillation periods decrease over the duration of the os-\ncillation in both cases. This can be explained by changing phys-\nical properties in the magnetic loop bundle, such as changing\nloop length or distribution of the plasma along the loop. Such\nevolution is not surprising in a dynamic environment like this, as\nthe length of the oscillating loops as well as the values of Alfvén\nspeed can change on timescales comparable to the duration of\nthe oscillation. The expected values of the oscillation periods es-\ntimated from P∼2L/vAusing the values for the average length\nand the Alfvén speed at the apex of the oscillating bundle loops\nat the beginning of each oscillation are 205 s and 175 s for os-\ncillation 1 and 2 respectively. This roughly agrees with the de-\ntected initial periods, the change in the loop properties during\nthe oscillation duration will however a ffect the oscillation peri-\nods. Changes in oscillation period over the duration of the oscil-\nlation due to changes in properties of the oscillating loop have\nboth been seen in observations and been reproduced by numer-\nical modelling (Kohutova & Verwichte 2017; Verwichte & Ko-\nhutova 2017; Su et al. 2018). Such shift in oscillation properties\nhas a diagnostic potential using coronal seismology methods. A\nmore detailed test of coronal seismology methods in convection-\nzone-to-corona simulations involving synthetic observables will\nbe addressed in a follow-up study. Oscillation damping times are\ncomparable to oscillation periods in both cases and in agreement\nwith the observations of damped transverse oscillations, which\nare typically observed to decay within 3-4 oscillation periods\n(Goddard et al. 2017). We do not analyse the detailed profile of\nthe oscillation damping, that is whether the damping profile is\nbetter represented by an exponential profile, Gaussian profile or\na transition from Gaussian to an exponential profile as described\nin Hood et al. (2013); Pascoe et al. (2016). Such a model is based\non an assumption of a monolithic cylindrical coronal loop with\na non-homogeneous boundary layer at the loop interface, which\nis not representative of coronal structures in the self-consistent\nconvection-zone-to-corona simulations.\n5.2. Collective behaviour\nThe bundle of magnetic loops in the simulation does not oscil-\nlate in isolation. It is in fact di fficult to isolate the oscillating\nstructures from a static, background plasma, as there is a large\ndegree of collective behaviour among the structures in the sim-\nulated corona. The evolution of the individual magnetic loops in\nthe bundle is not identical, but averaging the physical quantities\nat the bundle apex gives us the overall evolution of the bundle.\nThis implies that care should be taken when modelling individ-\nual coronal loops as isolated structures. As the coronal loop evo-\nlution is coupled to the environment, the arcade model described\nby Hindman & Jain (2021) is more representative of the evolu-\ntion of the coronal structures in our simulation. The observations\nalso suggest that the oscillations of individual coronal loops are\noften coupled to the oscillation of nearby magnetic structuresand rarely occur in isolation (Verwichte et al. 2004, 2009; Jain\net al. 2015).\nRegardless of the geometry, oscillating large scale structures\ncreate shear flows that lead to Kelvin-Helmholtz instability and\nthe subsequent development of vortices if the instability is not\ninhibited by the magnetic tension. The shear created by large-\nscale translational motions also contributes to the development\nof Kelvin-Helmholtz instability, provided the direction of the\nmotion is perpendicular to the coronal magnetic field (Hillier\net al. 2019; Barbulescu et al. 2019). The development of vor-\ntices in our simulation may therefore be a result of a com-\nplex interplay of oscillatory and translational motions of coronal\nstructures. The traditional models of Kelvin-Helmholtz unstable\nloops (e.g. Terradas et al. 2008; Antolin et al. 2014; Karampelas\net al. 2017) assuming well-defined oscillating cylindrical loops\nembedded in a static plasma are however too idealised, as there\nare no loops with clearly defined cross-sections in our simula-\ntion. Because of a lack of clear loop boundaries the vortices are\ninstead developed at the locations of maximum velocity shear.\n5.3. Oscillation damping and dissipation\nWe stress the importance of the oscillation damping time τas\nthis corresponds to the rate at which the wave energy is either\nconverted into another wave mode or dissipated. Empirical scal-\ning of the damping time with the loop oscillation period is com-\nmonly cited as an indirect evidence for resonant absorption being\nthe primary mechanism responsible for loop oscillation damp-\ning (see e.g. review by Nakariakov & Kolotkov (2020)). It has\nhowever been argued that the use of such scaling laws for dis-\ncriminating between di fferent damping mechanisms is question-\nable due to the inherent dependence of the damping time on the\nloop parameters (Arregui et al. 2008). We note that traditionally\nused scaling laws for resonant absorption assume large density\ncontrast between the loop and the surrounding plasma (Ofman\n& Aschwanden 2002), which is not the case in our simulation,\nas shown in Fig. 1. Oscillation damping times are further ex-\npected to depend on the magnetic Reynolds number given by\nRm=UL/η, where U is a typical velocity scale, L is a typical\nlength scale and ηis magnetic di ffusivity. Rmvaries in the sim-\nulation as Bifrost uses η(r,t) that is spatially and temporally de-\npendent; Rmin the simulation is however several orders of mag-\nnitude smaller than the estimated values in the solar corona.\nWe note that the traditional models for oscillation damping\ndue to resonant absorption rely on a presence of an inhomoge-\nneous layer at the boundary of a thin cylindrical loop with an\nAlfvén speed gradient (Ruderman & Roberts 2002). This is not a\nvalid approximation for the loops in our simulation, event though\nAlfvén speed gradients are abundant in the simulated corona. We\nsee no clear evidence of mode conversion in the oscillating bun-\ndle (that is, conversion to clearly identifiable spatially localised\nazimuthal oscillations); instead, the evolution of the physical\nquantities at the apex of the loop bundle suggests that the wave\nenergy is dissipated into heat through the development of shear\nflows, leading to an increase in the viscous and resistive dissi-\npation over the duration of the oscillation (in MHD models, the\nresistive and viscous dissipation terms are a parametrisation of\nprocesses operating on kinetic scales). Velocity shear leads to de-\nvelopment of Kelvin-Helmholtz vortices which are detectable in\nthe velocity field in the plane perpendicular to the bundle cross-\nsection. The increase of the ωxcomponent in the vicinity of the\nbundle apex during the first oscillation decay suggests that the\nshear due to the oscillatory motion drives the development of\nvortices which then dissipate, leading to subsequent ωxdecrease\nArticle number, page 7 of 9A&A proofs: manuscript no. waves\neffectively explaining oscillation damping and dissipation. How-\never, the absence of such a clear vorticity peak at the loop apex\nduring the second oscillation decay suggests the picture in this\ncase is not as clear and multiple processes can contribute to the\noscillation damping. The evolution of loop oscillations might be\nfurther a ffected by the motion of the loop footpoints, as these are\nnot static. We note that the wave dissipation is only one of the\nseveral possible mechanisms contributing to the large tempera-\nture increase seen at the apex of the oscillating loops.\nThe onset of Kelvin-Helmholtz instability in models of trans-\nversely oscillating loops is linked to the development of turbu-\nlence leading to the formation of small scales which allows for\nfast dissipation of the wave energy (Hillier et al. 2020). For a\nloop to be considered truly ’turbulent’ it is however necessary to\ndemonstrate a non-linear cascade of energy to small scales. We\ncannot draw any conclusions about the presence of turbulent be-\nhaviour /lack thereof in our simulation due to limits posed by spa-\ntial resolution as well as by the magnetic di ffusivity and viscos-\nity. We also note that low Reynolds numbers in self-consistent\nMHD simulations artificially restrict the cascade to small scales\n(Howson et al. 2017).\nKelvin-Helmholtz vortices developed during oscillation\ndamping have also been proposed as being responsible for coro-\nnal oscillations appearing as decayless in certain emission lines\n(Antolin et al. 2016). In such model, however, the KHI vortices\nwere formed at much smaller spatial scales due to the sharp den-\nsity (and Alfvén speed) contrast at the loop boundary which is\nnot the case for the loop bundle analysed in this work.\n5.4. Implication for coronal loop models\nWe find that oscillation parameters and evolution observed in\ncoronal loops are reproduced by self-consistent simulations\nwhich include complex magnetic field geometry and density\nstructuring and which do not contain well-defined coronal loops.\nThis approach provides insight into how does the oscillating loop\nbundle actually evolve in three dimensions, including the de-\ntailed evolution of the magnetic field, into the degree of collec-\ntive oscillation of the surrounding plasma and into the physical\nmechanisms associated with the oscillation damping and dissi-\npation. This type of dynamics is impossible to capture by ide-\nalised straight flux tube models. This might have widespread im-\nplications for the accuracy of the coronal seismology methods,\nwhich are mostly based on cylinder approximations for coro-\nnal loops. Even recent numerical studies of the evolution of ini-\ntially homogeneous coronal loops in response to transverse mo-\ntions suggest that the highly idealised picture of coronal loops as\nmonolithic plasma cylinders is unlikely to be realistic in the first\nplace (Antolin & Van Doorsselaere 2019). The question remains\nhow realistic our current self-consistent simulations really are\nwhen it comes to reproducing detailed characteristics of the so-\nlar corona. However, despite the complex collective behaviour in\nsuch self-consistent simulations some results from simple coro-\nnal loop models are reproduced, namely the generation of the\nKelvin-Helmholtz vortices by the transverse motions. The sim-\nple coronal loop models provide a lot of value for understanding\nfundamental physical processes at play. Caution should however\nbe taken when drawing conclusions from observations whether\nthe assumptions made in the models are still applicable in the\nanalysed scenario. Combination of simple and self-consistent\nmodels is necessary for detailed understanding of the oscillatory\nbehaviour in the corona.\nFinally, we note that the length of loops we can simulate in\nthis type of setup is limited by the size of the simulation domain,with the maximum length of magnetic loops in the simulation\nused in this work being 20 - 30 Mm. This a ffects the parameter\nspace that is accessible by such models, as several oscillation\nparameters scale with loop length. Larger domains are therefore\nneeded for more accurate one-to-one comparison of oscillation\nparameters with observations.\n6. Conclusions\nFor the first time, we analysed the damping of coronal oscilla-\ntions using a self-consistent 3D radiation-MHD model of the\nsolar atmosphere spanning from the convection zone into the\ncorona, the associated oscillation dissipation and heating, and\nfinally the physical processes responsible for the damping and\ndissipation. The simulated corona formed in such a model does\nnot depend on any prior assumptions about the shape of the coro-\nnal loops. Using magnetic field tracing we analysed the evolution\nof a bundle of magnetic loops in the centre of the simulation do-\nmain. The magnetic bundle shows dynamic evolution and a large\ndegree of collective behaviour of the individual loops in the bun-\ndle. We find that the bundle of magnetic loops shows damped\ntransverse oscillations in response to perturbations in two sepa-\nrate instances with oscillation periods of 177 s and 191 s, veloc-\nity amplitudes of 10 km s−1and 16 km s−1and damping times\nof 176 s and 198 s, respectively. The oscillation parameters and\nevolution observed are in line with the values typically seen in\nobservations of coronal loop oscillations. The oscillation peri-\nods decrease in both instances during the oscillations. We find\nthat transverse coronal oscillations lead to the development of\nvelocity shear in the simulated corona resulting in the formation\nof vortices with sizes around one Mm seen in the velocity field\ncaused by the Kelvin-Helmholtz instability, contributing to the\ndamping and dissipation of the transverse oscillations into heat.\nAssuming the structure of the corona in the self-consistent mod-\nels is indeed realistic, our models of monolithic and static coro-\nnal loops with constant lengths might need to be reevaluated in\nfavour of more realistic models accounting for loop properties\nchanging with time and relaxing the assumption of highly ide-\nalised waveguides.\nAcknowledgements. PK acknowledges funding from the Research Council of\nNorway, project no. 324523. 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S. & Verwichte, E. 2012, A&A, 537, A49\n¸ Sahin, S. & Antolin, P. 2022, ApJL, 931, L27\nArticle number, page 9 of 9"    },    {        "title": "2306.04617v2.Helicity_dependent_optical_control_of_the_magnetization_state_emerging_from_the_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "1  Helicity-dependent optical control of the magnetization state emerging from the Landau-Lifshitz-Gilbert equation  Benjamin Assouline, Amir Capua*  Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel   *e-mail: amir.capua@mail.huji.ac.il  Abstract:   It is well known that the Gilbert relaxation time of a magnetic moment scales inversely with the magnitude of the externally applied field, 𝑯, and the Gilbert damping, 𝜶. Therefore, in ultrashort optical pulses, where 𝑯 can temporarily be extremely large, the Gilbert relaxation time can momentarily be extremely short, reaching even picosecond timescales. Here we show that for typical ultrashort pulses, the optical control of the magnetization emerges by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. Surprisingly, when circularly polarized optical pulses are introduced to the LLG equation, an optically induced helicity-dependent torque results. We find that the strength of the interaction is determined by 𝜼=𝜶𝜸𝑯/𝒇𝒐𝒑𝒕, where 𝒇𝒐𝒑𝒕 and 𝜸 are the optical frequency and gyromagnetic ratio. Our results illustrate the generality of the LLG equation to the optical limit and the pivotal role of the Gilbert damping in the general interaction between optical magnetic fields and spins in solids.  2  The ability to control the magnetization order parameter using ultrashort circularly polarized (CP) optical pulses has attracted a great deal of attention since the early experiments of the all-optical helicity dependent switching (AO-HDS) [1-4]. This interaction was found intriguing since it appears to have all the necessary ingredients to be explained by a coherent transfer of angular momentum, yet it occurs at photon energies of 1\t−\t2\t𝑒𝑉, very far from the typical resonant transitions in metals. The technological applications and fundamental scientific aspects steered much debate and discussion [5,6], and the experiments that followed found dependencies on a variety of parameters including material composition [7-9], magnetic structure [10-12], and laser parameters [1,3,13], that were often experiment-specific [4]. Consequently, a multitude of mechanisms that entangle photons [14,15], spins [16,17], and phonons [18,19] have been discovered. References [4,20] provide a state of the art review of the theoretical and experimental works of the field. Ferromagnetic resonance (FMR) experiments are usually carried out at the 𝐺𝐻𝑧 range. In contrast, optical fields oscillate much faster, at ~\t400−800\t𝑇𝐻𝑧. Therefore, it seems unlikely that such fast-oscillating fields may interact with magnetic moments. However, the amplitude of the magnetic field in ultrashort optical pulses can, temporarily, be very large such that the magnetization may respond extremely fast. For example, in typical experiments having 40\t𝑓𝑠−1\t𝑝𝑠 pulses at 800\t𝑛𝑚, with energy of 0.5\t𝑚𝐽 that are focused to a spot size of ~0.5\t𝑚𝑚$, the peak magnetic flux density can be as high as ~\t5\t𝑇, for which the corresponding Gilbert relaxation time reduces to tens of picoseconds in typical ferromagnets. Here we show that ultrashort optical pulses may control the magnetization state by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. We find that the strength of the interaction is determined by 𝜂=𝛼𝛾𝐻/𝑓%&', where 𝑓%&' and 𝛼 are the angular optical frequency and the Gilbert damping, respectively, and 𝛾 is the gyromagnetic ratio. Moreover, we show that for circularly polarized (CP) pulses, the polarity of the optically induced torque is determined by the optical helicity. From a quantitative analysis, we find that a sizable effective out-of-plane field is generated which is comparable to that measured experimentally in ferromagnet/heavy-metal (FM/HM) material systems. 3  The LLG equation is typically not applied in the optical limit, and hence requires an alternative mathematical framework whose principles we adopt from the Bloch equations for semiconductor lasers [21,22]. We exploit the analogy between the magnetization state and the Bloch vector of a two-level system (TLS) [23,24] by transforming the LLG equation under a time-varying magnetic field excitation to the dynamical Maxwell-Bloch (MB) equations in the presence of an electrical carrier injection. In this transformation, the reversal of the magnetization is described in terms of population transfer between the states. The paper is organized as follows: We begin by transforming the LLG equation to the density matrix equations of a TLS. We then identify the mathematical form of a time-dependent magnetic field in the LLG equation, 𝐻AA⃗&()&↓↑, that is mapped to a time-independent carrier injection rate into the TLS. Such excitation induces a population transfer that varies linearly in time and accordingly to a magnetization switching profile that is also linear in time. The mathematical 𝐻AA⃗&()&↓↑ field emerges naturally as a temporal impulse-like excitation. We then show that when 𝛼 is sizable, 𝐻AA⃗&()&↓↑ acquires a CP component whose handedness is determined by the direction of the switching. By substituting 𝐻AA⃗&()&↓↑ for an experimentally realistic picosecond CP Gaussian optical magnetic pulse, we show that it can also exert a net torque on the magnetization. In this case as well, the helicity determines the polarity of the torque. Finally, we present a quantitative analysis that is based on experimental data. The LLG equation describing the dynamics of the magnetization, 𝑀AA⃗, where the losses are introduced in the Landau–Lifshitz form is given by [25]: 𝑑𝑀AA⃗𝑑𝑡=\t−𝛾1+𝛼$𝑀AA⃗×𝐻AA⃗−𝛾𝛼1+𝛼$1𝑀,𝑀AA⃗×𝑀AA⃗×𝐻AA⃗.(1) Here 𝑀, and 𝐻AA⃗ are the magnetization saturation and the time dependent externally applied magnetic field, respectively. We define 𝐻AA⃗-.. by: 𝐻AA⃗-..≜K𝐻AA⃗−\t𝛼𝑀,𝐻AA⃗×𝑀AA⃗\tL,(2) and in addition, 𝜅≜/012!O𝐻-..\t4−𝑗𝐻-..\t5Q/2 and 𝜅6\t≜/012!𝐻-..\t7, where 𝜅 and 𝜅6 can be regarded as effective AC and DC magnetic fields acting on 𝑀AA⃗, respectively. We 4  transform 𝑀AA⃗ to the density matrix elements of the Bloch state in the TLS picture and compare it to the Bloch equations describing a semiconductor laser that is electrically pumped [26]:  ⎩⎪⎨⎪⎧𝜌̇00=𝛬0−𝛾0𝜌00+𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇$$=𝛬$−𝛾$𝜌$$−𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇0$=\t−(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗(𝜌00−𝜌$$)𝑉0$\t.\t\t(3) In this reference model, 𝛬0 and 𝛬$ are injection rates of carriers to the ground and excited states of the TLS, respectively. They are assumed to be time independent and represent a constant injection of carriers from an undepleted reservoir [27]. 𝛾0 and 𝛾$ are the relaxation rates of the ground and excited states, and 𝛾;<= is the decoherence rate due to an inhomogeneous broadening. 𝑉0$ is the interaction term and 𝜔89: is the resonance frequency of the TLS. Figure 1(a) illustrates schematically the analogy between the magnetization dynamics and the electrically pumped TLS. We find the connection between the LLG equation expressed in the density matrix form and the model of the electrically pumped TLS: ]𝛬0−𝛾0𝜌00+[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]=\t−𝑗𝜅𝜌$0+𝑐.𝑐.𝛬$−𝛾$𝜌$$−[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]=\t𝑗𝜅𝜌$0+𝑐.𝑐.−(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗𝑀7𝑉0$=\t−𝑗𝜅6𝜌0$\t+𝑗𝜅𝑀7.(4) The pumping of the excited and ground states by the constant 𝛬0 and 𝛬$ rates implies that the reversal of the magnetization along the ∓\t𝑧̂ direction is linear in time. Using Eq. (4) we find 𝜅, and hence a field 𝐻AA⃗, that produces such 𝛬0 and 𝛬$. We define this field as 𝐻AA⃗&()&↓↑: 𝐻AA⃗&()&↓↑=\t±𝛬&𝑀,$−𝑀7$f𝑀5−\t𝑀40g.(5) 𝐻AA⃗&()&↓↑ depends on the temporal state of 𝑀AA⃗ while 𝛬&=𝛾𝛬0/(1+𝛼$) is the effective field strength parameter. 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ induce a linear transition of 𝑀AA⃗ towards the –𝑧̂ and +𝑧̂ direction, respectively. 5  Figure 1(b) presents the outcome of the application of 𝐻AA⃗&()&↓↑ by numerically integrating the LLG equation. The Figure illustrates 𝐻AA⃗(𝑡), 𝑀7(𝑡), and the 𝑧̂ torque, O−𝑀AA⃗×𝐻AA⃗Q7, for alternating 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ that switch 𝑀AA⃗ between ∓𝑀,𝑧̂. The magnitude of 𝛬& determines the switching time, 𝛥𝜏↓↑, chosen here to describe a femtosecond regime. Equation (4) yields 𝛥𝜏↓↑=(1+𝛼$)𝑀,/(\t𝛾𝛬&)≈𝑀,/𝛾𝛬& in which 𝑀7 is driven from 𝑀7=0 to 𝑀7≅±𝑀, (for derivation, see Supplemental Material Note 1). It is seen that O−𝑀AA⃗×𝐻AA⃗Q7 is constant when 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑ are applied so that the switching profile of 𝑀7 is linear in time. It is also seen that 𝐻AA⃗&()&↓↑ requires that m𝐻AA⃗m diverge as 𝑀7 approaches ±𝑀,, which is not experimentally feasible. To account for a more realistic excitation, in Fig. 1(c) we simulated a pulse whose trailing edge was taken as a reflection in time of 𝐻AA⃗&()&↓↑, and that is shorter by an order of magnitude as compared to the leading edge. In this case 𝑀AA⃗ remains in its final state when 𝐻AA⃗ is eventually turned off. The polarization state of 𝐻AA⃗&()&↓↑ is determined from the polarization state of the transverse components of 𝑀AA⃗. Next, we show that for larger 𝛼, 𝑀5(𝑡) becomes appreciable such that 𝐻AA⃗&()&↓↑ acquires an additional CP component. This result emerges naturally from the Bloch picture: we recall that the transverse components of 𝑀AA⃗ are expressed by the off-diagonal density matrix element. According to Eq. (3), 𝜌0$ oscillates at 𝜔89: and decays at the rate 𝛾;<=, whereas the sign of 𝜔89: determines the handedness of the transverse components of 𝑀AA⃗. Namely, the ratio between 𝜔89: and 𝛾;<= determines the magnitude of the circular component in the n𝑀4(𝑡),𝑀5(𝑡)o trajectory. Under the application of 𝐻AA⃗&()&↓↑, Eq. (4) yields 𝜔89:=±𝛾𝛬&𝛼𝑀,/[(𝑀,$−𝑀7$)(1+𝛼$)] and 𝛾;<==∓𝛾𝛬&𝑀7/[(𝑀,$−𝑀7$)(1+𝛼$)] readily showing that |𝜔89:/𝛾;<=|=𝛼𝑀,/𝑀7 increases with 𝛼, so that 𝐻AA⃗&()&↓↑ acquires an additional CP component (see Supplemental Note 2 for full derivation). Figure 2 illustrates these results. Panel (a) presents the components of 𝑀AA⃗(𝑡) for the same simulation in Fig. 1(b). It is seen that 𝑀5(𝑡) is negligible and thus 𝐻AA⃗&()&↓↑ remains linearly polarized. When 𝛼 is increased, an elliptical trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane emerges, while the constant transition rate of 𝑀7 persists as illustrated in Fig. 2(b). In this case, 6  𝐻AA⃗&()&↓↑ acquires a right-CP (RCP) or left-CP (LCP) component depending on the choice of 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑. The coupling between the handedness and reversal direction in a femtosecond excitation is reminiscent of the switching reported in AO-HDS experiments and emerges naturally in our model. These results call to examine the interaction of the CP magnetic field of a short optical pulse with 𝑀AA⃗. Figure 3(a) presents the calculation for experimental conditions [4]. The results are shown for an 800\t𝑛𝑚 optical magnetic field of an RCP Gaussian optical pulse 𝐻AA⃗%&'(𝑡). The pulse has a duration determined by 𝜏&, an angular frequency 𝜔%&', and a peak amplitude 𝐻&->? that is reached at 𝑡=𝑡&->?. In our simulations 𝜏&=3\t𝑝𝑠 and 𝑡&->?=10\t𝑝𝑠. The pulse energy was ~\t5\t𝑚𝐽 and assumed to be focused to a spot size of ~\t100\t𝜇𝑚$, for which 𝐻&->?=8⋅10@\t𝐴/𝑚. Here we take 𝛼=0.035 [28,29]. For such conditions, the Gilbert relaxation time corresponding to 𝐻&->? is 𝜏2=02/A\"#$%≈16\t𝑝𝑠 [30]. It is readily seen that for such 𝜏2 the magnetization responds within the duration of the optical pulse indicating that the interaction between the optical pulse and 𝑀AA⃗ becomes possible by the LLG equation. Following the interaction, 𝑀7=−5×10BC⋅𝑀:, namely a sizable net longitudinal torque results. In agreement with the prediction of the TLS model, pulses of the opposite helicity induce an opposite transition as shown in Fig. 3(b). The results are compared to the measured data discussed in Supplemental Material Note 3.  To this end we simulate the same conditions of the measurements including optical intensity and sample parameters. Accordingly, we find from our calculations an effective field which is of the same order of magnitude as measured.   For a given pulse duration, we define the interaction strength parameter 𝜂=2𝜋𝛼𝛾𝐻&->?/𝜔%&', which expresses the ratio between 𝜏2 and the optical cycle and is 2.5⋅10BC in Fig. 3(a). The principles of the interaction can be better understood at the limit where 𝜂→1 and for which the interaction can be described analytically. To this end, we set 𝜂=1. The higher optical magnetic fields required for this limit are achievable using conventional amplified femtosecond lasers, for example by focusing a ~\t5\t𝑚𝐽 pulse into a spot size of ~\t1\t𝜇𝑚$. Figure 3(c) illustrates the results for an RCP 𝐻AA⃗%&' pulse of a duration of 20\t𝑓𝑠 determined by the full width at half-maximum of the 7  intensity. The Figure reveals the different stages of the interaction. During the leading edge, for 𝑡<~\t40\t𝑓𝑠, the relative phase between 𝐻AA⃗%&' and 𝑀AA⃗ seems arbitrary. As 𝑡&->? is reached, the Gilbert relaxation time becomes as short as the optical cycle allowing 𝑀AA⃗ to follow 𝐻AA⃗%&' until it is entirely locked to 𝐻AA⃗%&'. In this case, 𝑀AA⃗ undergoes a right-circular trajectory about 𝑧̂. The switching of 𝑀AA⃗ takes place at the final stage of the interaction: During the trailing edge of the pulse, the amplitude of 𝐻AA⃗%&' reduces and 𝜏2 extends, thereby releasing the locking between 𝑀AA⃗ and 𝐻AA⃗%&'. In this case, the switching profile of 𝑀7 is monotonic linear-like in time, closely resembling the transition stemming from a constant carrier injection rate in the Bloch picture. The optically induced transition can be described analytically following the calculation presented in Supplemental Note 4, from which we find the transition rate:  𝛤/𝑀,=∓32√2𝑙𝑛K43L1𝜏&}~𝑙𝑛K𝐻&->?0.27𝐻'=L−~𝑙𝑛𝐻&->?𝐻'=/√2,(6) where 𝐻'==D&\"'$E/2 is the value of 𝐻&->? at 𝜂=1. The rate 𝛤/𝑀, is plotted as well in Fig. 3(c) and reproduces the numerical calculation. 𝛤 depends on the ratio between 𝐻&->? and 𝐻'= and is only weekly dependent on 𝐻&->?. Namely, when 𝐻&->?≫𝐻'=, the circular trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane persists longer after 𝑡&->?, but as the amplitude of the pulse decays below 𝐻'=/√2, 𝑀AA⃗ is driven out of the 𝑥−𝑦 plane and the reversal takes place (see Supplemental Material Note 5). This analysis also holds for LCP pulses, which result in an opposite reversal of 𝑀AA⃗, as shown in Fig. 3(d).  To summarize, in this work we demonstrated that the control of the magnetization by an optical field arises from first principles by introducing the magnetic part of the optical radiation to the LLG equation. This was seen from the comparison between the case where 𝜂≪1 and the case of 𝜂=1. Using the TLS model, we demonstrated the coupling between the optical helicity state and the polarity of the longitudinal torque. A quantitative analysis of the optically induced torque revealed that it can be comparable to that observed in experiments.   8  Figure 1 \n       Fig. 1. (a) Left panel: Illustration of 𝑴AAA⃗ on the Bloch sphere. Right panel: Illustration of the electrically pumped TLS. (b) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ of Eq. (5). The Figure illustrates the temporal plots of 𝑴𝒛/𝑴𝒔, 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑,𝒚 and O−𝑴AAA⃗×𝑯AAA⃗Q𝒛 normalized to unity. (c) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ and a more realistic trailing edge, for the same conditions in (b). Full lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and dashed lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↑.    \n9  Figure 2 \n Fig. 2. Temporal evolution of the components of 𝑴AAA⃗ under the influence of alternating 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑ for (a) small and (b) large damping. Black dashed lines indicate the alternation between 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑.   \n10   \n11  Fig. 3. (a) Magnetization reversal induced by an RCP Gaussian pulse for\t𝜼=𝟐.𝟓⋅𝟏𝟎B𝟒. Top and middle panels depict the temporal evolution of the 𝒙 and 𝒚 components of 𝑴AAA⃗ and 𝑯AAA⃗𝒐𝒑𝒕\tin normalized units. Bottom panel depicts 𝑴𝒛/𝑴𝒔. (b) 𝑴𝒛/𝑴𝒔, for the application of an LCP pulse. (c) Magnetization reversal induced by an RCP Gaussian pulse for\t𝜼=𝟏. Top panel presents the temporal behavior of m𝑯AAA⃗𝒐𝒑𝒕m and 𝝉𝜶, where 𝑯𝒄𝒓𝒊𝒕=𝑯𝒕𝒉/√𝟐 and 𝑯𝟏/𝟐=𝟎.𝟐𝟕𝑯𝒕𝒉. (d) 𝑴𝒛/𝑴𝒔, for the application of an LCP pulse. In (c) and (d), black solid lines represent the analytical solution of 𝜞/𝑴𝒔.   References  [1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, \"All-Optical Magnetic Recording with Circularly Polarized Light\", Physical Review Letters 99, 047601 (2007). [2] J. Hohlfeld, C. D. Stanciu, and A. Rebei, \"Athermal all-optical femtosecond magnetization reversal in GdFeCo\", Applied Physics Letters 94, 152504 (2009). [3] D. Steil, S. Alebrand, A. Hassdenteufel, M. Cinchetti, and M. 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Reithmaier, \"Coherent control in a semiconductor optical amplifier operating at room temperature\", Nature Communications 5, 5025 (2014). [22] A. Capua, O. Karni, G. Eisenstein, and J. P. Reithmaier, \"Rabi oscillations in a room-temperature quantum dash semiconductor optical amplifier\", Physical Review B 90, 045305 (2014). [23] R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, \"Geometrical Representation of the Schrödinger Equation for Solving Maser Problems\", Journal of Applied Physics 28, 49 (1957). [24] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997). [25] Alexander G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC Press, Boca Raton Florida, 1996). [26] L. Allen and J. Eberly, Optical Resonance and Two Level Atoms (Dover Publications, New York, 1987). [27] J. Yao, G. P. Agrawal, P. Gallion, and C. M. Bowden, \"Semiconductor laser dynamics beyond the rate-equation approximation\", Optics Communications 119, 246 (1995). [28] A. Capua, C. Rettner, S.-H. Yang, T. Phung, and S. S. P. Parkin, \"Ensemble-averaged Rabi oscillations in a ferromagnetic CoFeB film\", Nature Commun. 8, 16004 (2017). [29] N. Fujita, N. Inaba, F. Kirino, S. Igarashi, K. Koike, and H. Kato, \"Damping constant of Co/Pt multilayer thin-film media\", Journal of Magnetism and Magnetic Materials 320, 3019 (2008). [30] A. H. Morrish, The Physical Principles of Magnetism (Wiley-IEEE Press, 2001).  "    },    {        "title": "2306.13013v4.Gilbert_damping_in_metallic_ferromagnets_from_Schwinger_Keldysh_field_theory__Intrinsically_nonlocal_and_nonuniform__and_made_anisotropic_by_spin_orbit_coupling.pdf",        "content": "Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field theory:\nIntrinsically nonlocal and nonuniform, and made anisotropic by spin-orbit coupling\nFelipe Reyes-Osorio and Branislav K. Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n(Dated: March 1, 2024)\nUnderstanding the origin of damping mechanisms in magnetization dynamics of metallic ferro-\nmagnets is a fundamental problem for nonequilibrium many-body physics of systems where quantum\nconduction electrons interact with localized spins assumed to be governed by the classical Landau-\nLifshitz-Gilbert (LLG) equation. It is also of critical importance for applications as damping affects\nenergy consumption and speed of spintronic and magnonic devices. Since the 1970s, a variety of\nlinear-response and scattering theory approaches have been developed to produce widely used for-\nmulas for computation of spatially-independent Gilbert scalar parameter as the magnitude of the\nGilbert damping term in the LLG equation. The largely unexploited for this purpose Schwinger-\nKeldysh field theory (SKFT) offers additional possibilities, such as to rigorously derive an extended\nLLG equation by integrating quantum electrons out. Here we derive such equation whose Gilbert\ndamping for metallic ferromagnets is nonlocal —i.e., dependent on all localized spins at a given\ntime—and nonuniform , even if all localized spins are collinear and spin-orbit coupling (SOC) is\nabsent. This is in sharp contrast to standard lore, where nonlocal damping is considered to emerge\nonly if localized spins are noncollinear—for such situations, direct comparison on the example of\nmagnetic domain wall shows that SKFT-derived nonlocal damping is an order of magnitude larger\nthan the previously considered one. Switching on SOC makes such nonlocal damping anisotropic , in\ncontrast to standard lore where SOC is usually necessary to obtain nonzero Gilbert damping scalar\nparameter. Our analytical formulas, with their nonlocality being more prominent in low spatial\ndimensions, are fully corroborated by numerically exact quantum-classical simulations.\nI. INTRODUCTION\nThe celebrated Landau-Lifshitz equation [1] is the\nfoundation of standard frameworks, such as classical mi-\ncromagnetics [2, 3] and atomistic spin dynamics [4], for\nmodelling the dynamics of local magnetization within\nmagnetic materials driven by external fields or currents\nin spintronics [2] and magnonics [3]. It considers localized\nspins as classical vectors M(r) of fixed length normalized\nto unity whose rotation around the effective magnetic\nfieldBeffis governed by\n∂tM=−M×Beff+M×(D ·∂tM), (1)\nwhere ∂t≡∂/∂t. Although spin is a genuine quan-\ntum degree of freedom, such phenomenological equation\ncan be fully microscopically justified from open quantum\nmany-body system dynamics where M(r) tracks the tra-\njectories of quantum-mechanical expectation value of lo-\ncalized spin operators [5] in ferromagnets, as well as in\nantiferromagnets as long as the spin value is sufficiently\nlarge S >1. The presence of a dissipative environment in\nsuch justification invariably introduces damping mecha-\nnisms, which were conjectured phenomenologically in the\nearliest formulation [1], as well as in later renderings us-\ning the so-called Gilbert form of damping [6, 7] written as\nthe second term on the right-hand side (RHS) of Eq. (1).\nThe Gilbert damping Dwas originally considered as a\nspatially uniform scalar D ≡αG, or possibly tensor [8, 9],\n∗bnikolic@udel.edudependent on the intrinsic properties of a material. Its\ntypical values are αG∼0.01 in standard ferromagnetic\nmetals [10], or as low as αG∼10−4in carefully designed\nmagnetic insulators [11] and metals [12]. Furthermore,\nrecent extensions [13–21] of the Landau-Lifshitz-Gilbert\n(LLG) Eq. (1) for the dynamics of noncollinear magneti-\nzation textures find Dto be a spatially nonuniform and\nnonlocal tensor\nDαβ=αGδαβ+ηX\nβ′(M×∂β′M)α(M×∂β′M)β,(2)\nwhere ∂β′≡∂/∂β′, and α, β, β′∈ {x, y, z}.\nIt is generally believed that αGisnonzero only\nwhen SOC [22, 23] or magnetic disorder (or both) are\npresent [15, 24, 25]. For example, αGhas been ex-\ntracted from a nonrelativistic expansion of the Dirac\nequation [22, 23], and spin-orbit coupling (SOC) is vir-\ntually always invoked in analytical (conducted for sim-\nplistic model Hamiltonians) [26–28] or first-principles\ncalculations [24, 25, 29–33] of αGvia Kubo linear-\nresponse [9, 30, 34–36] or scattering [8] theory-based for-\nmulas.\nThe second term on the RHS of Eq. (2) is the\nparticular form [13] of the so-called nonlocal (i.e.,\nmagnetization-texture-dependent) and spatially nonuni-\nform (i.e., position-dependent) damping [13–21, 37]. The\nsearch for a proper form of nonlocal damping has a long\nhistory [19, 37]. Its importance has been revealed by ex-\nperiments [10] extracting very different Gilbert damping\nfor the same material by using its uniformly precessing\nlocalized spins versus dynamics of its magnetic domain\nwalls, as well as in experiments observing wavevector-\ndependent damping of spin waves [38]. Its particulararXiv:2306.13013v4  [cond-mat.mes-hall]  29 Feb 20242\nB\nL lead R leadxyz MnJsd\ne\nee\n(a)\n(b)\n(c)\nFIG. 1. Schematic view of (a) classical localized spins, mod-\neled by unit vectors Mn(red arrows), within an infinite metal-\nlic ferromagnet defined on a cubic lattice in 1D–3D (1D is\nused in this illustration); or (b) finite-size metallic ferromag-\nnet (central region) attached to semi-infinite NM leads termi-\nnating in macroscopic reservoirs, whose difference in electro-\nchemical potentials inject charge current as commonly done\nin spintronics. The localized spins interact with conduction\nelectron spin ⟨ˆs⟩(green arrow) via sd-exchange of strength\nJsd, while both subsystems can experience external magnetic\nfieldB(blue arrow). (c) Nonlocal damping λD\nnn′[Eq. (10)]\nobtained from SKFT vs. distance |rn−rn′|between two sites\nnandn′of the lattice for different dimensionality Dof space.\nform [13] in Eq. (2) requires only noncollinear and non-\ncoplanar textures of localized spins, so it can be nonzero\neven in the absence of SOC, but its presence can greatly\nenhance its magnitude [18] (without SOC, the nonlocal\ndamping in Eq. (2) is estimated [18] to be relevant only\nfor small size ≲1 nm noncollinear magnetic textures).\nHowever, recent quantum-classical and numerically ex-\nact simulations [39, 40] have revealed that αGcan be\nnonzero even in the absence of SOC simply because ex-\npectation value of conduction electron spin ⟨ˆs⟩(r) isal-\nways somewhat behind M(r). Such retarded response of\nelectronic spins with respect to motion of classical lo-\ncalized spins, also invoked when postulating extended\nLLG equation with phenomenological time-retarded ker-\nnel [41], generates spin torque ∝ ⟨ˆs⟩(r)×M(r) [42] and,\nthereby, effective Gilbert-like damping [39–41] that is\nnonzero in the absence of SOC and operative even if\nM(r) at different positions rarecollinear [40]. Including\nSOC in such simulations simply increases [43] the an-\ngle between ⟨ˆs⟩(r) and M(r) and, therefore, the effective\ndamping.\nTo deepen understanding of the origin of these phe-\nnomena observed in numerical simulations, which are\nanalogous to nonadiabatic effects discussed in diversefields where fast quantum degrees of freedom interact\nwith slow classical ones [44–47], requires deriving an an-\nalytical expression for Gilbert damping due to interac-\ntion between fast conduction electrons and slow local-\nized spins. A rigorous path for such derivation is offered\nby the Schwinger-Keldysh nonequilibrium field theory\n(SKFT) [48] which, however, remains largely unexplored\nfor this problem. We note that a handful of studies have\nemployed SKFT to study small systems of one or two\nlocalized spins [49–54] as they interact with conduction\nelectrons. While some of these studies [49, 53, 54] also\narrive at extended LLG equation with nonlocal damp-\ning, they are only directly applicable to small magnetic\nmolecules rather than macroscopic ferromagnets in the\nfocus of our study. It is also worth mentioning that an\nearly work [55] did apply SKFT to the same model we\nare using—electrons whose spins interact via sdexchange\ninteraction with many Heisenberg-exchange-coupled lo-\ncalized spins representing metallic ferromagnet in self-\nconsistent manner—but they did not obtain damping\nterm in their extended Landau-Lifshitz equation, and in-\nstead focused on fluctuations in the magnitude of Mn. In\ncontrast, the vectors Mnare of fixed length in classical\nmicromagnetics [2, 3] and atomistic spin dynamics [4], as\nwell as in our SKFT-derived extended LLG Eq. (9) and\nall other SKFT-based analyses of one or two localized\nspin problems [49–54].\nIn this study we consider either an infinite [Fig. 1(a)],\nor finite [Fig. 1(b)] but sandwiched between two semi-\ninfinite normal metal (NM) leads terminating in macro-\nscopic electronic reservoirs [8, 52, 53], metallic magnet\nwhose localized spins are coupled by ferromagnetic ex-\nchange in equilibrium. The setups in Fig. 1 are of di-\nrect relevance to experiments [10, 38] on external field\n[Fig. 1(a)] or current-driven dynamics [Fig. 1(b)] of lo-\ncalized spins in spintronics and magnonics. Our princi-\npal result is encapsulated by Fig. 1(c)—Gilbert damping,\ndue to conduction electron spins not being able to instan-\ntaneously follow changes in the orientation of classical\nlocalized spins, is always nonlocal and inhomogeneous,\nwith such features becoming more prominent in low-\ndimensional ferromagnets. This result is independently\nconfirmed [Fig. 2] by numerically exact simulations (in\none dimension) based on time-dependent nonequilibrium\nGreen’s function combined with LLG equation (TD-\nNEGF+LLG) scheme [40, 43, 56, 57].\nWe note that conventional linear-response formulas [9,\n30, 34–36] produce unphysical divergent Gilbert damp-\ning [33] in a perfectly crystalline magnet at zero tempera-\nture. In contrast to previously proposed solutions to this\nproblem—which require [58–60] going beyond the stan-\ndard picture of electrons that do not interact with each\nother, while interacting with classical localized spins—\nour formulas are finite in the clean limit, as well as in\nthe absence of SOC. The scattering theory [8] yields a\nformula for αGwhich is also always finite (in the absence\nof SOC, it is finite due to spin pumping [61]). However,\nthat result can only be viewed as a spatial average of our3\nnonlocal damping which cannot produce proper LLG dy-\nnamics of local magnetization [Fig. 3].\nThe paper is organized as follows. In Sec. II we for-\nmulate the SKFT approach to the dynamics of local-\nized spins interacting with conduction electrons within\na metallic ferromagnet. Sections III A and III B show\nhow this approach leads to nonlocal and isotropic, or\nnonlocal and anisotropic, damping in the presence or ab-\nsence of SOC, respectively. The SKFT-derived analyt-\nical results are corroborated by numerically exact TD-\nNEGF+LLG simulations [40, 43, 56, 57] in Sec. III C.\nThen, in Secs. III D and III E we compare SKFT-derived\nformulas with widely used scattering theory of conven-\ntional scalar Gilbert damping [8, 61, 62] or spin-motive\nforce (SMF) theory [13, 19] of nonlocal damping, respec-\ntively. Finally, in Sec. III F, we discuss how to com-\nbine our SKFT-derived formulas to first-principles calcu-\nlations on realistic materials via density functional theory\n(DFT). We conclude in Sec. IV.\nII. SCHWINGER-KELDYSH FIELD THEORY\nFOR METALLIC FERROMAGNETS\nThe starting point of SKFT is the action [48] of metal-\nlic ferromagnet, S=SM+Se,\nSM=Z\nCdtX\nnh\n∂tMn(t)·An− H[Mn(t)]i\n,(3a)\nSe=Z\nCdtX\nnn′h\n¯ψn(t)\u0000\ni∂t−γnn′\u0001\nψn′(t) (3b)\n−δnn′JsdMn(t)·sn′(t)i\n,\nwhere SMis contribution from localized spins and Seis\ncontribution from conduction electrons. The integrationR\nCis along the Keldysh closed contour C[48]. Here the\nsubscript nlabels the site of a D-dimensional cubic lat-\ntice;∂tMn·Anis the Berry phase term [63, 64]; H[Mn]\nis the Hamiltonian of localized spins; ψn= (ψ↑\nn, ψ↓\nn)T\nis the Grassmann spinor [48] for an electron at site\nn;γnn′=−γis the nearest-neighbor (NN) hopping;\nsn=¯ψnσψnis the electronic spin density, where σis\nthe vector of the Pauli matrices; and Jsdis the magni-\ntude of sdexchange interaction between flowing spins of\nconduction electrons and localized spins. For simplicity,\nwe use ℏ= 1.\nThe Keldysh contour C, as well as all functions defined\non it, can be split into forward (+) and backward ( −)\nsegments [48]. These functions can, in turn, be rewritten\nasM±\nn=Mn,c±1\n2Mn,qfor the real-valued localized spins\nfield, and ψ±\nn=1√\n2(ψ1,n±ψ2,n) and ¯ψ±\nn=1√\n2(¯ψ2,n±\n¯ψ1,n) for the Grassmann-valued fermion fields ψnand¯ψn.\nThe subscripts candqrefer to the classical and quantum\ncomponents of time evolution. This rewriting yields thefollowing expressions for the two actions\nSM=Z\ndtX\nnMα\nnq\u0000\nϵαβγ∂tMβ\nn,cMγ\nnc+Bα\neff[Mn,c]\u0001\n,(4a)\nSe=Z\ndtdt′X\nnn′¯ψσ\nn\u0000ˇG−1\nnn′δσσ′−JsdˇMα\nnn′σα\nσσ′\u0001\nψσ′\nn′,(4b)\nwhere subscript σ=↑,↓is for spin; summation over\nrepeated Greek indices is implied; ψ≡(ψ1, ψ2)T;\nBeff=−δH/δMis the effective magnetic field; ϵαβγis\nthe Levi-Civita symbol; and ˇOare 2×2 matrices in the\nKeldysh space, such as\nˇGnn′=\u0012\nGRGK\n0GA\u0013\nnn′,ˇMα\nnn′=\u0012\nMcMq\n2Mq\n2Mc\u0013α\nnδnn′.\n(5)\nHere GR/A/K\nnn′(t, t′) are electronic re-\ntarded/advanced/Keldysh Green’s functions (GFs) [48]\nin the real-space representation of sites n.\nThe electrons can be integrated out [49] up to the sec-\nond order in Jsdcoupling, thereby yielding an effective\naction for localized spins only\nSeff\nM=Z\ndtX\nnMα\nn,qh\nϵαβγ∂tMβ\nn,cMγ\nn,c+Bα\neff[Mn,c]\n+Z\ndt′X\nn′Mα\nn′,c(t′)ηnn′(t, t′)i\n, (6)\nwhere\nηnn′(t, t′) = iJ2\nsd\u0010\nGR\nnn′(t, t′)GK\nnn′(t′, t)\n+GK\nnn′(t, t′)GA\nnn′(t′, t)\u0011\n, (7)\nis the non-Markovian time-retarded kernel. Note that\nterms that are second order in the quantum fluctuations\nMn,qare neglected [48] in order to write Eq. (6). The\nmagnetization damping can be explicitly extracted by\nanalyzing the kernel, as demonstrated for different ferro-\nmagnetic setups in Secs. III A and III B.\nIII. RESULTS AND DISCUSSION\nA. Nonlocality of Gilbert damping in metallic\nferromagnets in the absence of SOC\nSince ηnn′(t−t′) depends only on the difference t−t′, it\ncan be Fourier transformed to energy ε. Thus, the kernel\ncan be written down explicitly for low energies as\nηnn′(ε) =J2\nsdiε\n2πX\nk,qeik·(rn−rn′)eiq·(rn−rn′)Ak(µ)Aq(µ),\n(8)\nwhere Ak(µ)≡i[GR\nk(µ)−GA\nk(µ)] is the spectral func-\ntion [52] evaluated at chemical potential µ;kis a4\nwavevector; and rnandrn′are the position vectors of\nsites nandn′. Equation (8) remains finite in the clean\nlimit and for low temperatures, so it evades unphysical\ndivergences in the linear-response approaches [58–60]. By\ntransforming it back into the time domain, we minimize\nthe effective action in Eq. (6) with respect to the quan-\ntum fluctuations to obtain semiclassical equations of mo-\ntion for classical localized spins. This procedure is equiv-\nalent to the so-called large spin approximation [65, 66] or\na one loop truncation of the effective action. The higher\norder terms neglected in Eq. (6) contribute a stochas-\ntic noise that vanishes in the low temperature and large\nspin limit. Although the fluctuating effect of this noise\ncan modify the exact dynamics [54, 65], the determinis-\ntic regime suffices for a qualitative understanding and is\noften the main focus of interest [66, 67].\nThus, we arrive at the following extended LLG equa-\ntion\n∂tMn=−Mn×Beff,n+Mn×X\nn′λD\nnn′∂tMn′,(9)\nwhere the conventional αGMn×∂tMnGilbert term\nis replaced by the second term on the RHS exhibit-\ning nonlocal damping λD\nnn′instead of Gilbert damping\nscalar parameter αG. A closed expression for λD\nnn′can\nbe obtained for one-dimensional (1D), two-dimensional\n(2D) and three-dimensional (3D) metallic ferromagnets\nby considering quadratic energy-momentum dispersion of\ntheir conduction electrons\nλD\nnn′=\n\n2J2\nsd\nπv2\nFcos2(kF|rn−rn′|) 1D ,\nk2\nFJ2\nsd\n2πv2\nFJ2\n0(kF|rn−rn′|) 2D ,\nk2\nFJ2\nsd\n2πv2\nFsin2(kF|rn−rn′|)\n|rn−rn′|2 3D.(10)\nHere kFis the Fermi wavevector of electrons, vFis their\nFermi velocity, and J0(x) is the 0-th Bessel function of\nthe first kind.\nB. Nonlocality and anisotropy of Gilbert damping\nin metallic ferromagnets in the presence of SOC\nTaking into account that previous analytical calcu-\nlations [26–28] of conventional Gilbert damping scalar\nparameter always include SOC, often of the Rashba\ntype [68], in this section we show how to generalize\nEq. (8) and nonlocal damping extracted in the presence\nof SOC. For this purpose, we employ the Rashba Hamil-\ntonian in 1D, with its diagonal representation given by,\nˆH=P\nkσεkσˆc†\nkσˆckσ, where ˆ c†\nkσ/ˆckσcreates/annihilates\nan electron with wavenumber kand spin σoriented along\nthey-axis, εkσ=−2γcosk+ 2σγSOsinkis the Rashba\nspin-split energy-momentum dispersion, and γSOis the\nstrength of the Rashba SOC coupling. By switching\nfrom second-quantized operators ˆ c†\nkσ/ˆckσto Grassmann-\nvalued two-component fields [64] ¯cσ\nn/cσ\nn, where cσ\nn=\nFIG. 2. (a) Time evolution of two localized spins Mn, lo-\ncated at sites n= 1 and n′= 3 within a chain of 19 sites\nin the setup of Fig. 1(b), computed numerically by TD-\nNEGF+LLG scheme [40, 43, 56, 57]. The two spins are\ncollinear at t= 0 and point along the x-axis, while mag-\nnetic field is applied along the z-axis. (b) The same infor-\nmation as in panel (a), but for two noncollinear spins with\nangle ∈ {0,45,90,135,180}between them. (c) and (d) Ef-\nfective damping extracted from TDNEGF+LLG simulations\n(red dashed line) vs. the one from SKFT [black solid line plots\n1D case in Eq. (10)] as a function of the site n′of the second\nspin. The two spins are initially parallel in (c), or antiparallel\nin (d). The Fermi wavevector of conduction electrons is cho-\nsen as kF=π/2a, where ais the lattice spacing.\n(cσ\n1,n, cσ\n2,n)T, we obtain for the electronic action\nSe=Z\ndtdt′X\nnn′¯cσ\nn\u0002\n(ˇGσ\nnn′)−1δσσ′−JsdˇMα\nnn′σβ\nσσ′\u0003\ncσ′\nn′.\n(11)\nHere ˇGσ\nnn′is diagonal, but it depends on spin through\nεkσ. In addition, ˇMx,y,z\nnn′, as the matrix which couples to\nthe same σx,y,zPauli matrix in electronic action without\nSOC [Eq. (3b)], is coupled in Eq. (11) to a different Pauli\nmatrix σy,z,x.\nBy integrating electrons out up to the second order in\nJsd, and by repeating steps analogous to those of Sec. II\nwhile carefully differentiating the spin-split bands, we\nfind that nonlocal damping becomes anisotropic\nλ1D\nnn′=\nα⊥\nnn′0 0\n0α∥\nnn′0\n0 0 α⊥\nnn′.\n. (12)5\nwhere\nα⊥\nnn′=J2\nsd\nπ\u0012cos2(k↑\nF|rn−rn′|)\nv↑\nF2+cos2(k↓\nF|rn−rn′|)\nv↓\nF2\u0013\n,\n(13a)\nα∥\nnn′=J2\nsd\nπ|v↑\nFv↓\nF|\u0010\ncos\u0002\n(k↑\nF+k↓\nF)|rn−rn′|\u0003\n(13b)\n+ cos\u0002\n(k↑\nF−k↓\nF)|rn−rn′|\u0003\u0011\n,\nandk↑/↓\nFandv↑/↓\nFare the Fermi wavevectors and veloc-\nities, respectively, of the Rashba spin-split bands. This\nmeans that the damping term in Eq. (9) is now given by\nMn×P\nn′λ1D\nnn′·∂tMn′.\nWe note that previous experimental [69], numeri-\ncal [9, 70], and analytical [26–28] studies have also found\nSOC-induced anisotropy of Gilbert damping scalar pa-\nrameter. However, our results [Eqs. (12) and (13)] ex-\nhibit additional feature of nonlocality (i.e., damping at\nsitendepends on spin at site n′) and nonuniformity (i.e.,\ndependence on |rn−rn′|). As expected from Sec. III A,\nnonlocality persists for γSO= 0, i.e., k↑\nF=k↓\nF=kF,\nwith λ1D\nnn′properly reducing to contain αnn′three di-\nagonal elements. Additionally, the damping component\nα∥\nnn′given by Eq. (13b) can take negative values, re-\nvealing the driving capability of the conduction electrons\n(see Sec. III C). However, for realistic small values of γSO,\nthe driving contribution of nearby localized spins is like-\nwise small. Furthermore, the decay of nonlocal damping\nwith increasing distance observed in 2D and 3D, together\nwith the presence of intrinsic local damping from other\nsources, ensures that the system tends towards equilib-\nrium.\nC. Comparison of SKFT-derived formulas with\nnumerically exact TDNEGF+LLG simulations\nAn analytical solution to Eq. (9) can be obtained in\nfew special cases, such as for two exchange-uncoupled lo-\ncalized spins at sites n= 1 and n′̸= 1 within 1D wire\nplaced in an external magnetic field Bext=Bextez, on\nthe proviso that the two spins are collinear at t= 0.\nThe same system can be simulated by TDNEGF+LLG\nscheme, so that comparing analytical to such numeri-\ncally exact solution for trajectories Mn(t) makes it pos-\nsible to investigate accuracy of our derivation and ap-\nproximations involved in it, such as: truncation to J2\nsd\norder; keeping quantum fluctuations Mn,qto first order;\nand low-energy approximation used in Eq. (8). While\nsuch a toy model is employed to verify the SKFT-based\nderivation, we note that two uncoupled localized spins\ncan also be interpreted as macrospins of two distant ferro-\nmagnetic layers within a spin valve for which oscillatory\nGilbert damping as a function of distance between the\nlayers was observed experimentally [71]. Note that semi-\ninfinite NM leads from the setup in Fig. 1(b), always usedin TDNEGF+LLG simulations to ensure continuous en-\nergy spectrum of the whole system [40, 56], can also be\nincluded in SKFT-based derivation by using self-energy\nΣR/A\nk(ε) [52, 72] which modifies the GFs of the central\nmagnetic region in Fig. 1(b), GR/A\nk= (ε−εk−ΣR/A\nk)−1,\nwhere εk=−2γcosk.\nThe TDNEGF+LLG-computed trajectory M1(t) of lo-\ncalized spin at site n= 1 is shown in Figs. 2(a) and\n2(b) using two localized spins which are initially collinear\nor noncollinear, respectively. For the initially parallel\n[Fig. 2(a)] or antiparallel localized spins, we can ex-\ntract Gilbert damping from such trajectories because\nMz\n1(t) = tanh\u0000¯λ1D\nnn′Bextt/(1 + ( ¯λ1D\nnn′)2)\u0001\n[4, 40], where\nthe effective damping is given by ¯λ1D\nnn′=λ1D\n00±λ1D\nnn′\n(+ for parallel and −for antiparallel initial condition).\nThe nonlocality of such effective damping in Figs. 2(c)\nand 2(d) manifests as its oscillation with increasing sep-\naration of the two localized spins. The same result\nis predicted by the SKFT-derived formula [1D case in\nEq. (10)], which remarkably closely traces the numeri-\ncally extracted ¯λ1D\nnn′despite approximations involved in\nSKFT-based analytical derivation. Note also that the\ntwo localized spins remain collinear at all times t, but\ndamping remains nonlocal. The feature missed by the\nSKFT-based formula is the decay of ¯λ1D\nnn′with increasing\n|rn−rn′|, which is present in numerically-extracted effec-\ntive damping in Figs. 2(c) and 2(d). Note that effective\ndrastically reduced for antiparallel initial conditions, due\nto the driving capabilities of the conduction electrons, in\naddition to their dissipative nature. For noncollinear ini-\ntial conditions, TDNEGF+LLG-computed trajectories\nbecome more complicated [Fig. 2(b)], so that we can-\nnot extract the effective damping λ1D\nnn′akin to Figs. 2(c)\nand 2(d) for the collinear initial conditions.\nD. Comparison of SKFT-derived formulas with the\nscattering theory [8] of uniform local Gilbert\ndamping\nThe scattering theory of Gilbert damping αGwas\nformulated by studying a single domain ferromagnet\nin contact with a thermal bath [8]. In such a setup,\nenergy [8] and spin [61] pumped out of the system\nby time-dependent magnetization contain information\nabout spin-relaxation-induced bulk [8, 62] and interfa-\ncial [61] separable contributions to αG, expressible in\nterms of the scattering matrix of a ferromagnetic layer\nattached to two semi-infinite NM leads. For collinear lo-\ncalized spins of the ferromagnet, precessing together as\na macrospin, scattering theory-derived αGis a spatially-\nuniform scalar which can be anisotropic [62]. Its expres-\nsion is equivalent [62] to Kubo-type formulas [9, 34–36]\nin the linear response limit, while offering an efficient al-\ngorithm for numerical first-principles calculations [24, 25]\nthat can include disorder and SOC on an equal footing.\nOn the other hand, even if all localized spins are ini-\ntially collinear, SKFT-derived extended LLG Eq. (9) pre-6\nFIG. 3. (a) Comparison of trajectories of localized spins\nMz\nn(t), in the setup of Fig. 1(b) whose central region is\n1D metallic ferromagnet composed of 5 sites, using LLG\nEq. (9) with SKFT-derived nonlocal damping (solid red lines)\nvs. LLG equation with conventional spatially-independent\nαG= 0.016 (black dashed line). This value of αGis ob-\ntained by averaging nonlocal damping over the whole ferro-\nmagnet. The dynamics of Mn(t) is initiated by an external\nmagnetic field along the z-axis, while all five localized spins\npoint along the x-axis at t= 0. (b) Comparison of spin cur-\nrentISz\nR(t) pumped [56, 57, 61] by the dynamics of Mn(t) for\nthe two cases [i.e., nonuniform Mn(t) for nonlocal vs. uniform\nMn(t) for conventional damping] from panel (a). The Fermi\nwavevector of conduction electrons is chosen as kF=π/2a.\ndicts that due to nonlocal damping each localized spin\nwill acquire a distinct Mn(t) trajectory, as demonstrated\nby solid red lines in Fig. 3(a). By feeding these trajec-\ntories, which are affected by nonlocal damping [1D case\nin Eq. (10)] into TDNEGF+LLG simulations, we can\ncompute spin current ISz\nR(t) pumped [56, 57] into the\nright semi-infinite lead of the setup in Fig. 1(b) by the\ndynamics of Mn(t). A very similar result for pumped\nspin current is obtained [Fig. 3(b)] if we feed identical\nMn(t) trajectories [black dashed line in Fig. 3(a)] from\nconventional LLG equation with Gilbert damping scalar\nparameter, αG, whose value is obtained by averaging the\nSKFT-derived nonlocal damping over the whole ferro-\nmagnet. This means that scattering theory of Gilbert\ndamping [8], which in this example is purely due to inter-\nfacial spin pumping [61] because of lack of SOC and dis-\norder (i.e., absence of spin relaxation in the bulk), would\npredict a constant αGthat can only be viewed as the\nspatial average of SKFT-derived nonlocal and nonuni-\nform λ1D\nnn′. In other words, Fig. 3 reveals that different\ntypes of microscopic magnetization dynamics Mn(t) can\nyield the same total spin angular momentum loss into\nthe external circuit, which is, therefore, insufficient on\nits own to decipher details (i.e., the proper form of ex-\ntended LLG equation) of microscopic dynamics of local\nmagnetization.\n1.5 2.0 2.5 3.0\nw/a024vDW(aJ/¯ h)×10−2\nαG= 0.1\nEq.(9)\nRef.[13] withη= 0.05\nRef.[19] withη= 0.05\nEq.(1) withη= 0(a)\n0 25 50 75\nSite i−1.0−0.50.00.51.0Mα(t)t = 410 ¯ h/J\nα=x,y,z(b)\n0 25 50 75\nSite i−2−101(Mn×/summationtext\nn/primeλd\nnn/prime·∂tMn/prime)α×10−2\n(c)\n0 25 50 75\nSite i−2−101(M×D·∂tM)α×10−3\n(d)FIG. 4. (a) Comparison of magnetic DW velocity vDWvs.\nDW width wextracted from numerical simulations using: ex-\ntended LLG Eq. (9) with SKFT-derived nonlocal damping\n[Eq. (10), red line]; extended LLG Eq. (1) with SMF-derived\nin Ref. [13] nonlocal damping [Eq. (2), blue line] or SMF-\nderived nonlocal damping (green line) in Ref. [19] [with ad-\nditional term when compared to Ref. [13], see Eq. (14)]; and\nconventional LLG Eq. (1) with local Gilbert damping [i.e.,\nη= 0 in Eq. (2), black line]. (b) Spatial profile of DW within\nquasi-1D ferromagnetic wire at time t= 410 ℏ/J, where Jis\nexchange coupling between Mnat NN sites, as obtained from\nSKFT-derived extended LLG Eq. (9) with nonlocal damping\nλ2D\nnn′[Eq. (10)]. Panels (c) and (d) plot the corresponding spa-\ntial profile of nonlocal damping across the DW in (b) using\nSKFT-derived expression [Eqs. (9) and Eq. (10)] vs. SMF-\nderived [13] expression [second term on the RHS of Eq. (2)],\nrespectively.\nE. Comparison of SKFT-derived formulas with\nspin motive force theory [13] and [19] of nonlocal\ndamping\nThe dynamics of noncollinear and noncoplanar magne-\ntization textures, such as magnetic DWs and skyrmions,\nleads to pumping of charge and spin currents assumed\nto be captured by the spin motive force (SMF) the-\nory [16, 73, 74]. The excess angular momentum of dy-\nnamical localized spins carried away by pumped spin cur-\nrent of electrons appears then as backaction torque [57]\nexerted by nonequilibrium electrons onto localized spins\nor, equivalently, nonlocal damping [13, 17–19]. From this\nviewpoint, i.e., by using expressions for pumped spin cur-\nrent [13, 17–19], a particular form for nonlocal damp-\ning [second term on the RHS of Eq. (2)] was derived in\nRef. [13] from the SMF theory, as well as extended in\nRef. [19] with an additional term, while also invoking a\nnumber of intuitively-justified but uncontrolled approxi-\nmations.\nIn this Section, we employ an example of a magnetic\nfield-driven DW [Fig. 4(b)] of width wwithin a quasi-7\n1D ferromagnetic wire to compare its dynamics obtained\nby solving extended LLG Eq. (1), which includes non-\nlocal damping tensor [Eq. (2)] of Ref. [13], with the\ndynamics obtained by solving SKFT-derived extended\nLLG Eq. (9) whose nonlocal damping is different from\nRef. [13]. By neglecting nonlocal damping in Eq. (2),\nthe ferromagnetic domain wall (DW) velocity vDWis\nfound [75] to be directly proportional to Gilbert damping\nαG,vDW∝ −BextwαG, assuming high external magnetic\nfieldBextand sufficiently small αG. Thus, the value of αG\ncan be extracted by measuring the DW velocity. How-\never, experiments find that αGdetermined in this fashion\ncan be up to three times larger than αGextracted from\nferromagnetic resonance linewidth measurement scheme\napplied to the same material with uniform dynamical\nmagnetization [10]. This is considered as a strong evi-\ndence for the importance of nonlocal damping in systems\nhosting noncollinear magnetization textures.\nIn order to properly compare the effect of two different\nexpressions for the nonlocal damping, we use αG= 0.1\nin Eq. (1) and we add the same standard local Gilbert\ndamping term, αGMn×∂tMn, into SKFT-derived ex-\ntended LLG Eq. (9). In addition, we set λ2D\n00=ηin\nEq. (10), so that we can vary the same parameter ηin all\nversions of extended LLG Eqs. (1), and (9). Note that\nwe use λ2D\nnn′in order to include realistic decay of nonlo-\ncal damping with increasing distance |rn−rn′|, thereby\nassuming quasi-1D wire. By changing the width of the\nDW, the effective damping can be extracted from the DW\nvelocity [Fig. 4(a)]. Figure 4(a) shows that vDW∝wre-\ngardless of the specific version of nonlocal damping em-\nployed, and it increases in its presence—compare red,\nblue, and green data points with the black ones obtained\nin the absence of nonlocal damping. Nevertheless, the\nclear distinction between red, and blue or green data\npoints signifies that our SKFT-derived nonlocal damping\ncan be quite different from previously discussed SMF-\nderived nonlocal damping [13, 19], which are compara-\nble regardless of the inclusion of the nonadiabatic terms.\nFor example, the effective damping extracted from blue\nor green data points is D= 0.17 or D= 0.15, respec-\ntively, while λ2D\nnn′= 0.48. This distinction is further clar-\nified by comparing spatial profiles of SKFT-derived and\nSMF-derived nonlocal damping in Figs. 4(c) and 4(d),\nrespectively, at the instant of time used in Fig. 4(b). In\nparticular, the profiles differ substantially in the out-\nof-DW-plane or y-component, which is, together with\nthex-component, an order of magnitude greater in the\ncase of SKFT-derived nonlocal damping. In addition,\nthe SKFT-derived nonlocal damping is nonzero across\nthe whole wire , while the nonlocal damping in Eq. (2)\nis nonzero only within the DW width, where Mnvec-\ntors are noncollinear [as obvious from the presence of\nthe spatial derivative in the second term on the RHS\nof Eq. (2)]. Thus, the spatial profile of SKFT-derived\nnonlocal damping in Fig. 4(c) illustrates how its nonzero\nvalue in the region outside the DW width does not re-\nquire noncollinearity of Mnvectors.Since SKFT-derived formulas are independently con-\nfirmed via numerically exact TDNEGF+LLG simula-\ntions in Figs. 2(c) and 2(d), we conclude that previously\nderived [13] type of nonlocal damping [second term on\nthe RHS of Eq. (2)] does not fully capture backaction of\nnonequilibrium conduction electrons onto localized spins.\nThis could be due to nonadiabatic corrections [16, 19, 74]\nto spin current pumped by dynamical noncollinear mag-\nnetization textures, which are present even in the ab-\nsence of disorder and SOC [43]. One such correction was\nderived in Ref. [19], also from spin current pumping ap-\nproach, thereby adding a second nonlocal damping term\nηX\nβ′h\n(M·∂β′∂tM)M×∂β′M−M×∂2\nβ′∂tMi\n,(14)\ninto the extended LLG Eq. (1). However, combined us-\nage [green line in Fig. 4(a)] of both this term and the one\nin Eq. (2) as nonlocal damping still does not match the\neffect of SKFT-derived nonlocal damping [compare with\nred line in Fig. 4(a)] on magnetic DW. As it has been\ndemonstrated already in Fig. 3, the knowledge of total\nspin angular momentum loss carried away by pumped\nspin current [Fig. 3(b)], as the key input in the deriva-\ntions of Refs. [13, 19], is in general insufficient to decipher\ndetails of microscopic dynamics and dissipation exhibited\nby localized spins [Fig. 3(a)] that pump such current.\nF. Combining SKFT-derived nonlocal damping\nwith first-principles calculations\nObtaining the closed form expressions for the nonlocal\ndamping tensor λnn′in Secs. III A and III B was made\npossible by using simplistic model Hamiltonians and ge-\nometries. For realistic materials and more complicated\ngeometries, we provide in this Section general formulas\nwhich can be combined with DFT quantities and evalu-\nated numerically.\nNotably, the time-retarded dissipation kernel in\nEq. (7), from which λnn′is extracted, depends on the\nKeldysh GFs. The same GFs are also commonly used\nin first-principles calculations of conventional Gilbert\ndamping scalar parameter via Kubo-type formulas [29–\n33]. Specifically, the retarded/advanced GFs are ob-\ntained from first-principles Hamiltonians ˆHDFTDFT as\nˆGR/A(ε) =\u0002\nε−ˆHDFT+ˆΣR/A(ϵ)\u0003−1. Here, ˆΣR/A(ε) are\nthe retarded/advanced self-energies [52, 72] describing es-\ncape rate of electrons into NM leads, allowing for open-\nsystem setups akin to the scattering theory-derived for-\nmula for Gilbert damping [8, 62] and its computational\nimplementation with DFT Hamiltonians [24, 25]. Since\nescape rates are encoded by imaginary part of the self-\nenergy, such calculations do not require iηimaginary pa-\nrameter introduced by hand when using Kubo-type for-\nmulas [29–33] (where η→0 leads to unphysical divergent\nresults [58–60]). Therefore, ˆHDFTcan be used as an\ninput to compute the nonlocal damping tensor, via the8\ncalculation of the GFs ˆGR/A(ε) and the spectral function\nˆA(ε) =i\u0002ˆGR(ε)−ˆGA(ε)\u0003\n.\nFor these purposes, it is convenient to separate the\nnonlocal damping tensor into its symmetric and anti-\nsymmetric components, λαβ\nnn′=λ(αβ)\nnn′+λ[αβ]\nnn′, where the\nparenthesis (brackets) indicate that surrounded indices\nhave been (anti)symmetrized. They are given by\nλ(αβ)\nnn′=−J2\nsd\n2πZ\ndε∂f\n∂εTrspin\u0002\nσαAnn′σβAn′n\u0003\n, (15a)\nλ[αβ]\nnn′=−2J2\nsd\nπZ\ndε∂f\n∂εTrspin\u0002\nσαReˆGR\nnn′σβAn′n\n−σαAnn′σβReˆGR\nn′n\u0003\n+J2\nsd\n2πZ\ndε(1−2f)\n×Trspin\u0002\nσαReˆGR\nnn′σβ∂An′n\n∂ε−σα∂Ann′\n∂εσβReˆGR\nn′n\u0003\n,\n(15b)\nwhere f(ε) is the Fermi function, and the trace is taken\nin the spin space. The antisymmetric component either\nvanishes in the presence of inversion symmetry, or is of-\nten orders of magnitude smaller than the symmetric one.\nTherefore, it is absent in our results for simple models\non hypercubic lattices. As such, the nonlocal damping\ntensors in Eqs. (10) and (13), are fully symmetric and\nspecial case of Eq. (15a) when considering specific energy-\nmomentum dispersions and assuming zero temperature.\nIV. CONCLUSIONS AND OUTLOOK\nIn conclusion, we derived a novel formula, displayed\nas Eq. (15), for magnetization damping of a metallic fer-\nromagnet via unexploited for this purpose rigorous ap-\nproach offered by the Schwinger-Keldysh nonequilibrium\nfield theory [48]. Our formulas could open a new route for\ncalculations of Gilbert damping of realistic materials by\nemploying first-principles Hamiltonian ˆHDFTfrom den-\nsity functional theory (DFT) as an input, as discussed\nin Sec. III F. Although a thorough numerical exploration\nof a small two-spin system based on SKFT was recently\npursued in Ref. [54], our Eqs. (15) are not only applica-\nble for large systems of many localized spins, but are also\nrefined into readily computable expressions that depend\non accessible quantities.While traditional, Kubo linear-response [9, 30, 34–\n36] or scattering theory [8] based derivations produce\nspatially uniform scalar αG, SKFT-derived damping in\nEqs. (15) is intrinsically nonlocal and nonuniform as it\ndepends on the coordinates of local magnetization at two\npoints in space rnandrn′. In the cases of model Hamil-\ntonians in 1D–3D, we reduced Eqs. (15) to analytical ex-\npressions for magnetization damping [Eq. (10)], thereby\nmaking it possible to understand the consequences of\nsuch fundamental nonlocality and nonuniformity on lo-\ncal magnetization dynamics, such as: ( i) damping in\nEq. (10) osc illates with the distance between xandx′\nwhere the period of such oscillation is governed by the\nFermi wavevector kF[Figs. 1(c), 2(c), and 2(d)]; ( ii)\nit always leads to nonuniform local magnetization dy-\nnamics [Fig. 3(a)], even though spin pumping from it\ncan appear [Fig. 3(b)] as if it is driven by usually an-\nalyzed [8, 61] uniform local magnetization (or, equiv-\nalently, macrospin); ( iii) when applied to noncollinear\nmagnetic textures, such as DWs, it produces an order\nof magnitude larger damping and, therefore, DW wall\nvelocity, than predicted by previously derived [13] non-\nlocal damping [second term on the RHS of Eq. (2)].\nRemarkably, solutions of SKFT-based extended LLG\nEq. (9) are fully corroborated by numerically exact TD-\nNEGF+LLG simulations [40, 43, 56, 57] in 1D, despite\nthe fact that several approximations are employed in\nSKFT-based derivations. 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It is observed that a series of stable lumps can be\ngenerated by an unstable magnetic soliton under Gaussian disturbance. Such magnetic lumps are highly stable\nand can maintain their shapes and velocities during evolution or collision. The interaction between lump and\nmagnetic soliton, as well as interaction between two lumps, are numerically investigated. We further discuss\nthe nonlinear motion of lumps in ferrites with Gilbert-damping and inhomogeneous exchange effects. The\nresults show that the Gilbert-damping effects make the amplitude and velocity of the magnetic lump decay\nexponentially during propagation. And the shock waves are generated from a lump when quenching the strength\nof inhomogeneous exchange.\nI. INTRODUCTION\nThe propagation of electromagnetic wave in ordered\nmagnetic materials, especially in a ferromagnetic medium,\nplays a vital role in faster and higher density storage fields [1–\n3]. In particular, magnetic soliton(MS), which exists in both\nferro- and antiferro-magnets, is becoming a very promising\ninformation carrier because of its particle-like behavior and\nmaneuverability [4–9]. In the past few decades, a wide range\nof soliton-type propagation phenomena has been theoretically\npredicted [10–13], and some of them have been confirmed\nexperimentally [14, 15].\nIndeed, wave propagation in ferromagnetic media is well-\nknown as a highly nonlinear problem. A complete description\nof all types of nonlinear excitations is governed by the\nMaxwell equations coupled with Landau-Lifschitz equation.\nFor this moment, let us notice that a fully nonlinear theory has\nnot been developed. But the linear theory for sufficiently small\namplitudes was established and validated experimentally [16].\nIn order to obtain results valid in nonlinear regimes, or at\nleast weakly nonlinear, one has to resort to intermediate\nmodels (by introducing a small perturbative parameter related\nto the soliton wavelength) [17]. These models include long-\nwave model [18–20], modulational asymptotic model [21],\nand short-wave model [22–25]. Both long-wave model and\nmodulational asymptotic model are mainly used to explain\nand predict the behavior of large-scale phenomena owing to\ntheir long-wave-type approximate condition [26]. However,\nthis condition is not always applicable because the scale of\nmagnetic materials and devices are getting more refined and\nmore sophisticated. Moreover, the main practical interest of\nferrites is that they propagate microwaves [27, 28]. On the\ncontrary, from the viewpoint of applied physics, the short-\nwave-type approximation is much more relevant to available\nexperiments than the former one.\nSince Kraenkel et al. first proposed the short-wave model\n[29], quite a few related nonlinear evolution equations have\nbeen derived, which belong to the Kraenkel-Manna-Merle\n(KMM) system [22, 23, 30–32]. Some significant works\n∗Corresponding author: linji@zjnu.edu.cnhave been devoted to searching and explaining different\nexcitation patterns of ferromagnetic insulators. As for (1+1)-\ndimensional KMM system, the existence of multi-valued\nwaveguide channel solutions has been verified, and the\nnonlinear interaction properties were investigated between\nthe localized waves alongside the depiction of their energy\ndensities [22]. By applying the Hirota bilinear transformation\nmethod, the one- and two-soliton solutions were constructed\nwhile studying in details the solitons scattering properties\n[23]. This system is also solvable using the inverse scattering\nmethod [25]. It is noteworthy that this system possesses\nthe loop-soliton and spike-like soliton [33, 34], and the\nmagnetic loop-soliton dynamics have been extensively studied\n[35–37]. The propagation of electromagnetic waves in\nhigher-dimensional ideal ferromagnets has also been studied,\ncorresponding to the (2+1)-dimensional KMM system [26, 31,\n38, 39]. The analytical one-line-soliton solution as well as its\ntransverse stability have been reported [26]. It has been shown\nthat these structures were stable under certain conditions.\nOn the other hand, most previous studies have only focused\non the propagation of MS in ideal ferrites, which means\nsome important properties of the magnetic material were\nneglected. The main reason is that the nonlinear wave\nequation describing the propagation of electromagnetic waves\nin non-ideal ferromagnetic materials is no longer integrable.\nHowever, the Gilbert-damping and inhomogeneous exchange\neffects are essential features in a real ferromagnetic film, and\ntheir connection with MS motion is an important issue that has\nnot been explored so far. In this paper, we aim to investigate\ntheoretically and numerically the dynamics of the MS in a\nferromagnetic film including damping and the inhomogeneous\nexchange effect. The rest of this paper is organized as follows.\nIn Section 2, we review the physical background and derive\na new (2+1)-dimensional short-wave asymptotic model in\nferromagnetic media. In Section 3, the bilinear-like form\nof the reduced system is constructed and the analytical MS\nsolutions are acquired. In Section 4, the transmission stability\nof the magnetic soliton is numerically explored. The results\nshow that an unstable MS will split to some magnetic lumps\nby a small perturbation. The motions of these lumps under\nthe influence of damping and inhomogeneous exchange are\nanalysed in detail. We end this work in Section 5 with a brief\nconclusion and perspectives.arXiv:2307.00903v1  [nlin.PS]  3 Jul 20232\nII. PHYSICAL BACKGROUND\nA. Basic equations\nThe physical system under consideration is a saturated\nmagnetized ferrite film lying in the x−yplane, as shown in\nFig. 1. Different from Ref. [32], we consider the external\nfield H∞\n0perpendicular to the film, i.e., M0= (0,0,m). So the\ntransverse drift is avoided. The typical thickness of the film\nis about 0.5mm, and the width is approximately 10mm. We\nassume the propagation distance is large enough with regard\nto the wavelength, say more than 50cm. The evolution of the\nmagnetic field Hand the magnetization density Mis governed\nby the Maxwell equations coupled with Landau-Lifschitz-\nGilbert equation, which read as\n−∇(∇·H)+∆H=1\nc2∂2\n∂t2(H+M), (1a)\n∂\n∂tM=−γµ0M×Heff+σ\nMsM×∂\n∂tM, (1b)\nwhere c=1/p\nµ0˜εis the speed of light with the scalar\npermittivity ˜εof the medium, γis the gyromagnetic ratio,\nµ0being the magnetic permeability of the vacuum, σis the\ndamping constant, and Msis the saturation magnetization. The\neffective field Heffis given by [30]\nHeff=H−βn(n·M)+α∆M. (2)\nHere αandβare the constants of the inhomogeneous\nFigure 1. Ferrite film under consideration. The sample is magnetized\nto saturation by long strong magnetic field H∞\n0applied in the\nz-direction. The x-direction of the short wave propagation is\nperpendicular to the direction of static magnetization.\nexchange and the magnet anisotropy ( β>0 corresponds to\nthe easy-plane case), respectively. For a simple tractability, the\nunit vector nof the anisotropy axis is assumed to be along the\nzaxis (i.e., n≡ez). In order to transform the above systems to\ndimensionless equation, we rescale the quantities M,H, and t\nintoµ0γM/c,µ0γH/c, and ct. Thus, the constants µ0γ/cand\ncin Eqs.(2) and (3) are replaced by 1, Msbym=µ0γMs/c,\nandσby˜σ=σ/µ0γ[30].\nB. Linear analysis\nTo study the linear regime we look at a small perturbation of\na given solution. Equations (1) are linearized about the steady\nstate:\nM0= (0,0,m),H0=µM0. (3)where µis the strength of the internal magnetic field. Before\nproceeding further we assume that the ferromagnetic materials\nhave weak damping ¯σ∼ε˜σ. The exchange interaction\nparameter αand anisotropy parameter βare of order ε2and\nε3, respectively (i.e. ¯α=ε2α,¯β=ε3β). Let us seek for\nthe plane wave perturbation solution propagating along the x-\ndirection such as\nM=M0+εmexp[i(kx+ly−ωt)],\nH=H0+εhexp[i(kx+ly−ωt)],(4)\nwhere kandlare the wave numbers in the xandydirections, ω\nis the frequency. Vectors m= (mx,my,mz)andh= (hx,hy,hz)\nare arbitrary real scalar quantities.\nSubstituting Eq. (4) into (1) and (2) in the linear limit, it is\nreduced to\n\nω20 0 ω2−l2kl 0\n0 ω20 kl ω2−k20\n0 0 ω20 0 ω2−k2−l2\n−iωmµ 0 0 −m 0\n−mµ−iω 0 m 0 0\n0 0 −iω 0 0 0\n·\nmx\nmy\nmz\nhx\nhy\nhz\n=0\nThen we obtain the following dispersion relation\nm2(µ+1)\u0002\nµ(k2+l2−ω2)−ω2\u0003\n−ω2(k2+l2−ω2) =0\n(5)\nNote that we focus on studying the short-wave approximation\nk→∞[2]. It comes k0∼ε−1through a small parameter ε≪1\nlinked to the magnitude of the wavelength. Consequently, the\nfrequency expands accordingly as\nω=ω−1ε−1+ω1ε+ω3ε3+.... (6)\nThis assumption guarantees the phase velocity ω(k)/kand\nthe group velocity ∂ω/∂kare always bounded [3]. Now,\nreplacing Eq. (6) into the dispersion relation above, we obtain\na set of equations:\n•At order of ε−4:ω−1=±k0\n•At order of ε−2:ω1=\u0002\n(µ+1)m2+l2\u0003\n/2k0\n•higher order equations which determines ω3,ω5,...\nThe direction of the wave propagation is assumed to be\nclose to the xaxis, thus yvariable gives only account of a\nslow transverse deviation[40, 41]. Therefore lis assumed\nto be very small with respect to kand we write l=l0of\norder 0 with respect to ε. The phase up to order εis thus\n(x−t)/ε+l0y−εω1t,which motivates the introduction of\nnew variables:\nζ=1\nε(x−Vt),y=y,τ=εt. (7)\nThe variable ζdescribes the shape of the wave propagating at\nspeed V; it assumes a short wavelength about 1 /ε. The slow\ntime variable τaccounts for the propagation during very long\ntime on very large distances with regard to the wavelength.\nThe transverse variable yhas an intermediate scale, as in KP-\ntype expansions [26, 41]\nC. Multiple scale approach\nIn order to derive the nonlinear model, fields MandHare\nexpanded in power series of εas\nM=M0+εM1+ε2M2+ε3M3+...,\nH=H0+εH1+ε2H2+ε3H3+....(8)3\nwhere M0,H0,M1,H1,...are functions of (ζ,y,τ).\nWe consider the boundary conditions: lim\nζ→−∞M0=\n(0,0,m),lim\nζ→−∞Mj=lim\nζ→−∞Hj=0,(j̸=0). We derive\nthe following expressions by substituting Expansions (8) into\nequation (1):\n•At order ε−2:\nM0is a constant vector M0=(0,0,m),\n•At order ε−1:\nHx\n0=0,My\n1=0,Mz\n1=0,\n•At order ε0:\nMx\n1ζ=mHy\n0,\nMx\n2ζζ=−Hx\n2ζζ−Hy\n1ζτ\nMy\n2ζζ=−Hx\n1ζy+Hx\n0ζy\nMz\n2ζζ=Hz\n2ζτ+Hz\nyy\n•At order ε1:\nMx\n2ζ=−mHy\n1\nMy\n2ζ=m¯αMx\n1ζζ+¯σM1ζx−Mx\n1Hz\n0+mHx\n1\nMz\n2ζ=Mx\n1Hy\n0\nlet us introduce some independent variables XandTdefined\nasX=−mζ/2,Y=my,T=mτ.\nAfter eliminating H2andM2, we finally obtain the (2+1)-\ndimensional KMM equation:\nCXT=−BBX+CYY,\nBXT=BCX+BYY−sBX+ρBXX,(9)\nwhere observables B,Cand constants s,ρare defined by\nC=−X−ZX\n(Hz\n0/m)dX,B=Mx\n1/2m,\ns=−¯σ/2,ρ=¯αm2/4.(10)\nThis equation is new, which describes the evolution of\nmagnetization field Mand magnetic field Hwithin a ferrite\nfilm in presence of Gilbert-damping and inhomogeneous\nexchange. The quantities H0andM1refer to the zeroth and\nfirst-order expansion coefficients of the external magnetic field\nand the magnetization, respectively. For some simplicity,\nin the next, the independent variables X,YandTwill be\nrewritten as their lower cases x,yandt, respectively.\nIII. HIROTA’S BILINEARIZATION AND SOLITON\nSOLUTIONS OF THE (2+1)-DIMENSIONAL KMM\nEQUATION\nTo explore soliton solutions for the (2+1)-dimensional\nKMM equation (9), we consider a specific dependent variable\ntransformation\nB=G\nF,C=δx−2(lnF)t−2(lnF)y, (11)\nwhere δis an arbitrary constant. Consequently, the bilinear-\nlike forms of the (2+1)-dimensional KMM equation can be\nderived as follow\nF·(DxDt+sDx−D2\ny)G·F+G·(DxDy+D2\ny)F·F=δF2G\n(12a)\n∂x\u0014G2\n2F2−(DyDt+D2\nt)F·F\nF2\u0015\n+∂y\u0014(DyDt+D2\nt)F·F\nF2\u0015\n=0\n(12b)where G,Fare all differential functions of (x,y,t)to be\ndetermined. The symbols Dx,Dtrefer to the Hirota’s operators\nwith respect to the variable x,t, respectively. In order to\nconstruct the solitary wave solutions of Eq.(6), we expand\nGandFwith respect to a formal expansion parameter as\nG=εG1+ε3G3+ε5G5+...,F=1+ε2F2+ε4F4+ε6F6+...,\nin which εis a perturbation parameter and functions Gi,Fi,(i=\n1,2,3,...)are expansion coefficients of the above series. The\none-soliton solution could be constructed by truncating the\nperturbation expansion of GandFas follow\nG=eη1,F=1+k2A2\n16δ2e2η1. (13)\nSubstituting these expressions into Eq.(9) and solving the\nbilinear system recursively, in the absence of damping,\nthe analytical one-soliton solution of the (2+1)-dimensional\nKMM equation can be transformed into\nB=2δ\nksech(η1+η0),C=δx−2δ\nk[tanh(η1+η0)+1],\n(14)\nwhere η1=kx+ly+ [(l2−kl)/2k]t,η0=ln(k/4δ),kandl\nare arbitrary real constants. It should be noted that this soliton\nsolution exists only when the damping is neglected (s=0).\nSimilar to the procedure for constructing one-soliton solution,\nthe two-soliton solution can be given by treating the truncated\nperturbation expansions of GandFas\nG=A1eξ1+A2eξ2+C12eξ1+2ξ2+C21e2ξ1+ξ2, (15a)\nF=1+B11e2ξ1+B22e2ξ2+B12eξ1+ξ2+E12e2ξ1+2ξ2,(15b)\nwhere A1,A2,k1,k2are real constants, ξi=kix+liy+\u0002\n(l2\ni+δ)/ki\u0003\nt,(i=1,2), and the remaining parameters have\nthe following forms:\nBii=A2\nik2\ni\n16δ2,B12=A1A2\n2δ2k2\n1k2\n2\nk2+,k1l2=k2l1,\nCi j=AiA2\nj\n16δ2k2\njk2\n−\nk2+,E12=A2\n1A2\n2\n256δ4k2\n1k2\n2k4\n−\nk4+,(16)\nwhere k+=k1+k2,k−=k1−k2. Parameters Ai,Aj,ki,kj\nandli,(i=1,2,j=3−i)are arbitrary real constants.\nIV . NUMERICAL INVESTIGATION OF LINE-SOLITON\nAND MAGNETIC LUMPS\nA. Unstable MS splits into lumps\nWe now turn to the stability and interactions between MSs\nin a ferromagnetic film. The initial data is a MS perturbed\nby some position-dependent Gaussian wave packets with the\nfollowing expression:\nf=bexp\"\n−\u0012x−x0\nxr\u00132\n−\u0012y\nyr\u00132#\n, (17)\nwhere b,xrandyrcorrespond to the shape of the wave packet\nandx0is related to the perturbation position.\nThe time evolution results clearly show the instability of\nthe MS. For small bi, the MS will break up and eventually4\n(a)\n (b)\n(c)\n (d)\nFigure 2. Propagation of MS perturbed by a Gaussian disturbance.\n(a) Component Hz, (b) Component Hy, (c) and (d) are enlarged views\nof the indicated areas circled in red and black, respectively. The\nparameters are chosen as A1=A2=1,δ=−1,l1=l2=0,k1=\n1,k2=2,x0=−29,b=0.1,xr=1.5,yr=2.5 in (16) and (17).\nevolve into some stable two-dimensionally localized lumps , as\ndisplayed in Figs. 2(a) and 2(b). We observe that most of the\nenergy is always propagated as a lump, even if its speed may\ndiffer from the input. Such a magnetic lump is a solitary wave\npacket that maintains its shape and speed during propagation\nor collision.\nA complete single lump of magnetic field component Hz\n(component Hy) is circled in red (black) in Fig.2. The enlarged\nviews (see Figs.2(c) and 2(d)) provide a clear picture of the\nshape and contour map of the lump. It can be found that\ncomponent Hzis a dipole-mode lump, whereas component\nHyis a standard KP-lump. We also show the vector field\nof the magnetic lump in Fig.3(a). Note that magnetic field\ncomponent Hxis zero, the magnetic field is always in the y−z\nplane, hence the lump can be regarded as a 360◦domain wall\nlocalized in xandydirections. Fig.3(b) presents the magnetic\nfield along y=0. The blue and red arrows correspond to the\nmagnetic field intensity of component Hz,Hy, respectively.\nThe rest of this work is concerned with the propagation and\ninteraction behavior of these lumps in ferrite medium.\n(a)\n (b)\nFigure 3. (a) The vector field of the magnetic lump. (b) The magnetic\nlump along y=0. The blue and red arrows correspond to the\nmagnetic field intensity of components Hz,Hy, respectively.B. Lump motion in ferromagnets with damping or\ninhomogeneous exchange effects\nFigure 4. Three dimensional projections of lump at t=0,HandW\nrepresent the definitions of lump height and width, respectively.\nThe evolution behavior of the magnetic lump in the ideal\nferrite is quite simple and imaginable. Each lump maintains\nits shape while it travels at a constant speed. However, in most\nof real ferromagnetic materials, we have to take the Gilbert-\ndamping into account . For instance, the dimensionless\ndamping constant sranges from 0.048 to about 0.385 in\ngarnet ferrite films. Here we are going to study the dynamics\nof magnetic lump in a damped ferrite film. The typical\nferromagnetic film under consideration is a garnet ferrite film\nwith the dimensionless damping constant s=0.1. For a\nclearer view of the change in shape of the lump, we define\nHandWas the height and width of the lump, which\nare the vertical distance between the highest point and the\nlowest point and the horizontal distance along the propagation\ndirection, respectively. All of these are summarized in Fig.4.\nThe propagation of a lump on the garnet ferrite film\nis presented in Fig.5. As shown in Fig.5(a), the lump\ntravels forward a visible distance in the damped ferrite.\nBeyond that, comparing the profiles of lump between t=0\nand t=10, we evidently observe that the lump becomes\nsmaller and narrower. Fig.5(b) shows the lump height\nand width exhibit a tendency of exponential decay. The\nsolid blue line is the exponential fitting curve to H(t),\nwith the function expression being H(t) = A0e−st. We\nconfirm the above-mentioned amplitude attenuation law is\nuniversal by simulating the motion of lump in ferrites with\nvirous damping factors. Moreover, a definite relationship\nbetween the amplitude and the localization region of solitons\nis important for the soliton excitations. We analyze different\nsizes of numerical lumps and mark the width and height of\nlumps in the phase diagram (see Fig. 5(c)). The results show\nthat for a magnetic lump excitation, its width and height meet\na linear relationship within the error range ( W/H∼0.305).\nSo the lump excitation, upon decay, retains a soliton form.\nTherefore, in this system, the Gilbert-damping plays a role of\ndissipating energy during the motion of magnetic lumps and\nit is characterized by decreasing the amplitude and width of\nlump.\nThe inhomogeneities otherwise referred to as deformities is\ninevitable in real magnetic materials, and it can be caused by\neither external fields or the presence of defects, voids and gaps\nin the material. It has already been reported that the MS may\nbe deformed by the presence of inhomogeneities, in particular5\n(a)\n (b)\n (c)\nFigure 5. Evolution of a magnetic lump in a damped ferrite film with dimensionless damping constant s=0.1. (a) Comparison picture of lump\nwave at t=0 and t=10. (b) The variation of lump height H, lump width Wand velocity V . (c) Numerical relationship between the width and\nheight of magnetic lump.\nits structure and speed [35, 42]. In this present system,\nthe inhomogeneous exchange process is unignorable when\nthe wavelength of lump is comparable to the characteristic\nexchange length.\n(a)\n (b)\n(c)\n (d)\nFigure 6. Propagation of lump with and without the inhomogeneous\ninteraction, respectively.\nWe now move to study the lump motion in the presence of\ninhomogeneous exchange effect. The initial data is the stable\nmagnetic lump shown in Fig.5. As can be observed from Fig.\n6(a) and 6(b), in ferrite without exchange interaction, the lump\nsolution propagates at a constant speed and along the previous\npath. We then consider the non-equilibrium dynamics of lump\nby performing a sudden interaction quench. The pictures\nof component Hyat dimensionless times t=2 and t=4.5\nare shown in Fig. 6(c) and 6(d). As we see, for a quench\nfrom the non-interacting to strong inhomogeneous exchange\nferrite film, the lump oscillates rapidly and diffracts alongthe propagation direction. A two-dimensional shock wave\nis generated and propagates forward. The shock wave front\ncontinues to propagate in the negative direction along x-axis.\nFinally, the energy of lump will be dissipated into numberless\ntiny waves. Accordingly, considering that the lump would be\ndestroyed by the inhomogeneous exchange process, one has to\nconsider keeping its wavelength away from the characteristic\nexchange length in the lump-based microwave applications.\nC. Some examples of excitations and interactions\nThe evolution pattern given in Fig.2 reveals that the lump\nmoves at a larger velocity than the broken MS in the\npropagation. The reason is that the velocity of soliton solution\nis proportional to the soliton amplitude. During the formation\nof the lump, the original MS will be destroyed, and most of the\nenergy is concentrated in some certain centers, which causes\nthe amplitude (and velocity) of the lump to be greater than that\nof MS. These lumps with various speeds enable us to explore\nthe interaction between lump and soliton, as well as between\ntwo lumps.\nA typical example of lump-MS collision is shown in\nFig.7(a). The MS begins to break up around at t=4.\nSubsequently, the splitting lump is going to catch up and\ncollide with the front-MS. After the collision, the front-\nMS is destroyed and broken into several lumps with various\nsizes. It is remarkable that the lump keep its localized\nform before and after the collision almost unchanged. This\nphenomenon implies such two-component lumps are natural\nresults from this nonlinear propagation equations. Further\nsimulation shows these lump structures could be generated\nby a MS with random disturbance. Fig.7(b) depicts a\ncharacteristic inelastic collision between two lumps. We\ninitially generate two adjoining lumps. They are emitted by\nMS at dimensionless time t=6.5. The merging process can\nbe performed as follows. From t=7.5 tot=9.5, two lumps\nmerge simultaneously together and give birth to a new lump\nwhose amplitude is significantly greater than the amplitude of\nprevious lumps. Obviously there is a weak attraction between\ntwo lumps which results in their fusion. In addition to the\nfusion of the two lumps, we also observed an extraordinary\npeak at a specific moment (about t=9.5), which looks like a6\n(a)\n(b)\nFigure 7. (a) Collision between lump and MS. (b) Mergence of two lumps and the formation of a second-order rogue wave-like structure.\nsecond-order rogue wave. It appears to be the result of the\ninteraction between the ripples surrounding the two lumps.\nAfter the fusion, the rouge wave-like structure disappears and\nthe dynamics of the output is determined mainly by a single\nhigh-amplitude lump.\nV . CONCLUSION\nAs a conclusion, the nonlinear propagation of MS in a\nsaturation magnetized ferromagnetic thick film is studied in\ndetail. In the starting point, we derive the (2+1)-dimensional\nKMM system that governs the evolution of short MS waves\nin a saturated ferromagnetic film. The bilinear form of the\nKMM system is constructed and the MS solutions are obtained\nanalytically.\nAfter that, numerical simulations are performed to analyse\nthe evolution behaviours of MS. A significant observation\nis that the unstable MS can be destroyed by Gaussian\nperturbation and broken into some stable magnetic lumps.\nThese lumps exhibit high stability during the propagation.\nFurthermore, some examples are given to analyse the collision\nbehaviours between lump and MS, and the interaction between\ntwo lumps. It is found the lump keeps its shape and speed in\nthe collision with MS. The results confirm that the lump is astable propagation mode in this system and, more to the point,\nthe velocity of lump can be adjusted by its amplitude. Their\nrobustness and controllability provide the possibility for future\ninformation memory and logic devices. We also study the\npropagation of such a lump in ferrites subjected to influence\nof damping and inhomogeneous exchange effects. When the\nGilbert-damping of ferrite is considered, the lumps undergo\nthe following changes: the amplitude and the speed of lump\nare decreased, and the width of lump along the propagation\ndirection is getting narrow. It would cause a strong diffraction\nof the lump if we quench the interaction strength.\nWe hope our work will invoke follow-up experimental\nstudies of lump-based microwave applications. Addition-\nally, since only one- and two-line-soliton are obtained,\nthe integrability of the (2+1)-dimensional system Kraenkel-\nManna-Merle (KMM) remains an open issue. The existence\nof the higher-dimensional evolution system as well as the\nbulk polariton solution is an intriguing avenue for future\nexploration.\nACKNOWLEDGMENT\nThis work was supported by the National Natural Science\nFoundation of China under Great Nos. 11835011; 11675146;\n11875220;.\n[1] M Daniel, V Veerakumar, and R Amuda. 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Angelo,\nI-80126 Napoli, Italy\n3)Centro de F´ ısica de Materiales, Centro Mixto CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5,\n20018 San Sebasti´ an, Spain\n4)Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebasti´ an,\nSpain\n5)CNR-SPIN c/o Universit´ a degli Studi di Salerno, I-84084 Fisciano (Sa), Italy\nWe investigate the switching current distributions of ferromagnetic anomalous Josephson junctions subjected\nto a linearly increasing bias current. Our study uncovers a significant correlation between the position\nof the switching current distributions and crucial system parameters, such as the strength of the spin-orbit\ncoupling and the Gilbert damping parameter. This indicates that these parameters can be directly determined\nthrough experimental measurements. By conducting a comprehensive analysis of the interplay among noise,\nmagnetization, phase dynamics, and the statistical properties of the switching current distribution, we\ndeepen our understanding of these intriguing cryogenic spintronics devices. These findings hold potential\nfor applications in the field of quantum computing architectures and information processing technologies.\nFerromagnetic anomalous Josephson junctions\n(FAJJs), formed by combining superconductivity with\nferromagnetic materials, have garnered significant\nattention due to their unique properties and potential\napplications in spintronics, quantum technologies, and\ninformation processing1–6. These junctions exhibit\nintriguing phenomena, including anomalous phase shifts\nand magnetization reversal induced by the flow of\nsupercurrent, making them promising candidates for a\nwide range of cutting-edge technological advancements.\nThe combination of superconductivity and\nferromagnetism in these junctions offers exciting\npossibilities for spintronics applications7. The\nexchange interaction between the superconducting\nand ferromagnetic regions enables the control of the\nmagnetization direction through the manipulation of\nsuperconducting phases. This unique functionality holds\ngreat promise for the development of spintronics memory\nelements, magnetic field sensors, and logic circuits with\nlow power consumption and high-speed operation.\nSince the work by Shukrinov et al.8, several\nstudies have explored the dynamics of FAJJs, for\ninstance focusing on the current-induced magnetization\nreversal phenomenon, in both short junctions9–19\nand superconducting quantum interference devices\n(SQUIDs)20–22. An interested reader may refer to\nRefs. 23 and 24 for a comprehensive review and further\ndetails.\nNoise influences the behavior of these systems,\nplaying an important role in both understanding their\na)Electronic mail: Author to whom correspondence should be\naddressed: cguarcello@unisa.it\nb)Electronic mail: fs.bergeret@csic.es\nc)Electronic mail: rocitro@unisa.itoperation and harnessing their full potential, although\nits effect has been little studied to date. Nonetheless,\nstochastic thermal fluctuations can profoundly affect\nthe behavior and the performance of these devices25,26.\nUnderstanding the interplay between noise and the\ncomplex dynamics of AFJJs is essential for optimizing\ntheir operation and ensuring the reliability of spintronics\nand quantum devices.\nIn the broader context of Josephson junctions (JJs),\nthe study of the switching current distribution (SCD)\nis of fundamental importance to characterize their\nbehaviour27. It provides insights into the dynamics of\nthe superconducting phase and the interplay between\nnoise and junction parameters, and it was throughly\ndiscussed, both theoretically and experimentally, in many\nworks about Josephson devices27–31. The SCD in\nconventional JJs can exhibit a wide range of behaviors,\nranging from narrow and well-defined distributions to\nbroad and multimodal distributions. In fact, noise\nsources introduce fluctuations in the superconducting\norder parameter, leading to variations in the switching\nprocess. Consequently, the shape of the SCD reflects\nthe intertwining of noise effects and the specific\nx yz\nImax\nFIG. 1. Sketch of the AFJJ driven by a linearly ramping bias\ncurrent, Ibias(t).arXiv:2307.07751v2  [cond-mat.supr-con]  11 Oct 2023SCD in anomalous JJs 2\ncharacteristics of the junction.\nCharacterizing the SCD and understanding the role of\nnoise in shaping its properties are crucial for designing\nand optimizing the performance of FAJJs for their\nvarious applications. In this Letter, we analyze the\nimpact of thermal noise on the behavior of FAJJs, with\na focus on the SCDs. In particular, we demonstrate how\nthey can reveal information on intrinsic characteristics\nof the junction, such as the Rashba coupling strength\nand the phenomenological Gilbert damping parameter.\nThis work adds a fundamental piece to the study of\nnoise-induced effects in magnetic Josephson systems,\nwhich plays a key role in understanding the actual\nresponse of a real device; indeed, we focus on the\nswitching dynamics under a linearly ramping bias\ncurrent, moving significantly away from the conventional\nscenarios considered so far, in which the phenomenon\nof pulse-controlled magnetization reversal8,12,14,19,25,26,32\nand the response to ac currents9,11,16had been largely\nconsidered.\nThe setup that we consider consists of a FAJJ, see\nFig. 1, with a thin ferromagnetic film with an out-of-\nplane magnetic anisotropy and a Rashba-like SOC25.\nDue to the interplay between the exchange field and\nthe SOC, the ferromagnetic junction presents a current-\nphase relation with the form33,34Iφ=Icsin(φ−φ0).\nHere, Icis the critical current of the junction, φis the\nJosephson phase difference, and φ0is the anomalous\nphase shift , which has been experimentally measured first\nin Josephson devices including topological insulators and\nAl/InAs heterostructures or nanowires35–38. The value\nofφ0depends on different system parameters, such as\nthe Rashba coupling α39,40, the transparency of S/F\ninterfaces, the spin relaxation, and the disorder degree.\nFor our purposes, the exact dependence of φ0on these\nparameters is not as relevant as the device geometry.\nIf we assume a two-dimensional SOC with magnetic\nmomenta in the plane of the F film (in particular, we are\nassuming a monodomain ferromagnetic arrangement),\nand the bias current flows in x-direction, the phase shift\nφ0is proportional to the y-component of the magnetic\nmoment according to33,34,41,42\nφ0=r my, (1)\nwhere my=My/M, with M=q\nM2x+M2y+M2z\nbeing the modulus of the magnetic moment, and the\nparameter rdepends on α(for the explicit dependence of\nron various system parameters, including the strength\nof the spin-orbit interaction, refer to Refs. 33 and 41).\nEquation (1) evidently establishes a direct coupling\nbetween the magnetic moment and the supercurrent.\nThe time evolution of the magnetic moment can be\ndescribed in terms of the Landau–Lifshitz–Gilbert (LLG)\nequation43,44\ndM\ndτ=γ\nM\u0012\nM×dM\ndτ\u0013\n−grM×Heff, (2)where grdenotes the gyromagnetic ratio. The first\nterm on the right-hand side accounts for the dissipation\nthrough the phenomenological dimensionless Gilbert\ndamping parameter γ, while the second term represents\nthe precession around Heff, which components can be\ncalculated as45\nHeff,i=−1\nV∂F\n∂Mi, with i=x, y, z. (3)\nHere, Vis the volume of the F layer and\nF=−EJφIbias+Es(φ, φ 0) +EM (4)\nis the free energy of the system. Here, EJ= Φ 0Ic/(2π)\n(with Φ 0being the flux quantum), Ibiasis the external\ncurrent in units of Ic,Es(φ, φ 0) =EJ[1−cos(φ−φ0)],\nandEM=−KV\n2\u0000Mz\nM\u00012is the magnetic energy that\ndepends on the anisotropy constant K. In the following,\nwe use the parameter ε=EJ/(KV) to indicate the ratio\nbetween the energy scales of the system. In Eq. (2)\nwe are neglecting a second-derivative term proportional\nto an angular momentum relaxation time τa46–49. On\ntimescales shorter than τa, this term is demonstrated\nto give nutation oscillations on top of the precession\nmotion. Instead, on timescales longer than τa, the usual\nLLG equation is expected to work well in describing the\nmagnetic evolution. This is exactly our case, being τa\nwell below both the inverse Josephson frequency and the\nramp time, tmax\nFrom Eq. (3), we obtain the effective magnetic field\nHeff=K\nM[εrsin (φ−rmy) ˆy+mzˆz], (5)\nwhere mx,y,z =Mx,y,z/M are the normalized\ncomponents of the magnetic moment that have to satisfy\nthe condition m2\nx+m2\ny+m2\nz= 1. The LLG equations can\nbe conveniently expressed in spherical coordinates50, so\nas to conserve the magnetic moment modulus even when\nstochastic noise contributions are taken into account.\nThen, mx,y,z can be written in terms of polar and\nazimuthal angles θandϕas\nmx(τ) = sin θ(τ) cosϕ(τ)\nmy(τ) = sin θ(τ) sinϕ(τ) (6)\nmz(τ) = cos θ(τ).\nWe can also define the θand ϕcomponents of the\nnormalized effective field as\neHeff,θ=εrsin(φ−rmy) cosθsinϕ−mzsinθ (7)\neHeff,ϕ=εrsin(φ−rmy) cosϕ. (8)\nThus, if the time is normalized to the inverse of the\nferromagnetic resonance frequency ωF=grK/M, that\nist=ωFτ, the LLG equations in spherical coordinates\nreduce to the following two coupled equations50\ndθ\ndt=1\n1 +γ2\u0010\neHeff,ϕ+γeHeff,θ\u0011\n(9)\nsinθdϕ\ndt=1\n1 +γ2\u0010\nγeHeff,ϕ−eHeff,θ\u0011\n. (10)SCD in anomalous JJs 3\n0.6 0.7 0.8 0.9 1.0010203040\niSWPDF\n0.60.70.80.91.0〈iSW〉\n0.005\nσiSW\n0.001-1.0-0.8-0.6-0.4-0.20.0\nDIS\n012345\nK0.60.70.80.91.0〈iSW〉\n0.0050.010.050.1\nσiSW\n0.001 0.005 0.005    0.01 0.05            0.1-1.0-0.8-0.6-0.4-0.20.0\nDIS\n012345\nK(a)\n(b)\n(c)\nγ\n0.01\n0.013\n0.016\n0.02\n0.025\n0.032\n0.04\n0.050.063\n0.079\n0.1\n0.13\n0.16\n0.2\n0.250.32\n0.4\n0.5\n0.63\n0.79\n0.75 0.8 0.85 0.9 0.95024681012\niSWPDF\n0.880.890.90.91〈iSW〉\n0.0340.0380.042\nσiSW\n0.01 0.05 0.1 0.5-1.2-1.1-1.0-0.9-0.8-0.7\nγS\n3.544.555.56\nK〈iSW〉\nσiSW\nKS\u000bG\f\n\u000bH\f\n\u000bI\f\n0.8 0.85 0.9 0.95024681012\niSWPDF\n0.870.880.890.90.91〈iSW〉\n0.0320.0360.040\nσiSW\n0.0 0.1 0.2 0.3 0.4 0.5-0.9-0.8-0.7\nrS\n3.644.44.8\nKr\n0.5 0r (g)\n(h)\n(i)DI\n0.001\n0.0014\n0.0019\n0.00270.0037\n0.0052\n0.0072\n0.010.014\n0.019\n0.027\n0.0370.052\n0.072\n0.1\n0.14\nFIG. 2. PDF of the switching currents iSW, see top panels, moments of the SCDs, i.e., mean values ( ⟨iSW⟩) and root mean\nsquare ( σiSW) in middle panels, whereas skewness ( S) and kurtosis ( K) in bottom panels, as a function of: (a-c) DIwith\n(r, γ) = (0 .25,0.1), (d-f) γwith ( DI, r) = (0 .05,0.1), and (i-l) r(from r= 0, red curve, to r= 0.5, purple curve, with\n∆r= 0.02) with ( DI, γ) = (0 .05,0.1). The dashed lines in panels (b-c) indicate the SCD moments obtained in the decoupled\ncase, i.e., r= 0. The legend in panel (e) [(f)] refers also to (b) and (h) [(c) and (i)].\nThe dynamics of the Josephson phase can be described\nin terms of the resistively and capacitively shunted\njunction (RCSJ) model51–54, which can be generalized\nto include the anomalous phase shift φ0=rmy55:\ndφ\ndt=ωχ+rdmy\ndt(11a)\ndχ\ndt=ω\nβc[Ibias(t) +Ith(t)−sin (φ−rmy)−χ].(11b)\nHere, time ( t) is still in units of the inverse of the\nferromagnetic resonance frequency, and ω=ωc/ωF, with\nωc= 2πIcR/Φ0being the characteristic frequency51of\nthe junction with a normal-state resistance R. The\ndamping is quantified by the McCumber parameter, βc=\n2πIcCR2/Φ0(Cis the junction capacitance), which is\nrelated to the so-called quality factor of the junction\naccording to βc=Q2. In this work, we consider a\nmoderately underdamped junction, i.e., we set Q= 5\njust in line with Refs. 13, 15, and 18. The values of\nthe other parameters are ε= 10 and ω= 1. The\nvalue of energy ratio εis expected to range from ∼100,\nfor a weak magnetic anisotropy, to ∼1, for a strong\nanisotropy8. We chose an intermediate value, in line with\nother theoretical work8, but to guess how a different value\naffects the system we observe that εcomes into play in\ni) the normalised effective field, see Eqs. (7-8) and in\nii) the thermal field intensity. Thus, we expect a larger\n(smaller) εto give faster (slower) magnetic dynamics\nand a more (less) relevant noise-induced effect, e.g., see\nRefs. 8, 10, 12, 23, and 24.\nThe noise term Ith(t) is a sort of “thermal current”\nwith the usual white-noise statistical properties that, innormalized units, can be expressed as51,56,57\n⟨Ith(t)⟩= 0 ⟨Ith(t)Ith(t′)⟩= 2DIδ(t−t′),\n(12)\nwith\nDI=kBT\nRωF\nI2c=1\nωkBT\nEJ, (13)\nbeing the dimensionless intensity of thermal current\nfluctuations. In this work we consider also magnetic field\nfluctuations, i.e., we take into account in Eq. (2) a delta-\ncorrelated stochastic “thermal field” Hthcontribution,\nwith an intensity DH= (γ εω)DI(see Ref. 25 for more\ndetails). This means that one can regulate the relative\nstrength of the two noisy mechanisms by changing the\nmagnetization energy, the Gilbert damping parameter,\nor the magnetic-resonance frequency. Thus, the system\nparameters can be optimized in such a way to make, for\ninstance, the impact of the thermal field negligible with\nrespect to the thermal current. Below, although we only\nspecify the value of DI, we point out that both thermal\ncurrent and thermal field are included.\nIn order to construct a SCD, we consider a linearly\nincreasing bias current ibias(t) =Ibias(t)/Ic=vbt, where\nvb=t−1\nmaxis the ramp speed. A measurement consists\nin slowly and linearly ramping the bias current in a time\ntmax, so that Imax≡Ib(tmax) =Ic, and to record the\ncurrent value, iSW, at which a switch occurs, namely, at\nwhich the phase particle leaves the initial potential well.\nAlternatively, one could look at the voltage drop. In\nthis way, it should be also possible to distinguish phase\ndiffusion events, in which the phase particle once escaped\nis retrapped in an adjacent minimum due to dampingSCD in anomalous JJs 4\naffecting its motion. In this case, the resulting SCD\nshould show opposite behavior with temperature, i.e., its\nwidth should increase as the temperature decreases58,59.\nIn this readout scheme, the noise influence is considered\nin the limit of the adiabatic bias regime, where the change\nof the slope of the potential induced by the bias current is\nslow enough to keep the phase particle in the metastable\nstate until the noise pushes out the particle. Since we are\ndealing with stochastic switching processes, we need to\nperform many independent experiments under the same\nconditions, in order to obtain comprehensive statistics.\nIn particular, sequences of N= 104independent\nnumerical experiments of maximum duration tmax =\n104are performed. Thus, the collection of these\naccumulated switching currents forms a SCD, whose\nmoments, i.e., mean, variance, skewness, and kurtosis,\ncan be investigated.\nWe assume that the magnetic moment initially points\ntowards the z-direction, that is M= (0,0,1) at t= 0,\nand that φ(0) = dφ(0)/dt= 0. With these initial\nconditions, we solve Eqs. (9)-(11) self-consistently, at\ndifferent system parameter values. Once the switching\nhas taken place, the Josephson phase undergoes fast\n2πrotations and a non-zero voltage drop appears. In\nthis condition, the instantaneous value of magnetic\nmoment is not so relevant, but rather the time-averaged\nvalue it takes after switching. Computing also the\nensemble average on the total amount of independent\nrealizations, we observe, in the regions of parameter\nspace under consideration, zero average after-switching\nmagnetic moments (data not shown). The readout of the\nmagnetic state of the system could be eventually achieved\nthrough a similar nondestructive scheme proposed in\nRef. 25.\nIn Fig. 2 we collect the probability distribution\nfunction (PDF) of switching currents iSW(see top panels)\nand the moments (i.e., mean value, ⟨iSW⟩, and root mean\nsquare, σiSW, in the middle panels, while skewness, S,\nand kurtosis, K, are shown in the bottom panels) of the\nSCDs, varying the main system parameters, ( DI, r, γ),\nin suitable ranges. In particular, we choose: in panels\n(a-c) DI∈[0.001−1.0] with ( r, γ) = (0 .25,0.1), in (d-f)\nγ∈[0.01−1.0] with ( DI, r) = (0 .05,0.1), and in (i-l)\nr∈[0−0.5] with ( DI, γ) = (0 .05,0.1). Skewness and\nkurtosis are obtained as\n˜S[X]=E\"\u0012X−µ\nσ\u00133#\nand ˜K[X]=E\"\u0012X−µ\nσ\u00134#\nwhere Xrepresents the random variable of the\nmeasurements, E[·] is the expectation operator, µ=\nE [X], and σ2= var ( X). These quantities can be\nestimated through the measured switching currents iSW,j\nas\nS=D\n(iSW,j− ⟨iSW⟩)3E\nσ3andK=D\n(iSW,j− ⟨iSW⟩)4E\nσ4,\nwhere ⟨iSW⟩andσare the estimates of the mean andthe standard deviation of the switching distribution,\nrespectively.\nWe first look in Fig. 2(a) at the behavior of the SCDs\nat different noise intensities DI, which is proportional\nto the temperature according to Eq. (13). We note\nthat the DIvalues are chosen in such a way that the\nmoments and the magnetic moments in Figs. 2(b-c)\nappear equally spaced on a logarithmic scale. The more\nthe noise intensity increases, the more the SCDs are\nlower, wider, and centered at smaller switching current\nvalues. The first two moments, i.e., mean value, ⟨iSW⟩,\nand root mean square, σiSW, see panel (b), show a\ntypical trend: in particular, ⟨iSW⟩tends to saturate\nat the level ∼1 at low noises and then it decreases,\nwhile σiSWincreases exponentially (i.e., linearly on a\nlog-log scale) as the noise increases. We stress that\nwe are assuming to work at temperatures above the so-\ncalled “crossover temperature”27,60, i.e., that value below\nwhich macroscopic quantum tunneling (MQT) effects\ndominate the switching dynamics: in this case, being the\nMQT probability independent of T, one would obtain\noverlapping SCDs, all centered around the crossover\ntemperature. In panel (c) we show the third and fourth\nmoments: we see that the skewness S(the kurtosis K)\ntakes the value S∼ −0.4 (K∼3) for DI= 0.001 and\ndecreases (increases) up to the value S∼ −0.8 (K∼4)\nforDI= 0.05. The obtained values are in line with\nthose already observed for other Josephson systems61.\nIn other words, increasing noise makes the SCDs more\nasymmetric and with tails more pronounced.\nIn order to understand the interplay between the\nJosephson and the magnetic systems, we enrich panels\n(b) and (c) with the SCD moments of a conventional JJ\n[i.e., setting r= 0 in Eq. (11) in order to decouple the\nJosephson and magnetic systems] marked with dashed\nlines. It is clear that all moments tend to diverge\nfrom the pure-thermal behavior for noises larger that\nDI≳0.05. The response of a FAJJ therefore cannot be\nmerely assimilated to that of a conventional JJ residing\nat a slightly higher temperature, i.e., subject to a larger\nnoise intensity. Indeed, in principle, one could assume an\neffective temperature such that the behavior of the SCDs\nin Fig. 2(a-c) could again be achieved in a conventional\nJJ. In particular, we could search for an ad-hoc additional\nnoise intensity that gives the values of ⟨iSW⟩(DI) that\nagree with those we obtained; however, by doing so,\nthe other moments would differ significantly from those\npresented in Fig. 2(b-c) (data not shown). Thus, one\ncannot merely reduce our results to what would be\nobserved considering a conventional JJ with a higher\neffective temperature.\nThe central panels of Fig. 2 report how the SCDs\ndepend on the Gilbert damping parameter, γ∈[0.01−\n1.0], imposing ( DI, r) = (0 .05,0.1). Looking at the\nPDF of the switching currents in Fig. 2(d), it is\nevident that by changing γboth the position and the\nshape of the SCDs change. Specifically, ⟨iSW⟩(σiSW)\nshow a monotonically decreasing (increasing) behaviorSCD in anomalous JJs 5\nincreasing γ, see Fig. 2(e). Instead, both SandKremain\npractically unchanged for γ≲0.2, while for higher γ\nthey tend to increase, the former, and to decrease, the\nlatter, see Fig. 2(f). The value of rsignificantly affects\nthe threshold γvalue above which the shape of the SCDs\nstarts to change appreciably. We stress that the scattered\nbehavior of the higher-moment curves can be smoothed\nout by increasing the number of numerical repetitions.\nFinally, Fig. 2(g-h) says that by increasing the SOC\nstrength, r, the SCDs tend to shift towards lower\nswitching currents. In particular, we observe that both\n⟨iSW⟩and skewness (root-mean-square and kurtosis)\ntend to decrease (increase) almost linearly by increasing\nr.\nFigure 2 says that the analysis of both the average and\nthe shape of the switching current distributions could be\nemployed to experimentally determine γorr, the other\nbeing known.\nIn conclusion, we discussed the behavior of the SCDs\nof a current-biased φ0-junction, that is a superconductor-\nferromagnet-superconductor JJ with a Rashba-like spin-\norbit coupling. We investigated the noise impact on both\nthe Josephson phase dynamics and the magnetization of\nthe system in the case of a slowly, linearly ramping bias\ncurrent. In particular, we showed how a change in the\nvarious characteristic parameters of the system impacts\nthe SCDs, also looking at the distribution moments, i.e.,\nmean values, root mean square, skewness, and kurtosis.\nIn particular, their analysis give information on the\nmain quantities of the system, i.e., the phenomenological\nGilbert damping parameter γand the SOC strength.\nHowever, the description of a concrete experiment should\nfirst proceed with a fine-tuning of the other parameters,\nsuch as Ic,ε, the saturation magnetization, ωc, and ωF\n(some values can be found in, e.g., Refs. 8–10, 17, 23, and\n24).\nThe ability to control and manipulate the\nmagnetization states through the superconducting\nphases opens up possibilities for encoding and processing\nquantum information, thus our results may contribute\nto the development of optimized spintronics.\nACKNOWLEDGMENTS\nF.S.B. acknowledges financial support from Spanish\nMCIN/AEI/ 10.13039/501100011033 through project\nPID2020-114252GB-I00 (SPIRIT) and TED2021-\n130292B-C42, and the Basque Government through\ngrant IT-1591-22. 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Box 513, 5600 MB Eindhoven, The Netherlands\n5Department of Physics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, Massachusetts 02467, USA\nSpin torque oscillators (STOs) are dissipative magnetic systems that provide a natural platform\nfor exploring non-Hermitian phenomena. We theoretically study a two-dimensional (2d) array of\nSTOs and show that its dynamics can be mapped to a 2d, non-Hermitian Su-Schrieffer-Heeger\n(SSH) model. We calculate the energy spectrum and identify the one-dimensional (1d) edge states\nof our model, corresponding to auto-oscillation of STOs on the boundary of the system while the\nbulk oscillators do not activate. We show that tuning the Gilbert damping, injected spin current,\nand coupling between STOs allows for exploring the edge state properties under different parameter\nregimes. Furthermore, this system admits 1d edge states with non-uniform probability density, and\nwe explore their properties in systems of different sizes. Additional symmetry analysis indicates\nthat these states are not topologically protected but are nevertheless confined to the edge of the\nsystem, as the bulk is protected by PT-symmetry. These results indicate that 2d arrays of STOs\nmay be useful to explore novel edge state behavior in dissipative systems.\nI. INTRODUCTION\nTopology and its connection to condensed matter sys-\ntems has been the subject of intense research for nearly\nfifty years, since the discovery of quantized Hall resis-\ntance and its topological origin [1, 2]. Topology is now\nunderstood as a critical underlying feature of many ma-\nterials that can affect global transport properties, result-\ning in, e.g., the quantum anomalous and spin Hall ef-\nfects [3, 4] and leading to entirely new classes of topo-\nlogical materials [5–7]. The effects of topology in sys-\ntems with dissipation, i.e. non-Hermitian systems, can be\nmarkedly different from their Hermitian counterparts [8–\n14]. Non-Hermitian systems can exhibit exceptional\npoints [15–18], i.e. the coalescence of two or more eigen-\nvectors, as well as the non-Hermitian skin effect, a phe-\nnomenon where bulk eigenstates localize on the edge of\nthe system [19–21]. The edge states in non-Hermitian\nsystems can also exist despite the breakdown of the bulk-\nboundary correspondence [9, 10] and exhibit lasing be-\nhavior [22, 23]. While several experimental realizations\nof non-Hermitian phenomena have been observed in pho-\ntonic [22], acoustic [24], and electronic circuits [25, 26],\ntheir experimental exploration in magnonic systems is\nstill in its infancy. Magnonic systems are, however, a nat-\nural platform in which to realize non-Hermitian physics\nbecause they are always coupled to a surrounding envi-\nronment and exhibit lossy dynamics [27, 28].\nSpin torque oscillators (STOs) have recently emerged\nas a promising platform for harboring non-Hermitian\nphenomena [27, 29, 30]. These magnetic nanopillars are\nnanometer sized devices that conduct spin currents via\nspin transfer torque [31]. STOs are dissipative systems\nbecause, like all magnetic systems, they are subject to\nubiquitous spin non-conserving interactions parameter-ized by Gilbert damping [32]. The magnetic dynamics\nof 1d STO arrays was successfully mapped to a non-\nHermitian Su-Schrieffer-Heeger (SSH) model with topo-\nlogically protected lasing edge states in Ref. [29]. In a\nsubsequent numerical study these edge states were shown\nto be robust in the presence of additional terms such\nas dipolar interactions and nonlinear STO behavior [33].\nHere, we examine whether these lasing edge states can\nbe realized in higher-dimensional arrays of STOs.\nThe 1d SSH model and it’s non-Hermitian variants are\nwidely used in condensed matter physics to study topo-\nlogical systems [10, 34]; 2d non-Hermitian SSH mod-\nels have been studied in a variety of contexts, where a\nnumber of interesting properties such as in-gap topolog-\nical states and non-trivial bulk-band topology have been\nfound [25, 26, 35–37]. However, most of these models lack\na clear experimental implementation, with topological\ncircuits being a notable exception [25, 26]. The sources of\nnon-Hermitian terms (i.e. energy non-conserving terms)\ncan be difficult to quantify and characterize in experi-\nmental platforms. Here, we propose STO arrays as a\nplatform for experimental realization of a non-Hermitian\n2d SSH model. Specifically, we focus on a geometry con-\nsisting of several 1d chains that are weakly coupled to\nform a 2d STO array. By introducing this new type of\n‘vertical coupling’, we derive a non-Hermitian 2d SSH\nmodel that exhibits 1d lasing edge states.\nThe manuscript is organized as follows: In Section II\nwe map the linearized Landau-Lifshitz-Gilbert (LLG)\nequation for the magnetization dynamics into a non-\nHermitian tight-binding Hamiltonian. In Sec. III we dis-\ncuss the properties of our model including the oscillatory\nedge states and the symmetry properties of the Bloch\nHamiltonian. Finally, we conclude and discuss directions\nfor future work in Sec. IV.arXiv:2307.13876v1  [cond-mat.mes-hall]  26 Jul 20232\nII. MODEL\nWe consider a 2d array of M×2NSTOs which could be\nfabricated from individual nanopillars [29] or a multilayer\nstructure [33]. Here, Mis the total number of rows in\nthe array and Nindicates the number of unit cells per\nrow, where there are two STOs per unit cell. A single\nSTO consists of a layer of fixed magnetic polarization\nand a ‘free’ magnetic layer without fixed polarization,\nseparated by a thin metallic spacer. By injecting spin\ncurrent into the free layer, the fixed layer is driven to\nprecess about its equilibrium direction, which is set by\nan external applied magnetic field. The dynamics of an\nisolated STO subjected to a magnetic field H0=H0ˆz\nand spin current JS\nη=JS\nηˆzare described by the LLG\nequation for the magnetization vector mη,ij[31]. Here,\nthe index η=A, B denotes the AandBsublattices and\nthe indices i, jlabel the sites using the (row, column)\nconvention. The resulting LLG equation is\n˙mη,ij|0=ωη,ijˆz×mη,ij+αη,ijmη,ij×˙mη,ij\n+JS\nηmη,ij×(mη,ij׈z). (1)\nThe STO ferromagnetic resonance frequency is given\nbyωη,ij=γη,ij(H0−4πMη,ij) where γη,ijis the gyro-\nmagnetic ratio and Mη,ijis the saturation magnetization;\nαη,ij≪1 is the dimensionless Gilbert damping parame-\nter. We assume the resonance frequency and the Gilbert\ndamping to the be same for all STOs in the array and\ndrop the subscripts going forward, i.e. ωη,ij→ωand\nαη,ij→α. The third term in Eq. (1) is the Slonczewski-\nBerger spin-transfer torque [38, 39]. The injected spin\ncurrent JS\nηis assumed to the be same for all sites in a\ngiven sublattice.\nInteractions between adjacent STOs are mediated by a\nreactive Ruderman- Kittel-Kasuya-Yosida (RKKY)-type\nmagnetic exchange coupling, which is given by [29]\n˙mA,ij|coup=−mA,ij×(JmB,ij+˜JmB,ij−1)\n−mA,ij×J2(mη′,i+1j+mη′,i−1j) (2)\n˙mB,ij|coup=−mB,ij×(JmA,ij+˜JmA,ij+1)\n−mB,ij×J2(mη′′,i+1j+mη′′,i−1j) (3)\nThe coupling strengths J,˜Jare the intracell and in-\ntercell coupling for the 1d unit cell, which contains two\nSTOs, and J,˜J, J 2>0 indicate ferromagnetic coupling.\nWe consider the geometry depicted in Fig. 1, for which\nη′=A, η′′=B.\nWe now derive the linearized equations of motion for\nthe STO magnetization mA(B). That is, we start in\nthe strong-field regime where the magnetic moment is\nmostly aligned along the ˆzdirection and the energy scale\ngiven by ωis the largest one in the problem. Starting\nFIG. 1. Schematic of the effective tight-binding model of a 2d\nSTO array, showing a system of size 4 ×8, where there are 4\nunit cells per row. Red (dark color) sites are the A sublattice\nand green (light color) sites are the B lattice.\nfrom Eq. (1), we can derive an effective Hamiltonian in\nthe following way: First, we linearize the equations of\nmotion about the equilibrium field direction by writing\nmη,ij= (mx\nη,ij, my\nη,ij,1)Twhere |mx\nη,ij| ≃ |my\nη,ij| ≪ 1,\nand|mz| ≃1 is assumed to be a constant. We then\nintroduce the variable 2 m−\nη,ij=mx\nη,ij−imy\nη,ij, and\nmake the Holstein-Primakoff approximation m−\nA(B),ij=\n⟨aij(bij)⟩e−iωtwhere the second quantized bosonic oper-\nators aij,bijannihilate a magnon on the STO at site\n(i, j) [40]. From the Heisenberg equations of motion\n˙aij=i/ℏ[H, a ij], we derive an effective Hamiltonian cor-\nresponding to the linearized equations of motion. In the\nfollowing we set ℏ= 1.\nFor the 2d STO array, H=P\nijHijis\nHij=ω(a†\nijaij+b†\nijbij)\n+i(JS\nA−αω)a†\nijaij+i(JS\nB−αω)b†\nijbij\n−J(a†\nijbij+h.c.)−˜J(a†\nijbij−1+h.c.)\n−J2(a†\nijai−1j+b†\nijbi−1j+h.c). (4)\nWe see that non-Hermiticity arises due to the onsite\nspin current injection and Gilbert damping, resulting\nin onsite terms ∝i(JS\nη−αω), and the degree of non-\nHermiticity can be tuned by balancing the injected spin\ncurrent and Gilbert damping. Given the non-Hermitian\nlattice model, we can now explore its energy spectrum\nand edge state properties. In this manuscript we use the\nterm ‘energy spectrum’ and the symbol Eto denote the\ncomplex eigenvalues of H; Re(E) can be thought of as an\nenergy while Im( E)>0 (<0) is an indication of lasing\n(damping).\nHere we consider the effect of different coupling\nstrengths J2. We confine the system to the parity-time\n(PT) symmetric regime where the injected spin current\nisJSA= 2αω,JSB= 0; further symmetry analysis is\nperformed in Sec. III C. This is a 2d extension of the\n1dPT-symmetric, non-Hermitian SSH model analyzed3\n−2.50.02.5Re(E) - ω\n [arb. units](a) (b) (c) (d)\n−2 −1 0 1 2\nJ/|̃J|−0.10.00.1Im(E) [arb. units]\n(e)\n−2 −1 0 1 2\nJ/|̃J|(f)\n−2 −1 0 1 2\nJ/|̃J|(g)\n−2 −1 0 1 2\nJ/|̃J|(h)\nFIG. 2. The dependence of the real (a-b) and imaginary (e-f) parts of the energy spectrum of HonJ/|˜J|for a system of 10 ×20\nSTOs for ω= 1˜J,α= 0.2, and J2= 0.01˜J(a,e) and J2= 0.1˜J(b,f). Modes with Im( E)>0 correspond to auto-oscillation of\nSTOs on the edge of the system, appearing for |J|<˜J. These modes correspond to the flat in-gap bands at Re( E)−ω≈0\nin the real spectrum. Panels (c-d, g-h) show the real (c-d) and imaginary (g-h) parts of the energy spectrum for a system of\n50×100 STOs, with J2= 0.01˜J(c,g) and J2= 0.1˜J(d,h). Apart from the different system size, all the other parameters are\nsame as in (a-b) and (e-f). In the larger system there is some activation of bulk STOs in the PT-broken regime, indicated in\n(g-h) by the additional states with Im( E)>0 in the region αω >|˜J−J|.\nin Ref. [29]. We note that other versions of the non-\nHermitian 2d SSH model studied in the literature have\nalternating A and B sublattice sites in the vertical di-\nrection [35–37], resulting in a four site unit cell. Here,\nhowever, we introduce the coupling such that each col-\numn is all A or B sublattice sites, thus reducing the unit\ncell to two STOs. Our reasoning for this geometry is that\nit would be easier to inject spin current into an entire col-\numn of STOs, for example using a metallic strip, rather\nthan having to individually address each A or B site STO\nin the grid.\nIII. RESULTS\nOur goal is to understand how the vertical coupling\nJ2between rows affects the properties of the original\n1d system, thus we briefly review the results for a 1d\nSTO chain [29, 33]. The real energy spectrum of the 1d\nSTO model Hamiltonian has two degenerate flat bands\nforJ <|˜J|and admits a real line gap in k-space [12].\nThe flat bands with degenerate energy eigenvalues are\nan indication of topologically protected edge states of\nthe Hamiltonian. The eigenstates corresponding to the\nflat-bands also have non-zero imaginary eigenvalues, in-\ndicating lasing (Im( E)>0) and damped (Im( E)<0)\nstates. The lasing states correspond to an auto oscilla-\ntion of STOs, which occurs only at the edge of the sys-\ntem for αω < |J−˜J|. In the regime αω > |J−˜J|, thebulk Hamiltonian also has complex eigenvalues which can\nlead to oscillation of the bulk STOs; this is the so-called\nPT-broken regime of the model where the Hamiltonian\nrespects PT-symmetry but its eigenstates do not [27].\nSymmetry analysis of the Bloch Hamiltonian for the 1d\nmodel confirms the auto-oscillation of the edge STOs to\nbe a zero-dimensional ‘edge state’ with topological pro-\ntection.\nIn the 2d case, the non-Hermitian Hamiltonian Eq. (4)\nalso has complex eigenvalues. The energy spectrum of\nthe 2d model looks similar to the 1d model in the case of\nweak vertical coupling J2/˜J≲0.01, but the degeneracy\nin the flat bands breaks immediately even with infinites-\nimal vertical coupling. In Fig. 2, we show the energy\nspectrum of the Hamiltonian considering different verti-\ncal coupling strengths J2/˜J= 0.01,0.1 for two different\nsystem sizes. We found that as J2increases with respect\nto˜J, the energy separation between flat bands increases\nand they hybridize with the bulk states. We simulated\nresults for a 2d array 20 sites wide and 10 sites in the ver-\ntical direction, as well as a larger system 100 sites wide\nand 50 sites in the vertical direction. In the upcoming\nsubsections, we respectively discuss the numerical results\nfor the edge states, analyze the Hamiltonian in momen-\ntum space, and present a symmetry analysis of the model.4\n0\n4\n8Site no.(a)\n|ψ|205\n(b)\n|ψ|205\n0 8 16\nSite no.0\n4\n8Site no.(c)\n|ψ|205\n0 8 16\nSite no.(d)\n|ψ|205\n0.000.050.100.15|ψ|2\n0 2 4 6 8\n Site no. 0.050.100.15|ψ|2(e)Re(E) -ω\n-0.19\n-0.17\n-0.13\n-0.08\n-0.03\n0.03\n0.08\n0.13\n0.17\n0.19\nFIG. 3. Spatial distribution of 1d edge modes in a 2d STO\narray. (a-d) Color density plots showing |ψ|2for four of the\nthe flat-band modes with Im( E)>0. The states are localized\non the left edge, with |ψ|2= 0 everywhere in the bulk. The\ninsets show the 1d distribution along the edge. (e) 1d plot\nshowing the spatial distribution of all 10 lasing edge states\nand the corresponding real part of the energy, Re( E)−ω.\nThe edge states are not uniform across the system due to the\ncoupling J2between rows. States appear in pairs where states\nwith equal magnitude and opposite sign of Re( E)−ωhave\nthe same spatial distribution (dashed and solid lines overlap).\nA. Edge States\nUsing exact diagonalization, we find that the real-\nspace Hamiltonian Eq. (4) exhibits one-dimensional ‘las-\ning’ edge states where Im( E)>0. In Fig. 3 we show\nthe spatial distribution of edge states for a system of\n10×20 STOs. Since there are 10 rows, the system\nexhibits 10 lasing edge modes, all with Im( E) = 0 .09.\nWe have confirmed that the corresponding damped edge\nmodes with Im( E) =−0.09 occur on the opposite edge\nof the system, as expected (not pictured here). We set\nthe intercell coupling to ˜J= 1 and consider the regime\nwhere intracell coupling J/˜J= 0.2 and vertical cou-\npling J2/˜J= 0.1. The physical manifestation of these\nedge states is an auto-oscillation of STOs that is non-\nuniform at the edge of the sample. Thus, the STOs will\nexhibit spatially varying microwave emission which can\nbe tuned based on the magnon population in each edge\nmode. We also diagonalized a larger STO array to\nsee how the results vary if we scale up the system size.\nThe Hamiltonian matrix has a block banded structure,\n0.000.010.020.03|ψ|2\n0.000.010.020.03|ψ|2\n0.000.010.020.03|ψ|2\n0.000.010.020.03|ψ|2\n0 20 40\nSite no.0.000.010.02|ψ|2\n0 20 40\nSite no.Re(E) -ω x 10−1\n0.06\n0.180.31\n0.430.55\n0.660.78\n0.891.0\n1.1FIG. 4. Example of the spatial distribution of 10 unique las-\ning edge states (out of 50) and the corresponding real part of\nthe energy, Re( E)−ω×10−1for a system of 50 ×100 sites.\nThe larger system shows greater heterogeneity in the spatial\ndistribution of edge modes. The top row of the legend corre-\nsponds to the left column and the bottom row corresponds to\nthe right column; values increase downward in each column.\nFor example Re( E)−ω= 1.0 corresponds to the bottom-left\npanel.\ncalled a block Toeplitz-tridiagonal (TT) matrix. For a\nlarger system i.e. 50x100 sites, the matrix becomes 5000\nx 5000 for which the diagonalization is computationally\nexpensive for different values of J2. To diagonalize this\nlarge sparse matrix efficiently, we used the method given5\nby Ref. [41]. The energy spectrum for the larger system -\nas shown in Fig. 2 (c-d), (g-h) - doesn’t show any devia-\ntion apart from very small bulk oscillations in the region\nwhere αω >|J−˜J|. This proves the 1d edge states can\nexist in a significantly larger system. As shown in Fig. 4,\nthe edge states in the larger system also exhibit a non-\nuniform spatial distribution, with increased oscillations\nand beating behavior visible in some modes. Here, we\nshow the first 10 out of 50 total edge states ranked by\nincreasing Re( E).\nB. Bloch Hamiltonian\nHere we consider the PT-symmetric regime where\nJSA= 2αωandJSB= 0, leading to balanced gain (loss)\nterms ±iαωon the A(B) sites. To analyze the Hamilto-\nnian in momentum space, we consider periodic boundary\nconditions and Fourier transform Eq. (4), which is writ-\nten as\nˆH=X\nk\u0000\na†\nkb†\nk\u0001\n(Hk−ω)\u0012\nak\nbk\u0013\n(5)\nwhere\nHk=\u0012\niαω−2J2cosky−J−˜Jeikx\n−J−˜Je−ikx−iαω−2J2cosky\u0013\n(6)\nThe resonant frequency simply provides an overall shift\nof the energy spectrum, therefore we redefine ωas the\nzero energy point.\nThe Bloch Hamiltonian can be written\nHk=d0(ky)1+d(kx)·σ (7)\nwhere 1is the 2 ×2 identity matrix and σ= (σx, σy, σz)\nis the vector of Pauli matrices. We define the functions\nd0(ky) =−2J2cos(ky) and\nd(kx) =\n−J−˜Jcos(kx)\n˜Jsin(kx)\niαω\n. (8)\nWe find the two-band energy spectrum\nϵ±(k) =−2J2cos(ky)±q\nJ2+˜J2+ 2J˜Jcos(kx)−α2ω2.\n(9)\nHere we see that the eigenvalues are real for αω <\n|J+˜Jeikx|, i.e. the system remains in the PT-unbroken\nregime exhibiting real eigenvalues as long as the Gilbert\ndamping is relatively small. This condition is satisfied for\nallkifαω < |J−˜J|. Furthermore, the spectrum is lin-\near in the vertical coupling J2; thus the effect of coupling\nadjacent 1d STO chains together is to shift the spectrum\naway from the resonant frequency ω. We can see this\nclearly for example in Fig. 2 (b), where the flat bands of\nadjacent 1d chains hybridize and are vertically shifted.\nFurthermore, J2can cause the energy gap to close, how-\never in the regime J2≲q\n(J−˜J)2−α2ω2the energy\ngap is open and the edge states remain well separated\nfrom the bulk.C. Symmetry Analysis\nSymmetry analysis can help determine whether the\n1d edge states displayed in Figs. 3 and 4 are topolog-\nically protected. We investigate the following symme-\ntries of the Bloch Hamiltonian, Eq. (6): chiral sym-\nmetry, chiral-inversion symmetry, sublattice symmetry,\nand parity-time ( PT) symmetry. Systems obeying chi-\nral, chiral-inversion, or sublattice symmetry can exhibit\ntopologically protected edge modes [11, 12, 42], therefore\nit is important to check whether these symmetries are\npreserved for our model. PT-symmetry ensures there is\nalways a regime in which the Hamiltonian has real eigen-\nvalues [43]. We note that for a Hermitian system, chi-\nral and sublattice symmetries are equivalent, however for\na non-Hermitian Hamiltonian this is no longer the case\nand some care must be taken. Here we use the symmetry\nnaming conventions from Ref. [12].\nTo have chiral symmetry (CS), the Hamiltonian must\nsatisfy the condition σzH†\nkσz=−Hk. We find that the\nvertical coupling term J2cos(ky) breaks chiral symmetry\nin general, however for the special values ky=±π/2 chi-\nral symmetry is preserved. To have chiral-inversion (CI)\nsymmetry, the Hamiltonian must satisfy the condition\nσyHkσy=−H−k. Like the case of CS, CI is in general\nbroken by J2and only preserved for ky=±π/2. For\nsublattice symmetry, the Hamiltonian must satisfy the\ncondition σzHkσz=−Hk. This model does not posses\nsublattice symmetry for any parameter regime due to the\nnon-Hermitian terms, as is the case in 1d [10, 29].\nTo have PT-symmetry, the Hamiltonian must satisfy\nthe condition σxH∗\nkσx=Hk. This condition is satisfied\nforJSA= 2αωandJSB= 0, thus as with the 1d case\nthe system can be tuned to the PT-symmetric regime by\naltering the injected spin current on A and B sublattice\nsites.PT-symmetry alone does not guarantee topological\nprotection [10], and the results from the symmetry anal-\nysis indicate that the edge states observed in this model\nare not topologically protected. However, the confine-\nment of the oscillatory modes to the edge can be under-\nstood as a result of PT-symmetry in the bulk, which is\nbroken spontaneously by the edge of the system. Fur-\nthermore, PT-symmetry guarantees that the bulk STOs\ndo not have any lasing modes as long as αω <|J−˜J|.\nIV. CONCLUSION AND OUTLOOK\nIn this work we have examined a novel realization of a\nnon-Hermitian 2d SSH model which can be constructed\nfrom an array of STOs. Using exact diagonalization and\nanalysis of the Bloch Hamiltonian, we have shown that\nthis model exhibits 1d lasing edge states with a non-\nuniform spatial distribution. The physical manifestation\nof these modes is an auto-oscillation of STOs along one\nedge of the system which is spatially varying. The exten-\nsion of the model from 1d to 2d via the addition of vertical\ncoupling between individual 1d STO chains breaks chiral-6\ninversion, chiral, and sublattice symmetry, indicating a\nloss of topological protection for these modes. However,\nthe vertical coupling preserves PT-symmetry for the bulk\nstates, thereby guaranteeing that the bulk oscillators do\nnot activate, even in the presence of spin current injected\ninto the bulk.\nHere we have considered an injected spin current such\nthat the system remains at the PT-symmetric point. Fu-\nture works could investigate the robustness of these edge\nstates in the presence of additional terms such as dipo-\nlar interactions as well as dissipative coupling between\nSTOs. However, results from studies of the analogous 1d\nmodel indicate that if these terms are small compared\nto the reactive RKKY coupling studied here, they would\nnot strongly affect the presence of edge states [29, 33].\nAnother interesting extension of this model would be\nto explore possible application in devices. The dispersionrelation in Eq. (9) indicates that the 1d edge states have\nnonzero group velocity due to the coupling J2. 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This study uses an associated Riccati equation to study the Ulam st ability of non-\nautonomous linear differential vector equations that model the da mped linear oscillator. In particu-\nlar, the best (minimal) Ulam constants for these non-autonomous lin ear differential vector equations\nare derived. These robust results apply to vector equations with s olutions that blow up in finite\ntime, as well as to vector equations with solutions that exist globally o n (−∞,∞). Illustrative,\nnon-trivial examples are presented, highlighting the main results.\n1.Introduction\nLetIbe an interval of R, and let n∈N. In this paper, we consider the second-order non-\nautonomous linear differential vector equation\nα(t)x′′+β(t)x′+γ(t)x=f(t), (1.1)\nwhereα,β,γ:I→Care continuous scalar functions, and f:I→Cnis a continuous vector\nfunction, and x∈C2(I,Cn). Moreover we assume that α(t)/negationslash= 0 for all t∈I. As is well known,\nwhen the coefficients α,β, andγof (1.1) are all positive constants, it is called a forced, damped\nlinear oscillator, and has various applications in for example mechanica l engineering. In many\nmathematical models, constant coefficients are determined throu gh statistical processing based on\nactualphenomena. However, itisalsotruethatconstantsmayno talwaysbesufficient duetofactors\nsuch as various changes in temperature, humidity, etc. due to sea sonal changes. In addition, there\nare phenomena for which variable coefficients are appropriate. For this reason, it can be expected\nthat considering the non-autonomous equation (1.1) will be useful in applications. So what is the\nbiggest problem when considering mathematical models? It goes with out saying that much of it has\nto do with perturbation. There will always be some kind of error betw een the actual phenomenon\nand the mathematical model, and depending on the situation, it may le ad to a large error as time\npasses. Therefore, ensuring that the error does not grow large is a very important application\nissue. Concepts such as “Ulam stability” originating from the field of f unctional equations (see,\n[1, 13, 14, 16, 17]) and “shadowing” originating from the field of dyna mical systems (see, [6, 7, 8, 9])\n2020Mathematics Subject Classification. 34D10, 34D20, 34A30.\nKey words and phrases. Ulam stability; best Ulam constant; damped linear oscillator; second- order linear differ-\nential equation; Lane-Emden differential equation; Riccati equat ion; variable coefficient.2 ANDERSON, ONITSUKA, AND O’REGAN\nareknown to tackle such problems. Bothof these are very similar, a ndthis study will use the former\ndefinition to proceed with the discussion.\nDefinition 1.1. Equation (1.1) is Ulam stable on I, if there exists a constant K >0 such that for\neveryε >0, and for every twice continuously differentiable function\nξ∈C2(I,Cn)\nsatisfying\nsup\nt∈I/bardblα(t)ξ′′+β(t)ξ′+γ(t)ξ−f(t)/bardbl ≤ε,\nthere exists a solution x∈C2(I,Cn) of (1.1) such that\nsup\nt∈I/bardblξ(t)−x(t)/bardbl ≤Lε.\nWe call such an Lan Ulam constant for (1.1) on I. Here/bardbly/bardblrepresents the norm of y∈Cn.\nSince the concept of Ulam stability was introduced into differential eq uations much research has\nbeen done. There are many results even limited to second-order diff erential equations, and they\nhave been actively studied in recent years. For example, see [19, 22 , 23, 24, 28, 29, 30]. Focusing\non second-order differential equations with constant coefficients , strong results have been obtained.\nIn 2020, Baias and Popa [10] investigated the details of the solutions of the constant coefficients\nequation\nx′′+a1x′+a2x= 0, (1.2)\nanditsperturbedequations, andderived theminimum Ulamconstant , wherea1anda2arecomplex-\nvalued constants, and( X,/bardbl·/bardbl)is a Banach space over the field C, andx∈C2(R,X). They analyzed\nthe Ulam stability based on case division according to the signs of the r eal parts of the roots of\nthe characteristic equations corresponding to (1.2). Finding the m inimum Ulam constant is a very\nimportant problem for applications. If we recall the words mathema tical model and phenomenon\nfrom earlier, the minimum Ulam constant means the minimum error betw een the mathematical\nmodel and phenomenon. Establishing Ulam stability remains an importa nt qualitative property,\nbut finding the minimum Ulam constant quantitatively specifies the minim um error between the\napproximate and true solutions. The minimum Ulam constant is usually c alled the best Ulam\nconstant, and research on the best Ulam constant for differentia l equations, difference equations,\nand functional equations with constant coefficients has progress ed in recent years (see, [2, 3, 4,\n11, 12, 27]). On the other hand, for equations with variable coefficie nts, there are few studies\nestablishing even the Ulam stability, let alone the best Ulam constants . As far as the authors are\naware, the best Ulam constants have been derived only for periodic equations (see, [5, 20, 21]).\nThe purpose of this paper is to derive the minimum (best) Ulam consta nts for the non-autonomous\nequation (1.1), which is not limited to periodic coefficients.BEST ULAM CONSTANTS FOR DAMPED LINEAR OSCILLATORS 3\nIn 2020, C˘ adariu, Popa and Ra¸ sa [15] investigated the Ulam stabilit y of the second-order non-\nautonomous linear differential scalar equation\nx′′+β(t)x′+γ(t)x= 0, (1.3)\nwhereβ∈C1(I,R),γ∈C(I,R) are real-valued scalar functions. They used the existence of\nsolutions to the initial value problem of a particular Riccati equation t o give a result that guarantees\nthe Ulam stability of (1.3). Later, this result was extended to the ca se where (1.1) was limited to\nscalar equations and real-valued coefficients (see, [26]). However, unfortunately, the best Ulam\nconstants were not obtained in the above two results. This study a ttempts to analyze (1.1) using\ntheir idea that the existence of solutions to a Riccati equation is use ful in analyzing the Ulam\nstability of second-order linear differential equations. It should be noted here that the present\nstudy does not extend their results. This work does not require co ntinuous differentiability for the\ncoefficient β. It is enough to assume continuity. In addition, the Riccati equatio n used in this study\nis also different from the one proposed by them, and the method is co mpletely different. Therefore,\nthe statements of the obtained theorems are also different from t heirs. In this study, by proposing\na new method, we succeeded in deriving the best Ulam constants for (1.1).\nThis paper is organized as follows. In Section 2, we show that when we guarantee the existence\nof a solution to a certain Riccati equation, we can use it to describe t he solution to the initial value\nproblem of (1.1). In Section 3, we give the main theorem and its proof . Section 4 gives the result of\nderiving the best Ulam constants, which is the goal of this study. In Section 5, we present various\nnon-trivial examples centering on Lane-Emden differential equatio ns.\n2.Representation of solution\nIn this section, we show that, given the existence of a solution to a c ertain Riccati equation, we\ncan use it to express the general solution of (1.1).\nLemma 2.1. Suppose that α(t)/negationslash= 0for allt∈I, and there exists a solution ρ∈C1(I,C)of the\nRiccati equation\nα(t)(ρ′+ρ2)+β(t)ρ+γ(t) = 0. (2.1)\nThen the solution of (1.1)withx(t0) =x0andx′(t0) =x′\n0is given by\nx(t) =/bracketleftBigg\nx0+/integraldisplayt\nt0/parenleftBigg\nx′\n0−ρ(t0)x0+/integraldisplays\nt0e/integraltextµ\nt0(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)f(µ)dµ/parenrightBigg\ne−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/bracketrightBigg\ne/integraltextt\nt0ρ(s)ds\nfort∈I, wheret0∈I.\nProof.Assume that α(t)/negationslash= 0 for all t∈I. Letρ(t) be a solution of (2.1) on I. Then we see that\nthe function\ny(t) =ce/integraltextt\nt0ρ(s)ds,c∈Cn4 ANDERSON, ONITSUKA, AND O’REGAN\nis a solution to the damped linear oscillator\nα(t)y′′+β(t)y′+γ(t)y=0\nonI. Now we use the reduction of order method. Letting\nx(t) =z(t)e/integraltextt\nt0ρ(s)ds,\nwe have\nx′(t) = (z′(t)+ρ(t)z(t))e/integraltextt\nt0ρ(s)ds,\nand\nx′′(t) =/bracketleftbig\nz′′(t)+2ρ(t)z′(t)+/parenleftbig\nρ′(t)+ρ2(t)/parenrightbig\nz(t)/bracketrightbig\ne/integraltextt\nt0ρ(s)ds.\nSubstituting these into (1.1) and using (2.1), we obtain\nα(t)z′′(t)+(2α(t)ρ(t)+β(t))z′(t) =f(t)e−/integraltextt\nt0ρ(s)ds.\nSinceα(t)/negationslash= 0 for all t∈I, we have\n/parenleftBig\nz′(t)e/integraltextt\nt0(2ρ(s)+β(s)\nα(s))ds/parenrightBig′\n=e/integraltextt\nt0(ρ(s)+β(s)\nα(s))ds\nα(t)f(t).\nThis implies that\nz′(t) =/parenleftBigg\nz′(t0)+/integraldisplayt\nt0e/integraltexts\nt0(ρ(µ)+β(µ)\nα(µ))dµ\nα(s)f(s)ds/parenrightBigg\ne−/integraltextt\nt0(2ρ(s)+β(s)\nα(s))ds,\nand that\nz(t) =z(t0)+/integraldisplayt\nt0/parenleftBigg\nz′(t0)+/integraldisplays\nt0e/integraltextµ\nt0(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)f(µ)dµ/parenrightBigg\ne−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds.\nFrom\nz(t0) =x(t0) =x0,z′(t0) =x′(t0)−ρ(t0)x(t0) =x′\n0−ρ(t0)x0,\nwe obtain\nx(t) =/bracketleftBigg\nx0+/integraldisplayt\nt0/parenleftBigg\nx′\n0−ρ(t0)x0+/integraldisplays\nt0e/integraltextµ\nt0(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)f(µ)dµ/parenrightBigg\ne−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/bracketrightBigg\ne/integraltextt\nt0ρ(s)ds\nfort∈I. This completes the proof. /square\nRemark 2.2. If we assume α(t)/negationslash= 0 almost everywhere t∈Iand with all the obvious functions in\nthe integrands in the expression two lines after (2.1) in L1, then we can get a solution of (1.1) in\nW1,1(I,Cn).BEST ULAM CONSTANTS FOR DAMPED LINEAR OSCILLATORS 5\n3.Ulam stability\nThe first theorem of this paper is as follows.\nTheorem 3.1. LetIbe either (τ,σ),(τ,σ],[τ,σ)or[τ,σ], where−∞ ≤τ < σ≤ ∞. Suppose that\nα(t)/negationslash= 0for allt∈I, and there exists a solution ρ:I→Cof(2.1). Letℜ(z)be the real part of\nz∈C. Then the following (i), (ii) and (iii) below hold:\n(i)if the functions\nf1(t) :=/integraldisplayσ\nte/integraltexts\ntℜ(ρ(µ)+β(µ)\nα(µ))dµ\n|α(s)|ds (3.1)\nand\nf2(t) :=/integraldisplayσ\nte−/integraltexts\ntℜ(ρ(µ))dµds (3.2)\nexist for all t∈I, andsupt∈If1(t)<∞andsupt∈If2(t)<∞hold. Then (1.1)is Ulam\nstable on I, with an Ulam constant\nL1:= sup\nt∈I/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne−/integraltexts\ntℜ(ρ(µ))dµds;\n(ii)if the functions f1(t)and\nf3(t) :=/integraldisplayt\nτe/integraltextt\nsℜ(ρ(µ))dµds (3.3)\nexist for all t∈I, andsupt∈If1(t)<∞andsupt∈If3(t)<∞hold, where f1(t)is given by\n(3.1). Then(1.1)is Ulam stable on I, with an Ulam constant\nL2:= sup\nt∈I/integraldisplayt\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsℜ(ρ(µ))dµds;\n(iii)if the functions f3(t)and\nf4(t) :=/integraldisplayt\nτe−/integraltextt\nsℜ(ρ(µ)+β(µ)\nα(µ))dµ\n|α(s)|ds (3.4)\nexist for all t∈I, andsupt∈If3(t)<∞andsupt∈If4(t)<∞hold, where f3(t)is given by\n(3.3). Then(1.1)is Ulam stable on I, with an Ulam constant\nL3:= sup\nt∈I/integraldisplayt\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsℜ(ρ(µ))dµds.\nProof.Assume that α(t)/negationslash= 0 for all t∈I. Assume also that there exists a solution ρ:I→Cof\n(2.1). Let ε >0 be given, and let the twice continuously differentiable function ξ:I→Cnsatisfy\nsup\nt∈I/bardblα(t)ξ′′+β(t)ξ′+γ(t)ξ−f(t)/bardbl ≤ε.6 ANDERSON, ONITSUKA, AND O’REGAN\nDefine\ng(t) :=α(t)ξ′′+β(t)ξ′+γ(t)ξ−f(t)\nfort∈I. Then we have supt∈I/bardblg(t)/bardbl ≤ε. Letp(t) be a solution to (1.1) on I, and let q(t) :=\nξ(t)−p(t) fort∈I. Thenq(t) is a solution to the equation\nα(t)q′′+β(t)q′+γ(t)q=g(t)\nfort∈I. Therefore, by Lemma 2.1, we see that the function q(t) is expressed as\nq(t) =/bracketleftBigg\nq0+/integraldisplayt\nt0/parenleftBigg\nq′\n0−ρ(t0)q0+/integraldisplays\nt0e/integraltextµ\nt0(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/bracketrightBigg\ne/integraltextt\nt0ρ(s)ds(3.5)\nfort∈I, wheret0∈I,q0=q(t0) =ξ(t0)−p(t0) andq′\n0=q′(t0) =ξ′(t0)−p′(t0). Hereafter, the\nproofs are given for each of the three cases (i)–(iii).\nCase (i). Assume that f1(t) andf2(t) given by (3.1) and (3.2) exist on I, and supt∈If1(t)<∞\nand supt∈If2(t)<∞are satisfied. Now we define\nc1:=q′\n0−ρ(t0)q0+/integraldisplayσ\nt0e/integraltextµ\nt0(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ. (3.6)\nNote that the integral contained in the right-hand side always conv erge. Actually, by using\nsupt∈If1(t)<∞, we can check that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayσ\nt0e/integraltextµ\nt0(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤/integraldisplayσ\nt0e/integraltextµ\nt0ℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|/bardblg(µ)/bardbldµ≤εf1(t0)<∞.\nThat is,c1is a well-defined constant vector. Therefore, (3.5) can be rewritt en as\nq(t) =/bracketleftBigg\nq0+/integraldisplayt\nt0/parenleftBigg\nc1−/integraldisplayσ\nse/integraltextµ\nt0(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/bracketrightBigg\ne/integraltextt\nt0ρ(s)ds\n=/bracketleftBigg\nq0+c1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds\n−/integraldisplayt\nt0/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne−/integraltexts\nt0ρ(µ)dµds/bracketrightBigg\ne/integraltextt\nt0ρ(s)ds(3.7)\nfort∈I. Moreover, we define\nc2:=q0−/integraldisplayσ\nt0/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne−/integraltexts\nt0ρ(µ)dµds.BEST ULAM CONSTANTS FOR DAMPED LINEAR OSCILLATORS 7\nSince\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayσ\nt0/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne−/integraltexts\nt0ρ(µ)dµds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n≤/integraldisplayσ\nt0/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublee−/integraltexts\nt0ℜ(ρ(µ))dµds\n≤/integraldisplayσ\nt0/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|/bardblg(µ)/bardbldµ/parenrightBigg\ne−/integraltexts\nt0ℜ(ρ(µ))dµds\n≤ε/integraldisplayσ\nt0/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne−/integraltexts\nt0ℜ(ρ(µ))dµds\n≤ε/parenleftbigg\nsup\nt∈If1(t)/parenrightbigg/integraldisplayσ\nt0e−/integraltexts\nt0ℜ(ρ(µ))dµds≤ε/parenleftbigg\nsup\nt∈If1(t)/parenrightbigg\nf2(t0)<∞\nholds, we can conclude that c2is well-defined. Hence, (3.7) is rewritten as\nq(t) =/bracketleftBigg\nc2+c1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds+/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne−/integraltexts\nt0ρ(µ)dµds/bracketrightBigg\ne/integraltextt\nt0ρ(s)ds\n=/parenleftbigg\nc2+c1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/parenrightbigg\ne/integraltextt\nt0ρ(s)ds+/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne−/integraltexts\ntρ(µ)dµds\nfort∈I.\nNext we consider the function\nw(t) :=/parenleftbigg\nc2+c1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/parenrightbigg\ne/integraltextt\nt0ρ(s)ds\nfort∈I. Note that this function is for g(t)≡0inq(t) above. So it is a solution of the differential\nequation α(t)w′′+β(t)w′+γ(t)w=0. Hence we see that the function\nx1(t) :=w(t)+p(t)\nis a solution of (1.1) for t∈I, wherep(t) is a solution of (1.1) given at the beginning of the proof.\nThis says that\nξ(t)−x1(t) =q(t)−w(t) =/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne−/integraltexts\ntρ(µ)dµds,\nand so that\n/bardblξ(t)−x1(t)/bardbl ≤ε/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne−/integraltexts\ntℜ(ρ(µ))dµds≤ε/parenleftbigg\nsup\nt∈If1(t)/parenrightbigg/parenleftbigg\nsup\nt∈If2(t)/parenrightbigg\n<∞8 ANDERSON, ONITSUKA, AND O’REGAN\nfort∈I. Thus, (1.1) is Ulam stable on I. Moreover,\nL1= sup\nt∈I/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne−/integraltexts\ntℜ(ρ(µ))dµds\nis an Ulam constant for (1.1).\nCase (ii). Assume that f1(t) andf3(t) given by (3.1) and (3.3) exist on I, and supt∈If1(t)<∞\nand supt∈If3(t)<∞hold. As in Case (i), we can rewrite q(t) as (3.7), using the constant c1\ndefined in (3.6). Now we define\nc3:=q0+/integraldisplayt0\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne/integraltextt0\nsρ(µ)dµds.\nSince\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt0\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne/integraltextt0\nsρ(µ)dµds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n≤/integraldisplayt0\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|/bardblg(µ)/bardbldµ/parenrightBigg\ne/integraltextt0\nsℜ(ρ(µ))dµds\n≤ε/parenleftbigg\nsup\nt∈If1(t)/parenrightbigg/integraldisplayt0\nτe/integraltextt0\nsℜ(ρ(µ))dµds≤ε/parenleftbigg\nsup\nt∈If1(t)/parenrightbigg\nf3(t0)<∞\nholds,c3is a well-defined constant. Thus, (3.7) is rewritten as\nq(t) =/bracketleftBigg\nc3+c1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds−/integraldisplayt\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne/integraltextt0\nsρ(µ)dµds/bracketrightBigg\ne/integraltextt\nt0ρ(s)ds\n=/parenleftbigg\nc3+c1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/parenrightbigg\ne/integraltextt\nt0ρ(s)ds−/integraldisplayt\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne/integraltextt\nsρ(µ)dµds\nfort∈I.\nNext we consider the solution of (1.1) given by\nx2(t) :=/parenleftbigg\nc3+c1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/parenrightbigg\ne/integraltextt\nt0ρ(s)ds+p(t)\nfort∈I. Then\nx2(t)−ξ(t) =/integraldisplayt\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne/integraltextt\nsρ(µ)dµds,\nand so that\n/bardblx2(t)−ξ(t)/bardbl ≤ε/integraldisplayt\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsℜ(ρ(µ))dµds≤ε/parenleftbigg\nsup\nt∈If1(t)/parenrightbigg/parenleftbigg\nsup\nt∈If3(t)/parenrightbigg\n<∞BEST ULAM CONSTANTS FOR DAMPED LINEAR OSCILLATORS 9\nfort∈I. Thus, (1.1) is Ulam stable on I. Moreover,\nL2= sup\nt∈I/integraldisplayt\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsℜ(ρ(µ))dµds\nis an Ulam constant for (1.1).\nCase (iii). Assume that f3(t) andf4(t) given by (3.3) and (3.4) exist on I, and supt∈If3(t)<∞\nand supt∈If4(t)<∞are satisfied. Now we define\nc4:=q′\n0−ρ(t0)q0−/integraldisplayt0\nτe−/integraltextt0\nµ(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ.\nSince supt∈If4(t)<∞holds, we see that\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt0\nτe−/integraltextt0\nµ(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤/integraldisplayt0\nτe−/integraltextt0\nµℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|/bardblg(µ)/bardbldµ≤εf4(t0)<∞,\nand that c4is a well-defined constant vector. Therefore, (3.5) can be rewritt en as\nq(t) =/bracketleftBigg\nq0+c4/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds+/integraldisplayt\nt0/parenleftBigg/integraldisplays\nτe−/integraltexts\nµ(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne−/integraltexts\nt0ρ(µ)dµds/bracketrightBigg\ne/integraltextt\nt0ρ(s)ds\nfort∈I. Moreover, we define\nc5:=q0−/integraldisplayt0\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµ(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne/integraltextt0\nsρ(µ)dµds.\nSince\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt0\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµ(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne/integraltextt0\nsρ(µ)dµds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\n≤/integraldisplayt0\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|/bardblg(µ)/bardbldµ/parenrightBigg\ne/integraltextt0\nsℜ(ρ(µ))dµds\n≤ε/parenleftbigg\nsup\nt∈If4(t)/parenrightbigg/integraldisplayt0\nτe/integraltextt0\nsℜ(ρ(µ))dµds≤ε/parenleftbigg\nsup\nt∈If4(t)/parenrightbigg\nf3(t0)<∞\nholds, we see that c5is well-defined. Hence, q(t) is rewritten as\nq(t) =/parenleftbigg\nc5+c4/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/parenrightbigg\ne/integraltextt\nt0ρ(s)ds+/integraldisplayt\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµ(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne/integraltextt\nsρ(µ)dµds\nfort∈I.\nNext we consider the solution of (1.1) given by\nx3(t) :=/parenleftbigg\nc5+c4/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/parenrightbigg\ne/integraltextt\nt0ρ(s)ds+p(t)10 ANDERSON, ONITSUKA, AND O’REGAN\nfort∈I. Then\nξ(t)−x3(t) =/integraldisplayt\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµ(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)g(µ)dµ/parenrightBigg\ne/integraltextt\nsρ(µ)dµds,\nand so that\n/bardblξ(t)−x3(t)/bardbl ≤ε/integraldisplayt\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsℜ(ρ(µ))dµds≤ε/parenleftbigg\nsup\nt∈If3(t)/parenrightbigg/parenleftbigg\nsup\nt∈If4(t)/parenrightbigg\n<∞\nfort∈I. Thus, (1.1) is Ulam stable on I. Moreover,\nL3= sup\nt∈I/integraldisplayt\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsℜ(ρ(µ))dµds\nis an Ulam constant for (1.1). The proof is now complete. /square\nConsider the constant coefficients linear oscillator\na0x′′+a1x′+a2x= 0, (3.8)\nwherea0,a1anda2are complex-valued constants, and a0/negationslash= 0. Then we obtain the following result.\nCorollary 3.2. LetI=R, andλ1andλ2be the roots of the characteristic equation\na0λ2+a1λ+a2= 0.\nIfa0ℜ(λ1)ℜ(λ2)/negationslash= 0, then(3.8)is Ulam stable on R, and an Ulam constant is1\n|a0ℜ(λ1)ℜ(λ2)|.\nProof.Assumea0/negationslash= 0. The proof is divided into three cases (i) ℜ(λ1)≥ ℜ(λ2)>0, (ii)ℜ(λ1)>\n0>ℜ(λ2), and (iii) 0 >ℜ(λ1)≥ ℜ(λ2). First, we notice that, since λ1andλ2are the roots of\nthe characteristic equation, they are constant solutions of (2.1) ; that is, we can choose ρ(t) =λ1or\nρ(t) =λ2for allt∈R. Moreover, if\nρ(t) =λ2=−a1+/radicalbig\na2\n1−4a0a2\n2a0,\nthen\nρ(t)+β(t)\nα(t)=λ2+a1\na0=−λ1.\nWe will use Theorem 3.1 with τ=−∞andσ=∞.\nCase (i). Suppose ℜ(λ1)≥ ℜ(λ2)>0. In this case, Theorem 3.1 (i) will be used. Set ρ(t) =λ2.\nFrom\nf1(t) =/integraldisplayσ\nte/integraltexts\ntℜ(ρ(µ)+β(µ)\nα(µ))dµ\n|α(s)|ds=/integraldisplay∞\nte−/integraltexts\ntℜ(λ1)dµ\n|a0|ds=1\n|a0|ℜ(λ1)\nand\nf2(t) =/integraldisplayσ\nte−/integraltexts\ntℜ(ρ(µ))dµds=/integraldisplay∞\nte−/integraltexts\ntℜ(λ2)dµds=1\nℜ(λ2)BEST ULAM CONSTANTS FOR DAMPED LINEAR OSCILLATORS 11\nfor allt∈R,f1(t) andf2(t) exist and are bounded on R. Hence, by Theorem 3.1 (i), (3.8) is Ulam\nstable on R, and an Ulam constant is\nsup\nt∈I/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne−/integraltexts\ntℜ(ρ(µ))dµds= sup\nt∈If1(t)f2(t) =1\n|a0|ℜ(λ1)ℜ(λ2).\nCase (ii). Suppose ℜ(λ1)>0>ℜ(λ2). Setρ(t) =λ2. Fromf1(t) =1\n|a0|ℜ(λ1)and\nf3(t) =/integraldisplayt\nτe/integraltextt\nsℜ(ρ(µ))dµds=/integraldisplayt\n−∞e/integraltextt\nsℜ(λ2)dµds=1\n−ℜ(λ2)\nfor allt∈R,f1(t) andf3(t) exist and are bounded on R. Hence, by Theorem 3.1 (ii), (3.8) is Ulam\nstable on R, and an Ulam constant is\nsup\nt∈I/integraldisplayt\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\nsℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsℜ(ρ(µ))dµds= sup\nt∈If1(t)f3(t) =1\n|a0ℜ(λ1)ℜ(λ2)|.\nCase (iii). Suppose 0 >ℜ(λ1)≥ ℜ(λ2). Setρ(t) =λ2. Fromf3(t) =1\n−ℜ(λ2)and\nf4(t) =/integraldisplayt\nτe−/integraltextt\nsℜ(ρ(µ)+β(µ)\nα(µ))dµ\n|α(s)|ds=/integraldisplayt\n−∞e/integraltextt\nsℜ(λ1)dµ\n|a0|ds=1\n−|a0|ℜ(λ1)\nfor allt∈R,f3(t) andf4(t) exist and are bounded on R. Hence, by Theorem 3.1 (iii), (3.8) is Ulam\nstable on R, and an Ulam constant is\nsup\nt∈I/integraldisplayt\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµℜ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsℜ(ρ(µ))dµds= sup\nt∈If3(t)f4(t) =1\n|a0ℜ(λ1)ℜ(λ2)|.\nTherefore, in any case, an Ulam constant is1\n|a0ℜ(λ1)ℜ(λ2)|. /square\n4.Minimum Ulam constants\nIn this section, we show that the Ulam constants given in Theorem 3.1 are the minimum Ulam\nconstants by restricting to real-valued scalar functions.\nTheorem 4.1. LetIbe either (τ,σ),(τ,σ],[τ,σ)or[τ,σ], where−∞ ≤τ < σ≤ ∞. Suppose that\nα,β,γ:I→Rare real-valued continuous functions, and α(t)/negationslash= 0for allt∈I, and there exists a\nreal-valued solution ρ:I→Rof(2.1). Then the following (i), (ii) and (iii) below hold:\n(i)suppose that f1(t)andf2(t)given by (3.1)and(3.2)exist for all t∈I, andsupt∈If1(t)<∞\nandsupt∈If2(t)<∞hold. If\nlim\nt→σ−/integraldisplayt\nt0ρ(s)ds=∞andlim\nt→σ−e/integraltextt\nt0ρ(s)ds/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds=∞, t0∈(τ,σ),(4.1)12 ANDERSON, ONITSUKA, AND O’REGAN\nthen(1.1)is Ulam stable on I, and the minimum Ulam constant is\nB1:= sup\nt∈I/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne−/integraltexts\ntρ(µ)dµds; (4.2)\n(ii)suppose that f1(t)andf3(t)given by (3.1)and(3.3)exist for all t∈I, andsupt∈If1(t)<∞\nandsupt∈If3(t)<∞hold. If\nlim\nt→τ+/integraldisplayt\nt0ρ(s)ds=∞andlim\nt→σ−e/integraltextt\nt0ρ(s)ds/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds=∞, t0∈(τ,σ),(4.3)\nthen(1.1)is Ulam stable on I, and the minimum Ulam constant is\nB2:= sup\nt∈I/integraldisplayt\nτ/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsρ(µ)dµds;\n(iii)suppose that f3(t)andf4(t)given by (3.3)and(3.4)exist for all t∈I, andsupt∈If3(t)<∞\nandsupt∈If4(t)<∞hold. If\nlim\nt→τ+/integraldisplayt\nt0ρ(s)ds=∞andlim\nt→τ+e/integraltextt\nt0ρ(s)ds/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds=−∞, t0∈(τ,σ),(4.4)\nthen(1.1)is Ulam stable on I, and the minimum Ulam constant is\nB3:= sup\nt∈I/integraldisplayt\nτ/parenleftBigg/integraldisplays\nτe−/integraltexts\nµ(ρ(ν)+β(ν)\nα(ν))dν\n|α(µ)|dµ/parenrightBigg\ne/integraltextt\nsρ(µ)dµds. (4.5)\nProof.Assume that α,βandγare real-valued continuous functions, and α(t)/negationslash= 0 for all t∈I. Let\nρbe a real-valued solution of (2.1). Throughout this proof, let f1,f2,f3andf4be the functions\ndefined by (3.1)–(3.4), respectively. Note that f1,f2,f3andf4are real-valued functions on I. Let\nt0∈(τ,σ).\nCase(i). Assumethat f1(t)andf2(t)existforall t∈I, andsupt∈If1(t)<∞andsupt∈If2(t)<∞\nhold. By Theorem 3.1, we see that (1.1) is Ulam stable on I, with an Ulam constant B1, whereB1\nis defined by (4.2). Let ε >0. Now we consider the function\nq(t) :=/parenleftbigg\nc2+c1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/parenrightbigg\ne/integraltextt\nt0ρ(s)ds+ε/bracketleftBigg/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)dµ/parenrightBigg\ne−/integraltexts\ntρ(µ)dµds/bracketrightBigg\nu\nfor allt∈I, wherec1andc2are well-defined constants given in the proof of Theorem 3.1 (i) and u\nis the unit vector. From the proof of Theorem 3.1 (i), we find that q(t) is a solution of the equation\nα(t)q′′+β(t)q′+γ(t)q=εu.\nLetp(t) be a solution to (1.1) on I, and let ξ(t) :=q(t)+p(t) fort∈I. Then\n/bardblα(t)ξ′′+β(t)ξ′+γ(t)ξ−f(t)/bardbl=ε (4.6)BEST ULAM CONSTANTS FOR DAMPED LINEAR OSCILLATORS 13\nis satisfied for t∈I. By the Ulam stability for (1.1), we find that there exists a solution x1:I→Rn\nof (1.1) such that\nsup\nt∈I/bardblξ(t)−x1(t)/bardbl ≤B1ε. (4.7)\nMore precisely, from the proof of Theorem 3.1 (i), we know that x1(t) is given as\nx1(t) =/parenleftbigg\nc2+c1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/parenrightbigg\ne/integraltextt\nt0ρ(s)ds+p(t).\nWe show that B1is the minimum Ulam constant by using the following two steps.\nStep 1. We first show that x1(t) is the unique solution of (1.1) satisfying (4.7). To show this fact\nusing contradiction, we assume that there exists a solution y1(t) of (1.1) such that y1(t)/negationslash=x1(t)\nfor allt∈I. That is, y1(t) is written as\ny1(t) =/parenleftbigg\nd2+d1/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/parenrightbigg\ne/integraltextt\nt0ρ(s)ds+p(t)\nwith (d1,d2)/negationslash= (c1,c2). Thus, we have\n/bardbly1(t)−x1(t)/bardbl ≤ /bardblξ(t)−y1(t)/bardbl+/bardblξ(t)−x1(t)/bardbl ≤2B1ε\nfor allt∈I. However, with (4.1), the following holds:\nlim\nt→σ−/bardbly1(t)−x1(t)/bardbl= lim\nt→σ−/vextenddouble/vextenddouble/vextenddouble/vextenddouble/bracketleftbigg\n(d2−c2)+(d1−c1)/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds/bracketrightbigg\ne/integraltextt\nt0ρ(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble=∞.\nThis contradicts the above inequality.\nStep 2. We next show that B1is the minimum Ulam constant. By way of contradiction, we\nassume that there exists 0 < U1< B1such that\nsup\nt∈I/bardblξ(t)−x1(t)/bardbl ≤U1ε.\nNote that ξ(t) satisfies (4.6), and that there is no other possible solution of (1.1) that satisfies this\ninequality other than x1(t) by Step 1. However, we see that\nsup\nt∈I/bardblξ(t)−x1(t)/bardbl= sup\nt∈I/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleε/bracketleftBigg/integraldisplayσ\nt/parenleftBigg/integraldisplayσ\nse/integraltextµ\ns(ρ(ν)+β(ν)\nα(ν))dν\nα(µ)dµ/parenrightBigg\ne−/integraltexts\ntρ(µ)dµds/bracketrightBigg\nu/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble=B1ε,\nand thus,\nsup\nt∈I/bardblξ(t)−x1(t)/bardbl ≤U1ε < B 1ε= sup\nt∈I/bardblξ(t)−x1(t)/bardbl.\nThis is a contradiction. Hence we can conclude that B1is the minimum Ulam constant.\nCases (ii) and (iii) can be shown by the same technique as in Case (i). Th e proof is now complete.\n/square\nConsider the linear oscillator (3.8) again, where a0,a1anda2are real-valued constants, and\na0/negationslash= 0. Theorem 4.1 implies the following result.14 ANDERSON, ONITSUKA, AND O’REGAN\nCorollary 4.2. LetI=R, andλ1andλ2be non-zero real roots of the characteristic equation\na0λ2+a1λ+a2= 0.\nIfa0/negationslash= 0, then(3.8)is Ulam stable on R, and the minimum Ulam constant is1\n|a0λ1λ2|.\nProof.The proof is divided into three cases (i) λ1≥λ2>0, (ii)λ1>0> λ2, and (iii) 0 > λ1≥λ2.\nRecall the proof of Corollary 3.2. Since λ1andλ2are the roots of the characteristic equation, they\nare constant solutions of (2.1), and if ρ(t) =λ2, thenρ(t)+β(t)\nα(t)=−λ1. We will use Theorem 4.1\nwithτ=−∞andσ=∞. Lett0∈(−∞,∞).\nCase (i). Suppose λ1≥λ2>0. Setρ(t) =λ2. From the facts obtained in Case (i) in the proof of\nCorollary 3.2, we have f1(t) =1\n|a0|λ1andf2(t) =1\nλ2, and thus, f1(t) andf2(t) exist and are bounded\nonR. Moreover,\nlim\nt→σ−/integraldisplayt\nt0ρ(s)ds= lim\nt→∞/integraldisplayt\nt0λ2ds=∞\nand\nlim\nt→σ−e/integraltextt\nt0ρ(s)ds/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds= lim\nt→∞e/integraltextt\nt0λ2ds/integraldisplayt\nt0e/integraltexts\nt0(λ1−λ2)dµds=∞\nare satisfied; that is, (4.1) holds. Hence, by Theorem 4.1 (i), (3.8) is Ulam stable on I, and the\nminimum Ulam constant is1\n|a0|λ1λ2.\nCase (ii). Suppose λ1>0> λ2. Setρ(t) =λ2. From the facts obtained in Case (ii) in the proof\nof Corollary 3.2, we have f1(t) =1\n|a0|λ1andf3(t) =1\n−λ2, and thus, f1(t) andf3(t) exist and are\nbounded on R. Moreover,\nlim\nt→τ+/integraldisplayt\nt0ρ(s)ds= lim\nt→−∞/integraldisplayt\nt0λ2ds=∞\nand\nlim\nt→σ−e/integraltextt\nt0ρ(s)ds/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds= lim\nt→∞e/integraltextt\nt0λ2ds/integraldisplayt\nt0e/integraltexts\nt0(λ1−λ2)dµds=∞\nare satisfied; that is, (4.3) holds. Hence, by Theorem 4.1 (ii), (3.8) is Ulam stable on I, and the\nminimum Ulam constant is1\n|a0λ1λ2|.\nCase (iii). Suppose 0 > λ1≥λ2. Setρ(t) =λ2. From the facts obtained in Case (iii) in the proof\nof Corollary 3.2, we have f3(t) =1\n−λ2andf4(t) =1\n−|a0|λ1, and thus, f3(t) andf4(t) exist and are\nbounded on R. Moreover, lim t→τ+/integraltextt\nt0ρ(s)ds=∞and\nlim\nt→τ+e/integraltextt\nt0ρ(s)ds/integraldisplayt\nt0e−/integraltexts\nt0(2ρ(µ)+β(µ)\nα(µ))dµds= lim\nt→−∞e/integraltextt\nt0λ2ds/integraldisplayt\nt0e/integraltexts\nt0(λ1−λ2)dµds=−∞\nare satisfied; that is, (4.4) holds. Hence, by Theorem 4.1 (iii), (3.8) is Ulam stable on I, and the\nminimum Ulam constant is1\n|a0λ1λ2|. /square\nRemark 4.3. Whena0= 0, Corollary 4.2 is completely consistent with the results given in [10, 2 5].BEST ULAM CONSTANTS FOR DAMPED LINEAR OSCILLATORS 15\n5.Examples\nWe now present some examples that utilize the main results of this wor k. In a few examples, we\napply the previous theorems directly to guarantee Ulam stability of t he given equation, and to find\nthe minimal Ulam constant. In other examples, we show how the crite ria of the theorems do not\nhold, in the cases of Ulam instability.\nExample 5.1. For dimension n= 1, consider (1.1) in the form of a homogeneous singular differ-\nential equation given by\nt(1−t)x′′(t)+(2−t)x′(t)+x(t) = 0, t∈I= (0,1), (5.1)\nwhereα(t) =t(1−t),β(t) = (2−t), andγ(t) = 1 are continuous scalar functions with α(t)/negationslash= 0 for\nallt∈I= (0,1). The associated Riccati equation (2.1) for (5.1) is\nt(1−t)/parenleftbig\nρ′+ρ2/parenrightbig\n+(2−t)ρ+1 = 0,\nwhich has as a solution the function\nρ(t) =−1\nt.\nWe then find that the general solution for (5.1) with x/parenleftbig1\n2/parenrightbig\n=x0andx′/parenleftbig1\n2/parenrightbig\n=x′\n0is\nx(t) = 2x0+x′\n0−2x0+3x′\n0\n8t−1\n2(2x0+x′\n0)t, t∈I= (0,1),\nfor arbitrary constants x0,x′\n0∈R. Using (3.3) and (3.4), we calculate that both\nf3(t) =/integraldisplayt\n0e/integraltextt\ns(−1\nµ)dµds=t\n2\nand\nf4(t) =/integraldisplayt\n0e−/integraltextt\ns(−1\nµ+2−µ\nµ(1−µ))dµ\n|s(1−s)|ds=/integraldisplayt\n0/parenleftbigg1\ns(1−s)/parenrightbigg/parenleftbiggs\n1−s/parenrightbigg/parenleftbigg1−t\nt/parenrightbigg\nds= 1\nare bounded on I= (0,1). Moreover, we have\nlim\nt→0+/integraldisplayt\nt0/parenleftbigg\n−1\ns/parenrightbigg\nds= lim\nt→0+lnt0\nt=∞\nand\nlim\nt→0+e/integraltextt\nt0(−1\ns)ds/integraldisplayt\nt0e−/integraltexts\nt0(−2\nµ+2−µ\nµ(1−µ))dµds= lim\nt→0+t0\nt/integraldisplayt\nt01−s\n1−t0ds=−∞,16 ANDERSON, ONITSUKA, AND O’REGAN\nwheret0∈I. Thus, (4.4) holds, so that Theorems 3.1 (iii) and 4.1 (iii) apply. It follow s that (5.1)\nis Ulam stable on I= (0,1) in this case, with minimum Ulam constant\nB3:= sup\nt∈I/integraldisplayt\n0/parenleftBigg/integraldisplays\n0e−/integraltexts\nµ(−1\nν+2−ν\nν(1−ν))dν\n|µ(1−µ)|dµ/parenrightBigg\ne/integraltextt\ns(−1\nµ)dµds\n= sup\nt∈I/integraldisplayt\n01e/integraltextt\ns(−1\nµ)dµds= sup\nt∈It\n2=1\n2\nby (4.5). This best constant from Theorem 4.1 (iii) can be verified dire ctly. Given ε >0, let\nx(t) =ε/parenleftbig\n1−t\n2/parenrightbig\nandξ(t)≡ε. Thenxis a solution of (5.1) and ξsatisfies\nt(1−t)ξ′′(t)+(2−t)ξ′(t)+ξ(t) =ε, t∈I= (0,1),\nwith\nsup\nt∈(0,1)|ξ(t)−x(t)|= sup\nt∈(0,1)tε\n2=1\n2ε.\nIn summary, (5.1) is Ulam stable with minimum Ulam stability constant B3=1\n2. As can be seen\nby looking at the general solution, our theorems have the strengt h to apply to solutions that blow\nup att= 0.\nThe following four examples all deal with the Lane-Emden differential equation, either of index\n0 or index 1. See also the recent paper [18].\nExample 5.2. Consider the Lane-Emden differential equation (1.1) given by\nx′′(t)+2\ntx′(t)+1 = 0 , t∈I= (1,∞), (5.2)\nwhereα(t) = 1,β(t) =2\nt,γ(t) = 0, and f(t)≡ −1 are continuous scalar functions with α(t)/negationslash= 0\nfor allt∈I= (1,∞). The associated Riccati equation (2.1) is\n1/parenleftbig\nρ′+ρ2/parenrightbig\n+2\ntρ= 0,\nwhich has as a solution the function\nρ(t) =−1\nt.\nWe then find that the general solution for (5.2) with x(1) =x0andx′(1) =x′\n0is\nx(t) =−2+3t−t3+6tx0−6x′\n0+6tx′\n0\n6t, t∈(1,∞),\nfor arbitrary constants x0,x′\n0∈R. Using (3.1) and (3.3), we calculate that\nf1(t) =/integraldisplay∞\nte/integraltexts\ntℜ(−1\nµ+2\nµ)dµ\n|1|ds=/integraldisplay∞\nts\ntds=∞\nand\nf3(t) =/integraldisplayt\n1e/integraltextt\nsℜ(−1\nµ)dµds=t\n2−1\n2tBEST ULAM CONSTANTS FOR DAMPED LINEAR OSCILLATORS 17\nis unbounded on I= (1,∞), so that Theorem 3.1 does not apply. Indeed, given an arbitrary ε >0,\nξ′′(t)+2\ntξ′(t)+1 =ε, t∈I= (1,∞)\nhas a solution ξ(t) =(ε−1)t2\n6, and thus\nsup\nt∈I|ξ(t)−x(t)|= sup\nt∈I/vextendsingle/vextendsingle/vextendsingle/vextendsingle2+6x′\n0−3t(1+2x0+2x′\n0)+t3ε\n6t/vextendsingle/vextendsingle/vextendsingle/vextendsingle=∞\nfor any choice of x0,x′\n0∈R, so that in fact (5.2) is not Ulam stable on (1 ,∞).\nExample 5.3. In this example we modify the interval Ifor the Lane-Emden equation (5.2) by\nconsidering\nx′′(t)+2\ntx′(t)+1 = 0 , t∈I= (0,σ),0< σ <∞. (5.3)\nIt is easy to verify that f3in (3.3) and f4in (3.4) have expression\nf3(t) =t\n2=f4(t), t∈(0,σ),\nasρ(t) =−1\nt,α(t) = 1, and β(t) =−2ρ(t). Moreover, we have\nlim\nt→0+/integraldisplayt\nt0/parenleftbigg\n−1\ns/parenrightbigg\nds= lim\nt→0+lnt0\nt=∞\nand\nlim\nt→0+e/integraltextt\nt0(−1\ns)ds/integraldisplayt\nt0e−/integraltexts\nt0(−2\nµ+2\nµ)dµds= lim\nt→0+t0\nt(t−t0) =−∞,\nwheret0∈I. Then (4.4) holds, so that (5.3) is Ulam stable on I= (0,σ) for any σ∈(0,∞), with\nminimum Ulam constant\nB3:= sup\nt∈I/integraldisplayt\n0/parenleftbigg/integraldisplays\n0e−/integraltexts\nµ(1\nν)dνdµ/parenrightbigg\ne/integraltextt\ns(−1\nµ)dµds\n= sup\nt∈(0,σ)/integraldisplayt\n0/parenleftBigs\n2/parenrightBig/parenleftBigs\nt/parenrightBig\nds= sup\nt∈It2\n6=σ2\n6\nby (4.5), after employing Theorems 3.1 (iii) and 4.1 (iii). In summary, (5 .3) is Ulam stable on (0 ,σ)\nfor anyσ∈(0,∞), with minimum Ulam stability constant B3=σ2\n6.\nExample 5.4. Consider (1.1) in the form of the Lane-Emden differential equation g iven by\nx′′(t)+2\ntx′(t)+x(t) = 0, t∈I= (0,σ), σ∈/parenleftBig\n0,π\n2/parenrightBig\n, (5.4)\nwhereα(t) = 1,β(t) =2\nt,γ(t) = 1, and f(t)≡0 are continuous scalar functions with α(t)/negationslash= 0 for\nallt∈I= (0,σ), where σ∈/parenleftbig\n0,π\n2/parenrightbig\n. The associated Riccati equation (2.1) is\n1/parenleftbig\nρ′+ρ2/parenrightbig\n+2\ntρ+1 = 0,18 ANDERSON, ONITSUKA, AND O’REGAN\nwhich has as a solution the function\nρ(t) =−tan(t)−1\nt, t∈(0,σ).\nWe then find that the general solution for (5.4) is\nx(t) =c1cost\nt+c2sint\nt, t∈(0,σ),\nfor arbitrary constants c1,c2∈R. Using (3.3) and (3.4), we calculate that both\nf3(t) =/integraldisplayt\n0e/integraltextt\ns(−tan(µ)−1\nµ)dµds=/integraldisplayt\n0scos(t)\ntcos(s)ds\nand\nf4(t) =/integraldisplayt\n0e−/integraltextt\ns(−tan(µ)+1\nµ)dµds= tan(t)+cos(t)−1\ntcos(t)\nare bounded on I= (0,σ), because\nlim\nt→0+/integraldisplayt\n0scos(t)\ntcos(s)ds= 0\nholds. Moreover, we have\nlim\nt→0+/integraldisplayt\nt0/parenleftbigg\n−tan(s)−1\ns/parenrightbigg\nds= lim\nt→0+lnt0cos(t)\ntcos(t0)=∞\nand\nlim\nt→0+e/integraltextt\nt0(−tan(s)−1\ns)ds/integraldisplayt\nt0e−/integraltexts\nt0[2(−tan(µ)−1\nµ)+2\nµ]dµds=t0cos(t)\ntcos(t0)/integraldisplayt\nt0cos2(t0)\ncos2(s)ds\n=sin(t−t0)\nt=−∞,\nwheret0∈I. Then (4.4) holds, so that Theorems 3.1 (iii) and 4.1 (iii) apply. It follows that (5.4)\nis Ulam stable on I= (0,σ) in this case, with minimum Ulam constant\nB3:= sup\nt∈I/integraldisplayt\n0/parenleftbigg/integraldisplays\n0e−/integraltexts\nµ(−tan(ν)−1\nν+2\nν)dνdµ/parenrightbigg\ne/integraltextt\ns(−tan(µ)−1\nµ)dµds\n= sup\nt∈I/integraldisplayt\n0/parenleftbigg\ntan(s)+cos(s)−1\nscos(s)/parenrightbigg/parenleftbiggscos(t)\ntcos(s)/parenrightbigg\nds\n= sup\nt∈I/parenleftbiggcos(t)\nt/parenrightbigg/parenleftbiggt−sin(t)\ncos(t)/parenrightbigg\n= 1−sin(σ)\nσ\nby (4.5). In summary, (5.4) is Ulam stable on (0 ,σ) for any σ∈/parenleftbig\n0,π\n2/parenrightbig\n, with minimum Ulam\nstability constant B3= 1−sin(σ)\nσ.BEST ULAM CONSTANTS FOR DAMPED LINEAR OSCILLATORS 19\nExample 5.5. In this example we modify the interval Ifor the Lane-Emden equation (5.4) by\nconsidering\nx′′(t)+2\ntx′(t)+x(t) = 0, t∈I= (τ,∞),0≤τ <∞. (5.5)\nOnce again, the corresponding Riccati equation has solution ρ(t) =−tan(t)−1\nt, which does not\nexist infinitely often on I= (τ,∞) due to the tangent function, so Theorems 3.1 and 4.1 cannot be\napplied. Given ε >0, consider\nξ(t) =ε\n8t/parenleftbig\ncos(t)+2tsin(t)−2t2cos(t)/parenrightbig\n, t∈(τ,∞).\nNote that\nsup\nt∈I/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ′′(t)+2\ntξ′(t)+ξ(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle= sup\nt∈I|εsin(t)|=ε.\nSince\nx(t) =c1cost\nt+c2sint\nt, t∈(τ,∞)\nis the general solution for (5.5), we have\nsup\nt∈I|ξ(t)−x(t)|=∞,\nmaking (5.5) unstable in the Ulam sense on I= (τ,∞) for any τ∈[0,∞).\nExample 5.6. In this example, we consider an extension of Example 5.3 to include mor e general\npower functions. We consider the second-order linear differential equation\nt1−ax′′(t)+bt−ax′(t)+(b−2)t−1−ax(t)+tb−2= 0, t∈I= (0,σ),0< σ <∞,(5.6)\nwhereaandbare real-valued constants with\n1−a < b≤2.\nIfa= 1 and b= 2, then this equation reduces to (5.3). It is easy to verify that ρ(t) =−1\ntis a\nsolution of the associated Riccati equation\nt1−a/parenleftbig\nρ′+ρ2/parenrightbig\n+bt−aρ+(b−2)t−1−a= 0,\nandf3in (3.3) and f4in (3.4) have expression\nf3(t) =t\n2, f4(t) =ta\na+b−1, t∈(0,σ),\nasα(t) =t1−a, andβ(t) =bt−a. Moreover, we have\nlim\nt→0+/integraldisplayt\nt0/parenleftbigg\n−1\ns/parenrightbigg\nds= lim\nt→0+lnt0\nt=∞\nand\nlim\nt→0+e/integraltextt\nt0(−1\ns)ds/integraldisplayt\nt0e−/integraltexts\nt0(−2\nµ+b\nµ)dµds= lim\nt→0+t0\nt/integraldisplayt\nt0/parenleftbiggs\nt0/parenrightbigg2−b\nds= lim\nt→0+tb−1\n0\n3−b/parenleftbigg\nt2−b−t3−b\n0\nt/parenrightbigg\n=−∞,20 ANDERSON, ONITSUKA, AND O’REGAN\nwheret0∈I. Then (4.4) holds, so that (5.6) is Ulam stable on I= (0,σ) for any σ∈(0,∞), with\nminimum Ulam constant\nB3:= sup\nt∈I/integraldisplayt\n0/parenleftBigg/integraldisplays\n0e−/integraltexts\nµ(−1\nν+b\nν)dν\n|s1−a|dµ/parenrightBigg\ne/integraltextt\ns(−1\nµ)dµds\n= sup\nt∈(0,σ)/integraldisplayt\n0/parenleftbiggsa\na+b−1/parenrightbigg/parenleftBigs\nt/parenrightBig\nds= sup\nt∈Ita+1\n(a+2)(a+b−1)=σa+1\n(a+2)(a+b−1)\nby (4.5), after employing Theorems 3.1 (iii) and 4.1 (iii). In summary, (5 .6) is Ulam stable on (0 ,σ)\nfor anyσ∈(0,∞), with minimum Ulam stability constant B3=σa+1\n(a+2)(a+b−1).\n6.Conclusions\nThis work investigated the Ulam stability of second-order linear differ ential vector equations with\nvariable coefficients. Sufficient conditions for Ulam stability and explicit Ulam stability constants\nare given. In particular, if restricted to real-valued coefficients, t he best Ulam constants are derived.\nTo the best of the authors’ knowledge, no best Ulam constants ar e known so far for second-order\nnon-autonomous equations other than periodic systems. Theref ore, this is the first study to derive\nbest Ulam constants for second-order non-periodic non-autono mous linear differential equations.\nVarious non-trivial examples, mainly Lane-Emden differential equat ions, were provided to illustrate\nthe results obtained. Some examples show that Ulam stability can be g uaranteed even for solutions\nthat blow up in finite time, andthat best Ulamconstants can bederive d. Note that it is emphasized\nhere that it is also the first time that a Ulam stability analysis of blow–up solutions of the second-\norder equation has been presented. In addition, examples of insta bility are also presented.\nAcknowledgments\nM. O. was supportedby theJapanSociety for thePromotionofScie nce (JSPS) KAKENHI(grant\nnumber JP20K03668).\nReferences\n[1] R. P. Agarwal, B. Xu and W. Zhang, Stability of functional equatio ns in single variable, J. Math. Anal. Appl.\n288 (2003), no. 2, 852–869.\n[2] D. R. Anderson and M. Onitsuka, Best constant for Hyers-Ulam stability of second-order h-difference equations\nwith constant coefficients, Results Math. 74 (2019), no. 4, Paper No. 151, 16 pp.\n[3] D. R. Anderson and M. Onitsuka, Hyers-Ulam stability and best co nstant for Cayley h-difference equations,\nBull. Malays. Math. Sci. Soc. 43 (2020), no. 6, 4207–4222.\n[4] D. R. Anderson and M. Onitsuka, Hyers-Ulam stability for differen tial systems with 2 ×2 constant coefficient\nmatrix, Results Math. 77 (2022), no. 3, Paper No. 136, 23 pp.\n[5] D. R. Anderson, M. Onitsuka and J. M. Rassias, Best constant f or Ulam stability of first-order h-difference\nequations with periodic coefficient, J. Math. Anal. 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Syst. 21 (2022), no. 2, Paper No. 54, 30 pp.\n[29] Y. Yang and F. Meng, A kind of stricter Hyers-Ulam stability of se cond order linear differential equations of\nCarath´ eodory type, Appl. Math. Lett. 115 (2021), Paper No. 1 06946, 7 pp.\n[30] A. Zada, L. Alam, J. Xu and W. Dong, Controllability and Hyers-Ula m stability of impulsive second order\nabstract damped differential systems, J. Appl. Anal. Comput. 11 ( 2021), no. 3, 1222–1239.\nDepartment of Mathematics, Concordia College, Moorhead, M N 56562 USA\nEmail address :andersod@cord.edu\nDepartment of Applied Mathematics, Okayama University of S cience, Okayama, 700-0005, Japan\nEmail address :onitsuka@ous.ac.jp\nSchool of Mathematical and Statistical Sciences, Universi ty of Galway, Galway, Ireland\nEmail address :donal.oregan@nuigalway.ie"    },    {        "title": "2308.05955v2.Dynamical_Majorana_Ising_spin_response_in_a_topological_superconductor_magnet_hybrid_by_microwave_irradiation.pdf",        "content": "Dynamical Majorana Ising spin response in a topological superconductor-magnet\nhybrid by microwave irradiation\nYuya Ominato,1, 2Ai Yamakage,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n2Waseda Institute for Advanced Study, Waseda University, Shinjuku, Tokyo 169-8050, Japan.\n3Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: March 20, 2024)\nWe study a dynamical spin response of surface Majorana modes in a topological superconductor-\nmagnet hybrid under microwave irradiation. We find a method to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjusting the external magnetic field angle and\nthe microwave frequency. This reflects the topological nature of the Majorana modes, enhancing\nthe Gilbert damping of the magnet, thereby, providing a detection method for the Majorana Ising\nspins. Our findings illuminate a magnetic probe for Majorana modes, paving the path to innovative\nspin devices.\nIntroduction.— The quest for Majoranas within matter\nstands as one of the principal challenges in the study of\ncondensed matter physics, more so in the field of quan-\ntum many-body systems [1]. The self-conjugate nature\nof Majoranas leads to peculiar electrical characteristics\nthat have been the subject of intensive research, both\ntheoretical and experimental [2]. In contrast, the focus of\nthis paper lies on the magnetic properties of Majoranas,\nspecifically the Majorana Ising spin [3–8]. A distinctive\ncharacteristic of Majorana modes, appearing as a surface\nstate in topological superconductors (TSC), is its exceed-\ningly strong anisotropy, which makes it behave as an Ising\nspin. In particular, this paper proposes a method to ex-\nplore the dynamical response of the Majorana Ising spin\nthrough the exchange interaction at the magnetic inter-\nface, achieved by coupling the TSC to a ferromagnet with\nferromagnetic resonance (FMR) (as shown in Fig.1 (a)).\nFMR modulation in a magnetic hybrid system has at-\ntracted much attention as a method to analyze spin ex-\ncitations in thin-film materials attached to magnetic ma-\nterials [9, 10]. Irradiating a magnetic material with mi-\ncrowaves induces dynamics of localized spin in magnetic\nmaterials, which can excite spins in adjacent thin-film\nmaterials via the magnetic proximity effect. This setup\nis called spin pumping, and has been studied intensively\nin the field of spintronics as a method of injecting spins\nthrough interfaces [11, 12]. Recent studies have theoret-\nically proposed that spin excitation can be characterized\nby FMR in hybrid systems of superconducting thin films\nand magnetic materials [13–18]. Therefore, it is expected\nto be possible to analyze the dynamics of surface Majo-\nrana Ising spins using FMR in hybrid systems.\nIn this work, we consider a TSC-ferromagnetic insula-\ntor (FI) hybrid system as shown in Fig. 1 (a). The FMR\nis induced by microwave irradiation on the FI. At the\ninterface between the TSC and the FI, the surface Ma-\n(b)\n(c)(a)\nFI~~\n~~Microwave\nϑS\nY, yX\nxZhdcHex\nTSC\n(d)\nhdchdc+δhα+δα\nHz\nFIG. 1. (a) The TSC-FI hybrid schematic reveals how,\nunder resonance frequency microwave irradiation, localized\nspins commence precessional motion, consequently initiating\nthe dynamical Majorana Ising spin response at the TSC inter-\nface. (b) In the TSC context, the liaison between a spin-up\nelectron and a spin-down hole with the surrounding sea of\nspin-triplet Cooper pairs drastically modulate their proper-\nties; notably, a spin-down hole can engage with a spin-triplet\nCooper pair, thereby inheriting a negative charge. (c) No-\ntably, spin-triplet Cooper pairs amass around holes and scat-\nter around electrons, thereby eroding the rigid distinction be-\ntween the two. (d) The interplay between the Majorana mode\nand the localized spin manipulates the FMR spectrum, trig-\ngering a frequency shift and linewidth broadening.\njorana modes interact with the localized spins in the FI.\nAs a result, the localized spin dynamics leads to the dy-\nnamical Majorana Ising spin response (DMISR), which\nmeans the Majorana Ising spin density is dynamically in-\nduced, and it is possible to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjust-\ning the external magnetic field angle and the microwave\nfrequency. Furthermore, the modulation of the localizedarXiv:2308.05955v2  [cond-mat.mes-hall]  19 Mar 20242\nspin dynamics due to the interface interaction leads to a\nfrequency shift and a linewidth broadening, which reflect\nthe properties of the Majorana Ising spin dynamics. This\nwork proposes a setup for detecting Majorana modes and\npaves the way for the development of quantum comput-\ning and spin devices using Majoranas.\nModel.— We introduce a model Hamiltonian Hconsist-\ning of three terms\nH=HM+HFI+Hex. (1)\nThe first, second, and third terms respectively describe\nthe surface Majorana modes on the TSC surface, the bulk\nFI, and the proximity-induced exchange coupling. Our\nfocus is on energy regions significantly smaller than the\nbulk superconducting gap. This focus allows the spin ex-\ncitation in the TSC to be well described using the surface\nMajorana modes. The subsequent paragraphs provide\ndetailed explanations of each of these three terms.\nThe first terms HMdescribes the surface Majorana\nmodes,\nHM=1\n2Z\ndrψT(r)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r),(2)\nwhere r= (x, y),ˆk= (−i∂x,−i∂y),vis a constant\nvelocity, and σ= (σx, σy, σz) are the Pauli matrices.\nThe two component Majorana field operator is given by\nψ(r) = ( ψ→(r), ψ←(r))T, with the spin quantization\naxis along the xaxis. The Majorana field operators sat-\nisfy the Majorana condition ψσ(r) =ψ†\nσ(r) and the an-\nticommutation relation {ψσ(r), ψσ′(r)}=δσσ′δ(r−r′)\nwhere σ, σ′=→,←. We can derive HMby using surface-\nlocalized solutions of the BdG equation based on the bulk\nTSC Hamiltonian. The details of the derivation of HM\nare provided in the Supplemental Material [19].\nA notable feature of the surface Majorana modes is\nthat the spin density is Ising like, which we call the Majo-\nrana Ising spin [3–8]. The feature follows naturally from\nthe Majorana condition and the anticommutation rela-\ntion. The Majorana Ising spin density operator is given\nbys(r) := ψT(r)(σ/2)ψ(r) = (0 ,0,−iψ→(r)ψ←(r))\n(See the Supplemental Material for details [19]). The\nanisotropy of the Majorana Ising spin is the hallmark of\nthe surface Majorana modes on the TSC surface.\nThe second term HFIdescries the bulk FI and is given\nby the ferromagnetic Heisenberg model,\nHFI=− JX\n⟨n,m⟩Sn·Sm−ℏγhdcX\nnSZ\nn, (3)\nwhere J>0 is the exchange coupling constant, Snis the\nlocalized spin at site n,⟨n, m⟩means summation for near-\nest neighbors, γis the electron gyromagnetic ratio, and\nhdcis the static external magnetic field. We consider the\nspin dynamics of the localized spin under microwave irra-\ndiation, applying the spin-wave approximation. This al-\nlows the spin excitation to be described by a free bosonic\noperator, known as a magnon [20].The third term Hexrepresents the proximity exchange\ncoupling at the interface between the TSC and the FI,\nHex=−Z\ndrX\nnJ(r,rn)s(r)·Sn=HZ+HT,(4)\nHZ=−cosϑZ\ndrX\nnJ(r,rn)sz(r)SZ\nn, (5)\nHT=−sinϑZ\ndrX\nnJ(r,rn)sz(r)SX\nn, (6)\nwhere the angle ϑis shown in Fig. 1 (a). HZis the\ncoupling along the precession axis and HTis the coupling\nperpendicular to the precession axis. In our setup, HZ\nleads to gap opening of the energy spectrum of the surface\nMajorana modes and HTgives the DMISR under the\nmicrowave irradiation.\nDynamical Majorana Ising spin response.— We con-\nsider the microwave irradiation on the FI. The coupling\nbetween the localized spins and the microwave is given\nby\nV(t) =−ℏγhacX\nn\u0000\nSX\nncosωt−SY\nnsinωt\u0001\n,(7)\nwhere hacis the microwave amplitude, and ωis the mi-\ncrowave frequency. The microwave irradiation leads to\nthe precessional motion of the localized spin. When the\nfrequency of the precessional motion and the microwave\ncoincide, the FMR occurs. The FMR leads to the DMISR\nvia the exchange interaction. The DMISR is character-\nized by the dynamic spin susceptibility of the Majorana\nmodes, ˜ χzz(q, ω), defined as\n˜χzz(q, ω) :=Z\ndre−iq·rZ\ndtei(ω+i0)tχzz(r, t),(8)\nwhere χzz(r, t) := −(L2/iℏ)θ(t)⟨[sz(r, t), sz(0,0)]⟩\nwith the interface area L2and the spin den-\nsity operator in the interaction picture, sz(r, t) =\nei(HM+HZ)t/ℏsz(r)e−i(HM+HZ)t/ℏ. For the exchange cou-\npling, we consider configuration average and assume\n⟨P\nnJ(r,rn)⟩ave=J1, which means that HZis treated\nas a uniform Zeeman like interaction and the interface\nis specular [21]. Using eigenstates of Eq. (2) and after a\nstraightforward calculation, the uniform spin susceptibil-\nity is given by\n˜χzz(0, ω)\n=−X\nk,λ|⟨k, λ|σz|k,−λ⟩|2f(Ek,λ)−f(Ek,−λ)\n2Ek,λ+ℏω+i0,\n→ −Z\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω+i0, (9)\nwhere |k, λ⟩is an eigenstate of HMwith eigenenergy\nEk,λ=λp\n(ℏvk)2+M2, (λ=±).M=J1Scosϑis\nthe Majorana gap, f(E) = 1 /(eE/kBT+ 1) is the Fermi3\ndistribution function, and D(E) is the density of states\ngiven by\nD(E) =L2\n2π(ℏv)2|E|θ(|E| − |M|), (10)\nwith the Heaviside step function θ(x). It is important to\nnote that the behavior of the uniform spin susceptibil-\nity is determined by the interband contribution, which is\nproportional to the Fermi distribution function, i.e., the\ncontribution of the occupied states. This mechanism is\nsimilar to the Van Vleck paramagnetism [22]. The con-\ntribution of the occupied states often plays a crucial role\nin topological responses [23].\nReplacing the localized spin operators with their statis-\ntical average values, we find the induced Majorana Ising\nspin density, to the first order of J1S, is given by\nZ\ndr⟨sz(r, t)⟩= ˜χzz\n0(0,0)J1Scosϑ\n+ Re[˜ χzz\n0(0, ω)]hac\nαhdcJ1Ssinϑsinωt, (11)\nwhere ˜ χzz\n0(0,0) is the spin susceptibility for M= 0. The\nfirst term originates from HZand gives a static spin den-\nsity, while the second term originates from HTand gives\na dynamic spin density. Figure 2 shows the induced Ising\nspin density as a function of time at several angles. As\nshown in Eq. (11), the Ising spin density consists of the\nstatic and dynamic components. The dynamic compo-\nnent is induced by the precessional motion of the local-\nized spin, which means one can induce the DMISR using\nthe dynamics of the localized spin.\nThe inset in Fig. 2 shows Im˜ χzz(0, ω) as a function of\nϑat a fixed frequency. When the frequency ℏωis smaller\nthan the Majorana gap, Im˜ χzz(0, ω) is zero. Once the\nfrequency overcomes the Majorana gap, Im˜ χzz(0, ω) be-\ncomes finite. The implications of these behaviors are that\nif the magnon energy is smaller than the Majorana gap,\nthere is no energy dissipation due to the DMISR. How-\never, once the magnon energy exceeds the Majorana gap,\nfinite energy dissipation associated with the DMISR oc-\ncurs at the surface of the TSC. Therefore, one can toggle\nbetween dissipative and non-dissipative Majorana Ising\nspin dynamics by adjusting the precession axis angle and\nthe microwave frequency.\nFMR modulation.— The retarded component of the\nmagnon Green’s function is given by GR(rn, t) =\n−(i/ℏ)θ(t)⟨[S+\nn(t), S−\n0(0)]⟩with the interaction picture\nS±\nn(t) =eiHFIt/ℏS±\nne−iHFIt/ℏ. The FMR signal is char-\nacterized by the spectral function defined as\nA(q, ω) :=−1\nπIm\"X\nne−iq·rnZ\ndtei(ω+i0)tGR(rn, t)#\n.\n(12)\nSSImχzz(0, ω) ˜⟨s   z⟩\n2\n1ωtϑ\nFInon-dissipativenon-dissipativedissipativedissipativeTSC\nFITSC000.00.51.0\nπ/4\nπ/2\n0 π/4 π/20\nϑ2π\nπFIG. 2. The induced Ising spin density, with a unit\n˜χzz\n0(0,0)J1S, is presented as a function of ωtandϑ. The\nfrequency and temperature are set to ℏω/J1S= 1.5 and\nkBT/J 1S= 0.1, respectively. The coefficient, hac/αhdc, is\nset to 0 .3. The static Majorana Ising spin density arises\nfrom HZ. When the precession axis deviates from the di-\nrection perpendicular to the interface, the precessional mo-\ntion of the localized spins results in the dynamical Majorana\nIsing spin response (DMISR). Energy dissipation due to the\nDMISR is zero for small angles ϑas the Majorana gap ex-\nceeds the magnon energy. However, once the magnon energy\novercomes the Majorana gap, the energy dissipation becomes\nfinite. Therefore, one can toggle between dissipative and non-\ndissipative DMISR by adjusting ϑ.\nFor uniform external force, the spectral function is given\nby\nA(0, ω) =2S\nℏ1\nπ(α+δα)ω\n[ω−γ(hdc+δh)]2+ [(α+δα)ω]2.\n(13)\nThe peak position and width of the FMR signal is given\nbyhdc+δhandα+δα, respectively. hdcandαcorre-\nspond to the peak position and the linewidth of the FMR\nsignal of the FI alone. δhandδαare the FMR modu-\nlations due to the exchange interaction HT. We treat\nHM+HFI+HZas an unperturbed Hamiltonian and HT\nas a perturbation. In this work, we assume the specular\ninterface, where the coupling J(r,rn) is approximated\nasDP\nn,n′J(r,rn)J(r′,rn′)E\nave=J2\n1. The dynamics\nof the localized spins in the FI is modulated due to the\ninteraction between the localized spins and the Majo-\nrana Ising spins. In our setup, the peak position and the\nlinewidth of the FMR signal are modulated and the FMR4\nmodulation is given by\nδh= sin2ϑSJ2\n1\n2NγℏRe˜χzz(0, ω), (14)\nδα= sin2ϑSJ2\n1\n2NℏωIm˜χzz(0, ω), (15)\nwhere Nis the total number of sites in the FI. These for-\nmulas were derived in the study of the FMR in magnetic\nmultilayer systems including superconductors. One can\nextract the spin property of the Majorana mode from the\ndata on δhandδα. Because of the Ising spin anisotropy,\nthe FMR modulation exhibits strong anisotropy, where\nthe FMR modulation is proportional to sin2ϑ.\nFigure 3 shows the FMR modulations (a) δαand (b)\nδh. The FMR modulation at a fixed frequency increases\nwith angle ϑand reaches a maximum at π/2, as can be\nread from Eqs. (14) and (15). When the angle ϑis fixed\nand the frequency ωis increased, δαbecomes finite above\na certain frequency at which the energy of the magnon\ncoincides with the Majorana gap. When ϑ < π/ 2 and\nℏω≈2M,δαlinearly increases as a function of ωjust\nabove the Majorana gap. The localized spin damping is\nenhanced when the magnon energy exceeds the Majorana\ngap. At ϑ=π/2 and ω≈0, the Majorana gap vanishes\nandδαis proportional to ω/T. In the high frequency\nregion ℏω/J 1S≫1,δαconverges to its upper threshold.\nThe frequency shift δhis almost independent of ωand\nhas a finite value even in the Majorana gap. This behav-\nior is analogous to the interband contribution to the spin\nsusceptibility in strongly spin-orbit coupled band insula-\ntors, and is due to the fact that the effective Hamiltonian\nof the Majorana modes includes spin operators. It is im-\nportant to emphasize that although the Majorana modes\nhave spin degrees of freedom, only the zcomponent of the\nspin density operator is well defined. This is a hallmark\nof Majorana modes, which differs significantly from elec-\ntrons in ordinary solids. Note that δhis proportional to\nthe energy cutoff, which is introduced to converge energy\nintegral for Re˜ χzz(0, ω). The energy cutoff corresponds\nto the bulk superconducting gap, which is estimated as\n∆∼0.1[meV] ( ∼1[K]). Therefore, our results are ap-\nplicable in the frequency region below ℏω∼0.1[meV]\n(∼30[GHz]). In addition, we assume that Majorana gap\nis estimated to be J1S∼0.01[meV] ( ∼0.1[K]).\nDiscussion.— Comparing the present results with spin\npumping (SP) in a conventional metal-ferromagnet hy-\nbrid, the qualitative behaviors are quite different. In con-\nventional metals, spin accumulation occurs due to FMR.\nIn contrast, in the present system, no corresponding spin\naccumulation occurs due to the Ising anisotropy. Also, in\nthe present calculations, the proximity-induced exchange\ncoupling is assumed to be an isotropic Heisenberg-like\ncoupling. However, in general, the interface interaction\ncan also be anisotropic. Even in such a case, it is no qual-\nitative change in the case of ordinary metals, although a\n0.00.5\n(a) (b)\nϑℏω/J1S 0\nπ/4\nπ/2024\nϑℏω/J1S 0\nπ/4\nπ/2024δ  α δ  h10\n0FIG. 3. The temperature is set to kBT/J 1S= 0.1. (a)\nThe damping modulation δαonly becomes finite when the\nmagnon energy exceeds the Majorana gap; otherwise, it van-\nishes. This behavior corresponds to the energy dissipation of\nthe Majorana Ising spin. (b) The peak shift is finite, except\nforϑ= 0, and is almost independent of ω. This behavior\nresembles the spin response observed in strongly spin-orbit\ncoupled band insulators, where the interband contribution to\nspin susceptibility results in a finite spin response, even within\nthe energy gap.\ncorrection term due to anisotropy is added [24]. There-\nfore, the Ising anisotropy discussed in the present work\nis a property unique to the Majorana modes and can\ncharacterize the Majorana excitations.\nLet us comment on the universal nature of the toggling\nbetween non-dissipative and dissipative dynamical spin\nresponses observed in our study. Indeed, such toggling\nbecomes universally feasible when the microwave fre-\nquency and the energy gap are comparable, and when the\nHamiltonian and spin operators are non-commutative,\nindicating that spin is not a conserved quantity. The\nnon-commutativity can be attributed to the presence of\nspin-orbit couplings [25–27], and spin-triplet pair corre-\nlations [28].\nMicrowave irradiation leads to heating within the FI,\nso that thermally excited magnons due to the heating\ncould influence the DMISR. Phenomena resulting from\nthe heating, which can affect interface spin dynamics, in-\nclude the spin Seebeck effect (SSE) [29], where a spin\ncurrent is generated at the interface due to a tempera-\nture difference. In hybrid systems of normal metal and\nFI, methods to separate the inverse spin Hall voltage due\nto SP from other signals caused by heating have been\nwell studied [30]. Especially, it has been theoretically\nproposed that SP and SSE signals can be separated us-\ning a spin current noise measurement [24]. Moreover, SP\ncoherently excites specific modes, which qualitatively dif-\nfers from SSE induced by thermally excited magnons [14].\nTherefore, even if heating occurs in the FI in our setup,\nthe properties of Majorana Ising spins are expected to\nbe captured. Details of the heating effect on the DMISR\nwill be examined in the near future.\nWe also mention the experimental feasibility of our the-\noretical proposals. As we have already explained, the\nFMR modulation is a very sensitive spin probe. Indeed,\nthe FMR modulation by surface states of 3D topological5\ninsulators [31] and graphene [32–36] has been reported\nexperimentally. Therefore, we expect that the enhanced\nGilbert damping due to Majorana Ising spin can be ob-\nservable in our setup when the thickness of the ferromag-\nnetic insulator is sufficiently thin.\nFinally, it is pertinent to mention the potential candi-\ndate materials where surface Majorana Ising spins could\nbe detectable. Notably, UTe 2[37], Cu xBi2Se3[38, 39],\nSrxBi2Se3and Nb xBi2Se3[40] are reported to be in a p-\nwave superconducting state and theoretically can host\nsurface Majorana Ising spins. Recent NMR measure-\nments indicate that UTe 2could be a bulk p-wave su-\nperconductor in the Balian-Werthamer state [41], which\nhosts the surface Majorana Ising spins with the per-\npendicular Ising anisotropy, as considered in this work.\nAxBi2Se3(A= Cu, Sr, Nb) is considered to possess in-\nplane Ising anisotropy [8], differing from the perpendic-\nular Ising anisotropy explored in this work. Therefore,\nwe expect that it exhibits anisotropy different from that\ndemonstrated in this work.\nConclusion.— We present herein a study of the spin\ndynamics in a topological superconductor (TSC)-magnet\nhybrid. Ferromagnetic resonance under microwave irra-\ndiation leads to the dynamically induced Majorana Ising\nspin density on the TSC surface. One can toggle between\ndissipative and non-dissipative Majorana Ising spin dy-\nnamics by adjusting the external magnetic field angle and\nthe microwave frequency. Therefore, our setup provides\na platform to detect and control Majorana excitations.\nWe expect that our results provide insights toward the\ndevelopment of future quantum computing and spintron-\nics devices using Majorana excitations.\nAcknowledgments.— The authors are grateful to R.\nShindou for valuable discussions. This work is partially\nsupported by the Priority Program of Chinese Academy\nof Sciences, Grant No. XDB28000000. We acknowl-\nedge JSPS KAKENHI for Grants (Nos. JP20K03835,\nJP21H01800, JP21H04565, and JP23H01839).\nSUPPLEMENTAL MATERIAL\nSurface Majorana modes\nIn this section, we describe the procedure for deriv-\ning the effective Hamiltonian of the surface Majorana\nmodes. We start with the bulk Hamiltonian of a three-\ndimensional topological superconductor. Based on the\nbulk Hamiltonian, we solve the BdG equation to demon-\nstrate the existence of a surface-localized solution. Us-\ning this solution, we expand the field operator and show\nthat it satisfies the Majorana condition when the bulk\nexcitations are neglected. As a result, on energy scales\nmuch smaller than the bulk superconducting gap, the\nlow-energy excitations are described by surface-localized\nMajorana modes. The above procedure is explained inmore detail in the following. Note that we use rfor three-\ndimensional coordinates and r∥for two-dimensional ones\nin the Supplemental Material.\nWe start with the mean-field Hamiltonian given by\nHSC=1\n2Z\ndrΨ†\nBdG(r)HBdGΨBdG(r), (16)\nwithr= (x, y, z ). We consider the Balian-Werthamer\n(BW) state, in which the pair potential is given by\n∆ˆk=∆\nkF\u0010\nˆk·σ\u0011\niσywith the bulk superconducting gap\n∆. Here, we do not discuss the microscopic origin of the\npair correlation leading to the BW state. As a result, the\nBdG Hamiltonian HBdGis given by\nHBdG=\nεˆk−EF 0 −∆\nkFˆk−∆\nkFˆkx\n0 εˆk−EF∆\nkFˆkx∆\nkFˆk+\n−∆\nkFˆk+∆\nkFˆkx−εˆk+EF 0\n∆\nkFˆkx∆\nkFˆk− 0 −εˆk+EF\n,\n(17)\nwith ˆk±=ˆky±iˆkz,ˆk=−i∇, and εˆk=ℏ2ˆk2\n2m. The four\ncomponent Nambu spinor ΨBdG(r) is given by\nΨBdG(r) :=\nΨ→(r)\nΨ←(r)\nΨ†\n→(r)\nΨ†\n←(r)\n, (18)\nwith the spin quantization axis along the xaxis. The\nmatrices of the spin operators are represented as\nσx=\u00121 0\n0−1\u0013\n, (19)\nσy=\u0012\n0 1\n1 0\u0013\n, (20)\nσz=\u00120−i\ni0\u0013\n. (21)\nThe fermion field operators satisfy the anticommutation\nrelations\n{Ψσ(r),Ψσ′(r′)}= 0, (22)\n{Ψσ(r),Ψ†\nσ′(r′)}=δσσ′δ(r−r′), (23)\nwith the spin indices σ, σ′=→,←.\nTo diagonalize the BdG Hamiltonian, we solve the BdG\nequation given by\nHBdGΦ(r) =EΦ(r). (24)\nWe assume that a solution is written as\nΦ(r) =eik∥·r∥f(z)\nu→\nu←\nv→\nv←\n, (25)6\nwithk∥= (kx, ky) and r∥= (x, y). If we set the four\ncomponents vector to satisfy the following equation (Ma-\njorana condition)\n\n0 0 1 0\n0 0 0 1\n1 0 0 0\n0 1 0 0\n\nu→\nu←\nv→\nv←\n=±\nu→\nu←\nv→\nv←\n, (26)\nwe can obtain a surface-localized solution. If we take a\npositive (negative) sign, we obtain a solution localized\non the top surface (bottom surface). As we will consider\nsolutions localized on the bottom surface below, we take\na negative sign. Finally, we obtain the normalized eigen-\nvectors of the BdG equation given by\nΦλ,k∥(r) =eik∥·r∥\n√\nL2fk∥(z)uλ,k∥, (27)\nwith\nfk∥(z) =Nk∥sin(k⊥z)e−κz, (28)\nNk∥=s\n4κ(k2\n⊥+κ2)\nk2\n⊥, (29)\nκ=m∆\nℏ2kF, (30)\nk⊥=q\nk2\nF−k2\n∥−κ2, (31)\nand\nu+,k∥=\nu+,→k∥\nu+,←k∥\nv+,→k∥\nv+,←k∥\n=1√\n2\nsinϕk∥+π/2\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\n,(32)\nu−,k∥=\nu−,→k∥\nu−,←k∥\nv−,→k∥\nv−,←k∥\n=1√\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\nsinϕk∥+π/2\n2\n.(33)\nThe eigenenergy is given by Eλ,k∥=λ∆k∥/kF. We can\nshow that the eigenvectors satisfy\nu−,−k∥=u+,k∥. (34)\nConsequently, the field operator is expanded as\nΨBdG(r) =X\nk∥\u0012\nγk∥eik∥·r∥\n√\nL2+γ†\nk∥e−ik∥·r∥\n√\nL2\u0013\n×fk∥(z)u+,k∥+ (bulk modes) ,(35)\nwhere γk∥(γ†\nk∥) is the quasiparticle creation (annihila-\ntion) operator with the eigenenergy E+,k∥. Substitutingthe above expression into Eq. (16) with omission of bulk\nmodes and performing the integration in the z-direction,\nwe obtain the effective Hamiltonian for the surface states\nHM=1\n2Z\ndr∥ψT(r∥)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r∥),(36)\nwhere v= ∆/ℏkFand we introduced the two component\nMajorana field operator\nψ(r∥) =\u0012ψ→(r∥)\nψ←(r∥)\u0013\n, (37)\nsatisfying the Majorana condition\nψσ(r∥) =ψ†\nσ(r∥), (38)\nand the anticommutation relation\nn\nψσ(r∥), ψσ′(r′\n∥)o\n=δσσ′δ(r∥−r′\n∥). (39)\nThe spin density operator of the Majorana mode is\ngiven by\ns(r∥) =ψ†(r∥)σ\n2ψ(r∥). (40)\nThexcomponent is given by\nsx(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00121/2 0\n0−1/2\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ→(r∥)−ψ†\n←(r∥)ψ←(r∥)\u0003\n=1\n2\u0002\nψ2\n→(r∥)−ψ2\n←(r∥)\u0003\n= 0. (41)\nIn a similar manner, the yandzcomponents are given\nby\nsy(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120 1/2\n1/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ←(r∥) +ψ†\n←(r∥)ψ→(r∥)\u0003\n=1\n2\b\nψ→(r∥), ψ←(r∥)\t\n= 0, (42)\nand\nsz(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120−i/2\ni/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=−i\n2\u0000\nψ†\n→(r∥)ψ←(r∥)−ψ†\n←(r∥)ψ→(r∥)\u0001\n=−iψ→(r∥)ψ←(r∥), (43)\nrespectively. As a result, the spin density operator is\ngiven by\ns(r∥) =\u0000\n0,0,−iψ→(r∥)ψ←(r∥)\u0001\n. (44)\nOne can see that the spin density of the Majorana mode\nis Ising like.7\nMajorana Ising spin dynamics\nIn this section, we calculate the Ising spin density in-\nduced on the TSC surface by the proximity coupling Hex.\nHexconsists of two terms, HZandHT.HZleads to the\nstatic spin density and HTleads to the dynamic spin\ndensity. First, we calculate the static spin density. Next,\nwe calculate the dynamic spin density.\nThe total spin density operator is given by\nsz\ntot=Z\ndr∥sz(r∥). (45)\nThe statistical average of the static spin density is calcu-\nlated as\n⟨sz\ntot⟩=−X\nk∥M\n2Ek∥\u0002\nf(Ek∥)−f(−Ek∥)\u0003\n→ −\u0012L\n2πℏv\u00132Z∆\nMEdEZ2π\n0dϕM\n2E[f(E)−f(−E)]\n=−Z∆\n0dED (E)f(E)−f(−E)\n2EM. (46)\nAt the zero temperature limit T→0, the static spin\ndensity is given by\n⟨sz\ntot⟩=1\n2L2\n2π(ℏv)2(∆−M)M≈˜χzz\n0(0,0)M, (47)\nwhere ˜ χzz\n0(0,0) = D(∆)/2 and we used ∆ ≫M.\nThe dynamic spin density is given by the perturbative\nforce\nHT(t) =Z\ndr∥sz(r∥)F(r∥, t), (48)\nwhere F(r∥, t) is given by\nF(r∥, t) =−sinϑX\nnJ(r∥,rn)\nSX\nn(t)\u000b\n≈ −sinϑJ1Sγhacp\n(ω−γhdc)2+α2ω2cosωt\n=:Fcosωt. (49)\nThe time dependent statistical average of the Ising spin\ndensity, to the first order of J1S, is given by\nZ\ndr∥\nsz(r∥, t)\u000b\n=Z\ndr∥Z\ndr′\n∥Z\ndt′χzz(r∥−r′\n∥, t′)F(r′\n∥, t−t′)\n= Re\u0002\n˜χzz(0, ω)Fe−iωt\u0003\n≈Re[˜χzz\n0(0, ω)]Fcosωt, (50)\nwhere we used Re˜ χzz\n0(0, ω)≫Im˜χzz\n0(0, ω). The real part\nof ˜χzz(0, ω) is given by\nRe˜χzz(0, ω) =−PZ\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω,\n(51)where Pmeans the principal value. When the integrand\nis expanded with respect to ω, the lowest order correc-\ntion term becomes quadratic in ω. In the frequency range\nconsidered in this work, this correction term is signifi-\ncantly smaller compared to the static spin susceptibility\nRe˜χzz(0,0). Therefore, the spin susceptibility exhibits\nalmost no frequency dependence and remains constant\nas a function of ω. The imaginary part of ˜ χzz(0, ω) is\ngiven by\nIm˜χzz(0, ω)\n=πD(ℏω/2)(ℏω/2)2−M2\n2(ℏω/2)2[f(−ℏω/2)−f(ℏω/2)].\n(52)\nFMR modulation due to the proximity exchange\ncoupling\nIn this section, we provide a brief explanation for the\nderivation of the FMR modulations δhandδα. The FMR\nmodulations can be determined from the retarded com-\nponent of the magnon Green’s function, which is given\nby\n˜GR(k, ω) =2S/ℏ\nω−ωk+iαω−(2S/ℏ)ΣR(k, ω),(53)\nwhere we introduce the Gilbert damping constant αphe-\nnomenologically. In the second-order perturbation calcu-\nlation with respect to HT, the self-energy is given by\nΣR(k, ω) =−\u0012sinϑ\n2\u00132X\nq∥|˜J(q∥,k)|2˜χzz(q∥, ω),(54)\nwhere ˜J(q∥,0) is given by\n˜J(q∥,k) =1\nL2√\nNZ\ndr∥X\nnJ(r∥,rn)ei(q∥·r∥+k·rn)\n(55)\nThe pole of ˜GR(k, ω) signifies the FMR modulations,\nincluding both the frequency shift and the enhanced\nGilbert damping. These are given by\nδh=2S\nγℏReΣR(0, ω), δα =−2S\nℏωImΣR(0, ω).(56)\nFrom the above equations and Eq. (54), it is apparent\nthat FMR modulations provide information regarding\nboth the properties of the interface coupling and the dy-\nnamic spin susceptibility of the Majorana modes.\nThe form of matrix element ˜J(q∥,0) depends on the\ndetails of the interface. In this work, we assume the\nspecular interface. |˜J(q∥,0)|2is given by\n|˜J(q∥,0)|2=J2\n1\nNδq∥,0. (57)8\nUsing Eq. (57), the self-energy for the uniform magnon\nmode is given by\nΣR(0, ω) =−\u0012sinϑ\n2\u00132J2\n1\nN˜χzz(0, ω). (58)\n[1] F. Wilczek, Nat. 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Jpn. 92, 063701 (2023)."    },    {        "title": "2308.07236v3.Temperature_Evolution_of_Magnon_Propagation_Length_in_Tm__3_Fe__5_O___12___Thin_Films__Roles_of_Magnetic_Anisotropy_and_Gilbert_Damping.pdf",        "content": "  \n1 \n Temperature Evolution of Magnon Propagation Length in \nTm 3Fe5O12 Thin F ilms: Roles  of Magnetic Anisotropy and Gilbert \nDamping  \n \nAmit Chanda1, Christian Holzmann2, Noah S chulz1, Aladin Ullrich2, Derick DeTellem1, \nManfred Albrecht2*, Miela J. Gross3, Caroline A. Ross3*, Dario  A. Arena1, Manh -Huong \nPhan1, and Hari haran  Srikanth1* \n1Department of Physics, University of South Florida, Tampa, Florida 33620, USA  \n2Institute of Physics, University of Augsburg, 86159 Augsburg, Germany  \n3Department of Materials Science and Engineering, Massachusetts Institute of Technology, \nCambridge, Massachusetts 02139, USA  \n \n*Corresponding authors: manfred.albrecht@physik.uni -augsburg.de ; caross@mit.edu ; \nsharihar@usf.edu  \n \nKeywords:  Longitudinal spin Seebeck effect, Inverse spin Hall effect, Magnon propagation length, \nGilbert damping, Magnetic anisotropy , Rare -earth iron garnet  \n \nABSTRACT  \nThe magnon propagation length , 〈𝜉〉 of a ferro -/ferrimagnet (FM) is one of the key factors that  \ncontrols the generation and propagation of thermally -driven magnonic spin current in FM/heavy \nmetal ( HM) bilayer based spincaloritronic devices . For the development of a complete physical \npicture of thermally -driven magnon transport in FM/HM bilayers over a wide temperature range,    \n2 \n it is of utmost importance to understand  the respective roles of temperature -dependent Gilbert \ndamping (𝛼) and effective magnetic anisotropy (𝐾𝑒𝑓𝑓) in controlling  the temperature evolution of \n〈𝜉〉. Here, we report  a comprehensive investigation  of the temperature -dependent longitudinal spin \nSeebeck effect (LSSE), radio frequency transverse susceptibility, and broadband ferromagnetic \nresonance measurements  on Tm 3Fe5O12 (TmIG )/Pt bilayers grown on different substrates. We \nobserve a remarkable drop  in the LSSE voltage below 200 K independent  of TmIG film thickness \nand substrate choice . This is attribute d to the noticeable increases in effective magnetic anisotropy \nfield, 𝐻𝐾𝑒𝑓𝑓 (∝𝐾𝑒𝑓𝑓) and 𝛼 that occur within the same temperature range. From the TmIG  \nthickness dependence of the LSSE voltage, we determined the temperature dependence of 〈𝜉〉 and \nhighlighted its correlation with the temperature -dependent  𝐻𝐾𝑒𝑓𝑓 and 𝛼 in TmIG/Pt  bilayers , which \nwill be beneficial for the development of rare -earth iron garnet -based efficient spincaloritronic \nnanodevices . \n \n \n \n \n \n \n \n \n   \n3 \n 1. INTRODUCTION  \nIn recent years, interface -engineered bilayer thin films have gained intense attention of the \nmaterials science community because of their multifunctionality and emergent  physical properties \nranging from  ferroelectricity1 and magnetism2 to spin -electronics3. Bilayers  comprised of  \ninsulating rare -earth iron garnet (REIG ) and heavy metal (HM) form  the most appealing platform  \nto generate, transmit , and detect pure spin currents  in the field of spin -based -electronics4–6. The \ninterplay of damping and magnon propagation length ( 〈𝜉〉) of the REIG layer and spin-orbit \ncoupling (SOC)  of the HM layer leads to a wide range of emergent  spintronic phenomena in t his \nfascinating class of heterostructures , including  the spin Hall effect7, spin -orbit torque8,9, spin-\npumping  effect  (SPE)10, and the longitudinal spin Seebeck effect  (LSSE)11–13.  The discovery of \nthe SSE14 instigated  a new generation of spintronic nano devices facilitating  electrical energy \nharvesting from renewable thermal energy wherein a magnonic  spin current is thermally generated \nand electrically detected by applying a temperature gradient across a magnetic insulator (MI) /HM \nbilayer15. Unlike magnetostatic spin waves with millimeter -range propagation lengths, 〈𝜉〉 for \nthermally generated magnons is significantly smaller, a few hundreds of nanometers16. In the \nframework of an atomistic spin model based on linear spin -wave theory, it was theoretically \nshown17,18 that thermally generated magnons have a broad frequency (𝑓) distribution with \n𝑓𝑚𝑖𝑛𝑖𝑚𝑢𝑚=2𝐾𝑒𝑓𝑓[ℎ(1+𝛼2)] ⁄  and 𝑓𝑚𝑎𝑥𝑖𝑚𝑢𝑚 =4𝐾𝑒𝑓𝑓[ℎ(1+𝛼2)] ⁄ , where ℎ is the Planck \nconstant, 𝐾𝑒𝑓𝑓 is the effective magnetic anisotropy constant and 𝛼 is the Gilbert damping \nparameter. While the high -f magnons experience stronger damping, low -f magnons possess a very \nlow group velocity, and hence, the majority of the thermally generated magnons become damped \non shorter length -scales17,18. Therefore, only the subthermal magnons, i.e., the low-f magnons   \n4 \n dominate the long -range thermo -spin transport19–21. Within this hypothesis, it was predicted that \n〈𝜉〉 is inversely proportional to both 𝛼 and √𝐾𝑒𝑓𝑓.17,18 \n \nY3Fe5O12 (YIG) has been a widely explored  MI for generating and transmitting pure spin \ncurrents due to its ultra -low damping  (𝛼 ≈ 10-4-10-5) and large  〈𝜉〉 (~100-200 nm) 11,13,17. This has \nled to a drastic  increase in research over the last few decade s, aimed at enhancing the spin current \ninjection efficiency across the MI/HM  interface  by reducing the conductivity mismatch between \nthe MI and HM layers  by introducing  atomically thin semiconducting interlayers22–28 and \nenhancing the  interfacial  spin-mixing conductance .29–31 Hariharan’s group has explored the roles \nof bulk and surface magnetic anisotrop ies in LSSE in different REIG -based MI/HM bilayers12,13, \nwhereas a recent study highlights  the influence of damping  on SPE and LSSE  in a compensated \nferrimagnetic  insulator .32 It has also been demonstrated that the LSSE in YIG/Pt bilayers varies \ninversely with intrinsic Gilbert damping of the YIG films, however, the LSSE coefficient does not \nshow any significant correlation with the enhanced damping due to SPE in YIG/Pt bilayers.33 All \nthese studies highlight the important role s of both magnetic anisotropy and Gilbert damping in \nthermally generated magnon propagation in MI/HM bilayers .  \n \nBy investigating  the YIG thickness dependence of the local LSSE measurements in YIG/Pt, \nGuo et al.34 determined the temperature ( T) dependence of 〈𝜉〉 and found a scaling behavior of \n〈𝜉〉 ∝ 𝑇−1. On the contrary , by employing non -local measurement geometries, Cornelissen et al. \ndemonstrated that the magnon diffusion length  for thermally driven magnonic spin currents (𝜆𝑡ℎ𝑚) \nof YIG decreases with decreasing temperature over a broad temperature range .35 Gomez -Perez et \nal. reported similar observation s and demonstrate d that the temperature dependen ce of  𝜆𝑡ℎ𝑚 is   \n5 \n independent of  the YIG thickness .19 The different trends of the temperature dependent \ncharacteristic critical length scales for thermally generated magnon propagation  in YIG observ ed \nby different groups indicate s distinct temperature evolutions of 𝛼 and 𝐾𝑒𝑓𝑓 in the YIG films grown \nby these groups. In other words, different thin film growth conditions and sample dependent \nchanges in the physical properties can give rise to different temperature dependences of both 𝛼 \nand 𝐾𝑒𝑓𝑓 and hence 〈𝜉〉. For the development of a complete physical picture of LSSE in these \nREIGs over a wide temperature range,  it is of utmost importance to comprehend  the respective \nroles of  both 𝛼 and 𝐾𝑒𝑓𝑓 simultaneously in determining the temperature evolution of  〈𝜉〉, which \nremains largely unexplored.   \n \nAlthough YIG is considered as a benchmark system for LSSE,11,34 there is only  a limited \nnumber of studies that explore temperature dependent LSSE in other iron garnets .12,32,36,37 For \nexample, Gd3Fe5O12 (GdIG) which is a ferrimagnetic insulator with magnetic compensation \ntemperature (𝑇𝐶𝑜𝑚𝑝 ) close to room temperature, shows a sign -inversion in the LSSE voltage12 as \nwell as in the spin -Hall anomalous Hall effect38 around its magnetic compensation. However, the \nGilbert damping in GdIG diverges over a broad temperature range around its  𝑇𝐶𝑜𝑚𝑝  which makes \nit difficult to probe the temperature evolution  of 𝛼 and its  contribution  towards 〈𝜉〉 over a wide \ntemperature range around the 𝑇𝐶𝑜𝑚𝑝 .32 Apart from YIG and GdIG , there has been a renaissance of \nresearch interest in another member of the REIG family: Tm 3Fe5O12 (TmIG) due to its  wide -\nranging extraordinary magnetic properties6 e.g., strain -tunable perpendicular magnetic anisotropy \n(PMA),39 chiral and topological spin textures,40 and interfacial Dzyaloshinskii -Moriya \ninteraction40,41 combined with low coercivity6 which make this system a promising candidate for \nnumerous efficient spintronic applications, such as spin -orbit torque induced magnetization   \n6 \n switching,8,42 current -induced domain -wall motion,43 and spin Hall –topological Hall effect s44,45. \nRecently, the LSSE has been investigated in TmIG/Pt bilayers with PMA at room temperature, \nand shown to exhibit high interfacial spin transparency and spin -to-charge conversion efficiency \nat the TmIG/Pt interface46. TmIG has a higher Gilbert damping parameter (≈10−2)6 compared to \nYIG, and unlike GdIG, TmIG does not exhibit any magnetic compensation in the temperature \nrange between 1.5 and 300 K47,48, which allows us to probe the relative contribution of 𝛼 towards  \nthe temperature evolution of  〈𝜉〉 and hence the LSSE  over a broad temperature range close to  the \nroom temperature . However, the temperature evolution of LSSE and hence 〈𝜉〉 as well as their \nrelation ship with 𝛼 and 𝐾𝑒𝑓𝑓 in TmIG/Pt bilayers are yet to be explored , which would be of critical \nimportance for REIG -based efficient magnonic device applications . Here , we have performed a \ncomprehensive investigation of the temperature -dependent LSSE, radio frequency (RF) transverse \nsusceptibility (TS), and broadband ferromagnetic resonance (FMR) of TmIG /Pt bilayers grown on \ndifferent substrates . From the TmIG thickness dependence of the LSSE voltage, we determined \nthe temperature dependence of 〈𝜉〉 and highlighted  its correlation with the temperature -dependent \neffective magnetic anisotropy field, 𝐻𝐾𝑒𝑓𝑓 (∝𝐾𝑒𝑓𝑓) and 𝛼 in TmIG/Pt  bilayers.  \n \n2. RESULTS AND DISCUSSION  \n2. 1. Structural Characterization  \nSingle -crystalline TmIG films with different thicknesses were grown on (111) -oriented \nGd3Sc2Ga3O12 (GSGG) and Gd 3Ga5O12 (GGG) substrates by pulsed laser deposition ( see \nMethods ). The high crystalline quality of the TmIG films was confirmed by X-ray diffraction \n(XRD). Figure 1 (a) shows the 𝜃−2𝜃 X-ray diffractograms of the GSGG/TmIG( 𝑡) films with \ndifferent TmIG film thickness  𝑡 (t = 236, 150, 89, 73, 46 and 28 nm) .    \n7 \n  \n \nFigure 1 . Structural and Morphological characterization.  (a) 𝜃−2𝜃 X-ray diffractogram  of \nthe GSGG/TmIG( 𝑡) films with different film thickness  𝑡 (t = 236, 150, 89, 73, 46 and 28 nm).  The \nreciprocal space maps recorded in the vicinity of the (642) reflection for (b) GSGG/TmIG( 30 nm) \nand (c) GSGG/TmIG(205 nm) films . For the thinner film (30  nm), the TmIG  film peak matches \nthe IP lattice constant of the GSGG substrate, whereas  for the thicker film (205nm), the TmIG film \nis largely relaxed.  \n \n The substrate choice and the TmIG film thickness influence  the strain state of the film . \nFigs. 1 (b) and ( c) show the reciprocal space maps in the vicinity of the (642) reflection for the \nGSGG/TmIG ( 30 nm) and GSGG/TmIG (205 nm) films, respectively . For the thinner film (30  \nnm), the TmIG film 𝑞𝑥 matches the in -plane (IP) lattice spacing  of the GSGG substrate  indicating \ncoherent growth , and the out -of-plane (OOP) lattice spacing is smaller than that of the substrate \n(higher 𝑞𝑍), consistent with the smaller unit cell volume for TmIG compared to GSGG. However,  \nthe thicker film (205  nm) is relaxed in plane with smaller IP and OOP lattice spacing than that of \nthe substrate, and its peak position is close to that of bulk TmIG. The 𝜃−2𝜃 scans show  a decrease \nin the OOP spacing (increase in 2𝜃) for thinner films . These trends are consistent with the TmIG \n  \n8 \n initially growing with an IP lattice match  to the substrate and hence a tensile IP strain (and a \nmagnetoelastic anisotropy favoring PMA), but the strain relaxes as the film thickness increases. \nThe thickest films, which are strain -relaxed, have a slightly higher OOP lattice spacing compared \nto bul k according to Fig. 1 (a) which suggests the presence of oxygen vacancies or Tm:Fe ratio \nexceeding 0.6, which can occur in thin films and raise the unit cell volume.  All the films show a \nsmooth surface morphology with a low root -mean -square roughness below 0.5nm, as visible in \natomic force microscopy (AFM) images for the GSGG/TmIG( 46nm), GGG/TmIG(44nm) and \nsGGG/TmIG(75nm) films shown in  the Supplementary Fig ure 1. \n \n A cross -section of an about 220 nm thick TmIG film on GSGG substrate, covered with \na 5 nm Pt layer, was analyzed by scanning transmission electron microscopy (STEM). Fig. 2 (a) \nshows a low magnification STEM image of the whole layer stack. An annular detector with a small \ncollector angle (24 -48 mrad) was used to highlight strain (Bragg) contrast over mass (Z) contrast \n49. The TmIG film shows columnar features attributed to strain contrast. An atomically resolved  \nSTEM image at the TmIG film -Pt interface ( Fig. 2 (b)) reveals a single crystalline TmIG film under \nthe polycrystalline Pt layer, with the bright spots indicating columns of Tm and Fe. The STEM \nimage of an area within the TmIG film close to the Pt interface shows the presence of a planar \ndefect in which s elected lattice planes are highlighted by colored lines in Fig. 2 (c). Such planar \ndefects could be associated with  partial dislocations or atomic level disorder, which are common \nin REIGs.50,51 \n \n   \n9 \n  \n \nFigure 2. Cross -sectional scanning transmission electron microscopy (STEM) analysis of the \nGSGG/TmIG(220 nm)/Pt(5nm) film.   (a) TEM image of the layer stack recorded by an annular \ndetector with a small collector angle (24 -48 mrad), highlighting strain (Bragg) contrast over mass \n(Z) contrast, (b) shows an atomic -resolution STEM image of the TmIG -Pt interface with [110] \nzone axis , while (c) shows an area within the TmIG film. The colored lines highlight a planar \ndefect . (d) electron energy loss spectroscopy ( EELS) scan at the Fe L3 and L2 edges. The measured \nenergy loss spectra are displayed as data points, exempl ified for positions close to  the garnet -\nsubstrate and garnet -Pt interfaces, with the fitted functions presented as colored lines. (e) The \nthickness dependent Fe L3 peak position and FWHM is extracted.  \n \n \n \n \n  \n10 \n 2. 2. Correlation between  Thermo -Spin Transport and Magnetism   \nFig. 3(a) shows the schematic illustration of our LSSE measurement  configuration . Simultaneous \napplication of a  vertical (+ z-axis) T-gradient ( 𝛁𝑻⃗⃗⃗⃗⃗ ) and an in-plane (x-axis) DC magnetic field \n(𝝁𝟎𝑯⃗⃗⃗⃗⃗⃗⃗⃗ ) across the TmIG fil m causes diffusion of thermally -excited magnons and develops a spatial \ngradient of magnon accumulation along the direction of 𝛁𝑻⃗⃗⃗⃗⃗ .52 The accumulated magnons close to \nthe TmIG/Pt  interface transfer spin angular momenta to the electrons of the adjacent Pt layer52. \nThe injected spin current density  is, 𝑱𝑺⃗⃗⃗ ∝−𝑆𝐿𝑆𝑆𝐸𝛁𝑻⃗⃗⃗⃗⃗ , where 𝑆𝐿𝑆𝑆𝐸 is the LSSE coefficient52,53. The \nspin current injected into the Pt layer along the z-direction is converted into a charge current , 𝑱𝑪⃗⃗⃗  =\n (2𝑒\nℏ)𝜃𝑆𝐻𝑃𝑡(𝑱𝑺⃗⃗⃗ × 𝝈𝑺⃗⃗⃗⃗⃗ ) along the y-direction via the inverse spin Hall effect (ISHE), where e, ℏ, 𝜃𝑆𝐻𝑃𝑡, \nand  𝝈𝑺⃗⃗⃗⃗⃗  are the electron ic charge, the reduced Planck’s constant , the spin Hall angle of Pt, and the \nspin-polarization vector, respectively . The corresponding LSSE voltage  is52,54,55  \n 𝑉𝐿𝑆𝑆𝐸= 𝑅𝑦𝐿𝑦𝐷𝑃𝑡(2𝑒\nℏ)𝜃𝑆𝐻𝑃𝑡| 𝐽𝑆|tanh(𝑡𝑃𝑡\n2𝐷𝑃𝑡),    (1) \nwhere, 𝑅𝑦,𝐿𝑦,𝐷𝑃𝑡,and 𝑡𝑃𝑡 represent the electrical resistance between the contact -leads, t he \ndistance between the contact -leads, the spin  diffusion length of Pt, and the Pt layer  thickness , \nrespectively .  \n \nFig. 3(b) shows the magnetic field  (H) dependen t ISHE  voltage,  𝑉𝐼𝑆𝐻𝐸(𝐻) for \nGSGG/TmIG(236  nm)/Pt (5 nm) for different values of the temperature  difference between the hot \n(𝑇ℎ𝑜𝑡) and cold ( 𝑇𝑐𝑜𝑙𝑑) blocks, ∆𝑇=(𝑇ℎ𝑜𝑡−𝑇𝑐𝑜𝑙𝑑), at a fixed average sample temperature 𝑇=\n 𝑇ℎ𝑜𝑡+𝑇𝑐𝑜𝑙𝑑\n2 = 295 K. For all Δ𝑇, 𝑉𝐼𝑆𝐻𝐸(𝐻) exhibit s a nearly square -shaped hysteresis  loop. The inset \nof Fig. 3(b) plots  the ∆𝑇-dependence of the background -corrected LSSE voltage,  𝑉𝐿𝑆𝑆𝐸(Δ𝑇)=  \n11 \n  [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑠𝑎𝑡,Δ𝑇)−𝑉𝐼𝑆𝐻𝐸(−𝜇0𝐻𝑠𝑎𝑡,Δ𝑇)\n2], where 𝜇0𝐻𝑠𝑎𝑡 is the  saturation field . Clearly,  𝑉𝐿𝑆𝑆𝐸 increases  \nlinearly with ∆𝑇 as expected from Eqn. 1 .12  \n \nFigure 3. Magnetism and longitudinal spin Seebeck effect (LSSE) in \nGSGG/TmIG(236nm)/Pt(5nm) film.  (a) Schematic illustration of the experimental configuration \nfor LSSE measurements.  A temperature gradient ( 𝛁𝑻⃗⃗⃗⃗⃗ ) is applied along the + z axis and an in -plane \n(IP) dc magnetic field ( 𝝁𝟎𝑯⃗⃗⃗⃗⃗⃗⃗⃗ ) is applied along the + x axis. The inverse spin Hall effect (ISHE) \ninduced voltage ( 𝑉𝐼𝑆𝐻𝐸) is measured along the y-axis. (b) 𝑉𝐼𝑆𝐻𝐸(𝐻) loops for different values of \nthe temperature difference ∆𝑇 at a fixed average sample temperature 𝑇 = 295 K. The inset shows \na linear ∆𝑇-dependence of the background -corrected LSSE voltage. (c) 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops \nmeasured at selected temperatures in the range 120 K ≤ T ≤ 295 K for Δ𝑇 = +10 K. (d) The IP \nM(H) hysteresis loops at selected temperatures.  \n \n  \n12 \n Fig. 3 (c) shows the 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops for GSGG/TmIG(236 nm)/Pt(5 nm) \nmeasured at selected temperatures for Δ𝑇= +10 K. Clearly, |𝑉𝐼𝑆𝐻𝐸(𝜇0𝐻𝑠𝑎𝑡)| significantly \ndecreases, and the hysteresis loop broadens at low temperatures, especially below 200 K. To \ncorrelate thermo -spin transport with the bulk magnetic properties, in Fig. 3 (d), we show the \nmagnetic field dependence of magnetization, 𝑀(𝐻) at selected temperatures for GSGG/TmIG(236 \nnm)/Pt(5 nm) measured while scanning an in -plane (IP) magnetic field. It is evident that with \nlowering the temperature, the saturation magnetization (𝑀𝑆) decreases and t he coercivity ( 𝐻𝐶) \nincreases with a corresponding increase in the magnetic anisotropy, especially below 200 K. This \nobservation is also in agreement with the T-dependent magnetic force microscopy (MFM) results \nshown in Supplementary Figure 2 , which clearly reveals that the root mean square (RMS) value \nof the phase shift, Δ𝜙𝑅𝑀𝑆 decreases significantly between 300 and 150 K indicating changes in the \nmagnetic domain structure at low -T. \n \nThe decrease in 𝑀𝑆 at low -T is well-known in TmIG48,56 and is a result of the increasing \nmoment of the Tm3+ ion at low -T, which competes with the net moment of the Fe3+ ions ( i.e., the \ndodecahedral Tm3+ moment opposes the net moment of the tetrahedral and octahedral Fe3+ \nmoments).  Based on the molecular -field-coefficient theory developed by Dionne57, we have \nperformed molecular -field simulations58,59 to determine 𝑀𝑆(T) for TmIG ( see Supplementary  \nFigure 3(n)) which is consistent with our experimental observation of the decrease in 𝑀𝑆 at low -\nT. It is apparent from Figs. 3 (c) and (d) that the temperature evolution of 𝑉𝐼𝑆𝐻𝐸 signal follows that \nof 𝑀𝑆. To further explore the correlation between 𝑉𝐼𝑆𝐻𝐸 and 𝑀𝑆, magnetometry and LSSE \nmeasurements were repeated on the GSGG/TmIG( t)/Pt(5 nm) sample series  with different TmIG \nfilm thicknesses (28 nm≤𝑡≤236 nm). Films with  46 nm≤𝑡≤236 nm possess IP easy -axes   \n13 \n while the 28 nm film has an OOP easy -axis of magnetization , which  was confirmed via IP -\nmagnetometry and OOP p-MOKE measurements (see Supplementary  Figure 3(e)). The total \nmagnetic anisotropy of a (111) -oriented TmIG fi lm, neglecting growth and interfacial anisotropies,  \nhas contributions from shape anisotropy ( 𝐾𝑠ℎ𝑎𝑝𝑒), cubic magnetocrystalline anisotropy ( 𝐾𝑚𝑐), and \nmagnetoelastic anisotropy ( 𝐾𝑚𝑒)47,49,60 i.e., 𝐾𝑒𝑓𝑓=𝐾𝑠ℎ𝑎𝑝𝑒+𝐾𝑚𝑐 + 𝐾𝑚𝑒=−1\n2 𝜇0𝑀𝑆2−𝐾1\n12−\n9\n4𝜆111𝑐44(𝜋\n2−𝛽), where K1 is the magnetocrystalline anisotropy coefficient, 𝜆111 is the \nmagnetostriction along the [111] direction,  𝑐44 is the shear modulus and 𝛽 is the cor ner angle of \nthe rhombohedrally -distorted unit cell. For a negative magnetostriction ( 𝜆111  = −5.2×10−6 for \nbulk TmIG47), the tensile IP  strain, which results from the difference in lattice parameters \n(𝑎𝐺𝑆𝐺𝐺=12.57 Å and 𝑎𝑇𝑚𝐼𝐺=12.32 Å) promotes PMA ( 𝐾𝑒𝑓𝑓>0).49,60,61  PMA is expected for \nfully -strained films (28  nm), but strain -relaxation in thicker films reduces the magnetoelastic \ncontribution , and the easy -axis reorients to IP  direction60. \n \nFigs. 4 (a) and (b) depict the  𝑉𝐼𝑆𝐻𝐸(𝐻) loop on the left y-scale and corresponding 𝑀(𝐻) \nloop on the right y-scale at 295 K for the thicknesses: 𝑡=236 and 28 nm,respectively . The \n𝑀(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis -loops for all other  thicknesses are shown in the Supplementary \nFigures 3 and 4. Clearly, the 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis -loops for all the thicknesses mimic the \ncorresponding 𝑀(𝐻) loops. Note that, unlike YIG -slab, there is no surface magnetic anisotropy \ninduced anomalous low field feature in the 𝑉𝐼𝑆𝐻𝐸(𝐻) loop for any of our TmIG thin films. This is \npossibly because the thickness of the TmIG films is smaller than their average magnetic domain \nsize62. This is why the YIG thin films also do not show any low field anomalous feature in the \n𝑉𝐼𝑆𝐻𝐸(𝐻) loops52. Additionally, the 𝑉𝐼𝑆𝐻𝐸(𝐻) loop for our TmIG film with PMA ( the 28 nm film)  \nat 295 K is quite similar to that of a TmIG thin film with PMA at room temperature reported in the   \n14 \n literature46. In Figs. 4 (c) and (d), we demonstrate the T-dependence of the background -corrected \nLSSE voltage,  𝑉𝐿𝑆𝑆𝐸(𝑇)=𝑉𝐼𝑆𝐻𝐸(𝑇,+𝜇0𝐻𝑠𝑎𝑡)−𝑉𝐼𝑆𝐻𝐸(𝑇,−𝜇0𝐻𝑠𝑎𝑡)\n2 for Δ𝑇= +10 K  on the left y-scale and \ncorresponding 𝑀𝑆(𝑇) on the right y-scale for GSGG/TmIG( 236 nm )/Pt(5 nm) and \nGSGG/TmIG( 28 nm )/Pt(5 nm) , respectively. Interestingly, 𝑉𝐿𝑆𝑆𝐸(𝑇) and 𝑀𝑆(𝑇) for both films \ndrop remarkably below the T-window of 180 -200 K. We observed a similar trend in 𝑉𝐿𝑆𝑆𝐸(𝑇) and \n𝑀𝑆(𝑇) for all GSGG/TmIG( t)/Pt(5 nm) films with other thicknesses (see Supplementary Figures \n3 and 4 ). These results indicate that this behavior is intrinsic to TmIG.  \n \nFigure 4. Longitudinal spin Seebeck effect in GSGG/TmIG( t)/Pt(5  nm) films.  The 𝑉𝐼𝑆𝐻𝐸(𝐻) \nhysteresis loops on the left y-scale and the IP 𝑀(𝐻) loops on the right y-scale at  T = 295 K for \nGSGG/TmIG( t)/Pt films for t = (a) 236 nm, and (b) 28 nm . The temperature dependence of the \nbackground -corrected LSSE voltage,  𝑉𝐿𝑆𝑆𝐸(𝑇) on the left y-scale and temperature dependence of \nsaturation magnetization, 𝑀𝑆(𝑇) on the right y-scale for the GSGG/TmIG( t)/Pt(5 nm) films  for t = \n(c) 236 nm and (d) 28 nm, for Δ𝑇 = +10 K . \n  \n15 \n Note that the yellow and grey background colors in all the  graphs throughout the \nmanuscript are used to highlight significant changes in physical parameters between  high (yellow)  \nand low (grey) temperature regions. Additionallly, we have used the sky blue background color in \nsome of the specific graphs (especially temperature dependence of 𝑉𝐿𝑆𝑆𝐸(𝑇) and 𝑀𝑆(𝑇)) to \nindicate considerable changes in the corresponding physical parameters occurring around the \nnarrow temperature window: 180 K ≤𝑇 ≤200 K. However, we have used a gradual transition \nfrom yellow to grey background in rest of the graphs where the changes in the physical parameters \nare less significant in the temperature window: 180 K ≤𝑇 ≤200 K. \n \nNext, we discuss  the additional voltage contribution s due to the magnetic proximity  effect \n(MPE) -induced anomalous Nernst effect (ANE) as well as MPE –induced LSSE in the Pt layer. \nThe MPE  leads to  a magnetic moment in a few atomic layers of Pt close to the TmIG/Pt \ninterface.63,64 In the presence of a vertical temperature gradient, a tra nsverse voltage is generated \nin the proximitized Pt layer  due to ANE which adds to the LSSE voltage.  Furthermore, due to the \ntemperature gradient, spin currents are generated inside the magnetized Pt layer, which induces an \nadditional IP charge current at the proximitized Pt/nonmagnetic Pt interface via the ISHE and \ntherefore contributes to the LSSE signa l.65 In an earlier study, Bougiatioti et al.,63 showed that the \nMPE -induced ANE in the proximitized Pt layer is only significant for a conducting FM/Pt bilayer \nbut negligible for semiconducting FM/Pt bilayers and becomes zero for insulating FM/Pt bilayers. \nSince TmIG is insulating, the contribution of the MPE -induced ANE in the proximitized Pt layer \ntowards the total LSSE signal can be neglected throughout the measured temperature range66. \nFurthermore , since  the LSSE voltage decreases  with decreasing thickness of the magnetic layer ,16 \nand the thickness of the proximitized Pt layer is very small, the MPE -induced LSSE contribution   \n16 \n due to the proximitized Pt layer  can also be neglected66. Therefore, the total voltage measured \nacross the TmIG/Pt bilayers is considered to be solely contributed by the intrinsic LSSE of the \nTmIG films.  \n \n2. 3. Analysis of the Thickness Dependent Longitudinal Spin Seebeck Effect  \nTo ascertain the origin of the decrease in 𝑉𝐿𝑆𝑆𝐸 below 180 -200 K in our TmIG  films, it is \nessential to determine the temperature evolution of 〈𝜉〉 which signifies the critical length -scale for \nthe thermally -generated magnons of a magnetic thin film16,18,34. For an effective determination of \nthe temperature dependence of 〈𝜉〉, the contributions of the the thermal resistances of the substrate \nand the grease layers as well as the interfacial thermal resistances need to be considered.67 To \nquantify the temperature evolution of 〈𝜉〉 for our TmIG/Pt bilayer films, we have employed a \nmodel proposed by Jimenez -Cavero et al.,68 according to which t he total temperature difference \n(Δ𝑇) across the GSGG/TmIG/Pt heterostructure can be expressed as a linear combination of \ntemperature drops in the Pt layer, at the TmIG/Pt interface, in the TmIG  layer, at the GSGG/TmIG  \ninterface and across the GSGG substrate  as well as in the N -grease layers (thickness ≈ 1 m) on \nboth sides of the GSGG/TmIG/Pt heterostructures as ,68 ∆𝑇= ∆𝑇𝑃𝑡+∆𝑇Pt\nTmIG+∆𝑇TmIG+\n ∆𝑇TmIG\nGSGG+∆𝑇GSGG+2.∆𝑇N−Grease  (see Fig. 5 (a)). Assuming negligible drops in ∆𝑇 in the Pt layer \nand at the GSGG/TmIG  and Pt/TmIG  interface ,68,69 the total temperature difference can be \napproximately written as, ∆𝑇= ∆𝑇Pt\nTmIG+∆𝑇TmIG+∆𝑇GSGG+2.∆𝑇𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 . Considering these \ncontributions, the temperature drops in the TmIG layer and at the TmIG/Pt interface can be written \nas,68,69 ∆𝑇TmIG = 𝛥𝑇\n [1+𝜅𝑇𝑚𝐼𝐺\n𝑡𝑇𝑚𝐼𝐺(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒\n𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝐺𝑆𝐺𝐺\n𝜅𝐺𝑆𝐺𝐺)] and ∆𝑇Pt\nTmIG= [(𝜅𝐺𝑆𝐺𝐺𝜅𝑇𝑚𝐼𝐺)𝑅𝑖𝑛𝑡\n(𝜅𝑇𝑚𝐼𝐺𝑡𝐺𝑆𝐺𝐺+𝜅𝐺𝑆𝐺𝐺𝑡𝑇𝑚𝐼𝐺)]𝛥𝑇, \nrespectively. The bulk (𝑉𝐿𝑆𝑆𝐸𝑏)and interfacial (𝑉𝐿𝑆𝑆𝐸𝑖) contributions to the LSSE voltage can then   \n17 \n be expressed as, 𝑉𝐿𝑆𝑆𝐸𝑏=𝑆𝐿𝑆𝑆𝐸𝑏.∆𝑇TmIG.𝐿𝑦=\n[(𝐴\n𝑡𝑇𝑚𝐼𝐺){cosh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)−1\nsinh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)}]{𝛥𝑇\n[1+𝜅𝑇𝑚𝐼𝐺\n𝑡𝑇𝑚𝐼𝐺(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒\n𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝐺𝑆𝐺𝐺\n𝜅𝐺𝑆𝐺𝐺)]}𝐿𝑦 and 𝑉𝐿𝑆𝑆𝐸𝑖=𝑆𝐿𝑆𝑆𝐸𝑖.∆𝑇Pt\nTmIG.𝐿𝑦=\n𝑆𝐿𝑆𝑆𝐸𝑖.[(𝜅𝐺𝑆𝐺𝐺𝜅𝑇𝑚𝐼𝐺)𝑅𝑖𝑛𝑡𝛥𝑇\n(𝜅𝑇𝑚𝐼𝐺𝑡𝐺𝑆𝐺𝐺+𝜅𝐺𝑆𝐺𝐺𝑡𝑇𝑚𝐼𝐺)]𝐿𝑦, respectively. Here,  𝑆𝐿𝑆𝑆𝐸𝑏=[(𝐴\n𝑡𝑇𝑚𝐼𝐺){cosh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)−1\nsinh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)}] and 𝑆𝑖𝑛𝑡 \ndenote  the bulk and interfacial LSSE coefficient s for TmIG and TmIG/Pt interface , respectively,  \n𝑡𝑇𝑚𝐼𝐺(𝑡𝐺𝑆𝐺𝐺) is the thickness of TmIG film (GSGG substrate), 𝜅𝑇𝑚𝐼𝐺  and 𝜅𝐺𝑆𝐺𝐺 are the thermal \nconductivity  of TmIG and GSGG respectively, 𝑡N−Grease  and 𝜅N−Grease  are the thickness and \nthermal conductivity of the N -grease layers, 𝑅𝑖𝑛𝑡 is the interfacial thermal -resistance at the \nTmIG/Pt interface  and 𝐴 is a constant .68. The approximate values of 𝜅N−Grease , 𝜅𝑇𝑚𝐼𝐺  and 𝜅𝐺𝑆𝐺𝐺 \nat different temperatures are obtained from the literature70–75. Note that , we have ignored the \ninterfacial thermal resistances between the N -grease and the hot/cold plates as well as between the \nsample and N -grease layers .76 Threfore , the total LSSE voltage across GSGG/TmIG/Pt can be \nexpressed as,68 \n𝑉𝐿𝑆𝑆𝐸(𝑡𝑇𝑚𝐼𝐺)= 𝑉𝐿𝑆𝑆𝐸𝑖(𝑡𝑇𝑚𝐼𝐺)+𝑉𝐿𝑆𝑆𝐸𝑏(𝑡𝑇𝑚𝐼𝐺)= [𝑆𝑖𝑛𝑡{(𝜅𝐺𝑆𝐺𝐺𝜅𝑇𝑚𝐼𝐺)𝑅𝑖𝑛𝑡\n(𝜅𝑇𝑚𝐼𝐺𝑡𝐺𝑆𝐺𝐺+𝜅𝐺𝑆𝐺𝐺𝑡𝑇𝑚𝐼𝐺)}𝐿𝑦𝛥𝑇+\n[(𝐴\n𝑡𝑇𝑚𝐼𝐺){cosh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)−1\nsinh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)}]{𝛥𝑇\n[1+𝜅𝑇𝑚𝐼𝐺\n𝑡𝑇𝑚𝐼𝐺(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒\n𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝐺𝑆𝐺𝐺\n𝜅𝐺𝑆𝐺𝐺)]}𝐿𝑦]     (2) \n \nIn Fig. 5 (b), we demonstrate the 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops at T = 295 K for the \nGSGG/TmIG/Pt films with different 𝑡𝑇𝑚𝐼𝐺  in the range 28 nm≤𝑡≤236 nm. Clearly, \n|𝑉𝐼𝑆𝐻𝐸(𝜇0𝐻𝑠𝑎𝑡)| decreases significantly with decreasing 𝑡𝑇𝑚𝐼𝐺 . Therefore, we fitted the thickness \ndependent LSSE voltage at different temperatures with Eqn. 2  to evaluate the temperature \ndependence of 〈𝜉〉 for our GSGG/TmIG/Pt films. It has recently been shown77 that the 𝑀𝑆 also   \n18 \n needs be considered to evaluate 〈𝜉〉 from the LSSE voltage by normalizing the LSSE voltage by \n𝑀𝑆. In Fig. 5 (c), we show the thickness -dependence of the background -corrected modified LSSE \nvoltage,𝑉𝐿𝑆𝑆𝐸(𝑡𝑇𝑚𝐼𝐺)\n∆𝑇.𝑀𝑆, at selected temperatures fitted to Eqn. 2 . From the fits, we obtained 〈𝜉〉 = 62 \n± 5 nm  for the TmIG film at 295 K, which is smaller than that of YIG thin films  grown by PLD  \n(90–140 nm)16, but higher than that for GdIG  thin films (45±8 nm)12.  \n \nFigure 5. Thickness D epende nt LSSE and Magnon Propagation L ength in \nGSGG/TmIG( t)/Pt(5  nm) films.  (a) Schematic illustration of heat flow across the \nGSGG/TmIG( t)/Pt(5 nm) films.  (b) The 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops for GSGG/TmIG( t)/Pt films \nwith different thicknesses at T = 295 K for Δ𝑇 = +10 K . (c) The thickness dependence of the \nnormalized background corrected LSSE voltage,  𝑉𝐿𝑆𝑆𝐸(𝑡)Δ𝑇.𝑀𝑆⁄  at three selected temperatures \nT = 295, 200 , and 140 K fitted with Eqn. (2) . (d) The temperature dependence of the magnon \npropagation length, 〈𝜉〉 obtained from the fits.  \n  \n19 \n Fig. 5 (d) demonstrates the T-dependence of 〈𝜉〉 obtained from the fit of 𝑉𝐿𝑆𝑆𝐸(𝑡𝑇𝑚𝐼𝐺)\n∆𝑇.𝑀𝑆 for \nGSGG/TmIG( t)/Pt(5nm)  films . Interestingly,  〈𝜉〉 decreases gradually with decreasing temperature \nand shows a comparatively faster decrease at low temperatures, especially below 200 K. Our \nobservation is strikingly different than that reported by Guo et al.34 for YIG/Pt bilayers. From the \nYIG thickness dependence of the local LSSE measurements in YIG/Pt, they determined the \ntemperature dependence of 〈𝜉〉 and found a scaling behavior of 〈𝜉〉 ∝ 𝑇−1.34 However, by \nemploying nonlocal measurement geometries, Cornelissen et al., demonstrated that 〈𝜉〉 (and hence, \nthe magnon diffusion length) for YIG/Pt decreases with decreasing temperature over a broad \ntemperature range35, similar to what we observed in our TmIG/Pt bilayers. Gomez -Perez et al.,19 \nalso observed similar behavior of the magnon diffusion length in YIG/Pt. However, none of these \nstudies indicated significant change in 〈𝜉〉 at low temperatures . Therefore, the observed \ntemperature evolution of 〈𝜉〉 presented in this study is intrinsic to TmIG. To rule out possible \neffects of strain o n 𝑉𝐿𝑆𝑆𝐸(𝑇), we performed LSSE measurements on TmIG films grown on \ndifferent substrates  (see Supplementary  Figures 5 and 6). It is evident that 𝑀𝑆(𝑇) and 𝑉𝐿𝑆𝑆𝐸(𝑇) \nfor the Gd3Ga5O12(GGG) /TmIG(44  nm)/Pt (5 nm) and \n(Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12(sGGG )/TmIG(40  nm)/Pt (5 nm) films (see Supplementary \nFigure 7) exhibit the same trend as GSGG/TmIG(46  nm)/Pt(5  nm). More specifically , both  \n𝑉𝐿𝑆𝑆𝐸(𝑇) and 𝑀𝑆(𝑇) drop be low 180 -200 K for all the TmIG films independent  of substrate choice.  \n \nTo interpret  the decrease in 〈𝜉〉 at lo w temperatures , we recall that 〈𝜉〉 of a magnetic \nmaterial with lattice constant 𝑎0 (considering simple cubic structure) is related to the Gilbert \ndamping parameter ( 𝛼), the effective magnetic anisotropy constant ( 𝐾𝑒𝑓𝑓), and the strength of the   \n20 \n Heisenberg exchange interaction between nearest neighbors ( 𝐽𝑒𝑥) through the relation17,18 〈𝜉〉=\n 𝑎0\n2𝛼.√𝐽𝑒𝑥\n2𝐾𝑒𝑓𝑓. As discussed before,  𝐾𝑒𝑓𝑓=𝐾𝑚𝑒−1\n2 𝜇0𝑀𝑆2−𝐾1\n12. Therefore, we can express 〈𝜉〉 as, \n〈𝜉〉= 𝑎0\n2𝛼.√𝐽𝑒𝑥\n2(𝐾𝑚𝑒−𝐾1\n12−1\n2 𝜇0𝑀𝑆2)                                 (3) \nEqn. 3 indicates that  (i) 〈𝜉〉∝ (1\n𝛼), and (ii) a decrease in 𝑀𝑆 also suppresses 〈𝜉〉. Since the \neffective anisotropy field, 𝐻𝐾𝑒𝑓𝑓 ∝𝐾𝑒𝑓𝑓, Eqn. 3 can be alternatively written as 〈𝜉〉=𝑎0\n2𝛼.√𝐽𝑒𝑥\n2𝐾𝑒𝑓𝑓 ∝\n1\n𝛼.(𝐻𝐾𝑒𝑓𝑓)1/2  , which indicates  that 〈𝜉〉 is inver sely proportional to the square -root of 𝐻𝐾𝑒𝑓𝑓. This \nimplie s that  the temperature evolution of  〈𝜉〉 is intrinsically dependent on both the physical \nquantities:  𝛼 and 𝐻𝐾𝑒𝑓𝑓. To determine the roles of 𝛼 and 𝐻𝐾𝑒𝑓𝑓in the temperature evolution of 〈𝜉〉, \nwe have performed radio frequency (RF) transverse susceptibility (TS) and broadband \nferromagnetic resonance (FMR) measurements, respectively on the TmIG films, which have been \ndiscussed in the following sections.  \n \n2. 4. Radio Frequency Transverse Susceptibility and Magnetic Anisotropy  \nRF TS measurements were performed  to determine the  temperature evolution of 𝐻𝐾𝑒𝑓𝑓 in \nthe TmIG films. The  magnetic field dependence ( 𝐻𝐷𝐶) of TS, 𝜒𝑇(𝐻𝐷𝐶), is known to exhibit \npeaks/cusps at the effective anisotropy fields, ±𝐻𝐾𝑒𝑓𝑓.78,79 The schematic illustration of our TS \nmeasurement configuration is shown in Fig. 6 (a). T he RF magnetic field, HRF is parallel to the film \nsurface and 𝐻𝐷𝐶 points perpendicular  to it. All the TS data in this paper are presented as the \nrelative change in  𝜒𝑇(𝐻𝐷𝐶), which we define as ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶)=𝜒𝑇(𝐻𝐷𝐶)−𝜒𝑇(𝐻𝐷𝐶=𝐻𝐷𝐶𝑠𝑎𝑡)\n𝜒𝑇(𝐻𝐷𝐶𝑠𝑎𝑡), where \n𝜒𝑇(𝐻𝐷𝐶= 𝐻𝐷𝐶𝑠𝑎𝑡) is the value of 𝜒𝑇(𝐻𝐷𝐶) at the saturation field (  𝐻𝐷𝐶𝑠𝑎𝑡). Bipolar field -scans of   \n21 \n ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) for the GSGG/TmIG( 236 nm )/Pt film at 295 and 100 K are shown in Fig. 6 (b), which \nclearly indicates an increase in 𝐻𝐾𝑒𝑓𝑓 at low -T.  \n \nFigure 6. RF Transverse Susceptibility and Magnetic Anisotropy in GSGG/TmIG( t)/Pt(5  \nnm) films. (a)  The schematic illustration of our RF transverse susceptibility measurement. (b) \nComparison of the bipolar field scans ( +𝐻𝐷𝐶𝑚𝑎𝑥→−𝐻𝐷𝐶𝑚𝑎𝑥→+𝐻𝐷𝐶𝑚𝑎𝑥) of transverse susceptibility at \nT = 295 and 100 K for the GSGG/TmIG( 236 nm)/Pt film measured with  configuration 𝐻𝐷𝐶⊥\nfilm surface  (IP easy axis) . (c) Fitting of our ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) data for the GSGG/TmIG( 236nm )/Pt film \nat 295 K with the Eqn. 4 . (d) Temperature dependence of the effective anisotropy field ( 𝐻𝐾𝑒𝑓𝑓) for \nthe GSGG/TmIG(236  nm)/Pt(5  nm) film  obtained from the transverse susceptibility (TS) \nmeasurements on the left y-scale and corresponding 𝑉𝐿𝑆𝑆𝐸(𝑇) for the same film on the right y-scale.  \n \n  \n22 \n For an accurate determination of 𝐻𝐾𝑒𝑓𝑓from the field dependent TS curves, we fitted the \nline shapes for the TS curves with the following expression,79,80  \n∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶)= ∆𝜒𝑆𝑦𝑚(∆𝐻\n2)2\n(𝐻𝐷𝐶−𝐻𝐾𝑒𝑓𝑓)2\n+(∆𝐻\n2)2+∆𝜒𝐴𝑠𝑦𝑚∆𝐻\n2(𝐻𝐷𝐶−𝐻𝐾𝑒𝑓𝑓)\n(𝐻𝐷𝐶 −𝐻𝐾𝑒𝑓𝑓)2\n+(∆𝐻\n2)2+∆𝜒0    (4) \nwhere, ∆𝐻 is the linewidth of the ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) spectrum,  ∆𝜒𝑆𝑦𝑚 and ∆𝜒𝐴𝑠𝑦𝑚  are the coefficients of \nsymmetric and antisymmetric Lorentzian functions and ∆𝜒0 is the constant offset parameter. Fig. \n6(c) shows the fitting of our ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) data for the GSGG/TmIG( 236 nm )/Pt film at 295 K with \nthe Eqn. 4 . As shown on the left y-scale of Fig. 6 (d), 𝐻𝐾𝑒𝑓𝑓(𝑇) increases throughout the measured \ntemperature range but the increase in 𝐻𝐾𝑒𝑓𝑓 is comparatively faster below the temperature range: \n180-200 K, which coincides with the remarkable drop in 𝑉𝐿𝑆𝑆𝐸. Similar behavior was also \nobserved for other film thicknesses (see Supplementary Figure 8 ). \n \nA significant increase  in magnetocrystalline anisotropy at low -T has been  reported in \nvarious REIGs, which was interpreted  in the framework of the single -ion anisotropy model \nconsidering the collective influence of the crystal and exchange fields of the REIG  on the energy \nlevels of the individual magnetic ions81. Typically,  𝐾1 increases by ≈ 80 -100% between 300 and \n150 K in most of the REIGs .81 Furthermore,  𝜆111 for TmIG increases from −5.2 ×10−6 at 300 K \nto −17.4 ×10−6 at 150 K which gives rise to enhanced contribution of  𝐾𝑚𝑒 towards 𝐾𝑒𝑓𝑓 in \nTmIG films at low temperatures.82,83 Shumate Jr. et al.,84 observed a rapid increase in 𝐻𝐾𝑒𝑓𝑓 and \ncoercive field at low temperatures in mixed REIGs. An increase in 𝐻𝐾𝑒𝑓𝑓and a corresponding \ndecrease in 𝑉𝐿𝑆𝑆𝐸 below 175 K was also observed in YIG/Pt13, which was attributed  to the single -\nion anisotropy of Fe2+ ions85. To gain knowledge on the oxidation state of Fe in our TmIG films,   \n23 \n electron energy loss spectroscopy (EELS) was conducted during the cross -sectional TEM study \ndescribed earlier. Fig. 2 (d) shows two EELS spectra, recorded at the Fe L3 and L2 edges, and at \npositions close to the film -substrate and the film -Pt interface. The spectra are fitted following86,87, \nshown by colored lines, using a Gauß ian profile and a combination of a power -law background \nand a double -step function (arctangent) with a fixed step -ratio. Fig. 2 (e) shows the extracted \nthickness -dependent Fe L3 peak position alongside the corresponding FWHM. While an exact \nquantification of the Fe oxidation state distribution using the EELS Fe L3 peak position or L3/L2 \nwhite -line ratio is challenging, the presence of different oxidation states can be  indicated \nqualitatively by a shift in the peak position  because  Fe2+ ions contribute at slightly lower energies \ncompared to Fe3+ ions86–89. However, in our measured spectra, a constant peak position at about \n710.1 eV and a constant FWHM of about 2.3 eV across the whole film thickness is observed. Our \nobservation strongly hints at the presence of only one Fe oxidation state, namely the Fe3+ ion and \nhence, we can rule out the contribution of single ion anisotropy of Fe2+ ions towards the increased \nmagnetic anisotropy. This is also in agreement with recent studies on Tb-rich TbIG thin films58,90 \nwhich reveal very low Fe2+ ion concentrations . Therefore, the increase in 𝐻𝐾𝑒𝑓𝑓 below 200  K in the \nTmIG films may arise from single -ion anisotrop ies of the Tm3+ and Fe3+ ions81 as well as  from the \nenhanced contributions of 𝐾1 and 𝐾𝑚𝑒 towards 𝐾𝑒𝑓𝑓 at low temperatures81–83.  \n \n2. 5. Magnetization Dynamics and Broadband Ferromagnetic Resonance  \nNext, we examine the temperature evolution of 𝛼 and its influence on 〈𝜉〉 through \nbroadband IP FMR measurements. Fig. 7 (a) shows the field -derivative of the microwave (MW) \npower absorption spectra (𝑑𝑃\n𝑑𝐻) as a function of the IP DC magnetic field for a fixed frequency f  = \n12 GHz at selected temperatures for the GSGG/ TmIG( 236nm ) film . As temperature decreases, the   \n24 \n (𝑑𝑃\n𝑑𝐻) lineshape noticeably broadens and the resonance field 𝐻𝑟𝑒𝑠 shifts to higher field values. The \nlinewidth of the (𝑑𝑃\n𝑑𝐻) lineshape becomes so broad at low temperatures that we were unable to \ndetect the FMR signal below 160 K. We observed the same behavior for the GSGG/TmIG( 236 \nnm)/Pt(5 nm)  film, as shown in the Supplementary Figure 9 . Fig. 7 (b) shows the (𝑑𝑃\n𝑑𝐻) lineshapes \nfor the GSGG/TmIG( 236 nm ) film for different frequencies in the range 6 GHz ≤𝑓 ≤20 GHz \nat 295K fitted with a linear combination of symmetric and antisymmetric Lorentzian function \nderivatives as,91 \n   𝑑𝑃\n𝑑𝐻= 𝑃𝑆𝑦𝑚∆𝐻\n2(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)\n[(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2\n]2+𝑃𝐴𝑠𝑦𝑚(∆𝐻\n2)2\n−(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2\n[(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2\n]2+𝑃0               (5) \nwhere, 𝐻𝑟𝑒𝑠 is the resonance field, ∆𝐻 is the linewidth of the 𝑑𝑃\n𝑑𝐻 lineshapes, 𝑃𝑆𝑦𝑚 and \n𝑃𝐴𝑠𝑦𝑚  are the coefficients of the symmetric and antisymmetric Lorentzian derivatives, respectively, \nand 𝑃0 is a constant offset parameter.  The fitted curves are shown by solid lines in Fig. 7 (b). Using \nthe values of 𝐻𝑟𝑒𝑠 obtained from the fitting of the 𝑑𝑃\n𝑑𝐻 lineshapes,  we fitted the f-𝐻𝑟𝑒𝑠 curves at \ndifferent temperatures using the Kittel equation for magnetic thin films with IP magnetic field,92 \nwhich is expressed as 𝑓= 𝛾𝜇0\n2𝜋√𝐻𝑟𝑒𝑠(𝐻𝑟𝑒𝑠+𝑀𝑒𝑓𝑓), where  𝑀𝑒𝑓𝑓 is the effective magnetization, \n𝛾\n2𝜋= 𝑔𝑒𝑓𝑓 𝜇𝐵\nℏ is the gyromagnetic ratio, 𝜇𝐵 is the Bohr magneton, 𝑔𝑒𝑓𝑓 is the effective Landé g-\nfactor, and ℏ is the reduced Planck’s constant. Fig. 7 (d) demonstrates the fitting of the f-𝐻𝑟𝑒𝑠 \ncurves at T = 295, 200, and 160 K. We found that 𝑔𝑒𝑓𝑓 = 1.642 ± 0.002 at T = 295 K for our \nGSGG/TmIG( 236 nm ) film, which is significantly lower than that of the free electron value ( 𝑔𝑒𝑓𝑓 \n= 2.002), but close to the bulk TmIG value ( 𝑔𝑒𝑓𝑓 = 1.63)93 as well as that for TmIG thin films \n(𝑔𝑒𝑓𝑓 ≈ 1.57)6,94. Furthermore, as shown in Fig. 7( e), 𝑔𝑒𝑓𝑓 for our GSGG/TmIG( 236 nm ) film    \n25 \n decreases gradually with decreasing temperature . We observed similar behavior of 𝑔𝑒𝑓𝑓 for the \nGSGG/TmIG( 236 nm )/Pt(5 nm) film, and well as for other TmIG film thicknesses (see \nSupplementary Figures 9 and 10 ). \n \nFigure 7. Broadband Ferromagnetic Resonance.  (a) The field derivative of microwave (MW) \npower absorption spectra ( 𝑑𝑃\n𝑑𝐻 line shapes) for the GSGG/TmIG( 236 nm ) film at a fixed \nfrequency ( f = 12 GHz) in the range 160 K ≤ T ≤ 295 K . (b) 𝑑𝑃\n𝑑𝐻 line shapes  at different frequencies \nbetween f = 6 - 20 GHz fitted with the linear combination of symmetric and anti -symmetric \nLorentzian function derivatives for the GSGG/TmIG( 236 nm ) film  at T = 295 K . (c) Frequency \ndependence of linewidth,  ∆𝐻 at different temperatures for the GSGG/TmIG( 236 nm ) film with \nlinear fit. (d) The f-𝐻𝑟𝑒𝑠 curves at T = 295, 200, and 160 K along with Kittel fits. (e) Temperature \ndependence of the Gilbert damping parameter,  𝛼𝑇𝑚𝐼𝐺 , the inhomogeneous broadening, ∆𝐻0 and \nthe effective Landé g-factor for the GSGG/TmIG( 236 nm) film.  \n \n  \n26 \n Finally, to quantify the temperature dependence of the Gilbert damping parameter  (𝛼𝑇𝑚𝐼𝐺 ), \nwe fitted the ∆𝐻-f curves at different temperatures using the expression ,95 ∆𝐻=∆𝐻0+4𝜋𝛼\n𝛾𝜇0𝑓, \nwhere  ∆𝐻0 is the  frequency -independent contribution to the linewidth, known as the \ninhomogeneous  broadening  linewidth . From the fits ( see Fig. 7(c)), we obtained   𝛼𝑇𝑚𝐼𝐺 = 0.0103 \n± 0.002 at 295  K for  our GSGG/TmIG( 236 nm) film which is close to the previously reported \nvalues of 𝛼 (≈ 0.0132 -0.0146) for TmIG films6,96. Most importantly,  𝛼𝑇𝑚𝐼𝐺  increases gradually \nwith decreasing temperature but shows a comparatively faster  increase at low temperatures, \nespecially below ≈ 200 K (Fig. 7(e)). A similar increase in 𝛼 at low-T has also been observed in \nGSGG/TmIG( 236 nm)/Pt(5 nm), GSGG /TmIG( 46 nm)/Pt(5 nm) and GGG/TmIG( 44 nm)/Pt(5 \nnm) films  (see Supplementary  Figures 9 and 10), indicating that this behav ior is independent of \nTmIG film thickness and substrate choice.  In compensated ferrimagnetic insulators, e.g., GdIG, 𝛼 \nincreases drastically  close to the magnetic compensation temperature32. However, most of the  \nearlier reports indicate that TmIG films do not show magnetic compensation in the temperature \nrange between 1.5 and 300 K.47,48 Since our TmIG  films also do not show magnetic compensation \nin the measured temperature range, the increased value of  𝛼𝑇𝑚𝐼𝐺  at low temperatures in our TmIG \nfilms has a different origin . Sizeable increase s in 𝛼 and ∆𝐻 at low  temperatures  were also  observed \nin YIG and different  REIGs92,97 –99 including TmIG46, which was primarily  attributed to Fe2+ and/or \nRE3+ impurity relaxation mechanisms. However, our EELS study  confirms the absence of Fe2+ \nions, and therefore, we can rule out the possibility of Fe2+ impurity relaxation in our TmIG films.  \nTherefore, t he increased damping at low temperatures in our TmIG films may be associated with \nenhanced magnon scattering by defects ,100–104 and slowly relaxing Tm3+ ions92,105. It is known that \nthe contribution of slowly relaxing RE impurity ions towards damping is proportional to the orbital \nmoment  (L) of the RE3+ ions105–107, suggesting  that this mechanism applies to Tm3+ (L = 5).    \n27 \n 2. 6. Correlating Magnon Propagation Length with Magnetic Anisotropy and Gilbert \nDamping   \nIn the previous sections, we have demonstrated that both 𝐻𝐾𝑒𝑓𝑓and  𝛼𝑇𝑚𝐼𝐺  for our TmIG \nfilms show  clear increases at low temperatures, especially below 200 K.  It is known that the \nmagnon energy -gap (ℏ𝜔𝑀) is related to 𝐾𝑒𝑓𝑓 through the expression: ℏ𝜔𝑀∝2𝐾𝑒𝑓𝑓17,18. \nTherefore, an increase in  𝐻𝐾𝑒𝑓𝑓 (and hence, 𝐾𝑒𝑓𝑓) below 200 K enhances ℏ𝜔𝑀 giving rise to only \nhigh-frequency  magnon propagation with shorter 〈𝜉〉. Since only the subthermal magnons, i.e., the \nlow frequency  magnons are primarily responsible for the long -range thermo -spin transport and \ncontributes towards LSSE19–21, the 𝑉𝐿𝑆𝑆𝐸 signal also decreases below 200 K in our TmIG films12,13. \nThis also explains the noticeable decrease in 〈𝜉〉 below 200 K, as the maximum value of the \nfrequency -dependent propagation length is 〈𝜉〉𝑚𝑎𝑥∝1\n√ℏ𝜔𝑀𝑚𝑖𝑛 , where ℏ𝜔𝑀𝑚𝑖𝑛 is the  minimum value \nof ℏ𝜔𝑀, and ℏ𝜔𝑀𝑚𝑖𝑛 ∝2𝐾𝑒𝑓𝑓.17 Therefore, according to the expression17,18 〈𝜉〉=𝑎0\n2𝛼.√𝐽𝑒𝑥\n2𝐾𝑒𝑓𝑓 ∝\n1\n𝛼.(𝐻𝐾𝑒𝑓𝑓)1/2 , the observed decrease in 〈𝜉〉 and hence, the 𝑉𝐿𝑆𝑆𝐸 signal at low temperatures, \nespecially below 200 K in our TmIG films has contributions from the temperature evolutions of \nboth 𝐻𝐾𝑒𝑓𝑓and  𝛼𝑇𝑚𝐼𝐺 . The roles of magnetic anisotropy and damping in LSSE in different REIG -\nbased MI/HM bilayers have been explored by different groups12,13,32,33. All these studies indicated \nthat the LSSE signal strength varies inversely with both magnetic anisotropy and damping. In this \nmanuscript, we have not only highlighted the roles of the temperature evolutions of both magnetic \nanisotropy and damping in cont rolling the temperature dependent LSSE effect in TmIG/Pt bilayers, \nbut also attempted to establish possible correlations between 〈𝜉〉, 𝐻𝐾𝑒𝑓𝑓and 𝛼. Since 〈𝜉〉 is intrinsic \nto a magnetic film and hence independent of the thickness of the magnetic film19, it is convenient   \n28 \n to directly correlate 〈𝜉〉 with the physical parameters 𝐻𝐾𝑒𝑓𝑓and 𝛼 of individual magnetic films with \ndifferent thicknesses. We display the temperature dependence of 〈𝜉〉 on the left -y scales, and the \ntemperature evolutions of 𝐻𝐾𝑒𝑓𝑓and  𝛼𝑇𝑚𝐼𝐺  are shown on the right y-scales of Figs. 8 (a) and (b), \nrespectively. It is evident that the prominent drop in 〈𝜉〉 below 200 K in the TmIG/Pt bilayers is \nassociated with the noticeable increases in 𝐻𝐾𝑒𝑓𝑓 and 𝛼 that occur within the same temperature \nrange.  \n \nFigure 8. Temperature evolution of magnon propagation length and its correlation with \nmagnetic anisotropy and Gilbert damping:  (a) and (b)  Temperature dependence of 〈𝜉〉 on the \nleft-y scales, and the temperature evolutions of 𝐻𝐾𝑒𝑓𝑓and  𝛼𝑇𝑚𝐼𝐺  are shown on the right y-scales , \nrespectively. 〈𝜉〉 as a function of  (c) √𝐻𝐾𝑒𝑓𝑓 and (d)  𝛼𝑇𝑚𝐼𝐺  for the GSGG/TmIG(236  nm) film \nobtained from the temperature evolutions of 〈𝜉〉, 𝐻𝐾𝑒𝑓𝑓 and  𝛼𝑇𝑚𝐼𝐺 . \n  \n29 \n For a clearer understanding of the direct correlation between 〈𝜉〉 and 𝐻𝐾𝑒𝑓𝑓 in our TmIG/Pt \nbilayer films , we have plotted 〈𝜉〉 as a function of √𝐻𝐾𝑒𝑓𝑓 for the GSGG/TmIG(236  nm)/Pt film in \nFig. 8 (c) obtained from the temperature evolutions of 〈𝜉〉 and 𝐻𝐾𝑒𝑓𝑓. 〈𝜉〉 varies inversely with \n√𝐻𝐾𝑒𝑓𝑓 in the measured temperature range,  which is consistent with the expression 〈𝜉〉 ∝\n1\n𝛼.(𝐻𝐾𝑒𝑓𝑓)1/2. Similarly, we have plotted 〈𝜉〉 as a function of  𝛼𝑇𝑚𝐼𝐺  for the GSGG/TmIG(236  nm)/Pt \nfilm in Fig. 8 (d) obtained from the temperature evolutions of 〈𝜉〉 and  𝛼𝑇𝑚𝐼𝐺 . An inverse \ncorrelation between 〈𝜉〉 and  𝛼𝑇𝑚𝐼𝐺  in the measured temperature range  is also apparent from this \nplot, and hence , in agreement  with the aforementioned theoretical expression. To establish a more \naccurate correlation between the parameters 〈𝜉〉, 𝐻𝐾𝑒𝑓𝑓and 𝛼, one needs to  fix 𝐻𝐾𝑒𝑓𝑓(𝛼), and then \nevaluate 〈𝜉〉 for different values of 𝛼 (𝐻𝐾𝑒𝑓𝑓). It is however challenging  to change 𝐻𝐾𝑒𝑓𝑓 of a \nmagnetic material without varying 𝛼 significantly. Nevertheless, we have observed concurrent \nremarkable drops in the LSSE voltage as well as 〈𝜉〉 below 200 K in our TmIG/Pt bilayers \nregardless of TmIG film thickness and substrate choice and correlated the temperature evolution \nof 〈𝜉〉 with the noticeable increases in 𝐻𝐾𝑒𝑓𝑓 and 𝛼 that occur within the same temperature range. \nIt is important  to note that FMR  probes  only the zone  center  magnons  in the GHz  range  whereas  \nthe subthermal  magnons  which  primarily  contribute  towards  the LSSE  signal  belong  to the THz \nregime34. Therefore,  the behavior  of LSSE  cannot  be determined  by FMR  excited  by GHz -range  \nmicrowaves.  As shown  by Chang  et al.,33 the LSSE  voltage  varies  inversely  with 𝛼. Furthermore,  \nthe correlation  between  〈𝜉〉 and 𝛼 was predicted  theoretically18 but never  shown  experimentally.  \nAs indicated  in this report,  an experimental  demonstration  of the correlation  between  〈𝜉〉, 𝐻𝐾𝑒𝑓𝑓 \nand 𝛼 would  be beneficial  to fabricate  efficient  spincaloritronic  devices  with higher  〈𝜉〉 by tuning    \n30 \n these  fundamental  parameters.  However,  for deeper  understanding  of LSSE  and 〈𝜉〉, their magnon  \nfrequency  dependences  need  to be highlighted . \n \n2. 7. Magnon Frequency Dependence s of the LSSE Voltage and Magnon Propagation Length  \n As discussed  before,  the low energy  subthermal  magnons  with longer  〈𝜉〉 are primarily  \nresponsible  for LSSE.  These  low frequency  magnons  are partially  frozen  out by the application  of \nexternal  magnetic  field because  of increased  magnon  energy  gap due to the Zeeman  effect.20,108 \nTherefore,  〈𝜉〉 and hence  the LSSE  signal  is strongly  suppressed  by the application  of high \nmagnetic  field.20,108 However,  the field induced  suppression  is dependent  on the thickness  of the \nmagnetic  film.34 If the film thickness  is lower  than 〈𝜉〉, the low frequency  subthermal  magnons  \ncannot  recognize  the local  temperature  gradient  and do not participate  in LSSE.  In that case,  only \nhigh frequency  magnons  with shorter  〈𝜉〉 and much  higher  energy  than the Zeeman  energy  \ncontribute  towards  the LSSE  signal  and hence  the field induced  suppression  of LSSE  becomes  \nnegligible.20,108 However,  if the film thickness  is higher  than 〈𝜉〉, most  of the low frequency  \nmagnons  contribute  towards  LSSE  and hence,  the field induced  suppression  becomes  more  \nsignificant.20,108 As we have  observed  in our TmIG  films  that the trend  of temperature  dependent  \nLSSE  signal  is nearly  independent  of the substrate  choice,  the temperature  dependent  〈𝜉〉 is not \nsupposed  to change  significantly  with the substrate  choice .    \n31 \n  \nFigure 9. Magnetic field induced suppression of the LSSE voltage : 𝑉𝐼𝑆𝐻𝐸(𝐻) loops for the \nsGGG/TmIG(40nm)/Pt nm film at T = (a) 295, (b) 200 and (c) 140 K measured up to high \nmagnetic field of 𝜇0𝐻=9 T. 𝑉𝐼𝑆𝐻𝐸(𝐻) loops for the sGGG/TmIG(75nm)/Pt nm film at T = (d) \n295, (e) 200 and (f) 140 K measured up to high magnetic field of 𝜇0𝐻=9 T. 𝑉𝐿𝑆𝑆𝐸(𝑇,𝜇0𝐻=9T) \nand 𝑉𝐿𝑆𝑆𝐸𝑚𝑎𝑥(𝑇)  for (g) 40 nm and (h) 75 nm films. (i) Temperature dependence of 𝛿𝑉𝐿𝑆𝑆𝐸(%) for \n40 nm and 75 nm films.  \n \n Therefore,  (i) to verify  the influence  of thickness  on the field induced  suppression  of the \nLSSE  signal  in TmIG  films  and (ii) to confirm  whether  〈𝜉〉 of the GSGG/TmIG/Pt films obtained \nby analyzing the low field LSSE signal matches closely with that of the sGGG/TmIG/Pt films , we \nperformed  the high field LSSE  measurements  on the sGGG/TmIG/Pt  films  with thicknesses  of 40 \nand 75 nm. Figs. 9 (a)-(c) demonstrate the  𝑉𝐼𝑆𝐻𝐸(𝐻) loops for the 40 nm film at T = 295, 200 and \n140 K measured up to high magnetic field of 𝜇0𝐻=9 T. It can be seen that the LSSE  signal  for \nthe 40 nm film does not show  prominent  suppression  at 9 T at 295 K . However,  as temperature  \ndecreases  below  200 K, the suppression  of the LSSE  signal  becomes  noticeable.  On the other  hand,  \n  \n32 \n as seen in Figs. 9 (d)-(f), the LSSE  signal  for the 75 nm film shows  significant  suppression  even  at \n295 K and the suppression  of LSSE  signal  enhances  with decreasing  temperature.  The more  intense  \nsuppression  of the LSSE  signal  in the 75 nm film compared  to the 40 nm film at all temperatures  \nbetween  295 and 140 K is also evident  from  𝑉𝐿𝑆𝑆𝐸(𝑇) for these two films shown in Figs. 9 (g) and \n(h). We have also estimated the percentage change in 𝑉𝐿𝑆𝑆𝐸 by the application of 9 T magnetic \nfield, which we define as, 𝛿𝑉𝐿𝑆𝑆𝐸(%)=[𝑉𝐿𝑆𝑆𝐸(9 T)−𝑉𝐿𝑆𝑆𝐸𝑚𝑎𝑥\n𝑉𝐿𝑆𝑆𝐸𝑚𝑎𝑥]×100% , where, 𝑉𝐿𝑆𝑆𝐸(9 T) is the \nabsolute value of 𝑉𝐿𝑆𝑆𝐸 at 9 T magnetic field and 𝑉𝐿𝑆𝑆𝐸𝑚𝑎𝑥 is the value of 𝑉𝐿𝑆𝑆𝐸 at the maximum point \nof the 𝑉𝐼𝑆𝐻𝐸(𝐻) loop. As shown in  Figs. 9 (i), |𝛿𝑉𝐿𝑆𝑆𝐸| for the 75 nm film is nearly 14% at 295 K \nbut increases to ≈ 32% at 140 K. On the other hand, |𝛿𝑉𝐿𝑆𝑆𝐸| for the 40 nm film is negligible at \n295 K but increases to ≈ 7% at 140 K. These  results  indicate  that 〈𝜉〉 for the sGGG/TmIG/Pt  films  \nat 295 K is between  40 and 75 nm, which  is close  to the value  of 〈𝜉〉 obtained  for the \nGSGG/TmIG/Pt  films.  Since  〈𝜉〉 decreases  at low temperatures  and becomes  smaller  than 40 nm \nbelow  150 K, the sGGG/TmIG(40nm)/Pt  film shows  significant  field induced  suppression  of 𝑉𝐿𝑆𝑆𝐸 \nat low temperatures.  Similarly,  since  〈𝜉〉 at low temperatures  is much  smaller  than 75 nm, the field \ninduced  suppression  of 𝑉𝐿𝑆𝑆𝐸 is also large at low temperatures for the 75 nm film. Note  that in case \nof YIG/Pt  films,  the magnetic  field induced  suppression  of the LSSE  signal  diminishes  with \ndecreasing  temperature,34 whereas,  an opposite  trend  has been  observed  in case of TmIG . Such  \nbehavior  can be explained  by different  trends  of the temperature  dependent  〈𝜉〉  in YIG and TmIG.  \nAs explained  by Guo et al.,34 the temperature  induced  enhancement  of 〈𝜉〉 neutralizes  the field \ninduced  suppression  of 〈𝜉〉, and because  of these  two competing  factors,  the field induced  \nsuppression  of the LSSE  voltage  is less prominent  at low temperatures  in YIG/Pt  films.  On the \ncontrary,  the combined  effects  of the temperature  induced  reduction  in 〈𝜉〉 observed  in our   \n33 \n TmIG/Pt  films  and field induced  suppression  of 〈𝜉〉 give rise to stronger  field induced  suppression  \nof the LSSE  voltage  at lower  temperatures.  \n \nFigure 10. Magnon frequency dispersion for TmIG : (a) Magnon frequency dispersion for TmIG \nfor 𝜇0𝐻=0 T and 9 T magnetic fields at T = 295 K. (b)  Comparison  of the magnon  frequency  \ndispersion  for YIG and TmIG  at room  temperature  for 𝜇0𝐻=0 T. Magnon frequency dispersion \nof TmIG at different temperatures for  (c) 𝜇0𝐻=0 T and (d) 9 T magnetic fields . \n \nNext, to have a qualitative understanding of the magnon frequency dependences of the \nLSSE signal and 〈𝜉〉, we have estimated the magnon frequency dispersion for TmIG in the absence \nand in presence of high magnetic field of 9T. According  to the classical  Heisenberg  ferromagnet  \nmodel  for spin waves,  the parabolic  magnon  frequency  dispersion  at low energies  can be expressed  \nas,17,20,109 ℏ𝜔𝑘= 𝑔𝑒𝑓𝑓𝜇𝐵𝐻+𝐷𝑆𝑊.𝑘2𝑎02+𝐸𝑎𝑛𝑖(𝐾𝑒𝑓𝑓), where  the first term represents  the \nZeeman  energy  gap due to the application  of external  magnetic  field,  the second  term is associated  \n  \n34 \n with spin wave  stiffness  (𝐷𝑆𝑊 is the spin wave  stiffness  constant),  and the third  term represents  \nthe contribution  of effective  magnetic  anisotropy  energy,  𝐸𝑎𝑛𝑖(𝐾𝑒𝑓𝑓). Here,  the value  of 𝐷𝑆𝑊 is \ntaken  as that of YIG,  i.e., 𝐷𝑆𝑊𝑎02=4.2 ×10−29 erg.cm2 at room  temperature.108 Using  the \nexpression  𝐾𝑒𝑓𝑓=𝐾𝑠ℎ𝑎𝑝𝑒+𝐾𝑚𝑐 + 𝐾𝑚𝑒, we determined the temperature dependence of \n𝐸𝑎𝑛𝑖(𝐾𝑒𝑓𝑓) for TmIG.  Here, the temperature variation of  𝐾𝑚𝑒 was obtained from the temperature \ndependence of 𝜆111 reported in the literature83 (see Supplementary Figures 8(e) ). Since the shear \nmodulus, 𝑐44 in REIGs is weakly dependent on the rare -earth species, the value of 𝑐44 is taken as \nthat of YIG (76.4 GPa at room temperature).47 The temperature dependence of 𝐾𝑠ℎ𝑎𝑝𝑒 was \nestimated from the temperature variation of 𝑀𝑆 (see Supplementary Figures 8(f) ). We assumed \nconstant value of 𝐾𝑚𝑐=0.058 kJ/m3 throughout the measured temperature range.6 As shown in \nSupplementary Figures 8(f),  𝐾𝑒𝑓𝑓 is positive for 𝑇≤300 K (𝐾𝑒𝑓𝑓=20 kJ/m3) and its absolute \nvalue increases considerably with decreasing temperature. Fig. 10 (a) shows the magnon frequency \ndispersion for TmIG for 𝜇0𝐻=0 T and 9 T magnetic fields at T = 295 K. Clearly,  the high \nmagnetic  field opens  a magnon  energy  gap in the low frequency  regime  (much  smaller  than \nthermal  energy  at room  temperature)  indicating  the suppression  of 〈𝜉〉 and hence  significant  \nreduction  of the LSSE  signal  at high magnetic  fields.  As shown  in Fig. 10 (b), we have  compared  \nthe magnon  frequency  dispersion  for YIG and TmIG  at room  temperature  for 𝜇0𝐻=0 T. It is \nevident  that the opening  of magnon  energy  gap in TmIG  is higher  than in YIG even  in the absence  \nof external  magnetic  field,  which  is mainly  caused  by the effective  magnetic  anisotropy.  Note  that \n𝐾𝑒𝑓𝑓 of TmIG is higher than that of YIG.110 The higher  value  of magnon  energy  gap in TmIG  \ncompared  to YIG thus indicates  the higher  possibility  of freezing  out of the low energy  subthermal  \nmagnons  in TmIG . This,  along  with higher  value  of 𝛼 in TmIG  contributes  to the lower  value  of \n〈𝜉〉 and hence  the LSSE  voltage  in TmIG  compared  to YIG16. Furthermore,  the value  of 𝑔𝑒𝑓𝑓 at   \n35 \n room  temperature  is lower  in TmIG  (≈1.63)93 than in YIG (≈ 2.046)111. Therefore,  for a given  \napplied  magnetic  field strength,  the magnitude  of the magnon  energy  gap due to Zeeman  effect  \nwill be different  in TmIG  than in YIG.  Moreover,  𝛼 in TmIG6,96 is nearly  two orders  of magnitude  \nhigher  than in YIG112. In other  words,  different  values  of 𝐾𝑒𝑓𝑓, 𝑔𝑒𝑓𝑓 and 𝛼 as well as their different \ntemperature dependences give rise to different temperature profiles of 〈𝜉〉 in TmIG and YIG. In \nFigs. 10 (c) and (d), we show the magnon frequency dispersion of TmIG at different temperatures \nfor 𝜇0𝐻=0 T and 9 T magnetic fields, respectively. Clearly, the magnon energy gap increases \nwith decreasing temperature due to enhanced magnetic anisotropy at low temperatures. \nApplication of 9 T magnetic field increases the magnon energy gap further due to Zeeman ef fect. \nThese results help explain  the observed decrease in 〈𝜉〉 and enhanced magnetic field induced \nsuppression of the LSSE signal at low temperatures in TmIG.20,108 \n \nWe believe that our findings will attract the attention of the spintronic community for \nfurther exploration of long-range thermo -spin transport in different REIG based magnetic thin \nfilms and heterostructures for tunable spincaloritronic efficiency by manipulating 𝐻𝐾𝑒𝑓𝑓 and 𝛼. For \nexample, 𝐻𝐾𝑒𝑓𝑓 of the REIG thin films grown on piezoelectric substrates can be modulated by \napplying a gate voltage,113 which can eventually influence 〈𝜉〉 and hence the spincaloritronic \nefficiency.  Therefore, our study also  provides a step towards the development of efficient \nspincaloritronic devices based on voltage controlled LSSE.  \n \n3. CONCLUSION  \nIn summary, we have performed a comprehensive investigation of the temperature \ndependent LSSE, RF transverse susceptibility, and broadband FMR measurements  on TmIG /Pt   \n36 \n bilayers grown on different substrates. The decrease  in the LSSE volta ge below 200 K independent  \nof TmIG film thickness and substrate choice is attribute d to the increases in 𝐻𝐾𝑒𝑓𝑓 and 𝛼 that occur \nwithin the same temperature range. From the TmIG thickness dependence of the LSSE voltage, \nwe determined the temperature dependence of 〈𝜉〉 and highlighted its correlation with the \ntemperature dependent 𝐻𝐾𝑒𝑓𝑓 and 𝛼 in TmIG/Pt  bilayers, which will be beneficial for the \ndevelopment of REIG -based spincaloritronic nanodevices . Furthermore, the enhanced suppression \nof the LSSE voltage by the application of high magnetic field at low temperatures together with \nthe temperature evolution of  magnon frequency dispersion in TmIG estimated from the \ntemperature dependent 𝐾𝑒𝑓𝑓 and 𝛼 support our observation of the decrement of 〈𝜉〉 at low \ntemperatures in the TmIG/Pt bilayers.  \n \n \n \n \n \n \n \n \n \n \n \n \n   \n37 \n 4. METHODS  \nThin film  growth  and structural/morphological characterization : Single -crystalline TmIG thin \nfilms were deposited by pulsed laser deposition (PLD), using two different PLD setups. The thin \nfilms were grown epitaxially on different (111) -oriented substrates, including GGG ( Gd3Ga5O12), \nGSGG ( Gd3Sc2Ga3O12), and sGGG ( (Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12). Substrates with (111) \norientation are chosen so that the magnetoelastic anisotropy of the TmIG films favors PMA.  Using \nthe first PLD setup, films with varying thickness between 28 nm and 236 nm were grown on GGG \nand GSGG substrates. A KrF excimer laser with a wavelength of 248 nm , a fluence of 3 -4 J/cm²,  \nand a repetition rate of 2 Hz is used. Before the first deposition, t he TmIG target was preablated \ninside the PLD chamber with more than 104 pulses.  All substrates were annealed for 8 h at 1250°C \nin oxygen atmosphere prior to the film deposition to provide a high substrate surface quality . \nGrowth conditions were selected to  achieve stoichiometric, single -crystalline thin films with a \nsmooth surface of about 0.2 -0.3 nm in root -mean square roughness (RMS).  For all films, the \nsubstrate was heated to 595°C  during the film deposition , monitored by a thermocouple inside the \nsubstrate holder. The TmIG thin films were grown  at a rate of 0.01 − 0.02 nm/s, in the presence of \nan oxygen background atmosphere of 0.05 mbar. After the deposition, the samples were  cooled to \nroom temperature at approximately 5 K/min , maintaining the oxygen atmosphere.  A layer of 5 nm \nPt was deposited at room temperature ex-situ on the garnet films by DC magnetron sputtering \nusing a shadow mask. The TmIG films were annealed at 400°C for 1 h inside the sputter chamb er \nprior to the Pt deposition to avoid surface contamination114.To complement these samples, TmIG \nfilms with thicknesses 75 and 40 nm were grown on sGGG substrates using a second  PLD setup . \nThe laser wavelength was 248 nm at 10 Hz, the fluence 1.3 J/cm2, and the substrate  temperature   \n38 \n was ~750 ˚C with an oxygen pressure of 0.2 mbar. Samples were cooled at 20 K/min in 0.2 mbar \noxygen.  \n \nThe film surface morphology was investigated by atomic force microscopy (AFM), while \nthe structural properties of the thin films were identified by x -ray diffraction (XRD) using \nmonochromatic Cu Kα radiation. The film thickness was evaluated from the Laue oscillations (for \nthe thinne r films) and by spectroscopic ellipsometry. Further, a cross -sectional high resolution \nscanning transmission electron microscopy (HR -STEM) was conducted, using a JEOL NEOARM \nF200 operated at an electron energy of 200 keV. Electron energy loss spectra (EELS) were \nobtained using a GATAN Continuum S EE LS spectrometer. The cross -sectional sample was \nprepared by mechanical dimpling and ion polishing.  Interdiffusion between the TmIG film and \nsubstrate is expected to be limited to a depth of order 1 -3 nm41 and its effects are neglected for the \nfilm thicknesses used in this study.  See Supplementary Figure 1 (e) for energy dispersive X -ray \nspectroscopy (EDX) using transmission  electron microscopy (TEM)  performed on the \nGSGG/TmIG(20 5nm) film.   \n \nTemperature dependent MFM measurements : Temperature dependent MFM measurements were \nperformed on a Hitachi 5300E system. All measurements were done  under high vacuum (P ≤ 10-6 \nTorr). MFM measurements utilized HQ: NSC18/Co -Cr/Al BS tips, which were magnetized out -\nof-plane with respect to the tip surface via a permanent magnet. Films were first magnetized to \ntheir saturation  magnetization by being placed in a 1T static magnetic field, in -plane with the film \nsurface. After that AC demagnetization of the film was implemented before init iating the MFM \nscans. After scans were performed, a parabolic background was subtracted, which arises from the   \n39 \n film not being completely flat on the sample stage. Then, line artifacts were subtracted before \nfinally applying a small Gaussian averaging/sharpening filter over the whole image. Phase \nstandard deviation was determined by fitting a Gaussian to the image p hase distribution and \nextracting the standard deviation from the fit parameters.  \n \nMagnetometry : The magnetic properties of the samples were measured using a superconducting \nquantum interference device - vibrating sample magnetometer (SQUID -VSM) at temperatures \nbetween 10 K and 350 K. A linear background stemming from the paramagnetic substrate was \nthereby subtracted. Due to a trapped remanent field inside the superconducting coils, the measured \nmagnetic field was corrected using a paramagnetic reference sample. Additionally, a polar \nmagneto -optical Kerr effect (MOKE) setup was used to record out -of-plane hysteresis loops at \nroom temperature.  The molecular field coefficient ( MFC ) model was a Python -coded version of \nDionne’s model115 using molecular field coefficients57. \n \nLongitudinal spin Seebeck effect measurements : The longitudinal spin Seebeck effect (LSSE) \nwas measured over a broad temperature window of  120 K ≤ T ≤ 295 K using a custom -built setup \nassembled on a universal PPMS sample puck. During the LSSE measurements, the films were \nsandwiched between two copper blocks, as shown in Fig. 3(a).  The s ame sample geometry was \nused for all films and the distance between the contact leads on the Pt surface were fixed at Ly = 3 \nmm for all films.  A single layer of thin Kapton tape was thermally affixed to the naked surfaces of \nthe top (cold) and bottom (hot) copper blocks. To ensure a good thermal link between the film \nsurface and the Kapton tape (thermally conducting and electrically insulating) attached to the top \nand bottom blocks, cryogenic Apiezon N -grease was used. Additionally, the Kapton tape   \n40 \n electrically insulated the cold (hot) blocks from the top (bottom) surface of the films . The \ntemperatures of both these blocks were controlled individually by two separate temperature \ncontrollers (Scientific Instruments Model no. 9700) to achieve an ultra -stable temperature \ndifference ( ∆𝑇) with [∆𝑇]𝐸𝑟𝑟𝑜𝑟  < ± 2 mK. The top block (cold) was thermally anchored to the  base \nof the  PPMS puck  using two molybdenum screws whereas a 4 -mm-thick Teflon block was \nsandwiched between the puck  base and the hot block (bottom) to maintain a temperature difference \nof ~ 10 K between the hot block and the PPMS base. A resistive chip -heater (PT -100 RTD sensor) \nand a calibrated Si -diode thermometer (DT-621-HR silicon diode sensor) were attached to each of \nthese blocks to efficiently control and sense the temperature. The heaters  and thermometers \nattached to the copper blocks were connected to the temperature controllers in such a manner that \na temperature gradient develops along the + z-direction that generates a temperature difference, ∆𝑇, \nbetween the top (cold) and bottom (hot) copper blocks. For a given temperature gradient, the in -\nplane voltage generated along the y-direction across the Pt layer due to the ISHE ( 𝑉𝐼𝑆𝐻𝐸) was \nrecorded by a Keithley 2182a nanovoltmeter while sweeping an external in -plane DC magnetic \nfield from positive to negative values along the x-direction. The Ohmic contacts for the voltage \nmeasurements were made by electrically anchoring  a pair of  ultra-thin gold wires (25 µm diameter) \nto the Pt layer  by high quality conducting silver paint (SPI Supplies ). \n \nTransverse susceptibility measurements : The temperature evolution of effective magnetic \nanisotropy in the GSGG/TmIG/Pt film was measured by employing a radio frequency (RF) \ntransverse susceptibility (TS) technique using a home -built self -resonant tunnel diode oscillator \n(TDO) circuit with a resonance frequency of 12 MHz and sensitivity of ±10 Hz. A physical \nproperty measurement system (PPMS) was employed as a platform to scan the external DC   \n41 \n magnetic field ( HDC) and temperature. Before the TS measurements, the film was mounted inside \nan inductor coil (L), which is a component of an LC tank circuit. The entire tank circuit was placed \noutside the PPMS except the coil , L, which was positioned at the base of the PPMS sample \nchamber using a multi -purpose PPMS probe insert ed in such a manner that the axial RF magnetic \nfield ( HRF) of amplitude ~ 10 Oe produced inside the coil was always parallel to the film surface, \nbut perpendicular to HDC. For the T mIG with IP easy axis, 𝐻𝐷𝐶⊥film surface , whereas for the \nfilms with OOP easy axis, 𝐻𝐷𝐶∥film surface. When the sample is subject to both HRF and HDC, \nthe dynamic susceptibility of the sample changes which in turn changes the inductance of the coil \nand, hence, the resonance frequency of the LC tank circuit. The relative change in the resonance \nfrequency is proportional to the relative change in the transverse susceptibility of the sample. \nTherefore, TS as a function of HDC was acquired by monitoring the shift in the resonance frequency \nof the TDO -oscillator circuit by employing an Agilent frequency counter . \n \nBroadband f erromagnetic resonance measurements : Broadband ferromagnetic resonance \n(FMR) measurements ( 𝑓 = 6-20 GHz) were performed using a broadband FMR spectrometer \n(NanOscTM Phase -FMR  Spectrometer , Quantum Design Inc., USA) integrated to a Dynacool \nPPMS. The TmIG film was firmly affixed  on the surface of a commercial 200-μm-wide coplanar \nwaveguide (CPW)  (also provided by  NanOscTM Phase -FMR Spectrometer, Quantum Design Inc., \nUSA ) using Kapton tape . The TmIG films were placed faced down on the CPW so that the CPW \ncan efficiently transmit the MW signal from the RF source over a broad f-range. The role of the \nKapton tape is to electrically insulate the films from the CPW. An in-plane  RF mag netic field, 𝐻𝑅𝐹 \nis generated in close vicinity to the CPW. In presence of an appropriate external in-plane  DC \nmagnetic field, 𝐻𝐷𝐶 provided by the superconducting magnet of the PPMS applied along the   \n42 \n direction of the MW current flowing through the CPW(𝐻𝐷𝐶⊥𝐻𝑅𝐹) and frequency, 𝐻𝑅𝐹 \nresonantly excites the TmIG film. The spectrometer employs lock -in detection and records the \nfield derivative of the power absorbed ( 𝑑𝑃/𝑑𝐻) by the film when it is excited by a  microwave \n(MW)  electromagnetic field generated by injecting a  MW current to the CPW . \n \nACKNOWLEDGEMENTS  \nFinancial support by the US Department of Energy, Office of Basic Energy Sciences, Division of \nMaterials Science and Engineering under Award No. DE -FG02 -07ER46438 at USF and by the \nGerman Research Foundation (DFG) within project No. 318592081AL618/37 -1 at U Augsburg \nare gratefully acknowledged. 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Appl. \nPhys.  1970 , 41 (12), 4874 –4881.  \n "    },    {        "title": "2308.08331v1.Discovery_and_regulation_of_chiral_magnetic_solitons__Exact_solution_from_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "Discovery and regulation of chiral magnetic solitons: Exact solution from Landau-Lifshitz-Gilbert\nequation\nXin-Wei Jin,1, 2Zhan-Ying Yang,1, 2,∗Zhimin Liao,3Guangyin Jing,1, †and Wen-Li Yang2, 4\n1School of Physics, Northwest University, Xi’an 710127, China\n2Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China\n3School of Physics, Peking University, Beijing, 100871,China\n4Insititute of Physics, Northwest University, Xi’an 710127, China\n(Dated: August 17, 2023)\nThe Landau-Lifshitz-Gilbert (LLG) equation has emerged as a fundamental and indispensable framework\nwithin the realm of magnetism. However, solving the LLG equation, encompassing full nonlinearity amidst\nintricate complexities, presents formidable challenges. Here, we develop a precise mapping through geometric\nrepresentation, establishing a direct linkage between the LLG equation and an integrable generalized nonlinear\nSchr ¨odinger equation. This novel mapping provides accessibility towards acquiring a great number of exact\nspatiotemporal solutions. Notably, exact chiral magnetic solitons, critical for stability and controllability in\npropagation with and without damping effects are discovered. Our formulation provides exact solutions for the\nlong-standing fully nonlinear problem, facilitating practical control through spin current injection in magnetic\nmemory applications.\nIntroduction.— The seminal 1935 work by Landau and Lif-\nshitz, which laid down the foundational dynamical equation\ngoverning magnetization based on phenomenological insights\n[1–3], and the subsequent introduction of a damping term by\nGilbert [4], the amalgamation of these concepts has given\nrise to the renowned Landau-Lifshitz-Gilbert (LLG) equation.\nOver the years, this equation has emerged as a fundamental\nand indispensable framework within magnetism field. Its con-\ntemporary significance has been amplified through remarkable\nadvancements, most notably the incorporation of an additional\nterm that facilitates the explication of spin torque phenomena\nin spintronics [5–10], spin waves [11–18], magnetic solitons\n[19–28], spatio-temporal patterns [29, 30], and even chaotic\nbehavior [31]. Further advancements have paved the way\nfor applications in next-generation magnetic storage [32–34],\nneural networks [35–37], and logic gates [38–42].\nDespite its deceptively simple appearance, solving the LLG\nequation poses an exceptional challenge [29, 30], rendering\nit a persistently unresolved problem for nearly nine decades.\nThis complexity emanates from its intricate nature as a vector-\nbased highly nonlinear partial differential equation. In real-\nworld scenarios, the LLG equation encompasses a myriad of\ncomplex interactions among the components of the magneti-\nzation vector [4]. Consequently, solutions often necessitate\nrecourse to linearization, approximations, and asymptotic\ntechniques such as the Holstein-Primakoff (HP) transforma-\ntion [43, 44], reductive perturbation scheme [45, 46], and long\nwavelength approximation. Nonetheless, these techniques\nprove utterly ineffectual in regions of large amplitudes or\nstrong nonlinearity. Therefore, exact solution of the LLG\nequation emerges as a potent bridge, overcoming these gaps\nand revealing profound revelations regarding magnetization\ndynamics, thereby furnishing insightful understandings for\nsimulating and comprehending intricate magnetic systems.\nIn this Letter, through a geometric representation [47],\nwe establish an exact mapping of the LLG equation onto\nan integrable generalized nonlinear Schr ¨odinger equation,\nfree of any approximation. This novel mapping provides\naccessibility towards acquiring a great number of exact\nspatiotemporal solutions of the original equation. Notably, we\nunveil an analytical formulation for chiral magnetic solitons,encompassing a spectrum ranging from left-handed, neutral to\nright-handed configurations, determined by a defined chirality\nfactor. The derived exact solution indicates the potential for\narbitrary manipulation of magnetic soliton motion through the\ninjection of spin current —a discovery that aligns seamlessly\nwith our numerical findings. To encapsulate the realism of\ndissipative devices, we incorporate Gilbert damping into the\ndynamics of these chiral magnetic solitons, thereby estimating\ntheir dynamic propagation.\nModeling.— We consider an isotropic ferromagnetic\nnanowire with spin-polarized current flowing along the axis\nof nanowire as depicted in Fig. 1. A “nanowire” as defined\nhere is a planar ferromagnetic stripe of length Lx, width\nLy, and thickness Lzalong ˆx,ˆy, and ˆz, respectively, with\nLx≫Ly>Lz.\nFigure 1: Schematic diagram of 1D ferromagnetic structure.\nMagnetic soliton excitation driven by spin-polarized currents.\nHere∆represents the width of magnetic solitons.\nThe magnetization dynamics is described by the famous\nLLG equation\n∂m\n∂t=−γm×Heff+α/parenleftbigg\nm×∂m\n∂t/parenrightbigg\n+τb, (1)\nwhere m=M/Ms= (mx,my,mz)is the unit magnetization\nvector with Msbeing the saturated magnetization. The first\nterm on the right-hand side represents the torque contributed\nby the effective field Heff(including applied, demagnetizing,\nanisotropy, and exchange fields), γis the gyromagnetic\nconstant. The second term describes the Gilbert damping\ntorque, parameterized by a dimensionless damping factor α.arXiv:2308.08331v1  [nlin.PS]  16 Aug 20232\nFigure 2: Spatial structure and classification of chiral magnetic solitons. (a) Vertical views of the left-handed and right-handed\nmagnetic solitons. (b) Schematic plot of the chirality defined by the azimuth angle change. The pair of red arrows delineate the\nazimuthal directional changes of the left and right chiral magnetic solitons across the distribution axis. Their discrepancies in\nazimuthal variation are denoted by ∆ϕ′\nLHand∆ϕ′\nRH. (c)-(e) Spatial spin structures of left-handed magnetic soliton, neutral\nmagnetic soliton, and right-handed magnetic soliton. (f)-(h) illustrate the azimuthal, polar angle, and phase gradient flow of the\nthree kinds of chiral solitons.\nThe last term τbrepresents the spin-transfer torque (STT),\nwhich comprises dual components that can be written as τb=\n−bJ(ˆJ·∇)M+βbJM×(ˆJ·∇)M. Here ˆJis the unit vector in\nthe direction of the current. These two components are most\ncommonly termed adiabatic and non-adiabatic spin torques,\nrespectively, with bJ=P jeµB/(eMs)andβdefined as the non-\nadiabatic torque coefficient. Wherein, Prepresents the spin\npolarization of current, jeis the electric current density, µBis\nthe Bohr magneton, and eis the magnitude of electron charge.\nIn what follows, we take only adiabatic STT into consideration\nfor two reasons: one is that the most widely agreed upon\ninteraction between a spin-polarized current and a magnetic\nsoliton is adiabatic STT; and the other is that the magnitude\nof the nonadiabatic spin torque is about 2 orders of magnitude\nsmaller than adiabatic torque (β≈10−2). Let us begin by\nexamining the most elementary effective field, encompassing\nsolely exchange fields, i.e. Heff= (2A/Ms)∇2m, where Ais\nthe exchange stiffness constant.\nThe spatiotemporal transformation τ=γµ0Mst/(1+α2)\nandζ=λex·xare introduced to recast the LLG equation\ninto the dimensionless Landau-Lifshitz form (Note that λex=/radicalbig\n2A/(µ0M2s)is the exchange length):\nmτ=−m×mζζ−αm×/parenleftbig\nm×mζζ/parenrightbig\n+Qmζ, (2)\nwhereQ=bJ(1+αβ)//radicalbig\n2Aγ2µ0, a dimensionless number\nmeasuring the ratio of external spin current over exchange\ninteraction strength. This dimensionless STT-LLG model\n(2) effectively describes the dynamics of nonlinear excita-\ntions, such as magnetic solitons, occurring in ferromagnetic\nnanowires upon spin injection. Moreover, it exhibits qualita-tive reproduction much of the behavior seen experimentally.\nFor a permalloy nanowire, the standard material parameters\nare:γ=1.76×1011rad/s·T,Ms=8×105A/m,A=1.3×\n10−11J/m,P=0.5. As a result, the units in time and space\nafter rescaling are 1 τ≈5.70 ps ,1ζ≈5.68 nm.\nChiral magnetic soliton.— The high nonlinearity of STT-\nLLG model (2) presents a great challenge for comprehensive\nanalytical research and restricts the exploration of novel\nspin textures to the realm of micromagnetic simulation or\nweak nonlinearity. In this context, to obtain an analytical\ndepiction of large-amplitude magnetic textures, we exactly\nmap the STT-LLG equation (2) into a generalized nonlinear\nSchr ¨odinger (GNLS) equation (See Supplementary Material\nfor further details on the spatial curve mapping procedure),\ndevoid of any approximations. For no damping, the GNLS\nequation reads\niΨτ+Ψζζ+2|Ψ|2Ψ−iQΨζ=0. (3)\nExact cycloidal chiral magnetic soliton solutions can be\nconstructed by applying the Darboux transformation (DT) [48,\n49]. Indeed, using the mapping relationship between equation\nsolutions, the specific expression of three components of\nmagnetization are obtained (See Supplementary Material for3\ndetailed calculations):\nmx=2a\na2+b2[asinh(Ξ)sin(Γ)−bcosh(Ξ)cos(Γ)]·sech2(Ξ),\nmy=2a\na2+b2[asinh(Ξ)cos(Γ)+bcosh(Ξ)sin(Γ)]·sech2(Ξ),\nmz=1−2a2\na2+b2sech2(Ξ),\n(4)\nwith\nΞ=2a/bracketleftbigg\nx+/integraldisplay\n(Q+4b)dt/bracketrightbigg\n,\nΓ=2b/bracketleftbigg\nx+/integraldisplay\n(Q−2(a2−b2)/b)dt/bracketrightbigg\n,\nwhere aandbdescribe the wave number and the velocity of\nthe magnetic soliton.\nFig. 2(a) depicts the spin textures of the obtained\nmagnetic solitons. Evidently, these solitons showcase a mirror\nsymmetry relative to the wave vector axis, indicating their\ninherent chirality. The chirality can be characterized by\nvariations of the azimuth angle. To clarify, we denote the polar\nand azimuthal angles of mbyθandϕ, respectively (as shown\nin Fig. 1), such that M+=mx+imy=M0sin(θ)exp(iϕ),mz=\nM0cos(θ). Note that the azimuthal angle exhibits a periodic\nbackground, which arises from the variation of magnetization\nin space, as revealed by the solution (4). The oscillation\nstructures present in the azimuthal angle profiles are related\nto the small oscillation of mν(ν=x,y), even though they\nmay not be readily visible in Fig. 2(a). By eliminating the\nmeaningless periodic background phase, the real phase jumps\nof the chiral magnetic solitons are obtained by calculating the\nintrinsic argument ϕ′(x) =argM′\n+, where M′\n+=M+exp(iΓ).\nAs a result, the azimuthal angles of both chiral solitons\nare demonstrated in Fig. 2(b). Two red arrows span\nbetween the blue dashed line representing negative infinity and\nthe corresponding positive infinity, delineating the azimuthal\nevolution of magnetic solitons across the distribution axis. The\ndistinction in azimuthal variation for chiral magnetic solitons\nare denoted as ∆ϕ′\nLHand∆ϕ′\nRH. Notably, the two classes\nof chiral magnetic solitons exhibit opposite phase jumps,\ncorresponding to two distinct chiralities.\nThe total phase change is defined as ∆ϕ′=ϕ′(x→+∞)−\nϕ′(x→ −∞). In general, the phase change of arbitrary\nmagnetic solitons can be determined by integrating the phase\ngradient flow. Insight can be gained from combining both\nargument and the phase gradient flow ∇ϕ′(x). Starting from\nM′\n+that constructed from exact solutions, we obtain\n∇ϕ′(x) =2b[sech(2Ξ)+1]/parenleftig\na2−b2\na2+b2/parenrightig\nsech(2Ξ)−1. (5)\nOne can observe that the denominator of the aforementioned\nexpression is consistently a non-positive value, which indi-\ncates that “ +” and “ −” families of phase gradient flow are\ncharacterized by the opposite signs of b. Here, we define\na chirality factor C=sgn(b) =±1, which determines the\nchirality of magnetic solitons. It is straightforward to verify\nthat the nonzero phase variation is characterized by a simple\nFigure 3: Coupling between chiral magnetic solitons and spin\ncurrent injection. (a) Velocities of three distinct classes of\nchiral magnetic solitons plotted against spin-polarized\ncurrents ( j). (b)-(d) Controlled manipulation of right-handed\nmagnetic solitons under varying current strengths, enabling\nforward, backward, and arrested motion.\nexpression: ∆ϕ′=2Carctan (|a/b|).Thus, the chirality of\nthe chiral magnetic soliton is entirely determined by this\nchirality factor C. When b=0, a special case naturally\noccurs, where the chirality factor cannot be defined, and\nchirality disappears, corresponding to the neutral cycloidal\nmagnetic solitons. Finally, we can now classify the exact\nsolution (4) into three categories based on the chirality factor,\ncorresponding to neutral, left-handed, and right-handed chiral\nmagnetic solitons. Figs. 2(c)–2(h) depict the typical spin\ntextures, azimuthal angles, polar angles, and phase gradient\nflow at t=0 when no spin current is applied.\nSpin-current coupling and damping effect.— We now move\nto study the coupling between chiral magnetic soliton and the\ninjection of spin current. It has been demonstrated that the\nspin transfer torque is capable of driving the domain wall\nor skyrmion [7, 26], eliciting their prompt movement at a\nconsiderable velocity upon the application of spin current.\nHere we report a comparable phenomenon on the chiral\nmagnetic soliton from both theoretical and simulation results.\nThe numerical simulation results depicting the relationship\nbetween the velocities of three categories of chiral magnetic\nsolitons and the injected spin currents are illustrated in Fig.\n3(a), and are in direct agreement with those obtained from\nthe analytical solutions (4). These linear correlations can be\nrealized from the equivalent GNLS equation (3), wherein the\nspin current term can be normalized to resemble the “driving\nvelocity”, as supported by the dimensional analysis of Q.\nFigs. 3(b)–3(d) exemplify the current manipulation for right-\nhanded magnetic soliton that comprise a series of transient\nsnapshots captured during the magnetization evolution process\n(the model parameters are shown in the caption). The above4\nFigure 4: Transmission of magnetic solitons in damped ferromagnetic nanowires and the anti-damping effect of non-adiabatic\nSTT. (a) Schematic diagram of magnetization dissipation under damping. (b) Propagation of right-handed magnetic solitons in\nferromagnetic nanowires with Gilbert-damping constant α=0.05. (c) Temporal evolution of the magnetization component mz\nin the absence of non-adiabatic STT, where damping constant α=0.01. (d) Temporal evolution of the magnetization component\nmzwith spin-polarized current j=3.7×107A·cm−2. (e) Gilbert-damping dependence of lifetime and moving distance.\nresults highlight two notable aspects. Firstly, the chiral\nmagnetic soliton possesses an inherent velocity linked to its\ninitial magnetization state. Secondly, the solitons’ motion can\nbe stimulated by a spin-polarized current, while preserving\ntheir chirality. The external injection of spin current offers a\nmeans to manipulate chiral magnetic solitons, granting control\nover their forward, backward, and frozen motion.\nUntil now, our analysis is based on a perfect ferromagnetic\nwire in the absence of damping. Strictly speaking, in\nrealistic nanowires, magnetic solitons cannot move over a\nlarge distance due to Gilbert damping. The existence of\ndamping introduces a small torque field, which dissipates the\nenergy of the system during magnetization dynamics, and\nleads to a helical precession of the magnetization towards\nthe direction of the effective field, i.e. the minimum energy\nstate (See Fig. 4(a)). To understand this damping effect in\ngreater detail we performed numerical simulations of single-\nsoliton dynamics. Fig. 4(b) shows the evolution of a right-\nhanded magnetic soliton in a nanowire with Gilbert-damping\nconstant α=0.05. It can be seen that the chiral magnetic\nsoliton degenerates to a homogeneous magnetized state after\npropagating for about 223.8 ps. During the whole process, the\nmagnetic soliton undergoes continuous deformation. This has\ntwo consequences: magnetic soliton spreading and slowing of\ninternal oscillations. In order to characterize the presence of a\nchiral magnetic soliton, we define the soliton polarization Ps=\n[1−min(mz)]\n2. The soliton is deemed to have dissipated when its\npolarization is lower than 5% in comparison to the maximum\nmagnetization. The movement of magnetic solitons within\nferromagnetic nanowires, subject to varying damping, resultsin distinct lifetimes. Fig. 4(e) depicts line graphs illustrating\nthe relationship between the damping coefficient, lifetime,\nand moving distance. The dissipation of solitons due to\ndamping is a challenge to circumvent, and one approach is to\nseek ferromagnetic materials with low damping coefficients.\nHere, we explored the potential anti-damping effect of non-\nadiabatic STT, as depicted in Figs. 4(c) and 4(d). In the\nabsence of external spin current, the mzcomponent of the\nmagnetic soliton diminishes during transmission. However,\nupon injecting an appropriate spin current, the incorporation\nof non-adiabatic STT enables the chiral magnetic soliton to\npropagate uniformly in its original velocity, resulting in a\nsignificantly extended lifetime.\nConclusions.— In this Letter, we have shown that the\ndimensionless LLG equation containing STT is entirely\nequivalent to the generalized nonlinear Schr ¨odinger equation\nwithout any approximation. This remarkable integrable\nsystem enables us to predict novel exact spatiotemporal\nmagnetic solitons. By applying the Darboux transformation,\nwe obtain exact solution of chiral magnetic solitons, emerging\nwithin an isotropic ferromagnetic nanowire. Our analytical\nformulation establishes a distinct correlation between chiral\nmagnetic solitons and the infusion of spin currents, corrobo-\nrating our numerical findings. This interrelation underscores\nthe potential for arbitrary manipulation of magnetic soliton\nmotion through spin current injection. The inherent chirality\nof the micromagnetic structure plays a pivotal role in soliton\nmotion: a reversal in chirality leads to a shift in motion\ndirection. To encapsulate the realism of dissipative devices,\nwe investigate the influence of Gilbert damping on the motion5\nof chiral magnetic solitons. The results reveal that in\nthe presence of damping, chiral magnetic solitons gradually\nevolve toward a uniformly magnetized state. This implies\na topological equivalence between the two magnetization\nstates. We propose selecting an appropriate spin current\nintensity to introduce an anti-damping effect, thereby ensuring\nthe long-distance transmission of the chiral magnetic soliton.\nThese results present new possibilities for developing chiral\nmagnetic soliton-based racetrack memory.\nThe authors thank Prof. H. M. Yu, Prof. L. C. Zhao, Prof. J.\nLiu and Prof. C. P. Liu for their helpful discussions. 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Guo, Communications in Nonlinear\nScience and Numerical Simulation 32, 285 (2016).Supplementary Materials for “Discovery and regulation of chiral magnetic solitons: Exact solution\nfrom Landau-Lifshitz-Gilbert equation”\nXin-Wei Jin and Zhan-Ying Yang∗\nSchool of Physics, Northwest University, Xi’an 710127, China and\nPeng Huanwu Center for Fundamental Theory, Xi’an 710127, China\nZhimin Liao\nSchool of Physics, Peking University, Beijing, 100871,China\nGuangyin Jing†\nSchool of Physics, Northwest University, Xi’an 710127, China\nWen-Li Yang\nPeng Huanwu Center for Fundamental Theory, Xi’an 710127, China and\nInsititute of Physics, Northwest University, Xi’an 710127, ChinaarXiv:2308.08331v1  [nlin.PS]  16 Aug 20232\nIn this supplementary material, we will show more details on the exact geometric mapping between Landau-Lifshitz-Gilbert\nand generalized nonlinear Schr ¨odinger equation, and calculation details of solving the magnetic solitons.\nA. Exact mapping between LLG and GNLS equation: Geometric Representation\nWe identify the magnetization state of ferromagnetic nanowire at any instant of time with a moving space curve in Euclidean\nthree-dimensional space E3. This is achieved by mapping the unit magnetization vector m(x,t)on the unit tangent vector e1\nassociated with the curve. Thus the dimensionless STT-LLG equation (in the absence of damping) becomes\ne1t=e1×e1xx+Qe1x. (A.1)\nIn the usual way, the normal and binormal vectors of the moving space curve are constructed by taking e2in the direction of e′1\nande3=e1×e2. The spatial variations of these orthogonal unit vectors is determined by the Serret-Frenet equations\n\ne1\ne2\ne3\n\nx=\n0κ0\n−κ0τ\n0−τ0\n\ne1\ne2\ne3\n, (A.2)\nwhere κ(x,t)andτ(x,t)are the curvature and torsion of the space curve. In view of (A.1) and (A.2) alongside the orthogonality\nof the three unit vectors, it is easy to obtain\n\ne1\ne2\ne3\n\nt=\n0 −κτ+Qκ κ x\nκτ−Qκ 0 −τ2+Qτ+κ−1κxx\n−κxτ2−Qτ−κ−1κxx 0\n\ne1\ne2\ne3\n. (A.3)\nThe compatibility conditions∂\n∂t/parenleftig\n∂ei\n∂x/parenrightig\n=∂\n∂x/parenleftig\n∂ei\n∂t/parenrightig\n,i=1,2,3, between Eqs. (A.2) and (A.3) lead to the following evolution\nequations for κandτ\nκt=−(κτx+2κxτ)−Qκx, (A.4a)\nτt=/parenleftbig\nκ−1κxx−τ2/parenrightbig\nx+κκx−Qτx. (A.4b)\nOn making the complex transformation Ψ=1\n2κexp/parenleftbig\ni/integraltext\nτdx/parenrightbig\n, we finally arrive at a generalized nonlinear Schr ¨odinger (GNLS)\nequation (it is easy to verify that the real and imaginary parts of (A.5) is equivalent to (A.4a) and (A.4b), respectively)\niΨτ+Ψζζ+2|Ψ|2Ψ−iQΨζ=0. (A.5)\nThus we have proved that the STT-LLG equation can be exactly mapped into the integrable GNLS equation.\nB. Lax Representation and Darboux Transformation\nWe now turn to establish the connection between the solutions of the LLG equation and the GNLS equation. Using the Pauli\nmatrices ( σ1,σ2,σ3), the LLG equation can be rewriten into the matrix form\n/hatwidemt=1\n2i[/hatwidem,/hatwidemxx]+Q/hatwidemx, (B.1)\nwhere/hatwidem=mxσ1+myσ2+mzσ3and[·,·]denotes the Lie bracket of the matrices. For this equation, the boundary condition is\ngiven by lim\nx→±∞/hatwidem=σ3, i.e., lim\nx→±∞m= (0,0,1). Considering the Lax representation of the GNLS (A.5)\n∂Φ\n∂x=UΦ,∂Φ\n∂t=VΦ, (B.2)\nwhere U=U0+λU1,V=V0+λV1+λ2V2andλis the spectral parameter,\nU0=/parenleftigg\n0Ψ\n−Ψ∗0/parenrightigg\n,U1=−σ3,V0=/parenleftigg\ni|Ψ|2iΨx+QΨ\niΨ∗\nx−QΨ∗−i|Ψ|2/parenrightigg\n,V1=−2iU0−Qσ3,V2=2iσ3. (B.3)3\nSuppose Φ1(x,t,λ)andΦ2(x,t,λ)are two linear independence eigenvectors of Lax pair (B.2), then Ω= (Φ1,Φ2)also satisfies\nEq. (B.2). Let g(x,t) =Ω(x,t,λ)/vextendsingle/vextendsingle\nλ=0, we have gx=U0g,gt=V0g. From transformation /hatwideΦ=g−1Φ, we obtain\n∂/hatwideΦ\n∂x=/hatwideU/hatwideΦ,∂/hatwideΦ\n∂t=/hatwideV/hatwideΦ, (B.4)\nwhere\n/hatwideU=g−1Ug−g−1gx=g−1(U−U0)g=λg−1U1g,\n/hatwideV=g−1Vg−g−1gt=g−1(V−V0)g=λg−1V1g+λ2g−1V2g,(B.5)\nLet/hatwidem=−g−1σ3gbe a solution of Eq. (B.1), then\n/hatwidem/hatwidemx=−g−1σ3U0σ3g+g−1σ2\n3U0g=2g−1U0g, (B.6)\nSubstitute (B.3) into (B.5), together with the definition of /hatwidemand (B.6), matrices (B.5) of new Lax pair (B.2) can be rewritten as\n/hatwideU=λ/hatwidem,/hatwideV=λQ/hatwidem−iλ/hatwidem/hatwidemx−2iλ2/hatwidem, (B.7)\nUsing the factor /hatwidem2=I, the compatibility condition /hatwideUt−/hatwideVx+ [/hatwideU,/hatwideV] =0 exactly yields the matrix form LLG equation (B.1).\nThus we proved the Lax gauge equivalence of the GNLS equation (A.5) and the dimensionless STT-LLG equation. Through the\nestablished gauge equivalence detailed above, it becomes evident that given a non-zero solution Φof the GNLS equation (A.5),\nthe corresponding eigenfunctions can be derived via Lax pair (B.2). This process thereby elucidates the determination of the\ninvertible matrices Ω(x,t,λ)andg(x,t). Further through the transformation /hatwideΦ=g−1Φand/hatwidem=−g−1σ3g, we are enabled to\nacquire the solution /hatwidemfor (B.1). Finally, the three components of magnetization, namely mx,my, and mz, can be obtained from\nthe definition of /hatwidem, constituting the non-trivial solution to the original STT-LLG equation.\nTo obtain the dynamical magnetic soliton in the ferromagnetic nanowire, we are going to construct the Darboux transformation\nof (A.5). Let Φ[0]\n1(x,t,λ)andΦ[0]\n2(x,t,λ)be the eigenfunction of the Lax pair (B.2) corresponding to the zero solution of\nthe GNLS equation (A.5). Demonstrating the reciprocity of the Lax pair solution with respect to spectral parameters, it is\nstraightforward to establish that if/parenleftbig\nΦ[0]\n1(x,t,λ1),Φ[0]\n2(x,t,λ1)/parenrightbigTrepresents the solution for Lax pair (B.2) corresponding to the\nspectral parameter λ1, then/parenleftbig\nΦ[0]∗\n2(x,t,λ∗\n1),−Φ[0]∗\n1(x,t,λ∗\n1)/parenrightbigTconstitutes the solution for the corresponding spectral parameter\nλ∗\n1. Denote\nH1=/parenleftigg\nΦ[0]\n1Φ[0]∗\n2\nΦ[0]\n2−Φ[0]∗\n1/parenrightigg\n,Λ1=/parenleftigg\nλ10\n0λ∗\n1/parenrightigg\n, (B.8)\nwhere/parenleftbig\nΦ[0]\n1,Φ[0]\n2/parenrightbigT=/parenleftbig\nexp[−λ1x+2iλ2\n1t−λ1/integraltext\nQdt],exp[λ1x−2iλ2\n1t+λ1/integraltext\nQdt]/parenrightbigT. The Darboux matrix is acquired through\nthe standard procedure\nT[1]=λI−H1Λ1H−1\n1, (B.9)\nleading to the solution Φ[1]=T[1]Φof new spectral problem. Therefore the Darboux transformation is written as\nΨ[1](x,t) =Ψ[0](x,t)−2(λ1+λ∗\n1)Φ[0]\n1Φ[0]∗\n2\n|Φ[0]\n1|2+|Φ[0]\n2|2. (B.10)\nTaking λ1=a+ibwe get the soliton solution of GNLS equation\nΨ[1](x,t) =−2asech[2a(x+4bt+Qt)]exp/bracketleftbig\n2i/bracketleftbig\n(2a2−2b2−Q)t−bx/bracketrightbig/bracketrightbig\n, (B.11)\nand the corresponding eigenfunction Φ[1](x,t,λ) = (Φ[1]\n1,Φ[1]\n2)T. Substitute the above results into Ω(x,t,λ) =/parenleftigg\nΦ[1]\n1Φ[1]∗\n2\nΦ[1]\n2−Φ[1]∗\n1/parenrightigg\n,\ng(x,t) =Ω(x,t,λ)/vextendsingle/vextendsingle\nλ=0,/hatwidem=−g−1σ3gin sequence, we finally obtain exact cycloidal chiral magnetic soliton solution in\nferromagnetic nanowires under the influence of spin current injection4\nmx=/hatwidem12+/hatwidem21\n2=2a\na2+b2[asinh(Ξ)sin(Γ)−bcosh(Ξ)cos(Γ)]·sech2(Ξ),\nmy=/hatwidem21−/hatwidem12\n2i=2a\na2+b2[asinh(Ξ)cos(Γ)+bcosh(Ξ)sin(Γ)]·sech2(Ξ),\nmz=/hatwidem11=1−2a2\na2+b2sech2(Ξ),(B.12)\nwith\nΞ=2a/bracketleftbigg\nx+/integraldisplay\n(Q+4b)dt/bracketrightbigg\n,Γ=2b/bracketleftbigg\nx+/integraldisplay\n(Q−2(a2−b2)/b)dt/bracketrightbigg\n,\nwhere aandbdescribe the wave number and the velocity of the chiral soliton.\nTo gain deeper insight into the interaction dynamics between two chiral magnetic solitons, we continue to utilize gauge\ntransformation (2.2.67) to construct two-soliton solutions based on the above single soliton solutions. The second-order Darboux\nmatrix is expressed as follows\nH2=/parenleftigg\nΦ[1]\n1Φ[1]∗\n2\nΦ[1]\n2−Φ[1]∗\n1/parenrightigg\n,Λ2=/parenleftigg\nλ20\n0λ∗\n2/parenrightigg\n,T[2]=λI−H2Λ2H−1\n2, (B.13)\nand the specific Darboux transformation form of the two-soliton solution is subsequently obtained\nΨ[2](x,t) =Ψ[1](x,t)−2(λ2+λ∗\n2)Φ[1]\n1Φ[1]∗\n2\n|Φ[1]\n1|2+|Φ[1]\n2|2. (B.14)\nTaking λ1=a1+ib1,λ2=a2+ib2, after tedious simplification, we get two-soliton solution of GNLS equation\nΨ[2](x,t) =4η1eiβ2cosh(α2)+η2eiβ1cosh(α1)+iη3/parenleftbig\neiβ1sinh(α1)−eiβ2sinh(α2)/parenrightbig\nη4cosh(α1+α2)+η5cosh(α1−α2)+η6cos(β1−β2), (B.15)\nwhere\nα1=2a1(x+4b1t+/integraldisplay\nQdt),β1=4(a2−b2)t−2b2(x+/integraldisplay\nQdt),\nα2=2a2(x+4b2t+/integraldisplay\nQdt),β2=4(a2−b2)t−2b2(x+/integraldisplay\nQdt),\nη1= [(a2\n2−a2\n1)−(b2−b1)2]a1,η4=−(a2−a1)2−(b2−b1)2,η3=2a1a2(b2−b1),\nη2= [(a2\n1−a2\n2)−(b1−b2)2]a2,η5=−(a2+a1)2−(b2−b1)2,η6=4a1a2.\nContinuing with the same approach in the previous text, we are able to provide a precise expression for the three-component of\nthe magnetization mfor the dynamic chiral magnetic two-solitons. Owing to the complexity of its explicit expression, we opt\nto omit it and solely showcase the corresponding figure. Two typical solutions for the interaction between two chiral magnetic\nsolitons are shown in Fig.(I).\nC. Chirality of cycloidal chiral magnetic soliton in Bloch sphere\nAs shown in the main text, we denote the polar and azimuthal angles of the vector masθandϕ, respectively. This notation\nallows us to express M+=mx+imy=M0sin(θ)exp(iϕ)andmz=M0cos(θ). By employing the three-component analytical\nformulation, we can infer the inverse solution for θandϕ, which in turn can be mapped onto the Bloch unit sphere. This\napproach yields a trajectory map delineating the movement of chiral magnetic solitons across the unit sphere. Consequently,\nwithin the magnetization unit sphere, a chiral magnetic soliton traces a closed curve encompassing a single pole. The trajectories\nof motion for the two distinct types of chiral magnetic soliton solutions on the Bloch spheres can indirectly manifest their\nchirality. Commencing from negative infinity, which corresponds to the pole of ground state, the left and right-handed chiral5\n(a)\n (b)\nFigure I: The interaction between two chiral magnetic solitons. (a) Interaction between left-handed and right-handed magnetic\nsoliton. (b) Bound states formed by two right-handed magnetic Solitons.\nFigure II: Trajectories of chiral magnetic solitons on the Bloch sphere at time t=0. (a) Left-handed chiral magnetic soliton\na=1,b=1, (b) Right-handed chiral magnetic soliton a=1,b=−1.\nmagnetic solitons will give rise to enclosed paths, one proceeding in a clockwise direction and the other counterclockwise. This\nmotion pattern eventually in mirror-symmetrical trajectories.\n∗zyyang@nwu.edu.cn\n†jing@nwu.edu.cn"    },    {        "title": "2309.09755v1.Coherent_Tunneling_and_Strain_Sensitivity_of_an_All_Heusler_Alloy_Magnetic_Tunneling_Junction__A_First_Principles_Study.pdf",        "content": "Coherent Tunneling and Strain Sensitivity of an All −Heusler Alloy Magnetic\nTunneling Junction: A First-Principles Study\nJoydipto Bhattacharya1,2, Ashima Rawat3, Ranjit Pati3, Aparna Chakrabarti†1,2, Ravindra Pandey3\n1Raja Ramanna Centre for Advanced Technology, Indore 452013, India\n2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India and\n3Department of Physics, Michigan Technological University, Houghton, Michigan 49931, USA\nHalf-metallic Co-based full Heusler alloys have captured considerable attention of the researchers\nin the realm of spintronic applications, owing to their remarkable characteristics such as exceptionally\nhigh spin polarization at Fermi level, ultra-low Gilbert damping, and high Curie temperature.\nIn this comprehensive study, employing density functional theory, we delve into the stability and\nelectron transport properties of a magnetic tunneling junction (MTJ) comprising a Co 2MnSb/HfIrSb\ninterface. Utilizing a standard model given by Julliere, we estimate the tunnel magnetoresistance\n(TMR) ratio of this heterojunction under external electric field, revealing a significantly high TMR\nratio (≈500%) that remains almost unaltered for electric field magnitudes up to 0.5 V/ ˚A. In-\ndepth investigation of K-dependent majority spin transmissions uncovers the occurrence of coherent\ntunneling for the Mn-Mn/Ir interface, particularly when a spacer layer beyond a certain thickness\nis employed. Additionally, we explore the impact of bi-axial strain on the MTJ by varying the\nin-plane lattice constants between -4% and +4%. Our spin-dependent transmission calculations\ndemonstrate that the Mn-Mn/Ir interface manifests strain-sensitive transmission properties under\nboth compressive and tensile strain, and yields a remarkable three-fold increase in majority spin\ntransmission under tensile strain conditions. These compelling outcomes place the Co 2MnSb/HfIrSb\njunction among the highly promising candidates for nanoscale spintronic devices, emphasizing the\npotential significance of the system in the advancement of the field.\nI. INTRODUCTION\nIn recent years, significant advancements have been\nmade in controlling spin-dependent tunneling between\ntwo ferromagnetic electrodes separated by an insulating\nbarrier. These developments have a profound impact on\nvarious magnetic data storage technologies, particularly\ndue to the observation of exceptionally high tunneling\nmagnetoresistance (TMR) values1–3. Initially, the\nability to achieve substantially high TMR in magnetic\ntunneling junctions (MTJs) was limited by the use of\namorphous tunnel barriers. However, the landscape\nhas since evolved significantly, thanks to the theoretical\npredictions4–7and subsequent experimental realization\nof epitaxial MTJs8,9.\nThe fabrication of epitaxial Co-based MTJ, which\nexploits the coherent electronic tunneling phenomenon\nto produce large TMR, was a major breakthrough\nin the field of MTJs. TMR values of up to 220% at\nroom temperature and 300% at low temperatures for\nCoFeB/MgO/CoFeB based MTJ was reported by Parkin\nand his co-workers in 2004.10Till now, the highest TMR\nratio in Heusler alloy based MTJs has been observed\nfor Co 2MnSi/MgO/Co 2MnSi junction, which produced\na TMR ratio of 1995% at 4K11which reaches up to\n2610% with Mn-rich and highly Fe-doped electrode.12\nHowever, from ab-initio theory based calculations, TMR\nratio of about 105−108has been reported for MTJs\nwith half-metallic electrodes.13–19\nIn this context, the half-metallic (HM) materials,\nexhibiting metallic behavior for one of the spin-upand spin-down channels and semiconducting for the\nother, have long been expected to work well as a spin\nfilter or spin-injecting source capable of generating\nextremely highly spin-polarized current.20,21Among\nthe various HM materials that have been explored,\nHM Heusler alloys are regarded as one of the most\npromising materials for the electrode, due to their low\nGilbert damping factor, high Curie temperature and\nreasonably good lattice matching with the traditionally\nused semiconductor substrates, e.g.MgO, GaAs etc.5–7\nSince the prediction of first HM Heusler alloy by de\nGroot et.al22, many HM Heusler alloy materials have\nbeen proposed from first-principles calculations23–26and\nmany of these materials were discovered experimentally\nas well.27–29This gives us a wide range of materials to\nchoose from, suitability depends on their electronic and\ngeometric properties.\nThese HM materials being close to 100% spin-\npolarized at Fermi level (E F), provide an enormous\nadvantage over other ferromagnetic electrode materials,\nleading to its wide application in spintronic devices.\nHowever, interestingly, most of the half metals loose\ntheir unique character ( i.e∼100% spin polarization at\nEF), when they are embedded into heterostructures\nconstructed for the purpose of achieving a high TMR\nratio or efficient spin injection into semiconductor spacer\nlayers.30So far, there are quite a few studies on the\nelectronic properties of heterojunction interfaces based\non first-principles calculations.5–7,31It is known that HM\nproperties almost always get affected and completely\nlose the spin-polarized character at interfaces. However,\nthere are a few theoretical exceptions. For example,arXiv:2309.09755v1  [physics.app-ph]  18 Sep 20232\nNiMnSb/CdS, zinc-blende CrAs/GaAs and all Heusler\ninterfaces (Co 2MnSi/Fe 2TiSi, Co 2MnSb/TiCoSb,\nCoFeTiSi/Fe 2TiSi.13,31–33. It is still not well understood\nhow much the interfacial electronic structures affect the\nspin polarization. As long as the functioning of devices\nrelies on coherent tunneling, their performance seems to\nbe sensitive to the condition of the interfaces.31,34\nThe Heusler alloy family is particularly attractive for\nconstructing spintronic devices due to the wide range\nof electronic properties it offers, along with its suitable\ngeometry. Recent literature has highlighted the potential\nfor achieving high TMR values and interface spin polar-\nization at E F(henceforth denoted by SP) in all-Heusler\nbased magnetic tunneling junctions at low bias. Several\nstudies have reported promising findings.13,31,35In\nour present investigation, as in recent past31, we have\nprobed Co 2MnSb, a half-metallic full-Heusler alloy, as an\nelectrode in spintronic devices. This alloy demonstrates\nexceptional bulk properties, including a high Curie\ntemperature, low Gilbert damping, and defect tolerance,\nall of which are essential for successful applications in\nspintronics.\nAlthough MTJ based magnetoresistive random access\nmemory (MRAM) is quite area efficient, there have\nbeen intensive challenges that needed to be overcome\nas far as the technology is concerned. One of the key\nchallenges is to reduce the writing current density, while\nmaintaining the low resistance area (RA) product. So,\nto maintain a reasonably high TMR ratio and low RA\nproduct (by reducing the barrier height at the interface),\nit is therefore crucial to explore various material combi-\nnations. This can offer more flexibility and freedom to\nengineer the device properties, by tuning the interfaces\nand expanding the search for material combinations\nbeyond those that have already been adopted by the\ncommunity.\nHere, for the spacer material, we are focusing on the\nfamily of Heusler alloys with 18 valence electron, per\nunit cell, specifically ternary XYZ compounds. These\ncompounds have elements at X, Y, and Z sites in a 1:1:1\nstoichiometry and consist of diverse chemical groups\nwith atoms spanning columns 1-5, 9-17 in the Periodic\nTable, providing a vast range of materials. However, not\nall possible combinations result in stable compounds.\nOut of 483 possible combinations, only 83 have been\nexperimentally synthesized. By using ab-initio based\nelectronic structure calculations on the remaining 400\nunreported compounds, it is predicted that 50 more\ncompounds should be stable.36,37Further, based on\nthese predictions, 15 compounds were experimentally\ngrown. The XYZ compounds with heavy elements like\nSb, Bi, Sn, or Pb occupying the Z site tend to have\na stable cubic structure, whereas those with light Z\natoms (O, S, Se, N, P, As, C, Si, and Ge) tend to\nhave non-cubic structures and are often metallic36–38.Therefore, out of the 108 experimentally synthesized 18\nvalence electron compounds, we must select a material\nthat is formed in a cubic phase with a relatively low\nlattice mismatch with the chosen electrode. The spacer\nmaterial should also exhibit a relatively low direct\nbandgap (around 1-2 eV) at Γ, which is crucial in\nachieving a low resistance-area (RA) product and highly\ncoherent tunneling. As the spacer material, we have\nthus chosen the direct bandgap semiconductor, HfIrSb.\nThis semiconducting half-Heusler alloy exhibits the\nlowest decay rate at the Γ point and has recently been\nexperimentally synthesized in a single phase.36\nStrain engineering is another alternative method to en-\nhance the writing power, which has been demonstrated\nfor CoFe-based leads, showing possibilities to modify\nTMR ratio and magnetocrystalline anisotropy.39,40Sev-\neral studies have been proposed for the strain mediated\nswitching mechanisms to lower the energy barrier be-\ntween parallel (P) and antiparallel (AP) states.41It has\nbeen shown in earlier studies that strain can control the\nhalf-metallicity, spin polarization at E Fand magnetic\nmoments.42,43\nIn this work, we propose a combination of full and\nhalf Heusler-based MTJ with an interface derived by\nmixing of alloys. This new all-Heusler based junc-\ntion (Co 2MnSb/HfIrSb) possesses smoothly varying elec-\ntronic properties, ranging from half-metallic and mag-\nnetic electrode, to semiconducting and nonmagnetic\nspacer. To determine which stacking sequence conserves\nthe high SP, we perform electronic structure calculations\nwith various interface geometries. Our calculations sug-\ngest that the HM property is maintained for the Mn-Sb\nas well as (Mn-rich) Mn-Mn (001) surfaces of the elec-\ntrode material. Further, we investigate the spin transport\nproperties for heterojunctions with Mn-Sb/Ir and Mn-\nMn/Ir interfaces and find the tunneling to be highly co-\nherent. Additionally, we explore the influence of bi-axial\nstrain on the spin-dependent transmission which reveals\nthat the introduction of controlled mechanical strain has\na profound impact on the quantum transport behavior\nof MTJs. Specifically, we show that the majority spin\ntransmission can be enhanced up to threefold due to the\nmodified electronic structure of the junction induced by\nthe applied strain. These results have important implica-\ntions for the design and optimization of high-performance\nMTJs39,44,45.\nThe paper is organized as follows. After a brief descrip-\ntion of the methodology adopted in the paper, we first\ndiscuss about the electronic and magnetic properties of\nthe bulk electrode (Co 2MnSb) and spacer (HfIrSb) ma-\nterials by explaining the criteria that brought us to focus\nin this specific combination. Thereafter we discuss the\nelectronic and transport properties of the heterostruc-\ntures with different interfaces. Then we show the strain-\nsensitive transmission properties of the most energeti-\ncally stable interface (Mn-Mn/Ir) under both compres-\nsive and tensile strain. Finally we conclude our work.3\nII. METHODOLOGY\nTo perform electronic structure calculations, we utilize\nthe density functional theory (DFT) based Vienna Ab-\ninitio Simulation Package (VASP)46, using the projector\naugmented wave method implemented in VASP.47We\nemploy the Perdew-Burke-Ernzerhof48generalized gradi-\nent approximation (GGA) for the exchange-correlation\n(XC) functional. An energy cut-off of 500 eV has been\nused to expand the plane waves. The Brillouin zone was\nsampled using the Monkhorst-Pack scheme49with a mesh\nof 17×17×1 k-points. A convergence criterion for energy\nin the self-consistent-field cycle of 10−6eV is adopted. To\noptimize the geometry of the heterojunction systems, we\nfix the in-plane lattice constant of the electrode material.\nThe total force tolerance on each atom is set to be be-\nlow 0.02 eV/ ˚A. For calculating the density of states, we\nuse the tetrahedron method of integration scheme, imple-\nmented in the VASP package, with a mesh of 21 ×21×1\nk-points.\nThe value of spin polarization at E F(SP) has been\ncalculated as follows:\nSP=n↑(EF)−n↓(EF)\nn↑(EF)+n↓(EF)×100%\nHere n↑(EF) and n↓(EF) correspond to the majority and\nminority spin density of states (DOS) at E F, respectively.\nTo investigate the magnetoelectric (ME) property\nand to understand the effect of spin injection bias on\nthe TMR ratio, we have incorporated transverse electric\nfield in our calculations. We have considered series of\nexternal electric fields ranging from 0.01 V/ ˚A to 0.50\nV/˚A in the direction perpendicular to the heterojunction\ninterfaces. For the electric field dependent calculation 15\n˚A of vacuum has been added along the (001) direction\nwith the same in-plane lattice parameter as obtained\nin the previous interface. To mitigate any artificial\nCoulomb interaction resulting from the external electric\nfield, a dipole correction has been incorporated.\nIn order to calculate the TMR ratio for each applied\nelectric field, we have taken the help of the model given\nby Julliere, which is based on two-current model50.\nSince, this model assumes that spin is conserved in\ntunneling in a MTJ, the TMR ratio can be calculated as\nfollows:\nTMR =GP−GAP\nGAP(1)\nwhere G Pand G APcan be calculated from the projected\ndensity of states (PDOS) of the bottom and top mag-\nnetic contact in both parallel (P) and anti-parallel (AP)\nconfigurations as given by,\nGP=e2\nh(n↑\nbottomn↑\ntop+n↓\nbottomn↓\ntop) (2)GAP=e2\nh(n↑\nbottomn↓\ntop+n↓\nbottomn↑\ntop) (3)\nwhere, n↑/↓\nbottomandn↑/↓\ntoprepresent the PDOS value of\nmajority/minority spin carrier for bottom and top mag-\nnetic contact at E F, respectively and eandhare the\nelectronic charge and Plank’s constant respectively.\nFurthermore, the first-principles calculations of ballis-\ntic conductance have been carried out using PWCOND\ncode51as implemented in the Quantum ESPRESSO\n(QE) package52. After obtaining the converged ge-\nometry using VASP, we calculate the spin-dependent\ntransmission, Tσ(k||, E) using the method proposed by\nChoi and Ihm53, also using GGA exchange-correlation\nfunctionals48. The spin-dependent tunneling conduc-\ntance is obtained by using Landauer-B¨ uttiker formula51:\nGσ=e2\nhP\nk||Tσ(k||, EF)\nwhere, σ(=↑,↓), is the spin index and Tσ(k||, EF)\nis the spin-dependent transmission coefficient at the\nenergy EF, with k||= (kx, ky). We set the wave\nfunction and charge density cut-off energy to 60 and\n600 Ry, respectively, and use a 10 ×10×1 k-point mesh\nfor the heterojunction calculations. All calculations\nare converged to an accuracy of 10−8Ry. We resolve\nthe transmission with a large k-grid in the xandy\ndirections (100 ×100) to accurately capture fine spikes\nin transmission. To reduce the 2D plane wave basis set,\nwe use an energy window of 45 Ry. More information\nabout the method for calculating ballistic conductance\ncan be found in Ref.53.\nThe Crystal Orbital Hamilton Population (COHP)\nanalysis has been carried out using the LOBSTER\npackage54,55.The pbeVaspFit2015 basis with the follow-\ning basis functions: Co: 3d, 4s, and 3p; Mn: 4s, 3d and\n4p; Sb: 5s, 4d and 5p; Hf: 5d, 6s and 5p ; Ir: 5d, 6s\nand 4f, have been used for the orbital projection of plane\nwaves. The wavefunctions are obtained from the DFT\ncalculations.\nIII. RESULTS &DISCUSSION\nIII.1. Electronic and Magnetic properties of Bulk\nand Surfaces (001) of Co 2MnSb Alloy\nThe HM nature of the bulk is confirmed through com-\nprehensive electronic structure calculations, as presented\nin Table S1 and Fig. S1.56Analysis reveals the presence\nof ∆ 1and ∆ 5symmetric bands within the majority\nspin channel along the transport direction (from Γ to X\nshown in Fig. S156). These bands play a crucial role in\nfacilitating efficient spin-dependent symmetry filtering\ntransport. Supplementary Information56gives details of4\nthe electronic properties of the bulk electrode.\nWe further investigate the impact of bi-axial strain on\nthe electronic and magnetic properties of the bulk elec-\ntrode. Tensile strain preserves the half-metallic (HM)\nbehavior, while compressive strain destroys it (Table S2,\nFig. S256). The band dispersion and orbital character\nalong the transport direction remain largely unaffected\nby strain (Fig. S3, S4, S556). However, under compres-\nsive strain, we observe the appearance of ∆ 1symmetric\nbands in the minority spin states (Fig. S356), likely to\ninfluence the transmission behavior, which will lead to\nthe reduction of TMR ratio under compressive strain.\nTo investigate the interfacial properties of the ferro-\nmagnetic/semiconductor heterostructure, we initially\nexamined the free-standing surface slabs of Co 2MnSb\n(001). Three different atomic terminations were con-\nsidered: Co-Co, Mn-Sb, and Mn-excess Mn-Mn, each\nconsisting of 17 diatomic layers with a 15 ˚A vacuum\nalong the z direction to prevent any interaction between\nthe slabs along the (001) direction due to the periodic\narrangement of the same. The energetic, electronic,\nand magnetic properties of these surfaces are presented\nin Table S3.56Analysis of the projected density of\nstates confirmed the preservation of the HM property\nin case of the Mn-Sb and Mn-Mn terminated surfaces\n(Fig S856). As Heusler alloys with high surface SP are\ndesirable for spintronic devices, we selected these two\nsurface terminations for further investigations on the\nheterostructures with semiconductors.\nIII.2. Bulk properties of HfIrSb Alloy\nIn the MgAgAs-type crystal structure of HfIrSb, there\nare four interpenetrating FCC sublattices: a rock-salt\nstructure formed by a lattice of Hf atoms and a lattice\nof Sb atoms, and a lattice of Ir atoms occupying the\ncenter of every other Hf4Sb4 cube. The remaining\nHf4Sb4 cubes have vacant centers. The calculated\nlattice constant and band gap of HfIrSb (Table S156)\nagree well with other calculations performed with the\nGGA XC term.57,58The lattice constant of HfIrSb is\nabout 5% larger than that of Co 2MnSb. This difference\nhas implications for the local bonding between atoms,\nas well as for the resulting electronic and transport\nproperties, which will be discussed later.\nFig. 1 depicts the atom projected band structure\nalong the high symmetry path in the 1st Brillouin zone,\nalong with the total DOS for HfIrSb. The compound\nhas a direct bandgap of approximately 0.89 eV at the Γ\npoint, whereas the experimental band gap is found to be\n1.3 eV.57The atom-projected band structure suggests\nthat the top of the valence band is mainly composed\nof contributions from Hf and Ir atoms, whereas the\nconduction band is predominantly due to Sb atoms.Since Hf and Ir have delectrons in the valence, the\nvalence band is expected to be mostly delectron derived.\nFurther, it is crucial to understand the symmetries\nof the conduction band minima (CBM) and valance\nband maxima (VBM) of the spacer material because we\nexpect that the contribution of electrons for transport\nwith different symmetries also differs. As can be seen\nfrom the orbital projected band-structure in Fig. 1(d)\nto (f), there are two valence bands along the Γ to X\ndirection that touches the E F. One has predominantly\n∆1and ∆ 2orbital symmetry (Fig. 1(d), (e)) whereas\nthe other shows predominant ∆ 5orbital character (Fig.\n1 (f)). However, due to the presence of spelectrons\nin the valence shell of Sb atoms, the CBM exhibits\ndominant ∆ 1symmetry. This suggests both the ∆ 1\nand ∆ 2states are likely to contribute to tunneling. We\nextend our investigation to examine the impact of spin-\norbit coupling (SOC) interaction on the electronic band\nstructure of HfIrSb. In Figure S656, the band structure\nof HfIrSb with SOC is presented. The introduction of\nSOC breaks the degeneracy of the valence bands at the\nΓ point, leading to a reduction of the band gap to 0.67\neV. Furthermore, we observe the splitting of valence\nbands as we move away from Γ. Notably, conduction\nbands near the Γ point exhibits a Zeeman-like spin\nsplitting of bands. Analyzing the orbital-projected\nbands (Figure S6 (b) - (d)56) incorporating SOC, we\ndiscern that while the CBM is predominantly of ∆ 1\norbital character, the valence band maximum (VBM) is\nprimarily characterized by the ∆ 5orbital.\nThe periodicity of the bulk crystal requires that the\nBloch k-vectors are real, but in metal-semiconductor (or\nmetal-insulator) interfaces, the metal-induced gap states\n(MIGS) play a crucial role. These states are itinerant in\nthe metallic electrodes and exponentially decaying in the\ninsulator. Their solutions with complex-k vectors result\nin complex band structures for the insulators.59In this\nstudy, we compare the complex band structures of HfIrSb\nalong the (001) direction, which is the propagation direc-\ntion of electrons in the heterojunctions.\nTo investigate the tunneling behavior in HfIrSb, we\nperform band structure calculations of the spacer mate-\nrial. Both real and complex K-values at k ||= Γ were\nconsidered, as depicted in Fig. 1(c). Understanding the\ndecay rate in the barrier layer, where the complex band\nenergies intersect with the E F, represented by Im(k), is\ncrucial for comprehending how tunneling electrons ap-\nproach the barrier layer perpendicular to the surface.\nA lower decay rate suggests that the electrons travel a\nshorter effective distance before encountering the barrier\nlayer59. For HfIrSb, we observe that the complex bands\nintersected at the E Fat a specific k-point in the complex\nregion, resulting in a lower decay rate of Im(k) = 0.112π\na\nat the Γ point (Fig. 1(c)), in contrast to the case of MgO\n(0.212π\na).31In our previous study on a MTJ with TiCoSb\nas spacer layer, which is an indirect band gap semicon-5\nX Г X Г X∆1∆5 ∆2(d) (e) (f)\n(a)(c)\nГ Г\n(d) (e) (f)\n∆1∆5 ∆2\n(c)\n(b) (a)\nFigure 1. The electronic properties of the HfIrSb spacer ma-\nterial are presented. Panel (a) shows the atom-projected\nband structure, while panel (b) shows DOS. Panel (c) depicts\nthe complex band structure along the Γ to X direction. Fi-\nnally, panels (d)-(f) display the orbital-projected band struc-\ntures. Here ∆ 1, ∆2, ∆5represent the dz2, (dx2−y2, dxy) and\n(dxz, dyz) orbital characters, respectively.\nductor, we have found decay rates of 0.252π\naat the Γ\npoint and 0.142π\naat the X point.31Therefore, we antici-\npate a larger Γ-centric tunneling in case of HfIrSb spacer\nlayer compared to TiCoSb and MgO34.\nIII.3. Co 2MnSb/HfIrSb/Co 2MnSb Heterojunction\nIII.3.1. Electronic and Magnetic properties of the\nHeterojunction\nGeometric Structure and Stability: In this sec-\ntion, we examine the electronic and magnetic properties\nof Co 2MnSb/HfIrSb/Co 2MnSb MTJ having different\ninterfaces. Two atomic terminations for HfIrSb along\n(001) crystal orientation were considered: Ir termination\nand Hf-Sb termination. The original structures along\n(001) crystal orientation included Mn-Sb/Ir, Mn-Mn/Ir\nand Mn-Sb/Hf-Sb , Mn-Mn/Hf-Sb interfacial termina-\ntions, which can be divided into two groups: one where\ninterfacial atoms sit on top of Mn or Sb atoms (Top),\nand the other where interfacial atoms are located in the\nbridge site between Mn and Sb atoms (Hollow) (shown\nin Fig. S756). Table I shows the surface free energy\nfor all the various interfaces considered for the present\nstudy. For both terminations, the interface with Ir atoms\nsitting on the hollow side was found to be energetically\nfavorable. Our DFT calculations were performed by fix-\ning the in-plane lattice constant to the electrode lattice\nconstant, and allowing the hetero-junctions to relax in\nthe z-direction. We prepare the supercell of multilayer\ncontaining 17 atomic layers (ML) of Co 2MnSb and 13\nML of HfIrSb for the Mn-Sb and Mn-Mn terminated in-\nterfaces, respectively. The selection of the semiconductorlayer thickness has been made judiciously, taking into\naccount the smaller band-gap of HfIrSb in comparison\nto widely used semiconductors like MgO. This choice\nenables it to serve as an effective barrier layer while\nproviding a reasonably high conductance. We observe\nthat the off-stoichiometric interface Mn-Mn/Ir had\nslightly lower surface free energy than the stoichiometric\nMn-Sb/Ir interface (TableI), and the bond distances\nat the interface are found to be similar to the Mn-Ir\n(2.69 ˚A) and Ir-Sb (2.70 ˚A) bond lengths in the bulk\nIrMnSb60, as shown in Table I.\nCharge Density Difference (CDD): In order\nto understand the chemical bonding at the interface,\nit is crucial to have a comprehensive understanding\nof the charge transfer in the system. The transfer of\ncharge (∆ ρ) at the interface can be visualized in three\ndimensions, as well as two dimensions, as shown in Fig.\n2. The calculation of ∆ ρis carried out by subtracting\nthe spatial charge densities of the electrode and the\nspacer layers from that of the whole heterostructure,\nrepresented by ρElectrode, and ρSpacer, and ρMTJ,\nrespectively.\n∆ρ=ρMTJ-ρElectrode-ρSpacer\nThe value of ∆ ρis positive in the yellow-colored regions\nand negative in the blue-colored regions. A positive value\nindicates an accumulation of electronic charge, while a\nnegative value signifies depletion of electronic charge. In\nFig. 2(a) and (b), it is observed that charge is mostly\ntransferred from the interface Mn and Sb atoms and is\naccumulated around the interfacial region between Mn-\nSb and Ir planes. In Figure S1056, we present the charge\ndensity difference (∆ ρ) at two distinct interfaces (Mn-\nSb/Ir and Mn-Mn/Ir), taking into account the influence\nof spin-orbit coupling (SOC). Remarkably, we find that\nthe charge distribution at the interfaces remains largely\nunchanged with the inclusion of SOC. However, a notable\ndisparity is observed at the Mn-Mn/Ir interface, revealing\nan accumulation of charges around the adjacent Hf atom\nin the subsequent Hf-Sb plane, a phenomenon absent in\nthe absence of SOC.\nMagnetic Properties : In Fig. 2(c), the magnetic\nmoments of the interface Mn atoms are shown across the\njunction for both interfaces. It is observed that the mag-\nnetic moments of the interface Mn atoms exhibit a sud-\nden jump. Moreover, for the Mn-Mn/Ir interface, the\nmagnetic moments of the two interface Mn atoms (Mn 1\nand Mn 2) are different. The difference magnetization\ndensity plots in Fig. 2 (d) and (e) indicate that the\nchange in magnetization is mainly localized at the in-\nterface Mn atom, while the interface Ir atom acquires a\nsmall magnetic moment (-0.1 µB). Additionally, the dif-\nference magnetization density supports the inequivalence\nof the magnetization density of the Mn 1and Mn 2atoms\nfor the Mn-Mn/Ir interface.\nIt should be emphasized that the difference magne-\ntization density around the interface Mn atoms arises6\nprimarily due to the localized d yz,xz orbitals. Further-\nmore, the small magnetic moment of the Ir atom at the\ninterface induced by the proximity effect of the magnetic\nlayer arises mainly from the out-of-plane d z2orbitals, as\nshown in Fig. 2 (d), (e).\nBader Charge and COHP Bond Analysis: The\nnet charge transfer at Co 2MnSb/HfIrSb interfaces as\ncalculated from Bader charge analysis61is shown in\nFig.S956. For both the interfaces we observe that the Ir\natom at the interface loses significant amount of charge\n(-0.27e) compared to the interface Mn (-0.14e) and Sb\n(-0.07e) atom, that get accumulated in the interface\nregion. Similar trend can also be observed from our\nMulliken and Loewdin charge analysis ( Table S4.56) as\nobtained from LOBSTER package55. Additionally, our\norbital-decomposed charge density analysis, facilitated\nby the LOBSTER package55, highlights that out-of-\nplane orbitals ( sanddz2for Ir and pzfor Sb) witness\nthe most significant charge losses, whereas for Mn, it\nis the in-plane ( dx2−y2, dxz) orbital that experiences\nsubstantial charge loss. These findings corroborates\nwell with our charge density and magnetization density\ndifference analysis presented in Fig.2.\nTo gain deeper insights into the interface bonding,\nwe conducted the Crystal Orbital Hamilton Population\n(COHP) analysis between Mn-Ir and Sb-Ir atom pairs, as\ndepicted in Fig.3. Additionally, in Fig.3, we present inte-\ngrated COHP values (ICOHP) between these atom pairs,\nserving as an indicator of bond strength by integrating\nthe COHP values up to the Fermi energy (E F). For the\nMn-Sb/Ir interface (Fig.3 (a)), the COHP analysis be-\ntween Mn-Ir atom reveals presence of anti-bonding states\nnear E Fin the majority spin channels, which lowers the\nbonding interaction. However, the situation is different\nfor the minority-spin channel, where the whole valance\nband (VB) has bonding interaction. Similar observations\nhave also been made in some previous bonding analysis\ninvolving 3 dtransition metal atoms62–64. Because, of the\nexchange hole, the majority spin orbitals are more spa-\ntially contracted than the minority spin orbitals and con-\ntributes less to the bonding interactions62. The COHP\nbonding analysis between the Ir-Sb pair mostly revealed\nbonding interaction in the valance band for both the spin\nchannel, apart from the presence of small anti-bonding\ninteraction in the minority spin-channel in the VB away\nfrom E F, which is compensated by the strong bonding\ncharacter deep into the energy. The ICOHP values in\nFig.3 indicates the Sb-Ir (ICOHP:-2.35 eV) bonding is\nsignificantly stronger than the Mn-Ir (ICOHP:-0.79 eV )\nbonding at the interface.\nIn the case of the Mn-Mn/Ir interface, the COHP anal-\nysis uncovers anti-bonding states in the majority spin\nchannels near E Fbetween the both, Mn 1-Ir and Mn 2-Ir\nbonds, contributing to even weaker bonding compared\nto the Mn-Sb/Ir interface. In contrast, the minority-spin\ncontributes exclusively to the bonding interaction belowEF(Fig.3 (b)).\nThe weaker covalent bonding at the Mn-Mn/Ir inter-\nface compared to the Mn-Sb/Ir interface, as indicated\nby our COHP bonding analysis is further supported by\nthe Bader charge analysis, which suggests relatively less\ncharge transfer at the Mn-Mn/Ir interface compared to\nthe Mn-Sb/Ir Interface.\nElectronic Density of States &Band Structure:\nTo investigate the influence of interfacial interactions on\nelectronic behavior, we analyze the projected DOS of\nthe interface atoms, as depicted in Fig. 4. The atom\nprojected DOS of the interfacial Mn atoms for both the\nMn-Sb/Ir and Mn-Mn/Ir interfaces are shown in Fig. 4\n(a) and (d), respectively. Our analysis reveals that the\nhalf-metallic character of the bulk is disrupted for both\ninterfaces. However, the Mn-Sb/Ir interface still exhibits\na high degree of spin polarization (approximately 68%).\nThe DOS of the different interface Mn atoms for the\nMn-Mn/Ir interface significantly differ from each other.\nSpecifically, the Mn 1atom displays high majority spin\ndensity at around -1 eV, while the Mn 2atom exhibits\na slightly shifted density of states, which is towards the\nhigher binding energy side (approximately -1.8 eV). This\nresults in a higher exchange splitting energy and thus\nlarger magnetic moment for the Mn 2atom is observed.\nWe show various d-orbital contributions of the Mn\natom for the Mn-Sb/Ir and Mn-Mn/Ir interfaces in Fig.\n4(c) and (f), respectively. Here, d 1, d2, d3correspond\nto the d z2, (dxy, dx2−y2), and (d yz, dxz) orbitals of the\ninterface Mn atoms, respectively. Though the formation\nof the heterojunction breaks the periodicity along the\nz-direction, the d 1orbital retains the half-metallic\ncharacter of the bulk. However, the d 2and d 5orbitals\nare primarily responsible for the destruction of the\nhalf-metallicity at the interface. There is a significant\ndifference between the majority spin DOS at the E Ffor\nthe two interfaces. For the Mn-Sb/Ir interface, there\nis a dominant contribution from the d 5orbitals (Fig.\n4(c)), while for the Mn-Mn/Ir interface, both d 2and d 5\norbitals have a significant contribution. Since the orbital\ncharacters of the traveling electrons play a crucial role\nin transmission, one would expect differences in the\nspin-transmission behavior between these two interfaces,\nwhich are discussed in the subsequent section.\nThe Ir atom at the interface becomes metallic for both\ninterfaces, while the semiconducting behaviors persist in\nthe bulk region (Fig. 4 (b), (e)). Consequently, there is\na small gain in magnetic moments (approximately -0.1\nµB) for the interface Ir atoms for both interfaces.\nFigure. S1156shows a comparison of total DOS of the\ninterface atoms for each interfaces of Co 2MnSb/HfIrSb\nheterojunction, with considering the effect of SOC and\nwithout SOC. As depicted in Figure. S1156, the DOS\naround the E Fgets hardly affected by SOC. Therefor we\ndo not include the effect of SOC in the following part of7\nTable I. The calculated interface free energies of various optimized Co 2MnSb/HfIrSb interface heterostructures are presented,\nalong with the corresponding bond lengths between interfacial atoms and the observed spin-polarization (SP) at the interface\nof the heterostructure showing lowest surface free energy for each termination.\nSurface termination Atomic position Interfaced Surface free energy (in eV/ ˚A2) Bond Type Bond-length (in ˚A) Interface SP (in %)\nMn-Sb Top MnSb—Ir -6.8106\nMnSb—HfSb -6.8122\nHollow MnSb—Ir -6.9206 Mn-Ir, Sb-Ir 2.69, 2.70 68\nMnSb—HfSb -6.8482\nMn-Mn Top MnMn—Ir -6.9141\nMnMn—HfSb -6.8312\nHollow MnMn—Ir -6.9825 Mn1-Ir, Mn2-Ir 2.60, 2.57 48\nMnMn—HfSb -6.7456\nSpacer Layer(c)\n(Mn2)\n(Mn1)\n(d)\n(e)\nMinMax\nMn2 Mn2 Mn1Mn Mn SbCo\nCo\nIrIr\n(a)\n(b)IrMn Mn SbCo\nCo\nMn2Mn2Mn1\nIr IrIrMn Mn Sb\nMn1Mn2 Mn2\nMinMax\nFigure 2. The charge density difference (∆ ρ) of the heterostructure is plotted in two forms: a 3D visualization and a 2D\nvisualization in the yz plane. Panels (a) and (b) show the ∆ ρat the Mn-Sb/Ir interface and Mn-Mn/Ir interface, respectively,\nwhere blue and yellow colors in the 3D visualization indicate negative and positive ∆ ρ, respectively. The isosurface value is\nset to 0.0005 e/A3for both cases. Panel (c) presents the magnetic moment of the Mn atoms across the Co 2MnSb/HfIrSb\nheterojunction for both the Mn-Sb/Ir and Mn-Mn/Ir interfaces. (d) and (e) depict the magnetization density difference\nprojected in 2D yz plane for Mn-Sb/Ir and Mn-Mn/Ir interface, respectively.\nour discussion.\nTo achieve a high TMR ratio, it is essential to minimize\nthe current passing through the barrier when the magne-\ntization of the electrodes is anti-parallel. When perfect\nhalf-metallic electrodes are used, spin flipping and tun-\nneling to or from an interfacial state can produce current\nin an anti-parallel configuration. The tunneling proba-\nbility of carriers in various bands can vary significantly\ndepending on their band symmetry, as demonstrated in\nthe literature.5,7Electrons in states with ∆ 1orbital sym-\nmetry exhibit weak decay within the barrier material,\nwhereas the transmission of electrons in other symmetry\nstates is exponentially suppressed. Our prior research31\nillustrated the presence of ∆ 1symmetric bands in the\nmajority spin channel for Co 2MnSb. This suggests that\nit is easy for majority spin electrons to tunnel through\nthe barrier in a parallel spin configuration, which is a\nprerequisite for achieving a high TMR.\nNonetheless, it is equally important to reduce the tun-\nneling rate into minority interface states to suppress the\ncurrent for anti-parallel magnetization. We present theband structure of minority spin states for the heterojunc-\ntion with Mn-Sb/Ir and Mn-Mn/Ir interfaces in Fig. 5.\nIn Fig. 5(b) and (e), we demonstrate the contribution\nof interface atoms (i.e., Mn, Sb, and Ir) to the minority\nspin bands. The orbital projected band structures of the\ninterface atoms in Fig. 5(c) imply that the minority spin\nconduction bands for the Mn-Sb/Ir interface have a dom-\ninant ∆ 1character, suggesting a larger transmission for\nthe minority states. Conversely, for the Mn-Mn/Ir inter-\nface (Fig. 5(f)), mostly in-plane d orbitals dominate the\nminority spin states near E F, leading to poor coupling\nwith the ∆ 1type bands of the HfIrSb spacer material.\nIII.3.2. TMR Ratio and the Effect of Electric Field\nNext, we aim to investigate the influence of an exter-\nnal electric field on the TMR ratio of the heterojunction.\nOur objective is to explore the potential for achieving\nelectrical control of magnetic tunnel junctions (MTJs).\nThe TMR ratio has been calculated according to the Eq.8\n(a)\n(b)\nFigure 3. COHP analysis of the bonds between the interface\natoms for Co 2MnSb/HfIrSb heterojunction: (a) Mn-Sb/Ir ;\n(b) Mn-Mn/Ir interface, respectively. E Fhas been set to\n0 eV. The positive (negative) values at the x-axis indicates\nbonding (anti-bonding) feature. The sign ↑(↓) indicate ma-\njority (minority) spin contributing to the bonding.\n(a) (b)\n(c) (d)\nDOS (states/eV-atom)DOS (states/eV-atom)\nEnergy (eV) Energy (eV) Energy (eV)\nLDOS (states/eV-atom)LDOS (states/eV-atom)\nFigure 4. The local density of states (LDOS) for the interfa-\ncial Mn, Sb, and Ir atoms are shown in panels (a), (b), and\n(d), (e), respectively, for the heterojunctions with Mn-Sb/Ir\nand Mn-Mn/Ir interfaces. The LDOSs in the bulk region are\nalso presented as a reference using filled curves in each figure.\nPanels (c) and (f) display the orbital projected DOS of the\ninterfacial Mn atom for the heterojunctions with Mn-Sb/Ir\nand Mn-Mn/Ir interfaces, respectively, with d 1, d2, d3denot-\ning dz2, (dxy, dx2−y2), and (d yzdxz), respectively.\n(a) (b)\n(d)(c)\n(e)\n(f)\n(a) (b) (c)\n(d) (e) (f)\nX Γ M X Γ M X Γ M\nX Γ M X Γ M X Γ MEnergy (eV)\nEnergy (eV)Figure 5. (a) Minority spin band structure for the considered\nheterojunctions with Mn-Sb/Ir (Mn-Mn/Ir) interface on the\ntop (bottom) panel along the X-Γ-M high symmetry direc-\ntions of the 2D Brillouin zone; (b), (c) represent the atomic\nand orbital contribution of the inter-facial atoms on the mi-\nnority spin band structures, respectively. Here ∆ 1∆2and ∆ 5\nrepresent ( s, p z, dz2) , (d xy, dx2−y2), and (p x, py, dyzdxz)\norbitals respectively.\n[1] based on the standard model given by Julliere.50As\ndepicted in Fig. 6, we observe that the TMR values for\nboth the Mn-Sb/Ir and Mn-Mn/Ir interfaces exhibit min-\nimal sensitivity to the external electric field. Significant\nTMR values are observed for both interfaces. However,\nwe do not observe any magnetoelectric coupling, as has\nbeen reported in some other MTJs.65–67Additionally, the\napplication of an external electric field does not signifi-\ncantly affect the magnetic moments at the interfaces. In-\nstead, we observe a proximity effect where the interface Ir\natoms acquire small magnetic moments (approximately\n0.10µB), and the direction of these moments depends on\nthe magnetization direction of the adjacent Mn atoms. In\nFigure. S1256we have further shown the charge density\ndifference (∆ ρ) of Co 2MnSb/HfIrSb heterojunction, fea-\nturing the Mn-Sb/Ir interface, in the presence and ab-\nsence of an external electric field. This clearly depicts\nthat the charge distribution at the interface remains unaf-\nfected under the application of external electric field. We\nonly observe difference in charges at those layers, which\nare exposed to the vacuum. The absence of electric field\ncontrol over the magnetic properties in the all-Heusler\nalloy junctions can be attributed to the weaker covalent\nbonding between the interface atoms, as suggested by\nthe negligible interface buckling observed. This weaker\nbonding stands in contrast to other Heusler alloy and\noxide-based magnetic tunneling junctions, which exhibit\ngreater electric field control.67\nIII.3.3. Spin-Transport Properties\nOur focus now turns to examining the spin-resolved\ntransport properties of the heterojunction . We present9\nFigure 6. The change in tunnel magnetoresistance (TMR)\nratio of the heterojunction system as a function of an applied\nexternal electric field is depicted. The schematic represen-\ntation of the heterojunction includes a top and bottom mag-\nnetic layer. The left (right) panel illustrates the ferromagnetic\n(anti-ferromagnetic) alignment of spins within the magnetic\nlayer. The orientation of magnetization is denoted by black\nand blue arrows, while the red arrow indicates the direction\nof the electric field.\nthe results for the parallel (P) and anti-parallel (AP) spin\nalignments in Fig. 7. Fig.7 (a) shows the majority spin\ntransmittance (in log scale) as a function of spacer layer\nthickness for both interfaces. The exponential decay of\nthe transmittance with spacer layer thickness confirms\nthe tunneling behavior of the heterojunction34,68. Fig-\nures 7(b) and (c) illustrate the energy-dependent spin\ntransmission for the heterojunction with Mn-Sb/Ir and\nMn-Mn/Ir interfaces, in P and AP spin configuration of\nthe electrodes, respectively. Upon examining the major-\nity spin transmission for both the heterojunctions, we\nobserve that two distinct features emerge. Firstly, in the\nenergy range from ≈-0.61 eV to 0.15 eV, the transmis-\nsion decays, suggesting the tunneling of electrons with\nenergies lower than the barrier. Secondly, a sudden drop\nin transmission occurs around 1 eV, since the ∆ 1band of\nthe Co 2MnSb electrode, which is the primary contributor\nto electron tunneling, extends up to just below 1 eV (see\nFig. S1(d)56). Beyond this energy, it is mostly the bands\nwith ∆ 5symmetry that contribute to the tunneling. For\nboth spin channels and magnetic configurations in Fig.7\n(b), (c) E Fof the Co 2MnSb/HfIrSb/Co 2MnSb junction\nlies close to the valence band of the bandgap of HfIrSb.\nDue the HM nature of the electrode, we do not observe\nany transmission at E Ffor the minority spin channel as\nwell as for majority and minority spin channels for AP\nconfiguration of the electrodes . For the AP configura-\ntion, as expected, the transmission coefficients are nearly\n(not exactly, because the inversion symmetry is broken)\nspin-degenerate. The transmission behavior that we have\ndescribed indicates a significant level of spin-filtering in\nthe junction.In Fig. 8, we present the k-dependent majority spin\ntransmission in the 2D-Brillouin Zone at E Ffor two\nheterojunctions with varying spacer layer thicknesses.\nOur analysis of the k-dependent transmission reveals\nthat the transmission predominantly occurs around the\nlowest decay point, which is located at Γ for both the\ninterfaces. However, there are notable differences in\nthe transmission profiles for the two interfaces. For the\nMn-Sb/Ir interface, the transmission profile exhibits\n2-fold rotational symmetry, whereas for the Mn-Mn/Ir\ninterface, it shows a combination of 4-fold and 2-fold\nrotational symmetry. This distinction suggests that\nthe majority spin carriers for these two interfaces have\ndifferent orbital symmetries. We have shown the orbital\nprojected DOS for the majority spin states at E Fin Fig.\n3(c) and (f), which indicates that d xz,yz orbitals have a\nsignificant contribution for the Mn-Sb/Ir interface, while\nfor Mn-Mn/Ir, d xz,yz as well as d x2−y2,xyorbitals play an\nimportant role. This observation explains the difference\nin the transmission profiles for the two cases, where\nthe heterostructure with Mn-Sb/Ir interface exhibits\nmajority spin transmission dominated by electrons with\n∆5orbital symmetry, while for Mn-Mn/Ir, the ∆ 2and\n∆5states dominate. Previous studies have established\nthat electrons with ∆ 2orbital symmetric states decay\nfaster than those with ∆ 5states inside the barrier region.\nThis is further supported by the variation of the absolute\nsquare of the scattering wavefunction at E Fas a function\nof heterojunction layer thickness (as depicted in Figure.\nS13)56. This shows for the Mn-Mn/Ir interface the\nscattering states inside the barrier decays faster than of\nMn-Sb/Ir interface. Consequently, we observe that the\nmajority spin transmittance with Mn-Mn/Ir interface is\nsmaller than that with the Mn-Sb/Ir interface (Fig. 7\n(a)). Additionally, we notice that with increasing barrier\nthickness, the transmissions of tunneling electrons with\nfinite k ||values are highly suppressed due to their\nfaster decay compared to those with k ||=0 and the\ntransmission is mostly centered around the Γ (Fig. 8 (c)\nand (f)). We, further observe that with increasing spacer\nlayer thickness, the transmission due to ∆ 2orbital\nsymmetric bands at the Mn-Mn/Ir interface is getting\nsuppressed and transmission due to the ∆ 5orbitals\nsymmetric bands sustains (Fig.8 (e)).\nIII.3.4. Effect of Strain Engineering on spin-transport\nproperties\nStrain engineering offers a promising approach to con-\ntrol the electronic and magnetic properties of a mate-\nrial, which is essential for improving the performance\nof spintronic devices. Several studies in the literature\ndemonstrate that modifying the TMR and magneto-\ncrystalline anisotropy of MTJs using strain engineering\nis feasible.39,69It is also reported that strain-mediated\nswitching mechanisms can reduce the energy barrier be-10\n(a)\n(b)(c)\n(a)\n(b) (c)\nFigure 7. (a) Transmittance in parallel magnetization case for Co 2MnSb/HfIrSb/Co 2MnSb junction with varying thicknesses\nof HfIrSb layers; (b), (c) Energy-dependent spin-resolved transmission coefficients for the same with 13 ML of HfIrSb layer\nwith Mn-Sb/Ir and Mn-Mn/Ir interface, top (bottom) panel is for parallel (anti-parallel) spin configuration of the electrodes,\nrespectively. All the graphs are plotted in log scale. In panels (b), (c) E Fhas been set to zero.\n(a) (b) (c)\n(a) (b) (c)\nKX KX\n(d) (e) (f)\nFigure 8. Dependence of the majority spin transmission (val-\nues plotted in log scale) over the 2D Brillouin zone at E Fin\nparallel spin configuration with Mn-Sb/Ir (Mn-Mn/Ir) inter-\nface in top (bottom) panel. (a), (b), (c) ((d), (e), (f) ) are for\n9ML, 13Ml, 17ML of HfIrSb layer, respectively.\ntween parallel and antiparallel states.41,44However, pre-\nvious research on the effect of strain on Heusler alloy-\nbased MTJs mainly focuses on half-metallicity, spin po-\nlarization, and magnetic moment. In this study, we also\naim to investigate the impact of strain engineering on the\nspin-transmission properties of the heterojunction. This\nis especially relevant because the selected material in the\npresent study exhibits approximately +5% lattice mis-\nmatch, indicating that the investigation on the effect of\nstrain on transport properties might be crucial.\nTo achieve strain engineering, we chose a heterojunc-\ntion with an Mn-Mn/Ir interface that exhibits lower sur-\nface free energy (Table I). Further analysis of the elec-\ntronic properties of this heterostructure at the interface\nreveals competing contributions from ∆ 2and ∆ 5states\nin the majority spin states, indicating that the interface\nmight show sensitivity to the bi-axial strain. Therefore,we investigate the majority and minority spin transmis-\nsion of the heterojunction with Mn-Mn/Ir interface over\nthe 2D Brillouin zone under bi-axial strain ranging from\n-4% to 4% as shown in Fig. 9 and 10.\nThe bi-axial strain was applied by fixing the in-plane\nlattice constant and allowing the volume of the hetero-\njunction to relax. Our results show that bi-axial strains\nhave a significant effect on the majority spin-transmission\nproperty, with a change in the transmission profile from\n4-fold rotational symmetry to 2-fold rotational symmetry\nas bi-axial strain changes from compressive to tensile\n(Fig. 9). This implies that the ∆ 2bands dominate the\nmajority spin transmission under compressive strain,\nwhile ∆ 5bands dominate under tensile strain. Addition-\nally, we observe a three-fold increase in the conductance\nof the majority spin channel from 2.26 ×10−5(e2\nh) to\n6.94×10−5(e2\nh) as bi-axial strain changed from -4%\nto 4%, which is consistent with the weaker decay rate\nof ∆ 5bands inside the barrier compared to the ∆ 2bands.\nUnder compressive bi-axial strain, the HM property\nof the electrode is disrupted, resulting in the observation\nof transmission due to minority spin states (Fig. 10).\nSpecifically, for 4% compressive strain, transmission\nspots are detected around the Γ point and at the\ncorners of the two-dimensional Brillouin zone (Fig.\n10). However, these transmission spots diminish in\nsize as the compressive strain decreases, and at 1%\ncompressive strain, the minority transmission becomes\ncompletely absent. Notably, under compressive strain,\na considerable amount of minority spin transmit-\ntance is observed, ranging from approximately 10−5\nto 10−6. This can be attributed to the presence of\n∆1symmetric bands in the minority spin states of\nCo2MnSb under compressive strain (refer to Fig. S3\nin the Supplementary Information56). Consequently,\nthis will lead to further decrement of the TMR ratio\nof the MTJ under compressive strain due to these effects.11\n-4% -3% -2% -1%\n1% 2% 3% 4%\nFigure 9. Dependence of majority spin transmission (values plotted in log scale) over the 2D Brillouin zone at E Fin parallel\nspin configuration with Mn-Mn/Ir interface corresponding to a applied compressive (-4% to -1% ) and tensile (+1% to +4%)\nbi-axial strain. The spacer layer thickness has been chosen to 13 ML.\n-4% -3% -2%\nFigure 10. Dependence of minority spin transmission (values given in log scale) over the 2D Brillouin zone at E Fin parallel\nspin configuration with Mn-Mn/Ir interface corresponding to a applied (compressive) bi-axial strain of -4% to -2%, beyond that\nno transmission is observed for the minority spin states .\n-4%\n4%(a)\n(b)∆1\n∆1∆2 ∆2∆5\n∆5\n∆5∆2 ∆1\n∆2\n∆1 ∆5 ∆2ΓX P NΓM S\nΓX P NΓM SΓX P NΓM SΓX P NΓM SΓX P NΓM S ΓX P NΓM S ΓX P NΓM S\nΓX P NΓM S-4%\nFigure 11. The atom and orbital projected band structure of bulk HfIrSb under strain is investigated, specifically for (a) -4%\n(compressive) strain and (b) +4% (tensile) strain. In this analysis, the symbols ∆ 1, ∆2, and ∆ 5represent the following orbital\ncompositions, respectively: (s, p z, dz2), (d xy, dx2−y2), and (p x, py, dyz, dxz).12\nIn order to investigate the underlying mechanism of\nsuch orbital sensitive majority transmission, we have con-\nducted a detailed analysis of the electronic properties of\nthe bulk electrode and spacer material. Our previous dis-\ncussion has revealed that bi-axial strain induces changes\nin the spin polarization of the electrode, resulting in the\nloss of its HM property under compressive bi-axial strain\n(as shown in Fig. S256). However, we have not observed\nany significant changes in the orbital character of the\nbands under the entire range of applied strain along the\ntransport direction (Γ to X (Z), as depicted in Figures\nS3, S456. Somewhat rigid shift of the bands can only be\nseen.\nAdditionally, we have demonstrated the effect of bi-\naxial strain on the electronic properties of the bulk spacer\nmaterial HfIrSb, as illustrated in Fig. 11. Specifically,\nunder -4% bi-axial strain, HfIrSb becomes an indirect\nbandgap semiconductor, with an increase in the bandgap\nof 1.25 eV compared to the unstained structure, and the\ndegeneracy of the valence bands at the Γ point is also\nlifted. Furthermore, the atom-projected band structure\nanalysis suggests that both the VBM at Γ and CBM\nat the M point have a dominant contribution from the\nHf atoms (as shown in Fig. 11(a)), which is unlike the\ncase of the unstained structure (as shown in Fig. 1).\nThese changes are also reflected in the orbital-projected\nband structure (as presented in Fig. 11 (a)), where we\nobserve that the VBM and CBM have ∆ 2and ∆ 5orbital\ncharacters, respectively.\nOn the other hand, under +4% bi-axial strain, we ob-\nserve that the CBM is mostly Sb atom derived and the\nVBM has a contribution from both Hf and Ir atoms (as\nillustrated in Fig. 11(b)). The orbital-projected band\nstructure shows, the VBM is mostly dominated by the\n∆5orbitals. The comprehensive differences in the elec-\ntronic properties of the spacer layer under compressive\nand tensile bi-axial strain lead to the orbital sensitivity\nof the majority spin transmission.\nFinally, we analyze the results corresponding to the d\norbitals of the interfacial Mn atoms of the heterojunction\nwith Mn-Mn/Ir interface under -4% (compressive) and\n+4% (tensile) strain (Fig. S1456). Our results clearly\nindicate that under -4% strain, the dx2−y2,xyorbitals\nof the Mn atoms have a significant contribution to the\nmajority spin states at the Fermi level, whereas it is\nthedxz,yz orbitals of Mn atom that contributes more\nfor +4% strain. These observations collectively explain\nthe orbital sensitivity of the majority spin transmission\nunder strain.\nOur comprehensive observations indicate that the\ntransmission properties in this heterojunction can be ef-\nfectively tuned by strain. A significant discovery from\nour calculations is that tensile strain enhances the trans-\nmission of the majority spin states, while it considerably\nhinders the transmission of minority spin states. Consid-\nering the lattice mismatch of +5% between our electrodeand spacer materials, this phenomenon can be utilized to\nachieve even higher TMR ratios under tensile strain. In\nessence, the application of tensile strain can further opti-\nmize the performance of the heterojunction and enhance\nthe TMR ratio.\nIII.4. Conclusion\nUsing first-principles density functional theory calcu-\nlations, we have explored electronic and transport prop-\nerties of a Co 2MnSb/HfIrSb/Co 2MnSb all-Heusler mag-\nnetic tunneling junction. We demonstrate that the Mn-\nSb and Mn-Mn terminated surfaces of Co 2MnSb along\nthe (001) direction preserve the half-metallic properties\nof the bulk. From the surface free energies, we pro-\npose that heterojunctions of half-metallic and ferromag-\nnetic alloy Co 2MnSb with direct band gap semiconduc-\ntor HfIrSb is feasible with both Mn-Sb and Mn-Mn sur-\nface terminations. Further, the COHP bonding analysis\nat the interface suggesting the Mn-Ir bonding (ICOHP:-\n1.15, -1.46 eV ) at the Mn-Mn/Ir interface is significantly\nweaker than Sb-Ir bonding (ICOHP:-2.35 eV) at the Mn-\nSb/Ir interface.\nThe results of tunnel magnetoresistance ratios of the\nMn-Mn/Ir and Mn-Sb/Ir interfaces indicate a higher ra-\ntio for the latter compared to the earlier. This is due\nto the significantly less contribution from the minority\nstates in the case of Mn-Sb/Ir interface. However, exper-\nimentally achieved TMR ratio achieved in a heterojunc-\ntion may be affected by several factors and is limited by\nthe interface quality which is governed by the conditions\nof growth.\nTo study the magnetoelectric property and to under-\nstand the effect of spin injection bias on the TMR ratio,\na transverse electric field in the range of 0.01 to 0.5 V/ ˚A,\nin the direction perpendicular to the interfaces, has been\nincluded in our calculations. Utilizing the standard two\ncurrent model given by Julliere, we have calculated the\nTMR ratio of these heterojunctions under the external\nelectric field. Significantly high TMR ratios have been\nobtained for these junctions, which is found to remain\nunaffected by electric field of magnitude up to 0.5 V/ ˚A.\nFurthermore, we demonstrate that the\nCo2MnSb/HfIrSb junction displays remarkable strain-\nsensitive transmission, with a 3-fold increase in majority\nspin transmission and supression of minority spin-\ntransmission under a bi-axial tensile strain at the\nMn-Mn/Ir interface.\nBased on our findings, we predict that a carefully en-\ngineered Co 2MnSb/HfIrSb junction may have enormous\npotential for a range of spintronic applications, includ-\ning magnetic sensors, non-volatile memories, and logic\ncircuits - which awaits experimental validation.13\nIV. ACKNOWLEDGEMENTS\nAuthors thank the Director, RRCAT for facilities and\nencouragement. Authors thank Haiying He for scientificdiscussions. The computer division of RRCAT, Indore\nand MTU, USA is thanked for the help in installing and\nsupport in smooth running of the codes. 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When grown epitaxially as MgO(110)[001] ∥\nMn3Sn(0¯1¯10)[0001], Mn 3Sn experiences a uniaxial tensile strain, which changes the bulk six-fold anisotropy to\na two-fold perpendicular magnetic anisotropy (PMA). Here, we investigate the field-assisted spin orbit-torque\n(SOT)-driven dynamics in single-domain Mn 3Sn with PMA. We find that for non-zero external magnetic\nfields, the magnetic octupole moment of Mn 3Sn can be switched between the two stable states if the in-\nput current is between two field-dependent critical currents. Below the lower critical current, the magnetic\noctupole moment exhibits a stationary state in the vicinity of the initial stable state. On the other hand,\nabove the higher critical current, the magnetic octupole moment shows oscillatory dynamics which could,\nin principle, be tuned from the 100s of megahertz to the terahertz range. We obtain approximate analytic\nexpressions of the two critical currents that agree very well with the numerical simulations for experimentally\nrelevant magnetic fields. We also obtain unified functional form of the switching time versus the input current\nfor different magnetic fields. Finally, we show that for lower values of Gilbert damping ( α≲2×10−3), the\ncritical currents and the final steady-states depend significantly on α. The numerical and analytic results\npresented in our work can be used by both theorists and experimentalists to understand the SOT-driven order\ndynamics in PMA Mn 3Sn and design future experiments and devices.\nI. INTRODUCTION\nAntiferromagnets (AFMs) are a class of magnetic ma-\nterials that produce negligible stray fields, are robust to\nexternal magnetic field perturbations, and exhibit res-\nonant frequency in the terahertz (THz) regime. These\ndistinctive properties are a consequence of strong ex-\nchange interactions between the uniquely arranged spins\nof the neighboring atoms, and a negligible net macro-\nscopic magnetization.1–4AFMs are, therefore, consid-\nered as promising candidates for building next generation\nmagnonic devices, high-density memory devices, and ul-\ntrafast signal generators.5Among the various possible\nAFMs, noncollinear but coplanar metallic AFMs of the\nform Mn 3X, with a triangular spin structure, have re-\ncently been explored extensively owing to their intrigu-\ning magneto-transport characteristics, such as a large\nspin Hall effect (SHE),6anomalous Nernst effect (ANE),\nanomalous Hall effect (AHE)7–10and magneto-optical\nKerr effect (MOKE),11ferromagnet-like spin-polarized\ncurrents,12,13and a finite tunneling magnetoresistance\n(TMR).14,15These noncollinear AFMs are chiral in na-\nture and could be further classified as positive (X = Ir,\nPt, Rh) or negative (X = Sn, Ge, Ga) chirality materials\nbased on the type of spin interaction.16\nHere, we focus on thin-film Mn 3Sn, owing to its var-\nious technologically-relevant properties. Bulk Mn 3Sn,\nwhich is a six-fold spin-degenerate chiral antiferromag-\na)Electronic mail: ankits4@illinois.edu\nb)Electronic mail: siyuanq3@illinois.edu\nc)Electronic mail: rakheja@illinois.edunet with a small net magnetization, has a high N´ eel\ntemperature of approximately 420 −430 K.11,17,18Re-\ncent experiments have demonstrated that the magnetic\norder parameter in Mn 3Sn, referred to as the magnetic\noctupole moment, can be switched between the six sta-\nble states using spin-orbit torque (SOT) in a bilayer\nsetup of AFM and heavy metal (HM).19–23The criti-\ncal charge current density required to switch the mag-\nnetic octupole moment was found to be of the order of\n106−107A/cm2, which is smaller than or comparable to\nthat required to switch the magnetization in most ferro-\nmagnets ( ∼107−108A/cm2).24Some experiments have\nalso alluded to current-driven oscillations of the octupole\nmoment, with frequencies in the range of 100’s of MHz\nto a few GHz, in the bilayer setup.20,25In all of these\nexperiments, the changes in the octupole moment were\ndetected via the AHE since Mn 3Sn exhibits large anoma-\nlous Hall conductivity, ranging between 30 −40 Ω−1cm−1\nat 300 K, owing to the broken time-reversal symmetry\n(TRS).7,26,27The magnitude and sign of the AHE signal\ncan be further modulated by the application of small in-\nplane tensile or compressive uniaxial strain of the order of\n0.1%, as revealed recently.28Such uniaxial strains alter\nthe crystal symmetry, followed by the spin degeneracy,\nand hence the Hall conductivity.28,29\nThin films of Mn 3Sn, when grown epitaxially on\nMgO(110)[001] substrate, also experience in-plane tensile\nstrain, arising from the lattice mismatch between Mn 3Sn\nand MgO. Consequently, the six-fold spin-degenerate\nsystem reduces to a two-fold spin-degenerate system,\nwith a comparatively larger net magnetization, leading\nto perpendicular magnetic anisotropy (PMA) in such\nfilms.30,31The AHE measurements further revealed thatarXiv:2309.10246v2  [cond-mat.mtrl-sci]  5 Mar 2024Mn3Sn with strain 2\nthe magnetic octupole moment of the PMA Mn 3Sn\nfilms, used in a bilayer setup, can be deterministi-\ncally switched in the presence of a symmetry-breaking\nmagnetic field, which is parallel to the current direc-\ntion.30,31Recently, all antiferromagnetic tunnel junctions\ncomprising Mn 3Sn/MgO/Mn 3Sn, utilizing PMA Mn 3Sn,\nwere found to display non-zero TMR of about 2% at\nroom temperature—owing to the TRS breaking and the\nmomentum-dependent spin splitting of electronic band\nstructure.15,32These promising developments make thin-\nfilm PMA Mn 3Sn a strong candidate for future high-\ndensity memory and ultrafast nano-oscillator devices.\nFor a better understanding of the current-driven dy-\nnamics, on the theoretical front, Higo et al. presented\na brief numerical investigation of the different possible\nsteady-states in PMA Mn 3Sn, as a function of the ap-\nplied current and magnetic field.30Their study, however,\nis limited in its scope as it does not provide an insight\ninto the dependence of the different dynamical regimes\non the intrinsic energy scale of Mn 3Sn, or its material\nparameters. Analytic expressions of threshold currents,\nswitching times, and oscillation frequency as functions of\nmaterial parameters and input stimuli are also lacking.\nAnother recent work, focusing on Mn 3Sn with uniaxial\nstrain, numerically investigated only the field-free oscilla-\ntion and pulsed-SOT switching dynamics.29In their very\nrecent work, Yoon and Zhang et al. developed analytic\nmodels of the first- and second-harmonic Hall resistances\nand successfully validated them against experimental ob-\nservation.31Their analysis, however, was limited to the\nquasi-static regime. Previous theoretical works have also\nexplored current-driven switching and oscillation dynam-\nics in AFMs with two-fold spin degeneracy.33–35However,\nthose materials were not representative of Mn 3Sn with\nuniaxial strain since the net magnetization was consid-\nered zero. In this work, therefore, we address the existing\nlimitations and investigate, both numerically and analyt-\nically, the magnetic field-assisted SOT-driven determin-\nistic switching and oscillation dynamics in monodomain\nMn3Sn, with two stable states.\nFor the numerical investigation of the static and dy-\nnamic properties of single-domain Mn 3Sn with in-plane\nuniaxial strain, an energy interaction model is presented\nin Section II of this work. To elucidate the properties of\nthe ground states as well as their dependence on the ma-\nterial parameters, a simpler analytic model of the energy\ninteraction is perturbatively obtained in Section II A, and\nshown to agree well with the numerical results. Next,\nthe field-assisted SOT-driven dynamics of the magnetic\norder in PMA Mn 3Sn is analyzed in Section III. Ana-\nlytic models, pertinent to the current-driven dynamics\nsuch as threshold currents, stationary states, switching\ntime and oscillation frequency, are presented in detail.\nPreviously, we had utilized this framework to analyze\nfield-free SOT-driven dynamics in monodomain Mn 3Sn\nwith six states36. In our present work, however, we build\nmodels that elucidate the impact of strain as well as mag-\nnetic field on the dynamics. The impact of the Gilbertdamping constant on the dynamics is investigated and\nthe salient features are discussed in Section IV. The field-\nassisted SOT-driven dynamics in Mn 3Sn with no strain37\nand compressive strain is discussed in supplementary ma-\nterial, along with a brief discussion of the AHE and the\nTMR detection schemes.\nII. FREE ENERGY MODEL AND GROUND STATES\nBelow its N´ eel temperature, Mn 3Sn crystallizes into a\nhexagonal Kagome D019lattice and can only be stabi-\nlized in slight excess of Mn atoms. The Mn atoms are\nlocated at the corners of the hexagons whereas the Sn\natoms are located at their respective centers. Such lat-\ntices are stacked along the caxis ([0001] direction) in\nan ABAB arrangement. A simple representation of the\ncrystal structure is presented in Fig. 1. In each Kagome\nplane, the magnetic moments on the Mn atoms form\na geometrically frustrated noncollinear triangular spin\nstructure (Fig. 1(a)), with spins on the nearest neighbors\naligned at an angle of approximately 120◦with respect to\neach other.17These spins are canted slightly toward the\nin-plane easy axes, resulting in a small net magnetiza-\ntion, which is six-fold degenerate in the Kagome plane.26\nUnder the application of a small in-plane uniaxial strain,\nthe system becomes two-fold degenerate, and its net mag-\nnetization is also altered.28,30,31\nx:[𝟐𝟏$𝟏$𝟎]y:[𝟎𝟏𝟏$𝟎]z:[𝟎𝟎𝟎𝟏]\n𝐦oct\n𝐦𝟏\n𝐦𝟐\n𝐦𝟑\n(a)\nMnSn\nabc(b)z = 0z = 1/2\nFIG. 1. (a) Atomic and one of the six possible spin arrange-\nments in the basal plane of bulk Mn 3Sn. (b) Unit cell of\nMn3Sn.\nTo investigate the static and the current-driven dy-\nnamic behavior of a single-domain particle of uniaxially\nstrained-Mn 3Sn, comprising three interpenetrating sub-\nlattices, the free energy density is defined as28,30\nF(m1,m2,m3) =JE((1 + δE)m1·m2+m2·m3\n+m3·m1) +DMz·(m1×m2+m2×m3\n+m3×m1)−3X\ni=1\u0010\nKe(mi·ue,i)2+MsHa·mi\u0011\n,(1)\nwhere m1,m2, and m3are the magnetization vectors\ncorresponding to the three sublattices, while JE(>0),\nDM(>0), and Ke(>0) are the symmetric exchangeMn3Sn with strain 3\ninteraction constant, asymmetric Dzyaloshinskii-Moriya\ninteraction (DMI) constant, and single-ion uniaxial mag-\nnetocrystalline anisotropy constant, respectively. Each\nmagnetization vector has a constant saturation magneti-\nzation, Ms. Here, it is assumed that the uniaxial strain\nacts between m1andm2. The effect of this uniaxial\nstrain is included in the empirical parameter δE—a pos-\nitive (negative) value indicates a stronger (weaker) ex-\nchange interaction between m1andm2.28,30,31There-\nfore, a positive (negative) δEcorresponds to a shorter\n(longer) bond length and hence compressive (tensile)\nstrain. The last term in Eq. (1) represents the Zee-\nman energy due to the externally applied magnetic field\nHa. Finally, ue,iis the local easy axis correspond-\ning to mi. The easy axes are assumed to be ue,1=\n−(1/2)x+ (√\n3/2)y,ue,2=−(1/2)x−(√\n3/2)yand\nue,3=x. Mn 3Sn is an exchange dominant AFM such\nthat JE≫DM≫Ke. Typical values of the material\nparameters of Mn 3Sn, considered in this work, are listed\nin Table I. The ground states for different δEandHacan\nbe obtained by minimizing Eq. (1) with respect to mi.\nThe ground states of single-domain strained-Mn 3Sn,\nin the absence of any magnetic field ( Ha= 0), are shown\nin Fig. 2. In all the cases, m1,m2, and m3exhibit\na clockwise ordering with approximately 120◦angle be-\ntween them, all lying within the x-y plane. Compared to\nthe six-fold degeneracy observed in single-domain Mn 3Sn\nwith no strain,36,38a two-fold spin degeneracy is observed\nin strained-Mn 3Sn, where m3coincides with its easy axis\nin the case of compressive strain (Fig. 2(a, b)) while it\nis perpendicular to its easy axis in the case of tensile\nstrain (Fig. 2(c, d)). A small non-zero net magnetiza-\ntion, m=m1+m2+m3\n3, exists in single-domain Mn 3Sn\nwith no strain.36,38We find that strained-Mn 3Sn also\nhosts a net magnetization which is parallel (antiparal-\nlel) to m3in the case of the compressive (tensile) strain.\nFor|δE|= 10−3, which represents a uniaxial strain of\n0.1%, the norm of the net magnetization increases from\nthe bulk value of ∥m∥ ≈3.66×10−3to∥m∥ ≈3.95×10−3\u0000\n∥m∥ ≈3.92×10−3\u0001\nin the case of compressive (ten-\nsile) strain. The non-zero msuggests that the angle\nbetween the sublattice vectors is not exactly 120◦. In-\ndeed, in the case of compressive (tensile) strain, η12≈\n0.78◦(−0.78◦) and η23=η31≈ − 0.39◦(0.39◦), where\nηij= cos−1(mi·mj)−2π\n3. For both compressive and\ntensile strain, η12+η23+η31= 0. For δE= 0, our\ncalculations showed that the respective |ηij|were differ-\nent from the values reported above and depended on the\nground state under consideration.36The aforementioned\nresults with and without strain are as expected—strong\nexchange and DM interactions attempt enforcing a clock-\nwise ordering of m1,m2, and m3with exactly 120◦an-\ngle between them, within a plane perpendicular to the\nz-axis. However, the anticlockwise ordering of ue,1,ue,2,\nandue,3forces the sublattice vectors in the x-y plane\nwith a small deviation from the 120◦ordering. Uniaxial\nstrain leads to further increase in the deviation.\nExternal magnetic fields, when applied to a thinTABLE I. List of material parameters for the AFM, Mn 3Sn,\nand the heavy metal (HM), which is chosen as W, in the SOT\ndevice setup.\nParameters Definition Values Ref.\nJE(J/m3) Exchange constant 2 .4×10816\nD(J/m3) DMI constant 2 ×10716\nKe(J/m3) Uniaxial anisotropy constant 3 ×10616\nMs(A/m) Saturation magnetization 1 .3×10616\n|δE| Strain parameter 10−330\nα Gilbert damping 0 .003 19\nθSH Spin Hall angle for HM 0 .06 30\nFIG. 2. Equilibrium states in single-domain Mn 3Sn crystal\nunder (a, b) compressive and (c, d) tensile strains. The sub-\nlattice vectors lie in the Kagome plane which is assumed to\ncoincide with the x-y plane. (a, b) Only m3coincides with\nits easy axis. (c, d) None of the sublattice vectors coincide\nwith their local easy axes. Instead, tensile strain forces m3\nperpendicular to its easy axis. The two equilibrium states for\ncompressive as well as tensile strains are separated by 180◦\nwith respect to each other. A small in-(Kagome)-plane aver-\nage magnetization, m, exists parallel (antiparallel) to m3in\nthe case of compressive (tensile) strain. The magnitude of m\ndepends on the strength of the strain. Here, mis not drawn\nto scale but magnified by 100 ×for the purpose of clear rep-\nresentation.\nfilm of Mn 3Sn, change the energy of the system, and\ntherefore, the ground states. Here, we only consider\nan external magnetic field in the Kagome plane as\nHa=H0(cos ( φH),sin (φH),0), where φHis the an-\ngle between the magnetic field and the x-axis. Fig-\nure 3(a) shows the ground states of Mn 3Sn when Ha=\n(0,0.1 T,0) is applied to the equilibrium state of Fig. 2(a)\nwhereas Fig. 3(b) shows the ground state when Ha=\n(−0.1 T,0,0) is applied to the equilibrium state of\nFig. 2(c). In both the cases mtilts towards the mag-Mn3Sn with strain 4\nnetic field, while the sublattice vectors either tilt to-\nwards Haor away from it, in order to lower the energy\nof the system. Compared to the equilibrium states of\nFig. 2, the angles between the sublattice vectors change,\nalthough by a negligible amount. However, if the ap-\nplied field is large it could disturb the almost 120◦rel-\native orientation of the magnetic moments. Therefore,\nin this work, we consider relatively small magnetic fields\nthat are sufficient to aid the dynamics (discussed later in\nSec. III) without disturbing the antiferromagnetic order,\nviz.JE≫DM≫H0Ms.\nFIG. 3. Ground states of monodomain Mn 3Sn crystal,\nwith (a) compressive strain and (b) tensile strain, un-\nder the effect of in-plane external magnetic field, Ha=\nH0(cos ( φH),sin (φH),0). In both the cases mtilts towards\nHa. Here, ∥Ha∥= 0.1 T while mis magnified by 100 ×for\nthe purpose of clear representation in both the cases.\nA. Perturbative Analysis\nFor a clear understanding of the aforementioned\nground states of the monodomain strained-Mn 3Sn,\nwe consider the perturbative approach presented in\nRefs. [31, 38–40]. Firstly, we define the sublattice vector\nasmi=\u0010p\n1−u2\nicos (φi),p\n1−u2\nisin (φi), ui\u0011\n, where\nφianduiare its azimuthal angle and the z-component,\nrespectively. Secondly, we define the experimentally rele-\nvant cluster magnetic octupole moment31,40,41asmoct=\n1\n3Mzx\u0002\nR(2π\n3)m1+R(−2π\n3)m2+m3\u0003\n. Here, m1andm2\nare rotated by +2 π/3 and −2π/3, respectively, while the\ny-component of the resultant vector undergoes a mirror\noperation with respect to x-z plane.31,40This ensures\nthat the octupole and the sublattice vectors are copla-\nnar, and φoct=−φ1+φ2+φ3\n3, where φoctis the azimuthal\nangle of the magnetic octupole moment. Thirdly, we de-\nfineφi=−φoct−2πi\n3+ηi, where ηiis a small angle\n(ηi≪2π/3) that includes the effect of small deviation\nfrom the rigid 120◦configuration due to both the frus-\ntrated bulk structure and the strain. Here, ηiis linearly\nindependent of φoctandη1+η2+η3= 0. Finally, we use\nthe perturbative approach, as outlined in the supplemen-\ntary material, to arrive at an energy landscape, which isa function of φoct31,38–40and is given as\nF(φoct)≈ −3A\n2cos (2 φoct)−B\n2cos (6 φoct)\n−3MsH0(Ccos (φoct−φH) +Dcos (φoct+φH)),(2)\nwhere A=2JEδEKe\n3(JE+√\n3DM),B=(3JE+7√\n3DM)K3\ne\n9(JE+√\n3DM)3,C=\nKe\n3(JE+√\n3DM), and D=JEδE\n3(JE+√\n3DM). The constant terms\nare not shown here as they do not affect the ground states\nsolution.\nForH0= 0 in Eq. (2), the cos (2 φoct) term dominates\nover the cos (6 φoct) term, if A≫Band|A| ≫ 3Bin\nthe case of Mn 3Sn with compressive and tensile strains,\nrespectively. For the material parameters listed in Ta-\nble I, these conditions are equivalent to δE≫0.08×10−3\nand|δE| ≫ 0.24×10−3, respectively. Therefore, for\n|δE|= 10−3, compressive (tensile) strain leads to two\nminimum energy equilibrium states of the octupole mo-\nment, corresponding to φoct= 0 and π(φoct=π/2 and\n3π/2), as shown in Fig. 2. On the other hand, when a\nmagnetic field is turned on ( H0>0), the energy of the\nsystem changes, and two ground states of the octupole\nmoment, corresponding to the minimum of Eq. (2), are\npossible. We find that, if φHis 0◦or 180◦(90◦or 270◦)\nfor Mn 3Sn with compressive (tensile) strain, the possible\nground states are same as the initial equilibrium states.\nConversely, if φHis different from the equilibrium di-\nrection, the possible ground states are different from the\nequilibrium states (Fig. 3).\nIn the special case of Haperpendicular to the equi-\nlibrium direction, degenerate ground states with energies\nlower than that of the equilibrium states are obtained.\nThis is depicted in Fig. 4, where an external magnetic\nfield is applied in the negative x-direction ( φH= 180◦)\nto Mn 3Sn with tensile strain. As the strength of the\nmagnetic field increases, the energy of the ground states\ndecrease and they move away from the equilibrium states\nof 90◦and 270◦, towards 180◦. In addition, the en-\nergy barrier separating the two states reduces at 180◦\nbut increases at 360◦. The two ground states exist if\nH0≲2|A+3B|\nMs(C+D). Within this limit, ∥m∥was found to\nincrease with both δEandH0(see supplementary ma-\nterial). For higher H0,φoct=πbecomes a minimum\ntoo and ∥m∥decreases with δEat fixed H0(supplemen-\ntary material). In the case of Mn 3Sn with compressive\nstrain and Haperpendicular to the equilibrium states,\ntwo ground states exist if H0≲2(A+3B)\nMs(C−D). These limits\nsuggest that the maximum value of H0, which could be\napplied while maintaining two ground states, increases\nwith δE. This is because larger δEleads to higher intrin-\nsic energy barrier, which is given as |3A+B|, between\nthe two equilibrium states.\nIII. SOT-DRIVEN DYNAMICS\nTo investigate the dynamics of Mn 3Sn under the effect\nof spin current, we consider the spin-Hall effect (SHE)Mn3Sn with strain 5\nFIG. 4. Energy density as a function of the octupole angle of\na monodomain Mn 3Sn with tensile strain for different applied\nmagnetic field, H0, along the -x-direction ( φH= 180◦). Ex-\nternal magnetic field breaks the symmetry of the system, and\ntherefore, the barrier height reduces at φoct=πbut increases\natφoct= 0, as H0increases.\nsetup shown in Fig. 5. This setup resembles the ex-\nperimental designs from Refs. [30, 31], where Mn 3Sn\ngrown epitaxially on a (110)[001] MgO substrate exhibits\nuniaxial tensile strain in the x-direction resulting in a\nPMA energy landscape for the magnetic octupole mo-\nment.30,31Hereafter, we only focus on the dynamics of\nsingle-domain Mn 3Sn with tensile strain while the discus-\nsion on the dynamics of Mn 3Sn with compressive strain\nand no strain is relegated to supplementary material. In\nour convention, as mentioned previously, the Kagome\nplane of Mn 3Sn is assumed to coincide with the x-y plane\nwhile the z-axis coincides with [0001] direction. Charge-\nto-spin conversion in the HM, due to the flow of charge\ncurrent density, Jc, leads to the generation of a spin cur-\nrent density, Js, polarized along np, which is assumed to\ncoincide with z-axis. Previous works have shown that the\ncurrent required to induce dynamics in this setup, with\nnpperpendicular to the Kagome plane, is significantly\nsmaller than that required in the case where npis paral-\nlel to the Kagome plane.20,30,31Finally, the external field\nHais assumed to be applied in the negative x-direction,\norφH= 180◦.\nFor each sublattice of Mn 3Sn, the magnetization dy-\nnamics is governed by the classical Landau-Lifshitz-\nGilbert (LLG) equation, which is a statement of the con-\nservation of angular momentum. The LLG equations for\nthe three sub-lattices are coupled via the exchange inter-\nactions.16,35For sublattice i, the LLG equation is given\nas42\n˙mi=−γ\u0000\nmi×Heff\ni\u0001\n+α(mi×˙mi)\n−ℏ\n2eγJs\nMsdami×(mi×np),(3)\nwhere ˙mi=∂mi\n∂t,tis time in seconds, Heff\niis the effec-\ntive magnetic field experienced by mi,αis the Gilbert\ndamping parameter for Mn 3Sn, and dais the thickness\n𝐦oct\nJc\n𝐧!Js\n𝐇\"\nxy\nz\n𝐦𝟑\n𝐦𝟏\n𝐦𝟐\n[𝟏#𝟏𝟎][𝟏𝟏𝟎][𝟎𝟎𝟏#]\n[𝟏𝟏𝟏][𝟐𝟏𝟏][𝟎𝟏#𝟏]\n[𝟐#𝟏𝟏𝟎][𝟎𝟏𝟏#𝟎][𝟎𝟎𝟎𝟏#]FIG. 5. Spin-orbit torque device setup for manipulating the\nmagnetic state in Mn 3Sn. The MgO substrate leads to tensile\nstrain in the x-direction, and hence the PMA in Mn 3Sn. Spin-\nHall effect in the HM generates SOT when a charge current\nflows into it. JcandJsare the charge current density and the\nspin current density, respectively. Hais the external magnetic\nfield which is applied to aid the deterministic switching of the\nmagnetic octupole moment, moct, in strained-Mn 3Sn.\nof the AFM layer. Other parameters in this equation,\nviz. ℏ= 1.054561 ×10−34J s,e= 1.6×10−19C, and\nγ= 17.6×1010T−1s−1are the reduced Planck’s con-\nstant, the elementary charge of an electron, and the gy-\nromagnetic ratio, respectively. The spin current density\ndepends on the input charge current density and the spin-\nHall angle of the HM, θSH, asJs=θSHJc. The spin-Hall\nangle is associated with the efficiency of the SOT effect.\nHere, we consider the HM to be W since it has a large\nθSH19.\nThe effective magnetic field for sublattice ican be ob-\ntained by using Eq. (1) as\nHeff\ni=−1\nMs∂F\n∂mi=−JE\nMs(mj+mk)\n+DMz×(mj−mk)\nMs+2Ke\nMs(mi·ue,i)ue,i+Ha,\n(4)\nwhere ( i, j, k ) = (1 ,2,3),(2,3,1),or (3,1,2), respectively.\nEquations (3) and (4) are then solved simultaneously,\nfor a range of H0(>0) and Js, for both φinit\noct=π/2\nandφinit\noct= 3π/2 as the initial states. The steady-state\nresponse of the magnetic order of Mn 3Sn is found to be\ndependent on the initial ground states, magnitude of H0,\nand the direction and magnitude of the input current.\nThese differences in the steady state behavior are shown\nin Figs. 6 and 7 for H0= 0.1 T and φH= 180◦.\nFigure 6 shows the time dynamics of the magnetic oc-\ntupole moment and that of the out-of-(Kagome)-plane\ncomponent of the average magnetization, mz, for the cur-\nrent pulse shown in Fig. 6(b). It can be observed that\nfort <1 ns, where Js= 0, the magnetic octupole mo-\nment evolves to ground states (i) and (ii), for the equilib-\nrium states at φinit\noct=π/2 and φinit\noct= 3π/2, respectively.\nWhen Jsis increased to 1 .5 MA /cm2att= 1 ns, (i)\nand (ii) evolve to stationary steady-states (iii) and (iv),\nrespectively. Although the φoctcorresponding to bothMn3Sn with strain 6\n(iii) and (iv) are larger than those for (i) and (ii), they\nare still near the initial states, that is, 1 .5 MA /cm2is\nnot large enough to induce deterministic switching. Sta-\ntionary steady-states near the initial ground states are\nalso observed when Jsis decreased to −1.5 MA /cm2at\nt= 3 ns: (i) and (ii) evolve to (v) and (vi), respectively.\nIn this case, the φoctcorresponding to the stationary\nstates are smaller than those of the ground states. States\n(iii) and (v) return to the ground state (i) while (iv) and\n(vi) return to the ground state (ii), when the current\nis turned off. The final steady-states, therefore, depend\non both the magnitude and the direction of the input\ncurrent. As shown in Fig. 6(b), mzis zero in the steady\nstate. It changes negligibly, in the direction of the change\ninJs, when the current is turned on or off.\nIt can be further observed from Fig. 6(a) that the mag-\nnetic octupole moment in both the ground states (i) and\n(ii) evolve to the stationary steady-state (vii), when Jsis\nincreased to 2 .5 MA /cm2att= 5 ns. On the one hand,\nthis dynamics corresponds to the deterministic switching\nof the magnetic octupole moment in (i). On the other\nhand, for the magnetic octupole moment in (ii), (vii) is\njust a stationary steady-state with φoctgreater than that\nof (ii). However, when the current direction is reversed\nby lowering Jsto−2.5 MA /cm2att= 7 ns, the mag-\nnetic octupole moment in (ii) switches deterministically\nto (viii), which is near (i) but has a smaller correspond-\ningφoct. In this case, the magnetic octupole moment\nin (i) will not switch, but move to (viii). As shown in\nFig. 6(b) at t= 5 ns and t= 7 ns, deterministic switch-\ning is accompanied by a large spike in mz. The direction\nof change in mzdepends on the direction of the input\ncurrent—positive (negative) Jsleads to positive (nega-\ntive) mz.\nFurther increasing |Js|to 2.69 MA /cm2results in chi-\nral oscillations for the magnetic octupole moment in both\nthe ground states (i) and (ii), as shown in Fig. 7(a). For\npositive Js, the magnetic octupole moment in (i) deter-\nministically switches to (ii) in the first step; therefore,\nthe phase of (i) lags that of (ii). On the contrary, for\nnegative Js, the magnetic octupole moment in (ii) deter-\nministically switch to (i) in the first step; therefore, the\nphase of (ii) lags that of (i). The oscillation dynamics of\nthe magnetic octupole moment is accompanied by large\nmz, as shown in Fig. 7(b). Similar to the case of deter-\nministic switching dynamics, the direction of mzdepends\non the direction of current—positive (negative) Jsleads\nto positive (negative) mz. However, unlike the case of de-\nterministic switching, mzshows two spikes per oscillation\n(inset of Fig. 7(b)).\nDetailed numerical simulations revealed that for\nφinit\noct=π/2 (φinit\noct= 3π/2) and Js>0 (Js<0), the\nfinal steady-state of moctdepended on the magnitude of\nJswith respect to two threshold currents— Jth1\nsandJth2\ns,\nwhere Jth1\ns< Jth2\ns. As summarized in Fig. 8, if the in-\njected current density is smaller than the lower threshold\ncurrent, that is |Js|< Jth1\ns, the ground state of the AFM\nevolves to a non-equilibrium stationary steady-state in\nFIG. 6. Steady-state response of (a) the magnetic octupole\nmoment and (b) the out-of-(Kagome)-plane component of the\naverage magnetization, mz, as a function of time under the\neffect of a current pulse, which is turned on at t= 1 ns\nandt= 3 ns to |Js|= 1.5 MA /cm2while it is increased\nto|Js|= 2.5 MA /cm2att= 5 ns and t= 7 ns. Otherwise\nthe current is turned off. (a) (i) and (ii): Possible ground\nstates for H0= 0.1 T in the negative x-direction and Js= 0.\n(iii) and (iv): Js= 1.5 MA /cm2. No switching. Station-\nary states at angles larger than (i) and (ii), respectively. (v)\nand (vi): Js=−1.5 MA /cm2. No switching. Stationary\nstates at angles smaller than (i) and (ii), respectively. (vii):\nJs= 2.5 MA /cm2. Deterministic switching of (i). No switch-\ning for (ii). Stationary states at angles larger than (ii). (viii):\nJs=−2.5 MA /cm2. Deterministic switching of (ii). Station-\nary states at angles smaller than (i). (b) |mz|is zero in the\nsteady state but increases during the switching process. The\nchange in |mz|is negligible for the case of no switching.\nthe initial energy well. The case of |Js|= 1.5 MA /cm2\nandH0= 0.1 T, shown in Fig. 6(a), belongs to this\nregime. On the other hand, if Jth1\ns<|Js|< Jth2\nsthe\nmagnetic octupole moment deterministically switches to\na non-equilibrium stationary steady-state in the other\nenergy well ( |Js|= 2.5 MA /cm2andH0= 0.1 T in\nFig. 6(a)). Finally, when Jth2\ns<|Js|the magnetic oc-\ntupole moment exhibits steady-state chiral oscillations\n(|Js|= 2.69 MA /cm2andH0= 0.1 T in Fig. 7(a)),Mn3Sn with strain 7\nFIG. 7. Steady-state oscillation dynamics of (a) the magnetic\noctupole moment and (b) the out-of-(Kagome)-plane com-\nponent of the average magnetization, mz, as a function of\ntime under the effect of a current pulse. The pulse is turned\non at t= 1 ns ( t= 3 ns) to Js= 2.69 MA /cm2(Js=\n−2.69 MA /cm2) and turned off at t= 2 ns ( t= 4 ns). (a) (i)\nand (ii): Possible ground states for H0= 0.1 T in the negative\nx-direction. Oscillation dynamics for |Js|= 2.69 MA /cm2:\nthe octupole moment in both (i) and (ii) oscillate at a fre-\nquency of about 2 .5 GHz in a direction decided by the di-\nrection of the input current. When the current is turned off,\nthe octupole moment probabilistically settles into either of\n(i) or (ii). (b) Non-zero mzfacilitates high frequency chiral\noscillations in the steady state due to the strong exchange in-\nteraction between out-of-(Kagome)-plane components of the\nsublattice vectors.\nwhose frequency could be tuned from the 100s of MHz to\nthe 10s of GHz range by varying |Js|. The three regimes\nof operation are marked as I, II, and III for no switching,\ndeterministic switching, and chiral-rotation, respectively.\nThe overlaid dashed white lines represent Jth1\nsandJth2\ns.\nIt can be observed that Jth1\nsdecreases with an increase\ninH0while Jth2\nsincreases with H0. As a result, the\nrange of input currents where the system exhibits deter-\nministic switching increases with H0. For φinit\noct= 3π/2\n(φinit\noct=π/2) and Js>0 (Js<0), the magnetic oc-\ntupole moment displays a stationary steady-state in the\ninitial energy well, if |Js|< Jth2\ns(Fig. 6(a)), while it\nshows chiral oscillations for Jth2\ns<|Js|(Fig. 7(a)). In\nthe limiting case of H0= 0, Jth1\ns=Jth2\ns, and no deter-ministic switching of the magnetic octupole moment is\nobserved. Instead, the magnetic octupole moment dis-\nplays either a non-equilibrium stationary state in the ini-\ntial energy well or chiral oscillations. If the current is\nturned off during the oscillation, the magnetic octupole\nmoment probabilistically switches to either of the energy\nwells.20,29,36\nFIG. 8. Final steady-state as a function of the magnitude of\nthe input current, |Js|, for different applied magnetic fields\nH0. I, II, and III represent regions of no switching, switching,\nand chiral oscillation, respectively. The dashed white lines\nrepresent the two threshold currents, Jth1\nsandJth2\nsobtained\nfrom numerical solution of Eq. (6). This phase diagram is\napplicable if φinit\noct=π/2 (φinit\noct= 3π/2) and Js>0 (Js<0).\nForφinit\noct= 3π/2 (φinit\noct=π/2) and Js>0 (Js<0), region III\nwould still represents chiral oscillation, however, the regions\nencompassing I and II would both correspond to no switching.\nA. Stationary State and Threshold Current\nTo explore the dependence of the dynamics on the in-\ntrinsic energy scale of the system, obtain analytic ex-\npressions for the two threshold currents as a function of\nthe applied magnetic field and material parameters, and\nestablish scaling laws related to switching and chiral os-\ncillations, we evaluate the rate of change of the average\nmagnetization, ˙m=(˙m1+˙m2+˙m3)\n3, as\n˙m=1\n33X\ni=1\u0012\n−γ\u0000\nmi×Heff\ni\u0001\n+α(mi×˙mi)\n−ℏ\n2eγJs\nMsda(mi×(mi×z))\u0013\n,(5)\nwhere mi×Heff\ni=1\n3µ0Ms∂F\n∂φoctzwhile∂F\n∂φoctis obtained\nfrom Eq. (2) with φH=π.\nIn the stationary states, irrespective of φinit\noct, the net\ntorque on the magnetic octupole moment is zero since\nthe spin-orbit torque generated by the input current is\nbalanced by the torque due to the internal and external\nmagnetic fields. Consequently, we set the time deriva-\ntives ( ˙mand ˙mi) in Eq. (5) to zero. Our numericalMn3Sn with strain 8\nsimulations revealed that the z-component of all the sub-\nlattice vectors were zero in the stationary steady-state.\nSo, we set mi,z=ui= 0 in Eq. (5) to arrive at the torque\nbalance equation:\nAsin (2 φoct) +Bsin (6 φoct)−MsH0(C+D) sin ( φoct)\n=ℏ\n2eJs\nda.\n(6)\nForφinit\noct=π/2 and 0 ≤Js< Jth1\ns, the solution\nto Eq. (6) should lead to one stationary solution with\nGS1≤φoct< π, where GS 1∈(π/2, π) is the smaller\nof the two minima of Eq. (2). On the other hand, for\nJth1\ns≤Js< Jth2\ns, the magnetic octupole moment should\nswitch to a stationary state in the other energy well, and\nthe solution of Eq. (6) should lead to GS 2< φ oct<2π.\nHere, GS 2∈(π,3π/2) is the larger of the two minima of\nEq. (2). Indeed the same can be observed from Fig. 9,\nwhere the numerical solutions of the coupled LLG equa-\ntions (symbols) fit the solutions from Eq. (6) (lines) very\nwell in both the energy wells, for three different values\nofH0. Consequently, Jth1\nsis the minimum current for\nwhich Eq. (6) has no solution in (GS 1, π), but has a so-\nlution in (GS 2,2π). On the other hand, Jth2\nsis the min-\nimum current for which Eq. (6) has no solutions. The\nnumerical solution of the threshold currents for differ-\nentH0, obtained from Eq. (6), is shown by the dashed\nwhite lines overlaid on Fig. 8. It can be observed that\nthe solutions from Eq. (6) match the results from Eq. (3)\nvery well. If φinit\noct= 3π/2 and 0 ≤Js< Jth2\ns, the so-\nlution of Eq. (6) would lead to GS 2≤φoct<2π, as is\nshown in Fig. 9 for Jth1\ns≤Js< Jth2\ns. Although not\nshown here, reversing the direction of current ( Js<0)\nwith 0 ≤ |Js|< Jth2\nsleads to stationary steady-states\nin (0,GS1] for φinit\noct=π/2 while φinit\noct= 3π/2 exhibits a\nstationary state in ( π,GS2], if 0 ≤ |Js|< Jth1\ns, and in\n(0,GS1), ifJth1\ns≤ |Js|< Jth2\ns.\nFIG. 9. Stationary steady-states as a function of the applied\nspin current, Js, for a thin film of Mn 3Sn under tensile strain.\nNumerical result from the solution of Eq. (3) (symbols) agree\nvery well with the results obtained from Eq. (6) (lines)An exact expression of either of the two threshold cur-\nrents is cumbersome to obtain, however, in the limit of\nsmall magnetic fields that do not disturb the two-fold de-\ngeneracy of PMA Mn 3Sn, they can be approximated as\nJth1\ns=da2e\nℏ\u0012\n−A+B−MsH0√\n2(C+D)\u0013\n, (7a)\nJth2\ns=da2e\nℏ\u0012\n−A+B+MsH0√\n2(C+D)\u0013\n. (7b)\nIn the absence of an external magnetic field, the thresh-\nold current is the minimum current that provides just\nenough SOT to overcome the maximum torque due to\nthe effective in-plane anisotropy. This maximum occurs\natφoct= 45◦, 135◦, 225◦, and 315◦since they lead to\nsin (2 φoct) =±1 and sin (6 φoct) =∓1. For non-zero ex-\nternal magnetic field, first, we consider the effect of the\nin-plane anisotropy to be dominant while that of H0to\nbe small. We then evaluate Eq. (6) at φoct= 135◦and\nφoct= 315◦to obtain Jth1\nsandJth2\ns, respectively. Fig-\nure 10 compares the analytic expressions of Eq. (7) (lines)\nagainst the values of the threshold currents obtained from\nthe solution of Eq. (6) (symbols), for different values of\nH0. It can be observed that the analytic results match\nvery well against the numerical values. Since the torque\ndue to H0acts against (along) the torque due to the ef-\nfective in-plane anisotropy at φoct= 135◦(φoct= 315◦),\nlarger H0reduces (increases) Jth1\ns(Jth2\ns). Although the\nerror between the numerical and the analytic values of\nthe two threshold currents increases with an increase in\nH0, it is still smaller than 5% even for H0= 0.3 T. This\nlinear dependence of the threshold currents on the exter-\nnal field is similar to that in the case of a PMA ferro-\nmagnet driven by a SOT.43\nFIG. 10. Comparison of the analytic expressions of thresh-\nold currents (Eq. (7)), shown by lines, against the numerical\nvalues obtained from the solution of Eq. (6), represented by\nsymbols. The numerical and analytic values show excellent\nagreement for the values of H0considered here.Mn3Sn with strain 9\nB. Deterministic Switching Dynamics\nForH0>0 and Jth1\ns≤ |Js|< Jth2\ns, the time deriva-\ntives in Eq. (5) change to non-zero values ( |˙m|>0 and\n|˙mi|>0), if φint\noct= 90◦(φint\noct= 270◦) and Js>0\n(Js<0). Here, the external magnetic field assists the\nSOT in overcoming the maximum torque due to the in-\nternal magnetic fields. Consequently, the magnetic oc-\ntupole moment moves away from its initial stable state\nand switches over the energy barrier at φoct= 180◦to\nthe other energy well, while both |mz|and|ui|increase\nto non-zero values. In the second energy well, initially,\nthe torque due to the magnetic fields and the SOT act\nin the same direction, leading to further increase in |mz|\nuntil ˙mdecreases to zero due to the effect of the intrinsic\ndamping. Subsequently, |mz|decreases to zero while the\noctupole moment slows down and reaches a stationary\nstate. The SOT cannot overcome the torques due to the\nmagnetic fields anymore since the external field aids the\ninternal fields in the second energy well. If, however, the\ninput current is reversed, such that Jth1\ns≤ |Js|< Jth2\ns,\nfor the same H0, the magnetic octupole moment goes\nback to the initial energy well, by crossing the barrier\natφoct= 180◦as the external field assists the SOT.\nThis bidirectional switching behavior was clearly demon-\nstrated in Fig. 6 for |Js|= 2.5 MA /cm2andH0= 0.1 T.\nInstead of reversing Js, ifHawas reversed to the positive\nx-direction, the deterministic switching dynamics would\nhave proceeded by crossing the barrier at φoct= 360◦,\nprovided that Jth1\ns≤ |Js|< Jth2\ns.\nThe SOT-driven bidirectional deterministic switching\ndynamics in PMA Mn 3Sn could be useful for building\nnext-generation antiferromagnetic memory devices. In\nthis regard, the switching time, tsw, as a function of the\ninput current is an important metric. Figure 11 shows\nthetswas function of Jsfor two different magnetic fields.\nHere, tswis defined as the time taken by the magnetic\noctupole moment in the ground state φinit\noct=π/2 to go\nfrom φoct=π/2 toφoct=π. This is the minimum dura-\ntion of an input current pulse that can induce determin-\nistic switching. Such a pulse ensures that the magnetic\noctupole moment reaches the top of the energy barrier.\nThereafter, the current pulse is turned off and the torques\ndue to the magnetic fields assist in switching to the other\nenergy well. It can be observed from Fig. 11 that tswde-\ncreases with an increase in either JsorH0. For a fixed\nH0,tswdecreases with an increase in Jssince a higher\ninput current leads to a larger SOT on the magnetic oc-\ntupole moment. On the other hand, at a fixed Js,tsw\ndecreases with an increase in H0since it lowers the en-\nergy barrier at φoct= 180◦, as shown in Fig. 4.\nOur numerical simulations showed that both uiand\nmzwere relatively small as the magnetic octupole moved\nfrom φoct=π/2 toφoct=π. Therefore, the switchingtime was obtained from the z-component of Eq. (5) as\ntsw=αMs\nγ2e\nℏda\nJsZπ\nπ/2\n×dφ′\noct\n1−(Asin(2φ′\noct)+Bsin(6φ′\noct)−MsH0(C+D) sin(φ′\noct))\nℏ\n2eJs\nda,\n(8)\nwhere we neglected the rate of change of mzsince∂mz\n∂t≪\nα∂φoct\n∂t. It can be observed from Fig. 11 that the switch-\ning times obtained from the numerical integration of\nEq. (8) (lines) fit the data obtained from the solution\nof Eq. (3) (symbols) very well for the two magnetic fields\nconsidered here.\n0.0 0.1 0.2 0.3 0.4\ntsw(ns)1.52.02.53.0Js(MA/cm2)\nH0=0.1T\nH0=0.2T\nFIG. 11. Switching time, tsw, as a function of the applied spin\ncurrent, Js, and external magnetic field, H0, for a thin film of\nMn3Sn under tensile strain. In each case, the applied current\nis above Jth1\nsand below Jth2\ns.tswobtained from the solution\nof Eq. (8) (lines) fits the data obtained from the solution of\nEq. (3) (symbols) very well.\nC. Oscillation Dynamics\nForH0>0 and Jth2\ns≤ |Js|, the SOT overcomes\nthe maximum of the torques due to the internal and ex-\nternal magnetic fields, irrespective of the initial state or\nthe direction of input current, resulting in |˙m|>0 and\n|˙mi|>0. Consequently, the magnetic octupole moves\naway from its initial stable state, crosses the barrier at\nφoct= 360◦, and oscillates between the two energy wells\nwith frequencies ranging between 100’s of MHz to 10s of\nGHz, as shown in Fig. 12. Similar to the case of deter-\nministic switching dynamics, |mz|increases till ˙mreaches\nzero due to the intrinsic damping, following which |mz|\ndecreases. Here, however, |mz|increases again as ˙min-\ncreases, owing to the different direction of the torques\nin each energy well. Therefore, mzexhibits two peaks\nof varying magnitude in each oscillation cycle, as shown\nclearly in the inset of Fig. 7(b). The higher peak occursMn3Sn with strain 10\nafter the octupole moment crosses the energy barrier at\nφoct= 360◦whereas the lower peak occurs after it crosses\nthe energy barrier at φoct= 180◦.\nFIG. 12. Oscillation frequency as a function of the applied\nspin current, Js, and external magnetic field, H0, for a thin\nfilm of Mn 3Sn under tensile strain. In each case, the applied\ncurrent is above the respective Jth2\ns. The dash-dotted red line\nrepresents f=γ\n2παM sℏ\n2eJs\ndaand fits the numerical data very\nwell for large Js. The figure in the inset shows the variation\ninfwith Jsfor smaller input currents.\nFor medium to large currents, the oscillation frequency,\nf, is almost independent of H0and increases linearly with\nJs, as shown in Fig. 12. The dash-dotted red line, which\ncorresponds to f=γ\n2παM sℏ\n2eJs\nda, represents this behavior\nclearly. For such Js,uiincreases to larger values. Conse-\nquently, the effect of the out-of-(Kagome)-plane exchange\ninteraction on the dynamics is expected to become sig-\nnificant while that of the in-plane anisotropy and H0is\nexpected to reduce. The x- and z-components of mas\nfunctions of time, for Js= 8 MA /cm2and three different\nmagnetic fields, are shown in Fig. 13. Although the av-\nerage frequency in the three cases is the same, there are\nsubtle differences in the magnetization dynamics, owing\nto the symmetry-breaking magnetic field. In particular,\nmzis symmetric only for H0= 0 while it shows the ex-\npected asymmetry for non-zero H0. Since the difference\nbetween the barrier heights increases with an increase in\nH0(Fig. 4), the asymmetry in mzis more prominent for\nH0= 0.2 T. A small asymmetry can also be observed in\nthe sinusoidal mx, where the magnitude in the negative\n(positive) x-direction increases (decreases) with H0, ow-\ning to Habeing along the negative x-direction. Although\nnot shown here, non-zero magnetic fields have negligible\neffect on the y-component of m.\nFor small currents, on the other hand, Fig. 12 shows\nthatfincreases non-linearly with Jsand depends on H0.\nThis dependence of the oscillation dynamics on magnetic\nfields can also be observed from Fig. 14, which shows the\ndynamics of mxandmzfor three different magnetic fields\nnear their respective threshold currents. The strong in-\nplane anisotropy leads to non-sinusoidal mx, unlike the\ncase of large Js(Fig. 13(a)). It also leads to a spike-like\nFIG. 13. (a) The x-component and (b) z-component of the\naverage magnetization vector, m, as functions of time, for\nlarge current Js= 8 MA /cm2and three different values of\nH0. Although the average oscillation frequency is evaluated\nto be the same for such large current, both mxandmzshow\nthe effect of H0.\ndynamics of mz, where each oscillation of the magnetic\noctupole moment is accompanied by two spikes. Non-\nzero magnetic field breaks the symmetry of the system\nleading to asymmetric profiles of mxandmz. The spikes\ninmzare equally spaced in time for H0= 0 T. On the\nother hand, for H0>0, the two spikes of each oscillation\ncycle are close to each other but far from those of the\nprevious or next cycle. This is mainly due to the varying\neffects of the torque due to Haas the magnetic octupole\ntraverses the two energy wells.\nFinally, in the non-linear regime, higher H0leads to\nlower f, at a fixed Js, as depicted clearly in the inset of\nFig. 12. This is because the barrier height at φoct= 360◦\nincreases with an increase in H0, thereby requiring higher\ninput energy in order to achieve the same oscillation fre-\nquency. As current-driven oscillations are accompanied\nby large mz, a strong exchange field along the z-direction\naffects the dynamics. However, since such an exchange\nenergy interaction is not included in Eq. (2), and conse-Mn3Sn with strain 11\nFIG. 14. (a) The x-component and (b) z-component of the\naverage magnetization vector as functions of time at small\ncurrent ( Js≃Jth2\ns) for two different values of H0. The effect\nof the in-plane effective anisotropy is evident from the shape\nofmx. For non-zero H0, the mzshows two spikes of varying\namplitude due to energy barriers of different heights.\nquently in Eq. (5), our model cannot be used to obtain\na unified model of fas a function of H0andJs.\nIV. EFFECT OF DAMPING\nThe results presented in this work, so far, corre-\nspond to a Gilbert damping constant of α= 0.003,\nwhich has previously been used for numerical analy-\nsis in Refs. [19, 29, 36]. On the other hand, a lower\ndamping constant of α= 0.0007 was used in other re-\ncent works.21,30In particular in Ref. [30], it was shown\nthrough numerical simulations that for α= 0.0007, as\ncompared to α= 0.003, the lower limit of external mag-\nnetic field for deterministic switching was a non-zero\nvalue. That is, for low values of H0≳0, the magnetic\noctupole moment cannot be deterministically switched.\nInstead, it exhibits either a stationary steady-state or chi-\nral oscillation depending on the magnitude of the inputcurrent. This behavior is distinct from that presented in\nFigs. 8 and 10. Therefore, we numerically investigate the\ndependence of the final steady-states and the threshold\ncurrents on α.\nFigure 15 shows the three components of the average\nmagnetization vector, m, for different values of αbut\nthe same values of H0andJs. In the case of α= 5×\n10−3, for Js= 2 MA /cm2andH0= 0.1 T applied at\nt= 100 ps, the AFM magnetization switches to a steady\nstate in the other energy well. This is signified by a\nchange in the sign of my. As shown by the dashed blue\ncurve and dotted green curve, the final steady-state for\nα= 5×10−3is exactly same as that obtained for α=\n3×10−3. The switching time, however, is longer in the\ncase of higher damping since tswis directly proportional\ntoα(Eq. (8)). On the other hand, for the case of lower\ndamping, namely α= 7×10−4, the magnetic octupole\nmoment exhibits chiral oscillations when Js= 2 MA /cm2\nandH0= 0.1 T are applied at t= 100 ps. This suggests\nthat for lower damping, the threshold currents are lower\nthan those predicted by Eq. (7); and could be dependent\nonα. It can also be observed that for α= 7×10−4,\nthe oscillating mzis rather large. This suggests that\nthe out-of-(Kagome)-plane exchange interaction plays a\nmajor role in the oscillation dynamics, similar to that\ndiscussed in Section III C.\nTo further elucidate this dependence of the dynamics\nonα, we present the phase space of the steady-states as\na function of JsandH0for various αvalues in Fig. 16.\nNotably, the phase space analysis reveals an additional\ndynamical regime, labeled as IV, for α= 9×10−4and\n2×10−3, alongside the three previously identified regimes\nobserved for α= 3×10−3, and also found in the case of\nα= 0.01. As highlighted earlier for α= 0.003, region\nI represents stationary steady-states in the proximity of\nthe initial ground states, for φinit\noct= 90◦(φinit\noct= 270◦)\nandJs>0 (Js<0). Region II, on the other hand, corre-\nsponds to deterministic switching between the two stable\nstates, while the magnetic octupole moment exhibits chi-\nral oscillations in region III, regardless of the initial state\nor the direction of Js, as long as |Js|exceeds the highest\nthreshold current. Interestingly, in region IV, chiral os-\ncillations occur for φinit\noct= 90◦(φinit\noct= 270◦) and Js>0\n(Js<0), while only stationary states are observed for\nφinit\noct= 270◦(φinit\noct= 90◦). This scenario is vividly de-\npicted in Fig. 17 for α= 9×10−4,|Js|= 1.72 MA /cm2\nandH0= 0.1 T.\nIn contrast to the scenario with high damping, where\nthe three regimes were distinguished by two threshold\ncurrents, the presence of four regimes in the low αcase is\nmarked by three distinct threshold currents. Although\nall the three threshold currents seem to scale linearly\nwith H0forα≲2×10−3, it can be clearly observed\nthat they depend on the Gilbert damping constant, un-\nlike Eq. (7). Moreover, for low values of H0, deterministic\nswitching is not possible; instead the magnetic octupole\nmoment can only exhibit oscillation dynamics above the\nthreshold current. Deterministic switching between theMn3Sn with strain 12\nFIG. 15. The three components of the average magnetization, m, as a function of time for three different values of the Gilbert\ndamping constant, α. Here, H0= 0.1 T and Js= 2 MA /cm2are both applied to the equilibrium state of Fig. 2(c) at t= 100 ps.\nIn the case of α= 0.003 and 0 .005 the magnetic octupole moment switches to the same final steady-state in the other energy\nwell. Switching time increases with α. On the other hand, for α= 0.0007, chiral oscillations with large mzare observed.\nFIG. 16. Final steady-state as a function of the input current Jsand applied magnetic fields for different Gilbert damping\nconstant. Here, I and II represent the case of no switching and switching while both III and IV correspond to chiral oscillation.\nThe black dashed lines superimposed on the phase space, in the case of α= 0.01, are the two threshold currents, as discussed\nin the case of α= 3×10−3.\ntwo stable states of PMA Mn 3Sn becomes feasible again\nfor larger values of H0. This lower limit of H0for de-\nterministic switching dynamics decreases as αincreases.\nAlthough further analytic investigation is required to un-\nderstand the dependence of the dynamics on the damp-\ning constant, we suspect that in region I the net input\nenergy is low; therefore, the magnetic octupole moment\ncannot overcome the barrier at 180◦. On the other hand,\nin region III the net input energy is very high such that\nmagnetic octupole moment can exhibit sustained oscilla-\ntions. In region IV, for low H0, the barrier at 180◦is low-\nered which enables deterministic switching to the other\nenergy well. However, the low damping of the system\npossibly does not dissipate enough energy of the mag-\nnetic octupole moment, and therefore, it goes over the\nbarrier at 360◦due to its inertia. This leads to sustained\noscillations. For higher fields, the barrier at 360◦be-\ncomes significantly large and the magnetic octupole can-\nnot overcome it, such that deterministic switching be-\ncomes possible. Finally, we found that for α= 0.01 the\nanalysis presented in Section III holds true. Since this\nanalysis is true for both α= 3×10−3andα= 10−2,\nit is applicable for all other values of damping constants\nbetween them.V. CONCLUSION\nMn3Sn is a metallic antiferromagnet that shows large\nAHE, ANE, and MOKE signals. In addition, the\noctupole states can be detected via TMR in an all-\nantiferromagnetic tunnel junction comprising two layers\nof Mn 3Sn with an insulator layer sandwiched between\nthem. Bulk Mn 3Sn has a 120◦anti-chiral structure, how-\never, a competition between the local anisotropy and the\nDMI leads to the existence of a small net magnetiza-\ntion which is six-fold degenerate. Application of strain\nto bulk Mn 3Sn reduces its symmetry from six-fold to\ntwo-fold degenerate, and provides a way to control the\nstrength of the net magnetization as well as that of the\nAHE signal. In this work, we analyzed the case of both\nuniaxial compressive and tensile strains, and discussed\nthe dependence of the magnetic octupole moment on the\nstrain as well as on the external field. Since recent exper-\niment reported tensile strain in epitaxial Mn 3Sn grown\non (110)[001] MgO substrate, we numerically and ana-\nlytically explored the field-assisted SOT driven dynam-\nics in monodomain Mn 3Sn with tensile strain. We found\nthat the magnetic octupole moment exhibits either a sta-\ntionary state or chiral oscillations in the absence of a\nsymmetry-breaking field. On the other hand, when anMn3Sn with strain 13\nFIG. 17. Steady-state dynamics of the magnetic octupole mo-\nment as a function of time for α= 9×10−4. (i) and (ii): Pos-\nsible ground states for H0= 0.1 T in the negative x-direction.\n(a) The pulse is turned on at t= 1 ns to Js= 1.72 MA /cm2\nand turned off at t= 2 ns. Steady-state oscillations for (i)\nbut not for (ii). (b) The pulse is turned on at t= 3 ns to\nJs=−1.72 MA /cm2and turned off at t= 4 ns. Steady-state\noscillations for (ii) but not for (i).\nexternal field is applied, in addition to the stationary\nstate and chiral oscillations, the magnetic octupole mo-\nment can also be deterministically switched between the\ntwo stable states for a range of currents. We derived\nan effective equation which accurately predicts the sta-\ntionary states in both the energy wells. We also derived\nsimple analytic expressions of the threshold currents and\nfound them to agree very well against the numerical re-\nsults for small external magnetic fields. We obtained\nfunctional form of the switching time as a function of the\nmaterial parameters and the external stimuli and found\nit to match very well against numerical data. The fre-\nquency of chiral oscillations, which can be tuned from\n100s of MHz to 10s of GHz range, was found to vary non-\nlinearly closer to the threshold current and linearly for\nlarger input currents. Further, through numerical simu-\nlations, we showed that the order dynamics is dependenton the Gilbert damping for lower values of α. For the sake\nof a complete picture, we also explored the field-assisted\nswitching dynamics in thin films of Mn 3Sn with no strain\nas well as compressive strain, and presented the relevant\nresults in the supplementary document. 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The strength of magnetic damping is\nusually quantified by the Gilbert damping constant in the Landau-Lifshitz-Gilbert equation. Here\nwe propose a first-principles based approach to evaluate the Gilbert damping constant contributed\nby spin-lattice coupling in magnetic insulators. The approach involves effective Hamiltonian mod-\nels and spin-lattice dynamics simulations. As a case study, we applied our method to Y 3Fe5O12,\nMnFe 2O4and Cr 2O3. Their damping constants were calculated to be 0 .8×10−4, 0.2×10−4,\n2.2×10−4, respectively at a low temperature. The results for Y 3Fe5O12and Cr 2O3are in good\nagreement with experimental measurements, while the discrepancy in MnFe 2O4can be attributed\nto the inhomogeneity and small band gap in real samples. The stronger damping observed in Cr 2O3,\ncompared to Y 3Fe5O12, essentially results from its stronger spin-lattice coupling. In addition, we\nconfirmed a proportional relationship between damping constants and the temperature difference\nof subsystems, which had been reported in previous studies. These successful applications suggest\nthat our approach serves as a promising candidate for estimating the Gilbert damping constant in\nmagnetic insulators.\nI. INTRODUCTION\nRecent decades have witnessed rapid developments in\nmagnetics and spintronics [1–3]. A long-time pursuit in\nspintronics is to actively control and manipulate the spin\ndegrees of freedom in solid-state systems. Related fun-\ndamental studies involve spin transport, spin dynamics\nand spin relaxation [4]. Within these domains, magnetic\ndamping often plays a crucial role. Generally, stronger\ndamping enables a faster writing rate for magnetic mem-\nories, while lower damping leads to a longer propagation\ndistance of spin waves. Therefore, it is always essential\nto accurately evaluate the magnetic damping in different\nmaterials. For instance, yttrium iron garnet (YIG) is a\nhighly promising spintronic material due to its ultra-low\nmagnetic damping [5–7]. However, the intrinsic mecha-\nnism behind its unique property has yet to be fully eluci-\ndated, which partly motivates us to carry out this work.\nAt present, magnetic damping is typically represented\nby a phenomenological term in the well-known Landau-\nLifshitz-Gilbert (LLG) equation, which has been widely\nemployed to simulate magnetization dynamics [8, 9]. A\nbasic form of this equation can be written as,\n∂ ⃗ m\n∂t=−γ ⃗ m×⃗B+α\nm⃗ m×∂ ⃗ m\n∂t(1)\nwhere ⃗Brepresents the total magnetic field acting on the\nlocal dipole ⃗ m,mdenotes the norm of ⃗ m,γis the gyro-\nmagnetic ratio, and αis the Gilbert damping constant.\nThe second term on the right side, as we mentioned, leads\n∗hxiang@fudan.edu.cndirectly to the relaxation process, in which the rate of en-\nergy dissipation is determined by the damping constant.\nGiven the importance of αin magnetization dynamics,\nits origin has been extensively studied in the literature\n[10–13]. To the best of our knowledge, both intrinsic and\nextrinsic mechanisms contribute to the damping. Specif-\nically, the intrinsic factors include spin-lattice and spin-\nelectron couplings, while the extrinsic contributions pri-\nmarily involve lattice imperfections and eddy currents\n[14, 15].\nTwo types of first-principles based methods have been\ndeveloped to calculate the damping constants in the past.\nOne approach involves the breathing Fermi surface model\n[16, 17] and the torque correlation model [18, 19], while\nthe other is based on the scattering theory from linear\nresponse [20–22]. These methods have demonstrated re-\nmarkable success in studying the magnetic damping in\ntransition metals such as Fe, Co, and Ni. Despite be-\ning free from complicated experiments, which are mostly\nbased on ferromagnetic resonance, these theoretical ap-\nproaches still exhibit several limitations. Firstly, when\ndealing with complex systems, we often have to spend a\nsignificant amount of computing resources on the first-\nprinciples calculations. In addition, these methods are\nmore suitable for calculating the electronic contribution\nto Gilbert damping in metallic magnets, thus rarely tak-\ning the effect of spin-lattice coupling into consideration\n[14, 23].\nRecently, spin-lattice dynamics (SLD) simulations [24]\nhave been adopted as an alternative method to evaluate\nthe Gilbert damping parameters. In Ref. [23], the au-\nthors constructed an empirically parameterized Hamil-\ntonian model for a cobalt cluster. They coupled a pre-\nheated lattice with a fully ordered spin state, then per-\nformed SLD simulation. During the relaxation process,arXiv:2309.11152v1  [cond-mat.mtrl-sci]  20 Sep 20232\nthe energy of lattice and spin subsystems were recorded\nand fitted to the following logistic functions,\nUlat=Ulat\n0−∆U0\n1 + exp[ −η∆U0t−Θ](2)\nUmag=Umag\n0+∆U0\n1 + exp[ −η∆U0t−Θ](3)\nfrom which they extracted the relaxation rate Γ = η∆U0\nand calculated the damping constant α=ηµS/γ. Here,\nµSdenotes the magnitude of magnetic moments. In Ref.\n[25], the authors also built an empirical potential model\nfor a periodic bcc Fe system. They firstly applied an ex-\nternal magnetic field in the z-direction and thermalized\nthe system to a finite temperature. Then, the magnetiza-\ntion orientation of each atom was rotated artificially by\na same angle. Afterwards, the system would relax back\nto equilibrium, during which the averaged z component\nof atomic magnetization was recorded and fitted to the\nfollowing function,\nmz(t) = tanh\u0014α\n1 +α2γBext(t+t0)\u0015\n(4)\nwhere αwas exactly the Gilbert damping parameter to\nbe estimated. Since these works selected transition met-\nals as the research object, their results were both orders\nof magnitude smaller than the experimental values. In\naddition, the use of empirically parameterized models re-\nduced the accuracy of their simulated results.\nIn this work, we combine SLD simulations with first-\nprinciples based effective Hamiltonian models to evalu-\nate the damping constants in magnetic insulators, where\nthe dominant contribution results from spin-lattice cou-\nplings. Compared to the previous studies, our work has\nmade improvements mainly in two aspects. Firstly, the\nutilization of first-principles based Hamiltonian models\nin simulations enhances the reliability of our conclusions.\nBesides, the better choice of research objects allows for\ndemonstrating the superiority of SLD simulations. In\nparticular, the microscopic origin of low damping in YIG\nwill be investigated. The paper is organized as follows.\nIn Sec. II, we introduce our effective Hamiltonian model,\nparameterization methods, and a scheme for evaluating\nGilbert damping parameters. Then, both the validation\nand application of our method are presented in Sec. III.\nFinally, we summarize this work and give a brief outlook\nin Sec. IV.\nII. MODEL AND METHODS\nThis section is split into three parts. Firstly (in Sec.\nII A), we introduce a generic form of our effective Hamil-\ntonian model. Then, methods involving the calculation\nof model parameters are presented in Sec. II B. At the\nlast part (Sec. II C), we propose a novel scheme to de-\ntermine the Gilbert damping constant through dynamics\nsimulations.A. The Hamiltonian Model\nSince our purpose is to evaluate the contribution of\nspin-lattice coupling to magnetic damping, the effective\nHamiltonian model must incorporate both spin and lat-\ntice degrees of freedom. A concise and generic formula\nthat meets our basic requirements consists of the three\nterms as follows:\nH=HL({ui,α}) +HS({⃗ sj}) +HSLC({ui,α,⃗ sj}) (5)\nwhere αabbreviates three orthogonal axes, ui,αrepre-\nsents the displacement of atom i, and ⃗ sjis a unit vector\nthat represents the direction of spin j.\nThe first term HLin Hamiltonian model describes the\ndynamical behavior of individual phonons. Technically,\nwe take the atomic displacements as independent vari-\nables and expand the Hamiltonian to the second order\nwith Taylor series. Then, we have the form as,\nHL=1\n2X\nijX\nαβKij,αβui,αuj,β+1\n2X\ni,αMi˙ui,α˙ui,α(6)\nwhere Kij,αβ denotes the force constant tensor and Mi\nrepresents the mass of atom i.\nSimilarly, the second term HSdescribes the dynami-\ncal behavior of individual magnons. For simplicity but\nno loss of accuracy, we only considered the Heisenberg\nexchange interactions between neighbor magnetic atoms\nin this work, though more complex interactions could be\ntaken into account in principle. Therefore, this term can\nbe expressed as,\nHS=X\n⟨i,j⟩Jij⃗Si·⃗Sj (7)\nwhere Jijdenotes the isotropic magnetic interaction co-\nefficient.\nThe third term HSLCrepresents the coupling of spin\nand lattice subsystems, and is expected to describe the\nscattering process between phonons and magnons. As\nan approximation of the lowest order, this term can be\nwritten as,\nHSLC=X\n⟨i,j⟩X\nkα\u0012∂Jij\n∂uk,αuk,α\u0013\n⃗Si·⃗Sj (8)\nAccording to the theory of quantum mechanics, this\ncoupling term provides a fundamental description of the\nsingle-phonon scattering process, which is believed to be\ndominant among all scatterings in the low-temperature\nregion. This type of relaxation mechanism in ferromag-\nnetic resonance was systematically studied by Kasuya\nand LeCraw for the first time [26]. It’s worth noting that\na higher order of Taylor expansion could have been con-\nducted to improve the accuracy of Hamiltonian models\ndirectly. For instance, the scattering between individual\nphonons can be adequately described by the anharmonic\nterms. However, as one always has to make a trade-off3\nbetween the precision and complexity of models, in this\nwork we choose to neglect the high order terms since the\nanharmonic effects in current investigated systems are\nnot important.\nIn this study, we adopted the symmetry-adapted clus-\nter expansion method implemented in the Property Anal-\nysis and Simulation Package for Materials (PASP) [27]\nto build the Hamiltonian model presented above. This\npackage can identify the nonequivalent interactions and\nequivalent atom clusters in a crystal system by analyz-\ning its structural properties based on the group theory.\nA significant benefit of working with PASP is we are en-\nabled to describe the target system with the least number\nof parameters. In the next section, we will discuss how\nto calculate the model parameters for different materials.\nB. Calculation of Model Parameters\nFirstly, the Heisenberg exchange coefficients Jijand\nspin-lattice coupling constants ∂Jij/∂uk,αcan be calcu-\nlated with the four-state method [28, 29]. The basic flow\nis to construct four artificially designated spin states of\nthe target system, calculate the corresponding energies\nand forces based on the density functional theory (DFT),\nthen determine the parameters by proper combination of\nthose results. At the last step, the following formulas will\nbe used,\nJij=E↑↑+E↓↓−E↑↓−E↓↑\n4S2(9)\n∂Jij\n∂uk,α=F↑↑\nk,α+F↓↓\nk,α−F↑↓\nk,α−F↓↑\nk,α\n4S2(10)\nwhere Sis the spin quantum number of magnetic atoms,\nEis the total energy of system and Fk,αrefers to one\ncomponent of the force on atom k. The superscripts ( ↑↑,\n↓↓,↑↓,↓↑) specify the constrained spin states of system\nin the calculation. More technical information about the\nfour-state method can be found in the references [28, 29].\nCompared to other approaches, the four-state method of-\nfers an obvious advantage in that no additional DFT cal-\nculations are needed to determine the coupling constants\n∂Jij/∂uk,αonce the exchange coefficients Jijhave been\nobtained. This is because the energy and forces are typ-\nically provided simultaneously by one DFT calculation.\nSince atomic masses Mican be directly obtained from\nthe periodic table, more efforts are needed to deal with\nthe force constant tensor Kij,αβ. Currently, there are two\ncommonly adopted ways to calculate the force constant\ntensor: density functional perturbation theory (DFPT)\nand finite displacement method. Both of these methods\nare applicable to our task.\nHowever, we cannot directly take the force constant\ntensor obtained from first-principles calculations as the\nmodel parameter. This is because in dynamics simula-\ntions we usually expand crystal cells to reduce the un-\ndesired influence of thermal fluctuations, which leads toa conflict between the periodic boundary condition and\nthe locality (also known as nearsightedness [30, 31]) of\nmodels. To be more specific, when calculating the con-\ntribution of one atom or spin to the total energy, we tend\nto set a well designed cutoff radius and ignore the inter-\nactions beyond it. This step is essential when dealing\nwith a large-scale system, otherwise we will suffer from\nthe model complexity and the computational cost. Nev-\nertheless, if we set the elements of Kij,αβ that represent\nout-of-range interactions to be zero and leave the others\nunchanged, we may violate the so-called acoustic sum-\nmation rules:\nX\niKij,αβ = 0 for all j, α, β. (11)\nIt should be pointed out that a straightforward en-\nforcement of the acoustic summation rules, achieved by\nsubtracting errors uniformly from force constants, will\nbreak the inherent crystal symmetry inevitably, which is\nthe technique employed in phonopy [32]. To address the\nabove issues, we adopted a more appropriate method in\nthis work. Before a detailed introduction, it’s necessary\nto recall that not every element of the force constant ten-\nsor serves as an independent variable due to the crystal\nsymmetries. Taking the cubic cell of Y 3Fe5O12(contain-\ning 160 atoms) for example, there are 230400 elements in\nthe tensor. After symmetry analyses, we find that only\n597 independent variables {pn}are needed to adequately\ndetermine all the tensor elements {Kij,αβ({pn})}, where\nthe effect of locality is already considered. Afterwards,\nour method is to set a correction factor xnfor each vari-\nablepnand minimize the deviation of parameters under\nthe constraints of Eq. (11). A mathematical reformula-\ntion of this method can be written as,\nmin\n{xn}X\nn(xn−1)2,with\nX\niKij,αβ({xnpn}) = 0 for all j, α, β.(12)\nIn the case of Y 3Fe5O12, there are only 18 linearly inde-\npendent constraints, which allow the extremum problem\nto be solved rigorously. The modified force constant ten-\nsor restores positive definiteness and translational sym-\nmetry while maintaining the crystal symmetries. There-\nfore, the modified tensor meets the requirements for dy-\nnamics simulations. In Sec. III B, the effectiveness of this\napproximate method will be demonstrated through a spe-\ncific example.\nAll the first-principles calculations mentioned in this\nsection are carried out using the Vienna ab initial simu-\nlation package (VASP) [33–35]. The force constants and\nphonon spectra are obtained by phonopy [32]. The opti-\nmizations formulated in (12) are accomplished with the\nfunction optimize.minimize implemented in SciPy [36].4\nC. Evaluation of Damping Constants\nAfter the construction and parameterization of Hamil-\ntonian models, we are finally able to perform spin-lattice\ndynamics simulations. Before the evaluation of Gilbert\ndamping constants, we briefly introduce the framework\nof SLD to cover some relevant concepts. In practice, the\nmotion of magnetic moments follows the stochastic Lan-\ndau–Lifshitz–Gilbert (SLLG) equation [14],\nd⃗ mi\ndt=−γL⃗ mi×\u0010\n⃗Bi+⃗Bfl\ni\u0011\n−γLα⃗ mi\n|⃗ mi|×h\n⃗ mi×\u0010\n⃗Bi+⃗Bfl\ni\u0011i\n(13)\nwhere γLis the renormalized gyromagnetic ratio, ⃗Bi=\n−∂H/∂ ⃗ m iis the effective local magnetic field and ⃗Bfl\ni\nrefers to a stochastic field introduced by Langevin ther-\nmostat. At the same time, the motion of atoms obeys\nthe Newton’s equation,\nd˙ui,α\ndt=1\nMi\u0010\n⃗Fi,α+⃗Ffl\ni,α\u0011\n−ν˙ui,α (14)\nwhere νis the damping constant and ⃗Ffl\ni,αrefers to a\nstochastic force caused by thermal fluctuations. In this\nwork, ⃗Bfl\niand⃗Ffl\ni,αare modeled as normally distributed\nnoises with temperature-dependent variances,\nBfl\ni,β∼N\u0010\n0,p\n2αkBTS/γ|⃗ mi|δt\u0011\n(15)\nFfl\ni,β∼N\u0010\n0,p\n2νMikBTL/δt\u0011\n(16)\nwhere TSandTLrefer to the equilibrium temperature of\nspin and lattice subsystems respectively. During simula-\ntions, we can also measure the transient temperature of\neach subsystem with the following formulas [37],\nTS=P\ni|⃗ mi×⃗Bi|2\n2kBP\ni⃗ mi·⃗Bi, TL=1\n2kBNX\ni,αMi˙u2\ni,α (17)\nIn this work, the LLG equation is numerically solved\nwith the semi-implicit SIB method proposed by Mentink\net al. [38]. The Newton’s motion equation is integrated\nusing the Grønbech-Jensen-Farago Verlet-type method\n[39]. To ensure the stability of those algorithms, a step\nlength of 0 .5 or 0 .2 fs is adopted [40], where the shorter\none is used in energy-conserving simulations.\nBased on the combination of atomistic spin dynamics\n(ASD) and SLD simulations, a new scheme is proposed\nto evaluate the damping constant in magnetic materials.\nHere is the basic flow of this method and more details of\na specific application are presented in Sec. III B.\n1. Freeze the spin degree of freedom and thermalize\nthe lattice from 0 to TLin the simulation.\n2. Fix atomic positions and raise the temperature of\nspin to TS> TL. Compared to TL> TS, this type\nof nonequilibrium state is more common in actual\nscenarios.3. Perform an energy-conserving SLD simulation to\nrelax the system. Normally, the spin temperature\nwill decrease to the same as lattice and stay there\ntill the end.\n4. Conduct a series of ASD simulations with different\nGilbert damping constants. The initial states are\nthe same as in step 3 and the equilibrium temper-\natures are set to be TL.\n5. Compare the cooling rates ∂TS/∂tof spin system\nbetween SLD and ASD simulations to evaluate the\nequivalent Gilbert damping constant contributed\nby spin-lattice coupling.\nThe key point behind step 5 is that the cooling rates\nobserved in ASD simulations are related to the assigned\ndamping constants, while in SLD simulation the cooling\nrate is determined by the strength of spin-lattice cou-\npling. Note that the former relation can be viewed as a\nnatural deduction of the LLG equation,\n∂TS\n∂t=1\nCV∂Emag\n∂t∝ −1\nCV\u0012∂ ⃗ m\n∂t·⃗B\u0013\n∝ −1\nCV\u0014\u0012α\nm⃗ m×∂ ⃗ m\n∂t\u0013\n·⃗B\u0015\n∝α (18)\nwhere we have used Eq. (1) and simplified the formula of\nmagnetic energy as Emag∝ −⃗ m·⃗B.\nIII. RESULTS\nThis section is divided into four parts. In Sec. III A,\nseveral test results are presented to validate the accu-\nracy of SLD simulations, which are implemented in the\nPASP package. Subsequently, detailed calculations on\nthree magnetic materials, namely Y 3Fe5O12, MnFe 2O4\nand Cr 2O3, are discussed in the rest parts.\nA. Validations\nIn order to guarantee the reliability of our conclusions\nobtained from dynamics simulations, a series of pretests\nwere carried out. We select some representative results\nand present them in Fig. 1, where Cr 2O3is taken as the\nobject to be studied.\nFirstly, we set the ground state of Cr 2O3as the ini-\ntial state and performed a NVT simulation with Tset=\n150K. As shown in Fig. 1(a), the temperature of spin\nand lattice subsystems increased to 150 Kin less than 5\nps and stayed there till the end. Since we can approxi-\nmate Ek= 0.5ELandEp= 0.5EL+ES, Fig. 1(b) also\nindicates that the contribution of phonons and magnons\nto the excited state energy is around 87.5% and 12.5%\nrespectively. This result could be verified from another\nperspective. Note that there are totally 10 atoms in the5\nFIG. 1. NVT and NVE relaxations of a spin-lattice coupled system (Cr 2O3) within the framework of spin-lattice dynamics.\nThe top row plots the time evolution of temperatures and the bottom row shows the variation of potential, kinetic and total\nenergies. (a) & (b): NVT thermalization from TL=TS= 0KtoTL=TS= 150 K. (c) & (d): NVE relaxation with TL= 30K,\nTS= 175 Kinitially. (e) & (f): NVE relaxation with TL= 180 K,TS= 30Kinitially.\nunit cell of Cr 2O3, which contribute 30 kBto the heat ca-\npacity. Meanwhile, the 4 magnetic atoms will contribute\nanother 4 kBin the low temperature region. Therefore,\nwe can estimate that the contribution of magnons to the\ntotal heat capacity is close to 11.8%, which is consistent\nwith the result from dynamics simulations.\nIn Figs. 1(c) & 1(d), the initial state was set to be a\nnonequilibrium state with TL= 30KandTS= 175 K. As\nwe expected, the total energy was well conserved when\nthe system evolved to equilibrium. In addition, the final\ntemperature fell within the range of 48 K∼55K, which\nagrees with our previous analysis of the heat capacities.\nLastly, we simulated the relaxation process using an-\nother nonequilibrium excited state with TL= 180 Kand\nTS= 30Kas the initial state. As shown in Figs. 1(e) &\n1(f), the temperature of spin system increased gradually\nto equilibrium with the total energy conserved through-\nout the simulation. Also, the final temperature is around\n160K, which matches well with our analysis. It should be\npointed out that there exist two notable differences be-\ntween this case and the previous. Firstly, the subsystems\nultimately evolved to a same temperature in a finite time,which alleviated our concerns about the accuracy of SLD\nsimulations. Besides, the relaxation time ( τ2) was much\nlonger than that ( τ1) in Fig. 1(c). For this phenomenon,\na qualitative explanation is presented below.\nBased on the theory of second quantization, the Hamil-\ntonian model presented in Sec. II A can be expressed in\nthe following form [41, 42],\nHL=X\nqpℏωqp(b†\nqpbqp+ 1/2) (19)\nHS=X\nλϵλa†\nλaλ+Const. (20)\nHSLC=X\nλ,qpMλ,qpa†\nλ−qaλ\u0000\nb†\nqp−b−qp\u0001\n(21)\nwhere bqpdenotes the annihilation operator of phonons\nwith wave vector qin branch p, and aλrepresents the an-\nnihilation operator of magnons with wave vector λ. All\nthe parameters, namely ωqp,ϵλandMλ,qp, can be deter-\nmined from the effective Hamiltonian model in principle.\nAccording to the Fermi’s golden rule, we have\nW{nλ−q, nλ, Nqp→nλ−q+ 1, nλ−1, Nqp+ 1}=2π\nℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(Nqp+ 1)δ(ϵλ−q−ϵλ+ℏωqp) (22)\nW{nλ−q, nλ, N−qp→nλ−q+ 1, nλ−1, N−qp−1}=2π\nℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(N−qp)δ(ϵλ−q−ϵλ−ℏω−qp) (23)6\nFIG. 2. (a) The primitive cell of Y 3Fe5O12. The golden balls\nrepresent iron atoms, the cyan ball stand for yttrium atoms,\nand the red balls refer to oxygen atoms. (b) The magnetic\nground state of YIG. The arrows of different colors represent\nthe spin directions of Fe atoms. (c) The density of states ob-\ntained by DFT calculations. (d) The temperature dependence\nof average magnetization measured in MC and ASD simula-\ntions. For YIG, the phase transition point from ferrimagnetic\nto paramagnetic lies in 530 K approximately.\nwhere Wrepresents the probability of one-phonon emis-\nsion or absorption, nλdenotes the occupation number of\nmagnons and Nqpstands for phonons. Both nλandNqp\ncan be evaluated approximately using the Bose–Einstein\ndistribution. According to the above formulas, the scat-\ntering rate Wgrows linearly with Nand quadratically\nwith n. Compared to Fig. 1(c), there are more phonons\nbut fewer magnons in the case of Fig. 1(e), thus leading\nto a lower transition probability and a longer relaxation\ntime. More technical details about the second quantiza-\ntion of interactions between phonons and magnons can\nbe found in Ref. [41, 42].\nB. Damping constants in Y 3Fe5O12\nIn the field of spintronics, Y 3Fe5O12(yttrium iron gar-\nnet, YIG) has gained much attention due to its ultra-low\nmagnetic damping [5–7]. The unique property of this\nmaterial motivated us to investigate the intrinsic mecha-\nnism behind it. The crystal structure of YIG is presented\nin Fig. 3(a). There are totally 80 atoms in the primitive\ncell, of which 12 Fe ions are located in the center of oxy-\ngen tetrahedrons while the other 8 Fe ions are sited in\noxygen octahedrons. The magnetic ground state of YIG\nis illustrated in Fig. 3(b). The Fe ions situated in differ-\nent chemical environments contribute spins in opposite\ndirections, which makes YIG a typical ferrimagnetic ma-\nterial.TABLE I. The Heisenberg exchange coefficients J of YIG,\nwhere an effective spin S= 1 is adopted. For the FeO−FeO\npairs, the Greek letters ( α&β) refer to different chemical\nenvironments. All the results are calculated with the four-\nstate method.\nSpin Pair. Distance (Angst) J (meV)\n1NN FeT−FeO3.445 47.414\n1NN FeT−FeT3.774 2.399\n1NN FeO−FeO(α) 5.337 0.538\n1NN FeO−FeO(β) 5.337 5.055\n2NN FeT−FeO5.555 0.285\n2NN FeT−FeT5.765 3.437\nIn order to evaluate the Gilbert damping constants in\nYIG, our first step is to prepare an effective Hamilto-\nnian model. Considering the balance between precision\nand efficiency, the cutoff radius of interactions was set\nto be 11.0 Bohr for atomic pairs and 6.7 Bohr for 3-\nbody clusters. After symmetry analyses, we identified\n612 nonequivalent interactions in total, which included\n6 Heisenberg exchange terms and 9 spin-lattice coupling\nterms.\nTo determine the interaction parameters, we carried\nout a series of first-principles calculations, where a cu-\nbic cell was adopted to reduce the interference between\nadjacent cells caused by periodic boundary conditions.\nFollowing the settings in Ref. [43], we utilized the pro-\njector augmented-wave (PAW) method [44] and revised\nPerdew-Burke-Ernzerhof exchange-correlation functional\nfor solids (PBEsol) [45] in our calculations. Besides, the\nDFT+U method in its simplified form [46] was employed\nwhere the effective Hubbard U parameter was set to be\n4 eV for the 3 delectrons of Fe ions. In addition, a cutoff\nenergy of 520 eV for plane wave basis and a Γ-centered\n2×2×2 mesh of k-points were used in the DFT calcu-\nlations.\nIn Figure 2(c), we present the density of states (DOS)\nfor YIG. With a band gap of 1.863 eV, there is hardly\nany electric current occurring in the low temperature re-\ngion. Moreover, the Heisenberg exchange coefficients of\nYIG is listed in Table I. To verify the accuracy of these\nparameters, we conducted both Monte Carlo (MC) and\nASD simulations. The temperature dependence of aver-\nage magnetization is shown in Fig. 2(d), which reveals\nthe critical temperature of YIG to be 530 K. This result\nis slightly lower than the measured Curie temperature,\nTC= 560 K[5], but falls within our tolerance. The cal-\nculated results of coupling constants are provided in the\nsupplementary material.\nNext, we come to deal with the force constant tensor.\nIn order to demonstrate the impact of locality and val-\nidate the effectiveness of our optimization method, we\npresent some results pertaining to the tensor of YIG in\nTable II. Here we use “VASP” to tag the original tensor7\nTABLE II. The force constant tensor of YIG. The columns\nlabeled by A represent the sorted absolute values ofP\niKij,αβ\nand the columns labeled by B list the sorted eigenvalues of\nKij,αβ. For the cubic cell of YIG, we obtained the original\ntensor with the VASP package. Then, we eliminated the el-\nements that represent interactions beyond the cutoff radius.\nThis step was done by PASP. Finally, the tensor was modified\nto meet the requirement of translational symmetry through\nthe optimization formulated in (12).\nVASP PASP Modified\nNo. A B A B A B\n1 0.000 0.000 1.587 -0.102 0.000 0.000\n2 0.000 0.000 1.587 -0.102 0.000 0.000\n3 0.000 0.000 1.587 -0.102 0.000 0.000\n4 0.000 1.065 1.587 0.643 0.000 0.444\n5 0.000 1.065 1.587 0.643 0.000 0.444\n6 0.000 1.065 1.587 0.643 0.000 0.444\nobtained from DFT calculations, “PASP” to label the\nmodified tensor in which interactions beyond the cutoff\nradius are eliminated, and “Modified” to label the tensor\nafter optimization of independent variables. As shown in\nTable II, the “PASP” tensor violated the acoustic sum\nrule and was not positive semi-definite, whereas these is-\nsues were resolved for the “Modified” tensor. Although\nan obvious difference existed between the “PASP” and\n“Modified” tensor in terms of their eigenvalues, we still\nassumed the target system could be reasonably described\nby the “Modified” tensor and the validity of this assump-\ntion would be verified by the calculated results of damp-\ning constants. Additional details regarding the selection\nof tensor elements and the deviation of phonon spectra\nare provided in Fig. 3. According to figure 3(b) and 3(c),\nthe major deviation in phonon spectra resulted from the\nelimination of tensor elements, rather than the subse-\nquent modification.\nCompleting the preparation of Hamiltonian model, we\napplied the scheme proposed in Sec. II C to our first ob-\nject, Y 3Fe5O12. An instance is presented in Figure 4. We\nsetTL= 30K,TS= 180 Kfor the initial nonequilibrium\nstate and adopted an expanded supercell which contains\n12800 atoms in the simulation. Fig. 4(a) shows the time\nevolution of spin temperature in different types of simu-\nlations. By comparing the curves, we could roughly esti-\nmate that the equivalent damping constant in SLD simu-\nlation fell within the range of 10−3∼10−4. To make the\nestimation more precise, we calculated the initial cool-\ning rates ∂TS/∂t|t=0through polynomial (or exponen-\ntial) fittings and plotted them in Fig. 4(b). Afterwards,\na linear regression was performed to determine the quan-\ntitative relation between lg( −∂TS/∂t|t=0) and lg( α). As\nwe expected, the cooling rates in ASD simulations were\nproportional to the assigned damping constants. Then,\nwe combined the results of SLD and ASD simulations toevaluate the equivalent damping constant. This step was\naccomplished by identifying the intersection of red and\nblue lines in Figure 4(b). Finally, the damping constant\nwas determined to be αf= (2.87±0.13)×10−4in this\ncase. To verify our method and result, we present a com-\nparison between SLD and ASD (where we set α=αf)\nsimulations in Fig. 4(c). The curves agree well with each\nother in the initial stage but deviate in the second half.\nThis phenomenon is within our expectation, because in\nthe SLD simulation the lattice heats up as the spin cools\ndown, thereby slowing the energy transfer between two\nsubsystems.\nIn addition to the above case, we have measured the\nequivalent damping constants under different conditions\nto investigate the temperature dependence of magnetic\ndamping. The final results are summarized in Figure 5.\nDetails about the estimation of uncertainties are given in\nthe supplementary material. For Y 3Fe5O12, the damping\nconstants at different temperatures stay on the order of\n10−4, which is in good agreement with the experimental\nresults (3 .2×10−4[47], 2 .2×10−4[48], 1 .2–1.7×10−4\n[49]). For example, the damping constant in bulk YIG\nwas reported as 0 .4×10−4in Ref. [50]. Meanwhile, our\ncalculations yielded α= (2.8±0.3)×10−5at ∆T= 15\nK and α= (7.0±0.7)×10−5at ∆T= 30 K, where both\nTL= 0 K. Therefore, the experimental value corresponds\nroughly to the temperature region of ∆ T= 15∼30 K in\nour study. We believe such extent of thermal excitation\nis quite common in all kinds of spintronics experiments.\nMoreover, Fig. 5 indicates that αis approximately pro-\nportional to the temperature difference between subsys-\ntems. This outcome is also consistent with some com-\nputational works in the past [23, 25]. By comparing the\nsubfigures in Figure 5, we found that αhas little depen-\ndence on the lattice temperature, although here TLcould\nbe viewed to some extent as the ambient temperature of\nthe spin system.\nAs a supplement to Sec. III A, we further validate our\nsimulations by analyzing the measured cooling rates in\nFig. 5(a). By subtracting Eq. (23) from Eq. (22), the\ntransfer rate of energy between magnon and phonon sys-\ntems can be expressed as,\n˙Q=X\nqpℏωqp⟨˙Nqp⟩=X\nλ,qpTλ,qp (24)\nwhere Tλ,qpdenotes different transfer channels,\nTλ,qp∝(nλ−nλ−q)Nqp+nλ−qnλ+ 1 (25)\nAccording to the Bose–Einstein distribution, the number\nof magnons and phonons can be expressed as,\nnλ=1\neϵλ/kBTS−1, Nqp=1\neℏωqp/kBTL−1(26)\nWhen TSis high enough and TLis close to zero, we can\napproximate nλ=kBTS/ϵλ∝TSandNqpclose to zero.\nUnder these conditions, we have ˙Q∝T2\nS. This relation8\nFIG. 3. (a) The selection of force constant tensor elements for the cubic cell of YIG. An 160 ×160 zero-one matrix is used\nto show the result of selection, in which ’1’ denotes the interactions within cutoff radius and ’0’ represents the elements that\nare artificially eliminated. (b) The phonon spectrum calculated from the force constant tensor before and after the elimination\nof tensor elements. (c) The phonon spectrum calculated from the force constant tensor before and after the optimization of\nindependent variables.\nFIG. 4. (a) The time evolution of spin temperature in SLD and ASD simulations. The gray line represents the SLD simulation\nwhile the others refer to the ASD simulations with different damping constants. (b) The initial cooling rates ∂TS/∂t|t=0with\nrespect to the damping constants α, where the scaling of axis is set to be logarithm. The gray squares refer to the results of\nASD simulations and the blue line acts as the linear regression. The red circle is plotted by intersection of the blue line and\nthe horizontal red dash line, which represents the initial cooling rate in the SLD simulation. Then we can obtain the equivalent\ndamping constant from the abscissa of the red circle. (c) The comparison between ASD and SLD simulations. In the ASD\nsimulation, the Gilbert damping constant is set to be α= 2.87×10−4, which is exactly the result of our evaluation from the\nSLD simulation.\nFIG. 5. The temperature dependence of Gilbert damping constants for Y 3Fe5O12. The label of abscissa axis ∆ Trefers to\nTS−TLof the initial state in dynamical simulations. Measurements on the magnetic damping are performed under different\ninitial conditions of the lattice temperature: (a) TL= 0, (b) TL= 30K, (c)TL= 60K.9\nFIG. 6. The relation between damping constants αand spin-\nlattice coupling constants ∂Jij/∂uk,αin YIG. Through a lin-\near fitting, the slope is determined to be 2 .01, which agrees\nwell with our theoretical predictions.\nis well verified by linear regressions and the details are\nprovided in the supplementary material.\nFurthermore, the accuracy of our simulations can also\nbe proved from another perspective. According to Eqs.\n(22) and (23), the scattering rate Wgrows quadratically\nwith the coupling parameters Mλ,qp. Based on the theory\nof second quantization, Mλ,qpshall be proportional to\nthe coupling constants ∂Jij/∂uk,α. Therefore, under a\ndefinite condition of temperature, we have:\nα∝˙Q∝∆W∝M2\nλ,qp∝(∂Jij/∂uk,α)2(27)\nIn order to verify this relation, we adjusted the spin-\nlattice coupling constants of YIG coherently while keep-\ning the other model parameters unchanged. Then, SLD\nsimulations were carried out to evaluate the correspond-\ning damping constants. The result is plotted in Fig. 6,\nwhere the x-label “slcc” stands for the spin-lattice cou-\npling constants and the subscript “0” refers to the orig-\ninal situation. From a linear fitting, the slope is deter-\nmined to be 2 .01, which agrees well with our prediction.\nC. Damping constants in MnFe 2O4\nAfter the calculation on YIG, we applied our method\nto MnFe 2O4(MFO), which was reported to possess a\nlarge Gilbert damping constant in the literature [13, 51].\nAs shown in Fig. 7(a), MnFe 2O4has a typical structure\nof spinels, where A sites are surrounded by four oxygen\natoms and B sites are located in octahedrons. Generally,\nspinels can be classified into normal and inverse struc-\ntures according to the distribution of divalent and triva-\nlent cations between A/B sites. In experiments, MFO\nusually crystallizes into a mixed phase where the normal\nstructure occupies the major part (80% in bulk MFO\n[52]). Here, we only considered its normal structure in\nthis work. Also, the magnetic ground state of MFO is\nshown in Fig. 22(b), where the magnetic moments are\nantiparallel between A/B sites.\nFIG. 7. (a) The cubic cell of MnFe 2O4. The purple balls rep-\nresent manganese atoms, the golden balls refer to iron atoms,\nand the red balls stand for oxygen atoms. (b) The magnetic\nground state of MFO. The arrows of different colors repre-\nsent the spin directions of Mn and Fe atoms separately. (c)\nThe density of states obtained by DFT calculations. (d) The\ntemperature dependence of average magnetization measured\nin MC and ASD simulations. For MnFe 2O4, the phase tran-\nsition point from ferrimagnetic to paramagnetic lies in 730K\napproximately.\nFirstly, we started to construct an effective Hamilto-\nnian model for MFO. With the same cutoff settings for\nYIG, we found 105 nonequivalent interactions, including\n4 Heisenberg exchange terms and 10 spin-lattice coupling\nterms. Subsequently, DFT calculations were carried out\nto determine the interaction parameters. In these calcu-\nlations, we adopted a cubic cell containing 56 atoms and\na Γ-centered 4 ×4×4 grid mesh in the reciprocal space.\nBesides, UMn= 3.3 eV and UFe= 3.6 eV were used as the\neffective Hubbard parameters [52]. With the exception of\naforementioned settings, all the relevant first-principles\ncalculations were performed under the same conditions\nas in Sec. III B.\nThe DOS of MnFe 2O4is plotted in Fig. 7(c), yielding\na calculated band gap of 0.612 eV. This value does not\nmatch with the result of transport experiments, which re-\nported a much smaller band gap (0 .04–0.06 eV) [53]. In\naddition, MC and ASD simulations were performed using\nthe Heisenberg exchange coefficients listed in Table III.\nThe temperature dependence of average magnetization,\nshown in Fig. 7(d), suggests the critical temperature to\nbe around 730 K. This result is significantly higher than\nthe measured value of 573 K [54]. Both of the above dis-\ncrepancies may be attributed to the inevitable difference\nbetween the ideal normal spinel structure in calculations\nand the partially disordered samples in reality. Despite\nthis problem, we proceeded to describe the target system\nwith our Hamiltonian model and expected to see how far\nthe calculated results of damping constants would differ10\nTABLE III. The exchange coefficients J of MnFe 2O4, where\nan effective spin S= 1 is adopted.\nSpin Pair. Distance (Angst) J (meV)\n1NN Fe-Fe 3.003 6.835\n1NN Mn-Fe 3.521 33.224\n1NN Mn-Mn 3.667 3.956\n2NN Fe-Fe 5.201 0.929\nfrom experimental values.\nAfter the preparation of Hamiltonian model, we con-\nducted dynamics simulations to evaluate the equivalent\ndamping parameters in MFO at different temperatures.\nA supercell containing 13440 atoms was adopted in the\nsimulation, and the results are summarized in Fig. 10.\nThe average of calculated damping constants is around\n8×10−5, which is much smaller than the measured value,\n1.0×10−2[13, 51]. Two factors may account for this in-\nconsistency. Firstly, the inhomogeneity in real MnFe 2O4\nsamples greatly enhances the scattering of magnons and\nphonons, thereby increasing the damping constants. Ad-\nditionally, due to the narrow band gap observed in ex-\nperiments, eddy currents can arise at finite temperatures,\nwhich leads to a rapid loss of energy in the form of joule\nheat. As the result of these factors, we failed to obtain a\nreasonable estimation of Gilbert damping constants for\nMnFe 2O4with our methodology. On the other side, the\ncontribution of different relaxation mechanisms to FMR\nlinewidth has been studied comprehensively for MnFe 2O4\nin Ref. [53], which further confirms our analyses.\nD. Damping constants in Cr 2O3\nChromia (Cr 2O3) is a well-known collinear magneto-\nelectric antiferromagnet, which holds great prospects in\nthe field of spintronics [55–57]. As shown in Fig. 8(a),\nthe primitive cell of Cr 2O3contains 10 atoms, with each\nchromium atom bonded to the six oxygen atoms around\nit. Additionally, Fig. 8(b) displays the magnetic ground\nstate of Cr 2O3, where the spins of two nearest neighbor-\ning Cr atoms are oriented in opposite directions.\nAs a preliminary step in constructing the Hamiltonian\nmodel, we set the cutoff radius of interactions to be 11.0\nBohr for atomic pairs and 7.0 Bohr for 3-body clusters.\nThrough symmetry analyses, we identified 319 nonequiv-\nalent interactions, including 5 Heisenberg exchange terms\nand 21 spin-lattice coupling terms.\nAfterwards, a series of first-principles calculations were\nperformed to determine the model parameters. Following\nthe settings in Ref. [58], we adopted a hexagonal cell of\nCr2O3which contained a total of 90 atoms in the calcula-\ntions. Additionally, we used the LSDA+U method in its\nfull spherically symmetric form [59]. As to the Hubbard\nparameters, Jwas fixed at its recommended value of 0.6\nFIG. 8. (a) The primitive cell of Cr 2O3. The dark blue balls\nrepresent chromium atoms, and the red balls stand for oxygen\natoms. (b) The magnetic ground state. The arrows of differ-\nent colors represent the spin directions of Cr atoms. (c) The\ndensity of states obtained by DFT calculations. (d) The tem-\nperature dependence of sublattice magnetization measured in\nMC and ASD simulations. For Cr 2O3, the phase transition\npoint from ferrimagnetic to paramagnetic lies in 310K approx-\nimately.\nTABLE IV. The exchange coefficients J of Cr 2O3, in which\nan effective spin S= 1 is adopted.\nSpin Pair. Distance (Angst) J (meV)\n1NN Cr-Cr 2.640 44.778\n2NN Cr-Cr 2.873 29.269\n3NN Cr-Cr 3.411 -0.182\n4NN Cr-Cr 3.635 0.007\n5NN Cr-Cr 4.137 -0.500\neV, and Uwas adjusted to fit the N´ eel temperature ob-\nserved in experiments [60]. We found U= 2.0 eV was the\noptimal value for 3 delectrons of Cr ions. Except for the\nsettings specified above, all the DFT calculations were\nconducted under the same conditions as in Sec. III C.\nThe DOS of Cr 2O3is plotted in Fig. 8(c), which yields\na calculated band gap of 1.935 eV. This value indicates\nthat the energy dissipation of electric currents can be ne-\nglected in this system. Additionally, we list the Heisen-\nberg exchange coefficients of chromia in Table IV. Both\nMC and ASD simulations were performed to investigate\nthe temperature dependence of sublattice magnetization.\nAccording to Fig. 8(d), the critical point was determined\nto be 310 K approximately, which was quite consistent\nwith experimental observations. Also, the force constants\nof Cr 2O3went through the modification formulated in\nSec. II B, and the spin-lattice coupling parameters are\nprovided in the supplementary material.\nAfter the construction of Hamiltonian model, we con-\nducted a series of dynamics simulations to evaluate the11\nFIG. 9. (a) The 1NN FeT-FeOpair in Y 3Fe5O12. (b) The\n1NN Cr-Cr pair in Cr 2O3. The steel blue arrow stands for\nthe orientation of ∂J/∂u and the red number along with it\nrepresents the magnitude in unit of meV/Angst.\nequivalent damping parameters in Cr 2O3. An expanded\nhexagonal cell containing 14400 atoms was adopted for\nthe simulation, and the results are summarized in Fig. 11.\nAs two specific cases, our calculation yielded α= (1.31±\n0.14)×10−4at ∆T= 15 K and α= (2.7±0.3)×10−4\nat ∆T= 30 K, where both TL= 0 K. Therefore, the\ncalculated damping constants within ∆ T= 15∼30 K\nare quite close to 2 ×10−4, which is the estimated value\nreported in Ref. [61].\nFurthermore, the damping constants in Cr 2O3exhibit\na significant non-linear relation with the temperature dif-\nference of subsystems. Through logarithmic fittings, we\ncalculated the power exponents for Figures 11(a) to 11(c),\nand the results were 1.17, 1.62, 1.38. If we disregard the\ndifference between ∆ TandTfor the moment, these val-\nues are in good agreement with the theoretical prediction\nof Kasuya and LeCraw [26]. According to their study, the\nrelaxation rate varies as Tnwhere n= 1∼2 while n= 2\ncorresponds to a larger regime of temperatures.\nCompared to YIG, the greater magnetic damping ob-\nserved in chromia can be attributed to its significantly\nstronger spin-lattice coupling. As shown in Fig. 9, the\nmagnitude of principal spin-lattice coupling constant in\nCr2O3is two or three times larger than that in YIG. This\ncould be explained by the fact that direct exchange in-\nteraction between two magnetic atoms decreases rapidlywith their distance [62]. Therefore, owing to the shorter\ndistance of Cr-Cr pair, the direct exchange interaction\nbetween neighboring Cr atoms is believed to have a great\ncontribution to the spin-lattice coupling in Cr 2O3.\nIV. CONCLUSIONS\nIn summary, we propose a scheme to evaluate the con-\ntribution of spin-lattice coupling to the Gilbert damp-\ning in insulating magnetic materials. Our methodology\ninvolves first-principles based Hamiltonian models and\nspin-lattice dynamics simulations. Following a series of\nvalidations, we applied our method to three magnetic ma-\nterials, namely Y 3Fe5O12, MnFe 2O4and Cr 2O3. Their\ndamping constants were estimated separately, and the\nresults show that, in general, αis approximately propor-\ntional to the temperature difference between spin and\nlattice subsystems. Under the condition of ∆ T= 30\nK, the calculated damping constants are averaged to be\n0.8×10−4for YIG, 0 .2×10−4for MFO and 2 .2×10−4\nfor Cr 2O3. The results for YIG and Cr 2O3are in good\nagreement with experimental measurements, while the\ndiscrepancy for MFO can be attributed to the inhomo-\ngeneity and small band gap in real samples. Overall, the\napproach presented in this work holds great promise for\naccurately predicting the Gilbert damping constants for\nmagnetic insulators.\nACKNOWLEDGMENTS\nThis work is supported by the National Key R&D\nProgram of China (No. 2022YFA1402901 ), the Na-\ntional Natural Science Foundation of China (Grant Nos.\n11825403, 11991061, and 12188101), the Guangdong Ma-\njor Project of the Basic and Applied Basic Research\n(Future functional materials under extreme conditions–\n2021B0301030005).\n[1] S. Wolf, D. Awschalom, R. Buhrman, J. Daughton,\nS. von Molnar, M. Roukes, A. Chtchelkanova, and\nD. Treger, Spintronics: A spin-based electronics vision\nfor the future, SCIENCE 294, 1488 (2001).\n[2] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nAntiferromagnetic spintronics, NATURE NANOTECH-\nNOLOGY 11, 231 (2016).\n[3] L. Smejkal, Y. Mokrousov, B. Yan, and A. H. MacDon-\nald, Topological antiferromagnetic spintronics, NATURE\nPHYSICS 14, 242 (2018).\n[4] I. Zutic, J. 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Oppeneer1\n1Department of Physics and Astronomy, Uppsala University, P. O. Box 516, S-751 20 Uppsala, Sweden\n2Department of Physics, Freie Universit¨ at Berlin, Arnimallee 14, D-14195 Berlin, Germany\n(Dated: January 17, 2024)\nUltrafast laser excitation of ferromagnetic metals gives rise to correlated, highly non-equilibrium\ndynamics of electrons, spins and lattice, which are, however, poorly described by the widely-used\nthree-temperature model (3TM). Here, we develop a fully ab-initio parameterized out-of-equilibrium\ntheory based on a quantum kinetic approach–termed (N+2) temperature model –that describes\nmagnon occupation dynamics due to electron-magnon scattering. We apply this model to per-\nform quantitative simulations on the ultrafast, laser-induced generation of magnons in iron and\ndemonstrate that on these timescales the magnon distribution is non-thermal: predominantly high-\nenergy magnons are created, while the magnon occupation close to the center of the Brillouin zone\neven decreases, due to a repopulation towards higher energy states via a so-far-overlooked scattering\nterm. We demonstrate that the simple relation between magnetization and temperature computed\nat equilibrium does not hold in the ultrafast regime and that the 3TM greatly overestimates the\ndemagnetization. The ensuing Gilbert damping becomes strongly magnon wavevector dependent\nand requires a description beyond the conventional Landau-Lifshitz-Gilbert spin dynamics. Our ab-\ninitio -parameterized calculations show that ultrafast generation of non-thermal magnons provides\na sizable demagnetization within 200fs in excellent comparison with experimentally observed laser-\ninduced demagnetizations. Our investigation emphasizes the importance of non-thermal magnon\nexcitations for the ultrafast demagnetization process.\nI. INTRODUCTION\nThe discovery that magnetic order can be manipu-\nlated on sub-picosecond timescales by femtosecond laser\npulses [1–3] has fueled the emergence of intensive exper-\nimental and theoretical research efforts in the field of ul-\ntrafast magnetization dynamics. What makes this field\nparticularly interesting, apart from its technological po-\ntential in future memory and spintronic devices [4, 5], is\nthat many well-established physical paradigms cannot be\nsimply transferred from the equilibrium to the ultrafast\nregime, due to its highly non-equilibrium nature. Relat-\nedly, albeit more than 25 years of intense research, the\nunderlying mechanisms of ultrafast demagnetization are\nstill heavily debated [6–8]: while some works [9–14] lean\ntowards longitudinal excitations – i.e., the reduction of\nthe magnetic moment carried by each atom due to the de-\ncrease of exchange splitting – others [15–19] hint at trans-\nverse spin excitations – a reduction of the average magne-\ntization due to the mutual tilting of the moments carried\nby different atoms – as the main contribution. Non-local\ncontributions due to superdiffusive spin currents [20, 21]\nare relevant in certain situations [22–25]. However, it has\nbecome evident that they are most likely not the only\nmechanism of ultrafast demagnetization [26, 27].\nTheoretical models describing ultrafast magnetization\ndynamics typically rely on a separation of electronic,\nphononic and – if magnetization dynamics are to be con-\nsidered separately – spin degrees of freedom. Beaurepaire\net al. [1] introduced the three-temperature model (3TM)\nto explain the flow of the energy transferred by the laser\n∗markus.weissenhofer@fu-berlin.deby assuming that each subsystem is internally in thermal\nequilibrium and the system can hence be described by\nthree temperatures (for electrons, phonons and spins), to-\ngether with the respective distributions (Fermi-Dirac and\nBose-Einstein). However, it was pointed out in numer-\nous investigations that the distributions are non-thermal\non ultrafast timescales [28–37]. Also, the 3TM discards\ncompletely the transfer of angular momentum due to de-\nmagnetization, which, according to recent experiments\n[38, 39], appears to be primarily to the lattice.\nTransverse demagnetization is often studied using\natomistic spin dynamics simulations based on the\nstochastic Landau-Lifshitz-Gilbert (LLG) equation to-\ngether with an extended Heisenberg model [40–42], which\ncan successfully reproduce experimentally measured de-\nmagnetization curves [43, 44]. The stochastic LLG is\na Langevin-type equation with a coupling to a heat\nbath with given temperature via a single parameter, the\nGilbert damping parameter. This parameter includes all\npossible contributions – Fermi surface breathing, crystal\ndefects, coupling to phonons, s−dcoupling, etc. [45–52] –\nto damping and while it can in principle be obtained from\nab initio calculations, in practice it is typically taken from\nexperimental measurements of ferromagnetic resonance\n(FMR) [53]. On the one hand, this ensures the versatility\nof atomistic spin dynamics simulations, but on the other\nhand, it obscures the details of the underlying micro-\nscopic energy and angular momentum transfer processes\n- which are crucial for understanding the fundamentals\nof ultrafast demagnetization. For this reason, steps have\nbeen taken in recent years to explicitly consider the cou-\npling of spins to phonons [54–62] and electrons [63–65].\nAlso, due to the classical nature of the commonly used\nstochastic LLG, the equilibrium magnon occupations cal-arXiv:2309.14167v3  [cond-mat.mtrl-sci]  15 Jan 20242\nculated by it follow Rayleigh-Jeans rather than Bose-\nEinstein statistics, henceforth leading to the wrong tem-\nperature scaling of the magnetization [66, 67]. Implemen-\ntation of quantum statistics in the spin-dynamics simu-\nlations can however provide the correct low-temperature\nscaling of the magnetization [68, 69].\nIn this work, we investigate the laser-induced gener-\nation of magnons, the low energy transverse excitations\nof the spin system, due to electron-magnon scattering.\nWe develop a quantum kinetic approach, which will be\ntermed (N+2)-temperature model [(N+2)TM], to per-\nform quantitative simulations of the time evolution of\nthe non-thermal magnon dynamics in bcc iron. Being\nbased on ab initio parameters and considering also non-\nthermal magnon distributions, our work goes well beyond\nwhat has been done in Refs. [63, 64, 70] and the conven-\ntional 3TM. In addition, we show that the 3TM and its\nrelevant parameters can be obtained from our (N+2)TM\nand, with that, from ab initio calculations. Importantly,\nusing ab initio calculated input parameters, our quantum\nkinetic theory predicts a sizable and ultrafast demagne-\ntization of iron within 200 fs, in excellent agreement with\nexperiments [15].\nII. OUT-OF-EQUILIBRIUM MAGNON\nDYNAMICS MODEL\nTo describe the time evolution of the ultrafast non-\nthermal magnon occupation dynamics, we assume that\ntheir creation and annihilation is dominated by electron-\nmagnon scattering processes. In this work, we use the\nsp−dmodel [71, 72] to describe such processes. The\nbasic idea of both s−dmodel and sp−dmodel is the\nseparation of electrons in localized ( dband) electrons and\nitinerant ( sband or sandpbands) electrons. The mag-\nnetic moments of the delectrons make up the Heisenberg-\ntype [73] magnetic moments of constant length, the small\nenergy excitations of which are the magnons. The itin-\nerant electrons are described within a Stoner-type model\n[74]. While an unambiguous identification of spandd\nelectrons as localized and itinerant is strictly speaking\nnot possible, it has nonetheless been established in liter-\nature that these models provide a suitable framework for\nthe description of electron-spin interaction in many phe-\nnomena relevant for spintronics, e.g. magnetic relaxation\n[75–77], ultrafast demagnetization [63–65, 70, 78–80] and\nspin torques [81].\nWe assume local exchange between the itinerant and\nlocalized spins, as given by the Hamiltonian ˆHem∼PN\ni=1ˆsitin·ˆSloc\ni, with Nbeing the number of atoms,\nandˆsitinand ˆSloc\nithe spin operators for itinerant ( sp)\nelectrons and localized ( d) electrons at atom i. In sec-\nond quantization and second order in magnon variables(details in Method Section V A), the Hamiltonian reads\nˆHem≈ −∆X\nkν\u0010\nˆc†\nkν↑ˆckν↑−ˆc†\nkν↓ˆckν↓\u0011\n−∆r\n2\nSNX\nkνν′,q\u0010\nˆc†\nk+qν↑ˆckν′↓ˆb†\n−q+ ˆc†\nk+qν↓ˆckν′↑ˆbq\u0011\n+∆\nSNX\nkνν′,qq′\u0010\nˆc†\nk−q+q′ν↑ˆckν′↑−ˆc†\nk−q+q′ν↓ˆckν′↓\u0011\nˆb†\nqˆbq′.\n(1)\nHere, ∆ is the sp−dexchange parameter, Sis the ab-\nsolute value of the localized spins, kandqare vectors in\nreciprocal space, ˆ c(†)\nkνσis the fermionic electron annihila-\ntion (creation) operator for the itinerant electrons – with\nνbeing the band index and σ∈ {↑,↓}– and ˆb(†)\nqis the\nbosonic magnon annihilation (creation) operator. The\nfirst term in Equation (1) describes the spin-splitting of\nthe itinerant electrons due to the exchange with the lo-\ncalized magnetic moments, the second one the excitation\n(annihilation) of a magnon due to a spin flip process and\nthe third one the spin-conserving scattering of a magnon\nand an electron from one state to another. It is worth\nnoting that the second term leads to a transfer of both\nenergy and angular momentum (i.e., spin) – since it can\nchange the total number of magnons – while the third\nterm can only transfer energy. For this reason, this term\nwas discarded earlier works [63–65], however, our quanti-\ntative analysis reveals that the energy transferred by this\nterm can exceed the energy transferred by the term first\norder in magnon operators.\nWe complete our Hamiltonian H=ˆHe+ˆHm+ˆHem\nby considering ˆHe=P\nkνσεkνσˆc†\nkνσˆckνσand ˆHm=P\nqℏωqˆb†\nqˆbq, with εkνσ=εkν−2∆δσ↑being the mode\nand spin dependent electron energies that are calcu-\nlated from first-principles calculations and ℏωqbeing the\nmagnon energies. Note that we have absorbed the term\nzero-th order in magnon variables in Equation (1) in the\notherwise spin-independent ˆHe.\nNext, we use the Hamiltonian introduced above to\nconstruct a quantum kinetic approach for the descrip-\ntion of the out-of-equilibrium dynamics of electrons and\nmagnons. We define the rates of energy exchange be-\ntween both subsystems as\n˙Em=X\nqℏωq˙nq (2)\n˙Ee=X\nkνσεkνσ˙fkνσ=−X\nqℏωq˙nq. (3)\nwhere the dot represents temporal derivative and with\nthe electron ( fkνσ) and magnon ( nq) occupation num-\nbers. The equivalence in Equation (3) results from the\nconservation of total energy. The time derivatives of\nthe occupation numbers can be calculated by applying\nFermi’s golden rule to the scattering Hamiltonian (1).3\nTo simplify the calculations, we further assume a ther-\nmal electron distribution and can hence introduce a sin-\ngle electronic temperature Tethat relates to the occu-\npation of electronic states via the Fermi-Dirac distribu-\ntion. This allows us to apply and also extend (by in-\ncluding terms second order in the bosonic operators) the\nideas laid out in Allen’s seminal work on electron-phonon\ninteraction [82] to electron-magnon scattering, yielding\n˙nq=\u0002\nnBE(ωq, Te)−nq\u0003\nγq+P\nq′\u0002\n(nq+ 1)nq′nBE(ωq−\nωq′, Te)+(q↔q′)\u0003\nΓqq′, with nBE(ωq, Te) = [eℏωq\nkBTe−1]−1\nbeing the Bose-Einstein distribution evaluated at the\nelectron temperature. The scattering rates are given by\nγq=4π∆2\nSNωqI↑↓(Te)X\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓),\n(4)\nΓqq′=2π∆2\nS2N2(ωq−ωq′)X\nσIσσ(Te)\n×X\nkνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ),(5)\nwith εFbeing the Fermi energy. The functions Iσσ′(Te)\nhave the property lim Te→0Iσσ′(Te) = 1 and account\nfor the smearing of the Fermi-Dirac distribution at high\nelectron temperatures, similar to what has been derived\nfor electron-phonon scattering [35]. The expression for\nIσσ′(Te) and details of the derivation of Equations (4)–\n(5) are in the Method Section V A. Note that a com-\nparison with linear spin-wave theory in the framework\nof the Landau-Lifshitz-Gilbert equation [83] reveals that\nγq/ωq=αqcan be viewed as a mode-dependent Gilbert\ndamping parameter.\nDue to the assumption that the electron occupation\nnumbers follow the Fermi-Dirac distribution at all times,\nthe change in electron energy is determined by the\nchange in Te, i.e., ˙Ee=P\nkνσεkνσ(∂fkνσ/∂T e)˙Te=\nCe˙Te, with the electronic heat capacity Ce=P\nkνσεkνσ(∂fkνσ/∂T e). By additionally considering the\nabsorption of a laser pulse with power P(t) by the\nelectrons and a coupling of the electrons to a phonon\nheat bath as in the 2TM, we finally obtain our out-of-\nequilibrium magnon dynamics model:\n˙nq=h\nnBE(ωq, Te)−nqi\nγq\n+X\nq′h\n(nq+ 1)nq′nBE(ωq−ωq′, Te) + (q↔q′)i\nΓqq′,\n(6)\n˙Te=1\nCeh\n−X\nqℏωq˙nq+Gep(Tp−Te) +P(t)i\n, (7)\n˙Tp=−Gep\nCp(Tp−Te). (8)\nHere, Tp,Cpand Gepare the phonon temperature\nand heat capacity and electron-phonon coupling con-stant, respectively. Note that we do not consider di-\nrect magnon-phonon coupling, which has been shown\nto be a reasonable approximation for 3 dferromagnets\n[43, 44]. We would like to point out that the non-\nthermal magnon occupations nqcan be translated to\nmode-specific temperatures via the Bose-Einstein distri-\nbution, Tq:=ℏωq/(kBln(n−1\nq+ 1)). Based on this – and\nin distinction from the 3TM – we term the framework\nprovided by Equations (6)-(8) the (N+2)-temperature\nmodel ((N+2)TM). Below, we reveal by solving these\ncoupled equations numerically that they provide a vi-\nable framework to describe laser-induced ultrafast mag-\nnetization dynamics and the generation of non-thermal\nmagnons, going beyond the well-established 3TM.\nBefore doing so, we want to shortly discuss the relation\nbetween the (N+2)TM introduced here and the 3TM. Al-\nbeit their phenomenological nature, the 2TM ( TeandTp)\nand the 3TM ( Te,TpandTm) have been successfully ap-\nplied to explain a plethora of phenomena [84], perhaps\nmost prominently by Beaurepaire et al. to describe the\nultrafast demagnetization of Ni [1]. Allen [82] and Man-\nchon et al. [78] demonstrated that the 2TM and the 3TM\ncan be derived from a microscopic out-of-equilibrium ap-\nproach similar to the one used here. By assuming instan-\ntaneous relaxation of the magnon occupation numbers\nto the Bose-Einstein distribution with a single magnon\ntemperature Tm, our (N+2)TM reduces to the 3TM (in\nabsence of magnon-phonon coupling),\nCm˙Tm=Gem(Te−Tm),\nCe˙Te=Gem(Tm−Te) +Gep(Tp−Te) +P(t),\nCp˙Tp=Gep(Te−Tp),(9)\nwith the magnon heat capacity Cm=P\nqCq=P\nqℏωq(∂nq/∂T m) and the electron-magnon coupling\nconstant\nGem=X\nqCqh\nγq+X\nq′kBTm\nℏωqΓqq′i\n. (10)\nDetails of the derivation are found in Method Section\nV B. The above expression goes beyond what was derived\nin Ref. [78] by including terms second order in magnon\nvariables and allows us to determine the electron-magnon\ncoupling fully based on ab initio parameters. We would\nlike to point out that it can be extended further by going\nto higher order in the magnon variables.\nIII. RESULTS\nA. Magnon lifetimes and Gilbert damping\nWe apply the (N+2)TM model defined by Equations\n(6)-(8) to bcc iron. To obtain a full solution of the out-\nof-equilibrium dynamics, it is required to calculate ma-\nterial specific quantities. First, we estimate ∆ ≈0.75 eV\nfrom the band structure and with that we compute the4\nΓ H N Γ P H020406080100120magnon frequency (THz)\n100101102103\nlifetime (ps)\nFigure 1. Magnon dispersion of bcc iron with lifetimes γ−1\nq\ngiven as color code, shown along high-symmetry lines of the\nBZ. The lifetimes are due to the first-order contribution to\nthe electron-magnon scattering.\nquantities γq, Γqq′andIσσ′(Te), both using the full-\npotential linear augmented plane wave code ELK [85]\n(details can be found in the Method Section V C). For\nbcc iron it turns out that Iσσ′(Te) only scales weakly\nwith temperature and hence we use the low tempera-\nture limit Iσσ′(Te) = 1 hereinafter. The parameters gov-\nerning the magnon energies ℏωq=S(2d+P\njJij[1−\nexp(−iq·(rj−ri))]) were taken from earlier works: the\nexchange constants Jijare from first-principles calcula-\ntions [86] and the magneto-crystalline anisotropy energy\nd= 6.97µeV per atom is from experiments [87]. Further,\nwe used the saturation moment µs= 2.2µBand spin\nS= 2.2/2. Based on these parameters and the formulas\nderived above, we get Cm= 5.720×104Jm−3K−1and\nGem= 6.796×1017Wm−3K−1atTm= 300 K. Notably,\nthe term first order in magnon variables leads to a con-\ntribution to Gemthat is one order of magnitude smaller\nthan the second-order term. We further use the room-\ntemperature values Ce= 1.013×105Jm−3K−1,Cp=\n3.177×106Jm−3K−1andGep= 1.051×1018Wm−3K−1\nthat were obtained in Refs. [35, 37] from first-principles\ncalculations. The influence of a temperature dependent\nelectronic heat capacity Ceon the demagnetization is dis-\ncussed in the Supporting Information.\nBoth ωqand the inverse of γq, i.e., the lifetime of\nmagnons due to the contribution to electron-magnon\nscattering linear in the magnon variables, are shown in\nFigure 1 along high-symmetry lines of the Brillouin zone\n(BZ). It can readily be observed that the lifetimes of\nhigh-frequency magnons are drastically reduced as com-\npared to the low energy ones. The q-dependent lifetimes\ngive rise to mode-specific Gilbert damping αq(=ωq/γq).\nOur finding of mode-dependent Gilbert damping is con-\nsistent with experiments [88] and also with a recent field-\ntheory derivation [89]. The computed αqvalues, shown\nin Method Section V C, range between 1 .5×10−3and\n1.08×10−2. These values are close to the experimentally\nobtained ones (via FMR measurements) for Fe ranging\nfrom 1 .9×10−3to 7.2×10−3[90–95], however with a\n0 1 2 3300400500600700temperature T (K)(a)\n0 1 2 3\ntimet(ps)−1.0−0.50.0∆M/M 0(%) (b)Te\nTp\n/angbracketleftTq/angbracketrightT3TM\ne\nT3TM\np\nT3TM\nm\n0.0 0.2 0.4 0.6 0.8 1.0\nlaser fluence (mJ /cm2)0510demagnetization (%)\n(c)Figure 2. Laser-induced ultrafast non-equilibrium dynamics\nof iron calculated from an ab initio parameterized model.\n(a) Temporal evolution of electron temperature Te, phonon\ntemperature Tpand average magnon temperature ⟨Tq⟩=\n1/NP\nqTqobtained by the (N+2)TM (solid lines). The\nblue shaded region indicates the temperature range within\nwhich all magnon temperatures are contained. Dashed lines\nshow the results of the 3TM solved with ab initio calcu-\nlated input parameters. (b) Relative change of total mag-\nnetization of the localized magnetic moments ∆ M/M 0=P\nq(ninit\nq−nq)/(NS−P\nqninit\nq), with ninit\nq=nBE(ωq,300 K)\nbeing the occupation number before the laser pulse. (c) De-\nmagnetization max( |∆M/M 0|) versus laser fluence computed\nfor a ferromagnetic layer with a thickness of 20 nm. The dot-\nted line serves as a guide to the eye.\nsomewhat larger variation with qas compared to what\nwas reported in Ref. [83].\nWe note that the q-dependent Gilbert damping goes\nbeyond the conventional LLG description which assumes\none single damping parameter for all spin dynamics.\nMoreover, a further distinction between the current the-\nory and the LLG framework is that, in the latter, there\nis a single damping term that governs both the energy\nand angular momentum transfer [96], whereas the cur-\nrent theory has two terms [see Equation (1)], one that\ntransfers energy and angular momentum and one that\ntransfers only energy. As shown in the following, this 2nd\nterm is found to be important for non-thermal magnon5\ngeneration.\nB. Ultrafast dynamics\nBased on the above-given parameters, we calculate\nthe coupled out-of-equilibrium magnon, electron, and\nphonon dynamics induced by a Gaussian laser pulse\nP(t) = A/p\n2πζ2exp[−(t/ζ)2/2] with A= 9.619×\n107Jm−3andζ= 60 fs for N= 203magnon modes. Note\nthat this value of Atranslates to an absorbed fluence of\n0.19 mJ /cm2for a ferromagnetic layer with thickness of\n20 nm, which is a typical thickness in ultrafast demagne-\ntization experiments [1].\nFigure 2(a) depicts the time evolution of electron,\nphonon and average magnon temperature – together with\nthe temperature range of all magnon temperatures – cal-\nculated using the (N+2)TM. The electron temperature\nreaches a maximum of 685 K at around 52 fs after the\nmaximum of the laser pulse (located at t= 0) and con-\nverges to the phonon temperature in less than 1 .5 ps. The\nmaximum of the average magnon temperature of 520 K\nis reached only slightly after the electronic one at around\n136 fs, followed by a convergence to the electronic and\nphononic temperature to a final temperature of around\n329 K at 3 ps, in agreement with what can be estimated\nfrom the energy supplied by the laser pulse and the in-\ndividual heat capacities via ∆ T=A/(Cm+Ce+Cp) =\n28.8 K. Notably, the magnon temperatures still cover a\nrange of around 50 K at this point in time. Our results\nclearly demonstrate the shortcomings of the conventional\n3TM (shown as dotted lines): While the initial increase\nof temperatures is comparable to the (N+2)TM, magnon\nthermalization happens much faster in the 3TM.\nIn Figure 2(b), we show the laser-induced change in\nmagnetization (associated with the localized magnetic\nmoments) due to the creation of additional magnons. We\nobserve ultrafast transversal demagnetization of around\none percent in less than 300 fs, demonstrating that the\ntimescales obtained by our ab initio based calculations\nare in reasonable agreement with experimental measure-\nments (see, e.g., [15, 97–99]). Notably, the minimum\nof the magnetization and the maximum in the average\nmagnon temperature computed by the (N+2)TM are at\ndifferent points in time. Also, the drop in the (localized)\nmagnetization is much less pronounced than expected\nfrom the increase in average temperature: in thermal\nequilibrium, a temperature increase from 300 K to above\n500 K approximately leads a demagnetization of 20% for\niron [100]. These observations clearly demonstrate the\nshortcomings of the 3TM – where a thermal magnon dis-\ntribution at all times is assumed – and underline the im-\nportance of treating the full, non-thermal magnon distri-\nbution in the ultrafast regime.\nFigure 2(c) depicts the maximum of the demagnetiza-\ntion versus laser fluence for an iron layer of 20 nm. We\nfind a nonlinear dependence, which is a result of the non-\nlinearity of our (N+2)TM, and a substantial demagneti-\n0 2 4 6 8 10\ntimet(ps)0.850.900.951.00M/M 0\n(N+2)TM\nexperimentFigure 3. Comparison between experiment and the (N+2)TM\ntheory for ultrafast demagnetization in iron. The experimen-\ntal data (symbols) are those of Carpene et al. [15] and the\nsolid lines are calculated from the ab initio parameterized\n(N+2)TM.\nzation of around ten percent at 0 .95 mJcm−2. We note\nthat for high fluences, higher-order magnon-magnon scat-\ntering terms that are not included in the current model\ncould start to play a role.\nThe obtained amount of demagnetization and the mag-\nnetization decay time (below 200 fs) for this fluence are\ncomparable with experiments, which suggests that ultra-\nfast magnon excitation [15–17] provides a viable mech-\nanism for ultrafast laser-induced demagnetization. It\nis also consistent with time-resolved extreme ultraviolet\nmagneto-optical and photoemission investigations that\ndetected magnon excitations during ultrafast demagneti-\nzation of elemental ferromagnetic samples [18, 101].\nFor a more precise examination of the predictions\nof the (N+2)TM, we compare the calculated time-\ndependent demagnetization with experimental data for\nFe in Figure 3. The experimental data were measured\nby Carpene et al. [15] on a 7-nm thin film, using the\ntime-resolved magneto-optical Kerr effect for two differ-\nent pump laser fluences of 1 .5 mJcm−2and 3 mJcm−2.\nIn the calculations we used an absorbed laser fluence\nthat is about five times lower, as the exact value of the\nabsorbed fluence in experiments is difficult to estimate\n(due to influence of optical losses, sample reflection,\netc). Specifically, in the simulations we used absorbed\nlaser energies of 433 Jcm−3and 693 Jcm−3in a 7-nm Fe\nfilm. Figure 3 exemplifies that not only the amount of\ndemagnetization but also the full time dependence of\nthe demagnetization predicted by the (N+2)TM is in\nremarkable agreement with experiments.\nC. Non-thermal magnon dynamics\nNext, we analyze the non-thermal magnon dynam-\nics in more detail in Figure 4. There, we show the\nmagnon temperatures versus frequency (a) and along\nhigh-symmetry lines of the BZ (b) at different points in\ntime. The laser pulse primarily heats up high energy\nmagnons, while the temperature of low energy magnons6\n300500\n−250 fs\n300500\n0 fs\n300500temperature (K)\n250 fs\n300500\n500 fs\n0 20 40 60 80 100 120\nmagnon frequency (THz)300500\n1000 fs\nΓ H N Γ P H−200 fs−100 fs0 fs100 fs200 fs300 fs400 fs500 fs600 fs700 fs800 fs900 fs1000 fs\n300350400450500550600\ntemperature (K)(a) (b)\nFigure 4. Magnon temperatures of iron during ultrafast laser excitation at different points in time (w.r.t. the maximum of\nthe laser pulse) calculated from the ab initio parameterized (N+2)TM. (a) Magnon temperatures (dots) versus frequency. The\nsolid line indicates the electron temperature. (b) Magnon dispersion and their temperatures, depicted by the color code, shown\nalong high-symmetry lines of the BZ.\nbarely changes and even decreases slightly in the vicinity\nof the Γ point (the temperatures drop by up to around\n2.5 K). This surprising observation is caused by a\nredistribution of magnons from this region to other\nparts of the BZ due to the term second order in the\nmagnon operators in Equation (1); the effective second\norder scattering rate γ(2)\nq:=P\nq′Γqq′is negative for low\nmagnon frequencies (more details can be found in the\nMethod Section V C). It is also observed that although\nthe magnon temperatures reached after the laser pulse\nare generally higher at higher frequencies, however, there\nis not necessarily a monotonous increase of temperature\nwith frequency at all times: e.g., at 100 fs after the laser\npulse [Figure 4(b)], the temperatures at the points H, N,\nand P is higher than in between these points. Notably,\nthe position of the maximum magnon temperature in\nthe BZ also varies with time.\nD. Discussion\nDifferent physical mechanisms have been proposed for\nultrafast demagnetization in elemental 3 dferromagnets\n[9, 11, 15, 18]. The preeminent mechanisms are Elliott-\nYafet (EY) electron-phonon spin-flip scattering [11, 13]\nand ultrafast magnon generation [15]. In the former, a\nStoner-type picture is used to model the longitudinal re-\nduction of the atomic moment due to electron-phonon\nspin-flip scattering, whereas the latter is based on length-\nconserving transverse spin-wave excitations. Experimen-\ntal indications of electron-phonon scattering [38, 39] aswell as of electron-magnon scattering have been reported\n[18, 101].\nThe strength of the different demagnetization channels\nis an important issue in the on-going discussion on the\ndominant origin of ultrafast demagnetization [7]. Ab ini-\ntiocalculated quantities such as the EY spin-flip prob-\nability are essential to achieve reliable estimates [102–\n104]. Griepe and Atxitia [14] recently employed the\nmicroscopic 3TM [11] and obtained quantitative agree-\nment with measured demagnetizations for the elemental\n3dferromagnets. They compared the fitted EY spin-\nflip probability αsfwith ab initio calculated values [104]\nand found these to be in good agreement, in support of\nan electron-phonon mechanism of ultrafast demagnetiza-\ntion. A drawback of their employed approach is how-\never that only magnetization reducing spin flips are in-\ncluded. EY spin flips that increase the magnetization are\nalso possible and, including these would lead to a signifi-\ncantly smaller demagnetization amplitude [104]. This in\nturn would question again what amount of demagneti-\nzation is precisely due to EY electron-phonon spin flip\nscattering. Conversely, in our non-thermal magnon ap-\nproach we employ ab initio calculated quantities without\nfit parameter. We find that the ab initio predicted ultra-\nfast demagnetization agrees accurately with experiments,\nwhich provides a strong support for the prominence of the\nnon-thermal magnon channel to the ultrafast demagne-\ntization process.7\nIV. CONCLUSIONS\nWe have developed an ab initio parameterized quan-\ntum kinetic approach to study the laser-induced gen-\neration of magnons due to electron-magnon scattering,\nwhich we applied to iron. Our results clearly demon-\nstrate that on ultrafast timescales the magnon distribu-\ntion is non-thermal and that henceforth the simple re-\nlation between magnetization and temperature via the\nM(T) curves computed at equilibrium does not hold:\nsince predominantly high-energy magnons are excited the\nenergy transferred from the laser-excited electrons cre-\nates relatively few magnons and hence the demagneti-\nzation (proportional to the total number of magnons) is\nmuch less pronounced than expected from the increaseof the average magnon temperature. Notably, the num-\nber of magnons actually decreases near the center of the\nBrillouin zone, which is due to the scattering from low to\nhigh energy magnons by a previously neglected scattering\nterm that can transfer energy but not angular momen-\ntum. This term, which is not included in LLG simula-\ntions, is a crucial quantity for out-of-equilibrium magnon\ndynamics.\nOurab initio -based calculations of the induced demag-\nnetization in iron furthermore provide strong evidence\nthat non-thermal magnons are excited fast and lead to\na sizable demagnetization within 200 fs. The result-\ning time-dependent demagnetization agrees remarkably\nwell with experiments, which establishes the relevance of\nmagnon excitations for the process of ultrafast optically\ninduced demagnetization.\nV. METHOD\nA. Derivation of electron-magnon scattering rates\nIn this Method Section we derive the (N+2)TM for the description of non-thermal magnons from a microscopic\nHamiltonian for electron-magnon scattering. We start with a local sp−dmodel Hamiltonian,\nˆHem=−Jsp−dX\niδ(r−ri)ˆsitin·Sloc\ni, (11)\nwith Jsp−dbeing the sp−dvolume interaction energy, ˆsitin=ˆσbeing the spin operators of itinerant ( sandp)\nelectrons and Sloc\nibeing the localized ( d) spins located at ri. For now, we treat the latter as classical vectors. The\nexpectation value for a given spin wave function Ψ(r) is given by\n⟨ˆHem⟩=−Jsp−dX\niZ\nΨ†(r)δ(r−ri)ˆsitin·Sloc\niΨ(r)dr (12)\n=−Jsp−dX\niZ\nδ(r−ri)\u0000Ψ∗\n↑(r),Ψ∗\n↓(r)\u0001\b\nˆσxSx\ni+ ˆσySy\ni+ ˆσzSz\ni\t\u0012\nΨ↑(r)\nΨ↓(r)\u0013\ndr (13)\n=−Jsp−dX\niZ\nδ(r−ri)(\nΨ∗\n↑(r)Ψ↓(r)S−\ni+ Ψ∗\n↓(r)Ψ↑(r)S+\ni+ (Ψ∗\n↑(r)Ψ↑(r)−Ψ∗\n↓(r)Ψ↓(r))Sz\ni)\ndr.(14)\nHere, we have introduced S±\ni=Sx\ni±iSy\ni. Next, we perform a plane wave expansion of the wave functions (for a\nsingle band of itinerant electrons),\nΨσ(r) =1√\nVX\nkeik·rckσ, (15)\nand a Holstein-Primakoff transformation of the localized spins,\nS+\ni=p\n2S−b∗\nibibi, S−\ni=b∗\nip\n2S−b∗\nibi, Sz\ni=S−b∗\nibi, (16)\ntogether with introducing the Fourier transform of the magnon amplitudes\nb∗\ni=1√\nNX\nqe−iq·rib∗\nq, b i=1√\nNX\nqeiq·ribq. (17)8\nInsertion of (15)–(17) into (14) and keeping terms up to second order in magnon variables, we get\n⟨ˆHem⟩=−Jsp−d\nVX\niX\nkk′(r\n2S\nNX\nqe−i(k−k′+q)·ric∗\nk↑ck′↓b∗\nq+r\n2S\nNX\nqe−i(k−k′−q)·ric∗\nk↓ck′↑bq\n+Se−i(k−k′)·ri(c∗\nk↑ck′↑−c∗\nk↓ck′↓)−1\nNX\nqq′e−i(k−k′+q−q′)·ri(c∗\nk↑ck′↑−c∗\nk↓ck′↓)b∗\nqbq′) (18)\n=−Jsp−dSN\nVX\nk(c∗\nk↑ck↑−c∗\nk↓ck↓)−Jsp−dSN\nVX\nkqr\n2\nSN\u0010\nc∗\nk+q↑ck↓b∗\n−q+c∗\nk+q↓ck↑bq\u0011\n+Jsp−d\nVX\nkqq′\u0010\nc∗\nk−q+q′↑ck↑−c∗\nk−q+q′↓ck↓\u0011\nb∗\nqbq′.(19)\nFor multiple itinerant bands and in second quantization we obtain\nˆHem=−∆X\nkν(ˆc†\nkν↑ˆckν↑−ˆc†\nkν↓ˆckν↓)−∆r\n2\nSNX\nkνν′,q\u0010\nˆc†\nk+qν↑ˆckν′↓ˆb†\n−q+ ˆc†\nk+qν↓ˆckν′↑ˆbq\u0011\n+∆\nSNX\nkνν′,qq′\u0010\nˆc†\nk−q+q′ν↑ˆckν′↑−ˆc†\nk−q+q′ν↓ˆckν′↓\u0011\nˆb†\nqˆbq′.(20)\nwhere we have introduced ∆ =Jsp−dSN\nV. Note that due to the plane wave ansatz we have implicitly assumed that\nthe itinerant electrons are completely delocalized and interband scattering (from νtoν′̸=ν) fully contributes to the\nelectron-magnon scattering.\nNext, we use Fermi’s golden rule to get the change of the magnon occupation number nq=⟨ˆb†\nqˆbq⟩. Fermi’s golden\nrule computes the probability W(i→f) for a small perturbation term in the Hamiltonian, ˆH′(in our specific case,\nˆHem) via\nW(i→f) =2π\nℏ|⟨f|ˆH′|i⟩|2δ(Ef−Ei), (21)\nwhere |i⟩and|f⟩denote the initial and final state, respectively.\nWe start with the term first order in the magnon variables,\n˙n(1)\nq=W(nq→nq+ 1)−W(nq→nq−1)\n=2π\nℏ2∆2\nSNX\nkνν′\b\n(1−fk−qν↑)fkν′↓−(fk−qν↑−fkν′↓)nq\t\nδ(εkν′↓−εk−qν↑−ℏωq),(22)\nwith fkνσ=⟨ˆc†\nkνσˆckνσ⟩andεkνσandℏωqbeing the eigenenergies of electrons and magnons, respectively.\nHereinafter, we make the assumption that due to the fast equilibration processes for electrons, they always follow\nthe Fermi-Dirac distribution, fFD(εkνσ, Te) = [e(εkνσ−εF)/kBTe+ 1]−1,with a single electron temperature Te. Before\nwe continue we need the following relation,\nfFD(εkν′↓, Te)(1−fFD(εk−qν↑, Te))δ(εkν′↓−εk−qν↑−ℏωq) =\n(fFD(εk−qν↑, Te)−fFD(εkν′↓, Te))nBE(ωq, Te)δ(εkν′↓−εk−qν↑−ℏωq)(23)\nwith nBE(ωq, Te) = [eℏωq\nkBTe−1]−1being the Bose-Einstein distribution evaluated at the electron temperature. Now\nwe can simplify Equation (22), yielding\n˙n(1)\nq≈2π\nℏ2∆2\nSNX\nkνν′\u0002\nnBE(ωq, Te)−nq\u0003\n(fFD(εk−qν↑, Te)−fFD(εkν′↓, Te))δ(εkν′↓−εk−qν↑−ℏωq)\n=\u0002\nnBE(ωq, Te)−nq\u0003\nγq. (24)9\nWith γqbeing the linewidth – i.e., the inverse lifetime – of the magnon due to the first order contribution to electron-\nmagnon scattering. Following the ideas laid out by Allen [82] and Maldonado et al. [35], it can be computed as\nγq=2π\nℏ2∆2\nSNX\nkνν′[fFD(εk−qν↑, Te)−fFD(εkν′↓, Te)]δ(εkν′↓−εk−qν↑−ℏωq) (25)\n=2π\nℏ2∆2\nSNX\nkνν′Z\ndε δ(ε−εk−qν↑)Z\ndε′δ(ε′−εkν′↓)[fFD(ε, Te)−fFD(ε′, Te)]δ(ε′−ε−ℏωq) (26)\n≈2π\nℏ2∆2\nSNX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z\ndεZ\ndε′[fFD(ε, Te)−fFD(ε′, Te)]δ(ε′−ε−ℏωq)g↑(ε)g↓(ε′)\ng↑(εF)g↓(εF)\n(27)\n≈2π\nℏ2∆2\nSNℏωqX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z\ndε(−1)∂fFD(ε, Te)\n∂εg↑(ε)g↓(ε+ℏωq)\ng↑(εF)g↓(εF)(28)\n≈2π\nℏ2∆2\nSNℏωqX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z\ndε(−1)∂fFD(ε, Te)\n∂εg↑(ε)g↓(ε)\ng↑(εF)g↓(εF)(29)\n=4π∆2\nNSωqX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)I↑↓(Te) (30)\nwith εFbeing the Fermi energy, the spin-dependent density of states is gσ(ε) =P\nkνδ(ε−εkνσ) and the thermal\ncorrection factor given by\nIσσ′(Te) =Z\ndε(−1)∂fFD(ε, Te)\n∂εgσ(ε)g′\nσ(ε)\ngσ(εF)g′σ(εF). (31)\nIt is obvious that lim Te→0Iσσ′(Te) = 1. Note that we have used that the energy scale of magnons is much smaller\nthan the one of electrons, i.e., that ℏωq≪ε, ε′.\nThe contribution of the term second order in magnon variables to the occupation number can be calculated analogous\nand reads\n˙n(2)\nq=2π\nℏ\u0010∆\nSN\u00112X\nkνν′σ,q′n\n(nq+ 1)nq′\u0010\n(1−fFD(εk−q+q′νσ, Te))fFD(εkν′σ, Te)δ(ℏωq−ℏωq′+εk−q+q′νσ−εkν′σ)\u0011\n−\u0010\nq↔q′\u0011o\n(32)\n=2π\nℏ\u0010∆\nSN\u00112X\nkνν′σ,q′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te)\u0010\nfFD(εk−q+q′νσ, Te)−fFD(εkν′σ, Te)\u0011\n×\nδ(ℏωq−ℏωq′+εk−q+q′νσ−εkν′σ)−\u0010\nq↔q′\u0011o(33)\n≈2π\nℏ\u0010∆\nSN\u00112X\nkνν′σ,q′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te)(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)−\u0010\nq↔q′\u0011o\n(34)\n=2π\nℏ\u0010∆\nSN\u00112X\nq′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te) +\u0000\nq↔q′\u0001oX\nkνν′σ(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)\n(35)\n=2π\nℏ\u0010∆\nSN\u00112X\nq′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te) +\u0000\nq↔q′\u0001oX\nkνν′σ(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)\n(36)\n=X\nq′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te) +\u0000\nq↔q′\u0001o\nΓqq′(Te)(37)\nwith\nΓqq′(Te) =2π\nℏ\u0010∆\nSN\u00112\n(ℏωq−ℏωq′)X\nσIσσ(Te)X\nkνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ). (38)10\nB. Derivation of the three temperature model\nIn what follows, it is demonstrated that the three temperature model (3TM) can be obtained from the (N+2)-\ntemperature model derived in the main text,\n˙nq=h\nnBE(ωq, Te)−nqi\nγq+X\nq′h\n(nq+ 1)nq′nBE(ωq−ωq′, Te) + (q↔q′)i\nΓqq′,(39)\n˙Te=1\nCeh\n−X\nqℏωq˙nq+Gep(Tp−Te) +P(t)i\n, (40)\n˙Tp=−Gep\nCp(Tp−Te), (41)\nby assuming instantaneous relaxation of the magnon occupation numbers to the Bose-Einstein distribution with a\nsingle magnon temperature Tm, i.e., nq=nBE(ωq, Tm). For the sake of readability we rewrite nBE(ωq, Tm) =nq(Tm).\nWe start with the first order scattering term:\n˙n(1)\nq= [nq(Te)−nq(Tm)]γq≈(Te−Tm)∂nq(T)\n∂T\f\f\f\f\nT=Tmγq(Te) = (Te−Tm)Cqγq\nℏωq. (42)\nHere we have introduced the mode-dependent magnon heat capacity Cq=ℏωq∂nq(Tm)\n∂T.\nIn order to calculate the scattering term second order in the magnon variables, we first introduce the following\nrelation\n\u0000\nnq′(Tm) + 1\u0001\nnq(Tm) =\u0002\nnq′(Tm)−nq(Tm)\u0003\nnq−q′(Tm). (43)\nNow we calculate\n˙n(2)\nq=X\nq′\u0010\n(nq(Tm) + 1) nq′(Tm)nq−q′(Te) + (q↔q′)\u0011\nΓqq′ (44)\n=X\nq′\u0010\nnq′−q(Tm)nq−q′(Te)−(q↔q′)\u0011\n×\u0000\nnq(Tm)−nq′(Tm)\u0001\nΓqq′ (45)\n=X\nq′1\n2\u0012\ncoth\u0012ℏ(ωq′−ωq)\n2kBTe\u0013\n−coth\u0012ℏ(ωq′−ωq)\n2kBTm\u0013\u0013\u0000\nnq(Tm)−nq′(Tm)\u0001\nΓqq′ (46)\n≈X\nq′nq(Tm)−nq′(Tm)\nℏ(ωq′−ωq)kB(Te−Tm)Γqq′ (47)\n≈X\nq′∂nq(Tm)\n∂(ℏωq)kB(Tm−Te)Γqq′ (48)\n=X\nq′∂nq(T)\n∂T\f\f\f\f\nT=TmkBTm\nℏωq(Te−Tm)Γqq′ (49)\n=X\nq′CqkBTm\n(ℏωq)2(Te−Tm)Γqq′. (50)\nUsing the expressions for ˙ n(1)\nqand ˙n(2)\nq, the change in total energy of the magnons can then be calculated as\n∂Em\n∂t=∂Em\n∂Tm∂Tm\n∂t=X\nqℏωq∂nq(T)\n∂T|T=Tm\n| {z }\nCm∂Tm\n∂t= (Te−Tm)X\nqCq\u0010\nγq+X\nq′kBTm\nℏωqΓqq′\u0011\n.\n| {z }\nGem(51)\nWith that, the (N+2)TM transforms into the 3TM (in the absence of magnon-phonon coupling), which is given by\nCm˙Tm=Gem(Te−Tm),\nCe˙Te=Gem(Tm−Te) +Gep(Tp−Te) +P(t),\nCp˙Tp=Gep(Te−Tp).(52)11\nC.Ab initio calculations\nTo obtain a full solution of the (N+2)TM, it is necessary to compute the material specific quantities ∆, γq, Γqq′\nandIσσ(Te). For this purpose, we use the full-potential linear augmented plane wave code ELK [85].\nAs a first step, we determine the coupling parameter ∆ of the sp−dmodel, which sets the general scale of the\nelectron-magnon scattering. As shown in the main text, the first term (zeroth order in magnon variables) in the\nelectron-magnon scattering Hamiltonian reads ˆH(0)\nem=−∆P\nkν(ˆc†\nkν↑ˆckν↑−ˆc†\nkν↓ˆckν↓), with ν∈ {s, p}. Based on this,\n∆ can be estimated from the projected density of states (DOS), since it is one half of the spin-dependent energy\nsplitting of the s- and p-bands. In general, this splitting may vary for different electronic states. This is not accounted\nfor in the model used here, where instead a single parameter is used to model the spin splitting. We find, however,\nthat for bcc iron this is justified, since the shift in both s- and p-bands around the Fermi energy – the relevant\nregion for electron-magnon scattering – between spin up and down states is approximately constant with a value of\n∆≈0.75 eV, see left panel of Figure 5.\nNow we calculate the first and second order scattering rates using the formulas derived above,\nγq=4π∆2\nSNωqI↑↓(Te)X\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓), (53)\nΓqq′=2π∆2\nS2N2(ωq−ωq′)X\nσIσσ(Te)X\nkνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ). (54)\nThe calculation of both quantities requires a spin-dependent summation over the Fermi surface, analogous to what\nwas done in Ref. [103] for the evaluation of the spin-dependent Eliashberg function for electron-phonon scattering.\nAs in Ref. [103] we use a Gaussian broadening of the Dirac delta distributions by 0 .03 eV. Also, since we only include\nthe contribution of s- and p-states (indicated by ν, ν′) to the scattering, we have to project the Kohn-Sham states\n(indicated by n, n′) onto the spherical harmonics Ym\nlvia\nδ(εF−εkνσ)δ(εF−εk′ν′σ′) =X\nnn′Pnν\nkσPn′ν′\nk′σ′δ(εF−εknσ)δ(εF−εk′n′σ′), (55)\nwith Pnν\nkσbeing projector functions.\nThe functions Iσσ′(Te) describe corrections to the scattering rate at high electron temperatures and are given by\nIσσ′(Te) =Z\ndε(−1)∂fFD(ε, Te)\n∂εgσ(ε)g′\nσ(ε)\ngσ(εF)g′σ(εF), (56)\nwith gσ(ε) =P\nkνδ(ε−εkνσ) =P\nkνP\nnPnν\nkσδ(ε−εknσ) being the cumulative DOS of both s- and p-states. We\nfind that they increase monotonously with the electron temperature (see right panel of Figure 5). However, even\nfor temperature up to 2000 K, the Iσσ′(Te) functions are below two. Hence, we concluded that the approximation\nIσσ′= 1 is reasonable for the laser fluences – heating the electrons up to around 700 K – considered in the main text.\nFigure 6 depicts the numerically calculated scattering rates using Iσσ′= 1 and ∆ = 0 .75 eV as obtained above. In\nthe left panel, we show the scattering rate γqthat is first order in the magnon variables through color code on the\nmagnon dispersion. It is strictly positive and tends to increase with magnon frequency. The right panel shows the\neffective scattering rate γ(2)\nq=P\nq′Γqq′due to the scattering term second order in magnon variables. Notably, this\nquantity is negative for low frequencies and positive for high frequencies, indicating that it leads to a depopulation\nof magnons at low energies due a scattering from low to high energies (the total magnon number is kept constant).\nIn general, the values of the effective second order scattering rate are comparable to the one first order in magnon\nvariables. They are, however, distributed differently: e.g., for magnons close to the Γ point the second order scattering\nrate is by far the dominating one. This is the reason why, as demonstrated in the main text, a laser pulse can in fact\nlead to a cooling of low energy magnons, i.e., to a decrease of their occupation numbers.\nLastly, we show in Figure 7 the ab initio computed mode-dependent Gilbert damping, αq=ωq/γq. Interestingly,\nthe Gilbert damping αqis large ( ∼0.01) at the BZ center and at the high-symmetry points H, N and P at the\nBZ edge. There is also a noticeable directional anisotropy in the Gilbert damping for modes along Γ −H and Γ −P.\nWe emphasize that the Gilbert damping is here due to the electron-magnon scattering term that is first order in\nthe magnon variables. Other scattering mechanisms as phonon-magnon scattering could contribute further to the\nmode-specific Gilbert damping.12\n−10−5 0 5 10\nenergyε−εF(eV)−0.04−0.020.000.020.040.06projected DOS (eV−1)s\np\n500 1000 1500 2000\nelectron temperature Te(K)1.01.21.41.61.8thermal correction factorI↑↓\nI↑↑\nI↓↓\nFigure 5. Left: Projected spin-polarized DOS for bcc iron. Spin-minority density is shown by positive values, spin-majority\ndensity by negative values. The exchange splitting is 2∆ ≈1.5 eV in a large interval around the Fermi energy and for both s-\nandp-states. Right : Thermal correction factors Iσσ′versus electron temperature Tecalculated from the projected DOS.\nΓ H N Γ P H020406080100120magnon frequency (THz)\n12345\nscattering rate γq(THz)\nΓ H N Γ P H020406080100120magnon frequency (THz)\n−4−3−2−101\nscattering rate γ(2)\nq(THz)\nFigure 6. Magnon dispersion of bcc iron along high-symmetry lines of the Brillouin zone. The color coding describes ( left)\nthe scattering rates γqdue to the electron-magnon scattering term first order in magnon variables γqand ( right) the effective\nscattering rate γ(2)\nq=P\nq′Γqq′due to the term second order in magnon variables, calculated with Iσσ′= 1 and ∆ = 0 .75 eV.\nΓ H N Γ P H020406080100120magnon frequency (THz)\n0.0020.0040.0060.0080.010\nGilbert damping αq\nFigure 7. Calculated mode-specific Gilbert damping αq=ωq/γq, depicted by the color code on the magnon dispersion of bcc\niron. The mode-specific Gilbert damping αqis due to the electron-magnon scattering term first order in magnon variables.13\nACKNOWLEDGMENTS\nThe authors thank K. Carva for valuable discussions.\nThis work has been supported by the Swedish Re-\nsearch Council (VR), the German Research Foundation\n(Deutsche Forschungsgemeinschaft) through CRC/TRR\n227 “Ultrafast Spin Dynamics” (project MF, project-ID:\n328545488), and the K. and A. Wallenberg Foundation\n(Grant No. 2022.0079). 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Awad1,5,6, Nilamani\nBehera1, Roman Khymyn1, Saroj Prasad Dash2,7and Johan\nÅkerman1,5,6*\n1Department of Physics, University of Gothenburg, Göteborg,\n41296, Sweden.\n2Department of Microtechnology and Nanoscience, Chalmers\nUniversity of Technology, Göteborg, 41296, Sweden.\n3Department of Physics, Indian Institute of Technology Ropar,\nRoopnagar, 140001, India.\n4Research Center for Magnetic and Spintronic Materials, National\nInstitute for Materials Science, 1-2-1, Sengen, Tsukuba, 305-0047,\nIbaraki, Japan.\n5Center for Science and Innovation in Spintronics, Tohoku\nUniversity, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan.\n6Research Institute of Electrical Communication, Tohoku\nUniversity, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan.\n7Graphene Center, Chalmers University of Technology, Göteborg,\n41296, Sweden.\n*Corresponding author(s). E-mail(s): lakhan.bainsla@iitrpr.ac.in;\nSAKURABA.Yuya@nims.go.jp; johan.akerman@physics.gu.se;\n†These authors contributed equally to this work.\nAbstract\nTopological quantum materials can exhibit unconventional surface states\nand anomalous transport properties [1, 2], but their applications to\n1arXiv:2311.08145v2  [cond-mat.mes-hall]  22 Nov 20232\nspintronic devices are restricted as they require the growth of high-\nquality thin films with bulk-like properties [3]. Here, we study 10–30\nnm thick epitaxial ferromagnetic Co 2MnGa films with high structural\norder. Very high values of the anomalous Hall conductivity, σxy=\n1.35×105Ω−1m−1, and the anomalous Hall angle, θH= 15.8%,\nboth comparable to bulk values [4, 5]. We observe a dramatic crys-\ntalline orientation dependence of the Gilbert damping constant of a\nfactor of two and a giant intrinsic spin Hall conductivity, σSHC =\n(6.08±0.02)×105(ℏ/2e)Ω−1m−1, which is an order of mag-\nnitude higher than literature values of single-layer Ni 80Fe20[6, 7],\nNi, Co, Fe [6], and multilayer Co 2MnGa stacks [8–10]. Theoretical\ncalculations of the intrinsic spin Hall conductivity, originating from\na strong Berry curvature, corroborate the results and yield values\ncomparableto the experiment. Ourresults open up forthe designof spin-\ntronic devices based on single layers of topological quantum materials.\nKeywords: Heusler alloys, magnetic Weyl semimetal, Berry curvature, spin\nHall conductivity, intrinsic spin-orbit torque.\nThe nontrivial topology in the electronic band structure of quantum mate-\nrials makes them potential candidates for emerging technologies such as\nspintronics [3], topological electronics [2], quantum computing [11], and ther-\nmoelectrics [12]. Weyl semimetals (WSMs) are one such class of materials,\nwhere the nontrivial topological properties arise from band touching points,\nso-called Weyl nodes, in the electronic band structure [1, 4]. WSMs host Weyl\nfermions in the bulk, as well as surface Fermi arcs connecting Weyl nodes of\nopposite chirality. Bulk Weyl nodes exist in WSMs that break time-reversal\nsymmetry (TRS) [1], the inversion symmetry (IS) [1, 13], or both [14]. WSMs\nthat break TRS have two Weyl points, while systems that break IS, or both,\nhave four [1]. WSMs that break TRS are particularly attractive as they allow\ninterplay between magnetism and topology, which can result in intriguing\nquantum states [1]. Due to the presence of only two Weyl nodes in magnetic\nWeyl semimetals, it is easier to observe the chiral anomaly in TRS-breaking\nWSMs [1], and these materials usually show a large anomalous Hall conductiv-\nity(AHC)andanomalousHallangleduetothelargeBerrycurvature [4,5,15],\nand therefore have great potential for spintronic applications.\nRecently, different ferromagnetic WSMs have been discovered using angle-\nresolved photoemission spectroscopy (ARPES), confirming their Weyl nodes\nand Fermi arc/surface states [4, 5, 15, 16]. One such material is the Heusler\nalloy Co 2MnGa (CMG), which has a cubic face-centered Bravais lattice,\nbelongs to the space group Fm3m(#225), has a room temperature satura-\ntion magnetization of 0.96–0.98 T, and stays ferromagnetic up to TC= 690\nK [4, 5, 12]. CMG single crystals show Weyl lines with drumhead surface states\nand a large anomalous Hall conductivity (AHC) due to the Berry curvature\nassociated with the Weyl lines. A large value of AHC ∼1.53×105Ω−1m−13\nwas obtained at 2 K for high-quality bulk samples [4], while a lower value of\n1.14×105Ω−1m−1was obtained for epitaxial 80 nm thick films [17].\nSpin-orbit torque (SOT) [18, 19] has emerged as the most promising and\nenergy-efficient way for switching [19, 20] and excitation of spin waves [21, 22],\nand the SOT efficiencies of several non-magnetic WSMs have been mea-\nsured[23–25].Veryrecently,theSOTefficiencyofCMGwasalsostudiedinthe\nCMG/Ti/CoFeBandCMG/Cu/CoTbsystemsandlargenegativevaluesofthe\neffective antidamping-like torque, ξeff\nAD= –0.078 and –0.118, respectively, were\nobtained [8, 9]. To use SOT from WSMs to manipulate ferromagnets (FM),\ntheyneedtobedepositedinsequence,eitherdirectly,asinWTe 2/NiFe[23,24],\nor with spacers, as in CMG/Ti/CoFeB [8]. Both material systems come with\nthe drawback of complicated interfaces, which can perturb or destroy the\ntopological states of the WSMs and hence remove any advantage of using\ntopological materials. Inspired by recent demonstrations of intrinsic SOT in\nsingle-layer FMs, such as CoPt [26] and NiFe [6, 27], self-induced intrinsic and\nstrongly anisotropic SOT was also very recently reported in B2ordered CMG\nthin films [10]. As the reported values of ξeff\nADare in the range of –0.15 to\n0.20 [8–10], and hence comparable to Pt [28], the reported CMG films does\nnot yet offer any particular advantage for applications. It is hence important\nto obtain high-quality CMG thin films to realize the full potential of its Berry\ncurvature-induced giant SOT for practical applications.\nHere, we study 10–30 nm thick epitaxial CMG films with a much improved\nstructural order that increases with film thickness. From transport measure-\nments on the blanket 30 nm film, we find very high values of the anomalous\nHall conductivity, σxy= 1.35×105Ω−1m−1, and the anomalous Hall angle,\nθH= 15 .8%, both comparable to bulk values [4, 5]. Ferromagnetic reso-\nnance measurements on the blanket films reveal a high effective magnetization\nbetween 0.75 T (10 nm) and 0.9 T (30 nm) and a dramatic (factor of two)\ncrystalline orientation dependence of the Gilbert damping constant in the 30\nnm film. By patterning the films into Hall bars and microstrips, we extract\na giant intrinsic spin Hall conductivity, σSHC= (6.08±0.02)×105(ℏ/2e)\nΩ−1m−1in the 30 nm film, which is an order of magnitude higher than litera-\nture values of single-layer Ni 80Fe20[6, 7], Ni, Co, Fe [6], and multilayer CMG\nstacks [8–10]. Theoretical calculations of the intrinsic spin Hall conductivity,\noriginating from a strong Berry curvature, corroborate the results and yield\nvalues comparable to the experiment.\nResults\nThe out-of-plane 2θ-θx-ray diffraction (XRD) measurements were performed\non CMG films with different thicknesses, as shown in Fig. 1a. The (002) super-\nlattice peaks and (004) fundamental peaks are clearly visible in the XRD\npatterns for all the samples, indicating that the films were grown with (001)\ncrystalline orientation and high B2structural ordering ( B2refers to Mn-Si\norder). Further, the epitaxial growth of the films was confirmed by performing4\nFig. 1 Growth of high-quality Co 2MnGa (CMG) thin films with strong\nanisotropic Gilbert damping and giant anomalous Hall conductivity. a , The crys-\ntal structure of L21-ordered Co 2MnGa Heusler alloy. b, Out-of-plane XRD patterns for\nCMG films with three different thicknesses. c, Structural ordering parameters SB2andSL21\nvs.film thickness; lines are guides to the eye. d, Schematic of the band structure showing\nthe presence of nodal line and drumhead surface states. e, Hall resistivity, ρxy,vs.applied\nmagnetic field, µ0Ha, at 50 and 300 K for the 20 and 30 nm films. f, Temperature-dependent\nanomalous Hall conductivity, σxy, and anomalous Hall angle, θH, for the 20 and 30 nm films;\nlines are guides to the eye. g, Ferromagnetic resonance spectra for 30 nm film. h, Ferromag-\nnetic resonance linewidth, ∆H,vs. ffor CMG 30 nm film. i, Gilbert damping constant as\na function of CMG thickness. g,h,i, Solid and open symbols represent the experimental\ndata points when applied magnetic field, µ0Ha, is parallel to CMG (110) and (001) planes,\nrespectively. g,h, solid lines are fit to the experimental data [29], while solid lines in i, are\njust guide to eye.\nϕscans for the CMG (220) fundamental peak. A clear 4-fold rotation sym-\nmetry confirms the epitaxial growth for all films (see supplementary Fig. 1a).\n2θ-θscans were also obtained for the CMG (111) superlattice peak, confirming\naL21structural order ( L21refers to Co-Mn order). The (111) peak is present\nonly for the 20 and 30-nm films, indicating the absence of L21ordering in the\n10-nm film (see supplementary Fig. 1b). The B2andL21structural ordering\nparameters SB2andSL21wereestimatedusingtheexperimentalandcalculated\nXRD intensity ratios [30], and the estimated values are plotted in Fig. 1b. The\n30-nm film shows a slightly higher value of SL21compared to the 20-nm film.5\nThe obtained values of SB2andSL21confirm full B2and partial L21ordering,\nsubstantially better than earlier thin film reports on this material [8–10].\nVibrating sample magnetometer measurements showed that the films have\nan in-plane easy axis and very low coercivity, as shown in the supplemen-\ntary Fig. 2. The highest saturation magnetization of 4πMS= 0.84 ±0.02T\nwas obtained for the 30 nm film, which is comparable to earlier reports on\nthin films [8, 9], but less than the bulk values, 4πMS= 0.96–0.98 T[4, 31].\nBroadband ferromagnetic resonance (FMR) measurements were performed in\nthe frequency range f= 3−38GHz with the magnetic field, Ha, applied\nparallel to the CMG (110) and (001) planes, as given in Fig. 1g for 30 nm\nfilm. The resonance field ( HR) and the linewidth ( ∆H) were extracted by\nfitting the experimental data to a sum of symmetric and anti-symmetric\nLorentzian derivatives [29]. The effective magnetization, µ0Meff, was obtained\nfrom fits of f vs. HRto Kittel’s equation [29, 32]; µ0Meffranges from 0.75\nto 0.9 T for the three films (see the supplementary Figure 3). The effective\nGilbert damping constant, α, was obtained from linear fits of ∆Hvs. fto\n∆H= ∆H0+(4παf)/γµ 0(Fig. 1h). The obtained αvalues are slightly higher\nthanpreviouslyreportedvalues[33],butshowaveryinterestinghithertounex-\nplored crystalline orientation dependence, as α001=0.025±0.001 ( Haapplied\nalong the (001) plane) is almost twice as large as α110=0.014±0.001 ( Ha\napplied along the (110) plane) for the 30 nm film; see Fig. 1i.\nAHC values of about 0.45×105and1.35×105Ω−1m−1were obtained at\na temperature of 50 K for the 20 and 30-nm CMG films, as shown Fig. 1e and\n1f, respectively. These values are higher than those reported for thick ( ∼80\nnm; 1138 S/cm at 2 K) CMG films [17] and comparable to the bulk case [4].\nThe origin of the large AHC was discussed in an earlier work by comparing\nmeasured angle-resolved photoemission spectroscopy (ARPES) data to the\nBerry curvature obtained from density functional theory (DFT) calculations\nand was attributed to the presence of a large berry curvature associated with\nWeyl lines in the CMG electronic band structure [4, 16]. Our high value of\nAHC for the 30 nm film suggests the presence of Weyl lines and corresponding\nBerry curvature. The anomalous Hall angle, θH=σAHE\nxy/σxx, which reflects\nthe ability of a material to deviate the electron flow from the direction of the\nlongitudinal electric field, was estimated using transport measurements (see\nthe supplementary Fig. 4 for electrical conductivity, σxx, measurements), and\na record high-value of θH= 15.8%was obtained for the 30 nm film.\nHarmonic Hall measurements were performed on the 20 and 30 nm CMG\nfilms, as shown in the schematic in Fig.2a. The SOT efficiency was estimated\nusing fits of the second harmonic ( R2ω) Hall resistance vs.ϕH, to [34, 35]\nR2ω=−(RAHEHAD\nHa−Heff\nk+RT) cosϕH\n+ 2RPHEHFL+HOe\nHa(2 cos3ϕH−cosϕH)(1)6\nFig. 2 Second harmonic Hall measurements of the giant intrinsic spin-orbit\ntorque. a , Schematic of the measurement set up showing the directions of alternating\ncurrent, IAC, and applied magnetic field, Hawith coordinates. b, Second harmonic Hall\nresistance, R2ω, of the 30 nm CMG film vs.in-plane magnetic field angle, ϕH, in an applied\nmagnetic field µ0Ha=0.2 T, for seven different alternating current values IAC=0.3–1.5\nmA, in steps of 0.2 mA. Open symbols are the experimental data points; solid lines are fits\nto Eq. 1.c, The cosϕcontribution to R2ω(Rcosϕ\n2ω) as a function of 1/(µ0Ha−µ0Heff\nk)at the\nseven different current values. Filled symbols are the experimental data points; solid lines\nare linear fits. d,e, The anti-damping-like field µ0HADvs.current density Jfor 20, and 30\nnm films, respectively. Solid symbols are the experimental data points; solid lines are linear\nfits. The extracted effective anti-damping-like torque value, ξeff\nAD, is shown in the figures.\nwhere RAHEandRPHEare the anomalous and planar Hall resistances, HFL,\nHAD,HOe, and Haare the effective field of the field-like torques, the effective\nfield of the antidamping-like torques, the current induced Oersted field, and\nthe applied magnetic field, respectively. RTis the second harmonic Hall resis-\ntance signal due to the thermo-electric effects including the anomalous Nerst\neffect and spin Seebeck effect. The measured R2ωvs.ϕHwas fitted with Eq. 1\nand separated into cosϕ Hand2cos3ϕH−cosϕ Hcontributions. The cosϕ Hcon-\ntribution of R2ω(Rcosϕ\n2ω)vs.1/(µ0Ha−µ0Heff\nk)for the 30 nm CMG film is\nshown in Fig. 2c; HADandRTvalues were obtained from the slope and y-\nintercept of the linear fit of the data, respectively. We assume no net Oersted\nfield contribution in our case, as we are working with a single CMG layer.\nFig. 2d and 2e show the obtained HADvs.J(current density) for the 20 and\n30 nm films, respectively. The slope of HADvs.Jwas obtained from the linear\nfit and used to evaluate the effective antidamping-like torque efficiency, ξeff\nAD,\nwith the relation [35],\nξeff\nAD=2emS\nℏδµ0HAD\nδJ(2)7\nwhere, eis the elementary charge, ℏis the reduced Plack’s constant, and\nmSis the saturation magnetization per unit area. ξeff\nADvalues of 0.24 ±0.01\nand 0.82 ±0.02 are obtained for the 20 and 30 nm films, respectively (see the\nsupplementary Fig. 5 for CMG 20 nm data). The obtained value of ξeff\nAD=\n0.82±0.02is an order of magnitude higher than the previously reported values\ninsystems suchasCMG/Ti/CoFeBasmentionedintheintroduction [8,9]and\nabout 5 times the value reported for single layer (110) oriented CMG films [10].\nThe effective spin Hall conductivity, σSHC=σxxξeff\nAD= (1.24±0.01)×105and\n(6.08±0.02)×105(ℏ/2e)Ω−1m−1were estimated for the 20 and 30 nm CMG\nfilms, respectively. The σSHCvalue for the 30 nm CMG film is at least an order\nof magnitude higher than reported values for other single layer magnets such\nas Ni 80Fe20[6, 7, 27], Ni, Fe, and Co [6].\nFig. 3 ST-FMR measurements of the giant intrinsic spin-orbit torque. a ,\nSchematic of the ST-FMR measurement setup. b, Five representative STFMR curves in the\nfrequency range f=7–15 GHz. Solid symbols are the experimental data; solid lines are fit\nto,Vmix=SFS(Ha) +AFA(Ha)[28, 36].c,d, Frequency f vs.resonance field HRand\nresonance linewidth ∆Hvs.frequency f, respectively. Here, the solid symbols show the val-\nues obtained by fitting the V mixsignal, and the solid lines show the fit to the data. The\nobtained values of effective magnetization, µ0Meffand the Gilbert damping constant αare\nshown in Fig. 3c and 3d, respectively. e, Resonance linewidth ∆Hvs.dc bias Idcmeasured\nat a frequency of 8 GHz, here solid symbols are the data points obtained by fitting the V mix\nsignal and the solid line is a linear fit to the obtained data. b-e, All the measurements are\ndone with ϕH= 60◦.\nTo further estimate the SOT efficiency, spin-torque ferromagnetic reso-\nnance (ST-FMR) measurements were performed on rectangular 4 ×14µm2\nmicrostrips fabricated with the longer axis along the CMG(110) plane (mea-\nsurement schematic is given in Fig. 3a), as a recent study found the largest8\nSOT efficiency when the current flows along the CMG (110) plane [10], and\nusing FMR measurements on blanket films we found that αis minimum\nwhen we measure the films with Haalong the CMG (110) plane. Anisotropic\nmagnetoreristance (AMR) measurements were performed on the ST-FMR\nmicrostrips and negative AMR values in the range from –0.37 to –0.41 were\nobtained, as shown in supplementary Fig. 6. A representative ST-FMR sig-\nnal (Vmix) for the 30 nm film is shown in Fig. 3b; we did not observe a clear\nsignal for the 20 and 10 nm films. The obtained Vmixwas fitted to a single\nLorentzian function, which is the sum of symmetric and antisymmetric com-\nponents [28, 36, 37]. µ0Meff, and αvalues are obtained in the same way as\nmentioned in the FMR analysis and shown in Fig. 3c and Fig. 3d, respectively.\nThe obtained values µ0Meff= 0.79 ±0.04, and α= 0.018±0.001are compara-\nble to the values obtained with FMR on blanket films when Haapplied along\nthe (110) plane.\nThe current dependent ST-FMR measurements were carried out at a fixed\nfrequency to estimate the effective antidamping-like torque efficiency ( θeff\nAD).\nThe change in ∆Hvs.applied dc current, Idc, at a frequency of 8 GHz is\nshown in Fig. 3e. The slope ( δ∆H/δI dc) of linearly fit ∆Hvs.Idcfrom Fig.\n3e indicates the SOT efficiency, which is extracted using [28, 36, 37],\nθeff\nAD=2e\nℏ(Ha+ 0.5Meff)µ0MstCMG\nsinϕHγ\n2πfδ∆H\nδIdcAc (3)\nwhere, tCMGisthethicknessoftheCMGlayer, γ/2πtheeffectivegyromagnetic\nratio, Acthe cross-sectional area of the ST-FMR microbars, and ϕHthe angle\nbetween the applied magnetic field and the rf/dc current. A θeff\nADvalue of\n1.58±0.40, and an effective spin Hall conductivity, σSHC=σxxθeff\nAD, value of\n(1.12±0.30)×106(ℏ/2e)(Ωm)−1were estimated for the 30 nm CMG film,\ncorroborating the very high value achieved above using the harmonic Hall\nmeasurements.\nTo gain further theoretical insight into the giant spin Hall effect, we cal-\nculated the spin Hall conductivity σs,xyinL21-ordered CMG by combining\nfirst-principles calculation with the Kubo formula. The spin Hall conductivity\nσs,xycan be calculated in the same way as the anomalous Hall conductivity\nσxyexcept that the usual momentum operator pyis replaced by ps\ny={py, sz}.\nFig. 4a shows the energy dependence of the anomalous Hall conductivity σxy,\nwhere ϵ= 0corresponds to the Fermi level in the present system. We obtained\na large value of σxy(∼1.2×105Ω−1m−1) atϵ= 0, consistent with previ-\nous studies [5, 16, 38]. It is known that L21-ordered CMG has several mirror\nsymmetries in its crystal structure, and these provide Weyl nodal loops in the\nki= 0plane ( i=x, y, z) in the Brillouin zone, as shown in the schematic\nof Fig. 1d [4, 16, 31, 38, 39]. When the spin-orbit interaction is taken into\naccount, some of these nodal loops are gapped and yield a large value of the\nBerry curvature, which is the reason for the very large σxy[16, 31, 38]. In\nFig. 4b, we show the energy dependence of the spin Hall conductivity σs,xy.9\nFig. 4 The physical origin of the giant intrinsic spin Hall conductivity in\nCo2MnGa. a ,b, The calculated anomalous Hall and spin Hall conductivities as a function\nofϵbeing the energy relative to the Fermi energy.\nWe obtained a large σs,xyof∼1.3×105(ℏ/e) Ω−1m−1atϵ= 0, which is com-\nparable to the experimentally obtained large values. Since σs,xyis given as the\nintegral of the spin Berry curvature in the Brillouin zone, the gapped nodal\nloops are also considered to be the origin of the large spin Hall conductiv-\nity. These results clearly indicate that the experimentally obtained large spin\nHall effect can be understood by the intrinsic mechanism originating from the\nelectronic structure of L21-ordered CMG.\nConclusion\nHigh-quality epitaxial thin films of the ferromagnetic Heusler alloy Co 2MnGa\nwere grown with high structural ordering and their intrinsic spin-orbit torque\nwas studied both experimentally and theoretically. High, bulk-like, values of\ntheanomalousHallconductivity, σxy= 1.35×105Ω−1m−1,andtheanomalous\nHall angle, θH= 15.8%were obtained, confirming the high film quality. The\nSOT efficiency was measured using both second harmonic Hall resistance and\nST-FMRmeasurements,anda σSHCvalueof (6.08±0.02)×105(ℏ/2e)Ω−1m−1\nwas obtained for the 30 nm CMG film, which is an order of magnitude higher10\nthan values reported in the literature for any single layer magnets and for\nmultilayer Co 2MnGa stacks. Theoretical calculations explain the origin of the\ngiant intrinsic SOT as due to the large Berry curvature. Our study opens\nup a new research direction to design spintronic devices based on single-layer\nmagnetic Weyl semimetals.\nMethods\nSample fabrication\nEpitaxial thin films of CMG ( t= 10, 20, and 30 nm) are grown on 0.5 mm\nthick single crystalline MgO(001) substrates using ultra-high vacuum mag-\nnetron sputtering with a base pressure of less than 2×10−10torr. All films\nare deposited at room temperature and followed by post-annealing at 550◦C\nfor 30 minutes. After cooling down the samples to room temperature, a 2 nm\nthick Al capping layer is deposited to protect the films from oxidation. Thin\nfilms are then used to fabricate cross Hall bars of width 500 nm and 4 ×14µm2\nST-FMR microstrips by e-beam lithography and argon ion milling using a neg-\native e-beam resist as the etching mask. Optical lithography is used to define\nthe top coplaner waveguide contacts for electrical measurements, followed by\na lift-off process of 780 nm of copper and 20 nm of platinum.\nCharacterization of Co 2MnGa (CMG) films\nThe composition of the CMG films is determined using X-ray fluorescence\nspectroscopy. The structural analysis is done using X-ray diffraction (XRD)\nmeasurements for different CMG atomic planes using different tilt angles χ.\nThe longitudinal and anomalous Hall resistivities are measured using a physi-\ncal property measurement system (PPMS; Quantum Design) at temperatures\n50-300K.Magnetizationmeasurementsaredoneusingavibratingsamplemag-\nnetometer at room temperature. Broadband ferromagnetic resonance (FMR)\nmeasurementsaredoneusingaNanOscPhaseFMR-40systemwithaco-planar\nwaveguide for broadband microwave field excitation at room temperature.\nMicrowave excitation fields hrfwith frequencies up to 38 GHz are applied in\nthe film plane and perpendicular to the applied in-plane dc magnetic field H.\nHarmonic Hall measurements\nThe effective fields of field-like ( HFL) and antidamping-like ( HAD) SOTs are\nevaluated using extended harmonic Hall measurements, excluding the ther-\nmoelectric effects originating due to the anomalous Nernst and spin Seebeck\neffects [34, 35]. The schematic of the harmonic Hall measurement setup is\nshown in Fig. 2a, where a 213 Hz alternating current ( IAC) is applied to the\nchannel in the presence of a fixed magnetic field, µ0Ha. The first and second\nharmonic Hall voltages ( VωandV2ω) are measured at room temperature using\na lock-in-amplifier while sweeping the in-plane angle, ϕH, between the IACand\nµ0Ha, as shown in Fig. 2a.11\nMagnetoresistance and spin torque ferromagnetic\nresonance (ST-FMR) measurements\nIn-plane angular dependent anisotropic magnetoresistance measurements are\nperformed on 4 ×14µm2ST-FMR microstrips at room temperature using\na rotatable projected vector field magnet with a magnetic field magnitude\nof 0.1 T and an applied dc current of 0.5 mA. Room-temperature ST-FMR\nmeasurements are performed by injecting a radio-frequency (rf) current to the\nmicrostrip through a high-frequency bias-T at a fixed frequency (ranging from\n7 to 17 GHz) with an input power of P=4 dBm. The rf current generates\nantidamping-like and field-like torques in the presence of an applied magnetic\nfield µ0Ha, and the resultant torques excite the magnetization procession of\nthe CMG film, which leads to a time-dependent change in the device resistance\ndue to the magnetoresistance of the CMG [28, 36]. The oscillating resistance of\nthe device mixes with the rf current and results in a dc mixing voltage, V mix,\nwhich is then measured using a lock-in-amplifier. ST-FMR measurements are\nperformed witha fixed in-plane angle, ϕH= 60◦, between the applied magnetic\nfield and input rf/dc current.\nTheoretical calculations of anomalous Hall and spin Hall\nconductivities\nThe anomalous Hall and spin Hall conductivities of CMG are calculated\nby combining first-principles calculations and the Kubo formula. First, we\ncalculated the electronic structure of L21-ordered CMG (Fig. 1a) based on\ndensity-functional theory, including the spin-orbit interaction, which is imple-\nmented in the Vienna ab initio simulation program (VASP) [40]. The lattice\nconstant is set to a typical experimental value of 5.755Å [16]. We adopted the\ngeneralized gradient approximation for the exchange-correlation energy and\nused the projected augmented wave pseudopotential to treat the effect of core\nelectrons properly. A cut-off energy of 337eV is employed, and the Brillouin-\nzone integration is performed with 91 ×91×91 k points. The convergence\ncriteria for energy and force are set to 10−5eV and 10−4eV/Å, respectively.\nUsing the obtained electronic structure, we calculated the anomalous Hall and\nspin Hall conductivities using the following expressions derived from the Kubo\nformula:\nσxy(ϵ) =−e2\nℏZd3k\n(2π)3Ωc\nxy(k), (4)\nσs,xy(ϵ) =−e\nℏZd3k\n(2π)3Ωs\nxy(k), (5)\nΩα\nxy(k) =−ℏ2\nm2X\nnf(En,k, ϵ)X\nn′̸=n2 Im⟨ψn,k|px|ψn′,k⟩⟨ψn′,k|pα\ny|ψn,k⟩\n(En′,k−En,k)2,(6)12\nwhere σxy(ϵ)andσs,xy(ϵ)are the anomalous Hall and spin Hall conductivities,\nrespectively, as a function of ϵbeing the energy relative to the Fermi energy.\nThese conductivities are given by integrating the charge Berry curvature Ωc\nxy\nand the spin Berry curvature Ωs\nxy, where the generalized momentum operator\npα\nyis defined as pc\ny=pyandps\ny={py, sz}with the spin operator sz=\nσz/2[41]. In Eq. (6), |ψn,k⟩is the eigenstate with the eigenenergy En,kfor\nthe band nand the wave vector k, and f(En,k, ϵ)is the Fermi distribution\nfunction. In our calculations, the direction of the magnetization is fixed to the\n[001] direction, consistent with the experimental setup.\nDeclarations\nFunding. Lakhan Bainsla thanks MSCA - European Commission for the\nMarie Curie Individual Fellowship (MSCA-IF Grant No. 896307) and the\nScience and Engineering Research Board (SERB) India for the Ramanujan fel-\nlowship. This work was partially supported by the Swedish Research Council\n(VR Grant No. 2016-05980), the Horizon 2020 research and innovation pro-\ngramme(ERCAdvancedGrantNo.835068\"TOPSPIN\"),FLAG-ERAproject\n2DSOTECH (VR No. 2021-05925), and EU project 2DSSPIN-TECH.\nConflict of interest. The authors declare no competing interests.\nAvailability of data and materials. The data is available upon reasonable\nrequest from corresponding authors.\nAuthors’ contributions. L.B., Y.S., and J.Å. conceived the idea and\nplanned the study. 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Titov1\n1Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, the Netherlands\n2Qblox, Delftechpark 22, 2628 XH Delft, the Netherlands\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden\n(Dated: March 29, 2024)\nA finite spin life-time of conduction electrons may dominate Gilbert damping of two-dimensional\nmetallic anti-ferromagnets or anti-ferromagnet/metal heterostructures. We investigate the Gilbert\ndamping tensor for a typical low-energy model of a metallic anti-ferromagnet system with honeycomb\nmagnetic lattice and Rashba spin-orbit coupling for conduction electrons. We distinguish three\nregimes of spin relaxation: exchange-dominated relaxation for weak spin-orbit coupling strength,\nElliot-Yafet relaxation for moderate spin-orbit coupling, and Dyakonov-Perel relaxation for strong\nspin-orbit coupling. We show, however, that the latter regime takes place only for the in-plane\nGilbert damping component. We also show that anisotropy of Gilbert damping persists for any\nfinite spin-orbit interaction strength provided we consider no spatial variation of the N´ eel vector.\nIsotropic Gilbert damping is restored only if the electron spin-orbit length is larger than the magnon\nwavelength. Our theory applies to MnPS 3monolayer on Pt or to similar systems.\nI. INTRODUCTION\nMagnetization dynamics in anti-ferromagnets con-\ntinue to attract a lot of attention in the context\nof possible applications1–4. Various proposals utilize\nthe possibility of THz frequency switching of anti-\nferromagnetic domains for ultrafast information storage\nand computation5,6. The rise of van der Waals magnets\nhas had a further impact on the field due to the pos-\nsibility of creating tunable heterostructures that involve\nanti-ferromagnet and semiconducting layers7.\nUnderstanding relaxation of both the N´ eel vector and\nnon-equilibrium magnetization in anti-ferromagnets is\nrecognized to be of great importance for the function-\nality of spintronic devices8–13. On one hand, low Gilbert\ndamping must generally lead to better electric control of\nmagnetic order via domain wall motion or ultrafast do-\nmain switching14–16. On the other hand, an efficient con-\ntrol of magnetic domains must generally require a strong\ncoupling between charge and spin degrees of freedom due\nto a strong spin-orbit interaction, that is widely thought\nto be equivalent to strong Gilbert damping.\nIn this paper, we focus on a microscopic analysis of\nGilbert damping due to Dyakonov-Perel and Elliot-Yafet\nmechanisms. We apply the theory to a model of a two-\ndimensional N´ eel anti-ferromagnet with a honeycomb\nmagnetic lattice.\nTwo-dimensional magnets typically exhibit either\neasy-plane or easy-axis anisotropy, and play crucial\nroles in stabilizing magnetism at finite temperatures17,18.\nEasy-axis anisotropy selects a specific direction for mag-\nnetization, thereby defining an axis for the magnetic or-\nder. In contrast, easy-plane anisotropy does not select a\nparticular in-plane direction for the N´ eel vector, allowing\nit to freely rotate within the plane. This situation is anal-\nogous to the XY model, where the system’s continuous\nsymmetry leads to the suppression of out-of-plane fluc-\ntuations rather than fixing the magnetization in a spe-\ncific in-plane direction19,20. Without this anisotropy, themagnonic fluctuations in a two-dimensional crystal can\ngrow uncontrollably large to destroy any long-range mag-\nnetic order, according to the Mermin-Wagner theorem21.\nRecent density-functional-theory calculations for\nsingle-layer transition metal trichalgenides22, predict the\nexistence of a large number of metallic anti-ferromagnets\nwith honeycomb lattice and different types of magnetic\norder as shown in Fig. 1. Many of these crystals may\nhave the N´ eel magnetic order as shown in Fig. 1a and are\nmetallic: FeSiSe 3, FeSiTe 3, VGeTe 3, MnGeS 3, FeGeSe 3,\nFeGeTe 3, NiGeSe 3, MnSnS 3, MnSnS 3, MnSnSe 3,\nFeSnSe 3, NiSnS 3. Apart from that it has been predicted\nthat anti-ferromagnetism can be induced in graphene by\nbringing it in proximity to MnPSe 323or by bringing it\nin double proximity between a layer of Cr 2Ge2Te6and\nWS224.\nPartly inspired by these predictions and recent\ntechnological advances in producing single-layer anti-\nferromagnet crystals, we propose an effective model to\nstudy spin relaxation in 2D honeycomb anti-ferromagnet\nwith N´ eel magnetic order. The same system was studied\nby us in Ref. 25, where we found that spin-orbit cou-\npling introduces a weak anisotropy in spin-orbit torque\nand electric conductivity. Strong spin-orbit coupling was\nshown to lead to a giant anisotropy of Gilbert damping.\nOur analysis below is built upon the results of Ref. 25,\nand we investigate and identify three separate regimes\nof spin-orbit strength. Each regime is characterized by\nqualitatively different dependence of Gilbert damping on\nspin-orbit interaction and conduction electron transport\ntime. The regime of weak spin-orbit interaction is dom-\ninated by exchange field relaxation of electron spin, and\nthe regime of moderate spin-orbit strength is dominated\nby Elliot-Yafet spin relaxation. These two regimes are\ncharacterized also by a universal factor of 2 anisotropy\nof Gilbert damping. The regime of strong spin-orbit\nstrength, which leads to substantial splitting of electron\nFermi surfaces, is characterized by Dyakonov-Perel relax-\nation of the in-plane spin component and Elliot-Yafet re-arXiv:2311.16268v2  [cond-mat.dis-nn]  28 Mar 20242\nFIG. 1. Three anti-ferromagnetic phases commonly found\namong van-der-Waals magnets. Left-to-right: N´ eel, zig-zag,\nand stripy.\nlaxation of the perpendicular-to-the-plane Gilbert damp-\ning which leads to a giant damping anisotropy. Isotropic\nGilbert damping is restored only for finite magnon wave\nvectors such that the magnon wavelength is smaller than\nthe spin-orbit length.\nGilbert damping in a metallic anti-ferromagnet can be\nqualitatively understood in terms of the Fermi surface\nbreathing26. A change in the magnetization direction\ngives rise to a change in the Fermi surface to which the\nconduction electrons have to adjust. This electronic re-\nconfiguration is achieved through the scattering of elec-\ntrons off impurities, during which angular momentum is\ntransferred to the lattice. Gilbert damping, then, should\nbe proportional to both (i) the ratio of the spin life-time\nand momentum life-time of conduction electrons, and (ii)\nthe electric conductivity. Keeping in mind that the con-\nductivity itself is proportional to momentum life-time,\none may conclude that the Gilbert damping is linearly\nproportional to the spin life-time of conduction electrons.\nAt the same time, the spin life-time of localized spins is\ninversely proportional to the spin life-time of conduc-\ntion electrons. A similar relation between the spin life-\ntimes of conduction and localized electrons also holds\nfor relaxation mechanisms that involve electron-magnon\nscattering27.\nOur approach formally decomposes the magnetic sys-\ntem into a classical sub-system of localized magnetic mo-\nments and a quasi-classical subsystem of conduction elec-\ntrons. A local magnetic exchange couples these sub-\nsystems. Localized magnetic moments in transition-\nmetal chalcogenides and halides form a hexagonal lat-\ntice. Here we focus on the N´ eel type anti-ferromagnet\nthat is illustrated in Fig. 1a. In this case, one can de-\nfine two sub-lattices A and B that host local magnetic\nmoments SAandSB, respectively. For the discussion of\nGilbert damping, we ignore the weak dependence of both\nfields on atomic positions and assume that the modulus\nS=|SA(B)|is time-independent.\nUnder these assumptions, the magnetization dynamics\nof localized moments may be described in terms of two\nfields\nm=1\n2S\u0000\nSA+SB\u0001\n,n=1\n2S\u0000\nSA−SB\u0001\n, (1)\nwhich are referred to as the magnetization and staggeredmagnetization (or N´ eel vector), respectively. Within the\nmean-field approach, the vector fields yield the equations\nof motion\n˙n=−Jn×m+n×δs++m×δs−, (2a)\n˙m=m×δs++n×δs−, (2b)\nwhere dot stands for the time derivative, while δs+and\nδs−stand for the mean staggered and non-staggered non-\nequilibrium fields that are proportional to the variation of\nthe corresponding spin-densities of conduction electrons\ncaused by the time dynamics of nandmfields. The en-\nergy Jis proportional to the anti-ferromagnet exchange\nenergy for localized momenta.\nIn Eqs. (2) we have omitted terms that are propor-\ntional to easy axis anisotropy for the sake of compact-\nness. These terms are, however, important and will be\nintroduced later in the text.\nIn the framework of Eqs. (2) the Gilbert damping can\nbe computed as the linear response of the electron spin-\ndensity variation to a time change in both the magneti-\nzation and the N´ eel vector (see e. g. Refs.25,28,29).\nIn this definition, Gilbert damping describes the re-\nlaxation of localized spins by transferring both total and\nstaggered angular momenta to the lattice by means of\nconduction electron scattering off impurities. Such a\ntransfer is facilitated by spin-orbit interaction.\nThe structure of the full Gilbert damping tensor can be\nrather complicated as discussed in Ref. 25. However, by\ntaking into account easy axis or easy plane anisotropy we\nmay reduce the complexity of relevant spin configurations\nto parameterize\nδs+=α∥\nm˙m∥+α⊥\nm˙m⊥+αmn∥×(n∥×˙m∥),(3a)\nδs−=α∥\nn˙n∥+α⊥\nn˙n⊥+αnn∥×(n∥×˙n∥), (3b)\nwhere the superscripts ∥and⊥refer to the in-plane\nand perpendicular-to-the-plane projections of the corre-\nsponding vectors, respectively. The six coefficients α∥\nm,\nα⊥\nm,αm,α∥\nn,α⊥\nn, and αnparameterize the Gilbert damp-\ning.\nInserting Eqs. (3) into the equations of motion of\nEqs. (2) produces familiar Gilbert damping terms. The\ndamping proportional to time-derivatives of the N´ eel vec-\ntornis in general many orders of magnitude smaller than\nthat proportional to the time-derivatives of the magneti-\nzation vector m25,30. Due to the same reason, the higher\nharmonics term αmn∥×(n∥×∂tm∥) can often be ne-\nglected.\nThus, in the discussion below we may focus mostly on\nthe coefficients α∥\nmandα⊥\nmthat play the most important\nrole in the magnetization dynamics of our system. The\nterms proportional to the time-derivative of ncorrespond\nto the transfer of angular momentum between the sub-\nlattices and are usually less relevant. We refer to the\nresults of Ref. 25 when discussing these terms.\nAll Gilbert damping coefficients are intimately related\nto the electron spin relaxation time. The latter is rel-\natively well understood in non-magnetic semiconductors3\nwith spin-orbital coupling. When a conducting electron\nmoves in a steep potential it feels an effective magnetic\nfield caused by relativistic effects. Thus, in a disordered\nsystem, the electron spin is subject to a random magnetic\nfield each time it scatters off an impurity. At the same\ntime, an electron also experiences precession around an\neffective spin-orbit field when it moves in between the\ncollisions. Changes in spin direction between collisions\nare referred to as Dyakonov-Perel relaxation31,32, while\nchanges in spin-direction during collisions are referred to\nas Elliot-Yafet relaxation33,34.\nThe spin-orbit field in semiconductors induces a char-\nacteristic frequency of spin precession Ω s, while scalar\ndisorder leads to a finite transport time τof the con-\nducting electrons. One may, then, distinguish two limits:\n(i) Ω sτ≪1 in which case the electron does not have\nsufficient time to change its direction between consec-\nutive scattering events (Elliot-Yafet relaxation), and (ii)\nΩsτ≫1 in which case the electron spin has multiple pre-\ncession cycles in between the collisions (Dyakonov-Perel\nrelaxation).\nThe corresponding processes define the so-called spin\nrelaxation time, τs. In a 2D system the spin life-time\nτ∥\ns, for the in-plane spin components, appears to be dou-\nble the size of the life-time of the spin component that\nis perpendicular to the plane, τ⊥\ns32. This geometric ef-\nfect has largely been overlooked. For non-magnetic 2D\nsemiconductor one can estimate35,36\n1\nτ∥\ns∼(\nΩ2\nsτ,Ωsτ≪1\n1/τ, Ωsτ≫1, τ∥\ns= 2τ⊥\ns. (4)\nA pedagogical derivation and discussion of Eq. 4 can\nbe found in Refs. 35 and 36. Because electrons are con-\nfined in two dimensions the random spin-orbit field is\nalways directed in-plane, which leads to a decrease in the\nin-plane spin-relaxation rate by a factor of two compared\nto the out-of-plane spin-relaxation rate as demonstrated\nfirst in Ref. 32 (see Refs. 36–40 as well). The reason is\nthat the perpendicular-to-the-plane component of spin is\ninfluenced by two components of the randomly changing\nmagnetic field, i. e. xandy, whereas the parallel-to-the-\nplane spin components are only influenced by a single\ncomponent of the fluctuating fields, i. e. the xspin pro-\njection is influenced only by the ycomponent of the field\nand vice-versa. The argument has been further general-\nized in Ref. 25 to the case of strongly separated spin-orbit\nsplit Fermi surfaces. In this limit, the perpendicular-to-\nthe-plane spin-flip processes on scalar disorder potential\nbecome fully suppressed. As a result, the perpendicular-\nto-the-plane spin component becomes nearly conserved,\nwhich results in a giant anisotropy of Gilbert damping in\nthis regime.\nIn magnetic systems that are, at the same time, con-\nducting there appears to be at least one additional energy\nscale, ∆ sd, that characterizes exchange coupling of con-\nduction electron spin to the average magnetic moment of\nlocalized electrons. (In the case of s-d model descriptionit is the magnetic exchange between the spin of conduc-\ntionselectron and the localized magnetic moment of d\norfelectron on an atom.) This additional energy scale\ncomplicates the simple picture of Eq. (4) especially in the\ncase of an anti-ferromagnet. The electron spin precession\nis now defined not only by spin-orbit field but also by\n∆sd. As the result the conditions Ω sτ≪1 and ∆ sdτ≫1\nmay easily coexist. This dissolves the distinction between\nElliot-Yafet and Dyakonov-Perel mechanisms of spin re-\nlaxation. One may, therefore, say that both Elliot-Yafet\nand Dyakonov-Perel mechanisms may act simultaneously\nin a typical 2D metallic magnet with spin-orbit coupling.\nThe Gilbert damping computed from the microscopic\nmodel that we formulate below will always contain both\ncontributions to spin-relaxation.\nII. MICROSCOPIC MODEL AND RESULTS\nThe microscopic model that we employ to calculate\nGilbert damping is the so-called s–dmodel that couples\nlocalized magnetic momenta SAandSBand conducting\nelectron spins via the local magnetic exchange ∆ sd. Our\neffective low-energy Hamiltonian for conduction electrons\nreads\nH=vfp·Σ+λ\n2\u0002\nσ×Σ\u0003\nz−∆sdn·σΣzΛz+V(r),(5)\nwhere the vectors Σ,σandΛdenote the vectors of Pauli\nmatrices acting on sub-lattice, spin and valley space,\nrespectively. We also introduce the Fermi velocity vf,\nRashba-type spin-orbit interaction λ, and a random im-\npurity potential V(r).\nThe Hamiltonian of Eq. (5) can be viewed as the\ngraphene electronic model where conduction electrons\nhave 2D Rashba spin-orbit coupling and are also cou-\npled to anti-ferromagnetically ordered classical spins on\nthe honeycomb lattice.\nThe coefficients α∥\nmandα⊥\nmare obtained using linear\nresponse theory for the response of spin-density δs+to\nthe time-derivative of magnetization vector ∂tm. Impu-\nrity potential V(r) is important for describing momen-\ntum relaxation to the lattice. This is related to the an-\ngular momentum relaxation due to spin-orbit coupling.\nThe effect of random impurity potential is treated pertur-\nbatively in the (diffusive) ladder approximation that in-\nvolves a summation over diffusion ladder diagrams. The\ndetails of the microscopic calculation can be found in the\nAppendices.\nBefore presenting the disorder-averaged quantities\nα∥,⊥\nm, it is instructive to consider first the contribution\nto Gilbert damping originating from a small number of\nelectron-impurity collisions. This clarifies how the num-\nber of impurity scattering effects will affect the final re-\nsult.\nLet us annotate the Gilbert damping coefficients with\nan additional superscript ( l) that denotes the number\nof scattering events that are taken into account. This4\n01234\u0016\u000b(i)\n?[\"\u001c]\n\u0016\u000b(0)\n?\u0016\u000b(1)\n?\u0016\u000b(2)\n? \u0016\u000b(1)\n?\n10\u0000210\u00001100101\n\u0015\u001c01234\u0016\u000b(i)\nk[\"\u001c]\n\u0016\u000b(0)\nk\u0016\u000b(1)\nk\u0016\u000b(2)\nk\u0016\u000b(1)\nk\nFIG. 2. Gilbert damping in the limit ∆ sd= 0. Dotted (green)\nlines correspond to the results of the numerical evaluation of\n¯α(l)\nm,⊥,∥forl= 0,1,2 as a function of the parameter λτ. The\ndashed (orange) line corresponds to the diffusive (fully vertex\ncorrected) results for ¯ α⊥,∥.\nm.\nmeans, in the diagrammatic language, that the corre-\nsponding quantity is obtained by summing up the ladder\ndiagrams with ≤ldisorder lines. Each disorder line cor-\nresponds to a quasi-classical scattering event from a sin-\ngle impurity. The corresponding Gilbert damping coeffi-\ncient is, therefore, obtained in the approximation where\nconduction electrons have scattered at most lnumber\nof times before releasing their non-equilibrium magnetic\nmoment into a lattice.\nTo make final expressions compact we define the di-\nmensionless Gilbert damping coefficients ¯ α∥,⊥\nmby extract-\ning the scaling factor\nα∥,⊥\nm=A∆2\nsd\nπℏ2v2\nfS¯α∥,⊥\nm, (6)\nwhere Ais the area of the unit cell, vfis the Fermi ve-\nlocity of the conducting electrons and ℏ=h/2πis the\nPlanck’s constant. We also express the momentum scat-\ntering time τin inverse energy units, τ→ℏτ.\nLet us start by computing the coefficients ¯ α∥,⊥(l)\nm in the\nformal limit ∆ sd→0. We can start with the “bare bub-\nble” contribution which describes spin relaxation without\na single scattering event. The corresponding results read\n¯α(0)\nm,⊥=ετ1−λ2/4ε2\n1 +λ2τ2, (7a)\n¯α(0)\nm,∥=ετ\u00121 +λ2τ2/2\n1 +λ2τ2−λ2\n8ε2\u0013\n, (7b)\nwhere εdenotes the Fermi energy which we consider pos-\nitive (electron-doped system).In all realistic cases, we have to consider λ/ε≪1,\nwhile the parameter λτmay in principle be arbitrary. For\nλτ≪1 the disorder-induced broadening of the electron\nFermi surfaces exceeds the spin-orbit induced splitting.\nIn this case one basically finds no anisotropy of “bare”\ndamping: ¯ α(0)\nm,⊥= ¯α(0)\nm,∥. In the opposite limit of substan-\ntial spin-orbit splitting one gets an ultimately anisotropic\ndamping ¯ α(0)\nm,⊥≪¯α(0)\nm,∥. This asymptotic behavior can be\nsummarized as\n¯α(0)\nm,⊥=ετ(\n1 λτ≪1,\n(λτ)−2λτ≫1,(8a)\n¯α(0)\nm,∥=ετ(\n1 λτ≪1,\n1\n2\u0000\n1 + (λτ)−2\u0001\nλτ≫1,(8b)\nwhere we have used that ε≫λ.\nThe results of Eq. (8) modify by electron diffusion. By\ntaking into account up to lscattering events we obtain\n¯α(l)\nm,⊥=ετ(\nl+O(λ2τ2) λτ≪1,\n(1 +δl0)/(λτ)2λτ≫1,(9a)\n¯α(l)\nm,∥=ετ(\nl+O(λ2τ2) λτ≪1,\n1−(1/2)l+1+O((λτ)−2)λτ≫1,(9b)\nwhere we have used ε≫λagain.\nFrom Eqs. (9) we see that the Gilbert damping for\nλτ≪1 gets an additional contribution of ετfrom each\nscattering event as illustrated numerically in Fig. 2. This\nleads to a formal divergence of Gilbert damping in the\nlimit λτ≪1. While, at first glance, the divergence looks\nlike a strong sensitivity of damping to impurity scatter-\ning, in reality, it simply reflects a diverging spin life-time.\nOnce a non-equilibrium magnetization mis created it\nbecomes almost impossible to relax it to the lattice in\nthe limit of weak spin-orbit coupling. The formal diver-\ngence of α⊥\nm=α∥\nmsimply reflects the conservation law\nfor electron spin polarization in the absence of spin-orbit\ncoupling such that the corresponding spin life-time be-\ncomes arbitrarily large as compared to the momentum\nscattering time τ.\nBy taking the limit l→ ∞ (i. e. by summing up the\nentire diffusion ladder) we obtain compact expressions\n¯α⊥\nm≡¯α(∞)\nm,⊥=ετ1\n2λ2τ2, (10a)\n¯α∥\nm≡¯α(∞)\nm,∥=ετ1 +λ2τ2\nλ2τ2, (10b)\nwhich assume ¯ α⊥\nm≪¯α∥\nmforλτ≫1 and ¯ α⊥\nm= ¯α∥\nm/2\nforλτ≪1. The factor of 2 difference that we observe\nwhen λτ≪1, corresponds to a difference in the elec-\ntron spin life-times τ⊥\ns=τ∥\ns/2 that was discussed in the\nintroduction32.\nStrong spin-orbit coupling causes a strong out-of-plane\nanisotropy of damping, ¯ α⊥\nm≪¯α∥\nmwhich corresponds to5\na suppression of the perpendicular-to-the-plane damping\ncomponent. As a result, the spin-orbit interaction makes\nit much easier to relax the magnitude of the mzcompo-\nnent of magnetization than that of in-plane components.\nLet us now turn to the dependence of ¯ αmcoefficients on\n∆sdthat is illustrated numerically in Fig. 3. We consider\nfirst the case of absent spin-orbit coupling λ= 0. In\nthis case, the combination of spin-rotational and sub-\nlattice symmetry (the equivalence of A and B sub-lattice)\nmust make Gilbert damping isotropic (see e. g.25,41). The\ndirect calculation for λ= 0 does, indeed, give rise to the\nisotropic result ¯ α⊥\nm= ¯α∥\nm=ετ(ε2+∆2\nsd)/2∆2\nsd, which is,\nhowever, in contradiction to the limit λ→0 in Eq. (10).\nAt first glance, this contradiction suggests the exis-\ntence of a certain energy scale for λover which the\nanisotropy emerges. The numerical analysis illustrated\nin Fig. 4 reveals that this scale does not depend on the\nvalues of 1 /τ, ∆sd, orε. Instead, it is defined solely by\nnumerical precision. In other words, an isotropic Gilbert\ndamping is obtained only when the spin-orbit strength\nλis set below the numerical precision in our model.\nWe should, therefore, conclude that the transition from\nisotropic to anisotropic (factor of 2) damping occurs ex-\nactly at λ= 0. Interestingly, the factor of 2 anisotropy is\nabsent in Eqs. (8) and emerges only in the diffusive limit.\nWe will see below that this paradox can only be re-\nsolved by analyzing the Gilbert damping beyond the in-\nfinite wave-length limit.\nOne can see from Fig. 3 that the main effect of finite\n∆sdis the regularization of the Gilbert damping diver-\ngency ( λτ)−2in the limit λτ≪1. Indeed, the limit of\nweak spin-orbit coupling is non-perturbative for ∆ sd/ε≪\nλτ≪1, while, in the opposite limit, λτ≪∆sd/ε≪1,\nthe results of Eqs. (10) are no longer valid. Assuming\n∆sd/ε≪1 we obtain the asymptotic expressions for the\nresults presented in Fig. 3 as\n¯α⊥\nm=1\n2ετ(2\n3ε2+∆2\nsd\n∆2\nsdλτ≪∆sd/ε,\n1\nλ2τ2 λτ≫∆sd/ε,(11a)\n¯α∥\nm=ετ(2\n3ε2+∆2\nsd\n∆2\nsdλτ≪∆sd/ε,\n1 +1\nλ2τ2λτ≫∆sd/ε,(11b)\nwhich suggest that ¯ α⊥\nm/¯α∥\nm= 2 for λτ≪1. In the op-\nposite limit, λτ≫1, the anisotropy of Gilbert damping\ngrows as ¯ α∥\nm/¯α⊥\nm= 2λ2τ2.\nThe results of Eqs. (11) can also be discussed in terms\nof the electron spin life-time, τ⊥(∥)\ns = ¯α⊥(∥)\nm/ε. For the\ninverse in-plane spin life-time we find\n1\nτ∥\ns=\n\n3∆2\nsd/2ε2τ λτ ≪∆sd/ε,\nλ2τ ∆sd/ε≪λτ≪1,\n1/τ 1≪λτ,(12)\nthat, for ∆ sd= 0, is equivalent to the known result of\nEq. (4). Indeed, for ∆ sd= 0, the magnetic exchange\n10\u0000310\u0000210\u00001100101\n\u0015\u001c10\u00001101103105\u0016\u000bm;k;?[\"\u001c]\n\u0001sd=\"= 0:1\u0001sd=\"= 0\u0016\u000bm;k\n\u0016\u000bm;?FIG. 3. Numerical results for the Gilbert damping compo-\nnents in the diffusive limit (vertex corrected)as the function\nof the spin-orbit coupling strength λ. The results correspond\ntoετ= 50 and ∆ sdτ= 0.1 and agree with the asymptotic\nexpressions of Eq. (11). Three different regimes can be dis-\ntinguished for ¯ α∥\nm: i) spin-orbit independent damping ¯ α∥\nm∝\nε3τ/∆2\nsdfor the exchange dominated regime, λτ≪∆sd/ε, ii)\nthe damping ¯ α∥\nm∝ε/λ2τfor Elliot-Yafet relaxation regime,\n∆sd/ε≪λτ≪1, and iii) the damping ¯ α∥\nm∝ετfor the\nDyakonov-Perel relaxation regime, λτ≫1. The latter regime\nis manifestly absent for ¯ α⊥\nmin accordance with Eqs. (12,13).\nplays no role and one observes the cross-over from Elliot-\nYafet ( λτ≪1) to Dyakonov-Perel ( λτ≫1) spin relax-\nation.\nThis cross-over is, however, absent in the relaxation of\nthe perpendicular spin component\n1\nτ⊥s= 2(\n3∆2\nsd/2ε2τ λτ ≪∆sd/ε,\nλ2τ ∆sd/ε≪λτ,(13)\nwhere Elliot-Yafet-like relaxation extends to the regime\nλτ≫1.\nAs mentioned above, the factor of two anisotropy in\nspin-relaxation of 2 Dsystems, τ∥\ns= 2τ⊥\ns, is known in the\nliterature32(see Refs.36–38as well). Unlimited growth of\nspin life-time anisotropy, τ∥\ns/τ⊥\ns= 2λ2τ2, in the regime\nλτ≪1 has been described first in Ref. 25. It can be qual-\nitatively explained by a strong suppression of spin-flip\nprocesses for zspin component due to spin-orbit induced\nsplitting of Fermi surfaces. The mechanism is effective\nonly for scalar (non-magnetic) disorder. Even though\nsuch a mechanism is general for any magnetic or non-\nmagnetic 2D material with Rashba-type spin-orbit cou-\npling, the effect of the spin life-time anisotropy on Gilbert\ndamping is much more relevant for anti-ferromagnets. In-\ndeed, in an anti-ferromagnetic system the modulus of m\nis, by no means, conserved, hence the variations of per-\npendicular and parallel components of the magnetization\nvector are no longer related.\nIn the regime, λτ≪∆sd/εthe spin life-time is de-\nfined by exchange interaction and the distinction between\nDyakonov-Perel and Elliot-Yafet mechanisms of spin re-\nlaxation is no longer relevant. In this regime, the spin-\nrelaxation time is by a factor ( ε/∆sd)2larger than the\nmomentum relaxation time.\nLet us now return to the problem of emergency of the6\n10\u00006410\u00005410\u00004410\u00003410\u00002410\u000014\n\u0015\u001c12\u0016\u000bk=\u0016\u000b?n= 32\nn= 64n= 96\nn= 128\nFIG. 4. Numerical evaluation of Gilbert damping anisotropy\nin the limit λ→0. Isotropic damping tensor is restored only\nifλ= 0 with ultimate numerical precision. The factor of 2\nanisotropy emerges at any finite λ, no matter how small it\nis, and only depends on the numerical precision n, i.e. the\nnumber of digits contained in each variable during computa-\ntion. The crossover from isotropic to anisotropic damping can\nbe understood only by considering finite, though vanishingly\nsmall, magnon qvectors.\nfactor of 2 anisotropy of Gilbert damping at λ= 0. We\nhave seen above (see Fig. 4) that, surprisingly, there ex-\nists no energy scale for the anisotropy to emerge. The\ntransition from the isotropic limit ( λ= 0) to a finite\nanisotropy appeared to take place exactly at λ= 0. We\ncan, however, generalize the concept of Gilbert damping\nby considering the spin density response function at a\nfinite wave vector q.\nTo generalize the Gilbert damping, we are seeking a\nresponse of spin density at a point r,δs+(r) to a time\nderivative of magnetization vectors ˙m∥and ˙m⊥at the\npoint r′. The Fourier transform with respect to r−r′\ngives the Gilbert damping for a magnon with the wave-\nvector q.\nThe generalization to a finite q-vector shows that the\nlimits λ→0 and q→0 cannot be interchanged. When\nthe limit λ→0 is taken before the limit q→0 one\nfinds an isotropic Gilbert damping, while for the oppo-\nsite order of limits, it becomes a factor of 2 anisotropic.\nIn a realistic situation, the value of qis limited from\nbelow by an inverse size of a typical magnetic domain\n1/Lm, while the spin-orbit coupling is effective on the\nlength scale Lλ= 2πℏvf/λ. In this picture, the isotropic\nGilbert damping is characteristic for the case of suffi-\nciently small domain size Lm≪Lλ, while the anisotropic\nGilbert damping corresponds to the case Lλ≪Lm.\nIn the limit qℓ≪1, where ℓ=vfτis the electron mean\n\u00002 0 2\nk[a.u.]\u00002:50:02:5energy [a.u.]\u0015=\u0001sd= 4\n\u00002 0 2\nk[a.u.]\u0015=\u0001sd= 2\n\u00002 0 2\nk[a.u.]\u0015=\u0001sd= 1FIG. 5. Band-structure for the effective model of Eq. (5)\nin a vicinity of Kvalley assuming nz= 1. Electron bands\ntouch for λ= 2∆ sd. The regime λ≤2∆sdcorresponds to\nspin-orbit band inversion. The band structure in the valley\nK′is inverted. Our microscopic analysis is performed in the\nelectron-doped regime for the Fermi energy above the gap as\nillustrated by the top dashed line. The bottom dashed line\ndenotes zero energy (half-filling).\nfree path, we can summarize our results as\n¯α⊥\nm=ετ\n\nε2+∆2\nsd\n2∆2\nsdλτ≪qℓ≪∆sd/ε,\n1\n3ε2+∆2\nsd\n∆2\nsdqℓ≪λτ≪∆sd/ε,\n1\n2λ2τ2 λτ≫∆sd/ε,, (14a)\n¯α∥\nm=ετ\n\nε2+∆2\nsd\n2∆2\nsdλτ≪qℓ≪∆sd/ε,\n2\n3ε2+∆2\nsd\n∆2\nsdqℓ≪λτ≪∆sd/ε,\n1 +1\nλ2τ2λτ≫∆sd/ε,(14b)\nwhich represent a simple generalization of Eqs. (11).\nThe results of Eqs. (14) correspond to a simple behav-\nior of Gilbert damping anisotropy,\n¯α∥\nm/¯α⊥\nm=(\n1 λτ≪qℓ,\n2\u0000\n1 +λ2τ2\u0001\nqℓ≪λτ,(15)\nwhere we still assume qℓ≪1.\nIII. ANTI-FERROMAGNETIC RESONANCE\nThe broadening of the anti-ferromagnet resonance\npeak is one obvious quantity that is sensitive to Gilbert\ndamping. The broadening is however not solely defined\nby a particular Gilbert damping component but depends\nalso on both magnetic anisotropy and anti-ferromagnetic\nexchange.\nTo be more consistent we can use the model of Eq. (5)\nto analyze the contribution of conduction electrons to an\neasy axis anisotropy. The latter is obtained by expanding\nthe free energy for electrons in the value of nz, which has\na form E=−Kn2\nz/2. With the conditions ε/λ≫1 and\nε/∆sd≫1 we obtain the anisotropy constant as\nK=A\n2πℏ2v2(\n∆2\nsdλ 2∆sd/λ≤1,\n∆sdλ2/2 2∆ sd/λ≥1,(16)7\nwhere Ais the area of the unit cell. Here we assume\nboth λand ∆ sdpositive, therefore, the model natu-\nrally gives rise to an easy axis anisotropy with K > 0.\nIn real materials, there exist other sources of easy axis\nor easy plane anisotropy. In-plane magneto-crystalline\nanisotropy also plays an important role. For example,\nN´ eel-type anti-ferromagnets with easy-axis anisotropy\nare FePS 3, FePSe 3or MnPS 3, whereas those with easy\nplane and in-plane magneto-crystalline anisotropy are\nNiPS 3and MnPSe 3. Many of those materials are, how-\never, Mott insulators. Our qualitative theory may still\napply to materials like MnPS 3monolayers at strong elec-\ntron doping.\nThe transition from 2∆ sd/λ≥1 to 2∆ sd/λ≤1 in\nEq. (16) corresponds to the touching of two bands in the\nmodel of Eq. (5) as illustrated in Fig. 5.\nAnti-ferromagnetic magnon frequency and life-time in\nthe limit q→0 are readily obtained by linearizing the\nequations of motion\n˙n=−Jn×m+Km×n⊥+n×(ˆαm˙m), (17a)\n˙m=Kn×n⊥+n×(ˆαn˙n), (17b)\nwhere we took into account easy axis anisotropy Kand\ndisregarded irrelevant terms m×(ˆαn˙n) and m×(ˆαm˙m).\nWe have also defined Gilbert damping tensors such as\nˆαm˙m=α∥\nm˙m∥+α⊥\nm˙m⊥, ˆαn˙n=α∥\nn˙n∥+α⊥\nn˙n⊥.\nIn the case of easy axis anisotropy we can use the lin-\nearized modes n=ˆz+δn∥eiωt,m=δm∥eiωt, hence we\nget the energy of q= 0 magnon as\nω=ω0−iΓ/2, (18)\nω0=√\nJK, Γ =Jα∥\nn+Kα∥\nm (19)\nwhere we took into account that K≪J. The expression\nforω0is well known due to Kittel and Keffer42,43.\nUsing Ref. 25 we find out that α∥\nn≃α⊥\nm(λ/ε)2and\nα⊥\nn≃α∥\nm(λ/ε)2, hence\nΓ≃α∥\nm\u0012\nK+J/2\nε2/λ2+ε2τ2\u0013\n, (20)\nwhere we have simply used Eqs. (10). Thus, one may\noften ignore the contribution Jα∥\nnas compared to Kα∥\nm\ndespite the fact that K≪J.\nIn the context of anti-ferromagnets, spin-pumping\nterms are usually associated with the coefficients α∥\nnin\nEq. (3b) that are not in the focus of the present study.\nThose coefficients have been analyzed for example in Ref.\n25. In this manuscript we simply use the known results\nforαnin Eqs. (17-19), where we illustrate the effect of\nboth spin-pumping coefficient αnand the direct Gilbert\ndamping αmon the magnon life time. One can see from\nEqs. (19,20) that the spin-pumping contributions do also\ncontribute, though indirectly, to the magnon decay. The\nspin pumping contributions become more important in\nmagnetic materials with small magnetic anisotropy. The\nprocesses characterized by the coefficients αnmay also be\n10\u0000310\u0000210\u00001100101\n\u0015\u001c0:000:010:021=\u0016\u000bk\nm\u0015=\"= 0:04\n\u0015=\"= 0:02\n\u0015=\"= 0:01FIG. 6. Numerical evaluation of the inverse Gilbert damping\n1/¯α∥\nmas a function of the momentum relaxation time τ. The\ninverse damping is peaked at τ∝1/λwhich also corresponds\nto the maximum of the anti-ferromagnetic resonance quality\nfactor in accordance with Eq. (21).\ninterpreted in terms of angular momentum transfer from\none AFM sub-lattice to another. In that respect, the spin\npumping is specific to AFM, and is qualitatively differ-\nent from the direct Gilbert damping processes ( αm) that\ndescribe the direct momentum relaxation to the lattice.\nAs illustrated in Fig. 6 the quality factor of the anti-\nferromagnetic resonance (for a metallic anti-ferromagnet\nwith easy-axis anisotropy) is given by\nQ=ω0\nΓ≃1\nα∥\nmr\nJ\nK. (21)\nInterestingly, the quality factor defined by Eq. (21) is\nmaximized for λτ≃1, i. e. for the electron spin-orbit\nlength being of the order of the scattering mean free path.\nThe quantities 1 /√\nKand 1 /¯α∥\nmare illustrated in\nFig. 6 from the numerical analysis. As one would ex-\npect, the quality factor vanishes in both limits λ→0\nandλ→ ∞ . The former limit corresponds to an over-\ndamped regime hence no resonance can be observed. The\nlatter limit corresponds to a constant α∥\nm, but the reso-\nnance width Γ grows faster with λthan ω0does, hence\nthe vanishing quality factor.\nIt is straightforward to check that the results of\nEqs. (20,21) remain consistent when considering systems\nwith either easy-plane or in-plane magneto-crystalline\nanisotropy. Thus, the coefficient α⊥\nmnormally does not\nenter the magnon damping, unless the system is brought\ninto a vicinity of spin-flop transition by a strong external\nfield.\nIV. CONCLUSION\nIn conclusion, we have analyzed the Gilbert damping\ntensor in a model of a two-dimensional anti-ferromagnet\non a honeycomb lattice. We consider the damping mech-\nanism that is dominated by a finite electron spin life-time8\ndue to a combination of spin-orbit coupling and impu-\nrity scattering of conduction electrons. In the case of a\n2D electron system with Rashba spin-orbit coupling λ,\nthe Gilbert damping tensor is characterized by two com-\nponents α∥\nmandα⊥\nm. We show that the anisotropy of\nGilbert damping depends crucially on the parameter λτ,\nwhere τis the transport scattering time for conduction\nelectrons. For λτ≪1 the anisotropy is set by a geo-\nmetric factor of 2, α∥\nm= 2α⊥\nm, while it becomes infinitely\nlarge in the opposite limit, α∥\nm= (λτ)2α⊥\nmforλτ≫1.\nGilbert damping becomes isotropic exactly for λ= 0, or,\nstrictly speaking, for the case λ≪ℏvfq, where qis the\nmagnon wave vector.\nThis factor of 2 is essentially universal, and is a geomet-\nric effect: the z-component relaxation results from fluctu-\nations in two in-plane spin components, whereas in-plane\nrelaxation stems from fluctuations of the z-component\nalone. This reflects the subtleties of our microscopic\nmodel, where the mechanism for damping is activated\nby the decay of conduction electron momenta, linked to\nspin-relaxation through spin-orbit interactions.\nWe find that Gilbert damping is insensitive to mag-\nnetic order for λ≫∆sd/ετ, where ∆ sdis an effective\nexchange coupling between spins of conduction and local-\nized electrons. In this case, the electron spin relaxation\ncan be either dominated by scattering (Dyakonov-Perel\nrelaxation) or by spin-orbit precession (Elliot-Yafet re-\nlaxation). We find that the Gilbert damping component\nα⊥\nm≃ε/λ2τis dominated by Elliot-Yafet relaxation irre-\nspective of the value of the parameter λτ, while the other\ncomponent crosses over from α∥\nm≃ε/λ2τ(Elliot-Yafet\nrelaxation) for λτ≪1, to α∥\nm≃ετ(Dyakonov-Perel re-\nlaxation) for λτ≫1. For the case λ≪∆sd/ετthe spin\nrelaxation is dominated by interaction with the exchange\nfield.\nCrucially, our results are not confined solely to the N´ eel\norder on the honeycomb lattice: we anticipate a broader\napplicability across various magnetic orders, including\nthe zigzag order. This universality stems from our focus\non the large magnon wavelength limit. The choice of the\nhoneycomb lattice arises from its unique ability to main-\ntain isotropic electronic spectra within the plane, coupled\nwith the ability to suppress anisotropy concerning in-\nplane spin rotations. Strong anisotropic electronic spec-\ntra would naturally induce strong anisotropic in-plane\nGilbert damping, which are absent in our results.\nFinally, we show that the anti-ferromagnetic resonance\nwidth is mostly defined by α∥\nmand demonstrate that the\nresonance quality factor is maximized for λτ≈1. Our\nmicroscopic theory predictions may be tested for systems\nsuch as MnPS 3monolayer on Pt and similar heterostruc-\ntures.ACKNOWLEDGMENTS\nWe are grateful to O. Gomonay, R. Duine, J. Sinova,\nand A. Mauri for helpful discussions. This project has\nreceived funding from the European Union’s Horizon\n2020 research and innovation program under the Marie\nSklodowska-Curie grant agreement No 873028.\nAppendix A: Microscopic framework\nThe microscopic model that we employ to calculate\nGilbert damping belongs to a class of so-called s–dmod-\nels that describe the physical system in the form of a\nHeisenberg model for localized spins and a tight-binding\nmodel for conduction electrons that are weakly coupled\nby a local magnetic exchange interaction of the strength\n∆sd.\nOur effective electron Hamiltonian for a metallic\nhexagonal anti-ferromagnet is given by25\nH0=vfp·Σ+λ\n2[σ×Σ]z−∆sdn·σΣzΛz,(A1)\nwhere the vectors Σ,σandΛdenote the vectors of Pauli-\nmatrices acting on sub-lattice, spin and valley space re-\nspectively. We also introduce the Fermi velocity vf,\nRashba-type spin-orbit interaction λ.\nTo describe Gilbert damping of the localized field n\nwe have to add the relaxation mechanism. This is pro-\nvided in our model by adding a weak impurity potential\nH=H0+V(r). The momentum relaxation due to scat-\ntering on impurities leads indirectly to the relaxation of\nHeisenberg spins due to the presence of spin-orbit cou-\npling and exchange couplings.\nFor modeling the impurity potential, we adopt a delta-\ncorrelated random potential that corresponds to the\npoint scatter approximation, where the range of the im-\npurity potential is much shorter than that of the mean\nfree path (see e.g. section 3.8 of Ref. 44), i.e.\n⟨V(r)V(r′)⟩= 2πα(ℏvf)2δ(r−r′), (A2)\nwhere the dimensionless coefficient α≪1 characterizes\nthe disorder strength. The corresponding scattering time\nfor electrons is obtained as τ=ℏ/παϵ , which is again\nsimilar to the case of graphene.\nThe response of symmetric spin-polarization δs+to the\ntime-derivative of non-staggered magnetization, ∂tm, is\ndefined by the linear relation\nδs+\nα=X\nβRαβ|ω=0˙mβ, (A3)\nwhere the response tensor is taken at zero frequency25,45.\nThe linear response is defined generally by the tensor\nRαβ=A∆2\nsd\n2πSZdp\n(2πℏ)2\nTr\u0002\nGR\nε,pσαGA\nε+ℏω,pσβ\u0003\u000b\n,(A4)9\nwhere GR(A)\nε,pare standing for retarded(advanced) Green\nfunctions and the angular brackets denote averaging over\ndisorder fluctuations.\nThe standard recipe for disorder averaging is the diffu-\nsive approximation46,47that is realized by replacing the\nbare Green functions in Eq. (A4) with disorder-averaged\nGreen functions and by replacing one of the vertex op-\nerators σxorσywith the corresponding vertex-corrected\noperator that is formally obtained by summing up ladder\nimpurity diagrams (diffusons).\nIn models with spin-orbit coupling, the controllable dif-\nfusive approximation for non-dissipative quantities may\nbecome, however, more involved as was noted first in\nRef. 48. For Gilbert damping it is, however, sufficient to\nconsider the ladder diagram contributions only.\nThe disorder-averaged Green function is obtained by\nincluding an imaginary part of the self-energy ΣR(not\nto be confused here with the Pauli matrix Σ 0,x,y,z) that\nis evaluated in the first Born approximation\nIm ΣR= 2παv2\nfZdp\n(2π)2Im1\nε−H0+i0. (A5)\nThe real part of the self-energy leads to the renormaliza-\ntion of the energy scales ε,λand ∆ sd.\nIn the first Born approximation, the disorder-averaged\nGreen function is given by\nGR\nε,p=1\nε−H0−iIm ΣR. (A6)\nThe vertex corrections are computed in the diffusive\napproximation. The latter involves replacing the vertex\nσαwith the vertex-corrected operator,\nσvc\nα=∞X\nl=0σ(l)\nα, (A7)\nwhere the index lcorresponds to the number of disorder\nlines in the ladder.\nThe operators σ(l)\nαcan be defined recursively as\nσ(l)\nα=2ℏv2\nf\nετZdp\n(2π)2GR\nε,pσ(l−1)\nαGA\nε+ℏω,p, (A8)\nwhere σ(0)\nα=σα.\nThe summation in Eq. (A7) can be computed in the\nfull operator basis, Bi={α,β,γ}=σαΣβΛγ, where each\nindex α,βandγtakes on 4 possible values (with zero\nstanding for the unity matrix). We may always normalize\nTrBiBj= 2δijin an analogy to the Pauli matrices. The\noperators Biare, then, forming a finite-dimensional space\nfor the recursion of Eq. (A8).\nThe vertex-corrected operators Bvc\niare obtained by\nsumming up the matrix geometric series\nBvc\ni=X\nj\u00121\n1− F\u0013\nijBj, (A9)where the entities of the matrix Fare given by\nFij=ℏv2\nf\nετZdp\n(2π)2Tr\u0002\nGR\nε,pBiGA\nε+ℏω,pBj\u0003\n.(A10)\nOur operators of interest σxandσycan always be de-\ncomposed in the operator basis as\nσα=1\n2X\niBiTr (σαBi), (A11)\nhence the vertex-corrected spin operator is given by\nσvc\nα=1\n2X\nijBvc\niTr(σαBi). (A12)\nMoreover, the computation of the entire response tensor\nof Eq. (A4) in the diffusive approximation can also be\nexpressed via the matrix Fas\nRαβ=α0ετ\n8ℏX\nij[TrσαBi]\u0014F\n1− F\u0015\nij[TrσβBj],(A13)\nwhere α0=A∆2\nsd/πℏ2v2\nfSis the coefficient used in\nEq. (6) to define the unit of the Gilbert damping.\nIt appears that one can always choose the basis of\nBioperators such that the computation of Eq. (A13)\nis closed in a subspace of just three Bioperators with\ni= 1,2,3. This enables us to make analytical computa-\ntions of Eq. (A13).\nAppendix B: Magnetization dynamics\nThe representation of the results can be made some-\nwhat simpler by choosing xaxis in the direction of the\nin-plane projection n∥of the N´ eel vector, hence ny= 0.\nIn this case, one can represent the result as\nδs+=c1n∥×(n∥×∂tm∥) +c2∂tm∥+c3∂tm⊥+c4n,\nwhere ndependence of the coefficients cimay be param-\neterized as\nc1=r11−r22−r31(1−n2\nz)/(nxnz)\n1−n2z, (B1a)\nc2=r11−r31(1−n2\nz)/(nxnz), (B1b)\nc3=r33, (B1c)\nc4= (r31/nz)∂tmz+ζ(∂tm)·n. (B1d)\nThe analytical results in the paper correspond to the\nevaluation of δs±up to the second order in ∆ sdusing\nperturbative analysis. Thus, zero approximation corre-\nsponds to setting ∆ sd= 0 in Eqs. (A1,A5).\nThe equations of motion on nandmare given by\nEqs. (2),\n∂tn=−Jn×m+n×δs++m×δs−, (B2a)\n∂tm=m×δs++n×δs−, (B2b)10\nIt is easy to see that the following transformation leaves\nthe above equations invariant,\nδs+→δs+−ξn, δ s−→δs−−ξm, (B3)\nfor an arbitrary value of ξ.\nSuch a gauge transformation can be used to prove that\nthe coefficient c4is irrelevant in Eqs. (B2).\nIn this paper, we compute δs±to the zeroth order in\n|m|– the approximation which is justified by the sub-\nlattice symmetry in the anti-ferromagnet. A somewhat\nmore general model has been analyzed also in Ref. 25 to\nwhich we refer the interested reader for more technical\ndetails.\nAppendix C: Anisotropy constant\nThe anisotropy constant is obtained from the grand po-\ntential energy Ω for conducting electrons. For the model\nof Eq. (A1) the latter can be expressed as\nΩ =−X\nς=±1\nβZ\ndε g(ε)νς(ε), (C1)\nwhere β= 1/kBTis the inverse temperature, ς=±is\nthe valley index (for the valleys KandK′),GR\nς,pis the\nbare retarded Green function with momentum pand in\nthe valley ς. We have also defined the function\ng(ε) = ln (1 + exp[ β(µ−ε)]), (C2)\nwhere µis the electron potential, and the electron density\nof states in each of the valleys is given by,\nνς(ε) =1\nπZdp\n(2πℏ)2Im Tr GR\nς,p, (C3)\nwhere the trace is taken only over spin and sub-lattice\nspace,\nIn the metal regime considered, the chemical potential\nis assumed to be placed in the upper electronic band.\nIn this case, the energy integration can be taken only for\npositive energies. The two valence bands are always filled\nand can only add a constant shift to the grand potential\nΩ that we disregard.\nThe evaluation of Eq. (C1) yields the following density\nof states\nντ(ε) =1\n2πℏ2v2\nf\n\n0 0 < ε < ε 2\nε/2 +λ/4ε2< ε < ε 1,\nε ε > ε 1,(C4)where the energies ε1,2correspond to the extremum\npoints (zero velocity) for the electronic bands. These\nenergies, for each of the valleys, are given by\nε1,ς=1\n2\u0000\n+λ+p\n4∆2+λ2−4ς∆λnz\u0001\n, (C5a)\nε2,ς=1\n2\u0000\n−λ+p\n4∆2+λ2+ 4ς∆λnz\u0001\n(C5b)\nwhere ς=±is the valley index.\nIn the limit of zero temperature we can approximate\nEq. (C1) as\nΩ =−X\nς=±1\nβZ∞\n0dε(µ−ε)νς(ε). (C6)\nThen, with the help of Eq. (C1) we find,\nΩ =−1\n24πℏ2v2\nfX\nς=±\u0002\n(ε1,ς−µ)2(4ε1,ς−3λ+ 2µ)\n+(ε2,ς−µ)2(4ε2,ς+ 3λ+ 2µ)\u0003\n. (C7)\nBy substituting the results of Eqs. (C5) into the above\nequation we obtain\nΩ =−1\n24πℏ2v2\nfh\n(4∆2−4nz∆λ+λ2)2/3\n+(4∆2+ 4nz∆λ+λ2)2/3−24∆µ+ 8µ3i\n.(C8)\nA careful analysis shows that the minimal energy cor-\nresponds to nz=±1 so that the conducting electrons\nprefer an easy-axis magnetic anisotropy. 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Titov, EPL 111,\n37004 (2015)."    },    {        "title": "2312.07116v2.Sliding_Dynamics_of_Current_Driven_Skyrmion_Crystal_and_Helix_in_Chiral_Magnets.pdf",        "content": "Sliding Dynamics of Current-Driven Skyrmion Crystal and Helix in Chiral Magnets\nYing-Ming Xie,1Yizhou Liu,1and Naoto Nagaosa1,∗\n1RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: December 14, 2023)\nThe skyrmion crystal (SkX) and helix (HL) phases, present in typical chiral magnets, can each be\nconsidered as forms of density waves but with distinct topologies. The SkX exhibits gyrodynamics\nanalogous to electrons under a magnetic field, while the HL state resembles topological trivial\nspin density waves. However, unlike the charge density waves, the theoretical analysis of the sliding\nmotion of SkX and HL remains unclear, especially regarding the similarities and differences in sliding\ndynamics between these two spin density waves. In this work, we systematically explore the sliding\ndynamics of SkX and HL in chiral magnets in the limit of large current density. We demonstrate\nthat the sliding dynamics of both SkX and HL can be unified within the same theoretical framework\nas density waves, despite their distinct microscopic orders. Furthermore, we highlight the significant\nrole of gyrotropic sliding induced by impurity effects in the SkX state, underscoring the impact of\nnontrivial topology on the sliding motion of density waves. Our theoretical analysis shows that\nthe effect of impurity pinning is much stronger in HL compared with SkX, i.e., χSkX/χHL∼α2\n(χSkX,χHL: susceptibility to the impurity potential, α(≪1) is the Gilbert damping). Moreover,\nthe velocity correction is mostly in the transverse direction to the current in SkX. These results are\nfurther substantiated by realistic Landau-Lifshitz-Gilbert simulations.\nIntroduction.— Density waves in solids represent a\nprevalent phenomenon, particularly in low-dimensional\nsystems [1, 2]. They break the translational symmetry\nof the crystal, leading to the emergence of Goldstone\nbosons, i.e., phasons, which remain gapless when the pe-\nriod of density waves is incommensurate with the crystal\nperiodicity. The sliding motion of density waves under an\nelectric field Ehas been extensively studied. In this con-\ntext, the impurity pinning of phasons results in a finite\nthreshold field [1, 2]. In general, exploring the dynam-\nics of pinning and depinning offers valuable insights into\nunderstanding the behavior of density waves.\nThe skyrmion crystal (SkX) and helix (HL) phases\nin chiral magnets can be recognized as periodic density\nwaves of spins, as depicted in Figs. 1 (a) and (b). The\nHL phase is stabilized in chiral magnet at small mag-\nnetic field regions, with spins of neighboring magnetic\nmoments arranging themselves in a helical pattern. SkX\nis a superposition of three phase-locked HL and comprises\narrays of magnetic skyrmions, nanoscale vortex-like spin\ntextures characterized by a non-zero skyrmion number\nNsk=1\n4πR R\nd2rs·(∂xs×∂ys) (sbeing the unit vec-\ntor of spin). Theoretically proposed magnetic skyrmions\n[3–5] were initially observed in the chiral magnet MnSi\nunder magnetic fields [6–8], wherein the skyrmion lattice\nstructure produces a six-fold neutron scattering pattern.\nSince then, the chiral magnetic states encompassing SkX\nand HL states have been the focus of extensive research\n[9–13].\nThe dynamics of SkX in a random environment, specif-\nically the pinning effects from impurities, are manifested\nthrough the topological Hall effect. The current depen-\ndence of topological Hall resistivity ρxywas initially ex-\nplored theoretically by Zang et al [14] and experimentally\n∗nagaosa@riken.jpby Schulz et al. [15]. To illustrate, a schematic plot is\npresented in Fig. 1(c). Typically, there are three distinct\nregions characterizing the dynamics of SkX: the pinned,\ncreep, and flow regions. The topological Hall resistiv-\nity decreases when SkX is depinned because the motion\nof SkX induces temporal changes in the emerging mag-\nnetic fields Be, subsequently generating emergent elec-\ntric fields Eeand an opposing Hall contribution. Theo-\nretically, the pinning problem of both SkX and HL was\ninvestigated in terms of replica symmetry breaking [16],\nrevealing a distinct difference in glassy states between\nSkX and HL. The key factor lies in the nontrivial topol-\nogy of SkX, contrasting with the trivial topology in HL\nand most density wave states. However, this difference\nhas not been theoretically explored in the context of slid-\ning/moving density wave states for chiral magnets.\nIn this work, we systemically study the current-driven\nsliding dynamics of the SkX and HL in chiral magnets.\nWe employ the methodology proposed by Sneddon et al.\n[17] in their investigation of charge density waves and\napply it to magnetic materials. This method allows us\nto investigate the current-driven dynamics of SkX and\nHL, considering both deformation and impurity pinning\neffects. Through this method, we reveal that the drift ve-\nlocity correction ∆ vddue to the impurity pinning effects\nversus the current density jsin the flow region, follows\n∆vd∝(vd0)d−2\n2(−e∥+G\nαDe⊥) for the SkX phase, while\n∆vd∝ −(vd0)d−2\n2e∥for the HL phase with the spatial di-\nmension denoted as d. Here, e∥represents the direction\nof the intrinsic drift velocity vd0(the magnitude of vd0is\nproportional to the current density jsdue to the univer-\nsal linear current-velocity relation [18]), G= 4πNsk,Dis\na form factor at the order of unity, α≪1 is the Gilbert\ndamping parameter so that G/αD ≫1. Although the\nscaling relation ( vd0)d−2\n2applies to both SkX and HL, we\ncan see that the gyrodynamics of the SkX state inducedarXiv:2312.07116v2  [cond-mat.mes-hall]  13 Dec 20232\n(a)\n(c)\nvd\nvs(b)\n(d)\nHL SkX\nρxy\nvdCreep Flow pinned\njs\nFIG. 1. (a), (b) The current-driven motion of the SkX and\nHL, respectively. (c) Schematic of the Hall resistivity ρxy\nand drift velocity vdversus current density jswith pinned\n(yellow), creeping (green), and flowing (purple) highlighted.\n(d) The collective flow motion of the SkX, where the center\nof each skyrmion (red dots) and the impurities (black crosses)\nare highlighted.\nby its nontrivial topology results in its sliding dynamics\nmore robust than in HL and mostly in the transverse di-\nrection. Finally, we explicitly conduct the micromagnetic\nsimulations on both the SkX and HL systems, aligning\nwell with our theoretical expectations.\nOur work demonstrates the unification of sliding dy-\nnamics between spin density waves and charge density\nwaves within the same theoretical framework. Our re-\nsults also vividly illuminate both the similarities and dif-\nferences in the sliding dynamics between SkX and HL\nphases. This insight significantly enhances our under-\nstanding of the sliding dynamics associated with topo-\nlogical density wave phenomena, which possesses pos-\nsible applications in areas, such as skyrmion-based de-\nvices [19–21], depinning dynamics [22–27], Hall responses\n[14, 15, 28], and current-driven motion of Wigner crystals\nunder out-of-plane magnetic fields [29–31].\nSliding dynamics for skyrmion crystals.— The current-\ndriven motion of SkX is described by the Thiele equation\nassuming that its shape does not change [18, 32, 33]:\nG×(vs−vd) +D(βvs−αvd) +F= 0. (1)\nHere, the first term on the left represents the Magnus\nforce, the second term is the dissipative force, and the\nlast term arises from deformation and impurity-pinning\neffects. Here, vsis the velocity of conduction electrons,\nαis the damping constant of the magnetic system, and β\ndescribes the non-adiabatic effects of the spin-polarized\ncurrent. The gyromagnetic coupling vector is denoted as\nG= (0,0,4πNsk), and the dissipation matrix Dij=δijD\nwhere i, j∈ {x, y}. It is noteworthy that the Thieleequation respects out-of-plane rational symmetry [Sup-\nplementary Material (SM) Sec. IA [34]].\nTo obtain the equation of motion of SkX, the displace-\nment vector field of skyrmions is defined as u(r, t) so\nthat the drift velocity vd=∂u(r,t)\n∂t, where ris the posi-\ntion vector, tis the time. The force Fcan be expressed\nwithu(r, t) asF(r, t) =Fimp+Fde, where the impurity\npinning force Fimp=−P\ni∇U(r+u(r, t)−ri)ρ(r) =\nfimp(r+u(r, t))ρ(r) and the deformation force Fde=R\ndr′D(r−r′)u(r′, t′). Here, U(r−ri) is the impu-\nrity potential around site ri,D(r−r′) characterizes the\nrestoration strength after deformation, and ρ(r) is the\nskyrmion density. Based on these definitions, the Thiele\nequation can be expressed as an equation of motion:\n∂u(r, t)\n∂t=ˆM0vs+ˆM1Z\ndr′D(r−r′)u(r′, t)\n+ˆM1fimp(r+u(r, t))ρ(r). (2)\nwhere ˆM0=1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\nand\nˆM1=1\nG2+α2D2\u0012\nαD G\n−G αD\u0013\n. Note that each skyrmion is\nnow considered as a center-of-mass particle, and these\nskyrmions form a triangular lattice and move collec-\ntively with scatterings from impurities, as illustrated in\nFig. 1(d).\nThe displacement vector can be expanded around the\nuniform motion,\nu(r, t) =vdt+˜u(r, t). (3)\nHere,vdis the dominant uniform skyrmion motion veloc-\nity,˜u(r, t) characterizes a small non-uniform part. Using\nthe Green’s function approach to solve the differential\nequation Eq. (2), ˜u(r, t) can be obtained as [17, 29, 34]\n˜u(r, t) =Z\ndr′Z\ndt′G(r−r′, t−t′){vd0−vd\n+M1fimp(r′+vdt′+˜u(r′, t′))ρ(r′)},(4)\nwhere the intrinsic drift velocity vd0=ˆM0vs, the Fourier\ncomponent of the Green’s function Gis given by\nG−1(k, ω) =−iω−ˆM1D(k). (5)\nHere,D(k) arises from the Fourier transformation of de-\nformation D(k) =R\ndd(r)D(r)e−ik·r(the spatial dimen-\nsion is denoted as d).\nIn the flow region, ˜u(r, t) in Eq. (4) can be solved\nperturbatively. Up to the second order, ˜u(r, t)≈\n˜u0(r, t) +˜u1(r, t) +˜u2(r, t), which, respectively, are ob-\ntained by replacing the terms in the brackets of Eq. (4) as\nM0vs−vd, M1fimp(r′+vdt′)ρ(r′), M1∇fimp(r′+vdt′)·\n˜u1(r, t)ρ(r′). Based on this approximation and making\nuse of ⟨˜u(r,t)\n∂t⟩= 0, the self-consistent equation for the3\nvelocity reads (for details see SM Sec. IB [34])\nvd=vd0+X\ngZddq\n(2π)d|ρ(g)|2Λ(q)ˆM1×\n\u0012q2\nxqxqy\nqxqyq2\ny\u0013\nIm [G(q−g,−q·vd)]ˆM1\u0012\nqx\nqy\u0013\n.(6)\nwhere ρ(g) is the Fourier component of ρ(r) with gas the\nreciprocal skyrmion lattice vectors, and Λ( q) arises from\nthe impurity average U(q1)U(q2) = (2 π)dΛ(q2)δ(q1+\nq2). Note that the crucial aspects for the above method\nto be valid are (i) the impurity strength is weak, (ii) the\ndrift velocity is large compared to the impurity effects\nand the SkX remains elastic, (iii) the deformation within\neach skyrmion is negligible so that each skyrmion can be\nregareded as a point object.\nTo proceed further, we adopt the following approxi-\nmations. The current-driven distortion is expected to be\nweak so that D(k) would be dominant by the long-wave\nlimit. In this case, D(k) can be expanded as Kxk2\nx+Kyk2\ny\nfor the 2D case and as Kxk2\nx+Kyk2\ny+Kzk2\nzfor the 3D\ncase. On the other hand, the characterized frequency\nthat enters into the Green’s function is q·vd∼vd/a.\nUsing a reasonable parameter vd= 10 m/s, the skyrmion\nlattice constant a= 25 nm, we estimate vd/a∼0.4 GHz.\nThis frequency is much smaller compared with the one of\nKj, which is roughly the scale of exchange energy J∼1\nmeV∼240 GHz [14, 18]. As a result, the dominant\ncontribution to the integral is given by the elastic modes\nvdgj≈ωk≈ D(k)/√\nG2+α2D2withk=q−g→0,\naround which the imaginary part of Green’s function is\nthe largest.\nWith the above approximations, we perform the in-\ntegral in Eq. (6) and sum over the smallest gvectors:\ngj=√\n3κ0(sin(j−1)π\n3,cos(j−1)π\n3) with jas integers from\n1 to 6 and κ0=4π\n3a. Since the Thiele equation exhibits\nout-of-plane rotational symmetry, without loss of gen-\nerality, we set vd0along x-direction here. After some\nsimplifications (for details see SM. Sec IB), we find the\ncorrection (∆ vd=vd−vd0) on the drift velocity due to\nthe impurity and deformation are given by\n∆vd≈χSkX\nd(vd0)d−2\n2(−e∥+G\nαDe⊥), (7)\nwhere the susceptibility to the impurity potential\nχSkX\nd =9κ3\n0|ρ1|2Λ0αD\n4√\nKxKy(G2+α2D2)ford= 2, while χSkX\nd =\n9√\n3κ7/2\n0Γ(G2+α2D2\n4α2D2)|ρ1|2Λ0(αD)3/2\nπ2√\nKxKyKz(G2+α2D2)ford= 3 (the function\nΓ(a) =R+∞\n0dxx6\n(x4−a)2+x4). Note that we have replaced\nρ(gj) =ρ1,Λ(gj) = Λ 0given the six-fold rotational sym-\nmetry of the skyrmion lattice.\nThe first important aspect in Eq. (7) is that the cor-\nrection ∆ vdis insensitive to vd0in 2D limit but follows\na square-root scaling: ( vd0)1/2in 3D limit. Similar to\nmany scaling phenomena, the dimension plays a critical\n0 0.005 0.01-1.5-1-0.50\n0 0.005 0.0100.510 0.005 0.01-1.5-1-0.50\n0 0.005 0.0100.511.5\n0 0.005 0.01-1.5-1-0.50\nSkyrmion 3DSkyrmion 2D, G < 0\ni=i=||Helix 2D\nSkyrmion 2D, G > 0(a) (b)\n(c) (d)\n(e) (f)-10 -8 -6 -4-2.5-2-1.5-1-0.50\nLog(vd0/K)Log(|∆vd|) Slope ≈ 0.5\n0°30°60°90°\n120°\n150°\n180°\n210°\n240°\n270°300°330°00.20.40.60.81\ni=i=||\ni=i=||\nHelix 3DFIG. 2. (a) and (b) The correction ∆ vdversus vd0(in units\nofK) for the 2D and 3D HL state, respectively. (c) and (d)\nThe correction ∆ vdversus vd0(in units of K) for the 2D\nSkX with G= 4πandG=−4π, respectively, where the\nlongitudinal (transverse) component is in blue (red). The 3D\nSkX case is plotted in (f) with G= 4π. (e) The angular-\ndependence of |∆vd(θ)|, where θis the angle of vd0. All the\nof ∆vdin these plots has been normalized. The parameters\nD= 5.577π,α= 0.04.\nrole here. The second important aspect is that the cor-\nrection along the transverse direction directly reflects the\nskyrmion topological number Gwith the ratio compared\nto the longitudinal one as G/αD . These interesting as-\npects embedded in the Eq. (7) will be further highlighted\nlater.\nHelix case.— It is straightforward to generalize the\nabove treatment to the helical spin order. The Thiele\nequation is reduced to one dimension:\nD(βvs−αvd) +F= 0. (8)\nThe essential difference here is the absence of gyrotropic\ncoupling ( G= 0). Following the same procedure [SM\nSec. II], the self-consistent equation for the drift velocity\nis given by\nvd=vd0+Zddq\n(2π)dX\ng|ρ(g)|2\nα2D2Λ(q)q3\nxIm[G(q−g,−qxvd)].\n(9)\nHere, vd0=β\nαvs, the flow direction of the HL is defined\nasx-direction. After adopting the approximation in the4\nprevious section, the analytical expression of the correc-\ntion ∆ vdof the helical magnetic state is\n∆vd≈ −χHL\nd(vd0)d−2\n2 (10)\nwhere χHL\nd =(KxKy)−1/2|ρ1|2Λ0g3\n0\n4αD,ford= 2\n(KxKyKz)−1/2|ρ1|2Λ0g7/2\n0\n2√\n2π(αD)1/2 ,ford= 3 with g0=π\na. Despite\ndifferent magnetic state nature, the ∆ vdas a function\nvd0in Eq. (10) for the HL displays a consistent scaling\nbehavior as the one of SkX shown in Eq. (7).\nNumerical evaluation.— To further justify our analyt-\nical results, we calculate the ∆ vdnumerically according\nto Eqs. (6) and (9). For simplicity, we set the elastic\ncoefficient Kjas isotropic with K≡Kj. Figs. 2(a) and\n(b) display the correction ∆ vdas a function of vd0of\nHL. Note that the zero drify velocity limit should be ig-\nnored since the impurity pinning effect would be dom-\ninant in practice. When the vdis beyond this limit so\nthat the pinning effect can be treated as a perturbation,\nwhich is true for the flow region, the plots clearly indi-\ncate ∆ vd∝(vd0)d−2\n2. The square root behavior in 3D\n(d= 3) is explicitly checked with the log-log plot (inset\nof Fig. 2(b)).\nFigs. 2(c) and (d) show the longitudinal component\n(blue) and transverse component (red) of the correction\nfor the SkX case with positive Gand negative G, respec-\ntively. It is consistent with Eq. (7) that the transverse\ncomponent is odd with respect to Gand is much larger\nthan the longitudinal component as G/αD ≫1. This\ngyrotropic type correction is inherited from the Magnus\nforces in the Thiele equation, and this correction also\nimplies that there exists a net change on the skyrmion\nHall angle due to the impurities. Moreover, the angu-\nlar dependence of the total correction |∆vd|is shown in\nFigs. 2(e), where the anisotropy is very small. The ∆ vd\nas a function vd0also displays distinct scaling behavior\nbetween 2D [Figs. 2(c),(d)] and 3D case [Fig. 2(f)].\nOverall, the scaling behavior of skyrmion similar to\nthat of the HL in Fig. 2, as expected from our theoretical\nanalysis. Moreover, the intrinsic drift velocity vd0is lin-\nearly proportional to the current density jsfor both SkX\nand HL ( vd0∝js) at large js. As a result, we can replace\nvd0with jsin the scaling relation, i.e., ∆ vd∝(js)d−2\n2. It\nis worth noting that the charge density wave also respects\nthis scaling relation [17], despite its distinct microscopic\nnature.\nPhysical interpretation.— Now we provide a physical\ninterpretation of the observed scaling behavior: ∆ vd∝\n(vd0)d−2\n2. As we mentioned earlier, the dominant con-\ntribution to the drift velocity correction arises from the\nexcitation of elastic modes. Hence, we expect the correc-\ntion to be proportional to the number of excited elastic\nmodes at a fixed vd0. For the SkX case, these modes\nfollow the dispersion: vd0|gj|=D(k)/√\nG2+α2D2,\nwhich can be rewritten as vd0=kd/2m′with m′=\n2π√\nG2+α2D2/(√\n3aK). Next, the problem is mapped\nto evaluate the density of states of free fermions with\n0.050.10.150.20.251234\n0.10.20.30.40.52D HL2D SkX0102030051015202530vd,x clean  \njs (1010 A/m2)vd (m/s)2D SkX, weak impuritiesvd,x weak impurities  vd,y defect  vd,y weak impurities  \n0102030051015202530cleanweak impurities2D HL, weak impuritiesvd (m/s)\njs (1010 A/m2)(a)(b)\n(d)∆vd (m/s)α(c)0102030012345|Δvd/<Δv∥>||Δv∥/<Δv∥>||Δv⊥/<Δv∥>|G/αD⃗vd⃗y⃗xΔ⃗v⊥Δ⃗v∥⃗vd0finalColumn 6Column 5|Δvd|FIG. 3. Simulation results of LLG equation. (a) and (c)\nThe drift velocity vdversus current density js(in units of\n1010A/m2) of the 2D SkX and HL at the clean and impurity\ncase. (b) The magnitdue of longitutinal (∆ d,∥) and trans-\nverse (∆ d,⊥) drift velocity correction versus the current den-\nsity (normalized the avegare value of ∆ d,∥), where the G/αD\nratio is highlighted (red dashed line). The coordinate relation\nbetween different vectors are shown in the inset. (d) the drift\nvelocity correction as a function of the damping parameter α\natjs= 2×1011A/m2(only vd,xis used for the SkX). For (a)\nto (c), α= 0.2 and β= 0.5αare employed in the simulations.\nan energy vd0. Recall that the density of states of\nfree fermion N(E)∝Ed−2\n2at energy E. Hence, it is\nexpected that the correction follows the same scaling:\n∆vd∝(vd0)d−2\n2according to this argument. We em-\nphasize that the microscopic nature of the density waves\nin this argument are not essential, which mainly stems\nfrom the long-wave characteristic of elastic modes. This\nexplains why the HL and charge density wave also follow\nthe same scaling behavior.\nMicromagnetic simulation.— We now further validate\nour theory through solving the Landau–Lifshitz–Gilbert\n(LLG) equation with the spin transfer torque effect [35–\n39] (for details see SM). The calculated drift velocity vd\nversus current density jscurves are shown in Fig. 3 for\nboth the SkX and HL. For simplicity, we mainly focus\non 2D SkX and HL with weak impurities here, where\nour analytical expressions from perturbation theory are\napplicable.\nFigs. 3(a) and (c) show vdin the clean and disordered\ncase with α= 0.2. The correction between these two\ncases at both SkX and HL is indeed insensitive to the\ncurrent density within the flow limit. It is noteworthy\nthat due to the gyrodynamcs, the SkX exhibits a much\nsmaller depinning critical current density. Fig. 3(b) is to\nshow that the correction along the transverse direction is\nobviously larger than the longitudinal one with the ratio5\n∼G/αD , being consistent with Eq. (7). Interestingly,\nthe longitudinal correction versus the damping parame-\nterαof 2D SkX and HL show a positive and negative\ncorrelation, respectively [Fig. 3(d)], which is also consis-\ntent with our analytical expressions [see χSkX\ndandχHL\nd\nin Eqs. (7) and (10)]. It can also be seen that the im-\npurity correction along longitudinal direction is typically\nmuch stronger in HL than in SkX as χSkX\nd/χHL\nd∼α2.\nThese distinct features between SkX and HL highlight\nthe importance of the nontrivial topology in the sliding\ndynamics of density waves.\nDiscussion.— We have provided a thorough analy-\nsis of the sliding dynamics exhibited by the SkX and\nHL phases, highlighting both their similarities and dif-\nferences in terms of density waves sliding with distinct\ntopologies. Our theory could have broader applica-\ntions. For instance, one can explore the relationship be-\ntween the topological Hall effect and the current den-\nsity in the flow region. In the clean limit, the uni-\nversal linear current-velocity relation vd0∝jsimplies\nthat the topological Hall resistivity ρxy, proportional to\n|(vs−vd0)×Be|/|vs|[15], is expected to exhibit a plateau\nin the flow region, as illustrated in Fig. 1(c). In thepresence of impurities, the topological Hall resistivity is\nmodified to ρxy∝ |(vs−vd0−∆vd)×Be|/|vs|. Consid-\nering that ∆ vd∝(vd0)(d−2)/2, we anticipate a modified\nrelation ρxy=a+bj−2+d/2\ns , where aandbremain inde-\npendent of the current magnitude js. The second term,\nbj−2+d/2\ns , represents the correction from impurities. 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Nagaosa, Sci Rep 8, 6328 (2018).1\nSupplementary Material for “ Sliding Dynamics of Current-Driven Skyrmion Crystal\nand Helix in Chiral Magnets ”\nYing-Ming Xie,1Yizhou Liu,1Nato Nagaosa,1\n1RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\nI. THE SLIDING DYNAMICS OF SKYRMION CRYSTAL WITH PINING AND DEFORMATION\nEFFECTS\nA. The skyrmion dynamics and Thiele equation\nFrom the Landau-Lifshitz-Gilbert equation, it was obtained that the current-driven skyrmion dynamics are captured\nby the Thiele equation:\nG×(vs−vd) +D(βvs−αvd) +F= 0. (S1)\nOne can rewrite the equation as\n−G(vsy−vdy) +D(βvsx−αvdx) +Fx= 0, (S2)\nG(vsx−vdx) +D(βvsy−αvdy) +Fy= 0. (S3)\nIn the matrix form:\n\u0012\nG αD\nαD−G\u0013\u0012\nvdx\nvdy\u0013\n=\u0012\nG βD\nβD−G\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nFy\nFx\u0013\n. (S4)\nThen,\n\u0012\nvdx\nvdy\u0013\n=\u0012\nG αD\nαD−G\u0013−1\u0012\nG βD\nβD−G\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nG αD\nαD−G\u0013−1\u0012\nFy\nFx\u0013\n. (S5)\n\u0012\nvdx\nvdy\u0013\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nG αD\nαD−G\u0013\u0012\n0 1\n1 0\u0013\u0012\nFx\nFy\u0013\u0015\n,\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nαD G\n−G αD\u0013\u0012\nFx\nFy\u0013\u0015\n(S6)\nWithout loss of generality, we can choose the current direction to be x-direction: vs= (vs,0). When the pinning\nforce is set to be F= 0, one can solve\nvd∥,0=G2+αβD2\nG2+α2D2vs,vd⊥,0=(α−β)GD\nG2+α2D2ˆz×vs. (S7)\nTherefore, the longitudinal drift velocity vdxis proportional to the electric current when the force Fis neglectable.\nIn a general direction, we can write the intrinsic drift velocity as\nvd0=1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n(S8)\n=s\n1 +β2γ2\n1 +α2γ2\u0012\ncosθSkH−sinθSkH\nsinθSkH cosθSkH\u0013\u0012\nvscosθs\nvssinθs\u0013\n(S9)\n=vd0\u0012\ncos(θs+θSkH)\nsin(θs+θSkH)\u0013\n. (S10)\nHere, the angle θsis to characterize the applied current direction, the skyrmion Hall angle θSkH= atanγ(α−β)\n1+αβγ2with\nγ=D\nG, and the magnitude of drift velocity vd0=|vd0|=vsq\n1+β2γ2\n1+α2γ2.2\nNow we show that the Thiele equation respects rotational symmetry with the principal axis along z-direction from\nEq. (S6). The rotational operator is defined as Rz=\u0012\ncosϕ−sin(ϕ)\nsin(ϕ) cos( ϕ)\u0013\nwith ϕas the rotational angle. Under this\nrotational operation, Eq. (S6) becomes\nRz\u0012\nvdx\nvdy\u0013\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nG αD\nαD−G\u0013\u0012\n0 1\n1 0\u0013\u0012\nFx\nFy\u0013\u0015\n,\n=1\nG2+α2D2\u0014\nRz\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\nR−1\nzRz\u0012\nvsx\nvsy\u0013\n+Rz\u0012\nαD G\n−G αD\u0013\nR−1\nzRz\u0012\nFx\nFy\u0013\u0015\n(S11)\nIt is easy to show\nRz\u0012\nA B\n−B A\u0013\nR−1\nz=\u0012\nA B\n−B A\u0013\n(S12)\nwith AandBas constant. The Eq. (S13) is simplified as\nRz\u0012\nvdx\nvdy\u0013\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nG αD\nαD−G\u0013\u0012\n0 1\n1 0\u0013\u0012\nFx\nFy\u0013\u0015\n,\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\nRz\u0012\nvsx\nvsy\u0013\n+\u0012\nαD G\n−G αD\u0013\nRz\u0012\nFx\nFy\u0013\u0015\n(S13)\nHence, we have shown that the Thiele equation respects out-of-plane rotational symmetry.\nB. The correction on the drifted velocity due to the pining and deformation effects\nLet us define the displacement of skyrmion lattice as u(r, t) so that the drift velocity vd(r, t) =∂u(r,t)\n∂t. The force\nis given by\nF(r, t) = Fimp+Fde (S14)\nFimp =−X\ni∇U(r+u(r, t)−ri)ρ(r) =fimp(r+u(r, t))ρ(r) (S15)\nFde=Z\ndr′D(r−r′)u(r′, t′), (S16)\nwhere Fimdescribes the pining effect from impurities and Fdearises from the deformation of skyrmion lattice, U(r−ri)\nis the impurity potential around the site ri.ρ(r) is the skyrmion densities.\n∂u(r, t)\n∂t=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nαD G\n−G αD\u0013Z\ndr′D(r−r′)\u0012\nux(r′, t)\nuy(r′, t)\u0013\u0015\n+1\nG2+α2D2\u0012\nαD G\n−G αD\u0013\nfimp(r+u(r, t))ρ(r). (S17)\nThe displacement vector can be expanded around the uniform motion,\nu(r, t) =vdt+˜u(r, t). (S18)\nHere, vdis the dominant uniform skyrmion motion velocity, ˜u(r, t) characterizes a small non-uniform motion. Then\nthe equation of motion is written as\n\u0014∂\n∂t−1\nG2+α2D2\u0012\nαD G\n−G αD\u0013Z\ndr′D(r−r′)\u0015\u0012\n˜ux(r′, t)\n˜uy(r′, t)\u0013\n=1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\n+1\nG2+α2D2\u0012\nαD G\n−G αD\u0013\nfimp(r+vdt+˜u(r, t))ρ(r).\n(S19)3\nHere, we have used D(q) = 0 in the long wave limit ( q→0) so thatR\ndr′D(r−r′) = 0.\nLet us try to solve the Green’s function of the operator at the left-hand side, which is given by\n\u0014∂\n∂t−1\nG2+α2D2\u0012\nαD G\n−G αD\u0013Z\ndr′D(r−r′)\u0015\nG(r′, t) =δ(t)δ(r)\u0012\n1 0\n0 1\u0013\n(S20)\nIt is more economical to work in the momentum space with\nG(r, t) =Zdω\n2πZddk\n(2π)de−iωt+ik·rG(k, ω). (S21)\nLet us define D(r) =Rddq\n(2π)deiq·rD(q), and then\nZ\ndr′Z\nD(r−r′)G(r′, t) =Z\ndr′Zddq\n(2π)deiq·(r−r′)D(q)Zddk\n(2π)dZdω\n2πG(k, ω)ei(k·r′−ωt)\n=Zddk\n(2π)dZdω\n(2π)G(k, ω)D(k)ei(k·r−ωt). (S22)\nIn the momentum space, we find\nG−1(k, ω) =−iω−1\nG2+ (Dα)2\u0012\nαD G\n−G αD\u0013\nD(k) (S23)\nTherefore, Eq. (S19) can be rewritten as\n˜u(r, t) =Z\ndr′Z\ndt′G(r−r′, t−t′){1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd (S24)\n+1\nG2+α2D2\u0012\nαD G\n−G αD\u0013\nfimp(r′+vdt′+˜u(r′, t′))ρ(r′)}.\nIn the flow limit, the perturbation from the deformation and impurity can be regarded as small in comparison with\nthe leading order term. As a result, the displacement vector can be expanded as\n˜u0(r, t) =Z\ndr′Z\ndt′G(r−r′, t−t′)\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n, (S25)\n˜u1(r, t) =1\nG2+α2D2Z\ndr′Z\ndt′G(r−r′, t−t′)\u0012\nαD G\n−G αD\u0013\nfimp(r′+vdt′)ρ(r′), (S26)\n˜u2(r, t) =1\nG2+α2D2Z\ndr′Z\ndt′G(r−r′, t−t′)\u0012\nαD G\n−G αD\u0013\n∇fimp(r′+vdt′)·˜u1(r′, t′)ρ(r′).(S27)\nHere, u0,u1, and u2are the leading, first, and second order terms, respectively. Next, let us evaluate the volume-\naverage velocity\n\u001c∂˜u(r, t)\n∂t\u001d\n=\u001c∂˜u0(r, t)\n∂t\u001d\n+\u001c∂˜u2(r, t)\n∂t\u001d\n(S28)\nNote the fact that under the impurity average fimp(r′+vt′) = 0 has been used so that u1(r, t) would not contribute\ndirectly. Since non-uniform motion must vanish over the volume average, we can obtain a self-consistent equation for\nthe velocity vd. Next, let us work out the self-consistent equation for vd.\nThe leading order\n∂u0(r, t)\n∂t=Z\ndr′Z\ndt′G(r−r′, t−t′)\n∂t\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n=Z\ndr′Z\ndt′Zddk\n(2π)dZdω\n2πeik·(r−r′)−iω(t−t′)(−iω)G(k, ω)\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n= lim\nω→0−iωG(k= 0, ω)\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n=\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n(S29)4\nAs mentioned theD\n∂˜u1(r,t)\n∂tE\nwould not contribute, now let us show it explicitly. Recall that\n˜u1(r, t) =1\nG2+α2D2Z\ndr′Z\ndt′G(r−r′, t−t′)\u0012\nαD G\n−G αD\u0013\nfimp(r′+vdt′)ρ(r′) (S30)\nThen,\n∂˜u1(r, t)\n∂t=Z\ndr′Z\ndt′ 1\nG2+α2D2G(r−r′, t−t′)\u0012\nαD G\n−G αD\u0013\nfimp(r′+vdt′)ρ(r′) (S31)\n=−1\nG2+α2D2Z\ndr′Z\ndt′Zddk\n(2π)dZdω\n2πeik·(r−r′)−iω(t−t′)G(k, ω)\u0012\nαD G\n−G αD\u0013\n× (S32)\nZddq\n(2π)d\u0012\niqx\niqy\u0013\nUqeiq·(r′+vdt′)ρ(r′) (S33)\n= 0, (S34)\nbecause after averaging over the impurity configurations, Uk= 0.\nNow let us look at the second-order term\n∂˜u2(r, t)\n∂t=1\nG2+α2D2Z\ndr′Z\ndt′∂G(r−r′, t−t′)\n∂t\u0012\nαD G\n−G αD\u0013\n∇fimp(r′+vdt′)·˜u1(r′, t′)ρ(r′).(S35)\nNote that\n∇fimp(r′+vdt′)·˜u1(r′, t′) =−Zddq\n(2π)dU(q)eiq·(r′+vdt′)\u0012q2\nxqxqy\nqxqyq2\ny\u0013\u0012\n˜u1x(r′, t′)\n˜u1y(r′, t′)\u0013\n(S36)\nSubstitute the form of ˜u1(r′, t),\n∂˜u2(r, t)\n∂t=−1\n(G2+α2D2)2Z\ndr′Z\ndt′∂G(r−r′, t−t′)\n∂t\u0012\nαD G\n−G αD\u0013\nρ(r′)×\nZddq1\n(2π)dU(q1)eiq1(·r′+vdt′)\u0012q2\n1xq1xq1y\nq1xq1yq2\n1y\u0013\n×\nZ\ndt′′Z\ndr′′G(r′−r′′, t′−t′′)\u0012\nαD G\n−G αD\u0013\nρ(r′′)Zddq2\n(2π)d\u0012\niq2x\niq2y\u0013\nU(q2)eiq2·(r′′+vdt′′)(S37)\nThen write the terms at the right-hand side of the equation with their Fourier components,\n∂˜u2(r, t)\n∂t=−1\n(G2+α2D2)2Z\ndt′Z\ndr′Zddk\n(2π)dZdω\n2πG(k, ω)(−iω)eik·(r−r′)−iω(t−t′)\u0012\nαD G\n−G αD\u0013\n×\nZddq1\n(2π)d\u0012q2\n1xq1xq1y\nq1xq1yq2\n1y\u0013\nU(q1)eiq1·(r′+vdt′)X\ng1ρ(g1)eig1·r′×\nZ\ndt′′Z\ndr′′Zddk′\n(2π)dZdω′\n2πG(k′, ω′)eik′·(r′−r′′)−iω′(t′−t′′)\u0012\nαD G\n−G αD\u0013X\ng2ρ(g2)eig2·r′′×\nZddq2\n(2π)d\u0012\niq2x\niq2y\u0013\nU(q2)eiq2·(r′′+vdt′′)(S38)\nWe can take integrals with respect to the space and time, and take the average over disorders, several delta functions5\nwould appear on the right-hand side:\nU(q1)U(q2) = (2 π)dΛ(q2)δ(q1+q2), (S39)Z\ndt′eiωt′eiq1·vdt′e−iω′t′= 2πδ(ω−ω′+q1·vd), (S40)\nZ\ndr′e−ik·r′eiq1·r′eig1·r′eik′·r′= (2π)dδ(k′−k+q1+g1), (S41)\nZ\ndt′′eiω′t′′eiq2·vdt′′= 2πδ(ω′+q2·vd), (S42)\nZ\ndr′′e−ik′·r′′eig2·r′′eiq2·r′′= (2π)dδ(g2+q2−k′). (S43)\nTake the volume average, and consider constraints from delta functions: q2=−q1=q,g1=−g2=g,ω′=−q·vd,\nwe find\n\u001c∂˜u2(r, t)\n∂t\u001d\n=1\n(G2+α2D2)2X\ngZddq\n(2π)d|ρ(g)|2Λ(q)\u0012\nαD G\n−G αD\u0013\u0012q2\nxqxqy\nqxqyq2\ny\u0013\n×\nIm [G(q−g,−q·vd)]\u0012\nαD G\n−G αD\u0013\u0012\nqx\nqy\u0013\n. (S44)\nTherefore,\n\u001c∂˜u(r, t)\n∂t\u001d\n=\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n+\n1\n(G2+α2D2)2X\ngZddq\n(2π)d|ρ(g)|2Λ(q)\u0012\nG αD\nαD−G\u0013\u0012q2\nxqxqy\nqxqyq2\ny\u0013\n×\nIm [G(q−g,−q·vd)]\u0012\nG αD\nαD−G\u0013\u0012\nqx\nqy\u0013\n(S45)\nSetD\n∂˜u(r,t)\n∂tE\n= 0, the self-consistent equation for the velocity is\nvd=1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\n1\n(G2+α2D2)2X\ngZddq\n(2π)d|ρ(g)|2Λ(q)\u0012\nαD G\n−G αD\u0013\u0012q2\nxqxqy\nqxqyq2\ny\u0013\n×\nIm [G(q−g,−q·vd)]\u0012\nαD G\n−G αD\u0013\u0012\nqx\nqy\u0013\n. (S46)\nAs argued in the main text, the largest imaginary part is contributed by k=q−gin long wave limit ( kis small).\nTo further proceed, let us evaluate Im[ G(k, ω)].\nG(k, ω) =1\n−iω−1\nG2+(Dα)2\u0012\nαD G\n−G αD\u0013\nD(k)=−G2+α2D2\nλ(k)iω−D(k)\nλ(k)\u0012\nαD−G\nG αD\u0013\n, (S47)\nwhere\nλ(k) = [D(k) +ω(iαD+G)][D(k) +ω(iαD−G)] =D(k)2−(G2+α2D2)ω2+ 2iαDωD(k). (S48)\nThe imaginary part of Green’s function is given by\nIm[G(k, ω)] = −(G2+α2D2)[D2(k)−(G2+α2D2)ω2]ω\n[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)\n+2αDωD2(k)\n[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)\u0012\nαD−G\nG αD\u0013\n. (S49)6\nThe largest imaginary part is given by the real mode ωk=D(k)/√\nG2+α2D2. As a result, the first term in Im[ G(k, ω)]\ncan be negligible. In 2D, we can expand\nD(k) =Kxk2\nx+Kyk2\ny. (S50)\nNow we can show that\nZd2k\n(2π)22αDωD2(k)\n[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)\n= (KxKy)−1/2Z+∞\n0dk′2πk′\n(2π)22αDωk′4\n(k′4−(G2+α2D2)ω2)2+ 4α2D2ω2k′4\n=(KxKy)−1/2sgn(ω)\n4. (S51)\nwhere the integralR+∞\n0dtt2\n(t2−√\nG2+α2D2\n2αD)2+t2=π\n2is used with t=k′2\n2αD|ω|. Note that αD̸= 0 is taken. The\nmultiplications between matrices give\n\u0012\nαD G\n−G αD\u0013\u0012q2\nxqxqy\nqxqyq2\ny\u0013\u0012\nαD−G\nG αD\u0013\u0012\nαD G\n−G αD\u0013\u0012\nqx\nqy\u0013\n= (G2+α2D2)(q2\nx+q2\ny)\u0012\nGqy+αDq x\n−Gqx+αDq y\u0013\n.(S52)\nFinally, we obtain\nδvd=(KxKy)−1/2\n4(G2+α2D2)X\ng|ρ(g)|2Λ(g)sgn(−g·vd0)|g|2\u0012\nGgy+αDg x\n−Ggx+αDg y\u0013\n(S53)\nWe have shown that δvdrespects out-of-plane rotational symmetry in the main text, which is inherited from the\nTheiele equation. Without loss of generality, let us set vd0to be along x-direction. In this case, after summing over\nthe six smallest gvectors: gj=√\n3κ0(sin(j−1)π\n3,cos(j−1)π\n3) with κ0=4π\n3a,jare integers from 1 to 6, we find\nδvd=9κ3\n0|ρ1|2Λ0αD\n4p\nKxKy(G2+α2D2)\u0012−1\nG\nαD\u0013\n(S54)\nwhere ρ1=ρ(gj),Λ0= Λ(gj). In the 3D case,\nD(k) =Kxk2\nx+Kyk2\ny+Kzk2\nz. (S55)\nThen,\nZd3k\n(2π)32αDωD2(k)\n[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)\n= (KxKyKz)−1/2Z+∞\n04πk′2dk′\n(2π)32αDωk′4\n(k′4−(G2+α2D2)ω2)2+ 4α2D2ω2k′4\n=(KxKyKz)−1/2Γ(G2+α2D2\n4α2D2)sgn( ω)p\n2αD|ω|\n2π2. (S56)\nwhere Γ( a) =R+∞\n0dxx6\n(x4−a)2+x4. Similarly, we can obtain\nδvd=(2αD)1/2(KxKyKz)−1/2Γ(G2+α2D2\n4α2D2)\n2π2(G2+α2D2)X\ng|ρ(g)|2Λ(g)sgn(−g·vd0)p\n|g·vd0||g|2\u0012\nGgy+αDg x\n−Ggx+αDg y\u0013\n.(S57)\nAfter summing over the six smallest gvectors, we find\nδvd=9√\n3κ7/2\n0Γ(G2+α2D2\n4α2D2)|ρ1|2Λ0αD√αDv d0\nπ2p\nKxKyKz(G2+α2D2)\u0012−1\nG\nαD\u0013\n. (S58)7\nII. HELICAL SPIN ORDER CASE\nIn this section, we consider the helical spin order case. Without loss of generality, we denote the helical spin order\nhas a variation along x-direction. The Thiele equation for the helical spin order would be\nD(βvs−αvd) +F= 0. (S59)\nFor simplicity, we have omitted the index xin the following. The equation of motion becomes\n∂u(r, t)\n∂t=β\nαvs+1\nαDZ\ndr′D(r−r′)u(r′, t′) +1\nαDfimp(r+u(r, t))ρ(r). (S60)\nSimilarly, by defining u(r, r) =vdt+ ˜u(r, t), the equation of motion can be rewritten as\n[∂\n∂t−1\nαDZ\ndr′D(r−r′)]˜u(r′, t) =β\nαvs+1\nαDfimp(r+u(r, t))ρ(r). (S61)\nLet us define the Green’s function G(r, t) so that\n[∂\n∂t−1\nαDZ\ndr′D(r−r′)]˜u(r′, t)G(r′, t) =δ(r)δ(t). (S62)\nIt is easy to obtain\nG−1(k, ω) =−iω−D(k)\nαD. (S63)\nThe displacement ˜ u(r, t) is given by\n˜u(r, t) =Z\ndr′Z\ndt′G(r−r′, t−t′)[β\nαvs−vd+1\nαDfimp(r′+u(r′, t′))ρ(r′)]. (S64)\nThen up to the second order,\n˜u0(r, t) =Z\ndr′dt′G(r−r′, t−t′)[β\nαvs−vd], (S65)\n˜u1(r, t) =1\nαDZ\ndr′Z\ndt′G(r−r′, t−t′)fimp(r′+vdt′)ρ(r′), (S66)\n˜u2(r, t) =1\nαDZ\ndr′Z\ndt′G(r−r′, t−t′)∂xfimp(r′+vdt′)˜u1(r′, t′)ρ(r′). (S67)\nFollowing a similar procedure in Sec. II, using ⟨∂u(r,t)\n∂t⟩= 0, we obtain\nvd=vd0+1\nα2D2X\ng|ρ(g)|2Zddq\n(2π)dΛ(q)q3\nxIm[G(q−g,−qxvd)]. (S68)\nwhere the intrinsic drift velocity vd0=β\nαvs. Consider the Im[ G(k, ω)] is dominant in the long wave limit ( k→0) and\nexpand D(k) =Kxk2\nx+Kyk2\nyind= 2, we find\nvd≈vd0−(KxKy)−1/2\n8αDX\ng|ρ(g)|2Λ(g)|gx|3(S69)\nand similarly in d= 3 case, D(k) =Kxk2\nx+Kyk2\ny+Kzk2\nz, we obtain\nvd≈vd0−(KxKyKz)−1/2\n4√\n2π(αD)1/2X\ng|ρ(g)|2Λ(g)|gx|3p\n|gxvd0|. (S70)\nAfter summing over the smallest reciprocal lattice vectors for the helix: g= (±1,0)g0with g0=π\na, the correction\n∆vd=β\nαvs−χHL\nd(vd0)d−2\n2 (S71)\nwhere\nχHL\nd=\n\n(KxKy)−1/2|ρ1|2Λ0g3\n0\n4αD, ford= 2\n(KxKyKz)−1/2|ρ1|2Λ0g7/2\n0\n2√\n2π(αD)1/2 ,ford= 3(S72)\nHere, we have set ρ(g0) =ρ1and Λ( g0) = Λ 0in this case.8\nIII. NUMERICAL METHOD DETAILS\nA. Details for the main text Fig.2\nThe main text Fig.2 is obtained from the main text Eq. (6) and (9) by numerically integrating qand summing over\nthe smallest reciprocal lattice gvectors. For the SkX, the 4000 ×4000 in-plane momentum grids of qare taken with\na hexagonal boundary (the boundary length is 4 κ0); while for the HL, the 1000 ×1000 in-plane momentum grids are\ntaken with a square boundary (the boundary length is 4 g0). In the 3D case, the 1000 out-of-plane momentum points\nofqwithin [ −2g0,2g0] are used for both SkX and HL in evaluating the integral. Also, we set the elastic coefficients\nKj= 10000, the lattice constant aas a natural unit of one, the damping parameter α= 0.04, the dissipative coefficient\nD= 5.577π, the additional parameter G=±4πfor the SkX.\nB. Micromagnetic simulations\nThe micromagnetic simulations were performed using MuMax3 [35]. The Landau-Lifshitz-Gilbert (LLG) equation\nis numerically solved\n˙s=−|γ0|s×Heff+αs×˙s+pa3\n2eS(js· ∇)s−pa3β\n2eM2ss×(js· ∇)s, (S73)\nwhere sis the unit vector of spin, γ0is the gyromagnetic constant, αis the Gilbert damping constant, Msis the\nsaturation magnetization, and Heff=−1\nµ0MsδH\nδSis the effective field. The spin transfer torque effect of the current\nis described by the last two terms [36–38]. βdescribes the non-adiabaticity of the spin transfer torque effect. The\ncurrent is applied along the x-direction in the simulations.\nA typical chiral magnet can be described by the following Hamiltonian density\nH=A(∇S)2−DS·(∇ ×S)−µ0MsB·S (S74)\nThe corresponding parameters and their values employed in the simulations are: the saturation magnetization\nMs= 111 kA/m, the exchange stiffness A= 3.645 pJ/m, and the Dzyaloshinskii-Moriya interaction strength\nD= 0.567 mJ/m2. An external magnetic field B= 0.3 T (with its direction perpendicular to the skyrmion plane) is\nused in the simulations to stabilize the skyrmions. The simulations for helical state are performed at zero-field. The\ncell size is 1 nm ×1 nm×1 nm.\nWe consider magnetic impurities with uniaxial magnetic anisotropy Himp=−KimpS2\nz, where the easy-axis is\nperpendicular to the skyrmion 2D plane. For the weak impurity case, an impurity concentration x= 0.1% and\nimpurity strength Kimp= 0.2A/l2(lis the cell size) were used. For the strong impurity case, an impurity concentration\nx= 0.5% and impurity strength Kimp= 0.6A/l2were used. The simulation results are averaged over 100 impurity\ndistributions. The skyrmion velocity is extracted by using the emergent electric field method [39]. For each current\ndensity, the emergent electric field is also averaged over 100 time steps in order to get the skyrmion velocity. For the\ntransverse correction, the damping value α= 0.2 is employed in the main text Figs. 3(a) and (b) for computational\nefficiency, as using smaller damping values results in significantly longer simulation times to obtain a reasonable\ncorrection along the transverse direction."    },    {        "title": "2312.08925v1.Smoluchowski_Kramers_diffusion_approximation_for_systems_of_stochastic_damped_wave_equations_with_non_constant_friction.pdf",        "content": "arXiv:2312.08925v1  [math.PR]  14 Dec 2023SMOLUCHOWSKI-KRAMERS DIFFUSION APPROXIMATION FOR SYSTEMS\nOF STOCHASTIC DAMPED WAVE EQUATIONS WITH NON-CONSTANT\nFRICTION\nSANDRA CERRAI AND ARNAUD DEBUSSCHE\nAbstract. We consider systems of damped wave equations with a state-de pendent damping coeffi-\ncientandperturbedbyaGaussian multiplicativenoise. Ini tially, weinvestigate theirwell-posedness,\nunder quite general conditions on the friction. Subsequent ly, we study the validity of the so-called\nSmoluchowski-Kramers diffusion approximation. We show tha t, under more stringent conditions\non the friction, in the small-mass limit the solution of the s ystem of stochastic damped wave equa-\ntions converge to the solution of a system of stochastic quas i-linear parabolic equations. In this\nconvergence, an additional drift emerges as a result of the i nteraction between the noise and the\nstate-dependent friction. The identification of this limit is achieved by using a suitable generaliza-\ntion of the classical method of perturbed test functions, ta ilored to the current infinite dimensional\nsetting.\nContents\n1. Introduction 2\n2. Notations and assumptions 7\n3. Statement of main results 13\n4. Proof of Theorem 3.1 15\n4.1. Proof of Proposition 4.1 16\n4.2. Proof of Proposition 4.2 19\n5. Uniform estimates in H1 23\n6. Uniform estimates in H2 29\n6.1. Proof of the H2-bounds in the case of bounded σ 30\n6.2. Proof of the H2-bounds in the case of unbounded σ 38\n7. Uniform bounds in H3 40\n8. Tightness 45\n9. About the limiting equation (3.12) and its truncated vers ion 48\n9.1. Properties of the mapping S 48\n9.2. Pathwise uniqueness of equation (3.12) 50\n10. Proof of Theorem 3.3 for the truncated equation (6.3) 52\n10.1. The construction of the corrector functions 52\n10.2. The identification of the limit 56\n10.3. Conclusion 58\n11. Proof of Theorem 3.3 61\nS. Cerrai was partially supported by the NSF grant DMS-19542 99 -Multiscale analysis of infinite-dimensional\nstochastic systems .\nA. Debussche benefits from the support of the French governme nt “Investissements d’Avenir” program integrated\nto France 2030, bearing the following reference ANR-11-LAB X-0020-01 and is partially funded by the ANR project\nADA.\n1References 63\n1.Introduction\nIn this paper, we are dealing with the following system of sto chastic non-linear wave equations,\ndefined on a boundeddomain O⊂Rdhaving a smooth boundaryand subjectto Dirichlet boundary\nconditions\n\nµ∂2\ntuµ(t) = ∆uµ(t)−γ(uµ(t))∂tuµ(t)+f(uµ(t))+σ(uµ(t))∂twQ\nt(t), t≥0,\nuµ(t,x) = 0, t≥0, x∈∂O,\nuµ(0,x) =uµ\n0(x), ∂tuµ(0,x) =vµ\n0(x), x∈O.(1.1)\nHereuµ(t,x) = (u1,µ(t,x),u2,µ(t,x),···,ur,µ(t,x)), for (t,x)∈[0,+∞)×O, and for some r≥2.\nThe stochastic perturbation wQ(t),t≥0, is given by a cylindrical Wiener process taking values\ninRr, which is white in time and colored in space, with reproducin g kernelHQ. The function γ\nmaps the space H1:=H1\n0(O;Rr) into the space of bounded linear operators on H:=L2(O;Rr),\nand satisfies suitable regularity properties, while f:H→Handσ:H1→L2(HQ,H1) are some\nLipschitz nonlinearities.\nAfter proving that, for every fixed µ>0 and every initial conditions ( uµ\n0,vµ\n0)∈H2:=H2(O)×\nH1\n0(O), system(1.1)hasauniquemildsolution uµ= (uµ,∂tuµ)∈L2(Ω;C([0,T];H2)), ourpurpose\nwill be studying the asymptotic behavior of the solution uµ, as the mass parameter µapproaches\nzero. Namely, we will prove that as µ↓0 the process uµconverges, in an appropriate sense, to\nthe solution of a system of stochastic quasi-linear parabol ic equations. Such limit is known in the\nliterature as the Smoluchowski-Kramers diffusion-approxim ation.\nWhilethispaperisthefirstonetoaddressthecaseof systems ofSPDEshavingastate-dependent\nfriction, a series of prior works have explored the validity of the Smoluchowski-Kramers approxi-\nmation. In the context of finite-dimensional systems, there exists a substantial body of literature.\nA simplified model of the Smoluchowski-Kramers phenomenon f or stochastic differential equations\nwas already investigated by Nelson in Chapters 9 and 10 of his book [38], and, more recently, it\nwas studied in the following references: [23], [24], [29], [ 31], [41], [51] and [52]. Additionally, in [6],\n[12], [25], and [34] an analogous problem was considered for systems subject to a magnetic field,\nand in [32], [39] and [48] related multi-scaling problems we re investigated.\nIn recent years, there has been a considerable research acti vity related to the Smoluchowski-\nKramers diffusion approximation for infinite-dimensional sy stems. The first results in this direction\ndealt with the case of constant damping term, with smooth noi se and regular coefficients (see [4],\n[5], [46], and [35]). More recently, the case of constant fri ction has been studied in [26] and [53] for\nequations perturbed by space-time white noise in dimension d= 2, and in [28] for equations with\nH¨ older continuous coefficients in dimension d= 1. In all these papers, the fact that the damping\ncoefficient is constant leads to a perturbative result, in the sense that, in the small-mass limit,\nthe solution uµof the stochastic damped wave equation converges to the solu tion of the stochastic\nparabolic problem obtained by taking µ= 0. The case of SPDEs with state-dependent damping\nwas considered for the first time in [13] (see also [14]). Nota bly, this scenario differs from the\nprevious one, as the non-constant friction leads to a noise- induced term in the small-mass limit.\nAn analogous phenomenon was identified in [2], where the case of SPDEs constrained to live on\nthe unitary sphere of the space of square-integrable functi onsL2(O) was considered. Actually,\n2also in this case the Smoluchowski-Kramers approximation l eads to a stochastic parabolic problem,\nwhose solution is constrained to live on the unitary L2(O)-sphere and where an additional drift\nterm appears. Somehow surprisingly, such extra drift does n ot account for the Stratonovich-to-Itˆ o\ncorrection term. Finally, in [11] systems subject to a magne tic field were studied and in this case\nthe small-mass limit was obtained only after a suitable regu larization of the problem and in the\nlimit a stochastic hyperbolic equation was obtained.\nThe study of the Smoluchowski-Kramers approximation exten ds beyond just proving the limit\nof the solutions uµ. In several applications, it is important to understand the stability of this\napproximation with respect to other significant asymptotic features exhibited by the two systems.\nTo this end, in [8],[4], and more recently in [40] and [15], it is demonstrated that the statistically\ninvariant states of the stochastic damped wave equation (in cases of constant friction) converge in\na suitable sense to the invariant measure of the limiting equ ation. In a similar vein, the papers\n[9], [10], and [14] focus on analyzing the interplay between the small-mass and small-noise limits.\nIn particular, [14] investigates the validity of a large dev iation principle for the trajectories of the\nsolution, while [9] and [10] delve into the study of the conve rgence of the quasi-potential, which\ndescribes, as known, the asymptotics of the exit times of the solutions from a functional domain\nand the large deviation principle for the invariant measure .\nAs wementioned above, thepresent paperis thefirstone todea l with thestudyof thesmall-mass\nlimit for systems of stochastic damped wave equations havin g a state-dependent friction coefficient.\nIn [13], the same problem was considered for one single equat ion. In that case, it has been shown\nthat ifγis inC1(R) and 0< γ0≤γ(s)≤γ1, for every s∈R, then for any initial condition\n(u0,v0)∈H1(O)×L2(O) and for any δ>0 andp<∞\nlim\nµ→0P/parenleftBig\n/ba∇dbluµ−u/ba∇dblC([0,T];H−δ(O))+/ba∇dbluµ−u/ba∇dblLp(O)>η/parenrightBig\n= 0, η>0,\nwhereuis the unique solution of the quasilinear stochastic parabo lic equation\n\n\n∂tu=1\nγ(u)∆u+f(u)\nγ(u)−γ′(u)\n2γ3(u)∞/summationdisplay\ni=1(σ(u)Qei)2+σ(u)\nγ(u)∂twQ, t>0, x∈O,\nu(0) =u0, u |∂O= 0.(1.2)\nIn particular, it has been shown how, as a consequence of the i nterplay between the non constant\ndamping and the noise, in the limit an extra drift term is crea ted.\nBeingγa scalar function, for every function u: [0,T]×O→Rit was possible to write\nγ(uµ(t,x))∂tuµ(t,x) =∂tρµ(t,x), t∈[0,T], x∈O, (1.3)\nwhereρµ=λ◦uµandλ′=γ, and it was shown that for every µ >0 the function ρµsolves the\nequation\nρµ(t)+µ∂tuµ(t) =g(u0)+µv0+/integraldisplayt\n0div[b(ρµ(s))∇ρµ(s)]ds\n+/integraldisplayt\n0fλ(ρµ(s))ds+/integraldisplayt\n0σλ(ρµ(s))dwQ(s),\nwhereb= 1/γ◦λ−1,fλ=f◦λ−1, andσλ(h) =σ(λ−1◦h). It turned out that working with\nthis equation, instead of (1.1) (with r= 1), made the proof of the small-mass limit more direct.\nActually, once tightness was proved, it was possible to get t he weak convergence of the sequence\n3{ρµ}µ∈(0,1)to someρthat solves the quasilinear parabolic SPDE\n\n\n∂tρ= div[b(ρ)∇ρ]+F(ρ)+σg(ρ)dwQ(t), t>0, x∈O,\nρ(0,x) =g(u0), ρ(t,x) = 0, x∈∂O.(1.4)\nThen, since it was possible to prove pathwise uniqueness for equation (1.4), the convergence in\nprobability followed from the weak convergence. Finally, a generalized Itˆ o formula allowed to get\nthe convergence of uµto the solution of equation (1.2), where the extra drift appe ars exactly as a\nconsequence of the Itˆ o formula.\nIf the case of systems, in general there is no way to define a new functionρµin such a way that\n(1.3) holds. Because of this all the arguments introduced in [13] and described above cannot be\nused, and the problem becomes much harder to study. Our aim in the present paper is finding\nan alternative approach to the proof of the validity of the Sm oluchowski-Kramers approximation\nthat applies to systems and provides a suitable generalizat ion of (1.2) for the limiting system of\nstochastic quasi-linear parabolic equations.\nAlready the well-posedness of system (1.1) for any fixed µ >0 is not known in the existing\nliterature and requires some work. We assume that γmapsH1intoL(H) and satisfies suitable\nconditions that are verified also in the local case, when d≤3 and\n[γ(h)k](x) =g(x,h(x))k(x), x∈O, (1.5)\nfor some bounded measurable function g:O×Rr→Rr×rsuch thatg(x,·) :Rr→Rr×ris Lipschitz\ncontinuous, uniformlywith respect to x∈O. By usinga generalized splitting methodthat allows to\nhandle the multiplicative noise, together with suitable lo calizations that allows to handle the local\nLipschitz continuity of the coefficients, we prove that for ev ery (uµ\n0,vµ\n0)∈H2and everyT >0 and\np≥1, there exists a uniquemild solution uµfor equation (1.1) such that uµ∈L2(Ω;C([0,T];H2)).\nOnce we have proven the well-posedness of system (1.1), we st udy the small-mass limit. The first\nfundamental step is proving a-priori bounds for uµand√µ∂tuµ. Due to the nature of the problem,\nwe need uniform bounds in H2and some other bounds in H3:=H3(O)×H2(O). The proofs\nof such bounds are quite challenging and intricate and requi re some more restrictive assumptions\nfor the friction coefficient γ. In particular, we cannot consider the local case (1.5), as w e need to\nassume that\n[γ(h)k](x) =g(h)k(x), x∈O,\nfor some differentiable mapping g:H1→Rr×r, which is bounded together with its derivative, and\nsuch that\ninf\nh∈H1/an}b∇acketle{tg(h)ξ,ξ/an}b∇acket∇i}htRr≥γ0/ba∇dblξ/ba∇dbl2\nRr, ξ∈Rr,\nfor someγ0>0.\nA fundamental consequence of such a-priori bounds is repres ented by the tightness of the family\n{uµ}µ∈(0,1)in suitable functional spaces, which, in turn, allows to get the existence of a weak\nlimit point for the family {uµ}µ∈(0,1)in the same functional spaces. The final crucial step is\nidentifying every possible limit point of {uµ}µ∈(0,1)with the solution of the same appropriate\nsystem of stochastic parabolic equations. To this purpose, for every fixed u,v∈H1we introduce\nthe problem\ndy(t) =−g(u)y(t)dt+σ(u)dwQ(t), y(0) =v, (1.6)\n4and we denote by yu,vits solution. Moreover, we denote by Pu\ntthe Markov transition semigroup\nassociated with equation (1.6), and by νuits unique invariant measure. This allows defining\nS(u) :=/integraldisplay\nH1/bracketleftbig\nDg−1(u)z/bracketrightbig\nzdνu(z), u∈H1,\nand introducing the problem\n\n\n∂tu(t,x) =g−1(u(t))∆u(t,x)+g−1(u(t))f(u(t,x))+S(u(t))+g−1(u(t))σ(u(t))∂twQ(t),\nu(0,x) =u0(x), x∈O, u(t,x) = 0, x∈∂O.\n(1.7)\nOur purpose is proving that if ( uµ\n0,vµ\n0)∈H3, for everyµ>0, with\nsup\nµ∈(0,1)/ba∇dbl(uµ\n0,√µvµ\n0)/ba∇dblH1<∞,lim\nµ→0µδ/ba∇dbl(uµ\n0,√µvµ\n0)/ba∇dbl2\nH2= 0,lim\nµ→0µ1+δ/ba∇dbl(uµ\n0,√µvµ\n0)/ba∇dbl2\nH3= 0,\nfor someδ∈(0,1/2), then, for any u0∈H1such that\nlim\nµ→0/ba∇dbluµ\n0−u0/ba∇dblH1= 0,\nand for every ̺<1,ϑ<2,p<2/(ϑ−1) andη,T >0 it holds\nlim\nµ→0P/parenleftBigg\nsup\nt∈[0,T]/ba∇dbluµ(t)−u(t)/ba∇dblH̺+/integraldisplayT\n0/ba∇dbluµ(t)−u(t)/ba∇dblp\nHϑdt>η/parenrightBigg\n= 0, (1.8)\nwhereu∈L2(Ω;C([0,T];H1)∩L2(0,T;H2)) is the solution of (1.7).\nAfter showing that S:H1→H1is well-defined and even differentiable, we prove the pathwise\nuniquenessforsystem(1.7)in L2(Ω;C([0,T];H1)∩L2(0,T;H2). Thus,ifwecanshowthatanyweak\nlimit point for the family {uµ}µ∈(0,1)solves (1.7) and belongs to L2(Ω;C([0,T];H1)∩L2(0,T;H2),\nwe can conclude that its limit is uniquely identified as the so lution of (1.7) and (1.8) holds. Our\nproof of the fact that any weak limit point of {uµ}µ∈(0,1)solves (1.7) is based on a generalization of\nthe classical perturbed test functions method first introdu ced in [43] and extended in recent years\nto treat the case of several different types of infinite dimensi onal systems (see e.g. [17], [18], [20],\n[21] and [22]). If we define vµ(t) :=√µ∂tuµ(t), system (1.1) can be written as\n\n\n∂tuµ(t) =1√µvµ(t)\n∂tvµ(t) =1√µ∆uµ(t)−1\nµγ(uµ(t))vµ(t)+1√µf(uµ(t))+1√µσ(uµ(t))∂twQ(t),(1.9)\nwith the initial conditions uµ(0) =uµ\n0andvµ(0) =√µvµ\n0. The Kolmogorov operator associated\nwith the system above is given by\nKµϕ(u,v) =1√µ(/an}b∇acketle{tDuϕ(u,v),v/an}b∇acket∇i}htH1+/an}b∇acketle{tDvϕ(u,v),∆u+f(u)/an}b∇acket∇i}htH1)\n−1\nµ/an}b∇acketle{tDvϕ(u,v),γ(u)v/an}b∇acket∇i}htH1+1\n2µTr/parenleftbig\nD2\nvϕ(u,v)[σ(u)Q][σ(u)Q]⋆/parenrightbig\n.\nIn particular, if we take\nϕµ(u,v) =/an}b∇acketle{tu,h/an}b∇acket∇i}htH+√µϕ1(u,v)+µϕµ\n2(u,v),(u,v)∈H1,\n5we have\nKµϕµ(u,v) =1√µ(Muϕ1(u,v)+/an}b∇acketle{th,v/an}b∇acket∇i}htH1)+Muϕµ\n2(u,v)+/an}b∇acketle{tDuϕ1(u,v),v/an}b∇acket∇i}htH1\n+/an}b∇acketle{tDvϕ1(u,v),∆u+f(u)/an}b∇acket∇i}htH1+√µ(/an}b∇acketle{tDvϕµ\n2(u,v),∆u+f(u)/an}b∇acket∇i}htH1+/an}b∇acketle{tDuϕµ\n2(u,v),v/an}b∇acket∇i}htH1),\nwhereMuis the Kolmogorov operator associated with equation (1.6). If we take ϕ1(u,v) :=\n/an}b∇acketle{tγ−1(u)v,h/an}b∇acket∇i}htHwe have\nMuϕ1(u,v)+/an}b∇acketle{th,v/an}b∇acket∇i}htH1= 0.\nMoreover, if we take some λ(µ)>0 such that λ(µ)↓0, asµ↓0, and define\nϕµ\n2(u,v) :=/integraldisplay∞\n0e−λ(µ)t(Pu\ntψ(u,·)(v)−/an}b∇acketle{tS(u),h/an}b∇acket∇i}htH)dt,\nwhereψ(u,v) :=/an}b∇acketle{tDuϕ1(u,v),v/an}b∇acket∇i}htH1, we get\nMuϕµ\n2(u,v) =λ(µ)ϕµ\n2(u,v)−(ψ(u,v)−/an}b∇acketle{tS(u),h/an}b∇acket∇i}htH).\nTherefore, if we apply the Itˆ o formula to ϕµand (uµ,vµ), we get\n/an}b∇acketle{tuµ(t),h/an}b∇acket∇i}htH=/an}b∇acketle{tuµ\n0,h/an}b∇acket∇i}htH+/integraldisplayt\n0/an}b∇acketle{tγ−1(uµ(s))∆uµ(s)+γ−1(uµ(s))f(uµ(s))+S(uµ(s)),h/an}b∇acket∇i}htHds\n+/integraldisplayt\n0/an}b∇acketle{tγ−1(uµ(s))σ(uµ(s))∂twQ(s),h/an}b∇acket∇i}htH+Rµ(t),\nwhere\nRµ(t) =√µ/an}b∇acketle{tϕ1(uµ\n0,vµ\n0)−ϕ1(uµ(t),vµ(t)),h/an}b∇acket∇i}htH+µ/an}b∇acketle{tϕµ\n2(uµ\n0,vµ\n0)−ϕµ\n2(uµ(t),vµ(t)),h/an}b∇acket∇i}htH\n+/integraldisplayt\n0(λ(µ)ϕµ\n2(uµ(s),vµ(s))+µ/an}b∇acketle{tDuϕµ\n2(uµ(s),vµ(s)),∂tuµ(s)/an}b∇acket∇i}htH1)dt\n+√µ/integraldisplayt\n0/an}b∇acketle{tDvϕµ\n2(uµ(s),vµ(s)),∆uµ(t)+f(uµ(s))/an}b∇acket∇i}htH1ds\n+√µ/integraldisplayt\n0/an}b∇acketle{tDvϕµ\n2(uµ(s),vµ(s)),σ(uµ(s))∂twQ(s)/an}b∇acket∇i}htH.\nOur goal will be proving that under suitable scaling conditi ons onλ(µ) the reminder Rµconverges\nto zero inL1(Ω;C([0,t])), asµ↓0. This will allow us to obtain that any weak limit of the famil y\n{uµ}µ∈(0,1)converges to the solution of (1.7). Notice that this task is f ar from trivial. Actually,\nfirst it requires to prove that the two functions ϕ1andϕµ\n2are sufficiently smooth. Then it requires\nto control the dependence of ϕ1andϕµ\n2and their derivatives on the parameter µand to show that\nthe bounds we find match appropriately with the a priori bound s we have previously found for uµ\nand∂tuµ. Also the reminder depends on derivatives of uµup to order 2. Since, Dvϕµ\n2acts onH1\nwe see that we need bounds on uµinH3. As it turns out, we will not be able to prove all these\nresults for system (1.9) and we will need to introduce a suita ble truncated system. The arguments\nwe have described above will allow us to prove (1.8) for the tr uncated problem and the truncated\nlimiting problem, and only at that point suitable a-priori b oundsfor the truncated limiting problem\nwill allow to obtain (1.8) for the original system.\n62.Notations and assumptions\nLetObe a bounded domain in Rd, withd≥1. In what follows, we shall denote by Hthe Hilbert\nspaceL2(O;Rr), endowed with the scalar product\n/an}b∇acketle{t(h1,...,h r),(k1,...,kr)/an}b∇acket∇i}htH=r/summationdisplay\ni=1/integraldisplay\nOhi(x)ki(x)dx,\nand the corresponding norm /ba∇dbl · /ba∇dblH. Moreover, for every p≥1, we will denote by Lpthe space\nLp(O;Rr).\nIfwedenoteby( ˆL,D(ˆL))therealization oftheLaplaceoperatorin L2(O), endowedwithDirichlet\nboundary conditions, and define\nLh:= (ˆLh1,...,ˆLhr), h∈D(L) =D(ˆL)×···×D(ˆL),\nit is possible to show that there exists a complete orthonorm al system {ei}i∈NinHand a non-\ndecreasing sequence of positive real numbers {αi}i∈Nsuch that\nLei=−αiei, i∈N.\nFor everyδ∈R, we define Hδto be the completion of C∞\n0(O;Rr) with respect to the norm\n/ba∇dblh/ba∇dblHδ=∞/summationdisplay\ni=1αδ\ni/an}b∇acketle{th,ei/an}b∇acket∇i}ht2\nH,\nand we define\nHδ:=Hδ×Hδ−1.\nIn the case δ= 0, we simply set H:=H0.\nThe cylindrical Wiener process wQis defined as the formal sum\nwQ(t) =∞/summationdisplay\ni=1Qeiβi(t), t≥0,\nwhereQ= (Q1,...,Q r) is a boundedlinear operator in L(H),{βi}i∈Nis a sequence of independent\nstandard Brownian motions defined on some filtered probabili ty space (Ω,F,{F}t≥0,P) and{ei}i∈N\nis the orthonormal basis of Hintroduced above.\nIn what follows we shall denote by HQthe setQ(H).HQis the so-called Reproducing Kernel of\nthe noisewQand is a Hilbert space, endowed with the scalar product\n/an}b∇acketle{th,k/an}b∇acket∇i}htHQ=/an}b∇acketle{tQ−1h,Q−1k/an}b∇acket∇i}htH.\nIn particular, the sequence {Qei}i∈Nis a complete orthonormal system for HQ.\nAs for the coefficient γ, we assume the following conditions.\nHypothesis 1. 1.The function γmapsH1intoL(H)with\nsup\nh∈H1/ba∇dblγ(h)/ba∇dblL(H)<+∞. (2.1)\nand\n/ba∇dbl[γ(h1)−γ(h2)]k/ba∇dblH≤c/ba∇dblh1−h2/ba∇dblH1/ba∇dblk/ba∇dblH1, (2.2)\nfor everyh1,h2,k,∈H1.\n2.The function γmapsH2intoL(H1)and for every h∈H2andk∈H1\n/ba∇dblγ(h)k/ba∇dblH1≤c/ba∇dblk/ba∇dblH1+c/ba∇dblh/ba∇dbl3/4\nH2/ba∇dblh/ba∇dbl1/4\nH1/ba∇dblk/ba∇dbl3/4\nH1/ba∇dblk/ba∇dbl1/4\nH. (2.3)\n7Notice that as a consequence of (2.3) we have\n/ba∇dblγ(h)/ba∇dblL(H1)≤c(1+/ba∇dblh/ba∇dblH2), h∈H2. (2.4)\nExample 2.1. For everyh,k:O→Rr, we define\n[γ(h)k](x) =g(x,h(x))k(x), x∈O,\nfor some bounded measurable function g:O×Rr→Rr×r. It is immediate to check that γmapsH1\nintoL(H) and (2.1) holds. Moreover, if we assume that g(x,·) :Rr→Rr×ris Lipschitz continuous,\nwith\n[g]Lip:= sup\nx∈Osup\nξ/ne}ationslash=η/ba∇dblg(x,ξ)−g(x,η)/ba∇dblRr×r\n/ba∇dblξ−η/ba∇dblRr<+∞,\nthen, for every h1,h2,k∈H1we have\n/ba∇dbl[γ(h1)−γ(h2)]k/ba∇dblH≤ /ba∇dblg(·,h1)−g(·,h2)/ba∇dblL4/ba∇dblk/ba∇dblL4≤[g]Lip/ba∇dblh1−h2/ba∇dblL4/ba∇dblk/ba∇dblL4.\nIf we assume that d≤4 we have that H1֒→L4and then we get (2.2).\nFinally, ifg:O×Rr→Rr×ris differentiable, with bounded derivative, for every h∈H2and\nk∈H1, we have\n/ba∇dblγ(h)k/ba∇dblH1≤ /ba∇dblDg(·,h)∇hk/ba∇dblH+/ba∇dblg(·,h)∇k/ba∇dblH≤ /ba∇dblDg/ba∇dbl∞/ba∇dbl∇h/ba∇dblL4/ba∇dblk/ba∇dblL4+c/ba∇dblk/ba∇dblH1.(2.5)\nIf we assume that d≤3, we have that H1֒→L6and by interpolation for every l∈H1we have\n/ba∇dbll/ba∇dblL4≤ /ba∇dbll/ba∇dbl3/4\nL6/ba∇dbll/ba∇dbl1/4\nH≤ /ba∇dbll/ba∇dbl3/4\nH1/ba∇dbll/ba∇dbl1/4\nH.\nThus, from (2.5) we get\n/ba∇dblγ(h)k/ba∇dblH1≤c/ba∇dblh/ba∇dbl3/4\nH2/ba∇dblh/ba∇dbl1/4\nH1/ba∇dblk/ba∇dbl3/4\nH1/ba∇dblk/ba∇dbl1/4\nH+c/ba∇dblk/ba∇dblH1,\nand (2.3) follows.\nExample 2.2. For everyh∈H1andk∈H, we define\n[γ(h)k](x) =g(h)k(x), x∈O,\nfor some bounded and Lipschitz function g:H1→Rr×r. It is easy to check that γsatisfies (2.1),\n(2.2) and (2.3). In fact, in this case (2.2) and (2.3) are cons iderably improved, as we have for every\ns∈R\n/ba∇dbl[γ(h1)−γ(h2)]k/ba∇dblHs≤c/ba∇dblh1−h2/ba∇dblH1/ba∇dblk/ba∇dblHs (2.6)\nand\n/ba∇dblγ(h)k/ba∇dblHs≤c/ba∇dblk/ba∇dblHs. (2.7)\nAs far as the non-linearity fis concerned, we make the following assumptions.\nHypothesis 2. The function f:H→His Lipschitz continuous. Moreover, fmapsH1into itself\nwith\n/ba∇dblf(h)/ba∇dblH1≤c(/ba∇dblh/ba∇dblH1+1), h∈H1, (2.8)\nand\n/ba∇dblf(h1)−f(h2)/ba∇dblH1≤c(1+/ba∇dblh1/ba∇dblH2)/ba∇dblh1−h2/ba∇dblH1. (2.9)\n8Remark 2.3. Assume that\nf(h)(x) =f(h(x)), x∈O,\nfor some Lipschitz continuous function f:Rr→Rr. Thenf:H→His Lipschitz continuous and\n(2.8) holds. Moreover, if fis differentiable and its derivative is Lipschitz continuous and bounded,\nwe have\n/ba∇dblf(h1)−f(h2)/ba∇dblH1≤ /ba∇dbl(Df(h1)−Df(h2))∇h1/ba∇dblH+/ba∇dblDf(h2)∇(h1−h1)/ba∇dblH\n≤[Df]Lip/ba∇dblh1−h2/ba∇dblL4/ba∇dbl∇h1/ba∇dblL4+/ba∇dblDf/ba∇dbl∞/ba∇dblh1−h2/ba∇dblH1,\nand ifd≤4, this implies (2.9).\n/square\nNext, we assume the following conditions for the diffusion coe fficient.\nHypothesis 3. 1.The function σmapsH1intoL2(HQ,H1), and\n/ba∇dblσ(h)/ba∇dblL2(HQ,H1)≤c(1+/ba∇dblh/ba∇dblH1), h∈H1. (2.10)\n2.It holds\nsup\nh∈H1/ba∇dblσ(h)/ba∇dblL2(HQ,H)<∞, (2.11)\nand for every h1,h2∈H1\n/ba∇dblσ(h1)−σ(h2)/ba∇dbl2\nL2(HQ,H)≤c/ba∇dblh1−h2/ba∇dbl2\nH1. (2.12)\n3.For everyh∈H2andk∈H1we have\n/ba∇dblσ(h+k)−σ(h)/ba∇dblL2(HQ,H1)≤c/ba∇dblk/ba∇dblH1(1+/ba∇dblh/ba∇dblH2). (2.13)\nRemark 2.4. 1. Assume that there exist some measurable mappings σi:O×Rr→Rrsuch\nthat for every h∈Handi∈N\n[σ(h)Qei](x) =σi(x,h(x)), x∈O.\nThen, if\nsup\nh∈H1∞/summationdisplay\ni=1/integraldisplay\nO|σi(x,h(x))|2dx<∞,\n(2.11) follows. Moreover, if there exists c>0 such that\nsup\nx∈O∞/summationdisplay\ni=1|σi(x,ξ)−σi(x,η)|2\nRr≤c|ξ−η|2\nRr, ξ, η ∈Rr,\nthen (2.12) holds.\n2. Ifσisconstant, then(2.11)meansthat σQisaHilbert-Schmidtoperatorin H. Equivalently,\nwhenσis the identity operator, (2.11) means that the noise wQlives inH, so that\nwQ∈C([0,T];H).\n3. Ifσis not constant, then (2.11) and (2.12) are satisfied when, fo r example,\nσi(x,ξ) =λ(ξ)Qei(x),(x,ξ)∈O×Rr, i∈N, (2.14)\nfor someλ:Rr→Rr×rbounded and Lipschitz continuous and for some Q∈L(H) such\nthat∞/summationdisplay\ni=1/ba∇dblQei/ba∇dbl2\nL∞(O)<∞. (2.15)\n9In caseQis diagonalizable with respect the basis {ei}i∈N, withQei=ϑiei, the condition\nabove reads\n∞/summationdisplay\ni=1ϑ2\ni/ba∇dblei/ba∇dbl2\nL∞(O)<∞. (2.16)\nUnder reasonable assumptions on the domain O, it holds that\n/ba∇dblei/ba∇dblL∞(O)≤ciα\nfor someα>0 depending on d, and (2.16) becomes\n∞/summationdisplay\ni=1ϑ2\nii2α<∞.\nIn particular, when d= 1 or the domain is a hyper-rectangle and d≥1, the eigenfunctions\n{ei}i∈Nare equi-bounded and (2.16) becomes/summationtext∞\ni=1ϑ2\ni<∞.\n4. Concerning condition (2.10), assume as above that all map pingsσiare given by (2.14).\nThen, if we assume that λis differentiable, with bounded derivative, for every h∈H2we\nhave\n/ba∇dblσi(·,h)/ba∇dblH1≤ /ba∇dblDλ(h)∇hQei/ba∇dblH+/ba∇dblλ(h)∇(Qei)/ba∇dblH\n≤ /ba∇dblDλ/ba∇dbl∞/ba∇dbl∇h/ba∇dblH/ba∇dblQei/ba∇dblL∞+/ba∇dblλ/ba∇dbl∞/ba∇dbl∇(Qei)/ba∇dblH.\nHence, if we assume (2.15) and\n∞/summationdisplay\ni=1/ba∇dblQei/ba∇dbl2\nH1<∞, (2.17)\nwe conclude that\n/ba∇dblσ(h)/ba∇dbl2\nL2(HQ,H1)=∞/summationdisplay\ni=1/ba∇dblσi(·,h)/ba∇dbl2\nH1≤c/parenleftbig\n1+/ba∇dblh/ba∇dbl2\nH1/parenrightbig\n.\nNotice that, if we assume as above that Qis diagonalizable with respect the basis {ei}i∈N,\nwithQei=ϑiei, then (2.17) follows once we assume\n∞/summationdisplay\ni=1ϑ2\niαi<∞.\n5. If we assume that λis twice differentiable, with bounded derivatives, and take d≤4, for\neveryh∈H2andk∈H1we have\n/ba∇dblσi(·,h+k)−σi(·,h)/ba∇dblH1≤ /ba∇dblDλ(h+k)∇kQei/ba∇dblH\n+/ba∇dbl(Dλ(h+k)−Dλ(h))∇hQei/ba∇dblH+/ba∇dbl(λ(h+k)−λ(h))∇(Qei)/ba∇dblH\n≤ /ba∇dblDλ/ba∇dbl∞/ba∇dblQei/ba∇dblL∞/ba∇dblk/ba∇dblH1+[Dλ]Lip/ba∇dblk/ba∇dblH1/ba∇dblh/ba∇dblH2/ba∇dblQei/ba∇dblL∞+[λ]Lip/ba∇dblk/ba∇dblH1/ba∇dbl∇(Qei)/ba∇dblL4.\nThis means that if we assume (2.15), (2.17), and\n∞/summationdisplay\ni=1/ba∇dbl∇(Qei)/ba∇dbl2\nL4<∞, (2.18)\n10we can conclude that (2.13) holds. If we assume, as above, Qei=ϑiei, sinceHd/6֒→L3,\nthen (2.18) holds if\n∞/summationdisplay\ni=1ϑ2\niα1+d\n6\ni<∞.\n/square\nIn the proof of the small-mass limit, we will need to assume so me further conditions on the\ncoefficients γ,fandσ.\nHypothesis 4. 1.There exists a differentiable mapping g:H1→Rr×r, which is bounded\ntogether with its derivative, such that for every h∈H1andk∈H\n[γ(h)k](x) =g(h)k(x), x∈O.\nMoreover there exists γ0>0such that\ninf\nh∈H1/an}b∇acketle{tg(h)ξ,ξ/an}b∇acket∇i}htRr≥γ0/ba∇dblξ/ba∇dbl2\nRr, ξ∈Rr. (2.19)\n2.The function fmapsH2into itself and\n/ba∇dblf(u)/ba∇dblH2≤c/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH2/parenrightbig\n, u∈H2.\n3.The function σmapsH2intoL2(HQ,H2)and there exists ¯κ<2such that\n/ba∇dblσ(h)/ba∇dblL2(HQ,H2)≤c/parenleftbig\n1+/ba∇dblh/ba∇dblH2+/ba∇dblh/ba∇dbl2\nH¯κ/parenrightbig\n. (2.20)\n4.Ifσ:H1→L2(HQ,H1)is not bounded, then\na)σmapsHρintoL2(HQ,Hρ)andfmapsHρintoHρ, for every ρ<1, with\n/ba∇dblσ(u)/ba∇dblL2(HQ,Hρ)≤cρ(1+/ba∇dblu/ba∇dblHρ),/ba∇dblf(u)/ba∇dblHρ≤cρ(1+/ba∇dblu/ba∇dblHρ), (2.21)\nb)there exists ¯s<1such that g:H¯s→Rr×ris differentiable, with\nsup\nh∈H1/ba∇dblDg(h)/ba∇dblL(H¯s,Rr×r)<+∞,inf\nh∈H¯s/an}b∇acketle{tg(h)ξ,ξ/an}b∇acket∇i}htRr≥γ0/ba∇dblξ/ba∇dbl2\nRr, ξ∈Rr,(2.22)\nc)there exists ¯κ<2such that\n/ba∇dblf(h)/ba∇dblH2≤c/parenleftbig\n1+/ba∇dblh/ba∇dblH2+/ba∇dblh/ba∇dbl2\nH¯κ/parenrightbig\n. (2.23)\nRemark 2.5. 1. There is no loss of generality in assuming that the constan t ¯κ<2 appearing\nin condition 3 and condition 4.c in Hypothesis 4 is the same.\n2. As we have seen in Example 2.2, if γsatisfies condition 1. in Hypothesis 4, then it satisfies\nconditions (2.1) and (2.3) in Hypothesis 1, even in the stron ger versions (2.6) and (2.7).\nMoreover, the fact that g:H1→Rr×ris differentiable with bounded derivative, implies\nthat condition (2.2) is satisfied as well, even in its stronge r version (2.6).\n3. Condition 2. on fis satisfied in the case of the example described in Remark 2.3 , if we\nassume that fis twice differentiable, with bounded derivatives.\n4. If for some s≤1, the matrix g(h) is invertible for every h∈Hsandg:Hs→Rr×ris\ndifferentiable, then we have\nDg−1(h)k=−g−1(h)[Dg(h)k]g−1(h), h,k∈Hs.\nIn particular, we have\nsup\nh∈Hs/ba∇dblDγ−1(h)k/ba∇dblRr×r≤c/ba∇dblk/ba∇dblHs, k∈Hs. (2.24)\n115. Assume that, as in Remark 2.4,\n[σ(u)h](x) =λ(u(x))h(x), x∈O,\nfor some bounded λ:Rr→Rr×r, which is twice differentiable, with bounded derivatives.\nIf we assume (2.15), (2.17) and (2.18), then, when d≤3 we have\n/ba∇dblσ(u)/ba∇dblL2(HQ,H2)≤c/parenleftbig\n1+/ba∇dblu/ba∇dblH2+/ba∇dbl∇u/ba∇dbl2\nL4/parenrightbig\n≤c/parenleftbig\n1+/ba∇dblu/ba∇dblH2+/ba∇dblu/ba∇dbl2\nH7/4/parenrightbig\n,\nso that (2.20) holds with ¯ κ= 7/4.\n6. By interpolation, for every r<¯κwe have\n/ba∇dblh/ba∇dblH¯κ≤ /ba∇dblh/ba∇dbl¯κ−r\n3−r\nH3/ba∇dblh/ba∇dbl3−¯κ\n3−r\nHr.\nThen, thanks to (2.20), for every R≥1 we get\n/ba∇dblh/ba∇dblHr≤R=⇒ /ba∇dblσ(h)/ba∇dblL2(HQ,H2)≤cR,r/parenleftbigg\n1+/ba∇dblh/ba∇dblH2+/ba∇dblh/ba∇dbl2(¯κ−r)\n3−r\nH3/parenrightbigg\n. (2.25)\n7. With similar arguments, we have that if fis given as in Remark 2.3, then it satisfies (2.23).\nIn particular, we have\n/ba∇dblh/ba∇dblHr≤R=⇒ /ba∇dblf(h)/ba∇dblH2≤cR,r/parenleftbigg\n1+/ba∇dblh/ba∇dblH2+/ba∇dblh/ba∇dbl2(¯κ−r)\n3−r\nH3/parenrightbigg\n. (2.26)\n8. Thanks to (2.19) we have that for every h∈H1the matrix g(h) is invertible and\ninf\nh∈H1/an}b∇acketle{tg−1(h)ξ,ξ/an}b∇acket∇i}htRr≥˜γ0/ba∇dblξ/ba∇dbl2\nRr, ξ∈Rr. (2.27)\n9. As shown in [44, Theorem 1, Section 5.3.6], if λ:Rr→Rris differentiable, with bounded\nderivative, then for any 0 <ρ<1\n/ba∇dblλ(u)/ba∇dblHρ≤c/ba∇dblDλ/ba∇dblL∞/ba∇dblu/ba∇dblHρ,\nso thatfsatisfies (2.21) if it is given as in Remark 2.4, for some fwhich is differentiable\nwith bounded derivative. Moreover, as proven in [44, Theore m 2, section 4.6.4.], we have\n/ba∇dblλ(u)Qei/ba∇dblHρ≤ /ba∇dblλ(u)/ba∇dblL∞/ba∇dblQei/ba∇dblHρ+/ba∇dblλ(u)/ba∇dblHρ/ba∇dblQei/ba∇dblL∞.\nhence, if in addition to (2.15) we assume\n∞/summationdisplay\ni=1/ba∇dblQei/ba∇dbl2\nHρ<∞,\nthenσsatisfies (2.21).\n/square\nThislast set ofconditions will berequiredwhenweintroduc ethenoise-induced driftthat appears\nin the limiting equation, and we need to prove its well-posed ness and identify the limit.\nHypothesis 5. 1.The mapping σ:H2→L2(HQ,H1)is differentiable and\n/ba∇dblDσ(u)h/ba∇dblL2(HQ,H1)≤c(1+/ba∇dblu/ba∇dblH2)/ba∇dblh/ba∇dblH1, u,h ∈H2. (2.28)\n2.The mapping g:H1→Rr×ris twice continuously differentiable, with\nsup\nu∈H1/ba∇dblD2g(u)/ba∇dblL(H1×H1;Rr×r)<∞.\n12Remark 2.6. Assume that, as in Remark 2.4,\n[σ(u)h](x) =λ(u(x))h(x), x∈O,\nfor some bounded λ:Rr→Rr×r. In this case, if λis differentiable, we have\n[(Dσ(u)k)Qei](x) = [Dλ(u(x))k(x)]Qei(x), x∈O.\nThus, if we assume that λis twice differentiable, with bounded derivatives, we have\n/ba∇dbl(Dσ(u)k)Qei/ba∇dblH1≤ /ba∇dblD2λ(u)(∇u,k)Qei/ba∇dblH+/ba∇dbl[Dλ(u)∇k]Qei/ba∇dblH+/ba∇dbl[Dλ(u)k]∇(Qei)/ba∇dblH\n≤c(/ba∇dbl∇u/ba∇dblL4/ba∇dblk/ba∇dblL4/ba∇dblQei/ba∇dblL∞+/ba∇dbl∇k/ba∇dblH/ba∇dblQei/ba∇dblL∞+/ba∇dblk/ba∇dblL4/ba∇dbl∇(Qei)/ba∇dblL4).\nTherefore, if d≤4 and we assume (2.15) and (2.18), we have\n∞/summationdisplay\ni=1/ba∇dbl(Dσ(u)k)Qei/ba∇dbl2\nH1≤c/parenleftbig\n/ba∇dblu/ba∇dbl2\nH2/ba∇dblk/ba∇dbl2\nH1+/ba∇dblk/ba∇dbl2\nH1/parenrightbig\n,\nand (2.28) follows.\n/square\nTo conclude this section, for every µ >0 andδ∈R, we define on Hδthe unbounded linear\noperator\nLµ(u,v) =1\nµ(µv,Lu),(u,v)∈Hδ+1.\nItcan beproven that Lµis thegenerator of a strongly continuous groupof boundedli near operators\n{Sµ(t)}t≥0on eachHδ(for a proof see [42, Section 7.4]. Moreover, for every z∈Hδ\n/ba∇dblSµ(t)z/ba∇dblHδ≤/radicalbig\nµ∨µ−1/ba∇dblz/ba∇dblHδ, t≥0. (2.29)\nFinally, for every µ>0 and (u,v)∈H1, we define\nBµ(u,v) =1\nµ(0,−γ(u)v+f(u)),Σµ(u,v) =1\nµ(0,σ(u)).\n3.Statement of main results\nIn what follows, for every function u: [0,T]×O→Rrwhich is differentiable with respect to t,\nwe will denote\nu= (u,∂tu).\nMoreover, for every T >0 we will denote\nKT:={u∈C([0,T];H2) such that u(·,x) : [0,T]→Rris differentiable, for a.e. x∈O,\nandu∈C([0,T];H2)}.(3.1)\nBy using the notations introduced in the previous section, e quation (1.1) can be rewritten as\n\n\nduµ(t) = [Lµuµ(t)+Bµ(uµ(t))]dt+Σµ(uµ(t))dwQ(t)\nuµ(0) = (u0,v0).(3.2)\nIn section 4 we will prove that this equation admits a mild sol ution and such solution is unique\nin a proper space of processes. Namely, we will prove the foll owing result\n13Theorem 3.1. Assume Hypotheses 1, 2 and 3, and fix µ>0and(uµ\n0,vµ\n0)∈H2. Then for every\nT >0andp≥1, there exists a unique adapted process uµtaking values in KT,P-a.s., such that\nuµ∈L2(Ω;C([0,T];H1))and\nuµ(t) = (uµ\n0,vµ\n0)+/integraldisplayt\n0Lµuµ(s)ds+/integraldisplayt\n0Bµ(uµ(s))ds+/integraldisplayt\n0Σµ(uµ(s))dwQ(s).(3.3)\nRemark 3.2. It is not difficult to prove that in fact we have uµ∈L2(Ω;C([0,T];H2)).\nOnce proved Theorem 3.1, we will study the limiting behavior ofuµ, asµ→0 when Hypothesis\n4 and Hypothesis 5 hold. Namely, we will prove that uµconverges in probability to some stochastic\nquasi-linear parabolic equation. In order to describe such limiting equation, we need to introduce\nsome notations and preliminary results.\nFor every fixed u,v∈H1we consider the problem\ndy(t) =−g(u)y(t)dt+σ(u)dwQ(t), y(0) =v, (3.4)\nwheregis the mapping defined on H1and taking values in Rr×rintroduced in Hypothesis 4, such\nthat [γ(u)k](x) =g(u)k(x), for every u,k∈H1andx∈O. Due to Hypotheses 3 and 4, the process\nyu,v(t) =e−g(u)tv+/integraldisplayt\n0e−g(u)(t−s)σ(u)dwQ(s), t≥0,\nis the unique solution of equation (3.4), belongs to Lp(Ω;C([0,+∞);H1)), for every p≥1, and\nsatisfies\nsup\nt≥0E/ba∇dblyu,v(t)/ba∇dblp\nH1≤cp/parenleftbig\n1+e−γ0t/ba∇dblv/ba∇dblp\nH1+/ba∇dblu/ba∇dblp\nH1/parenrightbig\n, t≥0. (3.5)\nNow, for every u∈H1, we denote by Pu\ntthe Markov transition semigroup associated with\nequation (3.4) and defined by\nPu\ntϕ(v) =Eϕ(yu,v(t)), v∈H1, t≥0,\nfor every function ϕ∈Bb(H1), whereBb(H1) is the space of Borel and bounded functions defined\nonH1and taking values in R. Moreover, we define\nΛu:=/integraldisplay∞\n0e−g(u)s[σ(u)Q][σ(u)Q]⋆e−gt(u)sds.\nDue to Hypotheses 3 and 4, we have that Λ u∈L+\n1(H)∩L+\n1(H1), with\nTrHΛu≤c,TrH1Λu≤c(1+/ba∇dblu/ba∇dblH1). (3.6)\nand the Gaussian measure νu:=N(0,Λu) is the unique invariant measure for the semigroup Pu\nt.\nNext, we define\nS(u) :=/integraldisplay\nH1/bracketleftbig\nDg−1(u)z/bracketrightbig\nzdνu(z), u∈H1. (3.7)\nIn Section 9, we will show that S:H1→H1is well-defined and we will study its differentiability\nproperties. Once introduced the mapping S, we can state our limiting result.\nTheorem 3.3. Assume Hypotheses 1 to 5 and fix an arbitrary ̺ <1andϑ∈[1,2). Moreover,\nassume that (uµ\n0,vµ\n0)∈H3, for every µ>0, with\nsup\nµ∈(0,1)/ba∇dbl(uµ\n0,√µvµ\n0)/ba∇dblH1<∞, (3.8)\n14and\nlim\nµ→0µδ/ba∇dbl(uµ\n0,√µvµ\n0)/ba∇dbl2\nH2= 0,lim\nµ→0µ1+δ/ba∇dbl(uµ\n0,√µvµ\n0)/ba∇dbl2\nH3= 0, (3.9)\nfor someδ∈(0,1/2). Then, ifu0∈H1is such that\nlim\nµ→0/ba∇dbluµ\n0−u0/ba∇dblH1= 0, (3.10)\nfor everyp<2/(ϑ−1)andη,T >0we have\nlim\nµ→0P/parenleftBigg\nsup\nt∈[0,T]/ba∇dbluµ(t)−u(t)/ba∇dblH̺+/integraldisplayT\n0/ba∇dbluµ(t)−u(t)/ba∇dblp\nHϑdt>η/parenrightBigg\n= 0, (3.11)\nwhereu∈L2(Ω;C([0,T];H1)∩L2(0,T;H2))is the unique solution of the problem\n\n\n∂tu(t,x) =g−1(u(t))∆u(t,x)+g−1(u(t))f(u(t,x))+S(u(t))+g−1(u(t))σ(u(t))∂twQ(t),\nu(0,x) =u0(x), x∈O, u (t,x) = 0, x∈∂O.\n(3.12)\nIn what follows, for the sake of simplicity of notations, we s hall denote\nΛ6µi:=/ba∇dbl(uµ\n0,√µvµ\n0)/ba∇dbl2\nHi, µ> 0, i= 1,2,3. (3.13)\n4.Proof of Theorem 3.1\nIn this section we study the well-posedness of system (3.2), with initial conditions ( uµ\n0,vµ\n0)∈H2,\nfor everyµ>0. Since here µis a fixed positive parameter, without any loss of generality we can\nconsider the case µ= 1 and the coefficients Lµ,Bµand Σµand the initial conditions uµ\n0andvµ\n0\nwill be just denoted by L,B, Σ,u0andv0.\nIn order to study the well-posedness of equation (3.2), we wi ll use a generalized splitting method,\nsimilar to the one introduced in [47] (see also [46]). For eve ryη∈KTwe consider the deterministic\nauxiliary problem\nd\ndtζ(t) =Lζ(t)+B(ζ(t)+η(t)),ζ(0) = (u0,v0). (4.1)\nA function ζ∈KTis a solution to problem (4.1) if the equality\nζ(t) = (u0,v0)+/integraldisplayt\n0Lζ(s)ds+/integraldisplayt\n0B(ζ(s)+η(s))ds\nholds in H1.\nProposition 4.1. Assume that Hypotheses 1 and 2 are satisfied. Then, for every (u0,v0)∈H2\nandη∈KT,problem(4.1)admits a unique solution ζ∈KT. Moreover, for every t∈[0,T]\nsup\ns∈[0,t]/ba∇dblζ(s)/ba∇dblH1≤cT/parenleftBigg\n1+ sup\ns∈[0,t]/ba∇dblη(s)/ba∇dblH1/parenrightBigg\n, (4.2)\nand\nsup\ns∈[0,t]/ba∇dblζ(s)/ba∇dblH2≤exp/parenleftBigg\nc/bracketleftBigg\n/ba∇dbl(u0,v0)/ba∇dblH1+1+ sup\ns∈[0,t]/ba∇dblη(s)/ba∇dblH2/bracketrightBigg\nt/parenrightBigg\n(/ba∇dbl(u0,v0)/ba∇dblH2+1).(4.3)\n15Thanks to Proposition 4.1, we can introduce the mapping Λ : KT→KTthat associates to every\nη∈KTthe solution Λ( η)∈KTof equation (4.1). With this definition, we consider the stoc hastic\nproblem\n\ndη(t) =Lη(t)dt+Σ(η(t)+Λ(η)(t))dwQ(t),\nη(0) = 0.\nProposition 4.2. Under Hypotheses 1, 2 and 3, for every T >0andp≥1there exists a unique\nadapted process ηsuch that η∈L2(Ω;C([0,T];H2))and\nη(t) =/integraldisplayt\n0Lη(s)ds+/integraldisplayt\n0Σ(η(s)+Λ(η)(s))dwQ(s). (4.4)\nOnce proved Proposition 4.2, we can conclude the proof of The orem 3.1. Actually, thanks to\nProposition 4.2, there exists ηsuch that ηbelongs toL2(Ω;C([0,T];H2)) and solves equation (4.4).\nThus, if we define\nu=η+Λ(η),\ndue to Theorem 4.1 it is immediate to check that utakes values in KT,P-a.s.,usatisfies equation\n(3.3) and belongs to L2(Ω;C([0,T];H1).\nFinally, concerning uniqueness, if uis any solution of equation (3.3), we consider the problem\ndη(t) =Lη(t)dt+Σ(u(t))dwQ(t),η(0) =0.\nThis problem admits a unique solution η∈Lp(Ω;C([0,T];H2)). Then, if we define ζ:=u−η, we\nhave that ζsolves equation (4.1). Since uniqueness holds for equation (4.1), we have that ζ= Λ(η)\nand henceu=η+Λ(η), withηbeing the unique solution of (4.4).\n4.1.Proof of Proposition 4.1. In this subsection, we assumethat Hypotheses 1 and 2 are alwa ys\nsatisfied. Moreover, ( u0,v0)∈H2andη∈KTare fixed once for all.\nAs a consequence of (2.1) and the Lipschitz continuity of finH, for everyζ= (ζ1,ζ2)∈H1we\nhave\n/ba∇dblB(t,ζ)/ba∇dblH1=/ba∇dbl−γ(ζ1+η(t))(∂tη(t)+ζ2)+f(η(t)+ζ1)/ba∇dblH\n≤c(/ba∇dbl∂tη(t)/ba∇dblH+/ba∇dblζ2/ba∇dblH)+c(1+/ba∇dblη(t)/ba∇dblH+/ba∇dblζ1/ba∇dblH),\nand this implies that B(t,·) :H1→H1is well defined, for all t∈[0,T], and\n/ba∇dblB(t,ζ)/ba∇dblH1≤c/ba∇dblζ/ba∇dblH1+c(1+/ba∇dblη(t)/ba∇dblH1).\nDue to (2.4) and (2.8), we have\n/ba∇dblB(t,ζ)/ba∇dblH2=/ba∇dbl−γ(ζ1+η(t))(∂tη(t)+ζ2)+f(η(t)+ζ1)/ba∇dblH1\n≤c(/ba∇dblζ1+η(t)/ba∇dblH2+1)(/ba∇dbl∂tη(t)/ba∇dblH1+/ba∇dblζ2/ba∇dblH1)+c(1+/ba∇dblη(t)/ba∇dblH1+/ba∇dblζ1/ba∇dblH1),\nso thatB(t,·) mapsH2intoH2, for everyt∈[0,T], and\n/ba∇dblB(t,ζ)/ba∇dblH2≤c/parenleftbig\n1+/ba∇dblη(t)/ba∇dbl2\nH2+/ba∇dblζ/ba∇dbl2\nH2/parenrightbig\n. (4.5)\nMoreover, due to (2.4) and (2.9) we have\n/ba∇dblB(t,ζ)−B(t,¯ζ)/ba∇dblH2=/ba∇dbl−γ(ζ1+η(t))(ζ2−¯ζ2)+f(η(t)+ζ1)−f(η(t)+¯ζ1)/ba∇dblH1\n≤c(1+/ba∇dblζ1/ba∇dblH2+/ba∇dblη(t)/ba∇dblH2)/ba∇dblζ2−¯ζ2/ba∇dblH1+c(1+/ba∇dblη(t)/ba∇dblH2+/ba∇dblζ1/ba∇dblH2)/ba∇dblζ1−¯ζ1/ba∇dblH1,\n16so that\n/ba∇dblB(t,ζ)−B(t,¯ζ)/ba∇dblH2≤c(1+/ba∇dblη(t)/ba∇dblH2+/ba∇dblζ1/ba∇dblH2)/ba∇dblζ−¯ζ/ba∇dblH2. (4.6)\nNow, for every ζ∈KTwe define\nA(ζ)(t) :=S(t)(u0,v0)+/integraldisplayt\n0S(t−s)B(s,ζ(s))ds, t∈[0,T].\nDue to to (2.29) and (4.5), we have\n/ba∇dblA(ζ)(t)/ba∇dblH2≤ /ba∇dbl(u0,v0)/ba∇dblH2+/integraldisplayt\n0/ba∇dblB(s,ζ(s))/ba∇dblH2ds\n≤ /ba∇dbl(u0,v0)/ba∇dblH2+c/parenleftBigg\nχt(η)+ sup\ns∈[0,t]/ba∇dblζ(s)/ba∇dbl2\nH2/parenrightBigg\nt,\nwhere\nχt(η) := sup\ns∈[0,t]/ba∇dblη(s)/ba∇dbl2\nH2.\nHence, if for any t∈(0,T] andR>0 we define\nKt(R) :=/braceleftBigg\nζ∈KT: sup\ns∈[0,t]/ba∇dblζ(s)/ba∇dblH2≤R/bracerightBigg\n, (4.7)\nwe have\nζ∈Kt(R) =⇒sup\ns∈[0,t]/ba∇dblA(ζ)(s)/ba∇dblH2≤ /ba∇dbl(u0,v0)/ba∇dblH2+c/parenleftbig\nχT(η)+R2/parenrightbig\nt.\nThis implies that, if we take\nR>/ba∇dbl(u0,v0)/ba∇dblH2, t 1≤R−/ba∇dbl(u0,v0)/ba∇dblH2\nc(χT(η)+R2),\nwe have that AmapsKt1(R) into itself. Moreover, due to (4.6)\nζ,¯ζ∈Kt(R) =⇒sup\ns∈[0,t]/ba∇dblA(ζ)(s)−A(¯ζ)(s)/ba∇dblH2≤c(χT(η)+R)tsup\ns∈[0,t]/ba∇dblζ−¯ζ/ba∇dblH2.\nHence, taking\n¯t:=t1∧(2c(χT(η)+R))−1,\nwe conclude that AmapsK¯t(R) into itself as a contraction. This means that there exists a unique\nζ∈K¯t(R) such that\nζ(t) =S(t)(u0,v0)+/integraldisplayt\n0S(t−s)B(s,ζ(s))ds, t∈[0,¯t].\nThis means that there exists a local solution to problem (4.1 ). Thus, in order to conclude the\nproof of Proposition 4.1, we have to prove that such solution can be extended to a unique global\nsolution defined in [0 ,T], satisfying (4.2) and (4.3).\nLetζbe a local solution defined on a maximal interval (0 ,T′). We show that it does not blow\nup atT′. For every t∈[0,T′), due to (2.1) we have\n1\n2d\ndt/ba∇dblζ(t)/ba∇dbl2\nH1=−/an}b∇acketle{tγ(ζ(t)+η(t))(∂tζ(t)+∂tη(t)),∂tζ(t)/an}b∇acket∇i}htH+/an}b∇acketle{tf(ζ(t)+η(t)),∂tζ(t)/an}b∇acket∇i}htH\n≤c/ba∇dblζ(t)/ba∇dbl2\nH1+c/parenleftbig\n1+/ba∇dblη(t)/ba∇dbl2\nH1/parenrightbig\n,\n17and this implies (4.2), that is\n/ba∇dblζ(t)/ba∇dblH1≤cT/parenleftBigg\n/ba∇dbl(u0,v0)/ba∇dblH1+1+ sup\nt∈[0,T]/ba∇dblη(t)/ba∇dblH1/parenrightBigg\n.\nMoreover, due to (2.3) and (2.8) we have\n1\n2d\ndt/ba∇dblζ/ba∇dbl2\nH2=−/an}b∇acketle{tγ(ζ(t)+η(t))(∂tζ(t)+∂tη(t)),∂tζ(t)/an}b∇acket∇i}htH1+/an}b∇acketle{tf(ζ(t)+η(t)),∂tζ(t)/an}b∇acket∇i}htH1\n≤c/ba∇dblζ(t)+η(t)/ba∇dbl1/4\nH1/ba∇dblζ(t)+η(t)/ba∇dbl3/4\nH2/ba∇dbl∂tζ(t)+∂tη(t)/ba∇dbl1/4\nH/ba∇dbl∂tζ(t)+∂tη(t)/ba∇dbl3/4\nH1\n+c(/ba∇dbl∂tζ(t)/ba∇dblH1+/ba∇dbl∂tη(t)/ba∇dblH1+/ba∇dblζ(t)/ba∇dblH1+/ba∇dblη(t)/ba∇dblH1+1)/ba∇dbl∂tζ(t)/ba∇dblH1,\nso that\nd\ndt/ba∇dblζ(t)/ba∇dbl2\nH2≤c/parenleftbig\n1+/ba∇dblη(t)/ba∇dbl2\nH2/parenrightbig\n/ba∇dblζ(t)/ba∇dbl2\nH2+c/ba∇dblη(t)/ba∇dbl2\nH2\n+c/ba∇dblζ(t)/ba∇dbl1/2\nH1/ba∇dblζ(t)/ba∇dbl1/2\nH2/ba∇dbl∂tζ(t)/ba∇dbl1/2\nH/ba∇dbl∂tζ(t)/ba∇dbl3/2\nH1.\nNow, thanks to (4.2) we have\n/ba∇dblζ(t)/ba∇dbl1/2\nH1/ba∇dblζ(t)/ba∇dbl1/2\nH2/ba∇dbl∂tζ(t)/ba∇dbl1/2\nH/ba∇dbl∂tζ(t)/ba∇dbl3/2\nH1\n≤c/parenleftBigg\n/ba∇dbl(u0,v0)/ba∇dblH1+1+ sup\ns∈[0,t]/ba∇dblη(s)/ba∇dblH1/parenrightBigg\n/ba∇dblζ(t)/ba∇dbl1/2\nH2/ba∇dbl∂tζ(t)/ba∇dbl3/2\nH1\n≤c/parenleftBigg\n/ba∇dbl(u0,v0)/ba∇dblH1+1+ sup\ns∈[0,t]/ba∇dblη(s)/ba∇dblH1/parenrightBigg\n/ba∇dblζ(t)/ba∇dbl2\nH2.\nThis implies\nd\ndt/ba∇dblζ(t)/ba∇dbl2\nH2≤c/parenleftBigg\n/ba∇dbl(u0,v0)/ba∇dblH1+1+ sup\ns∈[0,t]/ba∇dblη(s)/ba∇dbl2\nH2/parenrightBigg\n/parenleftbig\n/ba∇dblζ(t)/ba∇dbl2\nH2+1/parenrightbig\n,\nand hence\n/ba∇dblζ(t)/ba∇dbl2\nH2≤exp/parenleftBigg\nc/parenleftBigg\n/ba∇dbl(u0,v0)/ba∇dblH1+1+ sup\ns∈[0,t]/ba∇dblη(s)/ba∇dbl2\nH2/parenrightBigg\nt/parenrightBigg\n/parenleftbig\n/ba∇dbl(u0,v0)/ba∇dbl2\nH2+1/parenrightbig\n.\nFinally, as far as uniqueness is concerned, if ζ1andζ2are two solutions and if we denote ρ:=\nζ1−ζ2, due to (2.2) we have\n1\n2d\ndt/ba∇dblρ/ba∇dbl2\nH1\n=−/an}b∇acketle{tγ(ζ1(t)+η(t))∂tρ(t),∂tρ(t)/an}b∇acket∇i}htH−/an}b∇acketle{t[γ(ζ1(t)+η(t))−γ(ζ2(t)+η(t))]∂tζ2(t),∂tρ(t)/an}b∇acket∇i}htH\n−/an}b∇acketle{t[γ(ζ1(t)+η(t))−γ(ζ2(t)+η(t))]∂tη(t)+[f(ζ1(t)+η(t))−f(ζ2(t)+η(t))],∂tρ(t)/an}b∇acket∇i}htH\n≤c(1+/ba∇dbl∂tη(t)/ba∇dblH1+/ba∇dbl∂tζ2(t)/ba∇dblH1)/ba∇dbl∂tρ(t)/ba∇dblH/ba∇dblρ(t)/ba∇dblH1.\n18Therefore,\nd\ndt/ba∇dblρ/ba∇dbl2\nH1≤c/parenleftBigg\n1+ sup\ns∈[0,t]/ba∇dbl∂tη(s)/ba∇dblH1+ sup\ns∈[0,t]/ba∇dbl∂tζ2(s)/ba∇dblH1/parenrightBigg\n/ba∇dblρ(t)/ba∇dbl2\nH1,\nand this allows to conclude that ρ= 0, that is ζ1=ζ2.\n4.2.Proof of Proposition 4.2. As a consequence of Proposition 4.1, we can introduce the map -\nping Λ : KT→KTthat associates to every ηthe solution Λ( η) of equation (4.1). In what follows,\nwe want to prove that for every T >0 the stochastic equation\ndη(t) =Lη(s)dt+Σ(η(s)+Λ(η)(s))dwQ(s),η(0) = 0, (4.8)\nadmits a unique mild solution ηsuch that η∈Lp(Ω;C([0,T];H2)), for every p≥1.\nTo this purpose, we start by showing that the mapping Λ is Lips chitz continuous with respect\nto theH1-norm, locally in H2.\nLemma 4.3. For everyT,R >0there exists some LT,R>0such that if η1,η2∈KT(R)(as\ndefined in (4.7)) then for every t∈[0,T]\nsup\ns∈[0,t]/ba∇dblΛ(η1)(s)−Λ(η2)(s)/ba∇dblH1≤LT,R/integraldisplayt\n0/ba∇dblη1(s)−η2(s)/ba∇dblH1ds. (4.9)\nProof.We fixR >0 andη1,η2inKT(R) and we define ρ:= Λ(η1)−Λ(η2). We have ρ(0) =\n∂tρ(0) = 0 and\n1\n2d\ndt/ba∇dblρ(t)/ba∇dbl2\nH1=−/an}b∇acketle{t[γ(Λ(η1)(t)+η1(t))−γ(Λ(η2)(t)+η2(t))](∂tΛ(η1)(t)+∂tη1(t)),∂tρ(t)/an}b∇acket∇i}htH\n−/an}b∇acketle{tγ(Λ(η2)(t)+η2(t))∂tρ(t),∂tρ(t)/an}b∇acket∇i}htH−/an}b∇acketle{tγ(Λ(η2)(t)+η2(t))∂t(η1−η2)(t),∂tρ(t)/an}b∇acket∇i}htH\n+/an}b∇acketle{tf(Λ(η1)(t)+η1(t))−f(Λ(η2)(t)+η2(t)),∂tρ(t)/an}b∇acket∇i}htH.\nThen, in view of (2.1) and (2.2), we have\nd\ndt/ba∇dblρ(t)/ba∇dbl2\nH1≤c/ba∇dbl∂tρ(t)/ba∇dblH(/ba∇dblρ(t)/ba∇dblH1+/ba∇dbl(η1−η2)(t)/ba∇dblH1)(/ba∇dbl∂tη1(t)+∂tΛ(η1)(t)/ba∇dblH1)\n+c/ba∇dbl∂tρ(t)/ba∇dblH(/ba∇dblρ(t)/ba∇dblH+/ba∇dbl∂tρ(t)/ba∇dblH+/ba∇dbl(η1−η2)(t)/ba∇dblH+/ba∇dbl∂t(η1−η2)(t)/ba∇dblH).\nThis implies\nd\ndt/ba∇dblρ(t)/ba∇dbl2\nH1≤M(η1)/ba∇dblρ(t)/ba∇dbl2\nH1+M(η1)/ba∇dblη1(t)−η2(t)/ba∇dbl2\nH1,\nwhere\nM(η) =c/parenleftBigg\n1+ sup\nt∈[0,T]/ba∇dbl∂t(η+Λ(η))(t)/ba∇dblH1/parenrightBigg\n.\nAccording to (4.3), we have that there exists some constant mT,R>0 such that\nη∈KT(R) =⇒M(η)≤mT,R.\nHence\nd\ndt/ba∇dblρ(t)/ba∇dbl2\nH1≤mT,R/ba∇dblρ(t)/ba∇dbl2\nH1+mT,R/ba∇dblη1(t)−η2(t)/ba∇dbl2\nH1,\nso that\n/ba∇dblρ(t)/ba∇dbl2\nH1≤mT,RemT,RT/integraldisplayt\n0/ba∇dblη1(s)−η2(s)/ba∇dbl2\nH1ds,\n19and (4.9) follows.\n/square\nNow, for every R >0, we fix a smooth function Φ R, with 0 ≤ΦR≤1, such that Φ R(r) = 1, if\nr≤Rand ΦR(r) = 0, ifr≥R+1. Moreover, for every η∈KTandt∈[0,T] we define\nΨR(t,η) = ΦR/parenleftbigg\nsup\ns≤t/ba∇dblη(s)/ba∇dblH2/parenrightbigg\n.\nIt is immediate to check that there exists some c, independent of R, such that for every η1,η2∈KT\nandt∈[0,T]\n|ΨR(t,η1)−ΨR(t,η2)| ≤csup\ns≤t/ba∇dblη1(t)−η2(t)/ba∇dblH2. (4.10)\nNext, we introduce the regularized problem\ndη(t) =Lη(s)dt+Ψ2\nR(t,η)Σ(η(t)+Λ(η)(t))dwQ(t)η(0) = 0. (4.11)\nLemma 4.4. For everyT,R >0andp≥2, problem (4.11)admits a unique KT-valued weak\nsolutionηR, such that\nEsup\nt∈[0,T]/ba∇dblηR(t)/ba∇dblp\nH2≤cT,p, (4.12)\nfor some constant cT,p>0independent of R.\nProof.Equation (4.11), can be rewritten as the following integral equation\nη(t) =/integraldisplayt\n0S(t−s)BR(s,η)dwQ(s),\nwhereS(t) is the group generated by the operator LinH2and\nBR(t,η) = Ψ2\nR(t,η)Σ(η(t)+Λ(η)(t)).\nAs a first step, we prove that there exists cT>0, independent of R, such that for every η∈KT\nandt∈[0,T]\n/ba∇dblBR(t,η)/ba∇dblL2(HQ,H2)≤cT/parenleftbigg\n1+sup\ns≤t/ba∇dblη(s)/ba∇dblH1/parenrightbigg\n, (4.13)\nand there exists cT,R>0 such that for every η1,η2∈KTandt∈[0,T]\n/ba∇dblBR(t,η1)−BR(t,η2)/ba∇dblL2(HQ,H2)≤cT,Rsup\ns≤t/ba∇dblη1(s)−η2(s)/ba∇dblH2.(4.14)\nDue to (2.10) and (4.2), we have\n/ba∇dblBR(t,η)/ba∇dblL2(HQ,H2)= Ψ2\nR(t,η)/ba∇dblσ(η(t) +Λ(η)(t))/ba∇dblL2(HQ,H1)\n≤c(1+/ba∇dblη(t)+Λ(η)(t)/ba∇dblH1)≤cT/parenleftbigg\n1+sup\ns≤t/ba∇dblη(s)/ba∇dblH1/parenrightbigg\n,\n20and this gives (4.13). As for (4.14), we have\n/ba∇dblBR(t,η1)−BR(t,η2)/ba∇dblL2(HQ,H2)\n≤ |ΨR(t,η1)−ΨR(t,η2)|ΨR(t,η1)/ba∇dblσ(η1(t)+Λ(η1)(t))/ba∇dblL2(HQ,H1)\n+|ΨR(t,η1)−ΨR(t,η2)|ΨR(t,η2)/ba∇dblσ(η2(t)+Λ(η2)(t))/ba∇dblL2(HQ,H1)\n+ΨR(t,η1)ΨR(t,η2)/ba∇dblσ(η1(t)+Λ(η1)(t))−σ(η2(t)+Λ(η2)(t))/ba∇dblL2(HQ,H1)\n=:IR,1(t)+IR,2(t)+IR,3(t).\nThanks to (2.10) and (4.2), there exists cR>0 such that\nsup\nt∈[0,T]ΨR(t,ηi)/ba∇dblσ(ηi(t)+Λ(ηi)(s))/ba∇dblL2(HQ,H1)≤cR, i= 1,2.\nTherefore, thanks to (4.10), we get\nIR,1(t)+IR,2(t)≤cRsup\ns≤t/ba∇dblη1(s)−η2(s)/ba∇dblH2. (4.15)\nAs forIR,3(t), according to (2.13) we have\nIR,3(t)≤ΨR(t,η1)ΨR(t,η2) (/ba∇dblη1(t)−η2(t)/ba∇dblH1+/ba∇dblΛ(η1)(t)−Λ(η2)(t)/ba∇dblH1)\n×(1+/ba∇dblη2(t)/ba∇dblH2+/ba∇dblΛ(η2)(t)/ba∇dblH2),\nand then, from (4.3) and (4.9), we get\nIR,3(t)≤cT,Rsup\ns≤t/ba∇dblη1(s)−η2(s)/ba∇dblH1.\nCombining this together with (4.15), we obtain (4.14).\nNext, for every KT-valuedηsuch that η∈Lp(Ω;H2), we define\nFR(η)(t) =/integraldisplayt\n0S(t−s)BR(s,η)dwQ(s), t∈[0,T].\nIf we can prove that once fixed p≥2 there exists α∈(0,1) andt0>0 such that for every η1,η2\nEsup\nt≤t0/ba∇dblFR(η1)(t)−FR(η2)(t)/ba∇dblp\nH2≤αEsup\nt≤t0/ba∇dblη1(t)−η2(t)/ba∇dblp\nH2, (4.16)\nthen equation (4.11) has a unique solution ηR∈Lp(Ω;C([0,t0];H2)). Since we can repeat the\nsame argument in all intervals [ t0,2t0], [2t0,3t0] and so on, we get a unique solution ηRdefined in\nthe whole interval [0 ,T].\nAs shown for example in [16, Proposition 7.3], by using a stoc hastic factorization argument we\ncan prove that FR(η) is continuous in time with values in H2and for every t>0 andp>2\nEsup\ns≤t/ba∇dblFR(η1)(s)−FR(η2)(s)/ba∇dblp\nH2≤cT,pE/integraldisplayt\n0/ba∇dblBR(s,η1)−BR(s,η2)/ba∇dblp\nL2(HQ,H2)ds.\nHence, thanks to (4.14), we obtain\nEsup\ns≤t/ba∇dblFR(η1)(s)−FR(η2)(s)/ba∇dblp\nH2≤cT,R,ptEsup\ns≤t/ba∇dblη1(s)−η2(s)/ba∇dblp\nH2.\nThis implies that, if we fix some α<1 and take t0>0 such that\ncT,R,pt0≤α,\n21(4.16) follows. Finally, we obtain estimate (4.12) from (4. 13) and the Gronwall lemma, since\nEsup\ns≤t/ba∇dblηR(s)/ba∇dblp\nH2≤cT,pE/integraldisplayt\n0/ba∇dblBR(s,ηR)/ba∇dblp\nL2(HQ,H2)ds≤cT,p/integraldisplayt\n0/parenleftbigg\n1+Esup\nr≤s/ba∇dblηR(r)/ba∇dblp\nH1/parenrightbigg\nds.\n/square\nIn order to conclude the proof of Proposition 4.2, for every R>0 we define\nτR:= inf{t∈[0,T] :/ba∇dblηR(t)/ba∇dblH2>R},\nwith the convention that inf ∅= +∞. Due to (4.12), we have\nP(τR<+∞)≤P/parenleftBigg\nsup\nt∈[0,T]/ba∇dblηR(t)/ba∇dblH2≥R/parenrightBigg\n≤1\nR2Esup\nt∈[0,T]/ba∇dblηR(t)/ba∇dbl2\nH2≤cT\nR2,\nso that, if we set\nτ:= lim\nR→∞τR,\nwe getP(τ= +∞) = 1. Now, if we fix ω∈ {τ= +∞}andt∈[0,T], there exists some R >0\nsuch thatt≤τR(ω) and then we define\nη(t)(ω) =ηR(t)(ω).\nThis is a good definition, because we can prove that if t≤σ(ω) :=τR1(ω)∧τR2(ω), then\nηR1(t)(ω) =ηR2(t)(ω). (4.17)\nActually, if we assume that R1≤R2, for everyt∈[0,T] we have\nηR1(t∧σ)−ηR2(t∧σ) =/integraldisplayt\n0I{s≤σ}S(t−s)[BR1(s∧σ,ηR1)−BR2(s∧σ,ηR2)]dwQ(s).\nThen, by using (4.14) as in the proof of Lemma 4.4, we conclude that\nηR1(t∧σ) =ηR2(t∧σ), t∈[0,T],\nwhich implies (4.17). Once we have that ηis a well defined process and η∈L2(Ω,C[0,T];H2), by\nnoticing that\nBR(t,η) = Σ(η(t)+Λ(η)(t)), t≤τR,\nwe have that ηsolves equation (4.8).\nFinally, we show that such solution is unique. If η1andη2are two solutions, we have\nη1(t)−η2(t) =/integraldisplayt\n0S(t−s)[Σ(η1(s)+Λ(η1)(s))−Σ(η2(s)+Λ(η2)(s))]dwQ(s).\nThanks to (2.12), this yields\nEsup\ns∈[0,t]/ba∇dblη1(s)−η2(s)/ba∇dblp\nH1≤cT,p/integraldisplayt\n0E/ba∇dblσ(η1(s)+Λ(η1)(s))−σ(η1(s)+Λ(η1)(s))/ba∇dblp\nL2(HQ,H)ds\n≤cT,p/integraldisplayt\n0E/parenleftbig\n/ba∇dblη1(s)−η2(s)/ba∇dblp\nH1+/ba∇dblΛ(η1)(s)−Λ(η2)(s)/ba∇dblp\nH1/parenrightbig\nds.\nThus, thanks to (4.9) we get\nEsup\ns∈[0,t]/ba∇dblη1(s)−η2(s)/ba∇dblp\nH1≤cT,p/integraldisplayt\n0Esup\nr∈[0,s]/ba∇dblη1(r)−η2(r)/ba∇dblp\nH1ds,\nand we conclude that η1=η2.\n225.Uniform estimates in H1\nIn Theorem 3.1 we have seen that, under Hypotheses 1, 2 and 3, f or everyµ >0 and ev-\nery initial condition ( uµ\n0,vµ\n0)∈H2, equation (1.1) admits a unique solution uµ= (uµ,∂tuµ) in\nL2(Ω;C([0,T];H1)), taking values in the functional space KTdefined in (3.1), P-a.s. In particular,\nuµ∈C([0,T];H2),P-a.s. In what follows, by using the notations introduced in ( 3.13), we denote\nΛ1:= sup\nµ∈(0,1)Λ1.\nDue to (3.8), we have Λ 1<∞.\nIn this section assuming Hypothesis 4, we are going to prove t hat for every T >0 there exists a\nconstantcT>0 depending on Λ 1such that\nEsup\ns∈[0,T]/ba∇dbluµ(s)/ba∇dbl2\nH+/integraldisplayT\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1ds≤cT. (5.1)\nMoreover, we will prove that\nµ3Esup\nt∈[0,T]/ba∇dbl∂tuµ(t)/ba∇dbl4\nH+µEsup\nt∈[0,T]/ba∇dbluµ(t)/ba∇dbl4\nH1+µ/integraldisplayT\n0E/ba∇dbluµ(t)/ba∇dbl2\nH1/ba∇dbl∂tuµ(t)/ba∇dbl2\nHdt\n+µ2/integraldisplayT\n0E/ba∇dbl∂tuµ(t)/ba∇dbl4\nHdt+µ/integraldisplayT\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds≤cT.(5.2)\nLemma 5.1. Assume Hypotheses 1 to 4 and fix (uµ\n0,vµ\n0)∈H2satisfying condition (3.8). Then for\neveryT >0there exists cT>0, depending on Λ1, such that for every µ∈(0,1)andt∈[0,T]\nµ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl4\nH+Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl4\nH1+/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH/ba∇dbluµ(s)/ba∇dbl2\nHds\n+µ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl4\nHds≤cT\nµ/parenleftbigg/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1ds+1/parenrightbigg\n+cTµEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH.(5.3)\nProof.If we define K(v) =/ba∇dblv/ba∇dbl4\nH, we have\nDK(v) = 4/ba∇dblv/ba∇dbl2\nHv, D2K(v) = 8v⊗v+4/ba∇dblv/ba∇dbl2\nHI.\nIn particular, according to (2.11), we get\n∞/summationdisplay\nk=1/an}b∇acketle{tD2K(v)σ(u)Qek,σ(u)Qek/an}b∇acket∇i}htH= 8/ba∇dbl[σ(u)Q]⋆v/ba∇dbl2\nH+4/ba∇dblv/ba∇dbl2\nH/ba∇dblσ(u)/ba∇dbl2\nL2(HQ,H)≤c/ba∇dblv/ba∇dbl2\nH.\n23Hence, if we apply Itˆ o’s formula to Kand∂tuµ(t), we get\nd/ba∇dbl∂tuµ(t)/ba∇dbl4\nH≤4\nµ/ba∇dbl∂tuµ(t)/ba∇dbl2\nH/an}b∇acketle{t∂tuµ(t),∆uµ(t)−γ(uµ(t))∂tuµ(t)+f(uµ(t))/an}b∇acket∇i}htHdt\n+c\nµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nHdt+4\nµ/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH/ba∇dbl∂tuµ(t)/ba∇dbl2\nH\n≤ −2\nµ/ba∇dbl∂tuµ(t)/ba∇dbl2\nHd/ba∇dbluµ(t)/ba∇dbl2\nH1−4γ0\nµ/ba∇dbl∂tuµ(t)/ba∇dbl4\nHdt+c\nµ(1+/ba∇dbluµ(t)/ba∇dblH)/ba∇dbl∂tuµ(t)/ba∇dbl3\nHdt\n+c\nµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nHdt+4\nµ/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH/ba∇dbl∂tuµ(t)/ba∇dbl2\nH,\nso that\nd/ba∇dbl∂tuµ(t)/ba∇dbl4\nH+2γ0\nµ/ba∇dbl∂tuµ(t)/ba∇dbl4\nHdt+2\nµd/parenleftbig\n/ba∇dbl∂tuµ(t)/ba∇dbl2\nH/ba∇dbluµ(t)/ba∇dbl2\nH1/parenrightbig\n≤2\nµd/ba∇dbl∂tuµ(t)/ba∇dbl2\nH/ba∇dbluµ(t)/ba∇dbl2\nH1\n+c\nµ/parenleftbig\n1+/ba∇dbluµ(t)/ba∇dbl4\nH/parenrightbig\ndt+c\nµ3dt+4\nµ/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH/ba∇dbl∂tuµ(t)/ba∇dbl2\nH.(5.4)\nNow, thanks again to the Itˆ o formula, we have\nd/ba∇dbl∂tuµ(t)/ba∇dbl2\nH=2\nµ/an}b∇acketle{t∂tuµ(t),∆uµ(t)−γ(uµ(t))∂tuµ(t)+f(uµ(t))/an}b∇acket∇i}htHdt\n+1\nµ2/ba∇dblσ(uµ(t))/ba∇dbl2\nL2(HQ,H)dt+2\nµ/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH\n≤ −1\nµd/ba∇dbluµ(t)/ba∇dbl2\nH1−γ0\nµ/ba∇dbl∂tuµ(t)/ba∇dbl2\nHdt+c\nµ/parenleftbig\n1+/ba∇dbluµ(t)/ba∇dbl2\nH/parenrightbig\ndt\n+c\nµ2dt+2\nµ/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH,\nand this gives\n2\nµd/ba∇dbl∂tuµ(t)/ba∇dbl2\nH/ba∇dbluµ(t)/ba∇dbl2\nH1≤ −1\nµ2d/ba∇dbluµ(t)/ba∇dbl4\nH1−2γ0\nµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH/ba∇dbluµ(t)/ba∇dbl2\nH1dt\n+c\nµ2dt+c\nµ2/ba∇dbluµ(t)/ba∇dbl4\nH1dt+c\nµ3/ba∇dbluµ(t)/ba∇dbl2\nH1dt+4\nµ2/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH/ba∇dbluµ(t)/ba∇dbl2\nH1.\nTherefore, if we plug the inequality above into (5.4), we get\nd/ba∇dbl∂tuµ(t)/ba∇dbl4\nH+2\nµd/parenleftbig\n/ba∇dbl∂tuµ(t)/ba∇dbl2\nH/ba∇dbluµ(t)/ba∇dbl2\nH/parenrightbig\n+1\nµ2d/ba∇dbluµ(t)/ba∇dbl4\nH1+2γ0\nµ/ba∇dbl∂tuµ(t)/ba∇dbl4\nHdt\n+2γ0\nµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH/ba∇dbluµ(t)/ba∇dbl2\nH1dt≤c\nµ2/ba∇dbluµ(t)/ba∇dbl4\nH1dt+c\nµ3/ba∇dbluµ(t)/ba∇dbl2\nH1dt+c\nµ3dt\n+4\nµ2/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH/ba∇dbluµ(t)/ba∇dbl2\nH1+4\nµ/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH/ba∇dbl∂tuµ(t)/ba∇dbl2\nH.(5.5)\n24Thanks to (2.11), we have\n4\nµ2Esup\ns∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplays\n0/an}b∇acketle{tσ(uµ(r))dwQ(r),∂tuµ(r)/an}b∇acket∇i}htH/ba∇dbluµ(s)/ba∇dbl2\nH1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c\nµ2E/parenleftbigg/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH/ba∇dbluµ(s)/ba∇dbl4\nH1ds/parenrightbigg1\n2\n≤c\nµ2E/parenleftBigg\nsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH1/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH/ba∇dbluµ(s)/ba∇dbl2\nH1ds/parenrightBigg1/2\n≤γ0\nµ2E/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH/ba∇dbluµ(s)/ba∇dbl2\nH1ds+1\n2µ2Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl4\nH1+c\nµ2,\nand\n4\nµEsup\ns∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplays\n0/ba∇dbl∂tuµ(r)/ba∇dbl2\nH/an}b∇acketle{tσ(uµ(r))dwQ(r),∂tuµ(r)/an}b∇acket∇i}htH/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤c\nµE/parenleftbigg/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl6\nHds/parenrightbigg1\n2\n≤c\nµE/parenleftBigg\nsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl4\nHds/parenrightBigg1\n2\n≤γ0\nµE/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl4\nHds+c\nµEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH.\nHence, if we first integrate with respect to time both sides in (5.5), then take the supremum with\nrespect to time and finally take the expectation, we get\nEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl4\nH+1\n2µ2Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl4\nH1+γ0\nµ2/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH/ba∇dbluµ(s)/ba∇dbl2\nHds\n+γ0\nµ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl4\nHds≤ /ba∇dblvµ\n0/ba∇dbl4\nH+2\nµ/ba∇dbluµ\n0/ba∇dbl2\nH1/ba∇dblvµ\n0/ba∇dbl2\nH+1\nµ2/ba∇dbluµ\n0/ba∇dbl4\nH1\n+c\nµ2/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl4\nH1ds+c\nµ3/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1dr+c\nµ3t+c\nµEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH.\nThanks to (3.8) we have\n/ba∇dblvµ\n0/ba∇dbl4\nH+2\nµ/ba∇dbluµ\n0/ba∇dbl2\nH1/ba∇dblvµ\n0/ba∇dbl2\nH+1\nµ2/ba∇dbluµ\n0/ba∇dbl4\nH1≤c\nµ2Λ1.\nHence, the Gronwall lemma allows to conclude that\nEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl4\nH+1\nµ2Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl4\nH1+1\nµ2/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH/ba∇dbluµ(s)/ba∇dbl2\nHds\n+1\nµ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl4\nHds≤cT\nµ3+c\nµ2Λ1+cT\nµ3/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1ds+cT\nµEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH,\nand (5.3) follows, once we multiply both sides by µ2.\n/square\n25Lemma 5.2. Assume Hypotheses 1 to 4, and fix T >0and(uµ\n0,vµ\n0)∈H2satisfying (3.8). Then,\nthere exists cT>0, depending on Λ1, such that for every µ∈(0,1)andt∈[0,T]\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH+/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1ds\n≤cT/parenleftBigg\n1+µ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+µ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH/parenrightBigg\n.(5.6)\nProof.We have\n/an}b∇acketle{tη(uµ(t)),µ∂tuµ(t)/an}b∇acket∇i}htH=/an}b∇acketle{tη(u0),µv0/an}b∇acket∇i}htH+/integraldisplayt\n0/an}b∇acketle{t∂tη(uµ(s)),µ∂tuµ(s)/an}b∇acket∇i}htHds\n+/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),∆uµ(s)/an}b∇acket∇i}htHds−/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),γ(uµ(s))∂tuµ(s)/an}b∇acket∇i}htHds\n+/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),f(uµ(s))/an}b∇acket∇i}htHds+/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),σ(uµ(s))dwQ(s)/an}b∇acket∇i}htH\n=:/an}b∇acketle{tη(u0),µv0/an}b∇acket∇i}htH+4/summationdisplay\nk=1Ik(t),(5.7)\nwhere\nη(h) :=γ−1(h)h, h∈H,\nWe have\n∂tη(uµ(t)) = [Dγ−1(uµ(t))∂tuµ(t)]uµ(t)+γ−1(uµ(t))∂tuµ(t).\nThen, since\nDγ−1(uµ(t))∂tuµ(t)∈Rr×r, γ−1(uµ(t))∈Rr×r,\nand\nDg−1(u) =−g−1(u)Dg(u)g−1(u),\nwe have\n|I1(t)| ≤µ/integraldisplayt\n0/vextendsingle/vextendsingle/an}b∇acketle{t[Dγ−1(uµ(s))∂tuµ(s)]uµ(s),∂tuµ(s)/an}b∇acket∇i}htH+/an}b∇acketle{tγ−1(uµ(s))∂tuµ(s),∂tuµ(s)/an}b∇acket∇i}htH1/vextendsingle/vextendsingleds\n≤cµ/integraldisplayt\n0/ba∇dblDg(uµ(s))/ba∇dblL(H1,Rr×r)/ba∇dbl∂tuµ(s)/ba∇dblH1/ba∇dbl∂tuµ(s)/ba∇dblH/ba∇dbluµ(s)/ba∇dblHds+cµ/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds.\nHence, for every ǫ∈(0,1) there exists cǫ>0 such that\n|I1(t)| ≤ǫµ/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds+cǫµ/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds. (5.8)\nConcerning I2(t), due to (2.27), we have\nI2(t) =−/integraldisplayt\n0/an}b∇acketle{tγ−1(uµ(s))∇uµ(s),∇uµ(s)/an}b∇acket∇i}htHds≤ −˜γ0/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH1ds. (5.9)\n26As forI3(t), we have\nI3(t) =−/integraldisplayt\n0/an}b∇acketle{tuµ(t),∂tuµ(s))/an}b∇acket∇i}htHds=1\n2(−/ba∇dbluµ(t)/ba∇dbl2\nH+/ba∇dbluµ\n0/ba∇dbl2\nH). (5.10)\nNext, concerning I4(t), we have\n|I4(t)| ≤c/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nHds+ct. (5.11)\nFinally, we have\n|/an}b∇acketle{tη(uµ(t)),µ∂tuµ(t)/an}b∇acket∇i}htH| ≤cµ/ba∇dbluµ(t)/ba∇dblH/ba∇dbl∂tuµ(t)/ba∇dblH≤1\n4/ba∇dbluµ(t)/ba∇dbl2\nH+cµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH.(5.12)\nTherefore, collecting together (5.8), (5.9), (5.10), (5.1 1) and (5.12), from (5.7) we get\n1\n4/ba∇dbluµ(t)/ba∇dbl2\nH+ ˜γ0/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH1ds\n≤ /an}b∇acketle{tη(uµ\n0),µvµ\n0/an}b∇acket∇i}htH+1\n4/ba∇dbluµ\n0/ba∇dbl2\nH+ǫµ/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds+cǫµ/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds\n+c/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nHds+ct+cµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),σ(uµ(s))dwQ(s)/an}b∇acket∇i}htH/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nso that, thanks to Gronwall’s lemma and (3.8) there exists cT>0, depending on Λ 1, such that\n1\n2sup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH+˜γ0\n2/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH1ds\n≤cT+ǫcTµ/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds+cT,ǫµ/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds\n+cTµ2sup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH+cTsup\ns∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplays\n0/an}b∇acketle{tη(uµ(r)),σ(uµ(r))dwQ(r)/an}b∇acket∇i}htH/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(5.13)\nAccording to (2.11), we have\nEsup\nr∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n0/an}b∇acketle{tη(uµ(s)),σ(uµ(s))dwQ(s)/an}b∇acket∇i}htH/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤cE/parenleftbigg/integraldisplayt\n0/parenleftbig\n1+/ba∇dbluµ(s)/ba∇dbl2\nH/parenrightbig\nds/parenrightbigg1\n2\n≤˜γ0\n2E/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH1ds+cT,\nand then, if we take the expectation in (5.13), we get\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH+/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1ds≤cT+ǫcTµ/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds\n+cT,ǫµ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+cTµ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH.(5.14)\n27According to (5.3), we have\nµ/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds≤cT/parenleftbigg/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1ds+1/parenrightbigg\n+cTµ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH,\nand if we plug this inequality into (5.14) we get\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH+/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1ds\n≤cT+ǫcT/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1ds+cT,ǫµ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+cT,ǫµ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH.\nTherefore, if we pick ¯ ǫ∈(0,1) such that ¯ ǫcT≤1/2, we get (5.6).\n/square\nLemma 5.3. Under Hypotheses 1 to 4, for every T >0and(uµ\n0,vµ\n0)∈H2satisfying (3.8), there\nexists some constant cT>0, depending on Λ1, such that for every µ∈(0,1)andt∈[0,T]\nEsup\ns∈[0,T]/ba∇dbluµ(s)/ba∇dbl2\nH1+/integraldisplayT\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds+µEsup\ns∈[0,T]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH≤cT\nµ. (5.15)\nProof.We have\n1\n2d/parenleftbig\n/ba∇dbluµ(t)/ba∇dbl2\nH1+µ/ba∇dbl∂tuµ(t)/ba∇dbl2\nH/parenrightbig\n=−/an}b∇acketle{tγ(uµ(t))∂tuµ(t),∂tuµ(t)/an}b∇acket∇i}htHdt\n+/an}b∇acketle{tf(uµ(t)),∂tuµ(t)/an}b∇acket∇i}htHdt+1\n2µ/ba∇dblσ(uµ(t))/ba∇dbl2\nL2(HQ,H)dt+/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH.\nDue to (2.1) and (2.11), for every µ∈(0,1) this gives\n1\n2d/parenleftbig\n/ba∇dbluµ(t)/ba∇dbl2\nH1+µ/ba∇dbl∂tuµ(t)/ba∇dbl2\nH/parenrightbig\n≤ −γ0\n2/ba∇dbl∂tuµ(t)/ba∇dbl2\nHdt+c/ba∇dbluµ(t)/ba∇dbl2\nHdt+c\nµdt+/an}b∇acketle{t∂tuµ(t),σ(uµ(t))dwQ(t)/an}b∇acket∇i}htH,\nso that, thanks to (3.8),\n/ba∇dbluµ(t)/ba∇dbl2\nH1+µ/ba∇dbl∂tuµ(t)/ba∇dbl2\nH+γ0/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds\n≤Λ1\n2+cT\nµ+c/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nHds+/integraldisplayt\n0/an}b∇acketle{t∂tuµ(s),σ(uµ(s))dwQ(s)/an}b∇acket∇i}htH.(5.16)\nIn particular, since\nEsup\ns∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{t∂tuµ(s),σ(uµ(s))dwQ(s)/an}b∇acket∇i}htH/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤cE/parenleftbigg/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds/parenrightbigg1\n2\n≤γ0\n2/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds+cT,\n28by taking the supremum with respect to time and the expectati on in both sides of (5.16), we obtain\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH1+µEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH+γ0\n2/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nHds≤cT\nµ+c/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nHds,\nand the Gronwall lemma allows to obtain (5.15). /square\nFrom the combination of Lemma 5.2 and Lemma 5.3, we obtain (5. 1). Moreover, if we replace\n(5.1) and (5.15) into (5.3) we obtain (5.2).\n6.Uniform estimates in H2\nWe have different bounds in H2depending on the fact that the mapping σ:H1→L2(HQ,H1)\nis bounded or not and for this reason we will have to consider s eparately the two cases. First, we\nconstruct a family of suitable truncations for the coefficien tsγandσand introduce a family of\napproximating problems.\nFor everyR≥1, we fix a continuously differentiable function Φ R:R→[0,1] having a bounded\nderivative and such that Φ R(t) = 1, ift≤R,and,ΦR(t) = 0, ift≥R+1. Once introduced Φ R,\nwe fix\n¯r∈/parenleftbigg2¯s\n1+ ¯s∨(2¯κ−3)∨̺,1/parenrightbigg\n(6.1)\nwhere ¯s <1 and ¯κ <2 are the constants introduced in Hypothesis 4 and ̺ <1 is the constant\nintroduced in the statement of Theorem 3.3. Next, for every R≥1 andh∈H1we define\nγR(h) := ΦR(/ba∇dblh/ba∇dblH¯r)γ(h), f R(h) := ΦR(/ba∇dblh/ba∇dblH¯r)f(h), σ R(h) := ΦR(/ba∇dblh/ba∇dblH¯r)σ(h).\nClearly,\n/ba∇dblh/ba∇dblH¯r≤R=⇒γR(h) =γ(h), fR(h) =f(h), σR(h) =σ(h). (6.2)\nMoreover, it is immediate to check that all the mappings γR:H1→L(H) satisfy Hypothesis 1\nand Condition 1 in Hypothesis 4, with\ngR(h) :=γ0+ΦR(/ba∇dblh/ba∇dblH¯r)(g(h)−γ0), h∈H1,\nand all constants involved are independent of R≥1. In the same way, it is easy to check that all\nmappingsσR:H1→L2(HQ,H1) satisfy Hypotheses 3, with all constants that are independ ent\nofR≥1. Finally, if σR:H1→L2(HQ,H1) is not bounded, then σ:H1→L2(HQ,H1) is not\nbounded, so that, due to Condition 3 in Hypothesis 4, gandσsatisfy (2.22) and (2.20). We can\ncheck that this implies that also gRandσRsatisfy (2.22) and (2.20), with constants independent\nofR≥1. Finally, all fRsatisfy Hypothesis 2, and Condition 2 in Hypothesis 4, with a ll constants\nindependent of R≥1.\nNow, for every R≥1 and andµ>0, we consider the problem\n\n\nµ∂2\ntuR\nµ(t) = ∆uR\nµ(t)−γR(uR\nµ(t))∂tuR\nµ(t)+fR(uR\nµ(t))+σR(uR\nµ(t,x))dwQ\nt(t),\nuR\nµ(0,x) =uµ\n0(x), ∂tuR\nµ(0,x) =vµ\n0(x), uR\nµ(t,x) = 0, x∈∂O.(6.3)\nIn view of what we have discussed above for γR,fRandσR, we have that for every µ >0 and\n(uµ\n0,vµ\n0)∈H2, and for every R≥1,T >0 andp≥1 there exists a unique adapted process\nuR\nµtaking values in KT,P-a.s., and solving equation (6.3), such that uR\nµ∈L2(Ω;C([0,T];H1)).\nMoreover, all uR\nµsatisfy (5.1), for some constant cT>0 independent of R≥1. In what follows,\nwe shall denote\nΛ2:= sup\nµ∈(0,1)µδΛµ\n2,\n29where Λµ\n2has been defined in (3.13).\nIn this section, we are going to prove that if ( uµ\n0,vµ\n0) satisfy (3.8) and (3.9), then there exists\nsomecT,R>0, depending on Λ 1and Λ2, such that for every µ∈(0,1)\nEsup\ns∈[0,T]/ba∇dbluR\nµ(s)/ba∇dbl2\nH1+/integraldisplayT\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds≤cT,R, (6.4)\nand\nµEsup\ns∈[0,T]/ba∇dbluR\nµ(s)/ba∇dbl2\nH2+µ2Esup\ns∈[0,T]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1+µ/integraldisplayT\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds≤cT,R.(6.5)\nWe will also prove that for every µ∈(0,1)\n√µ/parenleftBigg\nEsup\nt∈[0,T]/ba∇dbluR\nµ(t)/ba∇dbl2\nH1+¯r+µEsup\nt∈[0,T]/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH¯r/parenrightBigg\n≤cT,R, (6.6)\nwhere ¯ris the constant introduced in (6.1).\n6.1.Proof of the H2-bounds in the case of bounded σ.Throughout this subsection, we\nassume that\nsup\nh∈H1/ba∇dblσ(h)/ba∇dblL2(HQ,H1)<+∞. (6.7)\nWe will prove that in this case uµitself satisfies estimates (6.4), and even a stronger versio n of\n(6.6). Namely\n√µ/parenleftBigg\nEsup\nt∈[0,T]/ba∇dbluµ(t)/ba∇dbl2\nH2+µEsup\nt∈[0,T]/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1/parenrightBigg\n≤cT.\nIn particular, all uR\nµsatisfy the same bounds as well, for some constant cT>0 independent of\nR≥1.\nMoreover, we will also prove that for all µ∈(0,1)\nµEsup\nt∈[0,T]/ba∇dbluµ(t)/ba∇dbl4\nH2+µ2Esup\nr∈[0,T]/ba∇dbl∂tuµ(r)/ba∇dbl2\nH1+µ3Esup\nt∈[0,T]/ba∇dbl∂tuµ(t)/ba∇dbl4\nH1\n+µ/integraldisplayT\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+µ2/integraldisplayT\n0E/ba∇dbl∂tuµ(t)/ba∇dbl4\nH1dt+µ/integraldisplayT\n0E/ba∇dbluµ(t)/ba∇dbl2\nH2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1dt≤cT.(6.8)\nWe start with the following lemma, that is an analogous to Lem ma 5.1.\nLemma 6.1. Assume Hypotheses 1 to 4 and fix T >0and(uµ\n0,vµ\n0)∈H2satisfying (3.8)and(3.9).\nThen, there exists cT>0, depending on Λ1andλ2, such that for every µ∈(0,1)andt∈[0,T]\nµ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl4\nH1+Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl4\nH2+µ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl4\nH1ds\n+/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds≤cT\nµ/parenleftbigg/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2ds+1/parenrightbigg\n+cTµEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1.(6.9)\nProof.If we define K(v) =/ba∇dblv/ba∇dbl4\nH1, we have\nDK(v) = 4/ba∇dblv/ba∇dbl2\nH1(−∆)v, D2K(v) = 8(−∆)v⊗(−∆)v+4/ba∇dblv/ba∇dbl2\nH1(−∆),\n30and hence\n∞/summationdisplay\nk=1D2K(v)(σ(u)Qek,σ(u)Qek) = 8/ba∇dbl[σ(u)Q]⋆(−∆)v/ba∇dbl2\nH+4/ba∇dblv/ba∇dbl2\nH1/ba∇dblσ(u)/ba∇dbl2\nL2(HQ,H1).\nFor everyh∈H, we have\n|/an}b∇acketle{t[σ(u)Q]⋆(−∆)v,h/an}b∇acket∇i}htH|=|/an}b∇acketle{tv,[σ(u)Q]h/an}b∇acket∇i}htH1| ≤ /ba∇dblσ(u)/ba∇dblL(HQ,H1)/ba∇dblv/ba∇dblH1/ba∇dblh/ba∇dblH,\nso that, thanks to (6.7), we get\nsup\nu∈H1/ba∇dbl[σ(u)Q]⋆(−∆)v/ba∇dblH≤c/ba∇dblv/ba∇dblH1,\nand\n∞/summationdisplay\nk=1D2K(v)(σ(u)Qek,σ(u)Qek)≤c/ba∇dblv/ba∇dbl2\nH1.\nNow, if we apply Itˆ o’s formula to Kand∂tuµ(t), by proceeding as in the proof of Lemma 5.1 we\nhave\nd/ba∇dbl∂tuµ(t)/ba∇dbl4\nH1+2γ0\nµ/ba∇dbl∂tuµ(t)/ba∇dbl4\nH1dt+2\nµd/parenleftbig\n/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1/ba∇dbluµ(t)/ba∇dbl2\nH2/parenrightbig\n≤2\nµd/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1/ba∇dbluµ(t)/ba∇dbl2\nH2\n+c\nµ/parenleftbig\n1+/ba∇dbluµ(t)/ba∇dbl4\nH1/parenrightbig\ndt+4\nµ/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH1/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1+c\nµ3dt.(6.10)\nNext, as we did in the proof of Lemma 5.1 for /ba∇dbl∂tuµ(t)/ba∇dbl2\nH, we apply Itˆ o’s formula to /ba∇dbl∂tuµ(t)/ba∇dbl2\nH1\nand we get\n2\nµd/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1/ba∇dbluµ(t)/ba∇dbl2\nH2≤ −1\nµ2d/ba∇dbluµ(t)/ba∇dbl4\nH2−2γ0\nµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1/ba∇dbluµ(t)/ba∇dbl2\nH2dt\n+c\nµ2/ba∇dbluµ(t)/ba∇dbl4\nH2dt+c\nµ3/ba∇dbluµ(t)/ba∇dbl2\nH2dt+4\nµ2/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH1/ba∇dbluµ(t)/ba∇dbl2\nH2.\nTherefore, we replace the inequality above into (6.10) and w e get\nd/ba∇dbl∂tuµ(t)/ba∇dbl4\nH1+2\nµd/parenleftbig\n/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1/ba∇dbluµ(t)/ba∇dbl2\nH2/parenrightbig\n+1\nµ2d/ba∇dbluµ(t)/ba∇dbl4\nH2\n+2γ0\nµ/ba∇dbl∂tuµ(t)/ba∇dbl4\nH1dt+2γ0\nµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1/ba∇dbluµ(t)/ba∇dbl2\nH2dt\n≤c\nµ2/ba∇dbluµ(t)/ba∇dbl4\nH2dt+c\nµ3/ba∇dbluµ(t)/ba∇dbl2\nH2dt+c\nµ3dt\n+4\nµ2/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH1/ba∇dbluµ(t)/ba∇dbl2\nH2+4\nµ/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH1/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1.(6.11)\n31Thanks to (2.10), we have\n4\nµ2Esup\ns∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplays\n0/an}b∇acketle{tσ(uµ(r))dwQ(r),∂tuµ(r)/an}b∇acket∇i}htH1/ba∇dbluµ(r)/ba∇dbl2\nH2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤c\nµ2E/parenleftbigg/integraldisplayt\n0/ba∇dbl∂tuµ(r)/ba∇dbl2\nH1/parenleftbig\n/ba∇dbluµ(r)/ba∇dbl2\nH1+1/parenrightbig\n/ba∇dbluµ(r)/ba∇dbl4\nH2dr/parenrightbigg1\n2\n≤γ0\nµ2/integraldisplayt\n0E/ba∇dbl∂tuµ(r)/ba∇dbl2\nH1/ba∇dbluµ(r)/ba∇dbl2\nH2dr+c\nµ2Esup\nr∈[0,t]/parenleftbig\n/ba∇dbluµ(r)/ba∇dbl2\nH1+1/parenrightbig\n/ba∇dbluµ(r)/ba∇dbl2\nH2,\n≤γ0\nµ2/integraldisplayt\n0E/ba∇dbl∂tuµ(r)/ba∇dbl2\nH1/ba∇dbluµ(r)/ba∇dbl2\nH2dr+1\n2µ2Esup\nr∈[0,t]E/ba∇dbluµ(r)/ba∇dbl4\nH2+c\nµ2.\nMoreover,\n4\nµEsup\ns∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplays\n0/ba∇dbl∂tuµ(r)/ba∇dbl2\nH1/an}b∇acketle{tσ(uµ(r))dwQ(r),∂tuµ(r)/an}b∇acket∇i}htH1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤c\nµE/parenleftbigg/integraldisplayt\n0/ba∇dbl∂tuµ(r)/ba∇dbl6\nH1/parenleftbig\n/ba∇dbluµ(r)/ba∇dbl2\nH1+1/parenrightbig\ndr/parenrightbigg1\n2\n≤c\nµE/parenleftBigg\nsup\nr∈[0,t]/ba∇dbl∂tuµ(r)/ba∇dbl2\nH1/parenleftbig\n/ba∇dbluµ(r)/ba∇dbl2\nH1+1/parenrightbig/integraldisplayt\n0/ba∇dbl∂tuµ(r)/ba∇dbl4\nH1dr/parenrightBigg1\n2\n≤γ0\n4µE/integraldisplayt\n0/ba∇dbl∂tuµ(r)/ba∇dbl4\nH1dr+c\nµEsup\nr∈[0,t]/ba∇dbl∂tuµ(r)/ba∇dbl2\nH1/parenleftbig\n/ba∇dbluµ(r)/ba∇dbl2\nH1+1/parenrightbig\n,\nso that we have\n4\nµEsup\ns∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplays\n0/ba∇dbl∂tuµ(r)/ba∇dbl2\nH1/an}b∇acketle{tσ(uµ(r))dwQ(r),∂tuµ(r)/an}b∇acket∇i}htH1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤γ0\nµE/integraldisplayt\n0/ba∇dbl∂tuµ(r)/ba∇dbl4\nH1dr+1\n2Esup\nr∈[0,t]/ba∇dbl∂tuµ(r)/ba∇dbl4\nH1+c\nµ2Esup\nr∈[0,t]/ba∇dbluµ(r)/ba∇dbl4\nH1+c\nµ2.\nIn particular, if, as we did in the proof of Lemma 5.1, we first i ntegrate with respect to time both\nsides in (6.11), then take the supremum with respect to time, take the expectation and finally apply\n32Gronwall’s lemma, thanks to (3.9) we get\nEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl4\nH1+1\nµ2Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl4\nH2+1\nµ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl4\nH1ds\n+1\nµ2/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds≤Λ2\n2\nµ2(1+δ)+cT\nµ3\n+cT\nµ3/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2ds+cT\nµEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1+cT\nµ2Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl4\nH1.(6.12)\nAccording to (5.3) and (5.1) we have\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl4\nH1≤cT\nµ/parenleftBigg\n1+µ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH/parenrightBigg\n.\nThus, recalling that δ <1/2, (6.9) follows once we replace the inequality above in (6.1 2) and\nmultiply both sides by µ2. /square\nLemma 6.2. Assume Hypotheses 1 to 4, and fix T >0and(uµ\n0,vµ\n0)∈H2satisfying (3.8)and\n(3.9). Then, there exists cT>0, depending on Λ1andΛ2, such that for every µ∈(0,1)and\nt∈[0,T]\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH1+/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2ds\n≤cT/parenleftBigg\n1+µ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+µ2Esup\nr∈[0,t]/ba∇dbl∂tuµ(r)/ba∇dbl2\nH1/parenrightBigg\n.(6.13)\nProof.We have\n/an}b∇acketle{tη(uµ(t)),µ∂tuµ(t)/an}b∇acket∇i}htH1=/an}b∇acketle{tη(u0),µv0/an}b∇acket∇i}htH1+/integraldisplayt\n0/an}b∇acketle{t∂tη(uµ(s)),µ∂tuµ(s)/an}b∇acket∇i}htH1ds\n+/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),∆uµ(s)/an}b∇acket∇i}htH1ds−/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),γ(uµ(s))∂tuµ(s)/an}b∇acket∇i}htH1ds\n+/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),f(uµ(s))/an}b∇acket∇i}htH1ds+/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),σ(uµ(s))dwQ(s)/an}b∇acket∇i}htH1\n=:/an}b∇acketle{tη(u0),µv0/an}b∇acket∇i}htH1+4/summationdisplay\nk=1Ik(t),(6.14)\nwhere, as in Lemma 5.2, we defined η(h) :=γ−1(h)hfor everyh∈H.\n33By proceeding as in the proof of Proposition 5.2, for every ǫ∈(0,1) there exists cǫ>0 such\nthat\n|I1(t)| ≤µ/integraldisplayt\n0/vextendsingle/vextendsingle/an}b∇acketle{t[Dγ−1(uµ(s))∂tuµ(s)]uµ(s),∂tuµ(s)/an}b∇acket∇i}htH1+/an}b∇acketle{tγ−1(uµ(s))∂tuµ(s),∂tuµ(s)/an}b∇acket∇i}htH1/vextendsingle/vextendsingleds\n≤ǫµ2/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl4\nH1ds+cǫ/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH1ds+cµ/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds.\n(6.15)\nForI2(t) andI3(t), we have\nI2(t)≤ −˜γ0/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH2ds, (6.16)\nand\nI3(t) =−/integraldisplayt\n0/an}b∇acketle{tuµ(t),∂tuµ(s))/an}b∇acket∇i}htH1ds=−/ba∇dbluµ(t)/ba∇dbl2\nH1+/ba∇dbluµ\n0/ba∇dbl2\nH1. (6.17)\nForI4(t), due Hypothesis 4 we have\nI4(t) =/integraldisplayt\n0/an}b∇acketle{tη(uµ(s)),f(uµ(s))/an}b∇acket∇i}htH1ds=−/integraldisplayt\n0/an}b∇acketle{tγ−1(uµ(s))∇uµ(s),∇[f(uµ(s))]/an}b∇acket∇i}htHds,\nand this yields\n|I4(t)| ≤c/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH1ds+ct. (6.18)\nTherefore, since\n|/an}b∇acketle{tη(uµ(t)),µ∂tuµ(t)/an}b∇acket∇i}htH1| ≤1\n2/ba∇dbluµ(t)/ba∇dbl2\nH1+cµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1,\ncollecting together (6.15), (6.16), (6.17), and (6.18), fr om (6.14) we get\n1\n2/ba∇dbluµ(t)/ba∇dbl2\nH1+ ˜γ0/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH2ds\n≤ /an}b∇acketle{tη(uµ\n0),µvµ\n0/an}b∇acket∇i}htH1+/ba∇dbluµ\n0/ba∇dbl2\nH1+cµ/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+ǫµ2/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl4\nH1ds\n+cǫ/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH1ds+cµ2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1+|I5(t)|.(6.19)\nAs a consequence of (2.10) and (5.1), we have\nEsup\nr∈[0,t]|I5(t)| ≤cE/parenleftbigg/integraldisplayt\n0/parenleftbig\n1+/ba∇dbluµ(s)/ba∇dbl2\nH1/parenrightbig\n/ba∇dbluµ(s)/ba∇dbl2\nH1ds/parenrightbigg1\n2\n≤1\n4Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH1+cT,\nand then, if we take first the supremum with respect to tand then the expectation in both sides of\n(6.19), thanks to (3.8), (3.9) and (5.1) we get\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH1+/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2ds\n≤cT+cµ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+ǫµ2/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl4\nH1ds+cµ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1,\n34for some constant cT>0 depending on Λ 1and Λ2. In view of (6.9), this implies that\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH1+/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2ds\n≤cTµ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+cTµ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH+ǫcT/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2ds+cT.\nThus, if we take ¯ ǫ∈(0,1) such that ¯ ǫcT<1/2, we get\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH1+/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2ds\n≤cTµ/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+cTµ2Esup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH+cT,\nand (6.13) follows.\n/square\nLemma 6.3. Under Hypotheses 1 to 4, for every T >0and(uµ\n0,vµ\n0)∈H2satisfying (3.8)and\n(3.9)there exists cT>0, depending on Λ1andΛ2, such that for every µ∈(0,1)andt∈[0,T]\nEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH2+µEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1+/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds≤cT\nµ. (6.20)\nProof.We have\n1\n2d/parenleftbig\n/ba∇dbluµ(t)/ba∇dbl2\nH2+µ/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1/parenrightbig\n=−/an}b∇acketle{tγ(uµ(t))∂tuµ(t),∂tuµ(t)/an}b∇acket∇i}htH1dt\n+/an}b∇acketle{tf(uµ(t)),∂tuµ(t)/an}b∇acket∇i}htH1dt+1\n2µ/ba∇dblσ(uµ(t))/ba∇dbl2\nL2(HQ,H1)dt+/an}b∇acketle{tσ(uµ(t))dwQ(t),∂tuµ(t)/an}b∇acket∇i}htH1.\nFor everyµ∈(0,1) this gives\n1\n2d/parenleftbig\n/ba∇dbluµ(t)/ba∇dbl2\nH2+µ/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1/parenrightbig\n≤ −γ0\n2/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1dt+c/ba∇dbluµ(t)/ba∇dbl2\nH1dt+c\nµdt+/an}b∇acketle{t∂tuµ(t),σ(uµ(t))dwQ(t)/an}b∇acket∇i}htH1,(6.21)\nso that, thanks to (3.9),\n/ba∇dbluµ(t)/ba∇dbl2\nH2+µ/ba∇dbl∂tuµ(t)/ba∇dbl2\nH1+γ0/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds\n≤Λ2\nµδ+c\nµt+c/integraldisplayt\n0/ba∇dbluµ(s)/ba∇dbl2\nH1ds+2/integraldisplayt\n0/an}b∇acketle{t∂tuµ(s),σ(uµ(s))dwQ(s)/an}b∇acket∇i}htH1.(6.22)\n35Thanks to (2.10), we have\nEsup\ns∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{t∂tuµ(s),σ(uµ(s))dwQ(s)/an}b∇acket∇i}htH1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤cE/parenleftbigg/integraldisplayt\n0/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1/parenleftbig\n1+/ba∇dbluµ(s)/ba∇dbl2\nH1/parenrightbig\nds/parenrightbigg1\n2\n≤γ0\n4/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds+1\n4Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH2+cEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH+cT.\nHence, by taking the supremum with respect to time and the exp ectation in both sides of (6.22),\ndue to (5.1) we obtain\n1\n2Esup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH2+µEsup\ns∈[0,t]/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1+γ0\n2/integraldisplayt\n0E/ba∇dbl∂tuµ(s)/ba∇dbl2\nH1ds\n≤cT\nµ+c/integraldisplayt\n0E/ba∇dbluµ(s)/ba∇dbl2\nH1ds+cEsup\ns∈[0,t]/ba∇dbluµ(s)/ba∇dbl2\nH≤cT,\nand (6.20) follows. /square\nFrom the combination of Propositions 6.2 and 6.3, we obtain t hat for every T >0 there exists a\npositive constant cT, depending on Λ 1and Λ2, such that\nEsup\ns∈[0,T]/ba∇dbluµ(s)/ba∇dbl2\nH1+/integraldisplayT\n0E/ba∇dbluµ(s)/ba∇dbl2\nH2ds≤cT,\nandweobtain (6.4). Inparticular, thanks to(6.20), thisim plies that thereexists apositive constant\ncT>0, depending on Λ 1and Λ2, such that (6.5) holds, for all µ∈(0,1). Finally, if we combine\n(6.4) and (6.5) together with (6.9), we get (6.8).\nBy a suitable modification of the argument used in the proof of Proposition 4.4 in [13] we can\nprove the following result.\nProposition 6.4. Assume Hypotheses 1 to 4 and fix T >0and(uµ\n0,vµ\n0)∈H2satisfying (3.8)and\n(3.9). Then there exists a constant cT>0, depending on Λ1andΛ2, such(6.6)holds.\nProof.Assume that (6.6) is not true. Then, there exists a sequence {µk}k∈N⊂(0,1) converging to\nzero such that\nlim\nk→∞√µk/parenleftBigg\nEsup\nt∈[0,T]/ba∇dbluµk(t)/ba∇dbl2\nH2+µkEsup\nt∈[0,T]/ba∇dbl∂tuµk(t)/ba∇dbl2\nH1/parenrightBigg\n=∞. (6.23)\nFor everyk∈N, we denote\nLk(t) :=/ba∇dbluµk(t)/ba∇dbl2\nH2+µk/ba∇dbl∂tuµk(t)/ba∇dbl2\nH1,\nand we fix a random time τk∈[0,T] such that\nLk(τk) = max\nt∈[0,T]Lk(t).\nAccording to (6.21), we have\ndLk(t)≤c/ba∇dbluµk(t)/ba∇dbl2\nH1dt+¯c\nµkdt+/an}b∇acketle{tσ(uµk(t))dwQ(t),∂tuµk(t)/an}b∇acket∇i}htH1.\n36Therefore, for every random time σ∈[0,T] such that P(σ≤τk) = 1, we have\nLk(τk)−Lk(σ)≤c/integraldisplayτk\nσ/ba∇dbluµk(s)/ba∇dbl2\nH1ds+¯c\nµk(τk−σ)+Mk(τk)−Mk(σ),\nwhere\nMk(t) :=/integraldisplayt\n0/an}b∇acketle{t∂tuµk(s),σ(uµk(s))dwQ(s)/an}b∇acket∇i}htH1.\nThus, if we define the two random variables\nU⋆\nk:=c/integraldisplayT\n0/ba∇dbluµk(t)/ba∇dbl2\nH1dt, M⋆\nk:= 2 sup\nt∈[0,T]|Mk(t)|,\nwe have\nLk(τk)−Lk(σ)≤U⋆\nk+M⋆\nk+¯c\nµk(τk−σ).\nAccording to (3.9), we have\nLk(0) =/ba∇dbluµk\n0/ba∇dbl2\nH2+µk/ba∇dblvµk\n0/ba∇dbl2\nH1≤Λ2\nµδ\nk.\nThen, if we take σ= 0, we get\nτk≥µk\n¯c(Lk(τk)−Lk(0)−U⋆\nk−M⋆\nk)≥µk\n¯c/parenleftbigg\nLk(τk)−Λ2\nµδ\nk−U⋆\nk−M⋆\nk/parenrightbigg\n=:µk\n¯cθk\nOn the set Ek:={θk>0}, if we fix any s∈[τk−µk\n2¯cθk,τk], we have\nLk(s)≥Lk(τk)−U⋆\nk−M⋆\nk−θk\n2=θk\n2+Λ2\nµδ\nk,\nso that\nIk:=/integraldisplayT\n0Lk(s)ds≥/integraldisplayτk\nτk−µk\n2¯cθkLk(s)ds≥µk\n4¯cθ2\nk+Λ2\n2¯cµ1−δ\nkθk.\nThus, by taking expectation on both sides, we get\nE(Ik)≥E(Ik;Ek)≥E/parenleftBigµk\n4¯cθ2\nk+c0\n2¯cµ1−δ\nkθk;Ek/parenrightBig\n≥1\n4¯cE/parenleftbig\nµkθ2\nk;Ek/parenrightbig\n.\nNow, thanks to (5.1) and (6.8) we have\nsup\nk∈NEU⋆\nk≤cT,sup\nk∈NEM⋆\nk≤cT√µk,\nand, since we are assuming δ<1/2, (6.23) yields\nlim\nk→∞√µkEθk= +∞.\nMoreover,√µkEθk≤/parenleftbig\n(E(µkθ2\nk;Ek)/parenrightbig1/2,\nso that\nlimsup\nk→∞E(Ik)≥1\n4¯climsup\nk→∞E(µkθ2\nk;Ek)≥1\n4¯clim\nk→∞(√µkEθk)2= +∞.\nHowever, this is not possible because, thanks to (6.4) and (6 .5), we have\nsup\nk∈NEIk≤cT.\nThis means that we get to a contradiction of (6.23) and we conc lude that (6.6) has to be true.\n37/square\n6.2.Proof of the H2-bounds in the case of unbounded σ.Since we are not assuming that\nσ:H1→L2(HQ,H1) is bounded, we cannot prove Lemma 6.1, not even for the trunc ated problem\n(6.3), so that we have to find a different way to prove an analogou s of Propositions 6.2 and 6.3,\nthat hold at least for each truncated problems.\nTo this purpose, we notice that, due to the unboundedness of σ, according to Hypothesis 4 we\ncan assume that there exists ¯ s<1 such that g:H¯s→Rr×ris differentiable, with\nsup\nh∈H1/ba∇dblDgR(h)/ba∇dblL(H¯s,Rr×r)<+∞,inf\nh∈H¯s/an}b∇acketle{tgR(h)ξ,ξ/an}b∇acket∇i}htRr≥γ0/ba∇dblξ/ba∇dbl2\nRr, ξ∈Rr.(6.24)\nLemma 6.5. Assume Hypotheses 1 to 4, and fix R≥1,T >0and(uµ\n0,vµ\n0)∈H2satisfying (3.8).\nThen, there exists cT,R>0, depending on Λ1andΛ2, such that for every µ∈(0,1)andt∈[0,T]\nEsup\ns∈[0,t]/ba∇dbluR\nµ(s)/ba∇dbl2\nH1+/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds\n≤cT,R/parenleftBigg\n1+µ/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds+µ2Esup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1/parenrightBigg\n.(6.25)\nProof.As in the proof of Lemma 6.2, we have\n/an}b∇acketle{tηR(uR\nµ(t)),µ∂tuR\nµ(t)/an}b∇acket∇i}htH1=/an}b∇acketle{tηR(uµ\n0),µvµ\n0/an}b∇acket∇i}htH1+/integraldisplayt\n0/an}b∇acketle{t∂tηR(uR\nµ(s)),µ∂tuR\nµ(s)/an}b∇acket∇i}htH1ds\n+/integraldisplayt\n0/an}b∇acketle{tηR(uR\nµ(s)),∆uR\nµ(s)/an}b∇acket∇i}htH1ds−/integraldisplayt\n0/an}b∇acketle{tηR(uR\nµ(s)),γR(uR\nµ(s))∂tuR\nµ(s)/an}b∇acket∇i}htH1ds\n+/integraldisplayt\n0/an}b∇acketle{tηR(uR\nµ(s)),fR(uR\nµ(s))/an}b∇acket∇i}htH1ds+/integraldisplayt\n0/an}b∇acketle{tηR(uR\nµ(s)),σR(uR\nµ(s))dwQ(s)/an}b∇acket∇i}htH1\n=:/an}b∇acketle{tηR(uµ\n0),µvµ\n0/an}b∇acket∇i}htH1+4/summationdisplay\nk=1IR,k(t),\nwhereηR(h) :=γ−1\nR(h)h, for everyh∈H.\nDue to (6.24) and an interpolation argument, we have\n/ba∇dbl[Dγ−1\nR(uR\nµ(s))∂tuR\nµ(s)]uR\nµ(s)/ba∇dblH1≤ /ba∇dblDγ−1\nR(uR\nµ(s))/ba∇dblL(H¯s)/ba∇dbl∂tuR\nµ(s)/ba∇dblH¯s/ba∇dbluR\nµ(s)/ba∇dblH1\n≤ /ba∇dblDγ−1\nR(uR\nµ(s))/ba∇dblL(H¯s)/ba∇dbl∂tuR\nµ(s)/ba∇dbl¯s\nH1/ba∇dbl∂tuR\nµ(s)/ba∇dbl1−¯s\nH/ba∇dbluR\nµ(s)/ba∇dbl1−¯r\n2−¯r\nH2/ba∇dbluR\nµ(s)/ba∇dbl1\n2−¯r\nH¯r.\nRecalling the definition (6.1) of ¯ r, we have ¯r≥2¯s/(1+ ¯s), so that\n1−¯r\n2−¯r≤1−¯s\n2.\nHence, the inequality above gives\n/ba∇dbl[Dγ−1\nR(uR\nµ(s))∂tuR\nµ(s)]uR\nµ(s)/ba∇dblH1≤cR/ba∇dbl∂tuR\nµ(s)/ba∇dbl¯s\nH1/ba∇dbl∂tuR\nµ(s)/ba∇dbl1−¯s\nH/ba∇dbluR\nµ(s)/ba∇dbl1−¯s\n2\nH2.\n38In particular, since we have\nIR,1(t) =µ/integraldisplayt\n0/an}b∇acketle{t[Dγ−1\nR(uR\nµ(s))∂tuR\nµ(s)]uR\nµ(s),∂tuR\nµ(s)/an}b∇acket∇i}htH1ds\n+/an}b∇acketle{tγ−1\nR(uR\nµ(s))∂tuR\nµ(s),∂tuR\nµ(s)/an}b∇acket∇i}htH1ds,\nfor everyǫ>0 we can find cR,ǫ>0 such that\n|IR,1(t)| ≤cRµ/integraldisplayt\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl1+¯s\nH1/ba∇dbl∂tuR\nµ(s)/ba∇dbl1−¯s\nH/ba∇dbluR\nµ(s)/ba∇dbl1−¯s\n2\nH2ds+cµ/integraldisplayt\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds\n≤ǫ/integraldisplayt\n0/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds+cR,ǫµ2/integraldisplayt\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nHds+cR,ǫµ/integraldisplayt\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds.\nAll the other terms IR,k(t), fork= 2,...,5 can be treated as in the proof of Lemma 6.2, so that\nby proceeding as in the proof of Lemma 6.2 we get\nEsup\ns∈[0,t]/ba∇dbluR\nµ(s)/ba∇dbl2\nH1+/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds≤ǫc/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds+cT+c/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH1ds\n+cR,ǫµ/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds+cR,ǫµ2/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nHds+cµ2Esup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1.\nThis means that if we pick ¯ ǫ>0 sufficiently small, in view of (5.3) we have\nEsup\ns∈[0,t]/ba∇dbluR\nµ(s)/ba∇dbl2\nH1+/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds≤cT+c/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH1ds\n+cR,¯ǫµ/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds+cµ2Esup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1,\nand we obtain (6.25).\n/square\nBy proceeding as in the proof of Lemma 6.3, we have that there e xistscT>0, depending on Λ 1\nand Λ2, such that for every µ∈(0,1) andt∈[0,T]\nEsup\ns∈[0,T]/ba∇dbluR\nµ(s)/ba∇dbl2\nH2+µEsup\ns∈[0,T]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1+/integraldisplayT\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds\n≤cT\nµ/parenleftbigg\n1+/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH1ds/parenrightbigg\n.(6.26)\nAs we did in Subsection 6.1, from the combination of (6.25) an d (6.26), we obtain (6.4).\nIn order to prove (6.6), we want to proceed as in the proof of Le mma 6.4. However, we see\nthat one of the fundamental facts in such proof is the fact tha tσ:H1→L2(HQ,H1) is bounded.\nIn order to make up with the fact that here σ:H1→L2(HQ,H1) is not bounded, we need the\nfollowing modification of Lemma 6.4.\nLemma 6.6. Assume Hypotheses 1 to 4 and (uµ\n0,vµ\n0)∈H2satisfying (3.8)and(3.9). Then, for\neveryR≥1andT >0there exists a constant cT,R>0, depending on Λ1andΛ2, such(6.6)holds.\n39Proof.We have\n1\n2d/parenleftbig\n/ba∇dbluR\nµ(t)/ba∇dbl2\nH1+¯r+µ/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH¯r/parenrightbig\n=−/an}b∇acketle{tγR(uR\nµ(t))∂tuR\nµ(t),∂tuR\nµ(t)/an}b∇acket∇i}htH¯rdt\n+/an}b∇acketle{tfR(uR\nµ(t)),∂tuµ(t)/an}b∇acket∇i}htH¯rdt+1\n2µ/ba∇dblσR(uR\nµ(t))/ba∇dbl2\nL2(HQ,H¯r)dt+/an}b∇acketle{tσR(uR\nµ(t))dwQ(t),∂tuR\nµ(t)/an}b∇acket∇i}htH¯r.\nThanks to (2.21), for every u∈H¯rwe have\n/ba∇dblfR(u)/ba∇dblH¯r≤cΦR(/ba∇dblu/ba∇dblH¯r)/ba∇dblu/ba∇dblH¯r≤cR,\nand\n/ba∇dblσR(u)/ba∇dblL2(HQ,H¯r)≤cΦR(/ba∇dblu/ba∇dblH¯r)(1+/ba∇dblu/ba∇dblH¯r)≤cR,\nand this allows to obtain\n1\n2d/parenleftbig\n/ba∇dbluR\nµ(t)/ba∇dbl2\nH1+¯r+µ/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH¯r/parenrightbig\n≤ −γ0\n2/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH¯r+cR\nµdt+|/an}b∇acketle{tσR(uR\nµ(t))dwQ(t),∂tuR\nµ(t)/an}b∇acket∇i}htH¯r|.\nThus, if we define\nLR\nµ(t) :=/ba∇dbluR\nµ(t)/ba∇dbl2\nH1+¯r+µ/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH¯r,\nfor every 0 ≤s≤t≤Twe have\nLR\nµ(t)−LR\nµ(s)≤cR\nµ(t−s)+2M⋆\nR,µ,\nwhere\nM⋆\nµ,R:= sup\nt∈[0,T]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{tσR(uR\nµ(s))dwQ(s),∂tuR\nµ(s)/an}b∇acket∇i}htH¯r/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nAt this point, the rest proof of the proof proceed exactly as i n the proof of Lemma 6.4.\n/square\n7.Uniform bounds in H3\nHere we do not prove uniform bounds in H3with respect to µ∈(0,1), as we did for the bounds\ninH1andH2. Nonetheless, we are able to control the growth of those boun ds with respect to\nµ∈(0,1) in a proper way. Namely, we are going to prove that\nEsup\ns∈[0,T]/ba∇dbluR(s)/ba∇dbl2\nH2+/integraldisplayT\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds≤ρT,R(µ)\nµδ, µ∈(0,1), (7.1)\nand\nµ3Esup\ns∈[0,T]/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nH1+µEsup\ns∈[0,T]/ba∇dbluR\nµ(s)/ba∇dbl4\nH2+µ2/integraldisplayT\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nH1ds\n+µ/integraldisplayT\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds≤ρT,R(µ)\nµδ, µ∈(0,1),(7.2)\nwhereδ∈(0,1/2) is the constant introduced in (3.9) and ρT,R: (0,1)→[0,+∞) is some function\nsuch that\nlim\nµ→0ρT,R(µ) = 0. (7.3)\n40Notice that, unlike in the case of the H2-bounds, here we do not need to distinguish the case σis\nbounded and the case it is not.\nIn Lemma 6.1 we have shown that if σis bounded in L2(HQ,H1), thenuµ, and hence all uR\nµ\nsatisfy estimate (6.9), which involves fourth moments of uµand∂tuµinH2. The same bounds\ncannot be proven in case of unbounded σ. However, going through all steps of the proof of the\nlemma, we see that even if σis not bounded, we still have the inequality\nEsup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nH1+1\nµ2Esup\ns∈[0,t]/ba∇dbluR\nµ(s)/ba∇dbl4\nH2+1\nµ/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nH1ds\n+1\nµ2/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds≤c\nµ1+δ+cT\nµ2/integraldisplayt\n0E/ba∇dblσR(uR\nµ(s))/ba∇dbl2\nL2(HQ,H1)/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds\n+cT\nµ3/integraldisplayt\n0E/ba∇dblσR(uR\nµ(s))/ba∇dbl2\nL2(HQ,H1)/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds\n+cT\nµ2+cT\nµEsup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1+cT\nµ2Esup\ns∈[0,t]/ba∇dbluR\nµ(ss)/ba∇dbl4\nH1.(7.4)\nNow, by interpolation for every u∈H2andv∈H1we have\n/ba∇dblσR(u)/ba∇dbl2\nL2(HQ,H1)/ba∇dblv/ba∇dbl2\nH1≤c/ba∇dblv/ba∇dbl2\nH1+cΦ2\nR(/ba∇dblu/ba∇dblH¯r)/ba∇dblu/ba∇dbl2\nH1/ba∇dblv/ba∇dbl2\nH1\n≤c/ba∇dblv/ba∇dbl2\nH1+cR/ba∇dblu/ba∇dblH2/ba∇dblv/ba∇dbl2\nH1≤cR/ba∇dblv/ba∇dbl2\nH1+1\n2/ba∇dblu/ba∇dbl2\nH2/ba∇dblv/ba∇dbl2\nH1,(7.5)\nand ifu∈H3, for everyǫ>0,\n/ba∇dblσR(u)/ba∇dbl2\nL2(HQ,H1)/ba∇dblu/ba∇dbl2\nH2≤c/ba∇dblu/ba∇dbl2\nH2+cΦ2\nR(/ba∇dblu/ba∇dblH¯r)/ba∇dblu/ba∇dbl2\nH1/ba∇dblu/ba∇dbl2\nH2\n≤c/ba∇dblu/ba∇dbl2\nH2+ΦR(/ba∇dblu/ba∇dblH¯r)/ba∇dblu/ba∇dblH3/ba∇dblu/ba∇dbl3\nH1≤c/ba∇dblu/ba∇dbl2\nH2+ǫ/ba∇dblu/ba∇dbl2\nH3+cR,ǫ/ba∇dblu/ba∇dbl6(1−¯r)\n2−¯r\nH2.\nSince without any loss of generality we can assume that ¯ r>1/2, we have that 6(1 −¯r)/(2−¯r)≤2,\nso that we conclude\n/ba∇dblσR(u)/ba∇dbl2\nL2(HQ,H1)/ba∇dblu/ba∇dbl2\nH2≤cR,ǫ/ba∇dblu/ba∇dbl2\nH2+ǫ/ba∇dblu/ba∇dbl2\nH3.\nBy using this and (7.5) into (7.4), we obtain\nEsup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nH1+1\nµ2Esup\ns∈[0,t]/ba∇dbluR\nµ(s)/ba∇dbl4\nH2+1\nµ/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nH1ds\n+1\n2µ2/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds≤cT,R\nµ2/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds+cT,R,ǫ\nµ3/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds\n+ǫcT\nµ3/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds+cT\nµ2+cT\nµEsup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1+cT\nµ2Esup\ns∈[0,t]/ba∇dbluR\nµ(s)/ba∇dbl4\nH1+c\nµ1+δ.\n41Therefore, recalling (5.2) and (6.4), after we multiply bot h sides byµ3we get\nµ3Esup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nH1+µEsup\ns∈[0,t]/ba∇dbluR\nµ(s)/ba∇dbl4\nH2+µ2/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nH1ds\n+µ/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH2/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds\n≤cT,R,ǫ+cT,R/parenleftBigg\n1+µ/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds+µ2Esup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1/parenrightBigg\n+ǫcT/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds.\n(7.6)\nLemma 7.1. Assume Hypotheses 1 to 4, and fix R≥1and(uµ\n0,vµ\n0)∈H3satisfying (3.8)and\n(3.9). Then, for every T >0there exists cT>0such that for every µ∈(0,1)andt∈[0,T]\nEsup\ns∈[0,t]/ba∇dbluR\nµ(s)/ba∇dbl2\nH2+/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds\n≤ρ1(µ)\nµδ+cTµ/parenleftBigg/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2ds+µEsup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2/parenrightBigg\n,(7.7)\nfor someρ1(µ)>0, depending on Λ1,Λµ\n2andΛµ\n3, such that\nlim\nµ→0ρ1(µ) = 0.\nProof.As in the proof of Lemma 5.2 and Lemma 6.2, we have\n/an}b∇acketle{tη(uR\nµ(t)),µ∂tuR\nµ(t)/an}b∇acket∇i}htH2=/an}b∇acketle{tη(u0),µv0/an}b∇acket∇i}htH2+/integraldisplayt\n0/an}b∇acketle{t∂tη(uR\nµ(s)),µ∂tuR\nµ(s)/an}b∇acket∇i}htH2ds\n+/integraldisplayt\n0/an}b∇acketle{tη(uR\nµ(s)),∆uR\nµ(s)/an}b∇acket∇i}htH2ds−/integraldisplayt\n0/an}b∇acketle{tη(uR\nµ(s)),γ(uR\nµ(s))∂tuR\nµ(s)/an}b∇acket∇i}htH2ds\n+/integraldisplayt\n0/an}b∇acketle{tη(uR\nµ(s)),f(uR\nµ(s))/an}b∇acket∇i}htH2ds+/integraldisplayt\n0/an}b∇acketle{tη(uR\nµ(s)),σR(uµ(s))dwQ(s)/an}b∇acket∇i}htH2\n=:/an}b∇acketle{tη(u0),µv0/an}b∇acket∇i}htH1+4/summationdisplay\nk=1IR\nk(t).(7.8)\nForIR\n1, we have\n|IR\n1(t)| ≤cµ/integraldisplayt\n0/ba∇dbl∂tuR\nµ(s)/ba∇dblH1/ba∇dbl∂tuR\nµ(s)/ba∇dblH2/ba∇dbluR\nµ(s)/ba∇dblH2ds+cµ/integraldisplayt\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2ds\n≤cµ/integraldisplayt\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1/ba∇dbluR\nµ(s)/ba∇dbl2\nH2+cµ/integraldisplayt\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2ds.(7.9)\nForIR\n2(t) andIR\n3(t), we have\nIR\n2(t)≤ −˜γ0/integraldisplayt\n0/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds, (7.10)\n42and\nIR\n3(t) =−/integraldisplayt\n0/an}b∇acketle{tuR\nµ(t),∂tuR\nµ(s))/an}b∇acket∇i}htH2ds=−/ba∇dbluR\nµ(t)/ba∇dbl2\nH2+/ba∇dbluµ\n0/ba∇dbl2\nH2. (7.11)\nForIR\n4(t), we have\n|IR\n4(t)| ≤ct+c/integraldisplayt\n0ΦR(/ba∇dbluR\nµ(r)/ba∇dblH¯r)/ba∇dbluR\nµ(s)/ba∇dbl3\nH2ds≤ct+cR/integraldisplayt\n0/ba∇dbluR\nµ(s)/ba∇dbl3(2−¯r)\n3−¯r\nH3ds.\nThen, since 3(2 −¯r)/(3−¯r)<2, we have\n|IR\n4(t)| ≤cT,R+˜γ0\n2/integraldisplayt\n0/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds (7.12)\nFinally, we have\n/vextendsingle/vextendsingle/an}b∇acketle{tη(uR\nµ(t)),µ∂tuR\nµ(t)/an}b∇acket∇i}htH2/vextendsingle/vextendsingle≤1\n2/ba∇dbluR\nµ(t)/ba∇dbl2\nH2+cµ2/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH2. (7.13)\nTherefore, due to (7.9), (7.10), (7.11), (7.12) and (7.13), if we integrate both sides in (7.8) and then\ntake the expectation, we get\n1\n2E/ba∇dbluR\nµ(t)/ba∇dbl2\nH2+˜γ0\n2/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds≤ /an}b∇acketle{tη(uµ\n0),µvµ\n0/an}b∇acket∇i}htH2+/ba∇dbluµ\n0/ba∇dbl2\nH2+cT,R\n+cµ/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2ds+cµ/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2+cµ2E/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH2.\nIn view of (3.9) we have\nµδ/parenleftbig\n/an}b∇acketle{tη(uµ\n0),µvµ\n0/an}b∇acket∇i}htH2+/ba∇dbluµ\n0/ba∇dbl2\nH2/parenrightbig\n≤(Λµ\n3Λµ\n2)1/2+Λµ\n2=:ρ1(µ)→0,asµ→0.\nTherefore, if in (7.6) we fix ¯ ǫ>0 sufficiently small so that ¯ ǫcT≤˜γ0/4 we have\n1\n2E/ba∇dbluR\nµ(t)/ba∇dbl2\nH2+˜γ0\n4/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds\n≤ρ1(µ)\nµδ+cT/parenleftBigg\n1+µ/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2ds+µ2Esup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2/parenrightBigg\n,\nand (7.7) follows.\n/square\nIn what follows, we shall denote\nΛ3:= sup\nµ∈(0,1)µ1+δΛµ\n3.\nLemma 7.2. Assume Hypotheses 1 to 4, and fix R≥1and(uµ\n0,vµ\n0)∈H3satisfying (3.8)and\n(3.9). Then, for every T >0there exists cT>0, depending on Λ1,Λ2andΛ3, such that for every\nµ∈(0,1)\nEsup\ns∈[0,T]/ba∇dbluR\nµ(s)/ba∇dbl2\nH3+µEsup\ns∈[0,T]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2+/integraldisplayT\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2ds\n≤ρ2,T,R(µ)\nµ1+δ+cR\nµ/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl4(¯κ−¯r)\n3−¯r\nH3ds,(7.14)\n43for someρ2,T,R(µ)>0, depending on Λ1,Λµ\n2andΛµ\n3, such that\nlim\nµ→0ρ2,T,R(µ) = 0.\nProof.Here, we assume that σis not bounded, so that fsatisfies (2.26). We omit the proof in the\ncaseσis bounded; it is analogous to the proof below and relies on es timate (7.6) instead of (2.26).\nWe have\n1\n2d/parenleftbig\n/ba∇dbluR\nµ(t)/ba∇dbl2\nH3+µ/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH2/parenrightbig\n=−/an}b∇acketle{tγR(uR\nµ(t))∂tuR\nµ(t),∂tuR\nµ(t)/an}b∇acket∇i}htH2dt\n+/an}b∇acketle{tfR(uR\nµ(t)),∂tuR\nµ(t)/an}b∇acket∇i}htH2dt+1\n2µ/ba∇dblσR(uR\nµ(t))/ba∇dbl2\nL2(HQ,H2)dt+/an}b∇acketle{tσR(uR\nµ(t))dwQ(t),∂tuR\nµ(t)/an}b∇acket∇i}htH2.\nNow, due to (2.25) and (2.26), for every µ∈(0,1) we have\n/ba∇dblfR(u)/ba∇dbl2\nH2+1\n2µ/ba∇dblσR(u)/ba∇dbl2\nL2(HQ,H2)≤cR\nµ/parenleftbigg\n1+/ba∇dblu/ba∇dbl2\nH2+/ba∇dblu/ba∇dbl4(¯κ−¯r)\n3−¯r\nH3/parenrightbigg\n,\nso that\n1\n2d/parenleftbig\n/ba∇dbluR\nµ(t)/ba∇dbl2\nH3+µ/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH2/parenrightbig\n≤ −γ0\n2/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH2dt+/ba∇dbluR\nµ(t)/ba∇dbl4\nH2dt\n+cR\nµ/ba∇dbluR\nµ(t)/ba∇dbl2\nH2dt+cR\nµdt+cR\nµ/ba∇dbluR\nµ(t)/ba∇dbl4(¯κ−¯r)\n3−¯r\nH3dt+/an}b∇acketle{t∂tuR\nµ(t),σR(uR\nµ(t))dwQ(t)/an}b∇acket∇i}htH1.\nIf we integrate both sides with respect to time, thanks to (3. 9) we get\n/ba∇dbluR\nµ(t)/ba∇dbl2\nH3+µ/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH2+γ0/integraldisplayt\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2ds≤Λµ\n3µ1+δ+cT,Rµδ\nµ1+δ\n+cR\nµ/integraldisplayt\n0/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds+cR\nµ/integraldisplayt\n0/ba∇dbluR\nµ(s)/ba∇dbl4(¯κ−¯r)\n3−¯r\nH3ds+2/integraldisplayt\n0/an}b∇acketle{t∂tuR\nµ(s),σR(uR\nµ(s))dwQ(s)/an}b∇acket∇i}htH2.\n(7.15)\nIn view again of (2.25), we have\nEsup\ns∈[0,t]/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{t∂tuR\nµ(s),σR(uR\nµ(s))dwQ(s)/an}b∇acket∇i}htH2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤cRE/parenleftbigg/integraldisplayt\n0/ba∇dbl∂tuR\nµ(r)/ba∇dbl2\nH2/parenleftbig\n1+/ba∇dbluR\nµ(s)/ba∇dbl2\nH2/parenrightbig\ndr/parenrightbigg1\n2\n≤µ\n4Esup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2+cR\nµ/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl4(¯κ−¯r)\n3−¯r\nH3ds+cT\nµ.\nHence, by taking the supremum with respect to time and the exp ectation in both sides of (7.15),\naccording to (6.4), we obtain\nEsup\ns∈[0,t]/ba∇dbluR\nµ(s)/ba∇dbl2\nH3+µEsup\ns∈[0,t]/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2+/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2ds\n≤1\nµ1+δ/parenleftBig\nΛµ\n3µ1+δ+cT,Rµδ/parenrightBig\n+cR\nµ/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl4(¯κ−¯r)\n3−¯r\nH3ds.\nThis implies (7.14), with ρ2,T,R(µ) := Λµ\n3µ1+δ+cT,Rµδ. /square\n44Now, thanks to (7.7), we have\nEsup\nr∈[0,t]/ba∇dbluR(s)/ba∇dbl2\nH2+/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds\n≤ρ1(µ)\nµδ+cTµ/parenleftBigg/integraldisplayt\n0E/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH2ds+µEsup\nr∈[0,t]/ba∇dbl∂tuR\nµ(r)/ba∇dbl2\nH2/parenrightBigg\n.\nHence, if we define\nˆρT,R(µ) :=ρ1(µ)+cTρ2,T,R(µ),\nwe have that ˆ ρT,R(µ)→0, asµ→0, and thanks to (7.14)\nEsup\nr∈[0,t]/ba∇dbluR(s)/ba∇dbl2\nH2+/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds≤ˆρT,R(µ)\nµδ+cT,R/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl4(¯κ−¯r)\n3−¯r\nH3ds.\nNow, because of the way we have chosen ¯ rin (6.1), we have 4(¯ κ−¯r)/(3−¯r)<2, so that\ncT,R/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl4(¯κ−¯r)\n3−¯r\nH3ds≤1\n2/integraldisplayt\n0E/ba∇dbluR\nµ(s)/ba∇dbl2\nH3ds+cT,R,\nso that (7.1) follows with\nρT,R(µ) := 2/parenleftBig\nˆρT,R(µ)+cT,Rµδ/parenrightBig\n.\nFinally, (7.2) follows from (7.6), due to (6.5) and (7.1).\n8.Tightness\nIn this section, by using the uniform bounds we have proved in Section 5 and Section 6, we will\nprove the tightness of the solution of the truncated problem 6.3 in appropriate functional spaces.\nProposition 8.1. Assume Hypotheses 1 to 4, and fix any R≥1andT >0and any initial\nconditions (uµ\n0,vµ\n0)∈H2satisfying conditions (3.8)and(3.9). Then, for any sequence (µk)k∈N\nconverging to zero and for every ̺ <1,ϑ∈[1,2)andp <2/(ϑ−1), the family of probability\nmeasures/parenleftbig\nL(uR\nµk)/parenrightbig\nk∈Nis tight in\nC([0,T];H̺)∩Lp(0,T;Hϑ).\nIn order to prove Proposition 8.1, we will need the following preliminary result.\nLemma 8.2. For everyR≥1,T >0andµ>0, we define\nΦR\nµ(t) :=uR\nµ(t)+µγ−1\nR(uR\nµ(t))∂tuR\nµ(t), t∈[0,T].\nThen, under the same assumptions of Proposition 8.1, there ex istθ∈(0,1)andcT,R>0such that\nfor every 0<h<T\nsup\nµ∈(0,1)h−θEsup\nt∈[0,T−h]/ba∇dblΦR\nµ(t+h)−ΦR\nµ(t)/ba∇dblHdt≤cT,R. (8.1)\nProof.For everyψ∈H2, we define\nFR\nµ(u,v) :=µ/an}b∇acketle{tγ−1\nR(u)v,ψ/an}b∇acket∇i}htH,(u,v)∈H1.\nWe have\nDuFR\nµ(u,v)h=µ/an}b∇acketle{t[Dγ−1\nR(u)h]v,ψ/an}b∇acket∇i}htH, D vFR\nµ(u,v)h=µ/an}b∇acketle{tγ−1\nR(u)h,ψ/an}b∇acket∇i}htH,\n45and the Itˆ o formula gives\ndFR\nµ(uR\nµ(t),∂tuR\nµ(t))\n=/an}b∇acketle{t[Dγ−1\nR(uR\nµ(t))µ∂tuR\nµ(t)]∂tuR\nµ(t),ψ/an}b∇acket∇i}htHdt+/an}b∇acketle{tγ−1\nR(uR\nµ(t))∆uR\nµ(t),ψ/an}b∇acket∇i}htHdt−/an}b∇acketle{t∂tuR\nµ(t),ψ/an}b∇acket∇i}htHdt\n+/an}b∇acketle{tγ−1\nR(uR\nµ(t))fR(uR\nµ(t)),ψ/an}b∇acket∇i}htHdt+/an}b∇acketle{tγ−1\nR(uR\nµ(t))σR(uR\nµ(t))dwQ(t),ψ/an}b∇acket∇i}htH.\nThus, integrating with respect to time, we obtain\n/an}b∇acketle{tΦR\nµ(t),ψ/an}b∇acket∇i}htH=/an}b∇acketle{tuR\nµ(t),ψ/an}b∇acket∇i}htH+FR\nµ(uR\nµ(t),∂tuR\nµ(t)) =/an}b∇acketle{tuµ\n0+µγ−1\nR(uµ\n0)vµ\n0),ψ/an}b∇acket∇i}htH\n+µ/integraldisplayt\n0/an}b∇acketle{t[Dγ−1(uR\nµ(s))∂tuR\nµ(s)]∂tuR\nµ(s),ψ/an}b∇acket∇i}htHds+/integraldisplayt\n0/an}b∇acketle{tγ−1\nR(uR\nµ(s))∆uR\nµ(s),ψ/an}b∇acket∇i}htHds\n+/integraldisplayt\n0/an}b∇acketle{tγ−1\nR(uR\nµ(s))fR(uR\nµ(s)),ψ/an}b∇acket∇i}htHds+/integraldisplayt\n0/an}b∇acketle{tγ−1\nR(uR\nµ(s))σR(uR\nµ(s))dwQ(s),ψ/an}b∇acket∇i}htH\n=:/an}b∇acketle{tuµ\n0+µγ−1\nR(uµ\n0)vµ\n0),ψ/an}b∇acket∇i}htH+4/summationdisplay\nk=1/an}b∇acketle{tIR\nµ,k(t),ψ/an}b∇acket∇i}htH.\nThanks to (2.24), we have\n|/an}b∇acketle{tIR\nµ,1(t+h)−IR\nµ,1(t),ψ/an}b∇acket∇i}htH| ≤c/integraldisplayt+h\ntµ/ba∇dbl∂tuR\nµ(s)/ba∇dblH/ba∇dbl∂tuR\nµ(s)/ba∇dblH1ds/ba∇dblψ/ba∇dblH\n≤c/parenleftbigg\nµ/integraldisplayT\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl2\nH1ds/parenrightbigg1\n2/parenleftbigg\nµ2/integraldisplayT\n0/ba∇dbl∂tuR\nµ(s)/ba∇dbl4\nHds/parenrightbigg1\n4\nh1\n4/ba∇dblψ/ba∇dblH,\nso that, in view of (6.5) and (5.2), we get\nsup\nµ∈(0,1)Esup\nt∈[0,T−h]/ba∇dblIR\nµ,1(t+h)−IR\nµ,1(t)/ba∇dblH≤cT,Rh1\n4/ba∇dblψ/ba∇dblH. (8.2)\nConcerning IR\nµ,2, we have\n|/an}b∇acketle{tIR\nµ,2(t+h)−IR\nµ,2(t),ψ/an}b∇acket∇i}ht| ≤c/integraldisplayt+h\nt/ba∇dbluR\nµ(s)/ba∇dblH2ds/ba∇dblψ/ba∇dblH≤c/parenleftbigg/integraldisplayT\n0/ba∇dbluR\nµ(s)/ba∇dbl2\nH2ds/parenrightbigg1\n2\nh1\n2/ba∇dblψ/ba∇dblH,\nand then, according to (6.4), we obtain\nsup\nµ∈(0,1)Esup\nt∈[0,T−h]/ba∇dblIR\nµ,2(t+h)−IR\nµ,2(t)/ba∇dblH≤cT,Rh1\n2/ba∇dblψ/ba∇dblH. (8.3)\nAs forIR\nµ,3, thanks to (2.8), and (6.4) we have\nsup\nµ∈(0,1)Esup\nt∈[0,T−h]/ba∇dblIR\nµ,3(t+h)−IR\nµ,3(t)/ba∇dblH≤cT,Rh/parenleftBigg\nEsup\nt∈[0,T]/ba∇dbluR\nµ(t)/ba∇dbl2\nH1+1/parenrightBigg\n≤cT,Rh.(8.4)\nTherefore, collecting together (8.2), (8.3), and (8.4), we have\nEsup\nt∈[0,T−h]/ba∇dblΦR\nµ(t+h)−ΦR\nµ(t)/ba∇dblH≤cT,Rh1\n4+Esup\nt∈[0,T−h]/ba∇dblIR\nµ,4(t+h)−IR\nµ,4(t)/ba∇dblH,(8.5)\nfor everyµ∈(0,1) andh∈(0,1).\n46By using a factorization argument as in [16, Theorem 5.11 and Theorem 5.15], due to the\nboundedness of σinHwe obtain that for some θ∈(0,1)\nsup\nµ∈(0,1)E/ba∇dblIR\nµ,4/ba∇dblCθ([0,T];H)<∞.\nThis and (8.5) allow to obtain (8.1).\n/square\nProof of Proposition 8.1. For every function f: [0,T]→Hand every 0 <h<T we define\nτhf(t) :=f(t+h), t∈[0,T−h].\nNext, for every L>0 we define\nKL,1:=/braceleftBigg\nf: [0,T]×O→Rr: sup\nt∈[0,T]/ba∇dblf/ba∇dblH1≤L/bracerightBigg\n,\nand\nKL,2:=/braceleftBigg\nf: [0,T]×O→Rr: sup\nt∈[0,T−h]/ba∇dblτhf(t)−f(t)/ba∇dblHdt≤Lhθ/bracerightBigg\n,\nwhereθ∈(0,1) is the same as the constant in (8.1). Since H1is compactly embedded in H̺, for\nevery̺<1, due to [49, Theorem 6] we have that KL,1∩KL,2is compact in C([0,T];H̺), for every\n̺<1.\nNow, according to (6.4) and (6.5), we have\nsup\nµ∈(0,1)Esup\nt∈[0,T]/ba∇dbluR\nµ(t)+µγ−1\nR(uR\nµ(t))∂tuR\nµ(t)/ba∇dblH1≤cT,R,\nand hence, for every ǫ>0 we can find Lǫ\n1,R>0 such that\ninf\nµ∈(0,1)P/parenleftBig\nuR\nµ+µγ−1\nR(uR\nµ)∂tuR\nµ∈K1,Lǫ\n1,R/parenrightBig\n≥1−ǫ. (8.6)\nMoreover, thanks to (8.1), we have that there exists Lǫ\n2,R>0 such that\ninf\nµ∈(0,1)P/parenleftBig\nuR\nµ+µγ−1\nR(uR\nµ)∂tuR\nµ∈K2,Lǫ\n2,R/parenrightBig\n≥1−ǫ. (8.7)\nNext, since we are assuming ¯ r≥ρ, thanks to (6.6), we have\nlim\nµ→0µEsup\nt∈[0,T]/ba∇dblγ−1\nR(uR\nµ(t))∂tuµ(t)/ba∇dblHρ= 0.\nThis means in particular that for every sequence ( µk)k∈Nconverging to zero, there exists a compact\nsetKR\n3,ǫinC([0,T];Hρ) such that\ninf\nk∈NP/parenleftbig\n−µkγ−1\nR(uR\nµk)∂tuR\nµk∈KR\n3,ǫ/parenrightbig\n>1−ǫ. (8.8)\nFrom (8.6), (8.7) and (8.8), we have that\ninf\nk∈NP/parenleftBig\nuR\nµk∈[K1,Lǫ\n1,R∩K2,Lǫ\n2,R]+KR\n3,ǫ/parenrightBig\n≥1−3ǫ. (8.9)\nIn particular, since\n[K1,Lǫ\n1,R∩K2,Lǫ\n2,R]+KR\n3,ǫ⊂C([0,T];H̺)\nis compact, due to the arbitrariness of ǫ >0 we have that the family ( L(uR\nµk))k∈Nis tight in\nC([0,T];H̺).\n47Now, if we define\nK4,L:=/braceleftbigg\nf: [0,T]×O→R:/integraldisplayT\n0/ba∇dblf(t)/ba∇dbl2\nH2dt≤L/bracerightbigg\n,\nthanks to (6.4) we have that there exists Lǫ\n4,R>0 such that\ninf\nµ∈(0,1)P/parenleftBig\nuR\nµ∈K4,Lǫ\n4,R/parenrightBig\n≥1−ǫ.\nThis, together with (8.9) implies that\ninf\nk∈NP/parenleftbig\nuµk∈KR\nǫ/parenrightbig\n≥1−4ǫ, (8.10)\nwhere\nKR\nǫ:=/parenleftBig\n[K1,Lǫ\n1,R∩K2,Lǫ\n2,R]+KR\n3,ǫ/parenrightBig\n∩K4,Lǫ\n4,R.\nSince the set [ K1,Lǫ\n1,R∩K2,Lǫ\n2,R] +KR\n3,ǫis compact in C([0,T];H̺), due to [49, Theorem 1] we\nhave that\nlim\nh→0sup\nf∈[K1,Lǫ\n1,R∩K2,Lǫ\n2,R]+KR\n3,ǫsup\nt∈[0,T−h]/ba∇dblτhf(t)−f(t)/ba∇dblH̺= 0.\nNow, for every ϑ∈[1,2), we have\nα:=ϑ−̺\n2−̺=⇒ /ba∇dblh/ba∇dblHϑ≤ /ba∇dblh/ba∇dblα\nH2/ba∇dblh/ba∇dbl1−α\nH̺.\nRecalling how K4,Lǫ\n4,Rwas defined, we have that KR\nǫis bounded in L2(0,T;H2). Therefore, due to\n[49, Theorem 7], we have that KR\nǫis compact in Lp̺,ϑ(0,T;Hδ), where\np̺,ϑ:=/parenleftbiggα\n2+1−α\n∞/parenrightbigg−1\n=2(2−̺)\nϑ−̺,\nand, in view of (8.10), this allows to conclude that/parenleftbig\nL(uR\nµk)/parenrightbig\nk∈Nis tight inLp̺,ϑ(0,T;Hδ). Now,\nsince̺<1 can be choosen arbitrarily close to 1, we obtain the tightne ss inLp(0,T;Hϑ), for every\np<2/(ϑ−1).\n/square\n9.About the limiting equation (3.12) and its truncated version\nWe recall that the function Swe have introduced in (3.7) is defined by\nS(u) =/integraldisplay\nH1[Dg−1(u)z]zdνu(z), u∈H1.\n9.1.Properties of the mapping S.\nLemma 9.1. Under Hypotheses 1, 3 and 4, we have that SmapsH1into itself, with\n/ba∇dblS(u)/ba∇dblH1≤c/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH1/parenrightbig\n,/ba∇dblS(u)/ba∇dblH≤c(1+/ba∇dblu/ba∇dblH1), u∈H1. (9.1)\nMoreover, if we also assume Hypothesis 5, we have that S:H2→H1is differentiable and\n/ba∇dblDS(u)h/ba∇dblH1≤c/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH2/parenrightbig\n/ba∇dblh/ba∇dblH1, u,h ∈H2. (9.2)\n48Proof.Due to (3.6), we have\n/ba∇dblS(u)/ba∇dblH1≤c/integraldisplay\nH1/ba∇dblz/ba∇dbl2\nH1dνu(z)≤cTrH1Λu≤c/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH1/parenrightbig\n, (9.3)\nand the first inequality in (9.1) follows. Moreover,\n/ba∇dblS(u)/ba∇dblH≤c/integraldisplay\nH1/ba∇dblz/ba∇dblH1/ba∇dblz/ba∇dblHdνu(z)≤c/parenleftbigg/integraldisplay\nH1/ba∇dblz/ba∇dbl2\nH1dνu(z)/parenrightbigg1\n2/parenleftbigg/integraldisplay\nH1/ba∇dblz/ba∇dbl2\nHdνu(z)/parenrightbigg1\n2\n≤c(TrH1ΛuTrHΛu)1\n2≤c(1+/ba∇dblu/ba∇dblH1),\nso that the second inequality in (9.1) follows.\nIn order to study the differentiability of S, for everyu∈H1we introduce the random variable\nξ(u) :=/integraldisplay0\n−∞eg(u)sσ(u)dˆwQ(s),\nwhere ˆwQ(t),t∈R, is the two-sided version of the cylindrical Wiener process wQ. As known, ξ(u)\nis distributed like νν, so that we can give an alternative representation of Sas\nS(u) =E/parenleftbig/bracketleftbig\nDg−1(u)ξ(u)/bracketrightbig\nξ(u)/parenrightbig\n. (9.4)\nDue to (9.3) we have that ξmapsH1intoL2(Ω;H1) and\nE/ba∇dblξ(u)/ba∇dbl2\nH1≤c/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH1/parenrightbig\n. (9.5)\nMoreover, we can show that ξ:H2→L2(Ω;H1) is differentiable. In order to prove that, for every\nu,h∈H1we consider the equation\ndx\ndt(t) =−g(u)x(t), x(0) =h.\nIts solution is given by xh(u,t) :=e−g(u)th, for everyt≥0, and\n/ba∇dblxh(u,t)/ba∇dblH1≤e−γ0t/ba∇dblh/ba∇dblH1, t≥0. (9.6)\nFor everyh∈H1andt≥0 the mapping u∈H1/ma√sto→xh(u,t)∈H1is differentiable and if we\ndenoteyh,k(u,t) :=Duxh(u,t)k, withk∈H1, we have that the function t/ma√sto→yh,k(u,t) satisfies the\nequation\ndy\ndt(t) =−g(u)y(t)−[Dg(u)k]·xh(u,t), y(0) = 0.\nThis means that\nyh,k(u,t) =−/integraldisplayt\n0e−g(u)(t−s)[Dg(u)k]xh(u,s)ds,\nand, thanks to (9.6),\n/ba∇dblyh,k(u,t)/ba∇dblH1≤/integraldisplayt\n0e−γ0(t−s)/ba∇dblxh(u,s)/ba∇dblH1ds/ba∇dblDg(u)/ba∇dblL(H1;Rr×r)/ba∇dblk/ba∇dblH1\n≤c/integraldisplayt\n0e−γ0tds/ba∇dblh/ba∇dblH1/ba∇dblk/ba∇dblH1=cte−γ0t/ba∇dblh/ba∇dblH1/ba∇dblk/ba∇dblH1.\nIn particular,\n/ba∇dblDu(eg(u)th)/ba∇dblH1≤ −cteγ0t/ba∇dblh/ba∇dblH1, t≤0. (9.7)\n49This, together with Hypothesis 5 implies that ξ:H2/ma√sto→ξ(u)∈L2(Ω;H1) is differentiable and\nDξ(u)h=/integraldisplay0\n−∞[Du(eg(u)sh)]σ(u)dwQ(s)+/integraldisplay0\n−∞eg(u)s[Dσ(u)h]dwQ(s), u,h ∈H2.\nMoreover, due to (2.10), (2.28) and (9.7), we have\nE/ba∇dblDξ(u)h/ba∇dbl2\nH1≤c/integraldisplay0\n−∞t2e2γ0tdt/ba∇dblσ(u)/ba∇dbl2\nL2(HQ,H1)/ba∇dblh/ba∇dbl2\nH1\n+c/integraldisplay0\n−∞e2γ0tdt/ba∇dblDσ(u)h/ba∇dbl2\nL2(HQ,H1)≤c/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH2/parenrightbig\n/ba∇dblh/ba∇dbl2\nH1.(9.8)\nThus, in view of (9.4) and Hypothesis 5, we have that Sis differentiable, with\nDS(u)h=E/parenleftbig\n[D2g−1(u)(h,ξ(u))]ξ(u)/parenrightbig\n+E/parenleftbig\n[Dg−1(u)(Duξ(u)h)]ξ(u)/parenrightbig\n+E/parenleftbig\n[Dg−1(u)ξ(u)]Duξ(u)h/parenrightbig\n,\nand\n/ba∇dblDS(u)h/ba∇dblH1≤cE/ba∇dblξ(u)/ba∇dbl2\nH1/ba∇dblh/ba∇dblH1+cE/ba∇dblDξ(u)h/ba∇dblH1/ba∇dblξ(u)/ba∇dblH1.\nThanks to (9.5) and (9.8), this implies (9.2).\n/square\n9.2.Pathwise uniqueness of equation (3.12). As a consequence of (9.2), we have that the\nmappingS:H2→H1is locally-Lipschitz continuous and\n/ba∇dblS(u1)−S(u2)/ba∇dblH1≤c/parenleftbig\n1+/ba∇dblu1/ba∇dbl2\nH2+/ba∇dblu2/ba∇dbl2\nH2/parenrightbig\n/ba∇dblu1−u2/ba∇dblH1. (9.9)\nThis allows to prove the following uniqueness result for equ ation (3.12).\nLemma 9.2. Under Hypotheses 1 to 5, equation (3.12)admits at most one solution uin the space\nL2(Ω;C([0,T];H1)∩L2(0,T;H2)).\nProof.Assumethat u1,u2∈L2(Ω;C([0,T];H1)∩L2(0,T;H2))aretwo solutionsofequation(3.12).\nIf we define ρ(t) :=u1(t)−u2(t), we have\n∂tρ(t,x) =/parenleftbig\ng−1(u1(t))∆ρ(t,x)+/bracketleftbig\ng−1(u1(t))−g−1(u2(t))/bracketrightbig\n∆u2(t,x)\n+[fg(u1(t))−fg(u2(t))]+[S(u1(t))−S(u2(t))])dt+[σg(u1(t))−σg(u2(t))]∂twQ(t),\nwhere\nfg(u) :=g−1(u)f(u), σ g(u) :=g−1(u)σ(u), u∈H1.\nIt is immediate to check that\n/ba∇dblfg(u1)−fg(u2)/ba∇dblH1≤c/ba∇dblu1−u2/ba∇dblH1(1+/ba∇dblu2/ba∇dblH1), (9.10)\nand\n/ba∇dblσg(u1)−σg(u2)/ba∇dblL2(HQ,H1)≤c/ba∇dblu1−u2/ba∇dblH1(1+/ba∇dblu2/ba∇dblH1). (9.11)\nMoreover, we have\n/angbracketleftbig/bracketleftbig\ng−1(u1(t))−g−1(u2(t))/bracketrightbig\n∆u2(t,x),ρ(t)/angbracketrightbig\nH1≤ /ba∇dblg−1(u1(t))−g−1(u2(t))/ba∇dblRr×r/ba∇dblu2(t)/ba∇dblH2/ba∇dblρ(t)/ba∇dblH2\n≤c/ba∇dblρ(t)/ba∇dblH1/ba∇dblu2(t)/ba∇dblH2/ba∇dblρ(t)/ba∇dblH2≤˜γ0\n2/ba∇dblρ(t)/ba∇dbl2\nH2+cρ(t)/ba∇dbl2\nH1/ba∇dblu2(t)/ba∇dbl2\nH2.(9.12)\n50Hence, according to (9.9), (9.10), (9.11) and (9.12), from t he Itˆ o formula we obtain\n1\n2d/ba∇dblρ(y)/ba∇dbl2\nH1≤ −˜γ0\n2/ba∇dblρ(t)/ba∇dbl2\nH2dt+c/ba∇dblρ(t)/ba∇dbl2\nH1/parenleftbig\n/ba∇dblu1(t)/ba∇dbl2\nH2+/ba∇dblu2(t)/ba∇dbl2\nH2+1/parenrightbig\ndt\n+/an}b∇acketle{t[σg(u1(t))−σg(u2(t))]dwQ(t),ρ(t)/an}b∇acket∇i}htH1.\nNext, for an arbitrary κ>0 we define\nΓκ(t) := exp/parenleftbigg\n−κ/integraldisplayt\n0/parenleftbig\n/ba∇dblu1(s)/ba∇dbl2\nH2+/ba∇dblu2(s)/ba∇dbl2\nH2+1/parenrightbig\nds/parenrightbigg\n,\nand apply Itˆ o’s formula to the process Γ κ(t)/ba∇dblρ(t)/ba∇dbl2\nH1. We obtain\nd/parenleftbig\nΓκ(t)/ba∇dblρ(t)/ba∇dbl2\nH1/parenrightbig\n≤Γκ(t)/parenleftbigg\n−˜γ0\n2/ba∇dblρ(t)/ba∇dbl2\nH2+¯c/ba∇dblρ(t)/ba∇dbl2\nH1/parenleftbig\n/ba∇dblu1(t)/ba∇dbl2\nH2+/ba∇dblu2(t)/ba∇dbl2\nH2+1/parenrightbig/parenrightbigg\ndt\n+Γκ(t)/an}b∇acketle{t[σg(u1(t))−σg(u2(t))]dwQ(t),ρ(t)/an}b∇acket∇i}htH1\n−Γκ(t)κ/parenleftbig\n/ba∇dblu1(t)/ba∇dbl2\nH2+/ba∇dblu2(t)/ba∇dbl2\nH2+1/parenrightbig\n/ba∇dblρ(t)/ba∇dbl2\nH1dt.\nIn particular, if we take κ:= ¯c, we get\nd/parenleftbig\nΓ¯c(t)/ba∇dblρ(t)/ba∇dbl2\nH1/parenrightbig\n≤Γκ(t)/an}b∇acketle{t[σg(u1(t))−σg(u2(t))]dwQ(t),ρ(t)/an}b∇acket∇i}htH1.\nTherefore, if we integrate in time and take the expectation o f both sides, since ρ(0) = 0 we get\nE/parenleftbig\nΓ¯c(t)/ba∇dblρ(t)/ba∇dbl2\nH1/parenrightbig\n≤0, t≥0.\nSinceu1,u2∈L2(Ω;L2(0,T;H2)), we have P(Γ¯c(t)>0, t≥0) = 1, and this implies that u1=u2.\n/square\nNow, for every R≥1 andu,v∈H1, we denote by yu,v\nRthe solution of problem (3.4), where the\nmappings gandσare replaced by gRandσR. Clearly, we have\nsup\nt≥0E/ba∇dblyu,v\nR(t)/ba∇dblp\nH1≤cp/parenleftbig\n1+e−γ0t/ba∇dblv/ba∇dblp\nH1+/ba∇dblu/ba∇dblp\nH1/parenrightbig\n, t≥0, R≥1. (9.13)\nNext, we define\nPR,u\ntϕ(v) =Eϕ(yu,v\nR(t)), v∈H, t≥0,\nfor every function ϕ∈Bb(H1), and\nΛR,u:=/integraldisplay∞\n0e−gR(u)s[σR(u)Q][σR(u)Q]⋆e−gt\nR(u)sds.\nAs we have seen for Λ u, we have that Λ R,u∈L+\n1(H)∩L+\n1(H1),νu\nR:=N(0,ΛR,u) is the unique\ninvariant measure for the semigroup PR,u\ntand for every p≥1\n/integraldisplay\nH1/ba∇dblz/ba∇dbl2p\nH1dνu\nR(z)≤cp(TrH1Λu)p≤cp/parenleftBig\n1+/ba∇dblu/ba∇dbl2p\nH1/parenrightBig\n, (9.14)\nfor some constant cindependent of R≥1.\nFinally, if we define\nSR(u) :=/integraldisplay\nH1/bracketleftbig\nDg−1\nR(u)z/bracketrightbig\nzdνu\nR(z), u∈H1,\n51it is immediate to check that Lemma 9.1 is true also for SR, so that\n/ba∇dblSR(u1)−SR(u2)/ba∇dblH1≤c/parenleftbig\n1+/ba∇dblu1/ba∇dbl2\nH1+/ba∇dblu2/ba∇dbl2\nH1/parenrightbig\n/ba∇dblu1−u2/ba∇dblH1, u1,u2∈H1,(9.15)\nfor some constant c>0 independent of R≥1. For every R≥1, we introduce the problem\n\n∂tuR(t,x) =g−1\nR(uR(t))∆uR(t,x)+g−1\nR(uR(t))fR(uR(t,x))\n+SR(uR(t))+g−1\nR(uR(t))σR(uR(t))∂twQ(t),\nuR(0,x) =u0(x), x∈O, uR(t,x) = 0, x∈∂O.(9.16)\nProposition 9.2 applies also to problem (9.16), so that for e veryR≥1 there exists at most one\nsolutionuR∈L2(Ω;C([0,T];H1)∩L2(0,T;H2)).\n10.Proof of Theorem 3.3 for the truncated equation (6.3)\nIn what follows we will prove the following analog of Theorem 3.3 for the truncated equation\n(6.3).\nTheorem 10.1. Assume Hypotheses 1 to 5 and fix an arbitrary ̺<1andϑ∈[1,2). Moreover,\nassume that (uµ\n0,vµ\n0)belong to H3, for every µ∈(0,1), and satisfy conditions (3.8),(3.9)and\n(3.10). Then, for every R≥1,p<2/(ϑ−1)andη,T >0\nlim\nµ→0P/parenleftBigg\nsup\nt∈[0,T]/ba∇dbluR\nµ(t)−uR(t)/ba∇dblH̺+/integraldisplayT\n0/ba∇dbluR\nµ(t)−uR(t)/ba∇dblp\nHϑdt>η/parenrightBigg\n= 0,\nwhereuR∈L2(Ω;C([0,T];H1)∩L2(0,T;H2))is the unique solution of problem (9.16).\nIn Proposition 8.1 we have seen that/parenleftbig\nL(uR\nµk)/parenrightbig\nk∈Nis tight inC([0,T];H̺)∩Lp(0,T;Hϑ), for\neveryR≥1. Hence, since equation (9.16) has pathwise uniqueness, Th eorem 10.1 follows once\nwe show that any weak limit point of {uR\nµ}µ∈(0,1)is a solution equation of (9.16). Notice that\nthroughout this section, we will assume that all Hypotheses , from 1 to 5 are satisfied.\n10.1.The construction of the corrector functions. Due to Hypotheses 4 and 5, the mapping\nu∈H1/ma√sto→yu,v\nR(t)∈L2(Ω;H1),\nis differentiable, for any fixed v∈H1andt≥0, and if we denote ηu,v\nR,k(t) :=Duyu,v\nR(t)·k, we have\nthat the process ηu,v\nR,kis the solution of the problem\ndη(t) =−/parenleftbig\ngR(u)η(t)+[DgR(u)k]yu,v\nR(t)/parenrightbig\ndt+[DσR(u)k]dwQ(t), η(0) = 0.\nIn particular, we have\nηu,v\nR,k(t) =/integraldisplayt\n0e−gR(u)(t−s)[DgR(u)k]yu,v\nR(s)ds+/integraldisplayt\n0e−gR(u)(t−s)[DσR(u)k]dwQ(s).\nThis implies that\nE/ba∇dblηu,v\nR,k(r)/ba∇dbl2\nH1≤c/parenleftBigg\nsup\ns∈[0,t]E/ba∇dblyu,v\nR(s)/ba∇dbl2\nH1/integraldisplayt\n0e−2γ0sds+c/integraldisplayt\n0e−2γ0(t−s)ds/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH2/parenrightbig/parenrightBigg\n/ba∇dblk/ba∇dbl2\nH1,\nso that, thanks to (9.13), we obtain that for every T >0 fixed\nsup\nt≥0E/ba∇dblηu,v\nR,k(t)/ba∇dbl2\nH1≤c/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH1/parenrightbig\n/ba∇dblk/ba∇dbl2\nH1, (10.1)\n52for some constant cindependent of R≥1.\nIt is immediate to check that PR,u\ntis a Feller contraction semigroup in Cb(H1). Moreover, it\nis weakly continuous in Cb(H1) (for the definition see [3, Appendix B]). In particular, thi s means\nthat there exists a closed operator MR,u:D(MR,u)⊆Cb(H1)→Cb(H1) such that for every λ>0\nandϕ∈C1\nb(H1)\n(λ−MR,u)−1ϕ(v) =/integraldisplay∞\n0e−λtPR,u\ntϕ(v)dt, v ∈H1.\nNotice that C2\nb(H1)⊂D(MR,u) andMR,uϕ=MR,uϕ, for every ϕ∈C2\nb(H1), where MR,uis the\nKolmogorov operator associated with equation (3.4)\nMR,uϕ(v) =1\n2TrH1/parenleftbig\nD2ϕ(v)[σR(u)Q][σR(u)Q]⋆/parenrightbig\n−/an}b∇acketle{tgR(u)v,Dϕ(v)/an}b∇acket∇i}htH1, ϕ∈C2\nb(H1).(10.2)\nLemma 10.2. Letϕ:H1→Rbe any continuous function such that\n|ϕ(v1)−ϕ(v2)| ≤cϕ/ba∇dblv1−v2/ba∇dblH1/parenleftbig\n1+/ba∇dblv1/ba∇dblκϕ\nH1+/ba∇dblv2/ba∇dblκϕ\nH1/parenrightbig\n, v1,v2∈H1,\nfor somecϕ>0andκϕ≥0. Then, there exists some other constant ¯cϕ>0such that for every\nu,v∈H1/vextendsingle/vextendsingle/vextendsingle/vextendsinglePR,u\ntϕ(v)−/integraldisplay\nH1ϕ(z)dνu\nR(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤¯cϕ/parenleftBig\n1+/ba∇dblv/ba∇dblκϕ+1\nH1+/ba∇dblu/ba∇dblκϕ+1\nH1/parenrightBig\ne−γ0t. (10.3)\nProof.Thanks to (9.13), for every v1,v2∈H1, we have\n|PR,u\ntϕ(v1)−PR,u\ntϕ(v2)| ≤cϕE/parenleftbig\n/ba∇dblyu,v1\nR(t)−yu,v2\nR(t)/ba∇dblH1/parenleftbig\n1+/ba∇dblyu,v1\nR(t)/ba∇dblκϕ\nH1+/ba∇dblyu,v2\nR(t)/ba∇dblκϕ\nH1/parenrightbig/parenrightbig\n≤cϕ/ba∇dble−g(u)t(v1−v2)/ba∇dblH1E/parenleftbig\n1+/ba∇dblyu,v1\nR(t)/ba∇dblκϕ\nH1+/ba∇dblyu,v2\nR(t)/ba∇dblκϕ\nH1/parenrightbig\n≤cϕe−γ0t/ba∇dblv1−v2/ba∇dblH1cκϕ/parenleftbig\n1+/ba∇dblu/ba∇dblκϕ\nH1+/ba∇dblv1/ba∇dblκϕ\nH1+/ba∇dblv2/ba∇dblκϕ\nH1/parenrightbig\n,\nlast inequality following from (3.5). Due to the invariance ofνu\nR, this implies that\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglePR,u\ntϕ(v)−/integraldisplay\nH1ϕ(z)dνu\nR(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplay\nH1/vextendsingle/vextendsingle/vextendsinglePR,u\ntϕ(v)−PR,u\ntϕ(z)/vextendsingle/vextendsingle/vextendsingledνu\nR(z)\n≤cϕcκϕe−γ0t/integraldisplay\nH1/ba∇dblv−z/ba∇dblH1/parenleftbig\n1+/ba∇dblu/ba∇dblκϕ\nH1+/ba∇dblv/ba∇dblκϕ\nH1+/ba∇dblz/ba∇dblκϕ\nH1/parenrightbig\ndνu\nR(z)\n≤cϕcκϕe−γ0t/parenleftbigg\n1+/ba∇dblu/ba∇dblκϕ+1\nH1+/ba∇dblv/ba∇dblκϕ+1\nH1+/integraldisplay\nH1/ba∇dblz/ba∇dblκϕ+1\nH1dνu\nR(z)/parenrightbigg\n,\nand thanks to (9.14) we obtain (10.3). /square\nIn what follows, we fix h∈Hand we introduce the function\nϕR\n1(u,v) :=/an}b∇acketle{tg−1\nR(u)v,h/an}b∇acket∇i}htH, u∈H1, v∈H.\nIt is immediate to check that that ϕR\n1(u,·)∈C2\nb(H) and\nMR,uϕR\n1(u,v) =−/an}b∇acketle{tv,h/an}b∇acket∇i}htH. (10.4)\nThanks to Hypothesis 4, for every fixed v∈Hwe have that ϕR\n1(·,v)∈C1\nb(H1) and\n/an}b∇acketle{tDuϕR\n1(u,v),k/an}b∇acket∇i}htH1=/an}b∇acketle{t[Dg−1\nR(u)k]v,h/an}b∇acket∇i}htH=−/an}b∇acketle{tg−1\nR(u)[DgR(u)k]g−1\nR(u)v,h/an}b∇acket∇i}htH, k∈H1.\n53Now, if we introduce the mapping\nψR(u,v) :=/an}b∇acketle{tDuϕR\n1(u,v),v/an}b∇acket∇i}htH1=/an}b∇acketle{t[Dg−1\nR(u)v]v,h/an}b∇acket∇i}htH, u,v ∈H1,\nwe have /integraldisplay\nH1ψR(u,z)dνu\nR(z) =/an}b∇acketle{tSR(u),h/an}b∇acket∇i}htH.\nLemma 10.3. For everyu∈H1, the function ψR(u,·) :H1→Rhas quadratic growth and is\nlocally Lipschitz-continuous, uniformly with respect to u∈H1. Namely\nsup\nu∈H1|ψR(u,v)| ≤c/ba∇dblh/ba∇dblH/ba∇dblv/ba∇dblH1/ba∇dblv/ba∇dblH, (10.5)\nand\nsup\nu∈H1|ψR(u,v1)−ψR(u,v2)| ≤c/ba∇dblv1−v2/ba∇dblH1(/ba∇dblv1/ba∇dblH1+/ba∇dblv2/ba∇dblH1)/ba∇dblh/ba∇dblH.(10.6)\nMoreover,ψR(u,·)is twice continuously differentiable in H1, with\nsup\nu∈H1/ba∇dblDvψR(u,v)/ba∇dblH1≤c/ba∇dblv/ba∇dblH1/ba∇dblh/ba∇dblH, (10.7)\nand\nsup\nu,v∈H1/ba∇dblD2\nvψR(u,v)/ba∇dblL(H1)≤c/ba∇dblh/ba∇dblH. (10.8)\nFinally, for every v∈H1the mapping ψR(·,v) :H1→Ris differentiable and\nsup\nu∈H1/ba∇dblDuψR(u,v)/ba∇dblH1≤c/ba∇dblv/ba∇dblH1/ba∇dblv/ba∇dblH/ba∇dblh/ba∇dblH. (10.9)\nProof.Bound (10.5) is obvious. Moreover, (10.6) follows immediat ely, as soon as we see that\n|ψR(u,v1)−ψR(u,v2)| ≤c(/ba∇dblv1−v2/ba∇dblH1/ba∇dblv1/ba∇dblH+/ba∇dblv1−v2/ba∇dblH/ba∇dblv2/ba∇dblH1)/ba∇dblh/ba∇dblH.\nAs for the differentiability of ψu, we have\n/an}b∇acketle{tDvψR(u,v),k/an}b∇acket∇i}htH1=−/an}b∇acketle{tg−1\nR(u)[DgR(u)k]g−1\nR(u)v+g−1\nR(u)[DgR(u)v]g−1\nR(u)k,h/an}b∇acket∇i}htH,\nand this immediately implies (10.7). Moreover, if we differen tiate once more, we get\n/an}b∇acketle{tD2\nvψR(u,v)k1,k2/an}b∇acket∇i}htH1=−/an}b∇acketle{tg−1\nR(u)[DgR(u)k1]g−1\nR(u)k2+g−1\nR(u)[DgR(u)k2]g−1\nR(u)k1,h/an}b∇acket∇i}htH,\nand (10.8) follows.\nFinally, since we are assuming that gRis twice differentiable and g−1\nRis bounded, we have that\nψR(·,v) is differentiable and for every k∈H1\n/an}b∇acketle{tDuψR(u,v),k/an}b∇acket∇i}htH1=/an}b∇acketle{tg−1\nR(u)[DgR(u)k]g−1\nR(u)[DgR(u)v]g−1\nR(u)v,h/an}b∇acket∇i}htH\n−/an}b∇acketle{tg−1\nR(u)/bracketleftbig\nD2gR(u)(v,k)/bracketrightbig\ng−1\nR(u)v,h/an}b∇acket∇i}htH+/an}b∇acketle{tg−1\nR(u)[DgR(u)v]g−1\nR(u)[DgR(u)k]g−1\nR(u)v,h/an}b∇acket∇i}htH.\nIn particular, (10.9) follows.\n/square\nNext, for every u,v∈H1andµ∈(0,1), we define\nϕR,µ\n2(u,v) :=/integraldisplay∞\n0e−λ(µ)t/parenleftBig\nPR,u\ntψR(u,·)(v)−/an}b∇acketle{tSR(u),h/an}b∇acket∇i}htH/parenrightBig\ndt,\n54whereλ: [0,1)→Ris some continuous increasing function such that λ(0) = 0, to be determined.\nAs a consequence of Lemma 10.2 and (10.6), we have\n/vextendsingle/vextendsingle/vextendsinglePR,u\ntψR(u,·)(v)−/an}b∇acketle{tSR(u),h/an}b∇acket∇i}htH/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsinglePR,u\ntψR(u,·)(v)−/integraldisplay\nH1ψR(u,z)dνu\nR(z)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤c/parenleftbig\n1+/ba∇dblv/ba∇dbl2\nH1+/ba∇dblu/ba∇dbl2\nH1/parenrightbig\n/ba∇dblh/ba∇dblHe−γ0t,\nso that the function ϕR,µ\n2:H1×H1→Ris well defined and\nsup\nµ∈(0,1)|ϕR,µ\n2(u,v)| ≤c/parenleftbig\n1+/ba∇dblv/ba∇dbl2\nH1+/ba∇dblu/ba∇dbl2\nH1/parenrightbig\n/ba∇dblh/ba∇dblH. (10.10)\nLemma 10.4. For every fixed µ∈(0,1)andu∈H1, the function ϕR,µ\n2(u,·) :H1→Ris twice\ncontinuously differentiable and\nMR,uϕR,µ\n2(u,v) =λ(µ)ϕR,µ\n2(u,v)−(ψR(u,v)−/an}b∇acketle{tSR(u),h/an}b∇acket∇i}htH), u,v∈H1. (10.11)\nMoreover, for every µ∈(0,1)andv∈H1, the function ϕR,µ\n2(·,v) :H2→Ris continuously\ndifferentiable, with\n|DuϕR,µ\n2(u,v)k| ≤c\nλ(µ)/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH1/parenrightbig\n/ba∇dblk/ba∇dblH1/ba∇dblh/ba∇dblH, k∈H2. (10.12)\nProof.For everyu∈H1andt≥0 the mapping\nv∈H1/ma√sto→yu,v\nR(t)∈L2(Ω;H1),\nis twice differentiable with\nDvyu,v\nR(t)k=e−g(u)tk, D2\nvyu,v\nR(t)(k1,k2) = 0.\nHence, since the function ψR(u,·) is twice continuously differentiable, we have that PR,u\ntψR(u,·) is\ntwice continuously differentiable in H1, with\n/an}b∇acketle{tDvPR,u\ntψR(u,·)(v),k/an}b∇acket∇i}htH1=E/an}b∇acketle{tDvψR(u,yu,v\nR(t)),e−g(u)tk/an}b∇acket∇i}htH1\nand\n/an}b∇acketle{tD2\nvPR,u\ntψR(u,·)(v)k1,k2/an}b∇acket∇i}htH1=E/an}b∇acketle{tD2\nvψR(u,yu,v\nR(t))e−g(u)tk1,e−g(u)tk2/an}b∇acket∇i}htH1.\nThese two equalities, together with (9.13), (10.7) and (10. 8), imply that\n/ba∇dblDvPR,u\ntψR(u,·)(v)/ba∇dblH1≤c/ba∇dblh/ba∇dblHE/ba∇dblyu,v\nR(t)/ba∇dblH1e−γ0t≤c/ba∇dblh/ba∇dblH(1+/ba∇dblv/ba∇dblH1+/ba∇dblu/ba∇dblH1)e−γ0t,\nand\n/ba∇dblD2\nvPR,u\ntψR(u,·)(v)/ba∇dblL(H1)≤c/ba∇dblh/ba∇dblHe−2γ0t.\nIn particular, the function ϕR,µ\n2(u,·) :H1→Ris twice continuously differentiable, with\n/ba∇dblDvϕR,µ\n2(u,v)/ba∇dblH1≤c(1+/ba∇dblv/ba∇dblH1+/ba∇dblu/ba∇dblH1)/ba∇dblh/ba∇dblH, (10.13)\nand\n/ba∇dblD2\nvϕR,µ\n2(u,v)/ba∇dblL(H1)≤c/ba∇dblh/ba∇dblH.\nFinally, since we have\nMR,uϕR,µ\n2(u,v) =λ(µ)ϕR,µ\n2(u,v)−(ψR(u,v)−/an}b∇acketle{tSR(u),h/an}b∇acket∇i}htH), u,v∈H1,\n(10.11) follows once we notice that, as ϕR,µ\n2(u,·) is twice continuously differentiable, it holds\nMR,uϕR,µ\n2(u,v) =MR,uϕR,µ\n2(u,v).\n55Now, we have already seen that the mapping SR:H2→H1is differentiable and\n/ba∇dblDSR(u)h/ba∇dblH1≤c(1+/ba∇dblu/ba∇dblH2)/ba∇dblk/ba∇dblH1, u,k∈H2. (10.14)\nHence, in order to study the differentiability of ϕR,µ\n2with respect to u∈H1, we need to study the\ndifferentiability of the mapping\nu∈H1/ma√sto→ΨR\nv,t(u) :=PR,u\ntψR(u,·)(v) =EψR(u,yu,v\nR(t))∈R,\nfor every fixed t≥0 andv∈H1.\nWe have\n/an}b∇acketle{tDuΨR\nv,t(u),k/an}b∇acket∇i}htH1=E/an}b∇acketle{tDuψR(u,yu,v\nR(t)),k/an}b∇acket∇i}htH1+E/an}b∇acketle{tDvψR(u,yu,v\nR(t)),ηu,v\nR,k(t)/an}b∇acket∇i}htH1.\nThus, thanks to (10.7) and (10.9), we obtain\n|/an}b∇acketle{tDuΨR\nv,t(u),k/an}b∇acket∇i}ht|H1≤c/parenleftBig\nE/ba∇dblyu,v\nR(t)/ba∇dbl2\nH1/ba∇dblk/ba∇dblH1+E/ba∇dblyu,v\nR(t)/ba∇dblH1/ba∇dblηu,v\nR,k(t)/ba∇dblH1/parenrightBig\n/ba∇dblh/ba∇dblH,\nand, as a consequence of (9.13) and (10.1), we conclude that\n/ba∇dblDuΨR\nv,t(u)/ba∇dblH1≤c/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH1/parenrightbig\n/ba∇dblh/ba∇dblH.\nThis, together with (9.2) and (10.14) , implies that ϕR,µ\n2(·,v) :H2→Ris differentiable, for every\nv∈H1, and\n|DuϕR,µ\n2(u,v)k| ≤c/integraldisplay∞\n0e−λ(µ)tdt/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH1/parenrightbig\n/ba∇dblk/ba∇dblH1/ba∇dblh/ba∇dblH,\nso that (10.12) follows. /square\n10.2.The identification of the limit. We fixh∈H, and for every R≥1 andµ∈(0,1) we\ndefine\nϕR,µ(u,v) =/an}b∇acketle{tu,h/an}b∇acket∇i}htH+√µϕR\n1(u,v)+µϕR,µ\n2(u,v),(u,v)∈H1.\nWe have seen that ϕR\n1∈C2\nb(H1×H1). Hence, due to Lemma 10.4, we argue that the function\nϕR,µbelongs toC2\nb(H1×H1). In particular, we can apply Itˆ o’s formula to ϕR,µand (uR\nµ,√µ∂tuR\nµ),\nand we have\ndϕR,µ(uR\nµ(t),√µ∂tuR\nµ(t))\n=KR,µϕR,µ(uR\nµ(t),√µ∂tuR\nµ(t))dt+1√µ/an}b∇acketle{tσR(uR\nµ(t))∂twQ(t),DvϕR,µ(uR\nµ(t),√µ∂tuR\nµ(t))/an}b∇acket∇i}htH1,(10.15)\nwhereKR,µis the Kolmogorov operator defined by\nKR,µϕ(u,v) =1√µ(/an}b∇acketle{tDuϕ(u,v),v/an}b∇acket∇i}htH1+/an}b∇acketle{tDvϕ(u,v),∆u+fR(u)/an}b∇acket∇i}htH1)\n−1\nµ/an}b∇acketle{tDvϕ(u,v),gR(u)v/an}b∇acket∇i}htH1+1\n2µTr/parenleftbig\nD2\nvϕ(u,v)[σR(u)Q][σR(u)Q]⋆/parenrightbig\n.\nIn particular, we have\nKR,µϕR,µ(u,v) =1√µ/parenleftbig\nMR,uϕR\n1(u,v)+/an}b∇acketle{th,v/an}b∇acket∇i}htH1/parenrightbig\n+MR,uϕR,µ\n2(u,v)+/an}b∇acketle{tDuϕR\n1(u,v),v/an}b∇acket∇i}htH1\n+/an}b∇acketle{tDvϕR\n1(u,v),∆u+fR(u)/an}b∇acket∇i}htH1+√µ/parenleftBig\n/an}b∇acketle{tDvϕR,µ\n2(u,v),∆u+fR(u)/an}b∇acket∇i}htH1+/an}b∇acketle{tDuϕR,µ\n2(u,v),v/an}b∇acket∇i}htH1/parenrightBig\n,\n56whereMR,uis the differential operator defined in (10.4).\nThanks to (10.4) we have\nMR,uϕR\n1(u,v)+/an}b∇acketle{tDuϕ(u),v/an}b∇acket∇i}htH1= 0. (10.16)\nMoreover, recalling that /an}b∇acketle{tDuϕR\n1(u,v),v/an}b∇acket∇i}htH1=ψR(u,v), thanks to (10.11) we have\nMR,uϕR,µ\n2(u,v)+/an}b∇acketle{tDuϕR\n1(u,v),v/an}b∇acket∇i}htH1=λ(µ)ϕR,µ\n2(u,v)+/an}b∇acketle{tSR(u),h/an}b∇acket∇i}htH. (10.17)\nTherefore, since\n/an}b∇acketle{tDvϕR\n1(u,v),k/an}b∇acket∇i}htH1=/an}b∇acketle{tg−1\nR(u)k,h/an}b∇acket∇i}htH,\nif we plug (10.16) and (10.17) into (10.15) and integrate bot h side with respect to time, we obtain\n/an}b∇acketle{tuR\nµ(t),h/an}b∇acket∇i}htH=/an}b∇acketle{tuµ\n0,h/an}b∇acket∇i}htH+/integraldisplayt\n0/an}b∇acketle{tg−1\nR(uR\nµ(s))∆uR\nµ(s)+fR,g(uR\nµ(s))+SR(uR\nµ(s)),h/an}b∇acket∇i}htHds\n+/integraldisplayt\n0/an}b∇acketle{tσR,g(uR\nµ(s))∂twQ(s),h/an}b∇acket∇i}htH+RR,µ(t),\nwhere\nRR,µ(t) =√µ/an}b∇acketle{tϕR\n1(uµ\n0,vµ\n0)−ϕR\n1(uR\nµ(t),√µ∂tuR\nµ(t)),h/an}b∇acket∇i}htH\n+µ/an}b∇acketle{tϕR,µ\n2(uµ\n0,vµ\n0)−ϕR,µ\n2(uR\nµ(t),√µ∂tuR\nµ(t)),h/an}b∇acket∇i}htH\n+/integraldisplayt\n0/parenleftBig\nλ(µ)ϕR,µ\n2(uR\nµ(s),√µ∂tuR\nµ(s))+µ/an}b∇acketle{tDuϕR,µ\n2(uR\nµ(s),√µ∂tuR\nµ(s)),∂tuR\nµ(s)/an}b∇acket∇i}htH1/parenrightBig\ndt\n+√µ/integraldisplayt\n0/an}b∇acketle{tDvϕR,µ\n2(uR\nµ(s),√µ∂tuR\nµ(s)),∆uR\nµ(t)+fR(uR\nµ(s))/an}b∇acket∇i}htH1ds\n+√µ/integraldisplayt\n0/an}b∇acketle{tDvϕR,µ\n2(uR\nµ(s),√µ∂tuR\nµ(s)),σR(uR\nµ(s))∂twQ(s)/an}b∇acket∇i}htH.\nLemma 10.5. Assume that\nlim\nµ→0λ(µ) = 0, lim\nµ→0µ1\n2−δ\nλ(µ)= 0, (10.18)\nwhereδ∈(0,1/2)is the constant introduced in (3.9). Then, under Hypotheses 1 to 5, for every\nT >0and(uµ\n0,vµ\n0)∈H2satisfying (3.8)and(3.9), we have\nlim\nµ→0Esup\nt∈[0,T]|RR,µ(t)|= 0. (10.19)\nProof.Since\n|ϕR\n1(u,v)| ≤c/ba∇dblv/ba∇dblH/ba∇dblh/ba∇dblH\ndue to (3.8) we have\n√µ|/an}b∇acketle{tϕR\n1(uµ\n0,vµ\n0)−ϕR\n1(uR\nµ(t),√µ∂tuR\nµ(t)),h/an}b∇acket∇i}htH| ≤c√µ/ba∇dblh/ba∇dblH+cµ/ba∇dbl∂tuR\nµ(t)/ba∇dblH/ba∇dblh/ba∇dblH,\nso that, from (6.6), we obtain\nlim\nµ→0√µEsup\nt∈[0,T]|/an}b∇acketle{tϕR\n1(uµ\n0,vµ\n0)−ϕR\n1(uR\nµ(t),√µ���tuR\nµ(t)),h/an}b∇acket∇i}htH|= 0. (10.20)\n57Next, according to (10.10),\n|ϕR,µ\n2(u,v)| ≤c/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nH1+/ba∇dblv/ba∇dbl2\nH1/parenrightbig\n/ba∇dblh/ba∇dblH,\nso that, thanks to (3.9), (6.4) and (6.6), we get\nlim\nµ→0µEsup\nt∈[0,T]|/an}b∇acketle{tϕR,µ\n2(uµ\n0,vµ\n0)−ϕR,µ\n2(uR\nµ(t),√µ∂tuR\nµ(t)),h/an}b∇acket∇i}htH|= 0, (10.21)\nand thanks to (5.1), (6.5) and (10.18) we get\nlim\nµ→0/integraldisplayT\n0λ(µ)E|ϕR,µ\n2(uR\nµ(t),√µ∂tuR\nµ(t))|dt= 0. (10.22)\nMoreover, according to (10.12), we have\nµ|/an}b∇acketle{tDuϕR,µ\n2(uR\nµ(t),√µ∂tuR\nµ(t)),∂tuR\nµ(t)/an}b∇acket∇i}htH1|\n≤c√µ\nλ(µ)/parenleftbig\n1+/ba∇dbluR\nµ(t)/ba∇dbl2\nH2+µ/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH1/parenrightbig√µ/ba∇dbl∂tuR\nµ(t)/ba∇dblH1/ba∇dblh/ba∇dblH\n≤c√µ\nλ(µ)/parenleftbig\n1+/ba∇dbluR\nµ(t)/ba∇dbl2\nH2+µ/ba∇dbluR\nµ(t)/ba∇dbl2\nH2/ba∇dbl∂tuR\nµ(t)/ba∇dbl2\nH1+µ2/ba∇dbl∂tuR\nµ(t)/ba∇dbl4\nH1/parenrightbig\n/ba∇dblh/ba∇dblH.\nIn view of (6.4), (7.2) and (10.18), this implies\nlim\nµ→0µ/integraldisplayT\n0E|/an}b∇acketle{tDuϕR,µ\n2(uR\nµ(t),√µ∂tuR\nµ(t)),∂tuR\nµ(t)/an}b∇acket∇i}htH1|dt= 0. (10.23)\nFinally, due to (10.13), we have\n|/an}b∇acketle{tDvϕ2R,µ(uR\nµ(t),√µ∂tuR\nµ(t)),∆uR\nµ(t)+f(uR\nµ(t))/an}b∇acket∇i}htH1|\n≤c/parenleftbig\n1+/ba∇dbluR\nµ(t)/ba∇dblH1+√µ/ba∇dbl∂tuR\nµ(t)/ba∇dblH1/parenrightbig/parenleftbig\n/ba∇dbluR\nµ(t)/ba∇dblH3+1/parenrightbig\n/ba∇dblh/ba∇dblH,\nand hence, thanks to (5.1), (6.5) and (7.1), we have\n/integraldisplayT\n0E|/an}b∇acketle{tDvϕR,µ\n2(uR\nµ(t),√µ∂tuR\nµ(t)),∆uR\nµ(t)+f(uR\nµ(t))/an}b∇acket∇i}htH1|dt≤cT/parenleftbiggρT,R(µ)\nµδ/parenrightbigg1/2\n.\nAccording to (7.3) and (10.18), this implies\nlim\nµ→0/integraldisplayT\n0√µE|/an}b∇acketle{tDvϕR,µ\n2(uR\nµ(t),√µ∂tuR\nµ(t)),∆uR\nµ(t)+fR(uR\nµ(t))/an}b∇acket∇i}htH1|dt= 0. (10.24)\nTherefore, combining together (10.20), (10.21), (10.22), (10.23) and (10.24), we obtain (10.19).\n/square\n10.3.Conclusion. All subsequences, filtered probability spaces and Wiener pr ocesses that we are\ngoing to introduce in what follows depend on R≥1. However, since we keep Rfixed in this\nsubsection, for the sake of simplicity of notations we do not emphasize their dependence on R.\nIn what follows, for every\n̺<1,1≤ϑ<2, p<2\nϑ−1,\nwe define\nXp\nϑ,̺:=C([0,T];H̺)∩Lp(0,T;Hϑ),\n58and\nKp\nϑ,̺:=/bracketleftBig\nXp\nϑ,̺×L∞(0,T;H̺)/bracketrightBig2\n×C([0,T];U),\nwhereUis a Hilbert space containing the reproducing kernel HQwith Hilbert-Schmidt embedding.\nIn Proposition 8.1 we have seen that the family {L(uR\nµ)}µ∈(0,1)is tight in Xp\nϑ,̺. Due to (6.6),\nthis implies that the family {L(uR\nµ,µ∂tuR\nµ)}µ∈(0,1)is tight in Xp\nϑ,̺×L∞(0,T;H̺). In particular,\nthanks to the Skorokhod theorem, for any two sequences {µ1\nk}k∈Nand{µ2\nk}k∈N, both converging\nto zero, there exist two subsequences, still denoted by {µ1\nk}k∈Nand{µ2\nk}k∈N, a sequence of random\nvariables\nYk:= ((̺1\nk,ϑ1\nk),(̺2\nk,ϑ2\nk),ˆwQ\nk), k∈N,\ninKp\nϑ,̺and a random variable Y= (̺1,̺2,ˆwQ) inXp\nϑ,̺×Xp\nϑ,̺×C([0,T];U), all defined on some\nprobability space ( ˆΩ,ˆF,ˆP), such that\nL(Yk) =L((uR\nµ1\nk,/radicalBig\nµ1\nk∂tuR\nµ1\nk),(uR\nµ2\nk,/radicalBig\nµ2\nk∂tuR\nµ1\nk), wQ), k∈N, (10.25)\nand, fori= 1,2,\nlim\nk→∞/ba∇dbl̺i\nk−̺i/ba∇dblXp\nϑ,̺+/radicalBig\nµi\nk/ba∇dblϑi\nk/ba∇dblL∞(0,T;H)+/ba∇dblˆwQ\nk−ˆwQ/ba∇dblC([0,T];U)= 0,ˆP−a.s. (10.26)\nIt is important to stress that, due to (6.4), the sequences {̺i\nk}k∈Nare bounded in the space\nL2(ˆΩ,C([0,T];H1)∩L2(0,T;H2)), fori= 1,2, and\n̺i∈L2(ˆΩ,C([0,T];H1)∩L2(0,T;H2)), i= 1,2.\nNext, a filtration ( ˆFt)t≥0is introduced in ( ˆΩ,ˆF,ˆP), by taking the augmentation of the canonical\nfiltration of the process ( ρ1,ρ2,ˆwQ), generated by its restrictions to every interval [0 ,t]. Due to this\nconstruction, ˆ wQis a (ˆFt)t≥0Wiener process with covariance Q∗Q(for a proof see [19, Lemma\n4.8]).\nNow, if we show that ̺1=̺2, we have that uR\nµconverges in probability to some\nuR∈Lp(0,T;Hϑ)∩C([0,T];H1)∩L2(0,T;H2),ˆP−a.s.\nActually, as observed by Gy¨ ongy and Krylov in [27], if Eis any Polish space equipped with the\nBorelσ-algebra, a sequence ( ξn)n∈NofE-valued random variables converges in probability if and\nonly if for every pair of subsequences ( ξm)m∈Nand (ξl)l∈Nthere exists an E2-valued subsequence\nηk:= (ξm(k),ξl(k)) converging weakly to a random variable ηsupported on the diagonal {(h,k)∈\nE2:h=k}.\nIn order to show that ̺1=̺2, we prove that they are both a solution of equation (9.16), wh ich,\nas proven in Theorem 9.2, has pathwise uniqueness in L2(Ω;C([0,T];H1)∩L2(0,T;H2)). Due to\n(10.25), we have that both ( ̺1\nk,ϑ1\nk) and (̺2\nk,ϑ2\nk) satisfy equation (6.3), with wQreplaced by ˆ wQ\nk.\nTherefore, we have\n/an}b∇acketle{t̺i\nk(t),h/an}b∇acket∇i}htH=/an}b∇acketle{tuµi\nk\n0,h/an}b∇acket∇i}htH+/integraldisplayt\n0/an}b∇acketle{tg−1\nR(̺i\nk(s))∆̺i\nk(s)+fR,g(̺i\nk(s))+SR(̺i\nk(s)),h/an}b∇acket∇i}htHds\n+/integraldisplayt\n0/an}b∇acketle{tσR,g(̺i\nk(s))∂tˆwQ\nk(s),h/an}b∇acket∇i}htH+Ri\nR,k(t),\n59where\nRi\nR,k(t) =/radicalBig\nµi\nk/an}b∇acketle{tϕR\n1(ζi\nk)−ϕR\n1(ζi\nk(t)),h/an}b∇acket∇i}htH+µi\nk/an}b∇acketle{tϕR,µ\n2(ζi\nk)−ϕR,µ\n2(ζi\nk(t)),h/an}b∇acket∇i}htH\n+/integraldisplayt\n0/parenleftbigg\nλ(µi\nk)ϕR,µ\n2(ζi\nk(s))+/radicalBig\nµi\nk/an}b∇acketle{tDuϕR,µ\n2(ζi\nk(s)),ϑi\nk(s)/an}b∇acket∇i}htH1/parenrightbigg\ndt\n+/radicalBig\nµi\nk/integraldisplayt\n0/an}b∇acketle{tDvϕR,µ\n2(ζi\nk(s)),∆̺i\nk(t)+fR(̺i\nk(s))/an}b∇acket∇i}htH1ds\n+/radicalBig\nµi\nk/integraldisplayt\n0/an}b∇acketle{tDvϕR,µ\n2(ζi\nk(s)),σR(̺i\nk(s))∂tˆwQ\nk(s)/an}b∇acket∇i}htH,\nand\nζi\nk:= (uµi\nk\n0,vµi\nk\n0), ζi\nk(t) := (̺i\nk(t),ϑi\nk(t)).\nAccording to (10.25) and (10.19), we have\nlim\nk→∞ˆEsup\nt∈[0,T]|Ri\nR,k(t)|= 0, i= 1,2. (10.27)\nIf we assume that h∈H1, we have\n/integraldisplayt\n0/an}b∇acketle{tg−1\nR(̺i\nk(s))∆̺i\nk(s),h/an}b∇acket∇i}htHds=−/integraldisplayt\n0/an}b∇acketle{t̺i\nk(s),[g−1\nR(̺i\nk(s))]th/an}b∇acket∇i}htH1ds.\nThus, thanks to (10.26), we have\nlim\nk→∞/integraldisplayt\n0/an}b∇acketle{tg−1\nR(̺i\nk(s))∆̺i\nk(s),h/an}b∇acket∇i}htHds=/integraldisplayt\n0/an}b∇acketle{tg−1\nR(̺i(s))∆̺i(s),h/an}b∇acket∇i}htHds,ˆP−a.s. (10.28)\nMoreover, due to (9.10), we have that fR,gis locally Lipschitz continuous from H1intoH, with\n/integraldisplayt\n0/vextendsingle/vextendsingle/an}b∇acketle{tfR,g(̺i\nk(s))−fR,g(̺i(s)),h/an}b∇acket∇i}htH/vextendsingle/vextendsingleds≤c/ba∇dbl̺i\nk−̺i/ba∇dblL2(0,T;H1)/parenleftbig\n1+/ba∇dbl̺i/ba∇dblL2(0,T;H1)/parenrightbig\n/ba∇dblh/ba∇dblH,\nand (10.26) implies\nlim\nk→∞/integraldisplayt\n0/an}b∇acketle{tfR,g(̺i\nk(s)),h/an}b∇acket∇i}htHds=/integraldisplayt\n0/an}b∇acketle{tfR,g(̺i(s)),h/an}b∇acket∇i}htHds,ˆP−a.s. (10.29)\nFinally, due to (9.15), we have\n/integraldisplayt\n0/vextendsingle/vextendsingle/an}b∇acketle{tSR(̺i\nk(s))−SR(̺i(s)),h/an}b∇acket∇i}htH/vextendsingle/vextendsingleds\n≤c/ba∇dbl̺i\nk−̺i/ba∇dblL2(0,T;H1)/parenleftBig\n1+/ba∇dbl̺i\nk/ba∇dbl2\nL4(0,T;H1)+/ba∇dbl̺i/ba∇dbl2\nL4(0,T;H1)/parenrightBig\n/ba∇dblh/ba∇dblH,\nand due to (10.26) we conclude that\nlim\nk→∞/integraldisplayt\n0/an}b∇acketle{tSR(̺i\nk(s)),h/an}b∇acket∇i}htHds=/integraldisplayt\n0/an}b∇acketle{tSR(̺i(s)),h/an}b∇acket∇i}htHds,ˆP−a.s. (10.30)\nNow, fori= 1,2 andt∈[0,T] we define\nMi\nR(t) :=/an}b∇acketle{t̺i(t),h/an}b∇acket∇i}htH−/an}b∇acketle{tu0,h/an}b∇acket∇i}htH−/integraldisplayt\n0/an}b∇acketle{tg−1\nR(̺i(s))∆̺i(s)+fR,g(̺i(s))+SR(̺i(s)),h/an}b∇acket∇i}htHds.\n60By proceeding as in the proof of [19, Lemma 4.9], in view of (10 .27), (10.28), (10.29) and (10.30),\nthanks to (3.10) we have for every t∈[0,T]\n/angbracketleftbigg\nMi\nR−/integraldisplay·\n0σR,g(̺i(s))∂tˆwQ(s)/angbracketrightbigg\nt= 0,\nwhere/an}b∇acketle{t·/an}b∇acket∇i}httis the quadratic variation process. This implies that both ̺1and̺2satisfy equation\n(9.16) and, as we have explained above, this allows to conclu de the proof of Theorem 10.1.\n11.Proof of Theorem 3.3\nAs a consequence of Theorem 10.1 and Proposition 9.2, for eve ryR≥1 andu0∈H1there\nexists a unique solution uR∈L2(Ω;C([0,T];H1)∩L2(0,T;H2)) for equation (9.16). We start by\nshowing that this allows to prove that equation (3.12) is wel l-posed inC([0,T];H1)∩L2(0,T;H2),\nas well.\nProposition 11.1. Under Hypotheses 1 to 5, for every u0∈H1equation (3.12)admits a unique\nsolutionu∈L2(Ω;C([0,T];H1)∩L2(0,T;H2)).\nProof.IfuRis the unique solution of equation (9.16) with initial condi tionu0, we have\n1\n2d/ba∇dbluR(t)/ba∇dbl2\nH1=/an}b∇acketle{tuR(t),g−1\nR(uR(t))∆uR(t)+fR,g(uR(t))+SR(uR(t))/an}b∇acket∇i}htH1dt\n+1\n2/ba∇dblσR,g(uR(t))/ba∇dbl2\nL2(HQ,H1)dt+/an}b∇acketle{tuR(t),σR,g(uR(t))dwQ(t)/an}b∇acket∇i}htH1.\nDue to (9.1) we have\n|/an}b∇acketle{tu,SR(u)/an}b∇acket∇i}htH1| ≤ /ba∇dblu/ba∇dblH2/ba∇dblSR(u)/ba∇dblH≤c/ba∇dblu/ba∇dblH2(1+/ba∇dblu/ba∇dblH1), u∈H2.\nHence, thanks to (2.27) we get\n1\n2d/ba∇dbluR(t)/ba∇dbl2\nH1≤ −˜γ0/ba∇dbluR(t)/ba∇dbl2\nH2dt+c/parenleftbig\n1+/ba∇dbluR(t)/ba∇dbl2\nH1/parenrightbig\ndt+\n+/ba∇dbluR(t)/ba∇dblH2/parenleftbig\n1+/ba∇dbluR(t)/ba∇dblH1/parenrightbig\ndt+/an}b∇acketle{tuR(t),σR,g(uR(t))dwQ(t)/an}b∇acket∇i}htH1\n≤ −˜γ0\n2/ba∇dbluR(t)/ba∇dbl2\nH2dt+c/parenleftbig\n1+/ba∇dbluR(t)/ba∇dbl2\nH1/parenrightbig\ndt+/an}b∇acketle{tuR(t),σR,g(uR(t))dwQ(t)/an}b∇acket∇i}htH1.\nAfter we first integrate both sides with respect to t, then take the supremum and finally take the\nexpectation, we get\nEsup\ns∈[0,t]/ba∇dbluR(s)/ba∇dbl2\nH1+ ˜γ0/integraldisplayt\n0E/ba∇dbluR(s)/ba∇dbl2\nH2ds≤ /ba∇dblu0/ba∇dbl2\nH1+cT+cT/integraldisplayt\n0E/ba∇dbluR(s)/ba∇dbl2\nH1ds,\nand Gronwall’s lemma gives\nsup\nR≥1/parenleftBigg\nEsup\ns∈[0,T]/ba∇dbluR(s)/ba∇dbl2\nH1+/integraldisplayT\n0E/ba∇dbluR(s)/ba∇dbl2\nH2ds/parenrightBigg\n≤cT. (11.1)\nNow, we introduce the stopping time\nτR:= inf/braceleftbig\nt∈[0,T] :/ba∇dbluR(t)/ba∇dblH¯r≥R/bracerightbig\n,\n61withtheconvention that inf ∅= +∞. Sincethefamily ofstoppingtimes {τR}R≥1isnon-decreasing,\nwe can define\nτ:= lim\nR→∞τR.\nAccording to (11.1) we have\nP(τ= +∞) = 1. (11.2)\nHence, if we define Ω′:={τ= +∞}, we have that for every t∈[0,T] andω∈Ω′there exists\nR≥1 such that t≤τR(ω) and we define\nu(t)(ω) =uR(t)(ω).\nDue to (6.2), by a uniqueness argument we have that\nR1≤R2=⇒uR2(t) =uR1(t), t≤τR1.\nThis implies that the definition of uis a good definition and uis a solution of equation (3.12).\nMoreover, in view of (11.1), we have that ubelongs toL2(Ω;C([0,T];H1)∩L2(0,T;H2)) and, in\nparticular, due to Proposition 9.2 is the unique solution of equation (3.12).\n/square\nNow, we are finally ready to conclude the proof of Theorem 3.3. In the same spirit of what we\nhave done above for uR, for everyµ∈(0,1) we define\nτR\nµ:= inf/braceleftbig\nt∈[0,T] :/ba∇dbluR\nµ(t)/ba∇dblH¯r≥R/bracerightbig\n,\nagain with the convention that inf ∅= +∞. For every η>0 we have\nP/parenleftBig\n/ba∇dbluµ−u/ba∇dblXp\nϑ,̺>η/parenrightBig\n≤P/parenleftBig\n/ba∇dbluµ−uR\nµ/ba∇dblXp\nϑ,̺>η/3;τR\nµ<∞/parenrightBig\n+P/parenleftBig\n/ba∇dbluR\nµ−uR/ba∇dblXp\nϑ,̺>η/3/parenrightBig\n+P/parenleftBig\n/ba∇dbluR−u/ba∇dblXp\nϑ,̺>η/3;τR<∞/parenrightBig\n≤P(τR\nµ<∞)+P(τR<∞)+P/parenleftBig\n/ba∇dbluR\nµ−uR/ba∇dblXp\nϑ,̺>η/3/parenrightBig\n.\nTherefore, since\nP/parenleftbig\nτR\nµ<∞/parenrightbig\n≤P/parenleftBigg\nsup\nt∈[0,T]/ba∇dbluR\nµ(t)−uR(t)/ba∇dblH¯r≥R/2/parenrightBigg\n+P/parenleftBigg\nsup\nt∈[0,T]/ba∇dbluR(t)/ba∇dblH¯r≥R/2/parenrightBigg\n≤P/parenleftBigg\nsup\nt∈[0,T]/ba∇dbluR\nµ(t)−uR(t)/ba∇dblH¯r≥R/2/parenrightBigg\n+P/parenleftBig\nτR/2<∞/parenrightBig\n,\nwe have\nP/parenleftBig\n/ba∇dbluµ−u/ba∇dblXp\nϑ,̺>η/parenrightBig\n≤2P(τR/2<∞)+P/parenleftBig\n/ba∇dbluR\nµ−uR/ba∇dblXp\nϑ,¯r≥R/2/parenrightBig\n+P/parenleftBig\n/ba∇dbluR\nµ−uR/ba∇dblXp\nϑ,̺>η/3/parenrightBig\n.\nThanks to (11.2) for every ǫ>0 we can fix Rǫ≥1 such that\nP(τRǫ/2<∞)<ǫ\n2,\n62and, as a consequence of Theorem 10.1, we get\nliminf\nµ→0P/parenleftBig\n/ba∇dbluµ−u/ba∇dblXp\nϑ,̺>η/parenrightBig\n≤ǫ+ lim\nµ→0P/parenleftBig\n/ba∇dbluRǫµ−uRǫ/ba∇dblXp\nϑ,¯r≥Rǫ/2/parenrightBig\n+ lim\nµ→0P/parenleftBig\n/ba∇dbluRǫµ−uRǫ/ba∇dblXp\nϑ,̺>η/3/parenrightBig\n=ǫ.\nDue to the arbitrariness of ǫ>0, we conclude that (3.11) holds and Theorem 3.3 follows.\nReferences\n[1] J. 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Simon, Compact sets in the space Lp(0,T;B), Annali di Matematica Pura ed Applicata 146 (1986), pp. 65– 96.\n64[50] M.Smoluchowski, Drei Vortage ¨ uber Diffusion Brownsche Bewegung und Koagulat ion von Kolloidteilchen , Physik\nZeit. 17 (1916), pp. 557-585.\n[51] K. Spiliopoulos, A note on the Smoluchowski-Kramers approximation for the La ngevin equation with reflection ,\nStochastics and Dynamics 7 (2007), pp. 141–152.\n[52] L. Xie, L. Yang, The Smoluchowski-Kramers limits of stochastic differential equations with irregular coefficients ,\nStochastic Processes and their Applications 150 (2022), pp . 91–115.\n[53] Y. Zine, Smoluchowski-Kramers approximation for the singular stoc hastic wave equations in two dimensions ,\narXiv:2206.08717.\nDepartment of Mathematics, University of Maryland,\nEmail address :cerrai@umd.edu\nUniv. de Rennes and IUF, CNRS, IRMAR - UMR 6625,\nEmail address :arnaud.debussche@ens-rennes.fr\n65"    },    {        "title": "2312.09140v1.Nonlocal_damping_of_spin_waves_in_a_magnetic_insulator_induced_by_normal__heavy__or_altermagnetic_metallic_overlayer__a_Schwinger_Keldysh_field_theory_approach.pdf",        "content": "Nonlocal damping of spin waves in a magnetic insulator induced by normal, heavy, or\naltermagnetic metallic overlayer: A Schwinger-Keldysh field theory approach\nFelipe Reyes-Osorio and Branislav K. Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n(Dated: December 15, 2023)\nUnderstanding spin wave (SW) damping, and how to control it to the point of being able to\namplify SW-mediated signals, is one of the key requirements to bring the envisaged magnonic tech-\nnologies to fruition. Even widely used magnetic insulators with low magnetization damping in their\nbulk, such as yttrium iron garnet, exhibit 100-fold increase in SW damping due to inevitable con-\ntact with metallic layers in magnonic circuits, as observed in very recent experiments [I. Bertelli\net al. , Adv. Quantum Technol. 4, 2100094 (2021)] mapping SW damping in spatially-resolved\nfashion. Here, we provide microscopic and rigorous understanding of wavevector-dependent SW\ndamping using extended Landau-Lifshitz-Gilbert equation with nonlocal damping tensor , instead\nof conventional local scalar Gilbert damping, as derived from Schwinger-Keldysh nonequilibrium\nquantum field theory. In this picture, the origin of nonlocal magnetization damping and thereby in-\nduced wavevector-dependent SW damping is interaction of localized magnetic moments of magnetic\ninsulator with conduction electrons from the examined three different types of metallic overlayers—\nnormal, heavy, and altermagnetic. Due to spin-split energy-momentum dispersion of conduction\nelectrons in the latter two cases, the nonlocal damping is anisotropic in spin and space, and it can\nbe dramatically reduced by changing the relative orientation of the two layers when compared to\nthe usage of normal metal overlayer.\nIntroduction. —Spin wave (SW) or magnon damping\nis a problem of great interest to both basic and ap-\nplied research. For basic research, its measurements [1–4]\ncan reveal microscopic details of boson-boson or boson-\nfermion quasiparticle interactions in solids, such as:\nmagnon-magnon interactions (as described by second-\nquantized Hamiltonians containing products of three or\nmore bosonic operators [5, 6]), which are frequently en-\ncountered in antiferromagnets [4, 5] and quantum spin\nliquids [7], wherein they play a much more important\nrole [8] than boson-boson interactions in other condensed\nphases, like anharmonic crystalline lattices or superflu-\nids [5]; magnon-phonon interactions [3], especially rel-\nevant for recently discovered two-dimensional magnetic\nmaterials [2]; and magnon-electron interactions in mag-\nnetic metals [1, 9–12]. For the envisaged magnon-\nbased digital and analog computing technologies [13–\n17], understanding magnon damping makes it possible\nto develop schemes to suppress [18] it, and, further-\nmore, achieve amplification of nonequilibrium fluxes of\nmagnons [19–22]. In fact, overcoming damping and\nachieving amplification is the keyto enable complex\nmagnon circuits where, e.g., a logic gate output must\nbe able to drive the input of multiple follow-up gates.\nLet us recall that the concept of SW was introduced by\nBloch [23] as a wave-like disturbance in the local mag-\nnetic ordering of a magnetic material. The quanta [6] of\nenergy of SWs of frequency ωbehave as quasiparticles\ntermed magnons, each of which carries energy ℏωand\nspin ℏ. As regards terminology, we note that in magnon-\nics [13] SW is often used for excitations driven by an-\ntennas [24–27] and/or described by the classical Landau-\nLifshitz-Gilbert (LLG) equation [9, 10, 28, 29], whereas\nmagnon is used for the quantized version of the same ex-\ne\nee\nee\neFIG. 1. (a) Schematic view of bilayers where a metallic over-\nlayer covers the top surface of magnetic insulator, as often\nencountered in spintronics and magnonics [13, 30]. Three\ndifferent energy-momentum dispersion of conduction elec-\ntrons at the interface are considered, with their Fermi sur-\nfaces shown in panel (b)—normal metal (NM); heavy metal\n(HM) with the Rashba SOC [31, 32], and altermagnetic metal\n(AM) [33, 34]—with the latter two being spin-split. The rel-\native alignment of the layers is labeled by an angle θ[33, 34],\nmeaning that the wavevector qof SWs within FI is at an an-\ngleθaway from the kx-axis.\ncitation [5], or these two terms are used interchangingly.\nIn particular, experiments focused on SW damp-\ning in metallic ferromagnets have observed [1] its de-\npendence on the wavevector qwhich cannot be ex-\nplained by using the standard LLG equation [28,\n29],∂tMn=−Mn×Beff\nn+αGMn×∂tMn(where ∂t≡\n∂/∂t), describing dynamics of localized magnetic mo-\nments (LMMs) Mnat site nof crystalline lattice (also\nused in atomistic spin dynamics [28]) viewed as classi-\ncal vectors of unit length. This is because αG, as the\nGilbert damping parameter [35, 36], is a local scalar (i.e.,\nposition-independent constant). Instead, various forms\nof spatially nonuniform (i.e., coordinate-dependent) and\nnonlocal (i.e., magnetization-texture-dependent) damp-\ning due to conduction electrons have been proposed [9,arXiv:2312.09140v1  [cond-mat.mes-hall]  14 Dec 20232\n10, 37–39], or extracted from first-principles calcula-\ntions [40], to account for observed wavevector-dependent\ndamping of SWs, such as ∝q2(q=|q|) measured in\nRef. [1]. The nonlocal damping terms require neither\nspin-orbit coupling (SOC) nor magnetic disorder scatter-\ning, in contrast to αGwhich is considered to vanish [41]\nin their absence.\nThus, in magnonics, it has been considered [30] that\nusage of magnetic insulators, such as yttrium iron gar-\nnet (YIG) exhibiting ultralow αG≃10−4(achieved on\na proper substrate [42]), is critical to evade much larger\nand/or nonlocal damping of SWs found in ferromagnetic\nmetals. However, very recent experiments [24–27] have\nobserved 100-fold increase of SW damping in the segment\nof YIG thin film that was covered by a metallic overlayer.\nSuch spatially-resolved measurement [24] of SW damp-\ning was made possible by the advent of quantum sensing\nbased on nitrogen vacancy (NV) centers in diamond [43],\nand it was also subsequently confirmed by other meth-\nods [25–27]. Since excitation, control, and detection of\nSWs requires to couple YIG to metallic electrodes [13],\nunderstanding the origin and means to control/suppress\nlarge increase in SW damping underneath metallic over-\nlayer is crucial for realizing magnonic technologies. To\nexplain their experiments, Refs. [24–27] have employed\nthe LLG equation with ad hoc intuitively-justified terms\n(such as, effective magnetic field due to SW induced eddy\ncurrents within metallic overlayer [24]) that can fit the\nexperimental data, which is nonuniversal and unsatisfac-\ntory (many other examples of similar phenomenological\nstrategy exist [1, 44]).\nIn contrast, in this Letter we employ recently derived\n∂tMn=−Mn×Beff\nn+Mn×X\nn′(αGδnn′+λR)·∂tMn′,\n(1)\nextended LLG equation with all terms obtained [45]\nmicroscopically from Schwinger-Keldysh nonequilibrium\nquantum field theory [46] and confirmed [45] via exact\nquantum-classical numerics [47–50]. It includes nonlo-\ncal damping as the third term on the right-hand side\n(RHS), where its nonlocality is signified by dependence\nonR=rn−rn′, where rnis the position vector of lat-\ntice site n. Equation (1) is applied to a setup depicted in\nFig. 1 where conduction electron spins from three differ-\nent choices for metallic overlayer are assumed to interact\nwith LMMs of ferromagnetic insulator (FI) at the inter-\nface via sdexchange interaction of strength Jsd, as well\nas possibly underneath the top surface of FI because of\nelectronic evanescent wavefunction penetrating into it.\nNote that FI/normal metal (NM) bilayer directly mod-\nels recent experiments [24] where FI was a thin film of\nYIG and NM was Au, and SW damping within FI was\nquantified using quantum magnetometry via NV centers\nin diamond. Next, the FI/heavy metal (HM) bilayer,\nsuch as YIG/Pt [18, 27], is frequently encountered in\n0 2 4\nK/J0.51.0q (1/a)\n kF= 0\n.92kF= 0\n.99kF= 1\n.08kF= 1\n.15kF= 1\n.22kF= 1\n.30kF= 1\n.38(a)\n1.0 1.2 1.4\nkF0.81.01.21.4qmax(1/a)\n∝kF(b)FIG. 2. (a) Wavevector qof SW generated by injecting spin-\npolarized current in TDNEGF+LLG simulations of NM over-\nlayer on the top of 1D FI [Fig. 1(a)] as a function of anisotropy\nK[Eq. (3)] for different electronic Fermi wavevectors kF. (b)\nMaximum wavevector qmaxof SWs that can be generated by\ncurrent injection [21, 57] before wavevector-dependent SW\ndamping becomes operative, as signified by the drop around\nkFin curves plotted in panel (a).\nvarious spintronics and magnonics phenomena [13, 30].\nFinally, due to recent explosion of interest in altermag-\nnets [33, 34], the FI/altermagnetic metal (AM) bilay-\ners, such as YIG/RuO 2, have been explored experimen-\ntally to characterize RuO 2as a spin-to-charge conver-\nsion medium [51]. The Schwinger-Keldysh field theory\n(SKFT), commonly used in high energy physics and cos-\nmology [52–54], allows one to “integrate out” unobserved\ndegrees of freedom, such as the conduction electrons in\nthe setup of Fig. 1, leaving behind a time-retarded dis-\nsipation kernel [48, 55, 56] that encompasses electronic\neffects on the remaining degrees of freedom. This ap-\nproach then rigorously yields the effective equation for\nLMMs only , such as Eq. (1) [45, 56] which bypasses the\nneed for adding [1, 24, 44] phenomenological wavevector-\ndependent terms into the standard LLG equation. In\nour approach, the nonlocal damping is extracted from\nthe time-retarded dissipation kernel [45].\nSKFT-based theory of SW damping in FI/metal bilay-\ners.—The nonlocal damping [45] λRin the third term\non the RHS of extended LLG Eq. (1) stems from back-\naction of conduction electrons responding nonadiabati-\ncally [48, 59]—i.e., with electronic spin expectation value\n⟨ˆsn⟩being always somewhat behind LMM which gener-\nates spin torque [60] ∝ ⟨ˆsn⟩×Mn—to dynamics of LMMs.\nIt is, in general, a nearly symmetric 3 ×3 tensor whose\ncomponents are given by [45]\nλαβ\nR=−J2\nsd\n2πZ\ndε∂f\n∂εTr\u0002\nσαAnn′σβAn′n\u0003\n. (2)\nHere, f(ε) is the Fermi function; α, β =x, y, z ;σα\nis the Pauli matrix; and A(ε) = i\u0002\nGR(ε)−GA(ε)\u0003\nis\nthe spectral function in the position representation ob-\ntained from the retarded/advanced Green’s functions\n(GFs) GR/A(ε) =\u0000\nε−H±iη\u0001−1. Thus, the calcula-\ntion of λRrequires only an electronic Hamiltonian H\nas input, which makes theory fully microscopic (i.e.,3\n−5 0 5\nX−505Y\nλNM\nR(a)NM\n−1 0 1\nλα\nR\n−5 0 5\nX−505Y\nλxx\nR(b)HM t SOC= 0.3t0\n−5 0 5\nX−505Y\nλzz\nR(c)\n−5 0 5\nX−505Y\nλ⊥\nR(d)AM t AM= 0.5t0\n0.0 0.5 1.0 1.5\nq (1/a)0123Γq≡Im(ωq) (J/¯ h)×10−1\nEq.(6)(e)\nη= 0.1\nη= 0\n−5 0 5\nX−505Y\nλyy\nR(f)\n−5 0 5\nX−505Y\nλxy\nR(g)\n−5 0 5\nX−505Y\nλ/bardbl\nR(h)\nFIG. 3. (a)–(d) and (f)–(h) Elements of SKFT-derived nonlocal damping tensor in 2D FI, λRwhere R= (X, Y, Z ) is the\nrelative vector between two sites within FI, covered by NM [Eq. (5)], HM [Eqs. (8)] or AM [Eqs. (9)] metallic overlayer. (e)\nWavevector-dependent damping Γ qof SWs due to NM overlayer, where the gray line is based on Eq. (6) in the continuous\nlimit [58] and the other two lines are numerical solutions of extended LLG Eq. (1) for discrete lattices of LMMs within FI. The\ndotted line in (e) is obtained in the absence of nonlocal damping ( η= 0), which is flat at small q.\nHamiltonian-based). Although the SKFT-based deriva-\ntion [45] yields an additional antisymmetric term, not\ndisplayed in Eq. (2), such term vanishes if the system\nhas inversion symmetry. Even when this symmetry is\nbroken, like in the presence of SOC, the antisymmet-\nric component is often orders of magnitude smaller [56],\ntherefore, we neglect it. The first term on the RHS of ex-\ntended LLG Eq. (1) is the usual one [28, 29], describing\nprecession of LMMs in the effective magnetic field, Beff\nn,\nwhich is the sum of both internal and external ( Bextez)\nfields. It is obtained as Beff\nn=−∂H/∂Mnwhere His\nthe classical Hamiltonian of LMMs\nH=−JX\n⟨nn′⟩Mn·Mn′+K\n2X\nn(Mz\nn)2−BextX\nnMz\nn.(3)\nHere we use g= 1 for gyromagnetic ratio, which sim-\nplifies Eq. (1); Jis the Heisenberg exchange coupling\nbetween the nearest-neighbors (NN) sites; and Kis the\nmagnetic anisotropy.\nWhen nonlocal damping tensor, λRis proportional\nto 3×3 identity matrix, I3, a closed formula for the\nSW dispersion can be obtained via hydrodynamic the-\nory [58]. In this theory, the localized spins in Eq. (1),\nMn= (Re ϕn,Imϕn,1−m)T, are expressed using com-\nplex field ϕnand uniform spin density m≪1. Then,\nusing the SW ansatz ϕn(t) =P\nqUqei(q·rn−ωqt), we ob-\ntain the dispersion relation for the SWs\nωq= (Jq2+K−B)\u0002\n1 +i(αG+˜λq)\u0003\n, (4)\nwhere qis the wavevector and ωis their frequency. Thedamping of the SW is then given by the imaginary part\nof the dispersion in Eq. (4), Γ q≡Imωq. It is comprised\nby contributions from the local scalar Gilbert damping\nαGand the Fourier transform of the nonlocal damping\ntensor, ˜λq=R\ndrnλrneiq·rn.\nResults for FI/NM bilayer. —We warm up by extract-\ning Γ qfor the simplest of the three cases in Fig. 1, a\none-dimensional (1D) FI chain under a 1D NM over-\nlayer with spin-degenerate quadratic electronic energy-\nmomentum dispersion, ϵkσ=t0k2\nx, where t0=ℏ2/2m.\nThe GFs and spectral functions in Eq. (2), can be\ncalculated in the momentum representation, yielding\nλ1D\nR=2J2\nsd\nπv2\nFcos2(kFR)I3, where vFis the Fermi velocity,\nR≡ |R|, and kFis the Fermi wavevector. Moreover, its\nFourier transform, ˜λq=2J2\nsd\nv2\nF[δ(q) +δ(q−2kF)/2], dic-\ntates additional damping to SWs of wavevector q=\n0,±2kF. Although the Dirac delta function in this ex-\npression is unbounded, this unphysical feature is an ar-\ntifact of the small amplitude, m≪1, approximation\nwithin the hydrodynamic approach [58]. The features of\nsuch wavevector-dependent damping in 1D can be cor-\nroborated via TDNEGF+LLG numerically exact simu-\nlations [47–50] of a finite-size nanowire, similar to the\nsetup depicted in Fig. 1(a) but sandwiched between two\nNM semi-infinite leads. For example, by exciting SWs\nvia injection of spin-polarized current into the metallic\noverlayer of such a system, as pursued experimentally in\nspintronics and magnonics [21, 57], we find in Fig. 2(a)\nthat wavevector qof thereby excited coherent SW in-4\ncreases with increasing anisotropy K. However, the max-\nimum wavevector qmaxis limited by kF[Fig. 2(b)]. This\nmeans that SWs with q≳kFare subjected to additional\ndamping, inhibiting their generation. Although our an-\nalytical results predict extra damping at q= 2kF, finite\nsize effects and the inclusion of semi-infinite leads in TD-\nNEGF+LLG simulations lower this cutoff to kF.\nSince SW experiments are conducted on higher-\ndimensional systems, we also investigate damping\non SWs in a two-dimensional (2D) FI/NM bilayer.\nThe electronic energy-momentum dispersion is then\nϵkσ=t0(k2\nx+k2\ny), and the nonlocal damping and its\nFourier transform are given by\nλNM\nR=k2\nFJ2\nsd\n2πv2\nFJ2\n0(kFR)I3, (5)\n˜λNM\nq=kFJ2\nsdΘ(2kF−q)\n2πv2\nFqp\n1−(q/2kF)2, (6)\nwhere Jn(x) is the n-th Bessel function of the first kind,\nand Θ( x) is the Heaviside step function. The nonlo-\ncal damping in Eqs. (5) and (6) is plotted in Fig. 3(a),\nshowing realistic decay with increasing R, in contrast to\nunphysical infinite range found in 1D case. Addition-ally, SW damping in Eq. (6) is operative for wavectors\n0≤q≤2kF, again diverging for q= 0,2kFdue to arti-\nfacts of hydrodynamic theory [58]. Therefore, unphysical\ndivergence can be removed by going back to discrete lat-\ntice, such as solid curves in Fig. 3(e) obtained for n=1–\n100 LMMs by solving numerically a system of coupled\nLLG Eq. (1) where λRin 2D is used [45]. In this numer-\nical treatment, we use kF= 0.5a−1where ais the lattice\nspacing; k2\nFJ2\nsd/2πv2\nF=η= 0.1;K= 0; Bext= 0.1J;\nandαG= 0.1.\nResults for FI/HM bilayer. —Heavy metals (such as of-\nten employed Pt, W, Ta) exhibit strong SOC effects due\nto their large atomic number. We mimic their presence at\nthe FI/HM interface [31] by using 2D energy-momentum\ndispersion ϵk=t0(k2\nx+k2\ny) +tSOC(σxky−σykx), which\nincludes spin-splitting due to the Rashba SOC [31, 32].\nUsing this dispersion, Eq. (2) yields\nλHM\nR=\nλxx\nRλxy\nR0\nλxy\nRλyy\nR0\n0 0 λzz\nR\n, (7)\nfor the nonlocal damping tensor. Its components are, in\ngeneral, different from each other\nλxx\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n+ cos(2 θ)\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8a)\nλyy\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n−cos(2 θ)\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8b)\nλzz\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n−\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8c)\nλxy\nR=−J2\nsdsin(2θ)\n4π\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\n, (8d)\nwhere kF↑andkF↓are the spin-split Fermi wavevec-\ntors [Fig. 1(b)], and θis the relative orientation angle\n[Fig. 1(b)] between the SW wavevector qand the kxdi-\nrection. Thus, the nonlocal damping tensor in Eq. (7)\ngenerated by HM overlayer is anisotropic in spin due to\nits different diagonal elements, as well as nonzero off-\ndiagonal elements. It is also anisotropic in space due to\nits dependence on the angle θ. Its elements [Eqs. (8)]\nare plotted in Figs. 3(b), 3(c), 3(f), and 3(g) using\ntSOC= 0.3t0. They may become negative, signifying the\npossibility of antidamping torque [21] exerted by conduc-\ntion electrons. However, the dominant effect of nearby\nLMMs and the presence of local scalar αGensures that\nLMM dynamics is damped overall. Although there is no\nclosed expression for the SW dispersion in the presence of\nanisotropic λHM\nR, we can still extract SW damping Γ qin-duced by an HM overlayer from the exponential decay of\nthe SW amplitude in numerical integration of extended\nLLG Eq. (1) using SW initial conditions with varying q.\nFor an HM overlayer with realistic [31, 32] tSOC= 0.1t0\nthe results in Fig. (4)(a) are very similar to those ob-\ntained for NM overlayer with the same Fermi energy.\nAlso, the spatial anisotropy of λHM\nRdid not translate into\nθ-dependence of the SW damping.\nResults for FI/AM bilayer. —Altermagnets [33, 34] are\na novel class of antiferromagnets with spin-split elec-\ntronic energy-momentum dispersion despite zero net\nmagnetization or lack of SOC. They are currently in-\ntensely explored as a new resource for spintronics [51,\n61, 62] and magnonics [63, 64]. A simple model for\nan AM overlayer employs energy-momentum dispersion\nϵkσ=t0(k2\nx+k2\ny)−tAMσ(k2\nx−k2\ny) [33, 34], where tAMis5\n0.0 0.5 1.0 1.5\nq (1/a)1.01.52.02.5Γq≡Im(ωq) (J/¯ h)×10−1\n(a)NM\nHM\n0.0 0.5 1.0 1.5\nq (1/a)1234Γq≡Im(ωq) (J/¯ h)×10−1\n(b)NM\nAM :θ= 45◦\nAM :θ= 0◦\nFIG. 4. (a) Wavevector-dependent damping Γ qof SWs under\nNM or HM overlayer with the Rashba SOC of strength tSOC=\n0.1t0. (b) Γ qof SWs under AM overlayer with tAM= 0.8t0\nand for different relative orientations of FI and AM layers\nmeasured by angle θ[Fig. 1]. All calculations employ η= 0.1\nand Fermi energy εF= 0.25t0.\nthe parameter characterizing anisotropy in the AM. The\ncorresponding λAM\nR= diag( λ⊥\nR, λ⊥\nR, λ∥\nR) tensor has three\ncomponents, which we derive from Eq. (2) as\nλ⊥\nR=J2\nsd\n4πA+A−\u0014\nJ2\n0\u0012rϵF\nt0R+\u0013\n+J2\n0\u0012rϵF\nt0R−\u0013\u0015\n,\n(9a)\nλ∥\nR=J2\nsd\n2πA+A−J0\u0012rϵF\nt0R+\u0013\nJ0\u0012rϵF\nt0R−\u0013\n, (9b)\nwhere A±=t0±tAMandR2\n±=X2/A±+Y2/A∓is\nthe anisotropically rescaled norm of R. They are plot-\nted in Figs. 3(d) and 3(h), demonstrating that λAM\nRis\nhighly anisotropic in space and spin due to the impor-\ntance of angle θ[61, 65, 66]. Its components can also\ntake negative values, akin to the case of λHM\nR. It is inter-\nesting to note that along the direction of θ= 45◦[gray\ndashed line in Figs. 3(d) and 3(h)], λ⊥\nR=λ∥\nRso that\nnonlocal damping tensor is isotropic in spin. The SW\ndamping Γ qinduced by an AM overlayer is extracted\nfrom numerical integration of extended LLG Eq. (1) and\nplotted in Fig. (4)(b). Using a relatively large, but real-\nistic [33], AM parameter tAM= 0.8t0, the SW damping\nfor experimentally relevant small wavevectors is reduced\nwhen compared to the one due to NM overlayer by up to\n65% for θ= 0◦[Fig. 4(b)]. Additional nontrivial features\nare observed at higher |q|, such as being operative for a\ngreater range of wavevectors and with maxima around\n|q|= 2p\nϵF/t0and|q|= 3p\nϵF/t0. Remarkably, these\npeaks vanish for wavevectors along the isotropic direction\nθ= 45◦[Fig. 4(b)].\nConclusions. —In conclusion, using SKFT-derived non-\nlocal damping tensor [45], we demonstrated a rigorous\npath to obtain wavevector damping of SWs in magnetic\ninsulator due to interaction with conduction electrons\nof metallic overlayer, as a setup often encountered in\nmagnonics [13–17, 30] where such SW damping was di-\nrectly measured in very recent experiments [24–27]. Ouranalytical expressions [Eqs. (5), (7), and (9)] for nonlo-\ncal damping tensor—using simple models of NM, HM,\nand AM overlayers as an input—can be directly plugged\ninto atomistic spin dynamics simulations [28]. For more\ncomplicated band structures of metallic overlayers, one\ncan compute λRnumerically via Eq. (2), including com-\nbination with first-principles calculations [40]. 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Buda -Prejbeanu1, \nU. Ebels1 \n1Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP *, INAC, SPINTEC, F -38000 Grenoble, France  \n2GFMC, Departamento de Física de Materiales, Universidad Complutense, Madrid, Spain.  \n* Institute of Engineering Univ. Grenoble Alpes  \na) Electronic email: nmonteblanco@gmail.com  \n \n  \nWe report the dynamic characterization of the spin -torque -driven in- plane precession modes of a spin -torque nano-\noscillator based on two different synthetic ferrimagnets: a pinned one  characterized by a strong RKKY interaction \nwhich is exchange coupled to an  antiferromagnetic layer; and a second one, non -pinned characterized by weak \nRKKY coupling. The microwave properties associated with  the steady -state precession of both SyFs are \ncharacterized by high spectral purity and power spectral density. However, f requency dispersion diagrams of the \ndamped and spin transfer torque modes reveal drastically different dynamical behavior  and microwave emission \nproperties in both SyFs. In particular, the weak coupling between the magnetic layers of the non- pinned SyF raises \ndiscontinuous dispersion diagrams suggesting a strong influence of mode crossing. An interpretation of the \ndifferent dynamical features observed in the damped and spin torque modes of both SyF systems was obtained by \nsolving simultaneously, in a macros pin approach, a linearized version of the Landau -Lifshitz -Gilbert equation \nincluding the spin transfer torque term.  \n \n \nI. INTRODUCTION  \n \nThe exceptional and multi -functional properties of spin- torque1-2 nano- oscillators (STOs) made them \npromising candidates for a wide range of emerging technologies  which span from microwave emitters3 \nand detectors4-5 to neuromorphic  computing  systems6-10. These devices use the  transfer of angular \nmomentum from a  spin-polarized  current  to the local magnetization of a ferromagnetic layer11,12 to \ninduce self -sustained oscillations of the magnetization, which translate into a microwave signal whose  \nfrequency can be finely tuned with the applied direct current13-17. A widely studied structure of the STO \nis of the type (AF/SyF/MgO/SL)  with in -plane magnetization , including a synthetic ferrimagnet \nstructure (called  SyF- Polarizer) pinned via  exchange bias to an antiferromagnetic layer (AF) and a sin gle \nferromagnetic layer (SL)18-20. A standard SyF layer is composed of two ferromagnetic layers, a top (TL) \nand bottom (BL) layer coupled antiferromagnetically through a thin metallic layer via the Ruderman-\nKittel- Kasuya- Yosida (RKKY) interaction.21,22 In STO s, spin-polarized electrons affect damped \noscillations by modifying  damping and thus linewidth and amplitudes  and overcoming a critical current \ndensity (j c) polarized electrons can induce the steady -state oscillations (STT excitations) . Steady -state \noscillations can be obtained  in both parts,  SyF or SL, by changing the polarity of the current. The STT \nexcitations of a SyF structure  present some advantages in comparison with the SL excitations, as the \nhigher spectral purity (smaller linewidth), zero field excitations23,24, frequency tuning as a function of \nthe current with the possibility to change from redshift (df/dj app<0) to blueshift (df/dj app>0) applying an \nin-plane field and achieving large thermal stability25-29. However, since the SyF is pinned by an \nantiferromagnet, achieving  steady -state excitations require s a relatively large critical current. The use of \nexchange -coupled layers with perpendicular anisotropy30,31 or in- plane magnetized magnetic layers32-34 \nhave been shown to increase the magnetic stiffness and can potentially improve the performance of the \noscillator . Replacing  the standard free layer SL  by an unpinned SyF layer is thus of potential interest \ntowards  improving the microwave properties of  spin torque oscillators .  \n \nIn this article, we report the main static and dynamic features of a spin torque oscillator  based on two \nSyFs  with the following structure : IrMn(6.1)/SyF -Polarizer/MgO(0.9)/SyF -FL, w here numbers 2 \n represent thickness in nanometers. The composition of the SyF- Polarizer and SyF -FL are  \nCoFe(1.8)/Ru(0.4)/CoFe B(2) and CoFe(0.5)/CoFeB(3.4)/Ru/CoFe(3.6)  respectively. The structure of \nthis manuscript is the following: section II introduces the numerical techniques used to predict some \nfeatures of the STO dispersion diagrams and it is devoted to the macrospin analysis of the STO structure, \nsection III presents the experi mental characterization of the STO dynamics. Section IV presents the \ndiscussion and conclusions. In the following, the nano- oscillator based on a double  SyF will be called  \nD-SyF for simplicity.  \n \n \nII. NUMERICAL ANALYSIS  \n \nTwo types of numerical studies were performed in the framework of the macrospin approximation: (1) \ncomputation of the hysteresis loops (MH) and magnetoresistance loops (MR) using the minimization of \nthe energy and (2) stability analysis based on the linearization of the Landau- Lifshitz -Gilbert (LLG) \nequation enhanced with the spin transfer torque term (Slonczewski term) around the equilibrium \npositions, in order to find instabilities due to the applied current and fiel d. The D -SyF under  study \npresent s the following structure: AF/SyF -Polarizer/insulator barrier/SyF -Free layer, wh ere the bottom \nlayer of the SyF -Polarizer  is pinned into the positive x direction by an exchange bias field (H eb) induced \nby an antiferromagnetic layer, see Figure 1.  \n \nFig. 1. (a) Schematics of the double SyF oscillator structure with the labels used in this article. Negative applied \ncurrent corresponds to electrons flowing from SyF -FL to SyF- Polarizer.  (b) Schematic  of the standard RF setup.   \n  \nII.1. Simulation of static hysteresis loops  \n In order to understand the complex frequency dispersion diagrams of the D -SyF oscillator, its hysteresis \nloop has been simulated as the first step considering the different coupling between the ferromagnetic \nlayers. The total free energy density of the coupled system is written as 𝜎𝜎\n𝑡𝑡𝑡𝑡𝑡𝑡=𝛴𝛴\n𝑖𝑖𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡,𝑖𝑖=𝛴𝛴\n𝑖𝑖(𝜎𝜎𝑖𝑖𝑖𝑖𝑡𝑡,𝑖𝑖+\n𝜎𝜎𝑒𝑒𝑒𝑒𝑡𝑡,𝑖𝑖) where the internal and external component for each layer are defined as follow,  \n \n 𝜎𝜎𝑖𝑖𝑖𝑖𝑡𝑡,𝑖𝑖=𝜎𝜎𝑧𝑧𝑒𝑒𝑒𝑒𝑧𝑧 ,𝑖𝑖+𝜎𝜎𝑎𝑎𝑖𝑖,𝑖𝑖+𝜎𝜎𝑑𝑑,𝑖𝑖+𝜎𝜎𝑒𝑒𝑒𝑒,𝑖𝑖        (1) \n 𝜎𝜎𝑒𝑒𝑒𝑒𝑡𝑡,𝑖𝑖=𝜎𝜎𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ,𝑖𝑖+𝜎𝜎𝑑𝑑𝑖𝑖𝑑𝑑,𝑖𝑖𝑖𝑖        (2) \n \nThe magnetic layers are labelled as i, j =1 to 4 and i≠ j, corresponding to the structure on Figure 1. The \nmodel includes the different internal energies contributions such as the demagnetizing σ d and \nmagnetocrystalline anisotropies  σan, exchange bias σeb (only for the BL of the SyF pinned) and the \nZeeman terms σzeem. Also, we include the Ruderman- Kittel- Kasuya -Yosida (RKKY) interlayer \nexchange interaction  σRKKY (internal to each SyF) and the dipolar stray field35-36 σdip, see Eq. (2). The \nRKKY coupling is taken into account only between the layers 1- 2 and 3- 4 (σRKKY,1(3) = σRKKY,2(4) ) and the \n3 \n dipolar field between the four layers ( σdip,ij= σdip,ji). More details about each term of the equations are \nfound in Ref. 34 and 38. Two different RKKY coupling constants were considered: J RKKY =-0.1mJ/m2 \nand -1.5mJ/m2 for the SyF -FL (weak coupling)  and SyF -Polarizer  (strong coupling)  layer respectively . \nThe influence of a high or weak RKKY coupling in the hysteresis loops (MH) and magnetoresistance \ncurves (MR) has been shown in previous studies34. The structure considered is not compensated, i.e., \nthe product (M S*t*S) for the TL and BL of each SyF are not close ,  (M S*t*S)TL,SyF- FL= 35.12 μA nm2, \n(M S*t*S)BL,SyF- FL=28.79 μA nm2, (M S*t*S)TL,SyF -Polarizer =14.06 μA nm2, and (M S*t*S)BL,SyF -Polarizer =17.56 \nμA nm2. The net magnetic moment of the SyF -FL is 6.33  μA nm2 and for the SyF -Polarizer is 3.5  μA \nnm2 so we should consider  the stray magnetic field (dipolar field) between both SyFs.  This is \nfundamental to understanding the magnetization dynamics of this structure.  \n \n \n \n SyF Pinned Layer  SyF Free Layer  \nParameters  1 2 3 4 \nMS(kA/m)  1470  1060  1112.5  1470  \nK (J/m3) 7957.75  \nt (nm)  1.8 2.0 3.9 3.6 \nMS*t *S \n(µA.nm2) 17.56  14.06  28.79  35.12  \nNx \nNy \nNz 0.024952  \n0.046343  \n0.928705  0.027061  \n0.050280  \n0.922658  0.044620  \n0.083119  \n0.872260  0.042085  \n0.078372  \n0.879543  \np - -m3 m2 - \nη 0.3 \nα 0.02 \nS (nm2) 6636.63  \nHeb (kA/m)  79.5 0 0 0 \nJRKKY (mJ/m2) -1.5 -0.1 \n \nTable I. Parameters used in the numerical simulations . Here M S is the saturation magnetization, K u is a uniaxial \nmagneto crystalline anisotropy constant (//ox axis in the plane), t is the film thickness,  S is the surface,  Nx, Ny, and \nNz are demagnetization factors, α is the damping constant. H eb is the exchange bias field that acts on the BL of the \nSyF- Polarizer and  η is the spin efficiency.   \n \nNumerical simulation of the hysteresis loop and the magnetoresistance of the D -SyF oscillator are shown \nin Figure 2(a). These curves provide information about the relative orientation of the magnetizations of \nthe different layers, as a function of the applied field. The MR was calculated with the scalar product of \nthe magnetizations adjacent to the insulator barrier  i.e. layers 2 and 3. The parameters  are listed in Table \nI. In our samples, the anisotropy axis corresponds to the longer axis of the ellipse,  and it is parallel to \nthe X -axis. Arrows on Figure 2 represent the SyF -FL and SyF- Polarizer magnetization s respectively \nfollowing the color convention of the layers in Figure 1. For relatively large positive values of the \napplied field [100mT, 400mT] the magnetization of both layers of the SyF -FL are parallel to the external \nfield while the magnetization of the TL of the SyF -pinned is pointing in opposite direction, \ncorresponding to the antiparallel state in the magnetoresistance curve (Figure 2b). Upon sweeping the \nfield from positive to negative values,  a first magnetization switching is observed at μ oHsw ≈ +8.4 mT \n(Hsat+SyF- FLin Figure 2(a)). It corresponds to the  switching of the  magnetization of the BL of the SyF -FL, \nwhich presents a lower net magnetic moment in comparison with the TL.  The magnetization switching \nof the BL of the SyF -FL is accompanied by a change from the antiparallel (high resistance) to the parallel \n(low resistance) state in the magnetoresistance curve,  see Figure 2(b).  Upon sweeping further the applied \nfield towards negative values, the magnetization of the TL layer of the SyF -FL is inverted at Hsat-SyF- FL \n(Figure 2(a)), which does not affect  the magnetoresistance  (Figure 2(b)).  From now on, we  refer to th e \nregion of around 100mT between the two character istic saturati on field values ( Hsat-SyF-FL and Hsat+SyF-FL) \nas “plateau”. It corresponds to the P state in the RH curve. Sweeping the field in the opposite direction  \n(from negative to positive  values ) the bottom layer of the SyF -FL is reversed first again leading to a 4 \n similar scenario except because  the plateau region is shifted towards positive field  value s and now \ncorresponds to the antiparallel state (Figure 2(b)) . \n \n \nII.2. Dynamics features, the LLG equation  \n \nThe magnetization dynamics of the D -SyF structure is described in a macrospin approach solving the \ngeneralized Landau- Lifshitz -Gilbert (LLG) equations enhanced by the spin torque term34,38. The \nequation for each ferromagnetic layer is written as follow s, \n \n𝑑𝑑𝒎𝒎𝑖𝑖\n𝑑𝑑𝑡𝑡=−𝛾𝛾0(𝒎𝒎𝑖𝑖×𝑯𝑯𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒)+𝛼𝛼𝑖𝑖�𝒎𝒎𝑖𝑖×𝑑𝑑𝒎𝒎𝑖𝑖\n𝑑𝑑𝑡𝑡�+�𝑑𝑑𝒎𝒎𝑖𝑖\n𝑑𝑑𝑡𝑡�\n𝑆𝑆𝑆𝑆𝑆𝑆  (3) \n�𝑑𝑑𝒎𝒎𝑖𝑖\n𝑑𝑑𝑡𝑡�\n𝑆𝑆𝑆𝑆𝑆𝑆=𝛾𝛾0𝑗𝑗𝑎𝑎𝑑𝑑𝑑𝑑𝐺𝐺(𝜂𝜂)𝒎𝒎𝑖𝑖×(𝒎𝒎𝑖𝑖×𝐩𝐩𝑖𝑖)  (4) \n \nwhere the layers are identified by the number i =1 to 4, see structure in Figure 1(a). The vector m i=Mi/MSi \nis the normalized magnetization vector, M Si is the corresponding saturation magnetization, γ 0 is the \ngyromagnetic ratio, α i is the Gilbert damping constant. All the parameters are listed in Table I. \nHeffi=Hinti+Hcouplingi  is the effective field, composed of the internal field Hinti and the coupling field \nHcouplingi of the ith layer. The effective field is derived from the energy term of equations (1) and (2) for \neach layer. The last term is the spin torque term that affects the damping (second term in (3)).  For this \nmodel we have not taken into account the effects of a mutual spin through the Ru spacer  \n \nDuring simulations, three types of dynamic couplings are considered : dynamic RKKY interaction \nbetween  the two magnetic layers of each SyF, dynamic dipolar interaction between the four magnetic \nlayers of the oscillator , and the mutual spin transfer torque (MSTT) only between the BL of the SyF -FL \nand the TL of the SyF -Polarizer , see Eq (4) . Dynamic RKKY interaction and dynamic dipolar interaction \nare conservative couplings included in the precession term of the LLG equation (first term in Eq. (3)) , \nwithout an energy loss . The  Gilbert term and the MSTT , are dissipative couplings  considered by the \ndamping and the spin-transfer torque term  (second and last term in Eq. (3)) . Here j app is the applied \ncurrent density, the pre -factor G (𝜂𝜂) is given by Eq. 2 in Ref. 38, with the spin polarization efficiency 𝜂𝜂 \nand the unit vector p i represents the direction of the spin polarization vector of electrons. We are only \nconsidering the STT between layers 2 and 3 with the polarizer of layer 2 is being layer 3 and the opposite \nas shown in Table I.  The equation (3) should be read as follow s: the spin-polarized electrons from the \nmj layer reach the layer mi where the intrinsic damping starts to be counteracted  (japp<jc) (only for the \ncorrect sign of current ) due to the spin transfer torque. These damped excitations produced by the applied \ncurrent are called the STT damped modes in this manuscript . There are four STT  damped modes such \nas the result of the hybridization of the isolated SyFs modes, due to the dipolar coupling present between the layers in the structure. Overcoming a critical current value (j\nc) the damping is completely \ncounteracted, and the magnetization state is destabilized, leading to switching to another stable state or into steady st ate oscillations (STT excitations). All excitations involve all four layers of the oscillator.  \n   \nIn the case of positive applied current (j\napp>jc>0) the BL of the SyF -FL magnetization is driven by the \nspin transfer torque of spin -polarized  current and this leads to excitations dominated by the SyF -FL, in \nwhich the SyF -Polarizer participate following the dynamics due to the dipolar coupling. For the case of \nthe SyF -Polarizer dominant precession (j app<jc<0), it is the TL magnetization of the SyF -Polarizer which \nis controlled by the spin transfer torque. As this structure is coupled by the dipolar field, the four layers \nmove together at the same frequency. In the absence of current (j app=0) the spin torque term disappears, \nand we can simulate the excitation modes in the linear regime.  5 \n  \nFig. 2. (a) -(b) Hysteresis and magnetoresistance curves of the D -SyF oscillator. (c) Dispersion diagram  of \nhybridized damped modes  f1,f2, f3, and f4 for j app=0. The magnetic field has been swept from positive to negative \nvalues. The regions (i) -(iv) are defined by the crossing of modes.  \n \nTo calculate the STT damped modes (j app≠0), the  set of equations (3) are rewritten in the form of a \nmatrix, giving a periodic solution for mi and linearizing around an equilibrium position, \nmi=(M eq,0,0)+(0,n y,i,nz,i)ept. The  corresponding equilibrium positions were extracted from the hysteresis \nloop. The eigenvalue of the characteristic matrix will be the complex number p=λ+iω, with a real part λ \ncorresponding to the decay rate of the excitation modes and the imaginary part ω corresponding to the \n2π frequency of the excitations. The eigenvectors (0,n y,i,nz,i) define the character of the oscillation mode \nand the amplitude of oscillation for each layer. Two types of instabilities can be described using the \nlinearization. The first one corresp onds to spin wave mode softening (f>0) and will coincide with the \nmagnetization switching32. In the second type the real part of the characteristic eigenvalue can cross the \nzero axis.  To monitor such instability, we have calculated the decay rates  of the system as a function of \nthe current density and the applied field λ(j app,Happ). This provides us with the main criteria to find the \ncritical current density of the STT regime. A negative value means that the system relaxes  into a stable \nstate (damped regime), while a positive value indicates that the system becomes unstable, where one of \nthe possibilities is the  STT regime  (auto oscillations). The passage from negative to positive defi nes j c, \ni.e. the current where the decay rate  is zero: λ(j app,Happ)=0→ japp=jc. This method does not consider \ntemperature. Parameters used for the simulations were extracted from the experimental devices and correspond to SyF structure s, which are not magnetically compensated.  \n The dipolar field was calculated numerically for both SyFs in the AP magnetic configuration (H\napp= 0), \nfinding for the SyF -FL values around ±0.2 mT, for the BL and TL respectively. In the case of the SyF -\nPolarizer, the dipolar field calculated in the same X direction was - 2.1 mT and - 1.2 mT, for TL and BL \nrespectively. Moreover, when the SyF -FL is already saturated (positive X direction) the dipolar field in \nthe SyF -Polarizer increases up  to -21.9 mT and -18.9mT. Therefore, due to the non- compensated SyF \nstructures, we always expect  the influence of the dynamical dipolar coupling in the magnetization \ndynamics.  \n Figure 2(c) shows the calculated STT damped  mode  frequencies at j\napp=0 obtained by sweeping the \napplied field from positive to negative values. These modes are labeled  f1, f2, f3, and f4 from low to \nhigh frequency. The dispersion diagram  show s a well -defined mode splitting between modes 1 and 2, \nas well as two  mode s anti-crossings between modes 1 and 2, and 2 and 3 respectively. These effects are \nreminiscent of the splitting between the acoustic and optical -like modes provoked by the conservative \nRKKY inter action on single SyFs structures37. Here, the dipolar field between the four layers is \n6 \n responsible for the splitting and anti -crossings of the frequency dispersion diagram ,35 see Figure 2(c). \nDue to the crossing of modes, we define four regions, indicated in Figure 2(c) as (i) -(iv). The region (i)  \nis located for negative fields in the P state (low resistance state), and the three regions (ii) -(iv) on the AP \nstate (high resistance state of the structure and interesting region to study the STT modes), see MR curve \nin Figure 2(b). In the following section, we study the evolution of the damped  modes upon increasing \nthe applied current and their stability . \n \nII.3. Magnetization stability analysis  \n \nWe start the study of the stability of the magnetization precession around the equilibrium positions. \nUsing the decay rate  λ criteria, λ(j app,Happ)<0→STT damped modes, λ(j app,Happ)>0→STT modes and \nλ(japp,Happ)=0→ japp=jc introduced before, we can distinguish between stable and unstable regimes, by \ncomputing the critical current for each value of the applied field . Figure 3(a) shows the corresponding \ndecay rate s λ(0,H app) of the damped  modes f1,f2,f3,  and f4 respectively. We will respect the color of \nmodes defined in Figure 2(c). As it was expected for j app=0, the decay rate of the four modes remain \nnegative s, corresponding to the stability for the damped  modes. Since the frequency dispersion vs. field \nwas divided into  several regions (i)-(iv), due to the crossing of the damped modes, we notice that the \ndecay rates suffer inflections  or in some cases abrupt jumps from one of these regions to another one. It \nis important to remark that the mode f4 is less affected by the crossing of modes, showing the larger \ndecay rate and without distortions in the whole range of applied field.   \n \n \n \n \nFig. 3. Decay rate  versus applied field dispersion diagrams, for the corresponding hybridized damped modes f1, \nf2, f3, and f4. Colors correspond to the modes defined in Figure 2(c). In (a) below the critical current j app=0, in (b) \nfor the SyF -FL dominant precession, j app=1x1012 A/m2 and in (c) for the SyF -Polarizer  dominant precession, j app=-\n1x1012 A/m2. (d) State diagram H app vs j app. Positive current corresponds to an electron flow from the TL of the \nSyF- Polarizer (red arrow) to the SyF -FL (green arrows). The light grey  and black  regions correspond to the \n7 \n unstable region of excitations, and the dark grey to the stable region. The black region corresponds  to the switching \nof the SyF -FL. \n \nWhen the positive current density is increased up to japp=1x1012 A/m2, the decay rate tendencies of the \nsystem change, see Figure 3( b). It is noticed that the decay rate for the modes f1 and f2 (dominated by \nthe SyF -FL) are already positive λ>0, which is an indication that these modes evolve into the STT regime  \ndue to the damping compensation by the increase of the applied current. As this method is a linearization \nof the LLG equation, small magnetization precession, it is not possible to study the tendency of the \ndecay rate when the system is already in the steady state regime, large magnetization precession. \nApplying negative current density, j app=-1x1012 A/m2, the system reaches the SyF -Polarizer dominant \nprecession; see Figure 3( c). The decay rate is already positive for the modes f1, f2 and f3 (dominated by \nthe SyF -Polarizer), generating STT modes in the frequency field diagram. For both senses of current \ndensity, it is observed a crossing of the decay rate in the (i) region. In conclusion, it is possible to obtain \nthe critical current of the STT modes using the criteria λ(j app,Happ)=0 ( for positive and negative current \ndensity) and we can predict which of the modes will be stable or unstable. Sweeping the current density \nfor each value of magnetic field we obtain the critical lines to build the state diagram H app vs j app, shown \nin Fig ure 3(d).  \n \nThe magnetic field was swept from positive to negative values. i.e . from the positive saturation \nmagnetization of the SyF -FL until its plateau region in the P state (low resistance). The arrows \ncorrespond to the orientation of the magnetization of the four layers of the structure. The stable and \nunstable dynamical states are indicated by the dark and light grey regions, respectively. Excitations \ndominated by the SyF -FL (SyF- Polarizer) are observed in the region of positive (negative) current. \nYellow lines indicate the critical current densities, j c,SyF -FL and j c,SyF- Polari zer. The critical current values \nobtained and shown in Figure 3(d)  are referential due to the simplification of the model and to the \nparameters used in performing simulations (exchange bias, RKKY coupling, thicknesses, M S etc). The \ncritical currents should be taken as an approximation when compared to experimental results . The \ndashed yellow lines represent the border with the switching of the SyF -FL magnetizations, represented \nby the dark black  region. In the SyF -FL dominant precession region, two small def lections can be \nobserved, which corresponds to the splitting shown in Figure 1(c). The critical current for the SyF -FL \nin the saturated state  is around 0.9x1012 A/m2. \n \nThis numerical simulation framework  allows to  predict the magnetization dynamic behavior of the \ncoupled SyFs of the STO structure  and the critical current as a function of the applied field. As w e will \nverify  in the next section, this framework  provides useful information to understand the complex \nfrequency dispersion diagrams studied experimentally, a fundamental issue in designing STO devices.  \n \nIII. EXPERIMENTAL SECTION  \n In this section, we present  the static and dynamic  features of the D-SyF nano- oscillator . Measurements  \nwere carried out  using a standard microwave measurement setup.  \n  \nIII.1. Static measurements \n \nThe experimental results are obtained for magnetic tunnel junctions with the following structure: \nIrMn(6.1)/SyF -Polarizer/MgO(0.9)/SyF -FL, where numbers represent thickness  in nanometers. The \ncomposition of the SyF- Polarizer and SyF -FL are  CoFe(1.8)/Ru(0.4)/CoFe B(2) and \nCoFe(0.5)/CoFeB(3.4)/Ru/CoFe(3.6)  respectively. The thickness of Ru in SyF -FL was selected to \nachieve a weak  and negative RKKY coupling.  Samples were grown by sputter -deposition and patterned \ninto elliptical nanopillars  (130nm x 65nm ) with and area of 6636.63nm\n2. The  uniaxial shape anisotropy \nstabilizes the magnetization in the direction of the longest axis.  We have characterized tens of devices \nwhich we classified in two categories depending on whether the TMR is above or below 50% (High-TMR and Low -TMR devices respectively)\n16. The MgO barrier of the latter is considered to have \ninhomogeneities and/or pin holes either caused during deposition or electrical characterization. In this 8 \n work, we mainly focus on HTMR devices. However, due to the slightly  high critical current of STT \nexcitations dominated by SyF -Polarizer, those cannot be achieved  easily  in HTMR devices applying \nvoltages below 400 mV  (MgO barrier degradation) . Thus, LTMR devices characterized by a smaller \nresistance are used to characterize STT excitations dominated by the SyF -Polarizer . Figure 4(a) shows  \nthe MR curve s of a High -TMR (HTMR, red curve) device and a Low -TMR (LTMR, black curve) device \nrespectively. The MR curves are in good agreement with the numerical simulations (Figure 2(b) ), which \nprovide information about the magnetic orientations of the layers as a function of the applied field.  \n \nIt has been found during the optimization of D -SyF structures, (not shown in this article) that the \nroughness in a multilayer structure gives less control of the thicknesses of the layers on top, thus the \nRKKY coupling of the SyF -FL becomes weak, reducing the size of the plateau. As it was shown in \nprevious studies34, the size of the plateau region of a SyF is directly related to  the strength of its RKKY \ncoupling and increased by  the exchange bias field. The SyF -FL plateau is located between two \ncharacteris tic fields, the Hsat-SyF- FL and Hsat+SyF-FL=H sw, however , it is not evident in the MR curve due to \nthe weak RKKY coupling. The plateau size will be  estimated in the next section measuring the STT \ndamped modes. Both  magnetizations of the SyF -Polarizer remain in its AP configuration ( plateau  \nregion)  for a quite large range of applied field (>500mT) thus  we can consider this static configuration \nfor all our measurements, until the spin flop field (Hsf-+SyF- Polarizer ).  \n \n \nIII.2. Dynamic Measurements  \n \nIn this section, the dynamical features  of the D -SyF oscillator device are presented . Section ( A) shows \nthe study of the STT  damped modes  on the plateau region of the SyF -FL. Section (B) corresponds  to the \nstudy of the STT  damped modes  on the AP region (high resistance state) . Section  (C) is focused on the \nanalysis of the STT modes for SyF -FL and SyF- Polarizer dominant precession. The same sign \nconvention of the numerical simulations  is considered:  electrons flow from the SyF -Polarizer towards \nthe SyF -FL for positive applied current, promoting STT excitations in the SyF -FL. It is worth noting \nthat during the STT measurements, the STT damped modes  are also excited.  \n \nA. STT damped modes, positive current  \n \nFirst, the STT damped  modes were measured  for a HTMR device (TMR= 60%) around zero applied \nfield. The corresponding excitation frequency dispersion as a function of the applied field for positive \napplied current I app=1 mA is shown in Figure 4(b). The power spectral density (PSD) of the STT  damped \nmodes is  shown on a logarithmic scale. The magnetic field was swept from positive to negative values.  \nWe identify three different regimes in Figure 4(b) corresponding to the three different magnetic \nconfigurations of the SyF -FL (plateau  (i) and both saturation regions ) described by the numerical \nsimulations (Figure 2). Regions  (i) and (ii) in Figure 4(b) correspond to regions (i) and (ii) in Figure \n2(c). Experimentally, we identified region (i) between -27 mT<μ 0Happ<10 mT. Out of this range, the \nSyF- FL is saturated.  \n \n 9 \n  \nFig. 4. (a) MR curves of an elliptical device (130x65nm2) for HTMR device in  red (TMR  60%) and LTMR in blue \n(TMR  28%) respectively. (b) Frequency vs. applied field of the dB ( 10 log of power spectral density (nV2/Hz)) \nfor positive applied current (I app=1 mA). The regions  (i) and (ii) are identified in agreement with numerical \nsimulations and the arrows correspond to the magnetic direction of magnetizations. The STT  damped  fundamental \nmodes f1, f2 and f3 are identified,  and the higher order damped modes f 11, f21, and the harmonics 2f1, 2f2. The \ndashed lines are included to identify the modes only as a visual reference.  Linewidth as a function of the applied \nfield of the damped f1 and f 2 STT  damped  modes for ( c) positive and ( d) negative applied currents (I app =±1 mA).  \n \nBy comparing with the numerical simulations (Figure 2(c)), we can identify the STT  damped  modes f1, \nf2, f3 in regions  (i) and (ii) in Figure 4(b), as well as other harmonics  (2f1 is the 2nd harmonic of the f1 \nmode) . In region (i)  we observe the splitting and curvature of modes, which indicates  weak conservative \nRKKY coupling  of the SyF -FL, according to simulations. In region (ii)  we observe other  higher damped \nmodes  such as f11, f21, and the harmonics (2f1, 2f2) , in agreement  with results  given in Ref. 39, where \nthe finite size of the device generates quantized spin waves.  As we will see in the next section, the \nappearance of higher -order  modes has negative consequences on the microwave properties of  the D -\nSyF oscillator.  \n \nAs can be seen in Figure 4(b), the intensi ty of the different modes change s with the applied field (and \napplied current). As previously discussed , the amplitude of the magnetization precession and the \nlinewidth should be proportional to the absolute value of the decay rate  λ(japp,Happ) of the STT  damped \nmodes in the linear regime. The  tendency of the linewidth observed experimentally ( Figure s 4(c) and \n4(d)) agrees well  with the evolution of  modes 1 and 2 obtained by numerical simulations in the region \n(i) (Figure s 3(b) and 3(c) ), indicating a good correspondence between our model and experiments . Large \nvalues of the linewidth (≈400- 800 MHz ), for a quality factor Q=∆f/f ≈50-100, corresponding to STT \ndamped  modes  (linear regime)  were measured.  \n \n10 \n  \nFig. 5. Frequency vs. applied field diagram for regions (i) -(iv) for the device with TMR=60%, for I app=-1 mA. \nDashed lines correspond to the damped f1, f2, and f3 modes only as a visual reference. Arrows indicate the \nmagnetization directions on the magnetic layers for the different regions.  \n \n \nB. STT damped modes negative current  \n STT damped modes in both SyFs structures were measured by applying -1mA  and they are introduced \nin the frequency dispersion diagram in Figure 5. For applied magnetic fields >10mT the SyF -FL is \ncompletely saturated (green arrows) while the SyF pinned remains in the AP magnetic configuration. \nBy comparing with numerical simulations in Figure 2(c), we can also identify the regions (ii)-(iv) and \nmodes f1, f2, and f3. The experimental  STT damped  modes follow the frequency- field dependence  of \nthe simulated modes, and they are accompanied by additional  higher -order  modes\n27 (f11, f31, f22). At low \npositive fields, modes  f1 and f2 are separated 2.4GHz, as a result of RKKY and dipolar interactions  \nwithin the SyF -FL (conservative couplings). Two additional mode splits are observed  around 100mT, \nbetween modes  f1 and f2 and between modes  f2 and f3 respectively , in agreement  with the  numerical \nanalysis, see region (iii) in  Figure 2( c). In the next section, we discuss the evolution of the different STT  \ndamped  modes into STT excitations in the SyF -FL or the SyF- Polarizer upon increasing the applied \ncurrent for positive and negative values  respectively .  \n \nC. STT excitation modes  \n \nC.1 HTMR device  : SyF -FL dominantes  STT mode  \n \nSteady -state oscillations (STT modes) were first analyzed by applying positive current, which favors \nSTT modes dominated by the SyF -FL. Figure 6(a) shows the frequency -field dispersion curve with an \napplied current of I app=0.92 mA. The observed SyF -FL dominated STT modes correspond to the \nevolution of the STT  damped modes f1 and f2 on region (ii)  and (iii), as predicted by the stability \nanalysis (Figure 3) . The fundamental STT mode f1 shows several discontinuities around 40, 58 and 82 \nmT, in agreement with the macro -spin simulations  (Figure 2(c)).  These discontinuities are interpreted \nto be due to interactions between the steady state mode and other damped modes of the system through \nhigher harmonics29-35. Indeed, modes f 3, f11 and f 31 cross the second harmonic of the STT mode f1 (lines \n11 \n in Figure 5 and 6(a)). Non -linear mode interactions through higher harmonics can produce \ndiscontinuities and kinks in the fundamental STT mode f140-41.  \n  \nFigure 6(b) displays t he linewidth of the STT mode  f1. The linewidth of STT mode 1 increases each \ntime there is a discontinuity in the f -H dispersion, as expected from the interaction with other damped \nmodes via higher harmonics26. The STT mode f2 is characterized by a much lower linewidth, reaching \na minimum of 42 MHz , Q≈4.94, around 100 mT . At low fields (region (i) ), the STT damped  mode in  \nthe SyF -FL plateau  shows a continuous frequency- field dispersion, without jumps or kinks, since t here \nare no mode crossings  in this range of field. Exciting STT modes in this region would potentially offer \nexcitations at zero fields without linewidth enhancements due to undesired mode interactions. Due to \nthe switching of the SyF -FL for an  applied current below the critical current, it was not possible to obtain \nSTT in the region (i) . \n \n \nFig. 6. (a) Frequency dispersion as a function of the field for the SyF -FL dominant precession, I app=0.92  mA. The \ncircles indicate the kink and jumps in the STT f1 mode. (b) Frequency (black) and linewidth (blue) versus applied \nfield. Deviations in frequency correspond to an abrupt increase of ∆f. (c) Frequency dispersion as a function of the \napplied field for the SyF -FL dominant precession at 0. 6, 0.92 and 0.94 mA . Inset: peaks of modes interactions \naround the bi -stable region ( around 82 mT) . (d) Frequency dispersion as a function of the  applied current for three \nvalues of applie d field. Device HTMR ( TMR= 60%).   \n \nTo study the discontinuities of the STT modes , the frequency field dispersion is plotted for three \ndifferent values of the applied current (Figure 6(c)). The evolution from a continuous STT mode at \n0.6mA ( black curve ), into a discontinuous STT mode with jumps and kinks at 0.94mA ( blue curve). The \nPSD spectra at different fields around 82 mT are shown in the inset. The PSD spectra shows a region of \nbi-stability  where two modes co -exist . This behavior has been reported to  come from  the interaction \nbetween a STT mode with other damped modes of the system through higher harmonics37,39,41,42, where \nthe apparent mode co -existence is indeed thermally activated mode hopping. This kind of discontinuities  \n12 \n and jumps are observed only for large values of current, which implies  a large amplitude of the \nmagnetization precession  and thus  large dipolar interaction .  \n \nAn interesting feature of the magnetization dynamic on SyF pinned structures is the possibility to tune \nthe frequency -current  dependence  from redshift (df/dj app<0) to blueshift (df/dj app>0) by applying an in-\nplane field . Figure 6 (d) shows th e frequency as a function of the applied current for μ0Happ=20, 70 and \n85 mT  respectively. A transition from redshift to blueshift is observed upon increasing field from 20 to \n85 mT. Interestingly both curves are continuous with n o abrupt discontinuities . However, at μ 0Happ= 70 \nmT, a bi -stable region characterized by mode  is observed around I app=0.82 mA (see dashed line).   \n \n \nC.2. LTMR device   \n \nAchieving SyF- Polarizer dominant precession requires increasing the applied current  above the \nbreakdown voltage in HTMR devices. Thus, an LTMR device (TMR= 28%) is used to pursue the study \nof the STT excitations dominated by the SyF -Polarizer. Figure 7 displays the STT modes of a LTMR \ndevice for positive (Figure 7(a)) and negative (Figure 7(b)) currents ( ±1.4 mA). The SyF- FL dominant \nexcitations  (Figure 7(a)) exhibit  several discontinuities and kinks  in the STT modes f1 and f2 mode s and \nhigher  harmonic s (2f1). Indeed, more discontinuities than the equivalent  frequency- field dispersion of \na HTMR device ( Figure 6(a)) are observed . This is expected since the existence of pinholes in the tunnel \nbarrier increases interlayer interaction. The frequency -field dependence becomes flat in the field region \naround 100mT  because of  the interaction between higher harmonics f31 and 2f1. For  negative current , \nthe SyF -PL dominant precession shows a gap between the STT f3 and STT f1 mode s (Figure 7 (b)).  \n \n \nFig. 7. (a) SyF -FL dominant precession, I app=+1.4  mA and (b) SyF -PL dominant precession, for I app=-1.4 mA. \nInteractions between the harmonics of the STT f1 and f2 mode s with the damped or higher order modes will be \n13 \n transmitted as a form of kinks, deviations or jumps of the normal tendency of the frequency and linewidth versus \napplied field dispersion, generating discontinuities in the linewidth. Device with LTMR ( TMR= 28%).  \n \nThe general tendency of the linewidth  not shown  is to decrease upon increasing the  applied field , \nalthough several discontinuities (regions of linewidth enhancement) are observed corresponding to the interaction  of the S TT f1, f2 , and f3 modes with damped and higher order modes, as in the HTMR device \n(Figure 6(b)).  The local minimum linewidths measured for the SyF -FL dominant precession  at 1.4  mA, \nwere 20.5 MHz (Q≈1.73)  at 112 mT, 13.6 MHz  (Q≈1.1) at 194 mT and 12.9 MHz (Q≈1.17)  at 214  mT. \nIn the case of the  SyF- Polarizer dominant precession  at -1.4 mA , the minimum linewidths were 88MHz  \n(Q≈8) at 46 mT, 53 MHz (Q≈5.57) at 150 mT and 46 MHz  (Q≈5.41)  at 214 mT. Interestingly, t he \nminimum values of the linewidth  associated to the SyF-FL dominant  precession are smaller  than the \nminimum linewidth associated to the precession of a single free layer o n standard  in-plane magnetized \nspin torque oscillators\n22. For the case of the SyF -Polarizer, the minimal linewidth was around ≈46  MHz. \nWe note that continuous  excitation  branches can be found for specific conditions in the case of the SyF -\nPolarizer.  \n \nFinally, we analyze the redshift and blueshift regimes associated to  SyF-FL dominant excitations. We \nnote that the redshift and blueshift regimes have  been studied only on SyF pinned structures23,25 so far. \nFigure 8 displays the frequency- current  dependence at different magnetic fields for the SyF -FL \ndominant regime. In (a) -(b) (20 mT -plateau region and 54 mT -P state respectively) , the applied current \nremains below the critical  current, therefore STT damped  mode shown in (a)  the standard redshift \nbehavior  with large linewidth Δf≈395- 550 MHz  (Q≈79-110). In (b) t he STT damped mode shows the \nblueshift regime with a reduction of the linewidth by the current from Δf≈480 MHz  (Q≈80) for low \ncurrent for Iapp <1 mA to 125 MHz  (Q≈17.8)  for Iapp>1.2 mA. Upon increasing the field, the critical \ncurrent gets smaller. Thus, at μ 0Happ=70.7  mT in (c) the system evolves into the STT excitations. The \nfrequency dispersion shows discontinuities and a transition from a blueshift regime to a redshift regime upon increasing current , around 1.1 mA . Looking at Figures  7 and 5, we observe that this discontinuity \nis a product of the crossing of the STT mode f1 with the damped mode f3, however , this transition is \nopposite to the previous observed (redshift into blueshift) , and due to the complexity of these coupled \nsystems. The mode t ransition into the redshift regime is accompanied by a reduction of the linewidth \ninto Δf≈75 MHz (Q≈10) around 1.25 mA. At large fields (Figure 8(d) ) a redshift to blueshift transition \nis observed upon increasing current, and a very low linewidth of Δf≈28  MHz  (Q≈2.54) is obtained in \nthe blueshift regime . This region of  STT excitations at  large  fields and applied current in  LTMR devices \nis of potential interest for applications as it offers the possibility of selecting excitations in the redshift \nor in the  blueshift regime.  \n 14 \n  \nFig. 8. Frequency as a function of the current density for I app>0 for different field values (SyF -FL dominant \nprecession). (a) STT damped mode f1 on the SyF -FL plateau  region . (b) In the P state close to the switching field. \nThe system shows a blueshift regime and large linewidth . In (c) the system shows a blueshift regime until \nIapp=1.1mA. Overcoming this value of current , the regime changes towards a redshift regime. In (d) the STT f1 \nmode shows first a redshift regime and after the splitting a blueshift regime with a minimum local linewidth of 28  \nMHz at 1.25  mA. \n \n \nIV. C ONCLUSIONS  \n In this manuscript, we conducted a comprehensive investigation into the spin- transfer torque damped \nmodes  and steady- state oscillations of a spintronic nano -oscillator employing two SyF structures. The \nstudy involved both numerical simulations and experimental analyses.  Numerical simulation s were \ncarried out using two different values of R KKY coupling, J\nRKKY ≈-0.1mJ/m2 and J RKKY ≈-1.5 mJ/m2 for \nthe SyF -FL and SyF- Polarizer respectively. The small RKKY coupling of a SyF -FL eliminates the spin \nflop region, producing an abrupt switching of the magnetization layers  and introducing a small plateau \nregion (≈40 mT), in comparison with the corresponding of the pinned SyF- Polarizer (>600 mT). Static  \nand dynamical experimental measurement s confirmed the weak RKKY coupling of the SyF -FL on top \nof the structure. The experimental STT  damped  hybridized modes  have been identified using numerical \nmodeling. W e found that it is possible to find the tendency of the linewidth and the PSD of the STT  \ndamped  hybridized modes following the tendency of the decay rate (λ), also we can obtain the critical \ncurrent of the STT modes. This study shows that the analysis of the stability predicts the state diagram \nof different structures, with an arbitrary number of layers. The frequency versus field dispersion \ndiagrams for b oth devices (TMR 60% and 28%) show several discontinuities, attributed to the crossing \nof the STT mode or its harmonics with other damped hybridized or higher order modes, as was reported \n15 \n in Ref. 2 4, 36, 37 and 39. The introduction of a SyF -FL instead of a single layer in the STO based will \ngenerate more damped modes which produce more discontinuities  in the STT mode s. This last fact is  \nalso attributed to the dipolar coupling between the  magnetic layers. The frequency current tuning shows \ntwo regimes  for the SyF FL, the redshift (df/dI app<0) and a blueshift (df/dI app>0) which can be selected \nfor different values of applied field. In the case of the SyF -Polarizer  STT modes,  a flat regi on (df/dI app≈0) \nwith a small redshift and blueshift regimes was found. We found the minimum linewidth on a LTMR \ndevice, around ≈12.9 MHz  (Q≈1.17)  for SyF- FL and around ≈46 MHz (Q≈5.41)  for the SyF -Polarizer \ndominant precession, evidence of the stability of the system.  \n \nV. A CKNOWLEDGE  \n \nM.R. acknowledges financial support from Spanish MIC, AEI and FEDER through Grant No. PID2020-\n116181RB -C33 (MCI/AEI/FEDER, UE) and from Comunidad de Madrid (Atracción de Talento Grant \nNo. 2018- T1/IND -11935).  \n \n \nVI. REFERENCES  \n \n1 N. Locatelli, V. Cros, and J. Grollier, Nat Mater 13, 11 (2014).  \n2 Kent, A., Worledge, D. A new spin on magnetic memories. Nature Nanotech  10, 187–191 (2015)  \n3 A. Dussaux et al., Nat. 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Lett , 105, 182406 (2014)  \n "    },    {        "title": "2312.13093v1.An_effective_field_theory_of_damped_ferromagnetic_systems.pdf",        "content": "Prepared for submission to JHEP\nAn effective field theory of damped ferromagnetic\nsystems\nJingping Li\nDepartment of Physics, Carnegie Mellon University, Pittsburgh, PA 15213\nE-mail: jingpinl@andrew.cmu.edu\nAbstract: Using the in-in formalism, we generalize the recently constructed magnetoelastic\nEFT [1] to describe the damping dynamics of ferromagnetic systems at long wavelengths. We find\nthat the standard Gilbert damping term naturally arises as the simplest leading-order symmetry-\nconsistentnon-conservativecontributionwithinthein-inframework. TheEFTiseasilygeneralized\nto scenarios with anisotropy and inhomogeneity. In particular, we find the classic Landau-Lifshitz\ndamping term emerges when isotropy is broken by a constant external background field. This\nprovides a first principle explanation for distinguishing the two types of damping dynamics that\nwere originally constructed phenomenologically. Furthermore, the EFT framework could also in-\ncorporate intrinsic anisotropy of the material in a straightforward way using the spurion method.\nFor systems with inhomogeneity such as nontrivial spin textures, we find that the leading order\nderivative correction yields the generalized Gilbert damping equations that were found in con-\ndensed matter literature. This shows that the EFT approach enables us to derive the form of\nhigher-derivative-order corrections in a systematic way. Lastly, using the phonon-magnon cou-\npling deduced in the magnetoelastic EFT, we are able to make a prediction for the generic form\nof the phononic contribution to the damping equation.arXiv:2312.13093v1  [hep-th]  20 Dec 2023Contents\n1 Introduction 1\n2 Review of the magnon EFT and Schwinger-Keldysh formalism 2\n2.1 Symmetry breaking in magnetoelastic systems 2\n2.2 The magnon EFT and the conservative equation of motion 4\n2.3 The Schwinger-Keldysh formalism 4\n3 Gilbert damping term from EFT 6\n3.1 Coupling at the leading order 6\n3.2 Gilbert damping 6\n4 More general materials 9\n4.1 Anisotropic materials 9\n4.2 Inhomogeneous materials 10\n5 Magnon damping term from phonons 11\n6 Conclusion and discussions 12\n1 Introduction\nIt has long been established that the methodology of coset construction serves as a powerful tool\nof relativistic effective field theories (EFTs) of Goldstone bosons (e.g. pions from spontaneously\nbroken approximate chiral symmetry) [2–5]. In recent years, many works have demonstrated\nthat its versatility is extendable to condensed matter systems where we are interested in the\nmacroscopic behavior which are usually the massless low energy excitations [6, 7]. Using this\napproach, a recent paper [1] constructed an EFT of magnetoelastic systems where the phonons\nand magnons are considered Goldstones associated with translations spontaneously broken by\nthe ground state location of the material (the lattice) and an SO(3)symmetry of the magnetic\nmoments by their ground state orientations. The EFT approach provides a systematic way to\nunderstand phonon-magnon interactions from first principles and predict the forms of higher-order\ncorrections which has not been done previously.\nWhile the paper was focused on conservative dynamics, there has also been extensive study\non the theoretical description of non-conservative dynamics of damped magnetic systems since\nthe seminal work of Landau, Lifshitz, and later Gilbert [8, 9]. However, to our knowledge, the\nprior works were mostly model-dependent phenomenological descriptions. It is therefore desirable\nto have a first principle derivation from a similar many-body EFT perspective.\nOn the other hand, the Schwinger-Keldysh formalism [10, 11] (and the related in-in formalism\n[12, 13]) has been known to describe the quantum field theory of open systems and hence fully\n– 1 –capable of describing dissipative effects. In recent years, its power has been successfully extended\nto the EFT framework of dissipative dynamics in astrophysics and black holes [14, 15, 17] that\nsystematically derives dissipative equations of motion. Therefore, it is natural to consider its\nutility in describing deriving damping equations in condensed matter EFTs.\nIn this paper, combining the power of the two techniques, we apply the Schwinger-Keldysh\nformalism to incorporate dissipative effects into the EFT of magnons to reproduce the known\nresults of magnetic damping. In section 2, we review the coset construction of the magnon EFT\nas well as the techniques in Schwinger-Keldysh formalism to be applied in this paper. Section 3\nderives the original Gilbert damping equation for homogeneous and isotropic materials. In section\n4, we move on to more general materials and recover the Landau-Lifshitz damping equation for\nanisotropic systems and generalized Gilbert damping for spatially inhomogeneous materials. In\nsection 5, we derive the damping terms originating from the magnon-phonon interaction.\nConventions: we use natural units where ℏ= 1. Unless specified, the uppercase Latin indices\nA, B, C . . . denote the full internal spin symmetry space which runs over 1,2,3while the lower\ncase ones a, b, c . . . in the begining alphabet index the broken subspace 1,2. Those in the middle\nalphabet i, j, k . . . run over the three spatial dimensions (to generalize to higher dimensions, the\ninternal spin symmetry will have to be modified accordingly).\n2 Review of the magnon EFT and Schwinger-Keldysh formalism\nIn this section, we first provide a self-contained review on the symmetries and the corresponding\ncoset construction of magnon-phonon EFT proposed by [1]. Furthermore, we summarize the\nSchwinger-Keldysh formalism which is the central tool for deriving the dissipative equations of\nmotion.\n2.1 Symmetry breaking in magnetoelastic systems\nWe follow the derivations in [1]. The symmetries under consideration are the spatial Galilean\ngroup (generated by translations Pi, rotations, Li, boost Ki) and internal symmetries (internal\ntranslations Ti, internal rotations Qi, spin rotations SA). The algebra of the generators is given\nby\n[Li, Kj] =iϵijkKk,[Li, Pj] =iϵijkPk, (2.1)\n[Ki, H] =−iPk,[Ki, Pj] =−iMδ ij, (2.2)\n[Qi, Tj] =iϵijkTk,[Qi, Qj] =iϵijkQk, (2.3)\n[SA, SB] =iϵijkKk,[Li, Lj] =iϵijkLk. (2.4)\nIn particular, the TiandQigenerators generate translations and rotations on the “comoving”\ncoordinates ϕI(x)(or the Lagrangian coordinates in continuum mechanics which simply gives an\ninitial labeling to the continuum)\nϕI(x)→ϕI(x) +aI, ϕI(x)→RI\nJϕJ(x), (2.5)\nandSAgenerates the internal rotation on the orientation of the spin\nNA→OA\nBNB, (2.6)\n– 2 –where O=eiχaSaand the Néel vector ⃗Nis the order parameter for the spin orientation.\nIn the ground state, the order parameters gain vacuum expectation values (VEVs) which we\nchoose to be D\n⃗ϕ(x)E\n=⃗ x,D\n⃗NE\n= ˆx3. (2.7)\nThe VEVs are not invariant under the transformations and hence spontaneously break some of\nthe symmetries\nUnbroken =\n\nH\nPi+Ti≡¯Pi\nLi+Qi\nS3\nM, Broken =\n\nKi\nTi\nQi\nS1, S2≡Sa. (2.8)\nThe parametrization of the ground state manifold which is simply the broken symmetry transfor-\nmations plus the unbroken translations is given by\nΩ =e−itHeixi¯PieiηiKieiπiTieiθiQieiχaSa, (2.9)\nwhere ηi,θi,χa, and πi=ϕi−xiare the corresponding Goldstone fields.\nFor any Goldstone fields ψicorresponding to the broken generators Xi, their covariant deriva-\ntives are the basic building blocks of the low energy EFT. They are systematically computed by\nthe coset construction using the Mauer-Cartan forms of the broken group\nΩ−1∂µΩ⊃\u0000\n∇µψi\u0001\nXi, (2.10)\nby computing the coefficients of the broken generator Xi. In addition, in the case that one broken\ngenerator X′appears in the commutation algebra of the other X’ with the unbroken translations\n\u0002¯P, X\u0003\n⊃X′, (2.11)\nit means that the two Goldstones are not independent and one of them can be eliminated. This\nis known as the inverse Higgs phenomenon.\nThe result of this exercise is that ηi,θiare eliminated and the only independent degrees of\nfreedom are the magnons χaand phonons πi, and at leading order in derivatives, they appear in\nthe following combinations:\n∇(iπj)= (D√\nDTDD−1)ij−δij, (2.12)\n∇tχa=1\n2ϵaBCn\nO−1h\n∂t−∂tπk(D−1)j\nk∂ji\nOo\nBC, (2.13)\n∇iχa=1\n2ϵaBC\u0000\nO−1∂iO\u0001\nBC, (2.14)\nwhere Dij=δij+∂iπj. [1] found the most general action for ferromagnetic material in the form\nL=c1\n2det (D)ϵabh\u0000\nO−1∂tO\u0001\nab−∂tπk\u0000\nD−1\u0001j\nk\u0000\nO−1∂jO\u0001\nabi\n(2.15)\n−1\n2Fij\n2(∇(iπj))∇iχa∇jχa−1\n2F3(∇(iπj))∇tχa∇tχa, (2.16)\nwhere the first term is similar to a Wess-Zumino-Witten (WZW) term that differs by a total\nderivative under the symmetry transformation.\n– 3 –2.2 The magnon EFT and the conservative equation of motion\nTo derive the equations of motion for magnons, it is more convenient to express the magnon fields\nin the nonlinear form\nˆn=R(χ)ˆx3= (sin θcosϕ,sinθsinϕ,cosθ), (2.17)\nwhere the two magnon fields are related to the angular fields by\nχ1=θsinϕ, χ 2=θcosϕ. (2.18)\nPhysically, this unit vector represents the direction of the magnetic moment. Under this repre-\nsentation, the pure magnon Lagrangian (in the absence of phonon excitations) becomes\nL →c2\n2ϵab\u0000\nO−1∂tO\u0001\nab+c6\n2(∂tˆn)2−c7\n2(∂iˆn)2, (2.19)\nwhere F3(0) = c6andFij\n2(0) = c7δij.\nThe dispersion relation for the quadratic Lagrangian has two solutions [1]\nω2\n+=\u0012c2\nc6\u00132\n+O(k2), ω2\n−=\u0012c7\nc2\u00132\nk4+O(k6). (2.20)\nFor ferromagnetic materials, where c2(c6c7)3/4, the first mode is gapped around the EFT cutoff\nscale, while the second has ω∼k2scaling and exits the EFT. In the long wavelength limit, we\nmay assign the scaling ∂1/2\nt∼∂ito the derivatives for ferromagnets.\nTo derive the equation of motion, we notice that an action of this form has a symmetry under\nthe infinitesimal spin rotation δˆn=⃗ ω׈n, where ⃗ ωis the constant infinitesimal parameter. It can\nbe shown that the Wess-Zumino-Witten term contributes to a total derivative ∂µ⃗Fµunder this\ntransformation. Using the equation for Noether current in Lagrangian mechanics\n⃗Jµ= ˆn×∂L\n∂∂µˆn−⃗Fµ, (2.21)\nwe find the conserved current\n⃗J0=−c2ˆn−c6∂tˆn׈n,⃗Ji=c7∂in׈n. (2.22)\nThe continuity equation ∂µ⃗Jµ= 0is then explicitly\nc2∂tˆn=−\u0000\nc6∂2\ntˆn−c7∇2ˆn\u0001\n׈n, (2.23)\nwhich is the equation of motion for Landau-Lifshitz model of magnetism.\n2.3 The Schwinger-Keldysh formalism\nThe appropriate formalism for non-conservative system is the so-called in-in or Schwinger-Keldysh\nformalism [10–13]. The basic idea is that there is an external sector Xthat the energy is dissipated\ninto since the total energy needs to be conserved. The external sector could evolve into any final\nstate which we do not observe, so all the dynamics are inclusive of the final states in the Hilbert\nspace of XX\nXout⟨Xin|. . .|Xout⟩⟨Xout|. . .|Xin⟩ ≡ ⟨. . .⟩in, (2.24)\n– 4 –and depends only on the initial state (hence the name in-in). We can generate an effective action\nfor the in-in observables via the Schwinger-Keldysh closed time path integral\nexp\u0014\niΓ[q; ˜q]\u0015\n=Z\ninitialDXD˜Xexp\u0014\niS[q, X]−iS[˜q,˜X]\u0015\n, (2.25)\nwhere we are integrating over an additional copy of variables ˜Xwhich corresponds to evolving\nback to the boundary conditions fixed at the initial time.\nThe equation of motion for the degrees of freedom in the observed sector qcan be derived\nfrom the action functional Γ[q; ˜q]by\nδ\nδqΓ[q; ˜q]\f\f\f\f\f\nq=˜q= 0. (2.26)\nAny external sector operator O(X)coupled to some operator in the observable sector F(q)by\nthe interaction termR\ndxO(X(x))F(q(x′))(where xis the corresponding spacetime coordinates)\nwould enter the equations of motion in terms of\n⟨O(X(x))⟩in=Z\ninitialDXD˜Xexp\u0014\niS[q, X]−iS[˜q,˜X]\u0015\nO(X), (2.27)\nwhere we have abbreviated ⟨O(X)⟩in≡ ⟨Xin|O(X)|Xin⟩. Just as in the perturbative quantum\nfield theory correlation functions calculated by Feynman propagators, this can be similarly calcu-\nlated using the Schwinger-Keldysh propagators\n⟨Oa(x)Ob(x′)⟩= \n⟨TO(x)O(x′)⟩ ⟨O (x′)O(x)⟩\n⟨O(x)O(x′)⟩ ⟨˜TO(x)O(x′)⟩!\n, (2.28)\nwhere Tand ˜Trepresent time and anti-time orderings. The sub-indices label the first and the\nsecond copy, which determines the relative time-ordering of the operators.\nExplicitly, the linear response gives\n⟨O(x)⟩=iZ\ndx′{⟨TO(x)O(x′)⟩ − ⟨O (x′)O(x)⟩}F\u0000\nq(x′)\u0001\n+O(F2). (2.29)\nor equivalently\n⟨O(x)⟩=Z\ndx′GR(x, x′)F\u0000\nx′\u0001\n, (2.30)\nwith the retarded Green’s function given by\nGR(x, x′) =iθ(t−t′)⟨[O(x),O(x′)]⟩\n=i(⟨TO(x)O(x′)⟩ − ⟨O (x′)O(x)⟩). (2.31)\nTherefore, the exact form of the damping term in the equation of motion would depend on the\ndetailed structure of these retarded response functions.\n– 5 –3 Gilbert damping term from EFT\n3.1 Coupling at the leading order\nThe composite operators Or(X)that encapsulate the external sector transform under arbitrary\nrepresentations (labeled by r), provided they form invariants of the unbroken SO(2)with the\nmagnon χaand derivatives. In order to achieve this, the operators have to be dressed with\nthe broken SO(3)/SO(2)subgroup parametrized by the Goldstones TR(χ)in the corresponding\nrepresentation\n˜Or(X)≡ Rr(χ)Or(X), (3.1)\nsuch that they transform covariantly under the unbroken subgroup [16].\nIn the long wavelength regime, the theory is organized by a spatial derivative expansion.\nIn fact, the simplest invariant operator at zeroth-order in the derivative expansion is the singlet\naligned along the ground state orientation ˆx3\nSint=Z\nd4x˜O3(X)≡Z\nd4xˆx3·˜⃗O(X), (3.2)\n(note that we are adopting manifestly relativistic notations for spacetime and energy-momentum\nfor convenience, albeit the system may or may not be relativistic). Equivalently, we may write\nSint=Z\nd4xˆn·⃗O(X) (3.3)\nwhere ˆn=O(χ)ˆx3as defined previously.\nAt the same order in this expansion, there could be more operators that can be added, such\nas when the operator is a two-index tensor operator O(2)and we may have combinations of the\nform ˆn· O(2)·ˆn. However, as will be explained in the next subsection, due to the constraint\nˆn2= 1, these will not lead to any new contributions, and from the perspective of the EFT, they\nare redundant operators. Hence, Eq. (3.3) is the only non-trivial operator at this order.\nFurthermore, we observe that the form of the operator restricts the possible external sector\nthat can couple in this way. For example, one can form such operators from fermions, where\n⃗O(ψ) =ψ†⃗ σψ+ψ⃗ σψ†, (3.4)\nwhere ⃗ σare the Pauli matrices acting as the intertwiner between the spinor and SO(3)spin space.\nOn the other hand, phonons cannot form operators in this form, and hence will not contribute in\nthis way.\n3.2 Gilbert damping\nThe undamped equation of motion is derived from the continuity equation\n∂µ⃗Jµ= 0 (3.5)\ncorresponding to the spin rotation transformation δˆn=⃗ ω׈n. To derive the non-conservative\nequation of motion, we need to find out how the additional term Eq. (3.3) affects the continuity\nequation.\n– 6 –The spin rotation transformation on ˆnalone is itself a valid symmetry for the pure magnon\naction Eq. (2.19), but the interaction term Eq. (3.3) is not invariant if we keep the external sector\nfixed. A standard trick of Noether theorem is that for an arbitrary symmetry transformation\nϕ(x)7→ϕ(x) +f(x)ϵ, if we promote the global symmetry variation parameter to be an arbitrary\nlocal variation ϵ(x), the total variation takes the form\nδS=−Z\nd4xJµ∂µϵ, (3.6)\nsuch that when ϵis a constant, invariance under the symmetry transformation is guaranteed\nδS= 0even off-shell (equation of motion is not satisfied). Integrate by parts, we find that\non-shell\nδS=Z\nd4x(∂µJµ)ϵ. (3.7)\nHowever, for an arbitrary ϵ(x), this is also the equation of motion, since\nδS=Z\nd4xδS\nδϕf(x)ϵ(x). (3.8)\nTherefore, the effect of an additional term in the action ∆Sis adding a term to the current\ndivergence\n∂µJµ= 0→∂µJµ+δ∆S\nδϕf(x) = 0 . (3.9)\nFor the spin rotations, the variation of the pure magnon EFT with local ω(x)is given by\nδS=Z\nd4x\u0010\n∂µ⃗Jµ\u0011\n·⃗ ω. (3.10)\nCorrespondingly, the addition of Eq. (3.3) leads to the modification\nδS=Z\nd4x\u0010\n∂µ⃗Jµ+ ˆn×⃗O(X)\u0011\n·⃗ ω. (3.11)\nThus, the (non-)continuity equation becomes\n∂µ⃗Jµ=−ˆn×⃗O(X). (3.12)\nWhen we focus on the measurements of the magnons, the effect of the external sector enters as\nan in-in expectation valueD\n⃗OE\nin. This may be evaluated via the in-in formalism, and the leading\norder contribution is given by\n⃗ ω·Z\nd4x∂µ⃗Jµ=−⃗ ω·Z\nd4xˆn×D\n⃗OE\nin=−⃗ ω·Z\nd4xˆn×Z\nd4x′\u0000\nGR·ˆn′\u0001\n, (3.13)\nwhere GR(t′, ⃗ x′;t, ⃗ x)is the retarded response function of the operator ⃗O.\nIn frequency space, we have GR(t, ⃗ x) =Rd3⃗kdω\n(2π)4e−iωt+⃗k·⃗ x˜GR(ω,⃗k)and furthermore, using the\nspectral representation (making the spin space indices explicit temporarily)\n˜GAB\nR(ω,⃗k) =Z∞\n−∞dω0\nπi\nω−ω0+iϵρAB(ω0,⃗k), (3.14)\n– 7 –we can separate the prefactor using the identity\ni\nω−ω0+iϵ=πδ(ω−ω0) +Pi\nω−ω0, (3.15)\ninto a δ-function and a principal part. The dissipative part is captured by the former\n˜GAB\nR,diss (ω,⃗k) =Z∞\n−∞dω0δ(ω−ω0)ρAB(ω0,⃗k) =ρAB(ω,⃗k). (3.16)\nTheindicesofthisspectralfunctionlivesinthespin SO(3)spaceand, forisotropicsystems, should\nbe built from invariant tensors δAB,ϵABC. However, the latter could not neither form a two-index\nobject nor respect parity invariance by itself, so the only symmetry-consistent possibility is\nρAB(ω,⃗k) =f(ω,|⃗k|2)δAB, (3.17)\nwith fassumed to be an analytic function of its arguments such that it has a smooth limit as\nωgoes to zero. Dissipative dynamics is antisymmetric under time reversal, so it should be odd\nunder the simultaneous transformation ω↔ −ωand(A, B)↔(B, A), meaning that the leading\norder contribution is given by\nρAB(ω,⃗k) =−iCωδAB(3.18)\nor in real spacetime\nGAB\nR,diss (t, ⃗ x) =C∂\n∂tδ(t)δ3(⃗ x)δAB. (3.19)\nCcould be understood as a Wilson coefficient in this non-conservative sector.\nFrom this, we arrive at the equation\n∂µ⃗Jµ=−Cˆn×∂\n∂tˆn. (3.20)\nCombining with the conservative part of the continuity equation Eq. (2.23), we find the Gilbert\ndamping equation\n∂\n∂t⃗ m=−γ ⃗ m×∂\n∂t⃗ m+. . . , (3.21)\nwhere γ=C/(c2ms),msˆn=⃗ m(for uniform materials), and the higher-order terms on the right-\nhand side of the original equation of motion Eq. (2.23) are contained in . . .which will be omitted\nin the following.\nWhen we have another singlet of the form ˆn·O(2)·ˆn, the effect of the extra ˆns is a replacement\nofthespectraldensity ρAB→ρACBDˆnCˆnD. ThetensorbasisstillconsistsofKroneckerdeltasince\nthe only structure that Levi-Civita tensors could contract to ˆnand would vanish automatically.\nTherefore, any additional structures will appear in the form of the inner product ˆn·ˆndue to\ncontractions with the Kronecker delta. Consequently, they do not lead to anything new due to\nthe normalization condition ˆn2= 1.\nWe see that using the in-in formalism, the form of the damping equation and the coeffi-\ncients are completely fixed by the principles of EFT: the symmetries, power counting, and Wilson\ncoefficients.\n– 8 –4 More general materials\nBy choosing the spectral function to depend only on invariant tensors and the frequency, we\nnaturally arrive at the Gilbert damping Eq. (3.21) which applies to isotropic and homogeneous\nsystems. However, in generic materials, we may be interested in situations with more general\nmaterials which for instance have non-trivial spin textures or highly anisotropic lattices and thus\ninhomogeneous or anisotropic. The advantage of the EFT framework is that these generalizations\ncan be systematically incorporated by including additional couplings. In this section, we explore\nseveral possibilities along these lines.\n4.1 Anisotropic materials\nFor homogeneous systems, one can still have anisotropy due to a background field. The retarded\nresponse function can then depend on the background. Given a homogeneous background vector\nfield⃗heff, the Levi-Civita tensor can now be incorporated into the response ρAB=DϵAB\nChC\neff.\nSince dissipative effects need to be antisymmetric under the simultaneous transformation A↔B,\nω↔ − ωandA↔Bantisymmetry is already included in the Levi-Civita tensor structure,\nthe response function has to be symmetric under ω↔ −ω. This means that the leading order\ncontribution is now independent of ω.1The corresponding response function is\nGA,B\nR,diss(t, ⃗ x) =DϵABChC\neffδ(t)δ3(⃗ x) (4.1)\nand gives rise to the conservation equation\n∂\n∂t⃗ m=−λ⃗ m×(⃗ m×⃗heff). (4.2)\nFor systems with conserved parity, the external field ⃗heffis an effective magnetic field in the sense\nof being parity odd to ensure an even-parity response function.\nThis damping equation involving an effective external background magnetic field is known\nas the Landau-Lifshitz damping equation [8]. We observe that from the EFT point of view, the\nLandau-Lifshitz and Gilbert damping terms are distinguished by symmetries.\nFor completeness, we note that in the literature there are generalizations to Gilbert damping\nby introducing anisotropic damping tensors. In field theoretic language, anisotropy corresponds\nto explicitly breaking of SO(3)and is thus straightforwardly realized in the EFT by a spurion\ncondensation. Instead of introducing a symmetry-breaking VEV to an explicit spurion operator\nin the action, one may assume the 2-point functions acquire a VEV and the most general resulting\ndissipative response function at LO is given by\nGA,B\nR,diss(ω) =SABω+AAB, (4.3)\nwhere SABandAABare general symmetric and antisymmetric tensors which are not SO(3)\ninvariant. Thegeneralizeddampingequationforananisotropicbuthomogeneousmagneticsystem\nthen become∂\n∂t⃗ m=−⃗ m×S·∂\n∂t⃗ m+⃗ m×A·⃗ m. (4.4)\n1This is analogous to the dissipative EFT of a spinning black hole in which case the role of this background is\nplayed by the direction of the spin vector [17].\n– 9 –The exact form of these anisotropy tensors can then be extracted from the microscopic details of\nthe given full theory.\n4.2 Inhomogeneous materials\nFor inhomogenous materials, e.g. configurations with background spin textures, we may have\ncontributions from higher orders in (spatial) derivative expansion. The simplest possible operator\nis given by the coupling\nSint=Z\nd4x∂iχa˜Oai(X). (4.5)\nIn terms of the orientation vector, this is equivalent to\nSint≈Z\nd4x(ˆn×∂iˆn)·⃗Oi(X), (4.6)\nat leading order in χ-field.\nAgain promoting the spin rotation transformation δˆn=⃗ ω׈nto a local parameter ⃗ ω(x), we\nfind an additional contribution to the current divergence\nδSint=Z\nd4x⃗ ω·(2∂iˆnˆn+ (ˆnˆn−δ)·∂i)·⃗Oi, (4.7)\nwhere we have used the fact that ˆn·∂iˆn=1\n2∂iˆn2= 0to simplify the expression. For the spectral\nfunction of the form\nρiA,jB(ω) =EωδijδAB, (4.8)\nthis leads to the damping term\nc2∂tˆn=E\u0010\n2∂iˆnˆn·(∂tˆn×∂iˆn) + (ˆnˆn−δ)·∂t\u0010\nˆn×⃗∇2ˆn\u0011\u0011\n=E\u0010\n2ˆn×(ˆn×∂iˆn) (ˆn×∂iˆn)·∂tˆn+ (ˆnˆn−δ)·∂t\u0010\nˆn×⃗∇2ˆn\u0011\u0011\n, (4.9)\nwhere we have used that ˆn×(ˆn×∂iˆn) =−∂iˆnon the first term. More compactly, this is\n∂\n∂t⃗ m=⃗ m×A·∂\n∂t⃗ m+E(ˆnˆn−δ)\nc2ms·∂t\u0010\n⃗ m×⃗∇2⃗ m\u0011\n, (4.10)\nwhere the first term contains the generalized damping tensor\nA=2E\nc2ms(��� m×∂i⃗ m) (⃗ m×∂i⃗ m). (4.11)\nThis corresponds to the generalized Gilbert damping in the presence of non-trivial spin textures\n(i.e. when ∇⃗ m̸= 0) [18].\nWe note that Eq. (4.6) is only the leading order derivative correction to the damping dynam-\nics. The EFT framework is capable of systematically generating higher derivative corrections. For\nexample, other types of inhomogeneity may be attributed to interactions in the lattice model [19]\nby\n⃗ mi×X\nijGij·⃗ mj, (4.12)\n– 10 –where the i, jindices label the lattice sites associated with the magnetic moments. In the contin-\nuum field theory, the lattice variables become ⃗ mi7→⃗ m(⃗ xi)and the tensorial structure becomes\nthe response function Gij7→GR,diss (t, ⃗ xi−⃗ xj), except that unlike the one in Eq. (3.19), it is\nnon-local (no longer proportional to δ(⃗ x)). In the simpler case that the long-range coupling falls\noff sufficiently quickly, these terms are traded for a series expansion\n⃗ m×X\nnAi1...in∂n\ni1...in∂t⃗ m, (4.13)\nfor some coefficient tensors Ai1...in. In terms of the action, this means the spectral functions are\nnow dependent on the wave vectors\nρAB(ω,⃗k) =X\nn˜Ai1...inki1. . . k inωδAB. (4.14)\n5 Magnon damping term from phonons\nFrom the magnetoelastic EFT, the generic magnon-phonon couplings are given by [1]\nLph=−1\n2Fij\n2(∇(iπj))∂iˆn·∂jˆn+1\n2ρ˜F3(∇(iπj))Dtˆn·Dtˆn, (5.1)\nwhere Dt≡∂t+vi∂iwith the velocity of the material given by vi=−∂tϕ(D−1)i\nj. The \"full\ntheory\" (technically the EFT at the next level of the hierarchy) action constrains the form of the\ncouplings between magnons the external sector to be\nLph=1\n2∂iˆn·∂jˆnOij\n2(π) +1\n2∂tˆn·∂tˆnO3(π) +1\n2∂tˆn·∂iˆnOi\n4(π), (5.2)\nwhere the last term arises from the linear-in- vcontribution in the expansion Dtˆn·Dtˆn.\nFor ferromagnets, the dispersion relation dictates the first term to be dominant. After inte-\ngrating out the external sector using the Schwinger-Keldysh methodR\nDπD˜π, we find its contri-\nbution to the in-in equation of motion is given by\nc2∂tˆn= ˆn×∂i\u0010\n∂jˆnD\nOij\n2E\u0011\n, (5.3)\nwhere the in-in expectation value is given by\nD\nOij\n2(x)E\n=Z\nd4x′Gij,kl\nR,2(x−x′)∂kˆn(x′)·∂lˆn(x′). (5.4)\nFor 4-index tensor structures under SO(3), there are two invariant tensors corresponding to the\nsymmetric-traceless and trace irreps. Therefore, one may write the leading-order dissipative con-\ntribution to the retarded response function as\nGij,kl\nR,2(x)≃\u00121\n2δijδklC2+δi(kδl)jD2\u0013\nδ3(⃗ x)∂tδ(t), (5.5)\nwhere C2andD2are the independent (Wilson) coefficients.\n– 11 –Substituting the results, we find the damping equation in a similar form\nc2∂tˆn=1\n2C2ˆn×∂i\u0000\n∂iˆn∂t\u0000\n∂jˆn·∂jˆn\u0001\u0001\n+D2ˆn×∂i\u0000\n∂jˆn∂t\u0000\n∂iˆn·∂jˆn\u0001\u0001\n=C2ˆn×∂i\u0000\n∂iˆn∂jˆn·∂j∂tˆn\u0001\n+D2ˆn×∂i\u0000\n∂jˆn∂iˆn·∂j∂tˆn\u0001\n+D2ˆn×∂j\u0000\n∂iˆn∂iˆn·∂j∂tˆn\u0001\n,(5.6)\nor more compactly\n∂\n∂t⃗ m=⃗ m×D·∂\n∂t⃗ m, (5.7)\nwhere the \"damping tensor\" Dis given by\nD=1\nc2msh\n∂i\u0000\nC2∂iˆn∂jˆn+D2∂jˆn∂iˆn\u0001\n+D2∂j\u0000\n∂iˆn∂iˆn\u0001\n+\u0000\nC2∂iˆn∂jˆn+D2∂jˆn∂iˆn\u0001\n∂i+D2∂iˆn∂iˆn∂ji\n∂j. (5.8)\nWe notice that the form of the couplings restricts the damping tensor to appear at higher-\norders in derivative expansions and hence they are expected to be small compared with contribu-\ntions from fermions (e.g. electrons) in the long wavelength limit. However, for insulating materials\nthat have electron-magnon coupling suppressed, we expect their effects to be more significant.\n6 Conclusion and discussions\nIn this paper, we used the in-in (Schwinger-Keldysh) formalism to generalize the recently con-\nstructed EFT of magnetoelasticity [1] to describe damped magnetic dynamics. We discover that\nthe Gilbert damping term naturally arises as the simplest symmetry consistent dissipative cor-\nrection within the in-in formalism. Systematic generalizations to anisotropic and inhomogeneous\nsetups also yield desired results such as the Landau-Lifshitz magnetic damping equation. More-\nover, we are able to predict the form of phononic contribution to the damping dynamics. Thus\nwe have shown that this is a useful framework to derive dissipative dynamics from first principles\nand to predict the forms of higher-order corrections in a systematic way.\nIt would be interesting to investigate the explicit full theory model of the “external sector”\nsuch as the fermionic fields in Eq. (3.4) and extract the relevant Wilson coefficients by matching\nthe response functions. In this way, we may gain better insights into what controls the damping\nparameter and give more predictive power to the EFT approach. It would also be interesting to\nmatch the relevant coefficients in Eq. (4.14) to obtain an EFT framework for a generalized class\nof models.\nFurthermore, various applications of the magnetoelastic EFT [20, 21] have appeared more\nrecently. It would be interesting to investigate the effects of adding dissipative terms into these\nproblem. There are also further developments in the technical aspects of such EFTs [22, 23]. It\nis natural to consider their implications on the non-conservative sector. We leave these problems\nfor future works.\nAcknowledgement\nThe author thanks Ira Rothstein for advising throughout the project and a careful reading of the\nmanuscript. The author also thanks Riccardo Penco for important discussions, Shashin Pavaskar\n– 12 –for other useful discussions, and Witold Skiba for comments on the draft. This work is partially\nsupported by the grants DE- FG02-04ER41338 and FG02- 06ER41449.\nReferences\n[1] S. Pavaskar, R. Penco, I. Z. Rothstein, An Effective Field Theory of Magneto-Elasticity , SciPost\nPhys.12.5.155 (2022), arXiv:2112.13873 [hep-th].\n[2] S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological Lagrangians. 1. , Phys.Rev.\n177 2239 (1969).\n[3] J. Callan, Curtis G., S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological\nLagrangians. 2. , Phys.Rev. 177 2247 (1969).\n[4] D. V. Volkov, Phenomenological Lagrangians , Fiz. Elem. Chast. Atom. Yadra 4 3 (1973).\n[5] V. I. Ogievetsky, Nonlinear Realizations of Internal and Space-time Symmetries , Proc. of. X-th\nWinter. School of Theoretical Physics in Karpacz, Vol. 1, Wroclaw 227 (1974) .\n[6] M. Baumgart et. al., Snowmass White Paper: Effective Field Theories for Condensed Matter\nSystems, in2021 Snowmass Summer Study. 10, 2022, arXiv:2210.03199[hep-ph].\n[7] T. Brauner et. al., Snowmass White Paper: Effective Field Theories for Condensed Matter Systems ,\nin2022 Snowmass Summer Study. 3, 2022„ arXiv:2203.10110[hep-th].\n[8] L. D. Landau and E. M. Lifshitz, Theory of the dispersion of magnetic permeability in ferromagnetic\nbodies, Phys. Z. Sowjetunion. 8, 153 (1935).\n[9] T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials , IEEE\nTransactions on Magnetics, vol. 40, no. 6, (2004).\n[10] J. Schwinger, Brownian Motion of a Quantum Oscillator , J. Math. Phys. 2 407 (1961).\n[11] L. V. Keldysh, Diagram technique for nonequilibrium processes , Zh. Eksp. Teor. Fiz. 47 1515 (1964).\n[12] C. R. Galley, The classical mechanics of non-conservative systems , Phys. Rev. Lett. 110, 174301\n(2013), arXiv:1210.2745 [gr-qc].\n[13] C. R. Galley, D. Tsang, and L. C. Stein, The principle of stationary nonconservative action for\nclassical mechanics and field theories (2014), arXiv:1412.3082 [math-ph].\n[14] S. Endlich, R. Penco, An effective field theory approach to tidal dynamics of spinning astrophysical\nsystems, Phys. Rev. D. 93.064021 (2016), arXiv:1510.08889 [gr-qc].\n[15] W. D. Goldberger and I. Z. Rothstein, Horizon radiation reaction forces , JHEP 10 026 (2020),\narXiv:2007.00731[hep-th].\n[16] L. V. Delacrétaz, S. Endlich, A. Monin, R. Penco, F. Riva, (Re-)Inventing the Relativistic Wheel:\nGravity, Cosets, and Spinning Objects , JHEP 11 (2014) 008, arXiv:1405.7384 [hep-th].\n[17] W. D. Goldberger, J. Li, and I. Z. Rothstein, Non-conservative effects on spinning black holes from\nworld-line effective field theory , JHEP 06 053 (2021) arXiv:2012.14869[hep-th].\n[18] S. Zhang, S. S.-L. Zhang, Generalization of the Landau-Lifshitz-Gilbert Equation for Conducting\nFerromagnets , Phys. Rev. Lett. 102, 086601 (2009).\n[19] S. Brinker, M. dos Santos Dias, S. Lounis, Generalization of the Landau-Lifshitz-Gilbert equation by\nmulti-body contributions to Gilbert damping for non-collinear magnets , J. Phys.: Condens. Matter\n34 285802 (2022), arXiv:2202.06154 [cond-mat.mtrl-sci].\n– 13 –[20] A. Esposito, S. Pavaskar, Optimal anti-ferromagnets for light dark matter detection (2022),\narXiv:2210.13516 [hep-ph].\n[21] S. Pavaskar, I. Z. Rothstein, The Dynamics of Line Defects and Their Sensitivity to the Lattice\nStructure (2022), arXiv:2212.10587 [hep-th].\n[22] A. Nicolis, I. Z. Rothstein, Apparent Fine Tunings for Field Theories with Broken Space-Time\nSymmetries (2022), arXiv:2212.08976 [hep-th].\n[23] C. O. Akyuz, G. Goon, R. Penco, The Schwinger-Keldysh Coset Construction (2023),\narXiv:2306.17232 [hep-th].\n– 14 –"    },    {        "title": "2312.15973v1.All_solution_grown_epitaxial_magnonic_crystal_of_thulium_iron_garnet_thin_film.pdf",        "content": "All solution grown epitaxial magnonic crystal of thulium iron garnet thin film\nRajnandini Sharma,1Pawan Kumar Ojha,1Simran Sahoo,1Rijul Roychowdhury,2and\nShrawan Kumar Mishra1\n1)School of Materials Science and Technology, Indian Institute\nof Technology (Banaras Hindu University), Varanasi-221 005,\nIndia\n2)Surface Physics and Material Science Division, Saha Institute of\nNuclear Physics Kolkata, 1/AF Bidhannagar, Sector 1, Kolkata 700 064,\nIndia\n(*Electronic mail: shrawan.mst@iitbhu.ac.in)\n(*Electronic mail: rajnandinisharma.rs.mst18@itbhu.ac.in)\n(Dated: 27 December 2023)\nMagnonics has shown the immense potential of compatibility with CMOS devices and\nthe ability to be utilized in futuristic quantum computing. Therefore, the magnonic crys-\ntals, both metallic and insulating, are under extensive exploration. The presence of high\nspin-orbit interaction induced by the presence of rare-earth elements in thulium iron gar-\nnet (TmIG) increases its potential in magnonic applications. Previously, TmIG thin films\nwere grown using ultra-high vacuum-based techniques. Here, we present a cost-effective\nsolution-based approach that enables the excellent quality interface and surface roughness\nof the epitaxial TmIG/GGG. The deposited TmIG (12.2 nm) thin film’s physical and spin\ndynamic properties are investigated in detail. The confirmation of the epitaxy using X-ray\ndiffraction in φ-scan geometry along with the X-ray reflectivity and atomic force for the\nthickness and roughness analysis and topography, respectively. The epitaxial TmIG/GGG\nhave confirmed the perpendicular magnetic anisotropy utilizing the polar-magneto-optic\nKerr effect. Analyzing the ferromagnetic resonance study of TmIG/GGG thin films pro-\nvides the anisotropy constant K U= 20.6×103±0.2×103N/m2and the Gilbert damping\nparameter α= 0.0216 ±0.0028. The experimental findings suggest that the solution-\nprocessed TmIG/GGG thin films have the potential to be utilized in device applications.\n1arXiv:2312.15973v1  [cond-mat.mtrl-sci]  26 Dec 2023Magnonics is the study of spin waves-based information processing and transmission1. Magnons\nhave the potential to be utilized in more dense logic gates, along with the processing and trans-\nport of information simultaneously2. Magnon’s superposition ability makes it a potential can-\ndidate for its uses as a qubit in quantum computing3. There are various magnon carrier sys-\ntems; some are conducting, and others insulating2. Conducting magnonic crystals are CoFeB4,5,\nNiFe(permalloy)6,7, and Heusler compounds8,9. Iron garnets are one class of these insulating\nmagnonic crystals10. Initially, various fundamental understandings of magnon behaviours like\nmagnon-magnon scattering and magnetic resonance have been considered as possible micro-\nscopic origins using these ferrimagnetic insulators11,12. Recently, the heterostructure yttrium iron\ngarnet (YIG) found its application in spin pumping. Both exchange and dipolar spin waves are\nhigher-order spin waves in the single-crystal YIG thin films13. Soon after its discovery, the exper-\nimental study confirmed system has the lowest dissipation (lowest linewidth of the ferromagnetic\nresonance), making it a promising system for various applications14. A recent study shows that in\npulsed laser deposition (PLD) grown Pt/YIG, the interfacial spin Hall angle ( θSH) is 0.3315. But\nfurther advanced processing can be achieved by perpendicular magnetic anisotropy (PMA) in the\nsystem16. YIG has a low anisotropy constant K U= 1×103N/m2when deposited on Gd 3Ga5O12\n(GGG) substrate17. High spin-orbit coupling in rare-earth iron garnets has the potential to resolve\nthis18. Complete rare-earth series can form the iron garnets19. The rare-earth elements have their\nunique magnetic ordering so they contribute to the ferrimagnetic coupling. This contribution\ncauses compensation temperature, which is the lowest magnetization state. Thulium iron garnet\n(TmIG) is the rare-earth garnet with Curie temperature (T C≈550 K) that has the lowest compen-\nsation temperature ≈15 K and room temperature moderate saturation magnetization. Recently, the\nPt/TmIG heterostructure with PMA shows the magnetic switching and spin magnetoresistance20,\nand TmIG/Au/TmIG shows the spin valves properties21.\nIron garnet thin films are grown using ultra-high vacuum facilities and require extravagant\nfacilities like PLD, off-centred rf-sputtering, and liquid phase epitaxy (LPE). Few studies have\nproduced polycrystalline iron garnets using solution methods like spin-coating22–24. However, the\nepitaxial thin film growth using spin-coating is not reported to date. This article presents a cost-\neffective, all-solution-based spin-coating method that uses the substrate’s crystal structure to refer-\nence and grows an epitaxial thin film of TmIG/GGG. The epitaxial TmIG/GGG has been studied\nusing synchrotron Grazing Incident X-ray diffraction (GIXRD). The confirmation of the epitaxy\nis presented using GIXRD φ-scan. The topography and elemental analysis of TmIG magnonic\n2crystal are studied in detail. The magnetic study of the good interface quality epitaxial TmIG thin\nfilm is reported in the present article.\nTmIG thin film on single-crystal gadolinium gallium garnet (GGG) substrate of (111) orien-\ntation was deposited using all solution-based spin-coating. To prepare a solution, nitrates of iron\n(Fe(NO 3)3.9H 2O, (98% purity)) and thulium (Tm(NO 3)3.5H 2O (99.9% purity)) in 3:2 ratio was\namalgam in the 2-methoxyethanol with 400 mM concentration. The solution was stirred and aged\nfor three days to make it uniform and have a gel-like consistency. The substrate’s surface quality\nshould be excellent to deposit the thin film. To clean, the substrate (GGG) was ultra-sonicated\nwith de-ionized water, acetone, and 2-propanol for 30 minutes each. Further, the substrate was\nplasma-cleaned for 10 min at 10 W in an oxygen atmosphere. The uniformly stirred solution was\nstatically spin-coated on the cleaned substrate at 4000 rpm for 30 sec. The excellent interface and\nfilm quality are achieved by heating the spin-coated film in three stages. An excessive solvent was\ninitially evaporated at 363 K for 2 hours in the air on the hot plate. Organic solvent decomposition\nwas processed by heating the film further at 623 K for 30 minutes in a muffle furnace (in air). The\nfinal phase formation was achieved by annealing the prepared film at 1223 K for 3 hrs in a tubu-\nlar furnace with an oxygen environment. The crystal structure of GGG and TmIG is analogous;\ntherefore, the growth of the epitaxial TmIG becomes favorable.\nThe structural confirmation was done using synchrotron grazing-incident x-ray diffraction\n(GIXRD) using 10 KeV energy of INDUS-2 (BL-13) RRCAT, Indore. Thickness estimation uses\nX-ray reflectivity (XRR) utilizing Bruker D8 Diffractometer. XRR data was fitted using Par-\nrat32 software utilizing Parratt’s formalism25,26. The morphology of the thin films was observed\nusing atomic force microscopy (AFM) utilizing a Bruker nano IR microscope. The elemental\ncomposition study uses X-ray photoelectron spectroscopy (XPS) using Thermo Fisher Scientific\nmodel K alpha using aluminum K-alpha radiation. The magnetic study uses a white LED-based\nMagneto-optical Kerr effect (MOKE) microscope in polar mode along with room temperature\nferromagnetic resonance (FMR), utilizing broadband FMR of Quantum design Phase FMR.\nThe phase formation of the TmIG thin film deposited using sol-gel-based spin coating is per-\nformed using the GIXRD. As the mismatch between the substrate and the thin film is less than a\npercent, therefore, highly monochromatic 10 keV synchrotron X-rays have been utilized. Figure 2\n(a) presents the out-of-plane XRD of TmIG (444) reflection with the substrate GGG (444) highest\nintensity reflection. Inset Figure 2(a) shows the logarithmic plot of the intensity to show the ex-\ncellent interface quality that confirms high crystallinity due to the presence of Laue oscillations27.\n3FIG. 1. Schematic of the synthesis method of TmIG/GGG thin film.\nThe interplanar distance of GGG (444) and TmIG (444) are 1.7938 ±0.0085 Å, and 1.7778 ±\n0.0084 Å, respectively. The lattice constants are 12.4281 ±0.0116 Å, and 12.3145 ±0.0116 Å,\nfor the GGG and TmIG, respectively. Experimental data confirm the smoothness of the interface\nand the epitaxial growth between the substrate and thin film as shown in Figure 2 (b). The strain\nbecause of the mismatch between the two ∆ε=aGGG−aTmIG\naGGGis 0.88 % which shows the tensile strain\non the layer of TmIG thin film. The tensile strain is the cause of the presence of the PMA in the\nsample (discussed in further sections). Figure 2 (b) represents the φ-scan of the TmIG thin film.\nTheφ-scan is measured along the (008) Bragg reflection, which is ψ= 54.7◦from the (111) Bragg\nreflection28. The three-fold symmetry in the φ-scan can be observed in Figure 2 (c). The angle\ndifference of 1200between three-fold symmetry is experimentally observed in φ-scan, which con-\nfirms the epitaxy of the deposited thin film29. The stress ( σ) at the interface is calculated using the\nfollowing equation30:\nσ=Y\n1−ν∆ε (1)\nwhere, Y is Young’s modulus (2.00 ×1011N/m2) and the νis Poisson’s ratio (0.29) as present\nin literature19. The σcalculated is 2.573 ×109±2 N/m2.\nThe substrate film interface quality is essential for the magnonic application. Figure 3 illustrates\n4FIG. 2. Structural confirmation of the epitaxial TmIG thin film deposited using all-solution-based spin-\ncoating. (a) depicts the θ−2θscan of GIXRD of the substrate and the thin film with the logarithmic inset\nto present excellent crystallinity, (b) is the φ-scan to confirm the epitaxy with the three-fold symmetry of\nthe (008) plan, and (c) shows the schematic of the three-fold symmetry in (008) plane.\nthe topography and structural quality of the deposited thin film using AFM and XRR. Figure 3 (a)\nillustrates AFM showing smooth topography, estimating mean roughness is ≈0.8 nm. Figure 3\n(b) illustrates the XRR of the TmIG thin film fitted using Parratt’s formalism and that estimated\nthe thickness of the ≈12.2 nm, and the interfacial roughness ≈0.2 nm, which is excellent. The\ndegree of the crystalline can also be accessed using the presence of Laue oscillations in the inset\nof Figure 2(a). The surface roughness estimated is ≈0.4 nm, which is in order equivalent to\nsurface roughness estimation using AFM. The topography of the deposited thin film is smooth and\nsuggests homogeneous growth on the substrate. Low surface and interface roughness show the\npotential of the sol-gel-based spin-coating method to study further application possibilities.\nThe elemental composition is probed using the XPS. Figure 4: depicts the survey scan and the\nhigh-resolution XPS of the TmIG thin film. Figure 4 (a): plot the survey scan, gives the presence of\n5FIG. 3. Surface topography and the thickness and roughness study estimation using (a) atomic force mi-\ncroscopy and (b) x-ray reflectivity, respectively.\nO, C, Fe, Tm, and N in the thin film. The sample constituted O, Fe, and Tm, but the environmental\nexposure caused the C and N absorption. The sample is calibrated using the carbon at peak position\n284.2±0.1 eV . Figure 4 (b) illustrates the high-resolution spectra of thulium 4d 5/2core electrons.\nTm4d5/2is observed at 175.7 ±0.1 eV which is supported by the literature31. Thulium is present\nin Tm2+and Tm3+but the most stable valence state is Tm3+. The presence of a satellite peak at\n179.2±0.1 is similar to the literature and confirms Tm3+charge state32. Figure 4 (c) illustrates\nthe high-resolution spectra of the oxygen 1s core electrons. The main peak at binding energy 529.4\n±0.1 eV is because the O 1selectrons bind in the TmIG; along with this, the surface contribution\nof the oxygen is also there at 530.8 ±0.1 eV . Figure 4 (d) illustrates the high-resolution spectra of\nthe iron 2p core electrons. Iron is present in a 2:3 ratio of octahedral and tetrahedral coordinates in\nTmIG (space group Ia ¯3d). Therefore, the Fe 2p 3/2and 2p 1/2peak comprise two peaks each. The\nFe octahedral (Fe oct) peaks 2p 3/2at binding energy 709.8 ±0.1 eV and 2p 1/2at binding energy\n723.1±0.1 eV . The Fe tetrahedral (Fe tetra) peaks 2p 3/2at binding energy 711.2 ±0.1 eV and\n2p1/2at binding energy 724.5 ±0.1 eV . The theoretical ratio between the area of octahedral and\ntetrahedral Fe is 2:3. The experimental ratio of the area of peak 2p 3/2octahedral and tetrahedral\nFe is 0.72, and of peak 2p 1/2octahedral and tetrahedral Fe is 0.6733. The ratio is very close to\nthe theoretical ratio 2:3, confirming that the quality of the sample is excellent33. The difference\nbetween the core electron peak and the satellite peak is large and equivalent to the 8 eV , which\nfurther establishes that Fe is in a 3+ valence state and the stoichiometry is balanced23. The atomic\n6FIG. 4. XPS of the TmIG thin film has been presented. (a) survey scan from binding energy 0-1000 eV .\nHigh-resolution XPS of the elements (b) Tm3+, (C) O2−and (d) Fe3+are fitted and presented with peak\ncomponent with the Shirley background.\npercentage of the constituents Tm3+, Fe5+and O2−are 16 %, 26%, and 58%, respectively. The\natomic percent is calculated using the CASAXPS software34.\nAs the TmIG has application in magnonics, the magnetic properties of the deposited sample\ndetermine its application potential. Figure 5 presents the magnetic behaviour of the deposited all\nsolution-based epitaxial TmIG thin film. Figure 5 (a) illustrates the polar MOKE measurements.\nThe presence of out-of-plane uniaxial anisotropy gives the MOKE signal, which confirms the ex-\nistence of the perpendicular magnetic anisotropy (PMA) at room temperature in the thin film35.\nThe uniaxial anisotropy (K U) in the thin film is a combination of various components like stress-\ninduced anisotropy (K σ), magneto-crystalline anisotropy (K M), and shape anisotropy (K S)36. Fig-\nure 5 (b) illustrates the schematics of the PMA in the film. The stress-induced anisotropy is\n7calculated using the formula as follows:\nKσ=−3\n2λ111σ (2)\nwhere, λ111is the magnetostriction constant (-5.2 ×10−6) of the TmIG as present in literature19.\nThe value of the estimated stress-induced anisotropy (K σ) is 20.07 ×103±5.71 10−2N/m2. Shape\nanisotropy (K S) estimated is 0.49 ×103N/m2. The cubic anisotropy constant (K 1) value is taken\nfrom the literature35is -1.1×103N/m2. The final uniaxial anisotropy value is estimated as follows:\nKU=−K1\n12+Kσ+KS (3)\nwhere, K 1/12 is K M. The estimated K Ufrom the strain is 20.11 ×103N/m2.\nThe magnetic study of the TmIG is also performed using FMR. It probes the precession of the\nmoments along the external field, and this precession resonates with the applied frequency. The\nabsorption of that frequency at a particular magnetic field gives the resonance magnetic field and\nthe linewidth of the absorption, which signifies the moment’s precession and energy dissipation,\nrespectively. Figure 5: depicts the FMR results, (a) plots in-plane the resonance magnetic field\n(Hres) as a function of frequency and fitted using Kittel equation, (b) illustrates the linewidth as a\nfunction of the frequency. Kittel equation37,38is as presented below:\nf=\u0010γ\n2π\u0011q\nH(H+µ0Me f f) (4)\nThe effective magnetization ( µ0Me f f) estimated is -0.292 ±0.003 T. As the PMA is confirmed\nwith the polar MOKE, the negative value of µ0Me f fshows that anisotropy dominates the satura-\ntion magnetization. µ0Me f fis composed of saturation magnetization ( µ0MS)and the anisotropy\nfield (H U) in following equation36:\nµ0Me f f=µ0MS−HU (5)\nIn literature, the value of saturation magnetization ( µ0MS) of bulk TmIG is 0.1244 T39. The\nanisotropy Field ( HU) is estimated to be 0.4167 T. The anisotropy constant (K U) is calculated by\nsubstituting the anisotropy field and saturation magnetization as present in the following equation:\nKU=HU×MS\n2(6)\n8KUis estimated by substituting, the values is 20.6 ×103±0.2×103N/m2. K Uestimation\nby FMR is equivalent to the K Ucalculated with the strain in GIXRD. This value is higher than\nthe literature due to the self-growth of the TmIG/GGG and form a better interface. The value\nof the gyromagnetic ratio is 19.46 ±0.09 GHz/T, which is lower than the free electron because\nof the high spin-orbit coupling of the thulium ions. The Landé g-factor estimated is 1.391 ±\n0.006 from the gyromagnetic ratio. The Landé g-factor is smaller than the free electron as well as\nthe value present for TmIG in literature38. The presence of low Landé g-factor can be due to the\npresence of high anisotropy present in TmIG/GGG thin film. The uniform mode is generated while\nthe moment precession relaxes by the dissipating energy due to extrinsic and intrinsic factors.\nExtrinsic factors are like defects and electro-electron interaction, and intrinsic factors are like two-\nmagnon interaction and high spin-orbit interaction. These factors can be obtained from the linear\nfitting of the magnetic linewidth as a function of the applied frequency. Figure 5 (d) depicts the\nlinewidth ( ∆H) as a function of the applied frequency. Yellow dots are obtained by analyzing the\nexperimental FMR data as a function of the applied field (the intensity of the dP/dH is low, which\ncauses the deviation), and the green line fits them linearly. The relation of linewidth, intrinsic, and\nextrinsic damping parameters is as follows:\n∆H=∆H0+4πα\nγf (7)\nThe∆H0is the extrinsic part of the TmIG energy dissipation, and the fitted value is the ∆H0\nis 17.69 ±1.08 mT. The intrinsic dissipation is stated as the Gilbert damping parameter ( α) =\n0.0216 ±0.0028. Gilbert damping parameter is of the same order as the present in literature in\nwhich samples are prepared with sophisticated methods35.\nThis article presents a cost-effective method and to understand it the literature is compared with\nexperimental observation in table I. All the samples are grown on the GGG (111) substrate, and the\nthicknesses are different, but the Gilbert damping parameter is of the same order. The estimated\nKUis higher compared to the literature and is supported by the lower value of the Landé g-factor\nof the present work.\nIn conclusion, sol-gel-based spin coating is utilized to deposit the epitaxial thulium iron garnet\n(TmIG) thin film on the GGG substrate. The elemental analysis confirms the stoichiometric depo-\nsition with the low interface and the surface roughness. The presence of the perpendicular mag-\nnetic anisotropy of this all-solution method deposited TmIG due to the stress-induced anisotropy.\nDue to the presence of the high spin-orbit coupling gives rise to the lower gyromagnetic ratio and\n9FIG. 5. Magnetic Ferromagnetic resonance study of the TmIG thin film (a) Kittel fitting to the resonance\nmagnetic field to study effective magnetic field and gyromagnetic ratio. (b) Linear fit to the linewidth as a\nfunction of frequency\nTABLE I. The comparison of the present data with the literature.\nSubstrate Deposition Method Thickness Damping Parameter Anisotropy constant reference\n(nm) (103N/m2)\nGGG (111) Sol-gel-based Spin-coating 12.2 0.0216 20.6 present\nGGG (111) PLD 29 0.013 940\nGGG (111) PLD 20 - 300 0.02 1041\nGGG (111) sputtering 9 0.013 -39\nGGG (111) PLD 8 - 11.8835\nLandé g-factor, which is well matched with the literature. The intrinsic and extrinsic dissipation\nfactors of TmIG present a potential for the deposition method. With further improvements, this\n10cost-effective method of deposition has the potential to be used for magnonics applications.\nCREDIT AUTHORSHIP CONTRIBUTION STATEMENT\nRajnandini Sharma : Conceptualization (lead); Synthesis (equal); Data curation (lead); For-\nmal analysis (lead); Project administration (supporting); Validation (equal); Visualization (lead);\nWriting – original draft (lead); Writing – review & editing (supporting). Pawan Kumar Ojha :\nSynthesis (equal); Formal analysis (supporting). Simran Sahoo : Formal analysis (support-\ning); Writing – review & editing (supporting). Rijul Roychowdhury : Experimental contribution\n(GIXRD). Shrawan K. Mishra : Funding acquisition (lead); Project administration (lead); Super-\nvision (lead); Validation (equal); Visualization (supporting); Writing – original draft (supporting);\nWriting – review & editing (lead).\nDATA A VAILABILITY STATEMENT\nThe data that support the findings of this study are available from the corresponding author\nupon reasonable request.\nDECLARATION OF COMPETING INTEREST\nThe authors declare that they have no known competing financial interests or personal relation-\nships that could have appeared to influence the work reported in this paper.\nACKNOWLEDGEMENT\nThis work is financially supported by the Nano Mission program, DST, India project No.\nIIT(BHU)/R&D/SMST/18-19/09. RS acknowledges the DST INSPIRE for the INSPIRE fellow-\nship. 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Ehesan Ali 2, Subhankar Bedanta 1,3\n1 Laboratory for Nanomagnetism and Magnetic Materials (LNMM),\nSchool of Physical Sciences, National Institute of Science Education and Research (NISER),\nAn OCC of Homi Bhabha National Institute (HBNI), Jatni 752050, Odisha, India\n2 Institute of Nano Science and Technology, Knowledge City, Sector-81, Mohali, Punjab 140306, India and\n3 Center for Interdisciplinary Sciences (CIS), National Institute of Science Education and Research (NISER),\nAn OCC of Homi Bhabha National Institute (HBNI), Jatni, Odisha 752050, India\nMd. Ehesan Ali∗and Subhankar Bedanta†\nDevelopment of power efficient spintronics devices has been the compelling need in the post-CMOS\ntechnology era. The effective tunability of spin-orbit-coupling (SOC) in bulk and at the interfaces of\nhybrid materials stacking is a prerequisite for scaling down the dimension and power consumption of\nthese devices. In this work, we demonstrate the strong chemisorption of C 60molecules when grown\non the high SOC β-W layer. The parent CFB/ β-W bilayer exhibits large spin-to-charge intercon-\nversion efficiency, which can be ascribed to the interfacial SOC observed at the Ferromagnet/Heavy\nmetal interface. Further, the adsorption of C 60molecules on β-W reduces the effective Gilbert\ndamping by ∼15% in the CFB/ β-W/C 60heterostructures. The anti-damping is accompanied by a\ngigantic ∼115% enhancement in the spin-pumping induced output voltage owing to the molecular\nhybridization. The non-collinear Density Functional Theory calculations confirm the long-range en-\nhancement of SOC of β-W upon the chemisorption of C 60molecules, which in turn can also enhance\nthe SOC at the CFB/ β-W interface in CFB/ β-W/C 60heterostructures. The combined amplifica-\ntion of bulk as well interfacial SOC upon molecular hybridization stabilizes the anti-damping and\nenhanced spin-to-charge conversion, which can pave the way for the fabrication of power efficient\nspintronics devices.\nI. INTRODUCTION\nSpintronic logic and memory devices have proven to\nbe one of the most suitable research domains to meet\nthe ultra-low power consumption demand in the post-\nComplementary Metal Oxide Semiconductor (CMOS)\ntechnology era. Especially, with the advent of artificial\nintelligence and the Internet of Things (IoT), the further\nscaling down of CMOS technology can reach its physi-\ncal limits in size, speed, and static energy consumption.\nThe conceptualized spin orbit torque magnetic random\naccess memory (SOT-MRAM) devices which take the ad-\nvantage of spin Hall effect (SHE) can bring down the\nenergy consumption to femto Joule from the pico Joule\nscale [1, 2]. The SHE based magnetization switching\nmechanism in SOT-MRAMs also offers much improved\nendurance owing to the separation in data writing and\nreading paths. Though these potentials of SOT-MRAMs\nhave attracted major foundries, several challenges need\nto be addressed before the commercialization of SOT-\nMRAMs [1, 2]. The increase of writing efficiency to re-\nduce power consumption is one of those aspects which\nrequires significant consideration. In this context, the\nspin Hall angle, θSH(JS⁄JC) of the nonmagnetic layer\npresent in the SOT-MRAMs, where J Cand J Sare the\ncharge and spin current densities, respectively, plays a\n∗ehesan.ali@inst.ac.in\n†sbedanta@niser.ac.incritical role in determining the writing efficiency [3]. The\nefficient charge to spin interconversion can lead to the\nfaster switching of magnetization of the adjacent mag-\nnetic layer via SHE. Hence, various types of heavy metals\n(HMs), like Pt, Ta, W, Ir etc. have been investigated in\nthe past two decades to reduce the power consumption\nof future spintronic devices [4–6]. On a similar note, the\nRashba-Edelstein effect (REE) occurring at the interfaces\nwith spatial inversion symmetry breaking and high spin\norbit coupling (SOC) has also the potential for the man-\nifestation of efficient charge to spin interconversion[7–9].\nHence, the combination of SHE and REE can be the most\nsuitable alternative for the development of power efficient\nspintronics application.\nAmong all the heavy metals, highly resistive ( ρβ−W∼\n100−300µΩ cm) metastable β-W possesses the largest\nθSH∼-0.3 to -0.4 [10–14], which makes it a strong can-\ndidate for SOT-MRAM devices. Usually, additional re-\nactive gases, like O 2, N2, and F are employed to stabilize\nthe A15 crystal structure of β-W [11] and consequently, a\nlarger θSHis realized. For example, Demasius et al., have\nbeen able to achieve θSH∼-0.5 by incorporating the oxy-\ngen into the tungsten thin films [12]. Interface engineer-\ning also acts as a powerful tool for enhancing the writing\nefficiency in β-W based SOT-MRAM devices [15–17]. For\ninstance, the presence of an interfacial atomically thin\nα-W layer in CoFeB/ α-W/β-W trilayer suppresses the\nspin backflow current, resulting in a 45% increase in the\nspin mixing conductance [15]. Further, the REE evolved\nat the W/Pt interface owing to the charge accumulationarXiv:2401.00486v1  [cond-mat.mtrl-sci]  31 Dec 20232\ngenerates an additional spin orbit field on the adjacent\nferromagnet (FM) NiFe (Py) layer [18]. The coexistence\nof SHE and REE has also been reported in CoFeB/ β-Ta\nand NiFe/Pt bilayers, where the interfacial SOC arising\nat the FM/HM interface plays a vital role in the spin-\nto-charge interconversion phenomena [19, 20]. More in-\nterestingly, a recent theoretical work has predicted the\ninterfacial SOC mediated spin Hall angle of Pt can be 25\ntimes larger than the bulk value in NiFe/Pt heterostruc-\nture [21]. The interfacial SOC mediated spin accumula-\ntion has also been reported to occur at the Rashba-like\nβ-Ta/Py interface without flowing the DC current [22].\nThe spin pumping induced by the ferromagnetic reso-\nnance results in non-equilibrium spin accumulation at the\ninterface which consequently reduces the effective Gilbert\ndamping of the β-Ta/Py bilayer. The reduction in ef-\nfective damping, also termed as antidamping, is similar\nto the interfacial Rashba like SOT, observed in various\nHM/FM heterostructures [22]. The anti-damping phe-\nnomena without the requirement of DC current depends\non several factors, like SOC of HM, strength of built in\nelectric field at the interface, interface quality etc. Hence,\nthe interface engineering via tuning the interfacial SOC\ninβ-W based HM/FM heterostructures can be the path\nforward for developing power efficient SOT-MRAM de-\nvices.\nTill the date, most of the interface engineering re-\nsearch have been focused on employing an additional\nmetallic or oxide layer in the HM/FM system for the\nenhancement of spin-to-charge interconversion efficiency.\nWhereas, the organic semiconductors (OSCs) can also\nbe incorporated in the HM/FM system to fabricate hy-\nbrid power efficient spintronic devices owing to their\nstrong interfacial hybridization and charge transfer na-\nture at metal/OSC interface [23]. Recently, the SOC\nof Pt has been found to be enhanced due to the on-\nsurface physical adsorption of C 60(fullerene) molecules\nin YIG/Pt/C 60trilayer [24]. However, the θSHof Pt is\nusually found to be smaller compared to β-W and it is\nimportant to investigate the magnetization dynamics and\nspin to charge conversion phenomena in FM/ β-W/C 60\nheterostructures. Hence, in this article, we report the\neffect of molecular hybridization at β-W/C 60interface\non magnetization dynamics and spin-to-charge conver-\nsion phenomena in Co 20Fe60B20(CFB)/ β-W/C 60het-\nerostructures. The molecular hybridization reduces the\neffective Gilbert damping and also enhances the spin-to-\ncharge conversion efficiency owing to the enhanced SOC\nofβ-W and consequent strengthening of possible Rashba-\nlike interaction at the CFB/ β-W interface. The strong\nchemisorption at the β-W/C 60interface and evolution of\nenhanced SOC of β-W upon the molecular hybridization\nhave also been confirmed by the first principle density\nfunctional theory (DFT) based calculations.II. EXPERIMENTAL AND COMPUTATIONAL\nMETHODS\nFour different types of heterostructures with CFB (7\nnm)/ β-W (2.5, 5 nm) (Figure 1 (a)) and CFB (7 nm)/ β-\nW (2.5, 5 nm)/C 60(13 nm) (Figure 1 (b)) stackings were\nfabricated on Si/SiO 2(300 nm) substrates for the inves-\ntigation of magnetization dynamics and spin pumping\nphenomena. In addition, the CFB (7 nm)/ β-W (10, 13\nnm) heterostructures were also fabricated to reaffirm the\nstabilization of β-W. The heterostructure stackings and\ntheir nomenclatures are mentioned in Table I. The CFB\nandβ-W layers were grown by DC magnetron sputter-\ning, while the Effusion cell equipped in a separate cham-\nber (Manufactured by EXCEL Instruments, India) was\nused for the growth of the C 60over layers in the CFWC\nseries. While preparing the CFWC1 and CFWC2, the\nsamples were transferred in-situ into the chamber with\nEffusion cell in a vacuum of ∼10−8mbar for the de-\nposition of C 60. Before the fabrication of heterostruc-\ntures, thin films of CFB, β-W, and C 60were prepared for\nthickness calibration and study of magnetic and electrical\nproperties. The base pressure of the sputtering chamber\nand chamber with Effusion cells were usually maintained\nat∼4×10−8mbar and ∼6×10−9mbar, respectively.\nThe structural characterizations of individual thin films\nand heterostructures were performed by x-ray diffrac-\ntion (XRD), x-ray reflectivity (XRR), and Raman spec-\ntrometer. The magneto-optic Kerr effect (MOKE) based\nmicroscope and superconducting quantum interference\ndevice based vibrating sample magnetometer (SQUID-\nVSM) were employed for the static magnetization char-\nacterization and magnetic domain imaging. The mag-\nnetization dynamics was investigated by a lock-in based\nferromagnetic resonance (FMR) spectrometer manufac-\ntured by NanOsc, Sweden. The heterostructures were\nkept in a flip-chip manner on the co-planner waveguide\n(CPW). The FMR spectra were recorded in the 4-17 GHz\nrange for all the samples. The FMR spectrometer set-up\nis also equipped with an additional nano voltmeter using\nwhich spin-to-charge conversion phenomena of all the de-\nvices were measured via inverse spin Hall effect (ISHE)\nwith 5-22 dBm RF power. The contacts were given at\nthe two opposite ends of 3 mm ×2 mm devices using\nsilver paste to measure the ISHE induced voltage drop\nacross the samples. The details of the ISHE measure-\nment set-up are mentioned elsewhere [25, 26].\nDensity functional theory (DFT)-based electronic\nstructure calculations were performed in the Vienna Ab-\ninitio simulation package (VASP) [27, 28] to understand\nthe interface’s chemical bonding and surface reconstruc-\ntions. The plane wave basis sets expand the valance\nelectronic states, and the core electrons are treated\nwith the pseudopotentials. The core-valance interac-\ntions are considered with the Projected Augmented Wave\nmethod. The exchange-correlation potentials are treated\nwith Perdew, Bruke and Ernzerof (PBE) [29] functional\nwhich inherits the Generalized Gradient Approximation3\nTABLE I. Details of the heterostructures and their nomenclatures\nSl. No. Stacking Nomenclature\n1 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (2.5 nm) CFW1\n2 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (5 nm) CFW2\n3 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (2.5 nm)/C 60(13 nm) CFWC1\n4 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (5 nm)/C 60(13 nm) CFWC2\n(GGA). This functional produces a reliable understand-\ning of similar kinds of interfaces. The convergences in the\nself-consistent field iterations were ensured with a plane-\nwave cutoff energy of 500 eV and a tolerance of 10−6\neV/cycle. A D3 dispersion correction term, devised by\nGrimme, accounts for the long-range interaction terms\nwas employed in the calculations. The optimized unit\ncell parameter obtained from the aforementioned meth-\nods for the cubic A15 crystal of the β-W is 5.014 ˚A,\nwhich resembles the experimental parameter of 5.036 ˚A.\nA 5×2×1 repetition is used to construct the (210) surface\nunit cell of the β-W to model the surface supercell. The\nlower two atomic layers were fixed at the bulk, and the\nremaining three layers were allowed to relax during the\ngeometry optimization. The surface layer of β-W con-\ntains the C 60molecules. To understand the effect of the\nspin-orbit coupling interactions, we have performed the\nnon-collinear DFT calculations as implemented in VASP.\nThe E SOC calculated from these calculations quantifies\nthe strength of the SOC term in the Hamiltonian.\nIII. RESULTS AND DISCUSSION\nThe grazing incidence x-ray diffraction (GIXRD) was\nperformed for all the heterostructures. The XRD pat-\nterns of CFB/ β-W heterostructures with different thick-\nnesses of β-W are shown in Figure 1 (c). The presence of\n(200), (210) and (211) Bragg’s peaks of W at 35.5◦, 39.8◦\nand 43.5◦indicate the stabilization of metastable βphase\nof W (A-15 crystal structure) [5, 30]. In addition, we\nhave also observed the (320) and (321) Bragg’s peaks of\nW, which further suggests the growth of polycrystalline\nβ-W. The relative intensity of (320) and (321) Bragg’s\npeaks of W is lower compared to (200), (210) and (211)\nBragg’s peaks, consistent with previous reports [30]. The\nBragg’s peaks are more prominent for heterostructures\nwith thicker W layers as diffraction intensity increases\nwith the increase in W thickness. The XRD patterns for\nCFWC1 and CFWC2 are similar to that for CFW1 and\nCFW2, respectively, as the thickness of β-W are same.\nHere, we have not used the reactive gases like, O 2and\nN2for the growth of β-W unlike some previous report\n[11]. The resistivity of W films with thicknesses 2.5, 5,\n10 nm were measured by standard four probe methods.\nThe resistivity decreases with increase in thickness of W\nand were found in between ∼300-100 µΩ-cm, further\nconfirming the growth of βphase of W [5, 30]. We donot also observe the (110), (200), (210) Bragg’s peaks for\nthe bcc α-W in the XRD patterns and the α-W would\nhave also exhibited one order less resistivity compared to\nwhat we have observed [30]. The stabilization of pure β\nphase of W is quite important for future SOT device fab-\nrication and hence, we can expect a high spin-to-charge\nconversion efficiency in our CFB/ β-W heterostructures\nowing to high SOC of β-W [10, 30].\nThe XRR measurements were performed for all the\nsamples in both the CFW and CFWC series to confirm\nthe desired thickness of individual layers and to investi-\ngate the interface quality. Figure S1 (Supporting Infor-\nmation) shows the XRR patterns of all the heterostruc-\ntures considered for the present study. The experimental\ndata were fitted using GenX software and the simulated\npatterns are also shown in Figure S1 (red curves). The\npresence of Kiessig oscillations for all the films infer the\nabsence of a high degree of interfacial disorder and dis-\nlocations. The relative peak positions and intensity of\nthe simulated patterns agree quite well with the experi-\nmentally observed low angel XRR data. The fit provides\nthe anticipated thickness of individual layer in each sam-\nple as mentioned in Table I. The interface roughness for\nall the heterostructures were found in between 0.2-0.5\nnm, further inferring the high quality growth of both the\nseries of samples. Figure S2 (Supporting Information)\ndisplays the Raman spectra of 13 nm C 60film grown on\nSi/SiO 2(300 nm) substrate with the same growth con-\ndition as in the heterostructures. The presence of A g(2)\nand H g(8) Raman modes of C 60around ∼1460 cm−1\nand 1566 cm−1, respectively confirms the growth of C 60\nfilm [31, 32]. In addition, the Raman mode around ∼\n495 cm−1corresponding to A g(1) mode of C 60is also ob-\nserved in the Raman spectrum. The anticipated thick-\nness of C 60in the C 60thin film, CFWC1 and CFWC2\nhas also been confirmed from the XRR measurements.\nThe Raman spectrum of our C 60film grown by effusion\ncells are quite similar to those prepared by different so-\nlution methods in HCl or N 2atmosphere [31, 32]. The\nsaturation magnetization and the magnetic domain im-\nages of all the heterostructures are found to be similar\n(see Supporting Information) as the bottom CFB layer\nis same for all the heterostructures.\nThe magnetization relaxation and propagation of spin\nangular momentum in the CFB thin film and the het-\nerostructures in both the CFW and CFWC series were\nstudied to explore the effect of high resistive β-W and\nβ-W/C 60bilayer by in-plane FMR technique. The het-4\nFIG. 1. Schematics of (a) Si/SiO 2/CFB/ β-W and (b) Si/SiO 2/CFB/ β-W/C 60heterostructures, (c) GIXRD patterns of\nSi/SiO 2/CFB/ β-W heterostructures with various thicknesses of β-W.\nerostructures are placed in a flip-chip manner on CPW\nas shown in the schematics in Figure S4 (a) (Support-\ning Information). Figure S4 (c) shows the typical FMR\nspectra of CFW1 and CFWC1 heterostructures measuredin the 4-17 GHz range. All the FMR spectra were fit-\nted to the derivative of symmetric and antisymmetric\nLorentzian function to evaluate the resonance field ( Hres)\nand linewidth (∆ H) [33]:\nFMRSignal =K14(∆H)(H−Hres)\n[(∆H)2+ 4(H−Hres)2]2−K2(∆H)2−4(H−Hres)2\n[(∆H)2+ 4(H−Hres)2]2+Offset, (1)\nwhere K 1and K 2are the antisymmetric and symmetric\nabsorption coefficients, respectively. The extracted Hres\nand ∆ Hvalues at different resonance frequencies ( f) of\nall the heterostructures are shown in Figure 2 (a-b). The\nfvsHresof different samples in the CFW and CFWC\nseries are plotted in Figure 2 (a). The fvsHresplots\nare fitted by using equation 2 [33]:\nf=γ\n2πq\n(HK+Hres)(HK+Hres+ 4πMeff),(2)\nwhere\n4πMeff= 4πMS+2KS\nMStFM\nand H K, KS, and t FMare the anisotropy field, perpen-\ndicular surface anisotropy constant, and the thicknessof FM, respectively. Here, γis the gyromagnetic ra-\ntio and 4 πMeffrepresents the effective magnetization.\nThe 4 πMeffextracted from the fitting gives similar val-\nues as compared with the saturation magnetization value\n(4πMS) calculated from the SQUID-VSM. Further, the\neffective Gilbert damping constant ( αeff) and hence, the\nmagnetization relaxation mechanism are studied from\nthe resonance frequency dependent FMR linewidth be-\nhavior. The ∆ Hvsfplots are shown in Figure 2 (b).\nThe linear dependency of ∆ Honfindicates the mag-\nnetic damping is mainly governed by intrinsic mechanism\nvia electron-magnon scattering rather than the extrinsic\ntwo magnon scattering. The ∆ Hvsfplots are fitted by5\nFIG. 2. (a) Frequency ( f) versus resonance field ( Hres) and (b) linewidth (∆ H) versus frequency ( f) behaviour for various\nheterostructures. The solid lines are the best fits to equation 2 and 3.\nthe following linear equation [33] to evaluate the αeff.\n∆H= ∆H0+4παeff\nγf, (3)\nwhere the ∆ H0is the inhomogeneous linewidth broad-\nening. The αeffvalues for all the heterostructures and\nCFB thin film obtained from the fitting are shown in Ta-\nble II. The αeffvalue for CFW series ( ∼0.0075 ±0.0001\nfor CFW1 and ∼0.0080 ±0.0001 for CFW2) are found\nto be larger compared to that of the CFB thin film\n(∼0.0059 ±0.0001). The enhancement of αeffindicates\nthe possible evolution of spin pumping mechanism in the\nCFB/ β-W bilayers. Interestingly, the αeffdecreases to\n∼0.0065 ±0.0001 upon the deposition of C 60molecules\non CFB/ β-W bilayers in CFWC series. The signifi-\ncant change in αefffor the CFB/ β-W/C 60heterostruc-\ntures compared to CFB/ β-W bilayers infers the modi-\nfication of physical properties of β-W layer in CFB/ β-\nW/C 60. The deposition of C 60molecules can lead to the\nmetal/molecule hybridization at the β-W/C 60interface,\nwhich in turn can alter the properties of β-W.\nThe DFT based first principle calculations were per-\nformed to elucidate further the molecular hybridization\nat the β-W/C 60interface and its consequences on the\nmagnetization dynamics of CFB/ β-W/C 60heterostruc-\ntures. The extended simulation supercell for the C 60on\nβ-W(210) are shown in Figure 3 (a). The C 60molecule\nis observed as strongly chemisorbed onto the β-W (210)\nsurface with an adsorption energy of -253.5 kcal/mol.\nThe adsorption energy is quite high as compared to the\nother substrates. For example, the adsorption energy for\nCo/C 60was found to be -90 kcal/mol [34] while for the\nPt/C 60interface it is reported to be -115 kcal/mol [35].\nThe chemisorption in case of β-W/C 60is quite strong\nand induces distortion to the spherical shape of the ad-\nsorbed C 60. The distance between two carbon atoms\nfrom two opposite hexagons of adsorbed C 60is shorter\nalong one direction compared to the other measured in\nthe plane (left panel of Figure 3 (a)). The diameter of C 60molecules decreases by 0.3 ˚Awhen it is measured perpen-\ndicular to the β-W (210) surface (right panel of Figure\n3 (a)). This distortion can be attributed to the W-C\nbond formation due to the strong chemisorption at the\nβ-W/C 60interface. This chemisorption strongly alters\nthe electronic structure of the β-W and C 60molecule\n(Figure 3 (b)). The pzorbital, which accommodates the\nπ-electrons of the C 60, hybridizes with the d-orbitals of\ntheβ-W atom and forms the hybridised interfacial states.\nThe out-of-plane d-orbitals ( dxz,dyzanddz2orbitals) are\nstrongly hybridized with the pzorbital of the carbon\natom over a large energy window near the Fermi energy\nlevel (Figure 3 and Figure S5 (Supporting Information)).\nThe sharp peaks observed in the DOS of free C 60layer\ngets significantly broadened, flattened, and shifted for β-\nW/C 60stacking. The strong metallo-organic hybridiza-\ntion also modifies the PDOS of various d-orbitals of β-\nW. The various d-orbitals become flattened and spread\nover larger energy spectrum around the Fermi level upon\nmolecular hybridization. The formation of the W-C bond\nalso costs a transfer of 3.25e−from the interfacial layer of\ntheβ-W to C 60molecule (Figure 3 (c)). This is relatively\nhigher compared to the previously reported the 0.25e−\ntransfer from Pt (111) and 3e−transfer from Cu (111) to\nthe adjacent C 60molecule, inferring the metallo-organic\nhybridization is quite stronger in case of β-W/C 60in-\nterface [35].Hence, the molecular hybridization of β-W\nis expected to alter its physical properties with greater\neffect and can be considered as an important tool to op-\ntimize the spintronics device performances.\nThe modified electronic structure was found to carry\na long-range effect on the strength of the spin-orbit cou-\npling. The E SOC of bare 2.5 nm β-W and 2.5 nm β-\nW covered with C 60molecules, and the variation of the\nESOC(∆E SOC) due to β-W/C 60hybridization are shown\nin Figure 4. The interfacial W atoms involved in the hy-\nbridization with C 60show a decrease in the E SOC. The\nrest of the W atoms from the surface layer exhibit an\nincrease in the E SOC. The lower atomic layers of W6\nFIG. 3. (a) The extended simulation supercell for the C 60onβ-W(210) substrate. The left panel shows the top view of the\nsurface supercell (along the z-axis), and the right panel shows the side view of the same. The pink balls of larger size and cyan\nballs of smaller size represent the tungsten and carbon atoms, respectively. The yellow bonds highlight the part of the C 60\nwhich takes part in the interface formation. The double-headed dotted arrows quantify the diameter of the C 60spheres in two\ndirections. (b-c) The modification in the electronic structure due to chemisorption of the C 60molecule on the β-W. (b) The\natom projected orbital resolved partial density of states of β-W(210), C 60, and β-W(210)/C 60, and (c) The electron density\nredistribution due to chemisorption. The red and green iso-surfaces depict electron density depletion and accumulation of the\nelectron density at the interface, respectively. The bi-coloured arrow depicts the direction of the electron transfer process.\nalso show an increment in the E SOC. The W layer, far-\nthest from the β-W/C 60interface (nearer to the CFB/ β-\nW interface), exhibits the most increased E SOC. Hence,\nthe hybridization at the β-W/C 60interface increases the\noverall spin-orbit coupling strength of the β-W layer.\nMore importantly, the SOC at the CFB/ β-W interface\nis enhanced for CFB/ β-W/C 60stacking compared to the\nCFB/ β-W bilayer. The enhanced bulk SOC of β-W and\nthe interfacial SOC at CFB/ β-W interface can facilitate\nan efficient spin to charge conversion in CFB/ β-W/C 60\nheterostructures.\nThe decrease in damping, usually know as anti-\ndamping, has been observed previously in FM/HM bilay-ers [22, 26, 30]. In those systems, the effective damping\nvalues become lower than the αeffof the FM layer and\nthis phenomenon has been attributed to the formation\nof Rashba like interfacial states [22, 30]. Similar type of\nevolution of Rashba like states at the CFB/ β-W inter-\nface can be expected due to structural inversion asym-\nmetry and large SOC of β-W. The spin accumulation\nat the CFB/ β-W interface can lead to evolution of the\nnon-equilibrium spin states. The non-equilibrium spin\nstates along with the enhanced SOC at CFB/ β-W inter-\nface due to molecular hybridization as confirmed from the\nDFT calculations can generate an additional charge cur-\nrent due to IREE and can also induce the antidamping7\nFIG. 4. The effect of the chemisorption of the C 60molecule at the β-W(210) surface on the E SOCof various atomic sites. The\npercentage change in the E SOC (∆E SOC) is calculated in terms of the change in the E SOC of the bare β-W(210) substrate.\nLayer 5 is the interfacial layer that interacts with the C 60, and layer 1 is the opposite to the β-W/C 60interface layer.\ntorque on the magnetization of FM layer. The antidamp-\ning torque can make the magnetization precession rela-\ntively slower and thus decreasing the αeffof the CFB/ β-\nW/C 60heterostructures compared to the CFB/ β-W bi-\nlayer. The control of Gilbert damping of FMs by inter-\nfacing with adjacent non-magnetic metal/organic bilay-\ners can also provide an alternative to the search for low\ndamping magnetic materials. Especially, the low cost and\nabundant availability of carbon based organic molecules\ncan be commercially beneficial in optimizing the mag-\nnetic damping for spintronic applications. Further, the\nGilbert damping modulation can also control the effec-\ntive spin mixing conductance ( g(↑↓)\neff) of the heterostruc-\ntures which also plays a vital role for efficient spin current\ntransport across the interface. Hence, the g(↑↓)\neffof all the\nheterostructures was calculated from the damping con-\nstant measurement by equation 4 [33]:\ng(↑↓)\neff=4πMstCFB\ngµB(αCFB/NM −αCFB), (4)\nwhere g, µBandtCFB are the Land´ e g factor (2.1),\nBohr’s magnetron, and thickness of CFB layer, respec-\ntively. αCFB/NM is the damping constant of bilayer ortri-layers and αCFB is the damping constant of the ref-\nerence CFB thin film. The g(↑↓)\nefffor CFW1 and CFW2\n(Table II) are relatively higher compared to the previ-\nous reports on FM/ β-W bilayers. Especially, the g(↑↓)\neffof\nCFW2 is one order higher than that reported for Py/ β-W\nbilayer (1.63 ×1018m−2) [30], and 2 order higher com-\npared to that of the YIG/ β-W (5.98 ×1017m−2) [14].\nThis indicates the absence of any significant amount of\nspin back flow from β-W layer and high SOC strength\nof parent β-W layer in our system. However, the g(↑↓)\neff\nvalues decrease for the CFWC1 and CFWC2 tri-layers\nowing to anti-damping phenomena.\nThe ISHE measurements were performed for all the\nheterostructures in CFW and CFWC series to gain more\ninsights about the effect of molecular hybridization in\nCFB/ β-W/C 60on the magnetization dynamics and spin\nto charge conversion efficiency. Figure 5 shows the typi-\ncal field dependent DC voltage ( Vdc) measured across the\nCFB (7 nm)/ β-W (5 nm)/C 60(13 nm) heterostructure\nunder FMR conditions. In order to separate the symmet-\nric (VSY M) and asymmetric ( VASY M ) components, the\nVdcvsHplots were fitted with the following Lorentzian\nfunction:\nVdc=VSY M(∆H)2\n(∆H)2+ (H−Hres)2+VASY M(∆H)(H−Hres)\n(∆H)2+ (H−Hres)2(5)\nThe extracted field dependent VSY M andVASY M are also plotted in Figure 5. Similar type of field depen-8\nFIG. 5. VMEAS ,VSY M andVASY M versus Hfor CFB (7)/ β-W(5)/C 60(13) [CFWC2] heterostructure with ϕ∼(a) 180◦and\n(b) 0◦measured at 15 dBm RF power. The red curve is Lorentzian fit with equation 5 to VdcvsHplot.\nTABLE II. Effective Gilbert damping, spin mixing conduc-\ntance, and symmetric component of measured DC voltage for\ndifferent heterostructures.\nHeterostructures αeff(±0.0001) g(↑↓)\neff(1019m−2)VSY M(µV)\nCFB 0.0059 - -\nCFW1 0.0075 0.87 1.08\nCFW2 0.0080 1.13 1.25\nCFWC1 0.0064 0.27 2.32\nCFWC2 0.0065 0.32 1.78\ndent VMEAS ,VSY M, and VASY M are also observed for\nother samples in both CFW and CFWC series. The\nVSY M is mainly contributed by the spin pumping voltage\n(VISHE ) and the spin rectification effects arising from\nthe anisotropic magnetoresistance (AMR) [ VAMR] [33].\nWhereas, the asymmetric component of the measured\nvoltage arises solely due to anomalous Hall effect and\nAMR [33]. The sign of VSY M is reversed when ϕ(angel\nbetween the perpendicular direction to the applied mag-\nnetic field ( H) and direction of voltage measurement) is\nchanged from 0◦to 180◦(Figure 5), confirming the pres-\nence of ISHE in our heterostructures. The field depen-\ndent VSY M for all the four heterostructures are plotted\nin Figure 6 (a-b). Interestingly, the VSY M value at the\nresonance field for CFB/ β-W/C 60trilayers is found to be\nincreased compared to that for CFB/ β-W bilayers. Theincrement is ∼115% for β-W thickness 2.5 nm, while it\nbecomes ∼20% for β-W thickness 5 nm. The gigantic\nenhancement of VSY M for CFB (7)/ β-W(2.5)/C 60(13)\ninfers the modification of SOC of β-W when capped with\norganic C 60molecules and the presence of an additional\nspin to charge conversion effect in the heterostructures.\nThe power dependent spin-to-charge conversion measure-\nments were also performed to further confirm the en-\nhancement of VSY M. The spin pumping induced voltage\nincreases linearly with the RF power as shown in Fig-\nure 6 (c) for both CFW1 and CFWC1. The VSY M at\ndifferent RF power is found to be increased for CFWC1\ncompared to CFW1, which further confirms the molec-\nular hybridization induced enhanced spin-to-charge con-\nversion. As the thickness, magnetic properties of bot-\ntom CFB layer is same for all the heterostructures, the\ncontribution of VAMR is expected to be same for CFB\n(7 nm)/ β-W(2.5 nm)/C 60(13 nm) and CFB (7 nm)/ β-\nW(2.5 nm). Hence, the sizable increase in the measured\nvoltage can be attributed to the enhanced SOC of β-W\ndue to molecular hybridization and additional charge cur-\nrent flowing at the CFB/ β-W interface due to IREE as\nshown in the Figure 6 (d). In order to understand the en-\nhanced spin-to-charge conversion phenomena further, we\nalso calculated the θSHof the heterostructures by using\nequations 6 and 7 [14, 33]:\nJs=g(↑↓)\neffγ2h2\nrfℏ[γ4πMs+p\n(γ4πMs)2+ 4ω2]\n8πα2\neff[(γ4πMs)2+ 4ω2]×(2e\nℏ), (6)9\nFIG. 6. VSY M versus applied magnetic field with ϕ∼180◦for (a) CFB (7)/ β-W(2.5) [CFW1] and CFB (7)/ β-W(2.5)/C 60\n(13) [CFWC1] and (b) CFB (7)/ β-W(5) [CFW2] and CFB (7)/ β-W(5)/C 60(13) [CFWC2] heterostructures measured at 15\ndBm RF power, (c) Power dependent VSY M for CFW1 and CFWC1 (The solid line is the linear fit), (d) Schematic showing\nthe spin-to-charge conversion phenomena in CFB/ β-W/C 60heterostructures.\nVISHE =wyLρNM\ntNMθSHλNMtanh(tNM\n2λNM)Js, (7)\nwhere the ρNMis the resistivity of the β-W measured by\nfour-probe technique and Lis the length of sample. The\nRF field ( hrf) and the width of the CPW transmission\nline ( wy) in our measurements are 0.5 Oe (at 15 dBm RF\npower) and 200 µm, respectively. The λNMfor the β-W\nhas been taken as ∼3 nm from the literature [36]. Angel\ndependent ISHE measurements were performed to sepa-\nrate the AMR contribution from the VSY M. The contri-\nbution of VAMR was found to be one order smaller com-\npared to V ISHE . For example, the VAMR and V ISHE for\nCFW2 heterostructure are found to be ∼0.15µV and ∼\n1.25µV, respectively (See Supporting Information). The\nρNMfor 5 nm β-W is found to be 250 µΩ cm. Hence, the\nθSHfor CFB (7 nm)/ β-W (5 nm) bilayer estimated using\nequations 6 and 7 is found to be ∼-0.6±0.01. A similar\ntype of calculation for CFB (7 nm)/ β-W (2.5 nm) bilayer\nestimates the θSHto be∼-0.67±0.01. The observed θSHvalue is larger compared to that reported in the literature\n[10–12]. The high SOC of our β-W and higher spin mix-\ning conductance could be responsible for this enhanced\nθSH. Further, the interfacial SOC at CFB/ β-W interface\ncan also induce an additive spin-to-charge conversion ef-\nfect, contributing to the enhancement of θSH. Such type\nof interfacial SOC mediated enhanced spin-to-charge con-\nversion has been reported previously for NiFe/Pt and\nCFB/ β-Ta [19, 20]. Here, it is important to note that\nit is difficult to disentangle the IREE and ISHE effect in\nthese type of FM/HM systems. On the other hand, the\ng(↑↓)\nefffor CFWC1 and CFWC2 decreases by 70 % due to\nthe anti-damping phenomena and hence, the reduction\ninJsaccording to equation 6. However, the VISHE for\nthe CFWC1 and CFWC2 are found to be larger than\nCFW1 and CFW2, respectively (Figure 6). This leads to\ntheθSHvalue >1, calculated using the equation 6 and10\n7 for CFB/ β-W/C 60heterostructures. This type of gi-\ngantic enhancement of θSHcannot be explained by mere\nbulk ISHE in β-W. The enhanced θSHcan be partly at-\ntributed to the enhanced bulk SOC of β-W upon molec-\nular hybridization as predicted by the DFT calculations.\nFurther, our DFT calculations also predict the enhance-\nment of SOC of β-W layer closer to the CFB/ β-W inter-\nface due to the molecular hybridization in the CFB/ β-\nW/C 60heterostructures. The larger interfacial SOC and\ninversion symmetry breaking at the CFB/ β-W interface\nmakes the scenario favorable for realizing an enhanced\ninterfacial charge current due to the IREE as depicted in\nFigure 6 (d). Hence, the combination of bulk and interfa-\ncial SOC enhancement owing to the strong chemisorption\nof C 60onβ-W can attribute to the sizable increase in the\nθSHin CFB/ β-W/C 60heterostructures.\nThe enhanced output DC voltage due to the spin\npumping upon the C 60deposition on β-W is also con-\nsistent with the reduced effective damping value as dis-\ncussed earlier. The enhanced SOC of β-W and the struc-\ntural inversion asymmetry at the CFB/ β-W interface\ncan stabilize the Rashba like states at FM/HM inter-\nface [19, 20]. The IREE mediated spin to charge con-\nversion has received considerable interest after it was\ndiscovered at the Ag/Bi interface [7]. Till the date,\nmost of the IREE effects have been experimentally re-\nalized at the all inorganic metal/metal, metal/oxide or\noxide/oxide interfaces [9]. Our experiments and theoret-\nical calculations show that the molecular hybridization\nat the HM/OSC interface can also help in strengthen-\ning the Rashba spin-orbit coupling at the FM/HM in-\nterface. The Rashba interaction leads to the spin split-\nting of bands, whose magnitude is dependent on the SOC\nstrength at the interface. Upon the molecular hybridiza-\ntion, the SOC strength of β-W is further enhanced. This\ncould have lead for a larger Rashba coefficient αRand\nhence, a relatively larger IREE at the FM/HM inter-\nface. The simultaneous observation of ISHE and IREE\nby engineering the HM interface with OSC can help in\nreducing the power consumption of future SOT-MRAM\ndevices. As the CFB/ β-W stacking is employed for fab-\nrication of spin Hall nano oscillators (SHNOs) [37], theincorporation organic molecules can also significantly en-\nhance their efficiency. Hence, the HM/C 60interface can\nreduce the power consumption for data storage as well as\nfacilitate in performing efficient spin logic operations.\nIV. CONCLUSION\nIn conclusion, we present that a strong interfacial SOC\ncan lead to the larger spin Hall angle in CFB/ β-W bi-\nlayer. The thermally evaporated organic C 60molecules\non CFB/ β-W bilayer leads to a strong chemisorption at\ntheβ-W/C 60interface. The experimental and theoreti-\ncal calculations confirm that the molecular hybridization\nenhances the bulk as well as interfacial SOC in CFB/ β-\nW/C 60heterostructures. The strengthening of techno-\nlogically important SOC manifests an anti-damping phe-\nnomena and gigantic ∼115% increase in spin-pumping\ninduced output voltage for CFB/ β-W/C 60stacking. The\ncontrol of magnetization dynamics and output efficiency\nin spintronics devices by the molecular hybridization can\nbe a viable alternative to the other interface engineering\nand surface alloying techniques. The stabilization of the\nanti-damping and enhanced spin-to-charge conversion by\ntuning the bulk as well interfacial SOC via employing\nthe cost effective, abundant organic molecule can pave\nthe way for the fabrication of next generation power effi-\ncient spintronics devices.\nV. ACKNOWLEDGEMENT\nWe acknowledge the Department of Atomic En-\nergy (DAE), the Department of Science and Technol-\nogy (DST) of the Government of India, and SERB\nproject CRG/2021/001245. A.S. acknowledges the DST-\nNational Postdoctoral Fellowship in Nano Science and\nTechnology. 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Krishnamurthy1,\nSatadeep Bhattacharjee2,∗and Manish Jain1†\n1Centre for Condensed Matter Theory,\nDepartment of Physics,\nIndian Institute of Science,\nBangalore 560012, India\n2Indo-Korea Science and Technology Center,\nBangalore 560065, India\n(Dated: January 2, 2024)\n1arXiv:2401.00714v1  [cond-mat.mtrl-sci]  1 Jan 2024Abstract\nMagnetization dynamics in magnetic materials are well described by the modified semiclassical\nLandau-Lifshitz-Gilbert (LLG) equation, which includes the magnetic damping ˆαand the magnetic\nmoment of inertia ˆItensors as key parameters. Both parameters are material-specific and physi-\ncally represent the time scales of damping of precession and nutation in magnetization dynamics.\nˆαandˆIcan be calculated quantum mechanically within the framework of the torque-torque corre-\nlation model. The quantities required for the calculation are torque matrix elements, the real and\nimaginary parts of the Green’s function and its derivatives. Here, we calculate these parameters\nfor the elemental magnets such as Fe, Co and Ni in an ab initio framework using density functional\ntheory and Wannier functions. We also propose a method to calculate the torque matrix elements\nwithin the Wannier framework. We demonstrate the effectiveness of the method by comparing it\nwith the experiments and the previous ab initio and empirical studies and show its potential to\nimprove our understanding of spin dynamics and to facilitate the design of spintronic devices.\nI. INTRODUCTION\nIn recent years, the study of spin dynamics[1–5] in magnetic materials has garnered\nsignificant attention due to its potential applications in spintronic devices and magnetic\nstorage technologies[6–9]. Understanding the behaviour of magnetic moments and their\ninteractions with external perturbations is crucial for the development of efficient and reliable\nspin-based devices. Among the various parameters characterizing this dynamics, Gilbert\ndamping[10] and magnetic moment of inertia play pivotal roles. The fundamental semi-\nclassical equation describing the magnetisation dynamics using these two crucial parameters\nis the Landau-Lifshitz-Gilbert (LLG) equation[11, 12], given by:\n∂M\n∂t=M× \n−γH+ˆα\nM∂M\n∂t+ˆI\nM∂2M\n∂t2!\n(1)\nwhere Mis the magnetisation, His the effective magnetic field including both external\nand internal fields, ˆαandˆIare the Gilbert damping and moment of inertia tensors with\nthe tensor components defined as αµνand Iµν, respectively, and γis the gyromagnetic\n∗s.bhattacharjee@ikst.res.in\n†mjain@iisc.ac.in\n2ratio. Gilbert damping, ˆαis a fundamental parameter that describes the dissipation of\nenergy during the precession of magnetic moments in response to the external magnetic\nfield. Accurate determination of Gilbert damping is essential for predicting the dynamic\nbehaviour of magnetic materials and optimizing their performance in spintronic devices. On\nthe other hand, the magnetic moment of inertia, ˆIrepresents the resistance of a magnetic\nmoment to changes in its orientation. It governs the response time of magnetic moments to\nexternal stimuli and influences their ability to store and transfer information. The moment\nof inertia[13] is the magnetic equivalent of the inertia in classical mechanics[14, 15] and acts\nas the magnetic inertial mass in the LLG equation.\nExperimental investigations of Gilbert damping[16–23] and moment of inertia involve\nvarious techniques, such as ferromagnetic resonance (FMR) spectroscopy[24, 25], spin-torque\nferromagnetic resonance (ST-FMR), and time-resolved magneto-optical Kerr effect (TR-\nMOKE)[26, 27]. Interpreting the results obtained from these techniques in terms of the\nLLG equation provide insights into the dynamical behaviour of magnetic materials and can\nbe used to extract the damping and moment of inertia parameters. In order to explain\nthe experimental observations in terms of more macroscopic theoretical description, various\nstudies[28–34] based on linear response theory and Kambersky theory have been carried out.\nLinear response theory-based studies of Gilbert damping and moment of inertia involve\nperturbing the system and calculating the response of the magnetization to the pertur-\nbation. By analyzing the response, one can extract the damping parameter. Ab initio\ncalculations based on linear response theory[33] can provide valuable insights into the mi-\ncroscopic mechanisms responsible for the damping process. While formal expression for the\nmoment of inertia in terms of Green’s functions have been derived within the Linear response\nframework[11], to the best of our knowledge, there hasn’t been any first principle electronic\nstructure-based calculation for the moment of inertia within this formalism.\nKambersky’s theory[35–37] describes the damping phenomena using a breathing Fermi\nsurface [38] and torque-torque correlation model [39], wherein the spin-orbit coupling acts\nas the perturbation and describes the change in the non-equilibrium population of electronic\nstates with the change in the magnetic moment direction. Gilmore et al.[32, 34] have reported\nthe damping for ferromagnets like Fe, Ni and Co using Kambersky’s theory in the projector\naugmented wave method[40].\nThe damping and magnetic inertia have been derived within the torque-torque correlation\n3model by expanding the effective dissipation field in the first and second-time derivatives\nof magnetisation[29–31]. In this work, the damping and inertia were calculated using the\ncombination of first-principles fully relativistic multiple scattering Korringa–Kohn–Rostoker\n(KKR) method and the tight-binding model for parameterisation[41]. However, there hasn’t\nbeen any full ab intio implementation using density functional theory (DFT) and Wannier\nfunctions to study these magnetic parameters.\nThe expressions for the damping and inertia involves integration over crystal momentum\nkin the first Brillouin zone. Accurate evaluation of the integrals involved required a dense k-\npoint mesh of the order of 106−108points for obtaining converged values. Calculating these\nquantities using full ab initio DFT is hence time-consuming. To overcome this problem,\nhere we propose an alternative. To begin with, the first principles calculations are done on\na coarse kmesh instead of dense kmesh. We then utilize the maximally localised Wannier\nfunctions (MLWFs)[42] for obtaining the interpolated integrands required for the denser k\nmeshes. In this method, the gauge freedom of Bloch wavefunctions is utilised to transform\nthem into a basis of smooth, highly localised Wannier wavefunctions. The required real\nspace quantities like the Hamiltonian and the torque-matrix elements are calculated in the\nWannier basis using Fourier transforms. The integrands of integrals can then be interpolated\non the fine kmesh by an inverse Fourier transform of the maximally localised quantities,\nthereby enabling the accurate calculations of the damping and inertia.\nThe rest of this paper is organized as follows: In Sec. II, we introduce the expressions for\nthe damping and the inertia. We describe the formalism to calculate the two key quantities:\nGreen’s function and torque matrix elements, using the Wannier interpolation. In Sec. III,\nwe describe the computational details and workflow. In Sec. IV, we discuss the results for\nferromagnets like Fe, Co and Ni, and discuss the agreement with the experimental values\nand the previous studies. In Sec. V, we conclude with a short summary and the future\nprospects for the methods we have developed.\nII. THEORETICAL FORMALISM\nFirst, we describe the expressions for Gilbert damping and moment of inertia within the\ntorque-torque correlation model. Then, we provide a brief description of the MLWFs and\nthe corresponding Wannier formalism for the calculation of torque matrix elements and the\n4Green’s function.\nA. Gilbert damping and Moment of inertia within torque-torque correlation\nmodel\nIf we consider the case when there is no external magnetic field, the electronic structure\nof the system can be described by the Hamiltonian,\nH=H0+Hexc+Hso=Hsp+Hso (2)\nThe paramagnetic band structure is described by H0andHexcdescribes the effective lo-\ncal electron-electron interaction, treated within a spin-polarised (sp) local Kohn-Sham\nexchange-correlation (exc) functional approach, which gives rise to the ferromagnetism. Hso\nis the spin-orbit Hamiltonian. As we are dealing with ferromagnetic materials only, we can\nclub the first two terms as Hsp=H0+Hexc. During magnetization dynamics, (when the\nmagnetization precesses), only the spin-orbit energy of a Bloch state |ψnk⟩is affected, where\nnis the band index of the state. The magnetization precesses around an effective field\nHeff=Hint+Hdamp+HI, where Hintis the internal field due to the magnetic anisotropy\nand exchange energies, Hdampis the damping field, and HIis the inertial field, respectively.\nFrom Eqn. (1), we can see that the damping field Hdamp =α\nMγ∂M\n∂t, while HI=I\nMγ∂2M\n∂t2.\nEquating these damping and inertial fields to the effective field corresponding to the change\nin band energies as magnetization processes, we obtain the mathematical description of the\nGilbert damping and inertia. It was proposed by Kambersky [39] that the change of the\nband energies∂εnk\n∂θµ(θ=θˆndefines the vector for the rotation) can be related to torque\noperator or matrix depending on how the Hamiltonian is being viewed Γµ= [σµ,Hso], where\nσµare the Pauli matrices. Eventually, within the so-called torque-torque correlation model,\nthe Gilbert damping tensor can be expressed as follows:\nαµν=g\nmsπZ Z\u0012\n−d f(ϵ)\ndϵ\u0013\nTr[Γµ(IG)(Γν)†(IG)]d3k\n(2π)3dϵ (3)\nHere trace Tris over the band indices, and msis the magnetic moment. Recently, Thonig\net al [30] have extended such an approach to the case of moment of inertia also, where they\n5deduced the moment of inertia tensor components to be,\nIνµ=gℏ\nmsπZ Z\nf(ϵ)Tr[Γν(IG)(Γµ)†∂2\n∂ϵ2(RG)\n+Γν∂2\n∂ϵ2(RG)(Γµ)†(IG)]d3k\n(2π)3dϵ (4)\nHere f(ϵ) is the Fermi function, ( RG) and ( IG) are the real and imaginary parts of Green’s\nfunction G=(ϵ+ιη− H)−1with ηas a broadening parameter, mis the magnetization in\nunits of the Bohr magneton, Γµ= [σµ,Hso] is the µthcomponent of torque matrix element\noperator or matrix, µ=x, y, z .αis a dimensionless parameter, and I has units of time,\nusually of the order of femtoseconds.\nTo obtain the Gilbert damping and moment of inertia tensors from the above two k-\nintegrals calculated as sums of discrete k-meshes, we need a large number of k-points, such\nas around 106, and more than 107, for the converged values of αand I respectively. The\nreason for the large k-point sampling in the first Brillouin zone (BZ) is because Green’s\nfunction becomes more sharply peaked at its poles at the smaller broadening, ηwhich need\nto be used. For I, the number of k-points needed is more than what is needed for αbecause\n∂2RG/∂ϵ2has cubic powers in RGand/or IGas:\n∂2RG\n∂ϵ2= 2\u0002\n(RG)3−RG(IG)2−IGRGIG−(IG)2RG\u0003\n(5)\nWe note that to carry out energy integration in αit is sufficient to consider a limited\nnumber of energy points within a narrow range ( ∼kBT) around the Fermi level. This\nis mainly due to the exponential decay of the derivative of the Fermi function away from\nthe Fermi level. However, the integral for I involves the Fermi function itself and not its\nderivative. Consequently, while the Gilbert damping is associated with the Fermi surface, the\nmoment of inertia is associated with the entire Fermi sea. Therefore, in order to adequately\ncapture both aspects, it is necessary to include energy points between the bottom of the\nvalence band and the Fermi level.\n6B. Wannier Interpolation\n1.Maximally Localised Wannier Functions (MLWFs)\nThe real-space Wannier functions are written as the Fourier transform of Bloch wave-\nfunctions,\n|wnR⟩=v0\n(2π)3Z\nBZdke−ιk.R|ψnk⟩ (6)\nwhere |ψnk⟩are the Bloch wavefunctions obtained by the diagonalisation of the Hamiltonian\nat each kpoint using plane-wave DFT calculations. v0is the volume of the unit cell in the\nreal space.\nIn general, the Wannier functions obtained by Eqn. (6) are not localised. Usually, the\nFourier transforms of smooth functions result in localised functions. But there exists a phase\narbitrariness of eιϕnkin the Bloch functions because of independent diagonalisation at each\nk, which messes up the localisation of the Wannier functions in real space.\nTo mitigate this problem, we use the Marzari-Vanderbilt (MV) localisation procedure[42–\n44] to construct the MLWFs, which are given by,\n|wnR⟩=1\nNX\nqNqX\nm=1e−ιq.RUq\nmn|ψmq⟩ (7)\nwhere Uq\nmnis a (Nq×N) dimensional matrix chosen by disentanglement and Wannierisation\nprocedure. Nare the number of target Wannier functions, and Nqare the original Bloch\nstates at each qon the coarse mesh, from which Nsmooth Bloch states on the fine k-mesh\nare extracted requiring Nq>Nfor all q,Nis the number of uniformly distributed qpoints\nin the BZ. The interpolated wavefunctions on a dense k-mesh, therefore, are given via inverse\nFourier transform as:\n|ψnk⟩=X\nReιk.R|wnR⟩ (8)\nThroughout the manuscript, we use qandkfor coarse and fine meshes in the BZ, respec-\ntively.\n7FIG. 1: The figure shows the schematic of the localisation of the Wannier functions on a\nRgrid. The matrix elements of the quantities like Hamiltonian on the Rgrid are\nexponentially decaying. Therefore, most elements on the Rgrid are zero (shown in blue).\nWe can hence do the summation till a cutoff Rcut(shown in red) to interpolate the\nquantities on a fine kgrid.\n2.Torque Matrix elements\nAs described in the expressions of αµνand Iνµin Eqns. (3) and (4), the µthcomponent of\nthe torque matrix is given by the commutator of µthcomponent of Pauli matrices and spin-\norbit coupling matrix i.e.Γµ= [σµ,Hso]. Physically, we define the spin-orbit coupling (SOC)\nand spin-orbit torque (SOT) as the dot and cross products of orbital angular momentum and\nspin angular momentum operator, respectively, such that Hso=ξℓ.where ξis the coupling\namplitude. Using this definition of Hso, one can show easily that −ι[σ,Hso] = 2ξℓ×which\nrepresents the torque.\nThere have been several studies on how to calculate the spin-orbit coupling using ab\ninitio numerical approach. Shubhayan et al. [45] describe the method to obtain SOC matrix\nelements in the Wannier basis calculated without SO interaction, using an approximation of\nweak SOC in the organic semiconductors considered in their work. Their method involves\nDFT in the atomic orbital basis, wherein the SOC in the Bloch basis can be related to\nthe SOC in the atomic basis. Then, by the basis transformation, they get the SOC in\nthe Wannier basis calculated in the absence of SO interaction. Farzad et al. [46] calculate\nthe SOC by extracting the coupling amplitude from the Hamiltonian in the Wannier basis,\ntreating the Wannier functions as atomic-like orbitals.\n8We present a different approach wherein we can do the DFT calculation in any basis (plane\nwave or atomic orbital). Unlike the previous approaches, we perform two DFT calculations\nand two Wannierisations: one is with spin-orbit interaction and finite magnetisation (SO)\nand the other is spin-polarised without spin-orbit coupling (SP). The spin-orbit Hamiltonian,\nHsocan then be obtained by subtracting the spin-polarized Hamiltonian, Hspfrom the full\nHamiltonian, HasHso=H−H sp. This, however, can only be done if both the Hamiltonians,\nHandHsp, are written in the same basis. We choose to use the corresponding Wannier\nfunctions as a basis. It should however be noted that, when one Wannierises the SO and\nSP wavefunctions, one will get two different Wannier bases. As a result, we can not directly\nsubtract the HandHspin these close but different Wannier bases. In order to do the\nsubtraction, we find the transformation between two Wannier bases , i.e. express one set of\nWannier functions in terms of the other. Subsequently, we can express the matrix elements\nofHandHspin the same basis and hence calculate Hso. In the equations below, the Wannier\nfunctions, the Bloch wavefunctions and the operators defined in the corresponding bases in\nSP and SO calculations are represented with and without the tilde ( ∼) symbol, respectively.\nTheNSO Wannier functions are given by:\n|wnR⟩=1\nNX\nqNqX\nm=1e−ιq.RUq\nmn|ψmq⟩ (9)\nwhere Uq\nmnis a (Nq× N) dimensional matrix. The wavefunctions and Wannier functions\nfrom the SO calculation for a particular qandRare a mixture of up and down spin states\nand are represented as spinors:\n|ψnq⟩=\n|ψ↑\nnq⟩\n|ψ↓\nnq⟩\n |wnR⟩=\n|w↑\nnR⟩\n|w↓\nnR⟩\n (10)\nThe ˜NsSP Wannier functions are given by:\n|˜ws\nnR⟩=1\nNX\nq˜Ns\nqX\nm=1e−ιq.R˜Uqs\nmn|˜ψs\nmq⟩ (11)\nwhere s=↑,↓.˜Uqs\nmnis a ( ˜Ns\nqטNs) dimensional matrix. Since the spinor Hamiltonian\ndoesn’t have off-diagonal terms corresponding to opposite spins in the absence of SOC, the\nwavefunctions will be |˜ψs\nnq⟩=|˜ψnq⟩⊗|s⟩. The combined expression for ˜Uqfor˜N↑+˜N↓=˜N\n9FIG. 2: This figure shows the implementation flow chart of the theoretical formalism\ndescribed in Sec. II\nSP Wannier functions is:\n˜Uq=\n˜Uq↑0\n0˜Uq↓\n (12)\nwhere ˜Uqis˜NqטNdimensional matrix with ˜Nq=˜N↑\nq+˜N↓\nq. Dropping the sindex for\nSP kets results in the following expression for ˜NSP Wannier functions:\n|˜wnR⟩=1\nNX\nq˜NqX\nm=1e−ιq.R˜Uq\nmn|˜ψmq⟩ (13)\n10We now define the matrix of the transformation between SO and SP Wannier bases as:\nTRR′\nmn=⟨˜wmR|wnR′⟩\n=1\nN2X\nqq′˜Nq,Nq′X\np,l=1eι(q.R−q′.R′)˜Uq†\nmp⟨˜ψpq|ψlq′⟩Uq′\nln\n=1\nN2X\nqq′eι(q.R−q′.R′)[˜Uq†Vqq′Uq′]mn (14)\nHereVqq′\npl=⟨˜ψpq|ψlq′⟩. Eqn. (14) is the most general expression to get the transformation\nmatrix. We can reduce this quantity to a much simpler one using the orthogonality of\nwavefunctions of different q. Eqn. (14) hence becomes,\nTRR′\nmn=1\nN2X\nqeιq.(R−R′)[˜Uq†(NVq)Uq]mn\n=1\nNX\nqeιq.(R−R′)[˜Uq†VqUq]mn (15)\nwhere Vq\npl=⟨˜ψpq|ψlq⟩. Using this transformation, we write SP Hamiltonian in SO Wannier\nbases as:\n(Hsp)RR′\nmn=⟨wmR|Hsp|wnR′⟩\n=X\nplR′′R′′′⟨wmR|˜wpR′′⟩\n⟨˜wpR′′|Hsp|˜wlR′′′⟩⟨˜wlR′′′|wnR′⟩\n=X\nplR′′R′′′(T†)RR′′\nmp(˜Hsp)R′′R′′′\npl TR′′′R′\nln (16)\nSince Wannier functions are maximally localised and generally atomic-like, the major con-\ntribution to the overlap TRR′\nmnis for R=R′. Therefore, we can write TRR\nmn=T0\nmn. The\nreason is that it depends on relative R−R′, we can just consider overlaps at R=0. Eqn.\n16 becomes,\n(Hsp)RR′\nmn=X\npl(T†)0\nmp(˜Hsp)RR′\nplT0\nln (17)\nTherefore, we write the Hsoin Wannier basis as,\n(Hso)RR′\nmn=HRR′\nmn−(Hsp)RR′\nmn (18)\n11The torque matrix elements in SO Wannier bases are given by,\n(Γµ)RR′\nmn= (σµHso)RR′\nmn−(Hsoσµ)RR′\nmn (19)\nConsider ( σµHso)RR′\nmnand insert the completeness relation of the Wannier functions, and\nalso neglecting SO matrix elements between the Wannier functions at different sites because\nof their being atomic-like.\n(σµHso)RR′\nmn=P\npR′′(σµ)RR′′\nmp(Hso)R′′R′\npn\n= (σµ)RR′\nmp(Hso)0\npn (20)\n(σµ)RR′\nmpis calculated by the Fourier transform of the spin operator written in the Bloch\nbasis, just like the Hamiltonian.\n(σµ)RR′\nmp=1\nNX\nqe−ιq.(R′−R)\u0002\nUq†(σµ)qUq\u0003\nmp(21)\nWe interpolate the SOT matrix elements on a fine k-mesh as follows:\n(Γµ)k\nmn=X\nR′−Reιk.(R′−R)(σµ)RR′\nmn (22)\nThis yields the torque matrix elements in the Wannier basis. In the subsequent expres-\nsions, WandHsubscripts represent the Wannier and Hamiltonian basis, respectively. In\norder to rotate to the Hamiltonian gauge, which diagonalises the Hamiltonian interpolated\non the fine kmesh using its matrix elements in the Wannier basis.\n(HW)k\nmn=X\nR′−Reιk.(R′−R)HRR′\nmn (23)\n(HH)k\nmn=\u0002\n(Uk)†(HW)kUk\u0003\nmn(24)\nHere Uk(not to be confused with Uq) are the matrices with columns as the eigenvectors of\n(HW)k, and ( HH)k\nmn=ϵmkδmn. We use these matrices to rotate the SOT matrix elements\nin Eqn. (22) to the Hamiltonian basis as:\n(Γµ\nH)k\nmn=\u0002\n(Uk)†(Γµ\nW)kUk\u0003\nmn(25)\n123.Green’s functions\nThe Green’s function at an arbitrary kandϵon a fine k-mesh in the Hamiltonian basis\nis given by:\nGk\nH(ϵ+ιη) = (ϵ+ιη−(HH)k)−1(26)\nwhere ηis a broadening factor and is caused by electron-phonon coupling and is generally\nof the order 5 −10 meV. Gk\nH(ϵ+ιη) is aN × N dimensional matrix.\nTherefore, we can calculate RG,IGand∂2RG/∂ϵ2as defined in Eqn. (5) and hence, α\nand I.\nIII. COMPUTATIONAL DETAILS\nPlane-wave pseudopotential calculations were carried out for the bulk ferromagnetic tran-\nsition metals bcc Fe, hcp Co and fcc Ni using Quantum Espresso package[47, 48]. The\nconventional unit cell lattice constants ( a) used for bcc Fe and fcc Ni were 5.424 and 6.670\nbohrs, respectively and for hcp Co, a=4.738 bohr and c/a=1.623 were used. The non-\ncollinear spin-orbit and spin-polarised calculations were performed using fully relativistic\nnorm-conserving pseudopotentials. The kinetic energy cutoff was set to 80 Ry. Exchange-\ncorrelation effects were treated within the PBE-GGA approximation. The self-consistent\ncalculations were carried out on 16 ×16×16 Monkhorst-Pack Grid using Fermi smearing of\n0.02 Ry. Non-self-consistent calculations were carried out using the calculated charge den-\nsities on Γ-centered 10 ×10×10 coarse k-point grid. For bcc Fe and fcc Ni, 64 bands were\ncalculated and hcp Co 96 bands were calculated (because there are two atoms per unit cell\nfor Co). We define a set of 18 trial orbitals sp3d2,dxy,dxz, and dyzfor Fe, 18 orbitals\nper atom s,panddfor Co and Ni, to generate 18 disentangled spinor maximally-localized\nWannier functions per atom using Wannier90 package [43].\nFrom the Wannier90 calculations, we get the checkpoint file .chk, which contains all the\ninformation about disentanglement and gauge matrices. We use .spnand.eigfiles generated\nbypw2wannier90 to get the spin operator and the Hamiltonian in the Wannier basis. We\nevaluate the SOT matrix elements in the Wannier Basis.\nWe get αby simply summing up on a fine- kgrid with appropriate weights for the k-\nintegration, and we use the trapezoidal rule in the range [-8 δ,8δ] for energy integration\n1310−610−410−2100\n10−610−410−210010−610−410−210010\n10\n10\n10\n10\n101\n0\n1−2\n−2\n−310−410−310−210−1100101\n10−410−310−210−1100101\nFe Ni Co(a) (b) (c)\nFIG. 3: (a), (b) and (c) shows the αvsηfor Fe, Ni and Co, respectively. Damping\nconstants calculated using the Wannier implementation are shown in blue. Damping\ncalculated using the tight binding method based on Lorentzian broadening and Green’s\nfunction by Thonig et al[29] are shown in brown and green, respectively. Comparison with\ndamping constants calculated by Gilmore et al[32] using local spin density approximation\n(LSDA) are shown in red. The dotted lines are guides to the eye.\naround the Fermi level where δis the width of the derivative of Fermi function ∼kBT. We\nconsider 34 energy points in this energy range. We perform the calculation for T= 300K.\nFor the calculation of I, we use a very fine grid of 400 ×400×400k-points. For η >0.1,\nwe use 320 energy points between VBM and Fermi energy. For 0 .01< η < 0.1, we use\n3200-6400 energy points for the energy integration.\nTABLE I\nMaterial η(meV) -I (fs) α(×10−3) τ(ps)\nFe 6 0.210 3.14 0.42\n8 0.114 2.77 0.26\n10 0.069 2.51 0.17\nNi 10 2.655 34.2 0.48\nCo 10 0.061 1.9 0.21\n14FIG. 4: Schematic explaining the dependence of intraband and interband contribution in\nαwith η.\nIV. RESULTS AND DISCUSSION\nA. Damping constant\nIn this section, we report the damping constants calculated for the bulk iron, cobalt and\nnickel. The magnetic moments are oriented in the z-direction. For reference direction z, the\ndamping tensor is diagonal resulting in the effective damping constant α=αxx+αyy.\nIn Fig. 3, we report the damping constants calculated by the Wannier implementation as\na function of broadening, ηknown to be caused by electron-phonon scattering and scattering\nwith impurities. We consider the values of ηranging from 10−6to 2 eV to understand the\nrole of intraband and interband transitions as reported in the previous studies[29, 32]. We\nnote that the experimental range is for the broadening is expected to be much smaller with\nη∼5−10 meV. The results are found to be in very good agreement with the ones calculated\nusing local spin density approximation (LSDA)[32] and tight binding paramterisation[29].\nThe expression for Gilbert damping[3] is written in terms of the imaginary part of Green’s\nfunctions. Using the spectral representation of Green’s function, Ank(ω), we can rewrite Eqn.\n15(3) as:\nαµν=gπ\nmsX\nnmZ\nTµ\nnm(k)T∗ν\nnm(k)Snmdk (27)\nwhere Snm=R\nη(ϵ)Ank(ϵ)Amk(ϵ)dϵis the spectral overlap. Although we are working in the\nbasis where the Hamiltonian is diagonal, the non-zero off-diagonal elements in the torque\nmatrix lead to both intraband ( m=n) and interband ( m̸=n) contributions. For the\nsake of simple physical understanding, we consider the contribution of the spectral overlaps\nat the Fermi level for both intraband and interband transitions in Fig. 4. But in the\nnumerical calculation temperature broadening has also been considered. For the smaller\nη, the contribution of intraband transitions decreases almost linearly with the increase in\nηbecause the overlaps become less peaked. Above a certain η, the interband transitions\nbecome dominant and the contribution due to the overlap of two spectral functions at\ndifferent band indices m and n becomes more pronounced at the Fermi level. Above the\nminimum, the interband contribution increases till η∼1 eV. Because of the finite Wannier\norbitals basis, we have the accurate description of energy bands only within the approximate\nrange of ( ϵF−10, ϵF+ 5) eV for the ferromagnets in consideration. The decreasing trend\nafter η∼1 eV is, therefore, an artifact.\n10−2\n0.000.050.100.150.20\n10−1100-0.001 0.000 0.001 0.002\n10−1100\nFIG. 5: Plot showing moment of inertia, −I versus broadening, η. The moment of inertia\nin the range 0 .03−3.0 eV is shown as an inset. The values using the Wannier\nimplementation and the tight binding method[30] are shown in blue and green, respectively.\n160.020.040.060.08\nCo(a)\n10−210−11000.0010−1100−0.0075−0.0050−0.0050−0.0025 0.000\n10−210−1100−\n0.00.51.0 1.01.5\nNi(b)\n0.000.010.02\n100101\nFIG. 6: (a) and (b) show negative of the moment of inertia, −I versus broadening, ηfor\nCo and Ni, respectively. The values using the Wannier implementation and the tight\nbinding method are shown in magenta and cyan, respectively. The moment of inertia is\nshown as an inset in the range 0 .03−2.0 eV and 0 .03−3.0 eV for Co and Ni, respectively.\nB. Moment of inertia\nIn Fig. 5, we report the values for the moment of inertia calculated for bulk Fe, Co\nand Ni. Analogous to the damping, the inertia tensor is diagonal, resulting in the effective\nmoment of inertia I = Ixx+ Iyy.\nThe behaviour for I vs ηis similar to that of the damping, with smaller and larger ηtrends\narising because of intraband and interband contributions, respectively. The overlap term\nin the moment of inertia is between the ∂2RG/∂ϵ2andIGunlike just IGin the damping.\nIn Ref. [30], the moment of inertia is defined in terms of torque matrix elements and the\noverlap matrix as:\nIµν=−gℏ\nmsX\nnmZ\nTµ\nnm(k)T∗ν\nnm(k)Vnmdk (28)\nwhere Vnmis an overlap function, given byR\nf(ϵ)(Ank(ϵ)Bmk(ϵ) +Bnk(ϵ)Amk(ϵ))dϵand\nBmk(ϵ) is given by 2( ϵ−ϵmk)((ϵ−ϵmk)2−3η2)/((ϵ−ϵmk)2+η2)3. There are other notable\nfeatures different from the damping. In the limit η→0, the overlap Vmnreduces to 2 /(ϵmk−\nϵnk)3. For intraband transitions ( m=n), this leads to I → −∞ . In the limit η→ ∞ ,\nVmn≈1/η5which leads to I →0. The behaviour at these two limits is evident from Fig. 5.\nThe large τ(small η) behaviour is consistent with the expression I = −α.τ/2πderived by\n17104\n−10−310−2−10−1100\n10−2−10−110−1100101102\nFIG. 7: The damping, magnetic moment of inertia and relaxation rate are shown as a\nfunction of broadening, ηin blue, green and red, respectively. The grey-shaded region\nshows the observed experimental relaxation rate, τ, ranging from 0 .12 to 0 .47 ps. The\ncorresponding range of ηis shown in purple and is 6 −12 meV. This agrees with the\nexperimental broadening in the range of 5 −10 meV, arising from electron-phonon\ncoupling. The numbers are tabulated in Table I\nF¨ahnle et al. [49]. Here τis the Bloch relaxation lifetime. The behaviour of τas a function\nofηusing the above expression in the low ηlimit is shown in Fig. 7. Apart from these\nlimits, the sign change has been observed in a certain range of ηfor Fe and Co. This change\nin sign can be explained by the Eqn. (5). In the regime of intraband contribution, at a\ncertain η, the negative and positive terms integrated over ϵandkbecome the same, leading\nto zero inertia. Above that η, the contribution due to the negative terms decreases until the\ninterband contribution plays a major role leading to maxima in I (minima in −I). Interband\ncontribution leads to the sign change from + to - and eventually zero at larger η.\nThe expression I = −α.τ/2πderived from the Kambersky model is valid for η <10 meV,\nwhich indicates that damping and moment of inertia have opposite signs. By analyzing the\nrate of change of magnetic energy, Ref. [11] shows that Gilbert damping and the moment\nof inertia have opposite signs when magnetization dynamics are sufficiently slow (compared\ntoτ).\nExperimentally, the stiffening of FMR frequency is caused by negative inertia. The\nsoftening caused by positive inertia is not observed experimentally. This is because the\nexperimentally realized broadening, ηcaused by electron-phonon scattering and scattering\nwith impurities, is of the order of 5 −10 meV. The values of Bloch relaxation lifetime, τ\n18measured at the room temperature with the FMR in the high-frequency regime for Ni 79Fe21\nand Co films of different thickness, range from 0 .12−0.47 ps. The theoretically calculated\nvalues for Fe,Ni and Co using the Wannier implementation for the ηranging from 5 −10\nmeV are reported in Table. I and lies roughly in the above-mentioned experimental range\nfor the ferromagnetic films.\nV. CONCLUSIONS\nIn summary, this paper presents a numerical method to obtain the Gilbert damping\nand moment of inertia based on the torque-torque correlation model within an ab initio\nWannier framework. We have also described a technique to calculate the spin-orbit coupling\nmatrix elements via the transformation between the spin-orbit and spin-polarised basis. The\ndamping and inertia calculated using this method for the transition metals like Fe, Co and\nNi are in good agreement with the previous studies based on tight binding method[29, 30]\nand local spin density approximation[32]. We have calculated the Bloch relaxation time\nfor the approximate physical range of broadening caused by electron-phonon coupling and\nlattice defects. The Bloch relaxation time is in good agreement with experimentally reported\nvalues using FMR[27]. The calculated damping and moment of inertia can be used to study\nthe magnetisation dynamics in the sub-ps regime. In future studies, we plan to use the\nWannier implementation to study the contribution of spin pumping terms, arising from\nthe spin currents at the interface of ferromagnetic-normal metal bilayer systems due to\nthe spin-orbit coupling and inversion symmetry breaking to the damping. 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He, Two-dimensional\nferromagnetic materials: From materials to devices, Applied Physics Letters 121(2022).\n23"    },    {        "title": "2401.00758v1.Magnon_Damping_Minimum_and_Logarithmic_Scaling_in_a_Kondo_Heisenberg_Model.pdf",        "content": "Magnon Damping Minimum and Logarithmic Scaling in a Kondo-Heisenberg Model\nYuan Gao,1, 2,∗Junsen Wang,3, 2,∗Qiaoyi Li,2, 1Qing-Bo Yan,3, 4,†Tao Shi,2, 4,‡and Wei Li2, 4, 5,§\n1School of Physics, Beihang University, Beijing 100191, China\n2CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China\n3Center of Materials Science and Optoelectronics Engineering,\nCollege of Materials Science and Opto-electronic Technology,\nUniversity of Chinese Academy of Sciences, Beijing 100049, China.\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijng 100190, China\n5Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing 100191, China\n(Dated: January 2, 2024)\nRecently, an anomalous temperature evolution of spin wave excitations has been observed in a van der Waals\nmetallic ferromagnet Fe 3GeTe 2(FGT) [S. Bao, et al. , Phys. Rev. X 12, 011022 (2022)], whose theoretical un-\nderstanding yet remains elusive. Here we study the spin dynamics of a ferromagnetic Kondo-Heisenberg lattice\nmodel at finite temperature, and propose a mechanism of magnon damping that explains the intriguing experi-\nmental results. In particular, we find the magnon damping rate γ(T) firstly decreases as temperature lowers, due\nto the reduced magnon-magnon scatterings. It then reaches a minimum at T∗\nd, and rises up again following a log-\narithmic scaling γ(T)∼ln (T0/T) (with T0a constant) for T<T∗\nd, which can be attributed to electron-magnon\nscatterings of spin-flip type. Moreover, we obtain the phase diagram containing the ferromagnetic and Kondo\ninsulator phases by varying the Kondo coupling, which may be relevant for experiments on pressured FGT. The\npresence of a magnon damping minimum and logarithmic scaling at low temperature indicates the emergence\nof the Kondo e ffect reflected in the collective excitations of local moments in a Kondo lattice system.\nIntroduction.— The van der Waals ferromagnetic (FM)\nmetal Fe 3GeTe 2(FGT) has raised great research interest due\nto its tunable high-temperature FM order [1, 2] and anoma-\nlous Hall e ffect with a topological origin [3], placing it a po-\ntential spintronic material [2, 4, 5]. Besides, FGT also raises\nfundamental research interest due to its intriguing thermody-\nnamic, dynamical, and transport properties [6–12]. In particu-\nlar, FGT has been proposed to exhibit intriguing Kondo-lattice\nbehaviors [6–9], including the enhancement of Fermi surface\nvolume, large e ffective electron mass [7], Fano resonance [7],\nand the presence of Kondo holes in the vicinity of Fe defi-\nciencies [8], etc. These experimental progresses point to the\ndichotomy of locality and itinerancy of the 3 delectrons in this\nFM metal [7, 8, 10, 12].\nVery recently, an anomalous damping of spin wave in FGT\nhas been observed with inelastic neutron scattering (INS)\nmeasurements [10]. The magnon dispersions are found to be\nmore di ffusive at lower temperature (e.g., 4 K) than those at\nhigher values (e.g., 100 K). This is quite unusual for con-\nventional ferromagnets [13, 14], where the magnon excita-\ntions are expected to be more coherent as temperature de-\ncreases. The spontaneous magnon decay has been witnessed\nat low temperature for highly frustrated antiferromagnets [15–\n17], however, for such 2D ferromagnet the INS results with\nmagnon damping remain to be understood. Moreover, latest\nINS experiments reveal a damping minimum and logarithmic\nscaling at low temperature [18], which thus urgently calls for\nsystematic theoretical studies on this 2D van de Waals ferro-\nmagnet.\nTo address the anomalous magnon damping, here we con-\nsider a ferromagnetic Kondo-Heisenberg (FMKH) model for\nFGT [see Eq. (1)] and study the temperature evolution of\nspin dynamics. The FMKH model consists of both itin-erant electrons and local moments, due to the dual nature\nofd-orbital magnetism in the compound. With state-of-\nthe-art tensor renormalization group (TRG) and perturbative\nfield-theoretical calculations, we find the anomalous magnon\ndamping in the FMKH model. In the FM phase of the model,\nitinerant d-electron scatters the collective magnon excitation\nof local moments via the Kondo coupling, which is renormal-\nized in analogous to the original Kondo e ffect [19, 20]. It\ngives rise to the anomalous decay of magnon characterized by\na minimum in damping rate γ(T) atT∗\ndand a logarithmic tem-\nperature dependence γ∼ln(T0/T)forT<T∗\nd. Moreover, in\nthe large Kondo coupling regime ( JK≫1), the conduction\nelectrons can dynamically screen the local moments by form-\ning Kondo singlets, and in such a Kondo insulator (KI) phase\nFM order vanishes while a coherent triplon wave forms. The\nFM and KI phases are separated by a quantum critical point\n(QCP) [21–23] that may be relevant for high-pressure experi-\nments on FGT.\nModel and methods.— The FMKH model considered in the\npresent work reads\nH=−tX\nn,σ\u0010\nc†\nn+1,σcn,σ+H.c.\u0011\n+JHX\nnSn·Sn+1+JKX\nnSn·sn,\n(1)\nwhere c†\nn,σ(cn,σ) is the creation (annihilation) operator of the\nconduction d-electron at site nwith spinσ∈{↑,↓}, and Sn\n(sn) represents the spin operator of local moment (conduction\nelectron) [c.f., Fig. 1(b)].\nIn the numerical simulations, we compute the thermody-\nnamic and dynamical properties of 1D FMKH chain with the\nTRG methods. The equilibrium thermal density matrix and re-\nlated thermodynamic properties can be obtained by the matrix\nproduct operator (MPO) based TRG methods [24–28], whicharXiv:2401.00758v1  [cond-mat.str-el]  1 Jan 20242\nare highly e fficient and accurate for both quantum spin [29–\n35] and fermion many-body systems [28, 30, 36]. Given the\nMPO representation of the thermal density matrix, we gener-\nalize the tangent-space approach for real-time evolution [37–\n39] to the mixed states and study the temperature evolution\nof the spectral density (c.f. Supplemental Sec. I [40]). The\nKondo lattice models have been widely used in the studies\nof heavy-fermion systems [22, 41–43], and here we consider\nthe half-filled case in the present study. The FM coupling\nstrength JH=−1 is taken as the energy unit and the hop-\nping energy is set as t/|JH|=2 throughout the calculations.\nThe intriguing interplay between the local moments and con-\nduction electrons [19, 44, 45] gives rise to a rich phase dia-\ngram. Beyond the 1D Kondo lattices, the 2D FMKH mod-\nels, particularly defined on the triangular lattice, are analyzed\nwith field-theoretical approach [40], where we find consistent\nconclusions. It suggests that the main conclusion of magnon\ndamping minimum and logarithmic low-temperature scaling\nin damping rate is independent of specific lattice geometries\nor spatial dimensions.\nFinite-temperature phase diagram of the FMKH model.—\nWe obtain the finite-temperature phase diagram of 1D FMKH\nmodel [Eq. (1)] with the exponential TRG approach [26, 27,\n46]. The simulations are restricted within the 2-leg ladder\nstructure of size N=L×2 with the length Lup to 36, where\none leg forms a chain of local moments while the other rep-\nresents itinerant electrons [c.f. Fig. 1(b)]. The retained bond\ndimension is D=1600, which guarantees accurate results till\nlow temperature T/|JH|≃0.01, with the data convergence al-\nways checked [40]. At low temperature, the TRG calculations\nare found to converge towards the ground-state density matrix\nrenormalization group (DMRG) results.\nIn Fig. 1(a) we show the obtained phase diagram of the\nFMKH model, with the contour plot of the spin structure fac-\ntorSzz\nk=1\nLP\nm,n⟨Sz\nmSz\nn⟩ei(n−m)katk=0 (i.e., Γpoint) shown\nin the background. In the phase diagram we can recognize the\nexistence of a ferromagnetic (FM, red) and a Kondo insula-\ntor (KI, blue) phase. At zero temperature, the two phases are\nseparated by a quantum critical point (QCP), from which em-\nanating the quantum critical regime at low temperature. The\nQCP located at Jc\nK≃3.3 is determined by the DMRG calcula-\ntions, where we find an algebraic specific heat at low T[40].\nAs temperature elevates, the FM and KI phases evolve into the\nhigh-temperature paramagnetic (PM) regime.\nIn companion with TRG numerics, we also perform mean-\nfield calculations. By introducing the FM order parameter\nmc=\nsz\nn\u000bandmf=\nSz\nn\u000band the Kondo hybridization or-\nder parameter [47–49] V=D\nc†\nn↑fn↑+f†\nn↓cn↓E\n, where fn,σis the\npseudo-fermion operator for the local moment and Vcharac-\nterizes the KI phase, we treat the FM and KI phases on an\nequal footing. The original Hamiltonian is then decoupled\ninto a mean-field one that can be solved self-consistently [40].\nThe resulting phase boundary shown by the black dashed\nline in Fig. 1(a) is found to be in qualitative agreement with\nthe TRG results. This is witnessed by the nice agreement\nbetween the mean-field dashed lines with the scatters ( T∗\nC\n(a)\n(b) (c)12345610-210-1100 T\n00.20.40.60.811.2\nS zz\n \nFM KIPM\nS+\nnS−\nn+1\nc†\nn+1,/uni2191cn,/uni2191S+\nns−\nnMean fieldBO\nFIG. 1. (a) The finite-temperature phase diagram of FMKH model.\nThe contour background is the spin structure factor Szz\nk=Γ. The black\ndashed line is the phase boundary obtained by mean-field calcula-\ntions which separates the paramagnetic (PM), ferromagnetic (FM)\nand Kondo insulator (KI) phase. In the ground state, the FM and KI\nphases are separated by a QCP at Jc\nK/|JH|≈3.3 indicated by the red\ncircle obtained by DMRG, and the mean-field result shows a transi-\ntion point at ˜Jc\nK/|JH|≈2.75. The solid squares ( T∗\nC) and triangles\n(TK) are obtained by TRG. The blue hollow circle of TKis obtained\nfrom the bond operator (BO) theory. T∗\ndin the FM phase represents\nthe characteristic temperature with the damping rate minimum. (b)\nMagnon damping mechanism. The spin flip dissipates into the itiner-\nant electron host through the Kondo coupling. (c) Feynman diagram\nof magnon self-energy due to electron-magnon scattering with the\nrenormalized vertex responsible for the anomalous decay of magnon,\nwhere the double-dashed lines represent electron’s full propagator.\nandTK) determined from the peak location of the tempera-\nture derivatives of local moment correlations −dFnn/dT(with\nFnn=1\nL−1P\nn⟨Sn·Sn+1⟩) and the local Kondo correlations\n−dKnn/dT(with Knn=−1\nLP\nn⟨Sn·sn⟩) [40]. In Fig. 1(a), we\nalso represent the phase boundary determined by bond opera-\ntor theory in KI phase as the blue circle [40, 50–52], where\nqualitative agreements are also observed.\nDamping rate minimum and logarithmic scaling.— Given\nthe MPO representation of the thermal density matrix ρ(β)=\ne−βH(β≡1/T), the dynamical properties at temperature T\ncan be simulated through a successive real-time evolution.\nThe spectral function χ′′(k,ω)≈Rtmax\n0dtIm[g(k,t)] sin (ωt)·\nW(t/tmax) is of central interest in studying the dynamical prop-\nerties at finite temperature, where Wis the window function,\ntmaxis the evolution time, and the (dynamical) correlation\nfunction is g(k,t)=⟨Ak(t)A†\nk⟩β, with Ak=1√\nLPL\nnS+\nneiknthe3\n012\nk/\n0123\n012\nk/\n0123\n012\nk/\n0123\n012\nk/\n0123\n00.511.5\n''\n12300.511.5 ''(k=, )\n0.45\n0.29T = 0.49\n12300.511.5 ''(k=, )\n0.33\n0.094T = 0.12\n12300.511.5 ''(k=, )\n0.36\n0.035T = 0.03\n12300.511.5 ''(k=, )\n0.4\n0.007T = 0.008\n(a) (b) (c) (d)\n(f) (g) (h) (i)(e)\nHigh T Low T\nPM FM\nFIG. 2. (a-d) show the contour plots of spectral density χ′′at various temperatures computed in the N=16×2 system and with a retained bond\ndimension D=1600, and (e) shows the corresponding phase diagram. (f-i) show the comparisons of χ′′(k=π,ω) between the Heisenberg\ncase ( JK=0, calculated by ED) and Kondo-Heisenberg case ( JK=1,D=4800). The fitted damping rate γπis shown beside the peak (blue\nfor Kondo-Heisenberg case and gray for pure Heisenberg case).\nFourier transformed spin operator. Therefore, the problem re-\nsorts to the calculations of time-dependent correlation func-\ntiong(k,t), which we exploit the tangent-space approach to\ncompute [37–39]. In the real-time evolution calculations, we\nconsider a 16×2 lattice and evolution time up to tmax/|JH|=\n20, with D=4800 bond states retained (see the data conver-\ngence in the Supplemental Sec. I C [40]).\nIn Fig. 2(a-d), we show the spectral function χ′′(π,ω) of\nthe FMKH model ( JK=1) where the line width firstly de-\ncreases and reaches its minimum at around T∗\nd, while it be-\ncomes broadened again as Tfurther lowers [c.f., the mini\nphase diagram Fig. 2(e)]. This is in sharp contrast to the pure\nHeisenberg ( JK=0) case where the line width narrows mono-\ntonically as temperature decreases and shows a very clear dis-\npersion at low T[40].\nIt can be seen more clearly by plotting the χ′′vs.ωat\nfixed k=π[c.f., Fig. 2(f-i)], and compare the results to\nthe pure Heisenberg ferromagnet. For the latter case, we\nfind from the χ′′(ω) data that a more coherent peak devel-\nops as temperature lowers (i.e., absence of damping mini-\nmum); while for the JK=1 case Kondo couplings introduce\nhybridization between local moments and the itinerant elec-\ntron. With the spectral density results at k=π, we conduct a\nfitting with damped harmonic oscillator model [53], namely,\nχ′′(k,ω)∝γkωEk\n(ω2−E2\nk)2+(γkω)2∗W(ω) where Ekis the magnon en-\nergy,γkis the damping rate, and W(ω) is the convolution win-\ndow function (c.f. Supplemental Sec. II B [40]). Based on\nthe fittings of spectral density at k=πin Fig. 2(f-h), we col-\nlect the fitted damping rate γπand show the results in Fig. 3,whereγπclearly exhibits a damping minimum at T∗\nd. Re-\nmarkably, the numerical simulations find a logarithmic scal-\ningγπ(T)∼lnT0/Tbelow T∗\nd. Such a damping minimum and\nlogarithmic scaling can be observed in various momentum k\nand for di fferent Kondo couplings JK<Jc\nK, i.e., it is a univer-\nsal behavior due to the scattering between the collective ex-\ncitations (magnons) of local moments and itinerant electrons\n(c.f. Supplemental Sec. II C [40]).\nAs comparisons, we also show the damping rate γπof the\npure Heisenberg model ( JK=0) and that in the gapped KI\nphase with JK>Jc\nK[54]. As shown in Fig. 3, for either the\npure Heisenberg or the KI case, γπdecreases monotonically as\nTlowers and converges to a small value (i.e., long life time).\nIn the KI phase, the local moments and the itinerant electrons\nform a singlet and the triplon wave moves coherently with-\nout scattering the electrons, as verified both by numerics (c.f.,\nSupplemental Sec. II E [40]) and the bond-operator theory\n[50–52] (c.f., Supplemental Sec. V [40]). Overall, the sharp\ncontrasts in Fig. 3 reveal the peculiarity in the magnon damp-\ning minimum and logarithmic scaling, which roots deeply in\nthe Kondo physics.\nField-theoretical analysis of the magnon decay.— We now\nprovide a physical picture to explain this anomalous magnon\ndecay based on a field-theoretical analysis. In the weak-\ncoupling limit, JK≪|JH|,t, the spin part and fermion part can\nbe dealt with separately first. The former, via the Holstein-\nPrimako ffmapping Sz\ni=S−na\niandS+\ni≈√\n2S ai, can\nbe approximated by free magnons; while the latter is sim-\nply the free fermions. In this same representation, the Kondo\ncoupling term, treated as a small perturbation, approximately4\n10-210-1100\nT0.30.350.40.45\n00.20.40.60.81\nM\n~ln(T0/T)M\nFIG. 3. Damping rate γπfor the FMKH model in the FM ( JK=1)\nphase, as compared to that in the KI ( JK=4) phases and in pure\nHeisenberg model ( JK=0). In the FM phase of 1D FMKH model,\nthe damping rate exhibits a minimum at the characteristic tempera-\ntureT∗\nd, and below T∗\ndit shows a logarithmic scaling γπ∼ln(T0/T)\nwith T0≃0.27. The gray solid line ( ΓM) is the damping rate of\ntriangular-lattice FMKH with k=M≡(0,√\n3\n3π),JH=−1,t=2, and\nJK=10 obtained via field-theoretical calculations (c.f. Supplemen-\ntal Sec. III C [40]). In contrast, the spin excitations become more\ncoherent as Tlowers for either the Heisenberg or the KI case. The\nvertical blue dashed line shows the Kondo temperature TK≃0.3 of\nthe KI phase, as estimated in Fig. 1(a).\nreads JKP\nnSn·sn≈JK\n2P\ni(aic†\ni↓ci↑+H.c.) [55]. Thus the\nmagnon self-energy, according to the Dyson equation [56], is\ngiven diagrammatically in Fig. 1(c). In sharp contrast to the\nusual electron-phonon interaction where the vertex correction\nis negligible due to Migdal theorem [57], here the vertex cor-\nrection [shown as a gray area in Fig. 1(c)] due to the Kondo\ncoupling JKis crucial [19]. By incorporating this renormal-\nization e ffect into the calculation of the magnon self-energy, a\nlog(1/T)-like temperature dependence of self-energy at low-\nTis manifested (c.f. Supplemental Sec. III A [40]). In Fig. 3,\nfor triangular-lattice model we incorporate magnon-magnon\ninteractions that leads to a power-law divergence also at high-\nT, and find the magnon damping rate, proportional to the\nimaginary part of the self-energy therefore exhibits a mini-\nmum at an intermediate temperature and a logarithmic scal-\ning at low temperatures [c.f. more details in Supplemental\nSec. III C [40]].\nMoreover, in Fig. 4(a-b), we show imaginary part of the\nrenormalized magnon self-energy (omitting magnon-magnon\ninteractions for simplicity) at relatively high ( T=0.1) and\nlow ( T=0.01) temperatures, where −Im(Σ) increases as\ntemperature lowers. We also plot the free magnon dispersion\nas the gray dash-dotted lines, whose large |k|part is found\nto “merge” into the regime of large −Im(Σ) values. Conse-\nquently, in Fig. 4(c-d) we find the corresponding renormalized\nmagnon spectral density χ′′(ω)=−Im[Gk(iωn→ω+i0+)]\nsuffers heavier damping e ffect near the Brillouin zone bound-\nary. This property, which is in agreement with recent INS ex-\nMK012345\n00.10.20.30.4\n-Im( )\n012345(a)\n(c)\nT=0.1\n012345\nT=0.01\n012345\n0510\n''MK(b)\n(d)\nMK\nkMK\nkT=0.1 T=0.01\nMKFIG. 4. (a, b) the contour plot of the magnon self-energy −Im(Σ) of\nthe triangular-lattice FMKH model at various temperatures, where\nthe gray dashed lines are the free magnon dispersion. The spectral\ndensityχ′′is shown in (c) and (d), with the k-path in the 1st Brillouin\nzone shown in the inset. Other parameters used: t=1,JH=−1 and\nJK=1.\nperiments [18] and also shared by other lattice geometries, has\na kinematic origin — There are many more damping channels\nfor large|k|. In Fig. 4(c-d), we plot the magnon spectral den-\nsity at these two temperatures, and indeed find the dispersions\nget more blurred at a lower temperature T=0.01.\nDiscussion and Outlook.— In this work, we study the finite-\ntemperature dynamics of FMKH model with TRG calcula-\ntions for 1D chain, with conclusions further supported by\nfield-theoretical analysis for 2D lattice. A magnon damping\nminimum and universal logarithmic scaling at low temper-\nature are revealed, which are ascribed to the scatterings be-\ntween magnons and the itinerant electrons via the Kondo cou-\npling. The logarithmical divergence of the e ffective Kondo\ncoupling in the renormalization process (i.e., vertex correc-\ntion) is analogous to the renowned Kondo e ffect. Our results\nwell explain, from a theoretical point of view, the INS mea-\nsurements on FGT [10, 18].\nBased on the 1D model calculations, we uncover a\nQCP [21, 22, 43, 58] occurring at Jc\nK≃3.3 between the FM\nand KI phases in the phase diagram. We note in a recent ex-\nperiment the Curie temperature of FGT compound decreases\nfrom Tc≃200 K to 100 K as pressure increases, with local\nmoments also prominently suppressed [59]. The high sen-\nsitivity of Curie temperature of the FM order in FGT upon\ncompression is consistent with present FMKH model study.\nFurther experimental investigations on the pressurized FGT\nare thus called for according to our theoretical study here.\nAcknowledgments.— The authors are indebted to Song Bao,\nZhao-Yang Dong, Xiao-Tian Zhang, Wei Zheng, Kun Chen,\nJianxin Li, and Jinsheng Wen for helpful discussions. This5\nwork was supported by the National Natural Science Foun-\ndation of China (Grant Nos. 12222412, 11974036, 11834014,\nand 12047503), the Fundamental Research Funds for the Cen-\ntral Universities, and CAS Project for Young Scientists in Ba-\nsic Research (Grant No. YSBR-057). We thank the HPC-ITP\nfor the technical support and generous allocation of CPU time.\n∗These authors contributed equally to this work.\n†yan@ucas.ac.cn\n‡tshi@itp.ac.cn\n§w.li@itp.ac.cn\n[1] V . Y . Verchenko, A. A. Tsirlin, A. V . Sobolev, I. A. Presniakov,\nand A. V . Shevelkov, Ferromagnetic order, strong magnetocrys-\ntalline anisotropy, and magnetocaloric e ffect in the layered tel-\nluride Fe 3−δGeTe 2, Inorganic Chemistry 54, 8598 (2015).\n[2] Y . Deng, Y . Yu, Y . Song, J. Zhang, N. Z. Wang, Z. Sun,\nY . Yi, Y . Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and\nY . Zhang, Gate-tunable room-temperature ferromagnetism in\ntwo-dimensional Fe 3GeTe 2, Nature 563, 94 (2018).\n[3] K. Kim, J. Seo, E. Lee, K. 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To be specific, the Hilbert space is\ngenerated by the basis ( {|↑⟩,|↓⟩}s⊗{|↑↓⟩,|↑⟩,|↓⟩,|0⟩}f)⊗N/2where slabels the spin (local moment) site while flabels the fermion\nsite, Nis the total number of spin and fermion sites. The matrix product operator (MPO) representation of the Hamiltonian is\nshown in Fig. S1.\nWe perform exponential tensor renormalization group (XTRG) simulations [27] of the finite-temperature properties, and to\ngenerate the finite-temperature density matrix for real-time simulations. In practical calculations, we start from the initial density\nmatrixρ0(τ) of a very high temperature with τ=O(10−4) (T≡1/τ), represented in a MPO form via a series expansion [26]\nρ0(τ)=e−τH=∞X\nn=0(−τH)n\nn!≃NcutX\nn=0(−τH)n\nn!. (S1)\nFor a high temperature such as τ=2.5×10−4, the series expansion saturates to machine precision with cuto ffNcut=6-7.\nSubsequently, by keeping squaring the density matrix repeatedly, i.e., ρn(2nτ)·ρn(2nτ)→ρn+1(2n+1τ), we cool down the system\nalong a logarithmic inverse-temperature grid τ→2τ→22τ→...→2nτ. Such a cooling process reaches low temperature ex-\nponentially fast, and it reduces significantly the projection and truncation steps, rendering higher accuracy than traditional linear\nevolution scheme, and thus it constitutes a very powerful thermal tensor network method for ultra-low temperature simulations.\nFIG. S1. The MPO representation of the Kondo-Heisenberg model Hamiltonian. The red and blue sites present the spin and fermion site\nrespectively with the corresponding local basis {|↑⟩,|↓⟩}and{|↑↓⟩,|↑⟩,|↓⟩,|0⟩}. The red, blue and orange solid line represent the ferromagnetic\nexchange, hopping amplitude, and the Kondo coupling, respectively. The zigzag gray line shows the mapping between the ladder lattice and\nthe MPO representation.\nB. Real-time Evolution: TDVP for matrix product density operator\nIn Heisenberg picture, the real-time evolution of a given operator OreadsO(t)=eiHtO0e−iHtwith the corresponding Heisen-\nberg equation\nid\ndtO=−[H,O]=OH−HO. (S2)8\nIn practice, we introduce an auxiliary parameter t′such thatO(t,t′)=eiHtO0e−iHt′, thus we can split Eq. S2 into two part, namely\nid\ndtO=−HO, (S3a)\nid\ndt′O=OH. (S3b)\nIn the tensor-network language, we prepare the operator Oand the Hamiltonian Hin the MPO form. And Eq. S3 can be\nsolved by the time-dependent variational principle (TDVP) [37–39] by introducing a Choi transformation [60]\nid\ndt|O⟩⟩=−H⊗I|O⟩⟩, (S4a)\nid\ndt′|O⟩⟩=I⊗HT|O⟩⟩. (S4b)\nNoticing that eitH⊗I=eitH⊗Iande−it′I⊗H=I⊗e−it′H, we can solve Eq. S3 keeping the MPO form [28].\nIn this article, we consider the correlation function g(k,t) calculated at temperature β≡1/T, i.e.,\ng(k,t)=⟨Ak(t)A†\nk⟩β=1\nZtr[eiHte−β\n2HAke−iHtA†\nke−β\n2H], (S5)\nwhere Zis the partition function and\nAk=1√\nLLX\nn=1S+\nneikn,k=2πm\nL,m∈{0,1,···,L−1} (S6)\nWith the density matrix at inverse temperature β/2, the correlation function g(k,t) can be obtained by performing real-time\nevolution of e−β\n2HAkfollowing Fig. S2. In the present work, we perform calculations with retained bond dimension up to 4800,\nδt=0.1, and tmax=20.\n1st2nd\nFIG. S2. The workflow of real-time evolution with a small step δt, starting form the quenched finite temperature density matrix e−β\n2HAk. The\nevolution operator eiHδtande−iHδtare acted on e−β\n2HAkfollowing the order indicated by the arrows.\nC. Data convergence for real-time evolution\nIn Fig. S3, we show the real-time evolutions versus di fferent bond dimension D, where we find the simulated results have\nreached convergence for D≳4000.9\n05 10 15 20\nt-0.2-0.100.10.20.30.4Re g(k, t)\n-0.3-0.2-0.100.10.2\nIm g(k, t)D = 3200\n4000\n4800\n05 10 15 20\nt-0.3-0.2-0.100.10.20.30.4Re g(k, t)\n-0.3-0.2-0.100.10.2\nIm g(k, t)\n05 10 15 20\nt-0.3-0.2-0.100.10.20.30.4Re g(k, t)\n-0.3-0.2-0.100.10.2\nIm g(k, t)\nT = 0.5 T = 0.1 T = 0.03(a) (b) (c)\nFIG. S3. Real (blue) and imaginary (red) part of the real-time correlation function with di fferent bond dimensions for (a) T=0.5, (b) T=0.1,\nand (c) T=0.03. The system size is N=16×2, the hopping amplitude is t=2, the Heisenberg exchange is JH=−1 and the Kondo coupling\nJK=1.\nII. Additional Data of the Spectral Density\nA. Calculations of the spectral density\nThe spectral function in a time translation invariant system is defined as χ′′\nAB(t)=1\n2π⟨[A(t),B]⟩and below we consider the spin\nexcitation in the FMKH model, with A=S+andB=A†=S−. The correlation function g(k,t) is defined as\ng(k,t)=⟨S+\nk(t)S−\nk⟩β=1\nZtr[eiHte−β\n2HS+\nke−iHtS−\nke−β\n2H]=g(k,−t)∗. (S7)\nDue to the spatial inversion symmetry, g(k,t)=g(−k,t). Moreover, we have ⟨S+(t)S−⟩β=⟨S−(t)S+⟩βfor the present system.\nThus the spectral density reads\nχ′′(k,t)=1\n2π[g(k,t)−g(k,t)∗]=i\nπImg(k,t), (S8)\nand\nχ′′\ndef(k,ω)≡Z+∞\n−∞dt eitωi\nπImg(k,t)=−2\nπZ∞\n0dtImg(k,t) sinωt. (S9)\nSince the real-time simulation can only be performed for a finite period of time, the spectral function can be obtained approx-\nimately as\nχ′′(k,ω)≈−aZtmax\n0dtImg(k,t) sinωt·W(t\ntmax), (S10)\nwhere W(t) is a window function, and ais a nonsignificant constant number (we set a=1 in the follow part). In practice, the\nHanning window defined below is chosen,\nW(t)=1\n2+1\n2cos(πt),|t|≤1\n0,|t|>1.(S11)\nB. Kondo-Heisenberg model and the damped harmonic oscillator analysis\nWe fitχ′′by the damped harmonic oscillator (DHO) model defined as [53]\nχ′′\ndef(k,ω)∝γkωEk\n(ω2−E2\nk)2+(γkω)2(S12)\nwhere Ekis the magnon energy and γkis the damping rate. Since the time window of real-time evolution is limited by tensor\nnetwork calculations, we add a Hanning window for cut-o ff. In practice, the DHO fitting should take into account of the\ninfluences from the window.10\n(a) (b)\n(c) (d)1 1.5 2 2.5 30123''(k = , )T = 0.49\n0.12\n0.061\n0.031\n0.015\nDHO fitting\n1 1.5 2 2.5 300.511.5''(k = , )T = 0.49\n0.12\n0.061\n0.031\n0.015\n1 1.5 2 2.5 300.20.40.60.81''(k = , )T = 0.49\n0.17\n0.086\n0.043\n0 0.5 1 1.5 201234''(k = , )T = 0.49\n0.24\n0.061\nFIG. S4. The DHO fittings of χ′′(k,ω) results of (a) Heisenberg model with Kondo coupling JK=0 (ED), (b) JK=1 (D=4800), (c) JK=1.5\n(D=3200), and (d) JK=4 (D=1600). The solid lines are the calculated χ′′(k,ω) results at di fferent temperatures, and the correspond DHO\nfittings are shown with dashed lines and in the same color code. The fitting window is indicated by the two gray lines. In (a) for the Heisenberg\nferromagnet with JK=0, we take JH=−1 and the calculations are performed on a L=16 chain. For the JK>0 cases in (b,c,d) panels,\ninstead we take t=2,JH=−1 and the system size is N=16×2.\nFrom Eq. S10, one may immediately find that\nχ′′(ω)=1\n2πχ′′\ndef(ω)∗W(ω). (S13)\nThus the DHO fitting is\nχ′′(ω)∝γkωEk\n(ω2−E2\nk)2+(γkω)2∗W(ω), (S14)\nwhere W(ω)=π2sin(tmaxω)\n−π2ω+tmaxω3is the fourier transform of Hanning window.\nThe DHO fitting results are shown in Fig. S4 with the χ′′(k,ω) obtained by TDVP ( JK>0) and exact diagonalization ( JK=0).\nWithin the fitting range, i.e. the gray line in Fig. S4, the DHO fitting shows excellent agreement with χ′′(k,ω).\nC. Damping rate minima and logarithmic scaling for di fferent JK\nThe damping minimum and the logarithmic scaling can also be obtained with di fferent momentum kand Kondo coupling JK\nas shown in Fig. S5. Since the damping coe fficient is related to the magnon self-energy Im( Σ), the momentum k<1 is lesser\ninfluenced by the Kondo e ffect, i.e., the damping minimum is reached with a smaller temperature [see Fig. S5 (a)]. For JK=1.5,\nwe obtained the damping coe fficient with N=16×2 and D=3200. As shown in Fig. S5 (b), γπshows similar behavior to the\nmain text case. Increasing JK,T∗\ndalso becomes higher.\nTo confirm the low-temperature scaling of the damping constant, we fit the damping coe fficientγπunder di fferent retaining\nbond dimension by the model alog(b/T)+c. The rescaled damping coe fficientγ′\nπ=1\na(γπ−c)−log(b)is shown in Fig. S5(c),\nand the low-temperature part shows exactly same behavior.11\n(a) (b)\nT*d\n10-210-1100\nT23456'D = 3200\n4000\n4800(c)\n10-210-1\n T0.360.380.40.420.440.463/4\n10-1\n T0.660.680.70.720.740.76\nT*d\nFIG. S5. Supporting data of the damping coe fficientγkfor more coupling parameter JKand momentum k. (a) JK=1 (k=4π/3) and (b)\nJK=1.5 (k=π), simulated on a N=16×2 geometry. T∗\ndindicates the damping minimum of γkand the dashed line represents the logarithmic\nscaling. (c) shows the rescaled damping coe fficientγ′\nπwith di fferent bond dimension D, where the logarithmic scaling is found to be robust\nfor various D.\nD. Ferromagnetic Heisenberg chain: ED calculations\nAs for the pure Heisenberg case ( JK=0) with L=16, we obtain the dynamical properties by exact diagonalization (ED)\ncalculations. The contour plots of the spectral density are shown in Fig. S6, where the dispersion line shape becomes clearer as\ntemperature lowers.\n012\n k/\n0123\n012\n k/\n0123\n012\n k/\n0123\n012\n k/\n0123\n0123\n''(a) (b) (c) (d)\nFIG. S6. Contour plot of spectral density χ′′of a pure FM Heisenberg chain with L=16 at various temperatures.\nE. Coherent triplon excitations in the Kondo insulator phase\nFor su fficiently large Kondo coupling JK, the system enters the Kondo insulator phase with coherent triplon waves. As shown\nin Fig. S7, we calculate the spin excitation dispersions with tensor network approach, and find the triplon excitation shows a\nsquare form dispersion with a minimum value at k/π=1.\n0 0.511.5 2\nk/\n01234567\n00.511.522.533.5\n''\n0 0.511.5 2\nk/\n01234567\n00.511.522.533.5\n''(a) (b)\nFIG. S7. Triplon dispersion for (a) JK=4 and (b) JK=6, with D=1600, N=16×2 and T=0.03.12\nIII. Field Theoretical Analysis of the Magnon Damping Rate\nA. Explicit evaluation of the bare polarization bubble diagram for the 1D chain\nThe pair bubble diagram for the magnon self-energy, given in Fig. 1(c), using bare electron propagator and bare vertex , reads\nΣk(iωn)= Σ+−\nk(iωn)\n=\n=−J2\nK\n2Zddp\n(2π)dnc,p+k−nc,p\niωn−ϵc,p+k+ϵc,p, (S15)\nwhere nc,k=1/(eβϵc,k+1) is the Fermi function. For the 1D chain, after analytical continuation, i ωn→ω+i0+, we have\nΣk(ω+i0+)=−J2\nK\n2Zdp\n2πnc,p+k−nc,p\nω−ϵc,p+k+ϵc,p+i0+\n=−J2\nK\n2Zdp\n2πnc,p+k−nc,p\nω−ϵc,p+k+ϵc,p+i\u0014πJ2\nK\n2Zdp\n2π(nc,p+k−nc,p)δ(ω−ϵc,p+k+ϵc,p)\u0015\n. (S16)\nFrom\nχ′′\nk(ω)=−Im[GR\nk(ω+i0+)]\n=−Im1\nω−Ek−ReΣk(ω)−i ImΣk(ω)\n=−ImΣk(ω)\n[ω−Ek−ReΣk(ω)]2+[ImΣk(ω)]2, (S17)\nthe magnon damping rate is found to be\nγ(ω,k)=|ImΣk(ω)|\n=\f\f\f\f\f\fπK2\n2Zdp\n2π(nc,p+k−nc,p)δ(ω−ϵc,p+k+ϵc,p)\f\f\f\f\f\f\n=J2\nK\n8tX\np0\f\f\f\f\fnc,p0+k−nc,p0\nsin(p0+k)−sinp0\f\f\f\f\f\n=J2\nK\n16tX\np0\f\f\f\f\f\f\fnc,p0+k−nc,p0\ncos2p0+k\n2sink\n2\f\f\f\f\f\f\f, (S18)\nwhere the summation runs over all solutions of the following equation for p0∈[−π,π),\nω=ϵc,p0+k−ϵc,p0\n=4tsin2p0+k\n2sink\n2. (S19)\nwhereϵc,k=−2tcoskis the dispersion of free fermion chain. Focusing on the on-shell region, ω=ϵk=|JH|(1−cosk), Eq. (S19)\nalways has two solutions given by p0,1=−k\n2+arccsch\nαcsck\n2i\nandp0,2=−k\n2+π−arccsch\nαcsck\n2i\nwithα=2t/|JH|>1 for\nthe model parameters of Fe 3GeTe 2. Limiting cases of the bare damping rate can be understood from the analytical expression\nEq. (S18). First, it is vanishingly small for small k. Indeed, when expanded around k=0, it’s straightforward to show that\nγ∝k. Secondly, it should reach the maximal value for knearπ, where the numerator becomes largest. Lastly, to the crudest\napproximation and as a reminiscence of the single-impurity Kondo problem, the Kondo coupling JKis renormalized as JK→\nJren\nK=JK[1+JKg(ϵF) log\u0010Λ\nT\u0011\n], where g(ϵF)=1/(2πt) is the density of states per spin at the Fermi energy, and Λ = 4tis the\nband width. When comparing with our numerical calculations given in Fig. 3, it is found that these model parameters should be\nfurther renormalized as JH=−0.07 and JK=0.88, which may be due to higher order corrections.\nWe present numerical results of imaginary part of magnon self-energy in Fig. S8(a-b) and the corresponding spectral density\nin Fig. S8(c-d). It is found indeed that the imaginary part of the magnon self-energy increases at lower temperature and the\nmagnon spectral density becomes more damped. Once again, we note that within this simple one-loop calculations, the magnon\ndamping occurs only around k∼πand is absent for small |k|.13\nΓXM\n(a)\n(c)(b)\n(d)(e)\n(g)(f)\n(h)\nFIG. S8. First row: Contour plot of the imaginary part of magnon self-energy Im( Σ) for the Kondo-Heisenberg lattice model defined on the\none-dimensional chain (a-b) and the square lattice (e-f), where the gray dash-dotted line is the corresponding free magnon dispersion. Second\nrow: the corresponding magnon spectral density χ′′=ImGR, for the one-dimensional chain (c-d) and the square lattice (g-h) cases. The inset\nin (g) shows the momentum path used in the first Brillouin zone. Parameters used: t/|JH|=0.81 and JK/|JH|=0.21 for the one-dimensional\nchain case; t/|JH|=0.77 and JK/|JH|=0.077 for the square lattice case.\nB. The square lattice case\nFor the FMKH model defined on the square lattice, with the primitive lattice vectors given by u1=(1,0) and u2=(0,1), the\ncorresponding Brillouin zone is shown in the inset of Fig. S8(g). After using the Holstein-Primako fftransformation, the free\nmagnon dispersion is ϵk=J(2−coskx−cosky). Meanwhile, the free electron dispersion reads ϵc,k=−2t(coskx+cosky). Hence\nthe self-energy can be calculated numerically using Eq. (3) in the main text, and results are shown in Fig. S8. Note at half-filling,\nthe density of state of free fermion for the square lattice is divergent due to the presence of van Hove singularity, which is of\ncourse an artifact resulting from the approximation used in calculating the corresponding momentum integration. In practice, we\nsimply regularize this density of state to a finite one in comparable to the one-dimensional case. The two main features, which\nare also shared for the one-dimensional case and the triangular lattice case, are that (1) the imaginary part of magnon self-energy\nincreases at lower temperature, and the corresponding magnon spectral density is more damped; (2) damping phenomenon\noccurs primarily at large |k|and becomes absent for small |k|. The second feature can easily be understood from the kinematics:\nthe damping channel becomes negligible for small |k|, therefore the magnon remains to be well-defined in that case.\nC. Magnon damping minimum for the triangular lattice case\nThe calculations present in this section so far only produce log Tscaling of magnon damping at low temperature. While in\nexperiment it is found that there is also a magnon damping minimum at an intermediate temperature [18]. In this subsection,\nwe further consider magnon-magnon interactions beyond linear spin wave theory. It turns out that this additional contribution\ngives a power-law divergence at relatively high temperatures. Together with contributions from electron-magnon interactions,\nthe magnon damping minimum found in experiments is therefore reproduced.\nAs the minimal model Hamiltonian for Fe 3GeTe 2, the ferromagnetic Kondo-Heisenberg model on the triangular lattice reads\nH=Ht+HJ+HK, where\nHt=−tX\n⟨i,j⟩c†\ni,σcj,σ+H.c., (S20)\nHJ=−JX\n⟨i,j⟩Si·Sj, (S21)\nHK=JKX\niSi·si. (S22)14\n1st BZΓ(b)\nMK(a)\nFIG. S9. (a) The triangular lattice is generated by linear combinations of the basis vectors a1=(1,0)and a2=\u0010\n−1/2,√\n3/2\u0011\n. (b) The\ncorresponding reciprocal lattice is generated by the basis vectors b1=2π×(1,1/√\n3) and b2=2π×(0,1/√\n3). The first Brillouin zone (BZ)\nis shown as a hexagon, with the high symmetric path indicated by the green lines.\nUsing the Holstein-Primako ff(HP) transformation,\nS+\ni=q\n2S−a†\niaiai, (S23a)\nS−\ni=a†\niq\n2S−a†\niai, (S23b)\nSz\ni=S−a†\niai, (S23c)\nEquation (S21) can be rewritten as HJ=H(−2)+H(−1)+H(0)+O\u00101\nS\u0011\n. Here the superscript labels the corresponding order in 1 /S.\nExplicitly, we have\nH(−2)= 1\nS!−2\n×(−zNJ), (S24)\nH(−1)= 1\nS!−1\n×2zJX\nk(1−γk)a†\nkak, (S25)\nH(0)= 1\nS!0\n×J\n4X\n⟨i,j⟩\u0010\na†\nia†\niaiaj+a†\nja†\njaiaj+a†\nia†\njaiai+a†\nia†\njajaj−4a†\nia†\njaiaj\u0011\n(S26)\n=X\n{ki}′V4(k1,k2;k3,k4)a†\nk1a†\nk2ak3ak4. (S27)\nEquation (S24) gives the classical ground state energy for a pure ferromagnetic Heisenberg model on the triangular lat-\ntice, with z=3 half the coordination number and Nthe total lattice sites. Equation (S25) gives the free magnon dis-\npersionεk=2zS J(1−γk). For the triangular lattice, we have γk=1\nzPz\nl=1cos(al·k), with a1,2,3shown in Fig. S9(a).\nEquation (S26) gives the interaction between magnons, where the interaction vertex explicitly reads V4(k1,k2;k3,k4)=\nJ\n4N\u0000γk1+γk2+γk3+γk4−4γk2−k4\u0001. The primed summation in Eq. (S27) means that the total momentum before and after scat-\ntering is conserved, i.e., k1+k2=k3+k4. Using the same HP transformation, Equation (S22) can be written as\nH=JKX\ni(S−ni)(nf\ni,↑−nf\ni,↓)+r\nS\n2(aic†\ni,↓ci,↑+a†\nic†\ni,↑ci,↓)+O 1\nS!1/2. (S28)\nBased on Eq. (S27) and (S28), all contributions to magnon self-energy up to the zeroth order in 1 /Sand second order in Jand\nJKare listed in Fig. S10.\nFigure S10(a) is a Hartree contribution, which vanishes since ⟨nf\ni,↑⟩0=⟨nf\ni,↓⟩0. Namely,\nΣ(a)\nR(k)=0. (S29)\nFigure S10(b) explicitly reads Σ(b)(k,iω)=J2\nKS\n2χ0(k,iω), where the pair-bubble diagram is\nχ0(k,iω)=1\nNX\nq1\nβX\niω11\niω1−ξq1\niω+iω1−ξk+q(S30)\n=1\nNX\nqfF(ξq)−fF(ξk+q)\niω−(ξk+q−ξq), (S31)15\nFIG. S10. Contributions to magnon self-energy due to electron-magnon interaction [(a) and (b)] and magnon-magnon interaction (up to zeroth\norder in 1/Sand second order in J) [(c) and (d)]. Solid (dashed double) line represents free magnon (renormalized electron) propagators. For\nsimplicity, we replace the renormalized electron propagator by the bare one for all our calculations. The gray area in (b) indicates the usage of\nrenormalized vertex. Here we also simply use the replacement JK→J(ren)\nK=JK[1+JKg(ϵF) log\u0010\nΛ\nT\u0011\n], as in the case of single impurity Kondo\nmodel, which is responsible for the anomalous temperature dependence of magnon damping rate at low temperatures.\nandξk=−2tzγkis free electron’s dispersion. After analytical continuation, and using renormalization of Kondo coupling\nJK→J(ren)\nK=JK[1+JKg(ϵF) log\u0010Λ\nT\u0011\n] reminiscent of the singlet-impurity Kondo problem [19], the retarded self-energy reads\nΣ(b)\nR(k,ω)=J2\nK\"\n1+JKg(ϵF) log Λ\nT!#S\n2NX\nqfF(ξq)−fF(ξk+q)\nω−(ξk+q−ξq)+i0+. (S32)\nHere g(ϵF) is the density of states per spin at the Fermi energy, and Λis the band width.\nFigure S10(c) is also a Hartree contribution. We replace the product of two operators by their expectation values in Eq. (S27),\nwhich leads to four non-vanishing terms. After simplification, the upshot is simply a correction to the magnon dispersion\nrelation,\nΣ(c)\nR(k)=X\nq2J(γq+γk−γ0−γk−q)fB(εq). (S33)\nFigure S10(d) explicitly reads\nΣ(d)(k,iω)=J2\nN2β2X\n{qi,iωi}\f\f\fV4(k,q1;q2,k+q1−q2)\f\f\f2\n(iω1−εq2)(iω2−εk+q1−q2)[i(ω1+ω2−ω)−εq1](S34)\n=J2\nN2X\n{qi}\f\f\fV4(k,q1;q2,k+q1−q2)\f\f\f2\niω+εq1−εq2−εk+q1−q2Fk,q1,q2. (S35)\nwhere Fk,q1,q2=h\n1+fB(εq2)ih\n1+fB(εk+q1−q2)i\nfB(εq1)−fB(εq2)fB(εk+q1−q2)h\n1+fB(εq1)i\n. After analytical continuation, the\nretarded self-energy reads\nΣ(d)\nR(k,ω)=J2\nN2X\n{qi}\f\f\fV4(k,q1;q2,k+q1−q2)\f\f\f2\nω+εq1−εq2−εk+q1−q2+i0+Fk,q1,q2. (S36)\nThus the total retarded self-energy reads ΣR= Σ(a)\nR+ Σ(b)\nR+ Σ(c)\nR+ Σ(d)\nR. And its imaginary part, relating to the magnon damping\nrate, reads\nΓ(k,ω)=|ImΣR(k,ω)| (S37)\n=\f\f\fImΣ(b)\nR(k,ω)+ImΣ(d)\nR(k,ω)\f\f\f (S38)\n=|J2\nK\"\n1+JKg(ϵF) log Λ\nT!#Sπ\n2NX\nqh\nfF(ξq)−fF(ξk+q)i\nδ(ω−(ξk+q−ξq))\n+πJ2\nN2X\n{qi}\f\f\fV4(k,q1;q2,k+q1−q2)\f\f\f2Fk,q1,q2δ(ω+εq1−εq2−εk+q1−q2)|. (S39)16\nΓ M K Γ\n𝜉(k)\n-6-4-202K(b)(k), |ImΣ R(b)(k)|\n0.00.51.01.5K(b)(k)\n|ImΣR(b)(k)|\nΓ M K Γ\n𝜀(k)\n01234K(d)(k), |Im ΣR(d)(k)|\n0.00.51.01.5K(d)(k)\n|ImΣR(d)(k)|(a) (b)\nFIG. S11. Scattering density of states Kand imaginary part of the self-energy Im Σdue to (a) electron-magnon interactions and (b) magnon-\nmagnon interactions, together with free electron dispersion ξ(k) and free magnon dispersion ε(k), respectively, along the high symmetry\nmomentum path. Model parameters used: t/J=1,JK/J=10 and T=1. Error bars of numerical integration are smaller than the symbol size.\nT0.01 0.10 1. 00Imaginary part of sel f-energy\n0.00.51.01.5 |ImΣR(b)(K)|\n|ImΣR(d)(K)|\n|ImΣR(K)|\nT0.01 0.10 1. 00Imaginary part of sel f-energy\n0.00.51.0|ImΣR(b)(M)|\n|ImΣR(d)(M)|\n|ImΣR(M)|(a) (b)\nFIG. S12. Imaginary part of the self-energy from two types of interactions and their total contributions (namely, the magnon damping rate),\nas a function of temperature T, at the M (a) and K (b) point in the first BZ. The shaded rectangle with T∗indicates the regime where magnon\ndamping minimum occurs. Model parameters used: t/J=2,JK/J=12. Error bars of numerical integration are smaller than the symbol size.\nThere are two contributions: one results from the Kondo coupling, Σ(b)\nR, and the other is due to magnon-magnon interactions,\nΣ(d)\nR. Each of them is an integration, and the corresponding integrand consists of two parts, the scattering coe fficient and the\ndelta function. The former gives temperature dependence and overall relative strength among di fferent momenta; while the latter\nprovides the kinematic constraint. To understand this kinematic constraint more carefully, we define the scattering density of\nstates [61] for each contribution,\nK(b)(k)=1\nNX\nqδ(ω−(ξk+q−ξq)), (S40)\nK(d)(k)=1\nN2X\n{qi}δ(ω+εq1−εq2−εk+q1−q2). (S41)\nIn Fig. S11, we show the imaginary part of self-energy Im Σand the corresponding scattering density of states K, for each\ncontribution. The multi-dimensional integrations over internal momenta are performed using MCIntegration.jl [62]. It is\nfound that for the electron-magnon interactions, the scattering density of states is divergent around Γpoint and suppressed\notherwise; while for the magnon-magnon interactions, it peaks at Mpoint and reaches minimum at ΓandK. For the respective17\ncontributions to the magnon damping, namely the imaginary part of the self-energy, both become larger when going away from\nthe BZ center, and become largest near the BZ boundary. This behavior is consistent with the experimental observation in\nFe3GeTe 2[18].\nIn Fig. S12, we show the temperature dependence of the damping rate resulting from two types of interactions and their total\ncontribution. In the large and small Tlimit, imaginary part of Eq. (S32) and (S36) can be understood analytically: On the one\nhand, it can be shown straightforwardly that Im Σ(b)\nR∝1/TforT→∞ and Im Σ(d)\nRdecay exponentially for T→0, respectively.\nOn the other hand, Im Σ(d)\nR∝T2in the high- Tlimit based on a simple series expansion, while Im Σ(b)\nR∝logTin the low- Tlimit\ndue to Kondo renormalization. Therefore, there is indeed a magnon damping minimum at an intermediate temperature T∗, inside\nthe shaded regime in Fig. S12. Using the same model parameters as the ones given in Fig. 3 of the main text with JK=10, we\nobtain the gray dotted line given there.\nIV . Tensor-network and Mean-field Calculations of the Phase Diagram\nA. Quantum critical point between ferromagnetic and Kondo insulator phases\nIn this section, we show more detailed calculations on the quantum critical point (QCP) between ferromagnetic (FM) and\nKondo insulator (KI) phases. We compute both the ground state and low-temperature properties, e.g., the heat capacity, which\nwitness the existence of QCP.\nTo determine the ferromagnetic quantum critical point, we perform DMRG simulations on a N=36×2 geometry,\nwith D=1600 bond states retained and truncation error around ϵ≈5e−7. We also calculate the spin structure factor\nSzz\nΓ=1\nLP\nm,n⟨Sz\nmSz\nn⟩ei(n−m)k(k=0,Γpoint) and local Kondo correlation function Knn=−1\nLP\nn⟨Sn·sn⟩. As shown in Fig. S13(a),\nFM spin correlation decreases fast when increasing Jc\nKand converges to a small value in the KI phase, while the Kondo correla-\ntion monotonically increases and saturates instead.\nFM KI\n3 3.1 3.2 3.3 3.4 3.5 3.600.511.5\n00.20.40.60.8\nKnnS zz\n (a) (b)\n10-210-1100\nT10-1100101C/T T -3.5\nFIG. S13. (a) shows the ground-state spin structure at k=0 and Kondo correlation function Knn, as function of Kondo coupling JK, with\nJH=−1,t=2,N=36×2. (b) shows C/Tdata with N=36×2 and JK=3.3,3.5. At the quantum critical point, the lower part of C/T\ndiverges algebraically with α∼0.7.\nWe further perform XTRG simulation at the putative phase transition point Jc\nK=3.3. As shown in Fig. S13 (b), the low\ntemperature heat capacity displays algebraic behavior as C/T∼T−0.7, signalling quantum criticality.\nB. Determination of the crossover temperature\nAlthough there is no finite-temperature phase transition in 1D quantum system, the crossover temperature can be determined\nform the thermodynamic quantities and correlation function. In the FM phase, we obtain the crossover temperature T∗\nCfrom the\nthe temperature derivatives of local moment correlations - dFnn/dTwith Fnn=1\nL−1P\nn⟨Sn·Sn+1⟩. The corresponding result is\nshown in Fig. S14(a). As the Kondo coupling increasing, the crossover temperature decreases. In the Kondo insulator phase, we\ndetermine the crossover temperature by - dKnn/dT. As shown in Fig. S14(b), TKhas positive correlation to the Kondo coupling\nJK. All the above calculation are performed by XTRG with N=36×2 and D=1600 and the corresponding result is shown in\nFig. 1(a) of the main text.18\n10-210-1100101\nT00.20.40.60.811.2-dKnn/dT4\n5\n6(a) (b)\n10-1100101\nT00.050.10.150.2\n1.5\n2\n2.5\n-dFnn/dT\nFIG. S14. (a) shows the - dFnn/dTin the FM phase and (b) shows the - dKnn/dTin the KI phase with N=36×2 and D=1600. The dashed\nline indicates T∗\nCandTKrespectively.\nC. Finite-temperature properties in the FM phase\nHere we discuss the Kondo correlation Knnand FM spin structure factor Szz\nΓin the FM region (with the same model parameter\nin Fig. 3 of the main text). As shown in Fig. S15, Knnalso displays a minimum value around T∗\nd, while Szz\nΓbehaves similarly for\nboth JK=0 and JK,0 above T∗\nd. These results indicate that the Kondo coupling JKprominently changes the thermodynamical\nproperties below T∗\nd. Notably, Szz\nΓkeeps increasing as system cooling down even below T∗\ndwhich means that the local moments\nare not fully screened by the itinerant electrons in this case.\n10-210-1100\nT0.0440.0460.0480.050.052Knn\n0.20.40.60.811.21.41.6\nS zz\n \nFIG. S15. Kondo correlation Knn(red circle) and spin structure factor Szz\nΓ(blue solid line) with N=16×2,t=2,JH=−1,JK=1 and\nD=1600. The gray solid line shows Szz\nΓwith JK=0 and the gray shadow region shows the di fference between JK=0 and JK=1. The\nvertical point-solid line shows the minimum damping temperature T∗\nddetermine by γπ(see Fig. 3 in the main text).\nD. Determination of the mean-field phase diagram\nIn this section, we discuss an e ffective mean-field theory which can treat the FM order and local Kondo screening e ffect on\nan equal footing. Hence a complete phase diagram containing both the FM phase at small JKand KI phase at large JKcan be\nobtained. The starting point of this e ffective mean-field theory is to notice that the Kondo coupling term can be decomposed into\nlongitudinal and transversal parts [47],\nSn·sn=Sz\nnsz\nn−1\n4[(c†\nn↑fn↑+f†\nn↓cn↓)2+(c†\nn↓fn↓+f†\nn↑cn↑)2], (S42)\nwhere we have used the pseudo-fermion representation for the local spin Sn=1\n2P\nαβf†\nnασαβfnβ, with a local constraint\nX\nσf†\nnσfnσ=1, (S43)19\n246\nJ K10-210-1100TFM KIPM\nFIG. S16. Phase diagram of the 1D FMKH model ( t=2,JH=−1), based on the e ffective mean field theory by solving Eq. (S59) self-\nconsistently. The FM phase is identified with nonzero mcandmf. The KI phase is identified with nonzero V.\nenforced at each site n. Equation (S42) inspired one to define, besides the standard FM order parameters\nmc=\nSz\nn\u000b,mf=\nsz\nn\u000b, (S44)\nalso the Kondo hybridization order parameter as follows,\nV=D\nc†\nn↑fn↑+f†\nn↓cn↓E\n. (S45)\nUsing these order parameters, one can decouple the original Hamiltonian, H=Ht+HH+HKwith\nHt=−tX\nnc†\nncn+1+H.c., (S46)\nHH=JHX\nnSn·Sn+1, (S47)\nHK=JKX\nnSn·sn, (S48)\nto a quadratic form. Firstly, the hopping term, Ht, is unchanged. Secondly, the ferromagnetic Heisenberg term becomes\nHH→JHX\nnmf(f†\nn↑fn↑−f†\nn↓fn↓)−JHLm2\nf. (S49)\nAnd lastly, the Kondo coupling term becomes\nHK→JKLX\nn=1\u0014mc\n2(f†\nn↑fn↑−f†\nn↓fn↓)+mf\n2(c†\nn↑cn↑−c†\nn↓cn↓)−mcmf\u0015\n+JKLX\nn=1\"\n−1\n2V(c†\nn↑fn↑+f†\nn↓cn↓)−1\n2V(c†\nn↓fn↓+f†\nn↑ci↑)+1\n2V2#\n. (S50)\nWe also introduce a Lagrangian multiplier λto enforce the local constraint Eq. (S43),\nλLX\nn=1(f†\nnσfnσ−1), (S51)\nand a chemical potential µto fix the total cfermion number at half filling,\n−µLX\nn=1(c†\nncn−1). (S52)20\nHence the mean-field Hamiltonian in momentum space can be written in a compact form\nHMF=X\nk,σ(c†\nkσ,f†\nkσ) εkσ−1\n2JKV\n−1\n2JKVλσ! ckσ\nfkσ!\n+Lε0, (S53)\nwhere\nεkσ=−2tcosk+JKmf\n2σ−µ, (S54)\nλσ=λ+\u0012JKmc\n2+JHmf\u0013\nσ, (S55)\nε0=JKV2\n2−JKmcmf−λ+µ−JHm2\nf. (S56)\nThe quasiparticle excitation spectra are thus obtained by\nEkσl=1\n2h\nεkσ+λσ+lp\n(εkσ−λσ)2+(JKV)2i\n, (S57)\nwhich corresponds to four quasiparticle bands, with σ=±andl=±. The corresponding Helmholtz free energy density reads\nf=−1\nβLX\nkX\nl,σ=±log[1 +exp(−βEkσl)]+ε0. (S58)\nwhich should be minimized with respect to the five unknown parameters:\n0=∂mff\n=1\nLX\nkX\nl,σ=±e−βEkσl\n1+exp(−βEkσl)∂mfEkσl−JKmc−2JHmf\n=1\nLX\nkX\nl,σ=±n(Ekσl)σJH\n2+JK\n4+l\u0012JK\n4−JH\n2\u0013εkσ−λσp\n(εkσ−λσ)2+(KV)2−JKmc−2JHmf (S59a)\n0=∂mcf\n=1\nLX\nkX\nl,σ=±n(Ekσl)JKσ\n41−lεkσ−λσp\n(εkσ−λσ)2+(JKV)2−JKmf (S59b)\n0=∂Vf\n=1\nLX\nkX\nl,σ=±n(Ekσl)lJ2\nKV/2\np\n(εkσ−λσ)2+(JKV)2+JKV (S59c)\n0=∂λf\n=1\nLX\nkX\nl,σ=±n(Ekσl)1\n21−lεkσ−λσp\n(εkσ−λσ)2+(JKV)2−1 (S59d)\n0=∂µf\n=1\nLX\nkX\nl,σ=±n(Ekσl)1\n2−1−lεkσ−λσp\n(εkσ−λσ)2+(JKV)2+1, (S59e)\nwith n(Ekσl)=(eβEkσl+1)−1being the Fermi function. We numerically solve above five equations self-consistently. The resulting\nfinite temperature phase diagram for ( JH,t)=(−1,2) is shown in Fig. S16.\nV . Excitation Spectrum in the Strong-coupling Limit: Bond-operator Theory\nIn this appendix, we give a strong-coupling mean-field analysis of the FMKH model for K≫J,t, using bond-operator theory.\nThis approach was originally proposed by Sachdev and Bhatt for the dimerized spin systems [50]. Its generalization to the\nKondo lattice systems was firstly given by Jurecka and Brenig [51]. R. Eder recently gave an improved treatment to obtained\nboth the fermionic quasiparticle and spin excitation spectrum for the Kondo lattice [52]. We here generalize it to the FMKH\nmodel.21\nA. Bond-operator basis\nThe Kondo-coupling term leads to eight eigenstates, defined as follows [51]:\n|s⟩=s†|0⟩=1√\n2(f†\n↓c†\n↑+c†\n↓f†\n↑)|0⟩, (S60a)\n|tx⟩=t†\nx|0⟩=−1√\n2(c†\n↑f†\n↑−c†\n↓f†\n↓)|0⟩, (S60b)\n\f\f\ftyE\n=t†\ny|0⟩=i√\n2(c†\n↑f†\n↑+c†\n↓f†\n↓)|0⟩, (S60c)\n|tz⟩=t†\nz|0⟩=1√\n2(c†\n↑f†\n↓+c†\n↓f†\n↑)|0⟩, (S60d)\n|aσ⟩=a†\nσ|0⟩=f†\nσ|0⟩, (S60e)\n|bσ⟩=b†\nσ|0⟩=c†\n↑c†\n↓f†\nσ|0⟩. (S60f)\nNote the ffermion has to satisfy the local constraint at each site,\nf†\nnσfnσ=1,∀n. (S61)\nHereafter Einstein summation rule over repeated Greek alphabet (i.e., spin indices) is assumed. When the spin index is omitted,\nthe corresponding operators should be viewed as a two-component object. The first four particles in Eq. (S60) are bosons; and\nthe rest are fermions. They together satisfy the local constraint\ns†\nnsn+a†\nnσanσ+b†\nnσbnσ+t†\nn·tn=1,∀n (S62)\nwhich ensures Eq. (S61) automatically.\nB. Operators in bond-operator representation\nHere we give explicit form of relevant operators in this representation. First of all, the spin operator for itinerant electrons and\nlocal moments becomes, respectively,\nsσ\nn=1\n2(−t†\nnσsn−s†\nntnσ−iϵσαβt†\nnαtnβ), (S63)\nSσ\nn=1\n2(t†\nnσsn+s†\nntnσ−iϵσαβt†\nnαtnβ)+Sσ\na,n+Sσ\nb,n, (S64)\nwhere Sµ\na,n=1\n2a†\nnσµanandSµ\nb,n=1\n2b†\nnσµbn. Then the Kondo coupling term can be shown to be\nJKX\nnSn·sn=X\nn[3JK\n4(b†\nnσbnσ+a†\nnσanσ)+JKt†\nn·tn]−3JNK\n4, (S65)\nwhich is diagonal as expected. Next, the itinerant electron annihilation operator becomes\ncn=1√\n2[(−sn+tn·τ)iτya†\nn+(s†\nn+t†\nn·τ)bn], (S66)\nfrom which we can obtain the hopping term for itinerant electrons,\n−tX\nnc†\nncn+1+H.c.=Hc1+Hc2+Hc3+Hc4, (S67)22\nwith\nHc1=1\n2X\nn,mtn,m[(−s†\nnsma†\nman+sns†\nmb†\nnbm−(snsmb†\nniτya†\nm+H.c.)], (S68)\nHc2=1\n2X\nn,mtn,m[−t†\nn·tma†\nman+tn·t†\nmb†\nnbm+(tn·tmb†\nniτya†\nm+H.c.)], (S69)\nHc3=1\n2X\nn,mtn,m{(s†\nntm+smt†\nn)·(am,n+bm,n)+[(smtn−sntm)·π†\nn,m+H.c.]}, (S70)\nHc4=1\n2X\nn,mtn,m[−t†\nn×tm·i(am,n+bm,n)+(tm×tn·iπ†\nm,n+H.c.)], (S71)\nHere tm,n=−tfor the nearest-neighbors, and vanishing otherwise. And the following vectorized operators have been used:\nam,n=a†\nmτbn,bm,n=b†\nmτbnandπ†\nm,n=b†\nmτiτya†\nn. Interestingly, the hopping term, which is originally quadratic, now becomes\nquartic. Such behavior is common for strong-coupling approaches, see also [63]. Lastly, the Heisenberg coupling term for\nlocalized spins becomes\nJHX\nnSn·Sn+1=Hs1+Hs2+Hs3+Hs4,J>0, (S72)\nwhere\nHs1=1\n8X\nn,mJn,m(t†\nnsn+s†\nntn)·(t†\nmsm+s†\nmtm),\nHs2=−i\n4X\nn,mJn,m(t†\nnsn+s†\nntn)·t†\nm×tm,\nHs3=−1\n8X\nn,mJn,m[(t†\nn·t†\nm)(tn·tm)−t†\nnαtnβt†\nmβtmα],\nHs4=1\n2X\nn,mJn,m[(t†\nnsn+s†\nntn−it†\nn×tn)·(Sa,m+Sb,m)+(Sa,n+Sb,n)·(Sa,m+Sb,m)],(S73)\nwith Jn,m=JH<0 for nearest neighbors and vanishing otherwise.\nC. Simplest mean-field approximation\nIn the simplest mean-field approach, we condense singlets, s†\nn,sn→s, and collect terms up to quadratic order in the remaining\noperators. The Kondo coupling term is unchanged,\nJKX\nnSn·sn→X\nn[3JK\n4(b†\nnσbnσ+a†\nnσanσ)+JKt†\nn·tn]−3NJK\n4\n=X\nk[3JK\n4(b†\nkbk+a†\nkak)+JKt†\nk·tk]−3NJK\n4. (S74)\nThe hopping term for itinerant electrons becomes\n−tX\nnc†\nncn+1+H.c.→s2\n2X\nn,mtn,m[(−a†\nman+b†\nnbm−(b†\nniτya†\nm+H.c.)]\n=−ts2X\nk[cosk(−a†\nkak+b†\nkbk)−(coskb†\nkiτya†\n−k+H.c.)]. (S75)\nAnd finally the Heisenberg coupling term becomes\nJHX\nnSn·Sn+1→s2\n8X\nn,mJn,m(t†\nn+tn)·(t†\nm+tm)\n=s2JH\n4X\nk[2 cos kt†\nk·tk+(coskt†\nk·t†\n−k+H.c.)]. (S76)23\nWe further introduce two Lagrange multipliers, λandµ, to enforce Eq. (S62) globally,\n−λX\nnh\ns†\nnsn+t†\nn·tn+a†\nnan+b†\nnbn−1i\n(S77)\nand also to fix the cfermion number at half-filling,\n0=−µX\nn\u0010\ns†\nnsn+t†\nn·tn+2b†\nnbn−1\u0011\n=−µX\nn\u0010\nb†\nnbn−a†\nnan\u0011\n. (S78)\nThus the resulting mean-field Hamiltonian reads\nHMF=HF+HB−3NJK\n4+N(1−s2)λ, (S79)\nwhere\nHF=X\nk[(3JK\n4−λ+µ+s2tcosk)a†\nkak+(3JK\n4−λ−µ−s2tcosk)b†\nkbk+(s2tcoskb†\nkiτya†\n−k+H.c.)], (S80)\nHB=X\nk[(JK−λ−s2J\n2cosk)t†\nk·tk+(s2JH\n4coskt†\nk·t†\n−k+H.c.)]. (S81)\nD. Solving the fermionic part\nThe fermionic sector, Eq. (S80), can be written as\nHF=X\nk\u0010\na†\nkb†\nk\u0011\nAk ak\nbk!\n+\"1\n2\u0010\na†\nkb†\nk\u0011\nBk a†\n−k\nb†\n−k!\n+H.c.#\n, (S82)\nwhere\nAk= 3JK\n4−λ+µ+s2tcosk 0\n03JK\n4−λ−µ−s2tcosk!\n, (S83)\nBk= 0 s2tcoskiτy\ns2tcoskiτy0!\n. (S84)\nBy defining the Nambu spinor ψ†\nk=\u0010\na†\nkb†\nka−kb−k\u0011\n, Eq. (S82) can be further recast into the Bogoliubov-de Gennes (BdG)\nform, HF=1\n2P\nkψ†\nkMkψk+1\n2P\nkTr{Ak}, where the BdG matrix reads Mk= AkBk\nB†\nk−AT\n−k!\n. This matrix can be diagonalized by\nusing a unitary matrix Uk, such thatψk=Ukφk, and\n1\n2X\nkψ†\nkMkψk=1\n2X\nkφ†\nkDkφk\n=X\nkea,k˜a†\nk˜ak+eb,k˜bk˜bk−X\nkX\nl=a,bel,k. (S85)\nNote that there is a factor of two for the last term, which is due to double degeneracy. Here φ†\nk=\u0010\n˜a†\nk˜b†\nk˜ak˜bk\u0011\nis the Nambu\nspinor form of the fermionic quasiparticle, Dkis a diagonal matrix, and the eigenvalues are\nea(b),k=r\n(3JK\n4−λ)2+s4t2cos2k±µ±s2tcosk. (S86)\nHence Eq. (S80) becomes\nHF=X\nkea,k˜a†\nk˜ak+eb,k˜b†\nk˜bk−X\nkX\nl=a,bel,k+1\n2X\nkTr{Ak}. (S87)24\nE. Solving the bosonic part\nThe bosonic sector, Eq. (S81) can be further written as\nHB=X\nkt†\nkAktk+ 1\n2t†\nkBkt†\n−k+H.c.!\n, (S88)\nwhere\nAk=K−λ+s2JH\n2cosk, (S89)\nandBk=s2JH\n2cosk. By defining the Nambu spinor ψ†\nk=\u0010\nt†\nkt−k\u0011\n, Eq. (S88) can be written in the BdG form, Hb=\n1\n2P\nkψ†\nkMkψk−P\nk3\n2Tr{Ak}. Note that there is a factor of 3 for the last term, which is due to triple degeneracy. And the\nBdG matrix reads Mk= AkBk\nB†\nkAT\n−k!\n. This matrix is diagonalized by a pseudounitary matrix Uk= coshθksinhθk\nsinhθkcoshθk!\n, with\ntanh 2θk=−s2JH\n2cosk/(JK−λ+s2JH\n2cosk), such that ψk=Ukφk, and\n1\n2X\nkψ†\nkMkψk=1\n2X\nkφ†\nk ωk\nωk!\nφk=X\nkωk˜tk˜tk+3\n2X\nkωk. (S90)\nNote again the factor of 3 in the last term. Here φ†\nk=\u0010\n˜t†\nk˜t−k\u0011\nis the Nanbu spinor form of the triplon, with dispersion\nωk=p\n(JK−λ)2+(JK−λ)s2JHcosk, (S91)\nHence Eq. (S81) becomes\nHb=X\nkωk˜t†\nk˜tk+3\n2X\nkωk−3\n2X\nkTr{Ak}. (S92)\nF. Helmholtz free energy and its minimization\nBased on above analysis, we can now obtain the Helmholtz free energy density of the system,\nf=−2\nβNX\nkX\nl=a,blog[1 +exp\u0000−βel,k\u0001]+3\nβNlog[1−exp(−βωk)]\n−1\nNX\nkX\nl=a,bel,k+1\n2NX\nkTr\bAF,k\t)+3\n2NX\nkωk−3\n2NX\nkTr\bAB,k\t(S93)\n−3JK\n4+λ(1−s2), (S94)\nwhere AB,kandAF,kare given by Eq. (S89) and (S83), respectively.\nThere are three unknown parameters: s,λandµ, which have to be determined by minimizing the free energy density. For λ,\nwe have\n0=∂λF (S95)\n=3\nNX\nk\"\nfB(ωk)+1\n2#−JK+λ−s2JH\n2cosk\np\n(JK−λ)2+(JK−λ)s2JHcosk\n−2\nNX\nkX\nl=a,b\"\nfF(el,k)−1\n2# 3JK\n4−λ\nq\n(3JK\n4−λ)2+s4t2cos2k\n+1\n2−s2. (S96)25\ns = h s n i = h sy\nn i\n3 4 5\nJ K10-210-1100T\n00.20.40.60.81KIPM\nFIG. S17. Finite-temperature phase diagram for large JK, obtained by bond-operator theory, with other parameters chosen the same as Fig. 1(a).\nForµ, we have\n0=∂µf=2\nNX\nk[fF(ea,k)−fF(eb,k)]. (S97)\nAnd for s, we have\n0=∂sF (S98)\n=−3\nNX\nk\"\nfB(ωk)+1\n2#−(JK−λ)sJHcoskp\n(JK−λ)2+(JK−λ)s2JHcosk\n+2\nNX\nk\"\nfF(ea,k)−1\n2#2s3t2cos2kq\n(3JK\n4−λ)2+s4t2cos2k+2stcosk\n+2\nNX\nk\"\nfF(ea,k)−1\n2#2s3t2cos2kq\n(3JK\n4−λ)2+s4t2cos2k−2stcosk\n−2λs. (S99)\nHere fF(B)(x)=(eβx±1)−1is the Fermi (Bose) function.\nWe numerically solve these three coupled nonlinear equations, (S96), (S97) and (S99). The finite-temperature crossover from\nparamagnetic (PM) phase to the Kondo insulating (KI) phase can be characterized by the onset of a nonzero s. In Fig. S17, we\nplot the condensed singlet ⟨s⟩as a function of temperature Tand the Kondo coupling JK. It is found that this strong coupling\napproach is in excellent agreement with the XTRG result for large JK."    },    {        "title": "2401.09938v2.Real_space_nonlocal_Gilbert_damping_from_exchange_torque_correlation_applied_to_bulk_ferromagnets_and_their_surfaces.pdf",        "content": "Real-space nonlocal Gilbert damping from exchange torque correlation applied to\nbulk ferromagnets and their surfaces\nBalázs Nagyfalusi,1,2,∗László Szunyogh,2,3,†and Krisztián Palotás1,2,‡\n1Institute for Solid State Physics and Optics, HUN-REN Wigner Research Center for Physics,\nKonkoly-Thege M. út 29-33, H-1121 Budapest, Hungary\n2Department of Theoretical Physics, Institute of Physics,\nBudapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary\n3HUN-REN-BME Condensed Matter Research Group,\nBudapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary\n(Dated: February 29, 2024)\nIn this work we present an ab initio scheme based on linear response theory of exchange torque\ncorrelation, implemented into the real-space Korringa-Kohn-Rostoker (RS-KKR) framework to cal-\nculate diagonal elements of the atomic-site-dependent intrinsic Gilbert damping tensor. The method\nis first applied to bcc iron and fcc cobalt bulk systems. Beside reproducing earlier results from the\nliterature for those bulk magnets, the effect of the lattice compression is also studied for Fe bulk,\nand significant changes for the Gilbert damping are found. Furthermore, (001)-oriented surfaces\nof Fe and Co are also investigated. It is found that the on-site Gilbert damping increases in the\nsurface atomic layer and decreases in the subsurface layer, and approaches the bulk value moving\nfurther inside the magnets. Realistic atomic relaxation of the surface layers enhances the identified\neffects. Thefirst-neighbordampingparametersareextremelysensitivetothesurfacerelaxation. De-\nspite their inhomogeneity caused by the surface, the transverse Gilbert damping tensor components\nremain largely insensitive to the magnetization direction.\nI. INTRODUCTION\nIt is highly demanded to understand and control\nthe dynamical processes governing the manipulation\nof various magnetic textures, such as atomic chains1,2,\nmagnetic skyrmions3,4or domain walls5, which can\nbe potentially used in future magnetic recording and\nlogic devices. These processes are often described by\nthe phenomenological Landau-Lifshitz-Gilbert (LLG)\nequation6,7,\n∂ ⃗ mi\n∂t=−γ ⃗ mi×⃗Beff\ni+α\nmi⃗ mi×∂ ⃗ mi\n∂t,(1)\nwhere ⃗ miis the magnetic moment at site i,mi=|⃗ mi|\nis its length, and γis the gyromagnetic ratio. The\nfirsttermonthe rhsofEq.(1)describestheprecession\nof⃗ miaround the effective magnetic field ⃗Beff\ni, while\nthe second term is the Gilbert damping due to the\nenergy dissipation to the lattice. Clearly, this latter\nterm causes the relaxation of the magnetization to its\nequilibrium value, which is controlled by the damping\nconstant αand plays a crucial role in the realization\nof high-speed spintronic devices.\nThe Gilbert damping constant αcan be deter-\nmined experimentally from the ferromagnetic reso-\nnance (FMR) spectroscopy where the damping pa-\nrameter is related to the line-width in the measured\nspectra8. FMR spectroscopy is a well-established\nmethod for bulk materials9,10, but especially in the\nlow temperature measurement it is controversial be-\ncause the intrinsic Gilbert damping needs to be sepa-\nrated from various extrinsic sources of the line-width,\ne.g., two-magnon scattering, eddy-current damping,\nradiative damping, spin-pumping, or the slow relaxer\nmechanism11–16. The comparison of experimental\nmeasurement to theoretical calculations is also made\ndifficult bythe sampleproperties likethe exactatomic\nstructure.From a theoretical perspective the ultimate goal is\nto develop a method to calculate the Gilbert damp-\ning parameters from the electronic structure of the\nmaterial. In the last decades there have been sev-\neral efforts to understand the damping process. The\nfirst successful method was developed by Kamberský\nwhorelatedthedampingprocesstothespin-orbitcou-\npling (SOC) in terms of the breathing Fermi surface\nmodel17, while he also proposed the spin-orbit torque\ncorrelation model18,19. Later on several other meth-\nods were introduced such as the spin-pumping20and\nlinear-response approaches11,21,22. A recent summary\nof these methods was published by Guimarães et al.23\nDue to the increased interest in noncollinear mag-\nnetism Fähnle et al.24suggested an inhomogeneous\ntensorial damping. The replacement of a scalar αby\na damping matrix αmeans that the damping field in\nEq.(1)isnolongerproportionaltothetimederivative\nof⃗ mi, it becomes a linear function of ∂ ⃗ mi/∂t. More-\nover, nonlocality of the damping process implies that\nthe damping field at site iexperiences ∂ ⃗ mj/∂tfor any\nsitej. The LLG equation (1) is then replaced by the\nset of equations25,\n∂ ⃗ mi\n∂t=⃗ mi×\n−γ⃗Beff\ni+X\njαij1\nmj∂ ⃗ mj\n∂t\n,(2)\nwhere the damping term is unfolded to pairwise con-\ntributions of strength αij. The appearance of non-\nlocal damping terms was evidenced for magnetic do-\nmain walls26,27by linking the Gilbert damping to the\ngradients of the magnetization. In NiFe, Co, and\nCoFeB thin films Li et al.28measured wave-number-\ndependent dissipation using perpendicular spin wave\nresonance, validating thus the idea of nonlocal damp-\ning terms. Different analytical expressions for αijare\nalready proposed22,25,29,30, and the nonlocal damp-\ning is found for bulk materials25,31as well as its ef-arXiv:2401.09938v2  [cond-mat.mtrl-sci]  28 Feb 20242\nfect on magnon properties of ferromagnets have been\ndiscussed32. Recent studies went further and, anal-\nogously to the higher order spin-spin interactions in\nspin models, introduced multi-body contributions to\nthe Gilbert damping33.\nThe calculation of the Gilbert damping prop-\nerties of materials has so far been mostly fo-\ncused on 3D bulk magnets, either in chemically\nhomogeneous11,19,23,25,34–36or heterogeneous (e.g.\nalloyed)11,22,31forms. There are a few studies avail-\nable reporting on the calculation of the Gilbert damp-\ning in 2D magnetic thin films12,23,37,38, or at surfaces\nand interfaces of 3D magnets31,35,37. The calculation\nof the Gilbert damping in 1D or 0D magnets is, due\nto our knowledge, not reported in the literature. Fol-\nlowing the trend of approaching the atomic scale for\nfunctional magnetic elements in future spintronic de-\nvices, the microscopic understanding of energy dissi-\npation through spin dynamics in magnets of reduced\ndimensions is inevitable and proper theoretical meth-\nods have to be developed.\nOur present work proposes a calculation tool for\nthediagonalelementsofthenon-localintrinsicGilbert\ndamping tensor covering the 3D to 0D range of mag-\nnetic materials on an equal footing, employing a real-\nspace embedding Green’s function technique39. For\nthis purpose, the linear response theory of the Gilbert\ndamping obtained by the exchange torque correlation\nis implemented in the real-space KKR method. As a\ndemonstration of the new method, elemental Fe and\nCo magnets in their 3D bulk form and their (001)-\noriented surfaces are studied in the present work. Go-\ning beyond comparisons with the available literature,\nnew aspects of the Gilbert damping in these materials\nare also reported.\nThe paper is organized as follows. In Sec. II the\ncalculation of the Gilbert damping parameters within\nthe linear response theory of exchange torque corre-\nlation using the real-space KKR formalism is given.\nSec. III reports our results on bulk bcc Fe and fcc Co\nmaterials and their (001)-oriented surfaces. We draw\nour conclusions in Sec. IV.\nII. METHOD\nA. Linear response theory within real-space\nKKR\nThe multiple-scattering of electrons in a finite clus-\nter consisting of NCatoms embedded into a 3D or 2D\ntranslation-invariant host medium is fully accounted\nfor by the equation39\nτC=τH\u0002\nI−(t−1\nH−t−1\nC)τH\u0003−1,(3)\nwhere τCand τHare the scattering path operator\nmatrices of the embedded atomic cluster and the host,\nrespectively, tCandtHare the corresponding single-\nsite scattering matrices, all in a combined atomic site\n(j, k∈ {1, ..., N C}) and angular momentum ( Λ,Λ′∈\n{1, ...,2(ℓmax+ 1)2}) representation: τ={τjk}=\n{τjk\nΛΛ′}andt={tj\nΛΛ′δjk}, where ℓmaxis the angularmomentum cutoff in describing the scattering events,\nand for simplicity we dropped the energy-dependence\nof the above matrices.\nFor calculating the diagonal Cartesian elements\nof the nonlocal Gilbert damping tensor connecting\natomic sites jandkwithin the finite magnetic atomic\ncluster, we use the formula derived by Ebert et al.22,\nαµµ\njk=2\nπmj\nsTr\u0010\nTj\nµ˜τjk\nCTk\nµ˜τkj\nC\u0011\n, (4)\nwhere µ∈ {x, y, z}, the trace is taken in the\nangular-momentum space and the formula has to\nbe evaluated at the Fermi energy ( EF). Here,\nmj\nsis the spin moment at the atomic site j,\n˜τjk\nC,ΛΛ′= (τjk\nC,ΛΛ′−(τkj\nC,Λ′Λ)∗)/2i, and Tj\nµis the\ntorque operator matrix which has to be calculated\nwithin the volume of atomic cell j,Ωj:Tj\nµ;ΛΛ′=R\nΩjd3rZj\nΛ(⃗ r)×βσµBxc(⃗ r)Zj\nΛ′(⃗ r),wherethenotationof\nthe energy-dependence is omitted again for simplicity.\nHere, βis a standard Dirac matrix entering the Dirac\nHamiltonian, σµare Pauli matrices, and Bxc(⃗ r)is the\nexchange-correlation field in the local spin density ap-\nproximation (LSDA), while Zj\nΛ(⃗ r)are right-hand side\nregular solutions of the single-site Dirac equation and\nthe superscript ×denotes complex conjugation re-\nstricted to the spinor spherical harmonics only22. We\nshould emphasize that Eq. (4) applies to the diagonal\n(µµ) elements of the Gilbert tensor only. To calcu-\nlate the off-diagonal tensor elements one needs to use,\ne.g., the more demanding Kubo-Bastin formula40,41.\nNote also that in noncollinear magnets the exchange\nfield Bxc(⃗ r)is sensitive to the spin noncollinearity42\nwhich influences the calculated torque operator ma-\ntrix elements, however, this aspect does not concern\nour present study including collinear magnetic states\nonly.\nNote that the nonlocal Gilbert damping is, in gen-\neral, not symmetric in the atomic site indices, αµµ\njk̸=\nαµµ\nkj, instead\nαµµ\nkj=mj\ns\nmksαµµ\njk(5)\nholds true. This is relevant in the present work for\nthe ferromagnetic surfaces. On the other hand, in\nferromagnetic bulk systems αµµ\njk=αµµ\nkjsince mj\ns=\nmk\ns=msfor any pair of atomic sites.\nIn practice, the Gilbert damping formula in Eq. (4)\nis not directly evaluated at the Fermi energy, but a\nsmall imaginary part ( η) of the complex energy is ap-\nplied, which is called broadening in the following, and\nits physical effect is related to the scattering rate in\nother damping theories19,25,37,43. Taking into account\nthe broadening η, the Gilbert damping reads\nαµµ\njk(η) =−1\n4h\n˜αµµ\njk(E+, E+) + ˜αµµ\njk(E−, E−)\n−˜αµµ\njk(E+, E−)−˜αµµ\njk(E−, E+)i\n,(6)\nwhere E+=EF+iηandE−=EF−iη, and the3\nindividual terms are\n˜αµµ\njk(E1, E2) =\n2\nπmj\nsTr\u0010\nTj\nµ(E1, E2)τjk\nC(E2)Tk\nµ(E2, E1)τkj\nC(E1)\u0011\n(7)\nwith E1,2∈ { E+, E−}, and the ex-\nplicitly energy-dependent torque opera-\ntor matrix elements are: Tj\nµ;ΛΛ′(E1, E2) =R\nΩjd3rZj×\nΛ(⃗ r, E1)βσµBxc(⃗ r)Zj\nΛ′(⃗ r, E2).\nB. Effective damping and computational\nparameters\nEq.(6)givesthebroadening-dependentspatiallydi-\nagonal elements of the site-nonlocal Gilbert damping\ntensor: αxx\njk(η),αyy\njk(η), and αzz\njk(η). Since no longitu-\ndinal variation of the spin moments is considered, the\ntwo transversal components perpendicular to the as-\nsumed uniform magnetization direction are physically\nmeaningful. Given the bulk bcc Fe and fcc Co sys-\ntems and their (001)-oriented surfaces with C4vsym-\nmetry under study in the present work, in the follow-\ning the scalar αrefers to the average of the xxand\nyyGilbert damping tensor components assuming a\nparallel magnetization with the surface normal z[001]-\ndirection: αjk= (αxx\njk+αyy\njk)/2 =αxx\njk=αyy\njk. From\nthe site-nonlocal spatial point of view in this work we\npresent results on the on-site (\" 00\"), first neighbor\n(denoted by \" 01\") and second neighbor (denoted by\n\"02\") Gilbert damping parameters, and an effective,\nso-called total Gilbert damping ( αtot), which can be\ndefined as the Fourier transform of αjkat⃗ q= 0. The\nFourier transform of the Gilbert damping reads\nα⃗ q=∞X\nj=0α0jexp(−i⃗ q(⃗ r0−⃗ rj))\n≈X\nr0j≤rmaxα0jexp(−i⃗ q(⃗ r0−⃗ rj)),(8)\nwhere r0j=|⃗ r0−⃗ rj|and the effective damping is\ndefined as\nαtot=α⃗ q=⃗0=∞X\nj=0α0j≈X\nr0j≤rmaxα0j.(9)\nSince we have a real-space implementation of the\nGilbert damping, the infinite summation for both\nquantities is replaced by an approximative summation\nfor neighboring atoms upto an rmaxcutoff distance\nmeasured from site \"0\". Moreover, note that for bulk\nsystems the effective damping αtotis directly related\nto the ⃗ q= 0mode of FMR experiments.\nThe accuracy of the calculations depends on many\nnumerical parameters such as the number of ⃗kpoints\nused in the Brillouin zone integration, the choice of\nthe angular momentum cutoff ℓmax, and the spatial\ncutoff rmaxused for calculating α⃗ qandαtot. Previ-\nous research25showed that the Gilbert damping heav-\nily depends on the broadening η, so we extended ourstudies to a wider range of η= 1meV to 1 eV. The\nsufficient k-point sampling was tested at the distance\nofrmax= 7a0(where a0is the corresponding 2D lat-\ntice constant) from the reference site with the broad-\nening set to 1mRy, and the number of ⃗kpoints was\nincreased up to the point, where the 5th digit of the\ndamping became stable. Maximally, 320400 ⃗kpoints\nwere used for the 2D layered calculation but the re-\nquested accuracy was reached with 45150 and 80600\n⃗kpoints for bulk bcc Fe and fcc Co systems, respec-\ntively.\nThe choice of ℓmaxwas tested through the whole η\nrange for bcc Fe, and it was based on the comparison\nof damping calculations with ℓmax= 2andℓmax= 3.\nThe maximal deviation for the on-site Gilbert damp-\ning was found at around η= 5mRy, but it was still\nless than 10%. The first and second neighbor Gilbert\ndamping parameters changed in a more significant\nway (by ≈50%) in the whole ηrange upon changing\nℓmax, yet the effective total damping was practically\nunchanged, suggesting that farther nonlocal damping\ncontributions compensate this effect. Since αtotis the\nmeasurable physical quantity we concluded that the\nlower angular momentum cutoff of ℓmax= 2is suffi-\ncient to be used further on.\nThe above choice of ℓmax= 2for the angular mo-\nmentumcutoff, themathematicalcriterionofpositive-\ndefinite αjk(which implies α⃗ q>0for all ⃗ qvectors),\nand the prescribed accuracy for the effective Gilbert\ndamping in the full considered η= 1meV to 1 eV\nrange set rmaxto 20 a0for both bcc Fe and fcc Co. It\nis worth mentioning that the consideration of lattice\nsymmetries made possible to decrease the number of\natomic sites in the summations for calculating α⃗ qand\nαtotby an order of magnitude.\nIII. RESULTS AND DISCUSSION\nOur newly implemented method was employed to\nstudy the Gilbert damping properties of Fe and Co\nferromagnetsintheirbulkand(001)-orientedsurfaces.\nIn these cases only unperturbed host atoms form\nthe atomic cluster, and the so-called self-embedding\nprocedure44is employed, where Eq. (3) reduces to\nτC=τHforthe3Dbulkmetalsand2Dlayeredmetal-\nvacuum interfaces.\nA. Bulk Fe and Co ferromagnets\nFirst we calculate and analyze the nonlocal and ef-\nfective dampings for bulk bcc Fe by choosing a 2D\nlattice constant of a0= 2.863Å. The magnitude of\nthe magnetic moments are obtained from the self-\nconsistent calculation. The spin and orbital moments\narems= 2.168µBandmo= 0.046µB, respectively.\nThe broadening is set to η= 68meV. The inset of Fig.\n1a) shows the typical function of the nonlocal Gilbert\ndamping α0jdepending on the normalized distance\nr0j/a0between atomic sites \"0\" and \" j\". In accor-\ndance with Ref. 25 the nonlocal Gilbert damping\nquickly decays to zero with the distance, and can be4\na)\n5 10 15 2005\n5 10 15 20−505\nr0j/a0α0j[×10−4]\nr0j/a0α0j·(r0j/a0)2[×10−4]\nb)\n5 10 15 20−202468\nrmax/a0αtot[×10−3]\nFIG. 1. a) Nonlocal Gilbert damping in bulk bcc Fe as a\nfunction of distance r0jbetween atomic sites \"0\" and \" j\"\nshown upto a distance of 20 a0(the 2D lattice constant is\na0= 2.863Å): the black squares are calculated α0jval-\nues times the normalized squared-distance along the [110]\ncrystallographic direction, and the red line is the corre-\nsponding fitted curve based on Eq. (10). The inset shows\nthe nonlocal Gilbert damping α0jvalues in the given dis-\ntance range. b) Convergence of the effective damping pa-\nrameter αtot, partial sums of α0jupto rmaxbased on Eq.\n(9), where rmaxis varied. The broadening is chosen to be\nη=68 meV.\nwell approximated with the following function:\nα(r)≈Asin (kr+ϕ0)\nr2exp(−βr).(10)\nTo test this assumption we assorted the atomic sites\nlying in the [110] crystallographic direction and fit-\nted Eq. (10) to the calculated data. In practice, the\nfit is made on the data set of α0j(r0j/a0)2, and is\nplotted in Fig. 1a). Although there are obvious out-\nliers in the beginning, the magnitude of the Gilbert\ndamping asymptotically follows the ∝exp(−βr)/r2\ndistance dependence. The physical reason for this de-\ncay is the appearance of two scattering path operators\n(Green’s functions) in the exchange torque correlation\nformula in Eq. (4) being broadened due to the finite\nimaginary part of the energy argument.\nIn our real-space implementation of the Gilbert\ndamping, an important parameter for the effective\ndamping calculation is the real-space cutoff rmaxin\nEq. (9). Fig. 1b) shows the evolution of the ef-\nfective (total) damping depending on the rmaxdis-\ntance, within which all nonlocal damping terms α0jare summed up according to Eq. (9). An oscillation\ncan similarly be detected as for the nonlocal damping\nitself in Fig. 1a), and this behavior was fitted with\na similar exponentially decaying oscillating function\nas reported in Eq. (10) in order to determine the ex-\npected total Gilbert damping αtotvalue in the asymp-\ntotic r→ ∞limit. In the total damping case it is\nfound that the spatial decay of the oscillation is much\nslower compared to the nonlocal damping case, which\nmakes the evaluation of αtotmore cumbersome. Our\ndetailed studies evidence that for different broaden-\ningηvalues the wavelength of the oscillation stays\nthe same but the spatial decay becomes slower as\nthe broadening is decreased (not shown). This slower\ndecay together with the fact that the effective (to-\ntal) damping value itself is also decreasing with the\ndecreasing broadening results that below the 10meV\nrange of ηthe amplitude of the oscillation at the dis-\ntance of 20 a0is much larger than its asymptotic limit.\nIn practice, since the total damping is calculated as\nther→ ∞limit of such a curve as shown in Fig. 1b),\nthis procedure brings an increased error for αtotbelow\nη= 10meV, and this error could only be reduced by\nincreasing the required number of atomic sites in the\nreal-space summation in Eq. (10).\nFig. 2 shows the dependence of the calculated on-\nsite, first- and second-neighbor and effective total\nGilbert damping parameters on the broadening η.\nThe left column shows on-site ( α00) and total ( αtot)\nwhile the right one the first ( α01) and second ( α02)\nneighbor Gilbert dampings. We find very good agree-\nment with the earlier reported results of Thonig et\nal.25, particularly that the on-site damping has the\nlargest contribution to the total damping being in the\nsame order of magnitude, while the first and second\nneighbors are smaller by an order of magnitude. The\nobtained dependence on ηis also similar to the one\npublished by Thonig et al.25:α00andαtotare in-\ncreasing with η, and α01andα02do not follow a\ncommon trend, and they are material-dependent, see,\ne.g., the opposite trend of α02with respect to ηfor\nFe and Co. The observed negative values of some of\nthesite-nonlocaldampingsarestillconsistentwiththe\npositive-definiteness of the full (infinite) αjkmatrix,\nwhich has also been discussed in Ref. 25.\nThe robustness of the results was tested against a\nsmall change of the lattice constant simulating the ef-\nfect of an external pressure for the Fe bulk. These re-\nsults are presented in the second row of Fig. 2, where\nthe lattice constant of Fe is set to a0= 2.789Å. In this\ncase the magnetic moments decrease to ms= 2.066µB\nandmo= 0.041µB. It can clearly be seen that the on-\nsite, first and second neighbor Gilbert dampings be-\ncome smaller upon the assumed 2.5% decrease of the\nlattice constant, but the total damping remains prac-\ntically unchanged in the studied ηrange. This sug-\ngests that the magnitudes of more distant non-local\ndamping contributions are increased.\nThe third row of Fig. 2 shows the selected damp-\ning results for fcc Co with a 2D lattice constant of\na0= 2.507Å. The spin and orbital moments are\nms= 1.654µBandmo= 0.078µB, respectively. The\nincreaseofthetotal, theon-site, andthefirst-neighbor5\n10−310−210−11000246810Fe -a0= 2.863˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Fe -a0= 2.863˚A\nη(eV)α[×10−4]α01\nα02\n10−310−210−11000246810Fe -a0= 2.789˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Fe -a0= 2.789˚A\nη(eV)α[×10−4]α01\nα02\n10−310−210−11000246810Co -a0= 2.507˚A\nη(eV)α[×10−3]α00\nαtot\n10−310−210−1100−202468Co -a0= 2.507˚A\nη(eV)α[×10−4]α01\nα02\nFIG. 2. Left column: Local on-site ( α00, black square)\nand total ( αtot, red triangle) Gilbert damping as a func-\ntionofthebroadening ηforbccFe(001)with a0= 2.863Å,\nbcc Fe(001) with a0= 2.789Å, and fcc Co(001) with\na0= 2.507Å. Right column: Nonlocal first nearest neigh-\nbor (α01, black square) and second nearest neighbor ( α02,\nred triangle) Gilbert damping for the same systems.\ndampings with increasing ηis similar to the Fe case,\nand the on-site term dominates αtot. An obvious\ndifference is found for the second-neighbor damping,\nwhich behaves as an increasing function of ηfor Co\nunlike it is found for Fe.\nConcerning the calculated damping values, there is\na large variety of theoretical methods and calculation\nparameters, as well as experimental setups used in\nthe literature, which makes ambiguous to compare\nour results with others. Recently, Miranda et al.31\nreported a comparison of total and on-site damping\nvalues with the available theoretical and experimen-\ntal literature in their Table S1. For bcc Fe bulk they\nreported total damping values in the range of 1.3–\n4.2×10−3and for fcc Co bulk within the range of 3.2–\n11×10−3, and our results fit very well within theseranges around η≈100meV for Fe and for η >100\nmeV for Co. Moreover, we find that our calculated on-\nsite damping values for bcc Fe are larger ( >5×10−3)\nthan the reported values of Miranda et al.(1.6×10−3\nand 3.6 ×10−3), but for fcc Co the agreement with\ntheir reported total (3.2 ×10−3) and on-site damping\n(5.3×10−3) values is very good at our η= 136meV\nbroadening value.\n10−310−210−110010−510−410−310−2Fe\nη(eV)αtot\nαSOC=1\nαSOC=0\n10−310−210−110010−510−410−310−2Co\nη(eV)αtot\nαSOC=1\nαSOC=0\nFIG. 3. Effective (total) Gilbert damping for bcc Fe\n(left) and fcc Co (right) as a function of broadening ηon\na log-log scale. The error bars are estimated from the\nfitting procedure of Eq. (10). The red triangles show the\ncase with normal SOC ( αSOC=1), and the blue diamonds\nwhere SOC is switched off ( αSOC=0).\nNext, weinvestigatethespin-orbit-coupling-(SOC)-\noriginated contribution to the Gilbert damping. Our\nmethodmakesitinherentlypossibletoincludeaSOC-\nscaling factor in the calculations45. Fig. 3 shows the\nobtained total Gilbert damping as a function of the\nbroadening ηwith SOC switched on/off for bcc Fe\nand fcc Co. It can be seen that the effect of SOC\nis not dominant at larger ηvalues, but the SOC\nhas an important contribution at small broadening\nvalues ( η < 10−2eV), where the calculated total\nGilbert damping values begin to deviate from each\nother with/without SOC. As discussed in Ref. 23,\nwithout SOC the damping should go toward zero for\nzero broadening, which is supported by our results\nshown in Fig. 3.\nB. (001)-oriented surfaces of Fe and Co\nferromagnets\nIn the following, we turn to the investigation of the\nGilbertdampingparametersatthe(001)-orientedsur-\nfaces of bcc Fe and fcc Co. Both systems are treated\nas a semi-infinite ferromagnet interfaced with a semi-\ninfinite vacuum within the layered SKKR method46.\nIn the interface region 9 atomic layers of the ferromag-\nnet and 3 atomic layers of vacuum are taken, which is\nsandwiched between the two semi-infinite (ferromag-\nnet and vacuum) regions. Two types of surface atomic\ngeometries were calculated: (i) all atomic layers hav-\ning the bulk interlayer distance, and (ii) the surface\nand subsurface atomic layers of the ferromagnets have6\nTABLE I. Geometry relaxation at the surfaces of the fer-\nromagnets: change of interlayer distances relative to the\nbulk interlayer distance at the surfaces of bcc Fe(001) and\nfcc Co(001), obtained from VASP calculations. \"L1\" de-\nnotes the surface atomic layer, \"L2\" the subsurface atomic\nlayer, and \"L3\" the sub-subsurface atomic layer. All other\ninterlayer distances are unchanged in the geometry opti-\nmizations.\nL1-L2 L2-L3\nbcc Fe(001) -13.7% -7.7%\nfcc Co(001) -12.4% -6.4%\nbeenrelaxedintheout-of-planedirectionusingtheVi-\nenna Ab-initio Simulation Package (VASP)47within\nLSDA48. For the latter case the obtained relaxed\natomic geometries are given in Table I.\nFigure 4 shows the calculated layer-resolved on-\nsite and first-neighbor Gilbert damping values (with\nη= 0.68eV broadening) for the bcc Fe(001) and fcc\nCo(001) surfaces. It can generally be stated that the\nsurface effects are significant in the first 4 atomic lay-\ners of Fe and in the first 3 atomic layers of Co. We\nfind that the on-site damping ( α00) increases above\nthe bulk value in the surface atomic layer (layer 1:\nL1), and decreases below the bulk value in the sub-\nsurface atomic layer (L2) for both Fe and Co. This\nfinding is interesting since the spin magnetic moments\n(ms, shown in the insets of Fig. 4) are also consider-\nably increased compared to their bulk values in the\nsurface atomic layer (L1), and the spin moment enters\nthe denominator when calculating the damping in Eq.\n(4).α00increases again in L3 compared to its value in\nL2, thus it exhibits a nonmonotonic layer-dependence\nin the vicinity of the surface. The damping results ob-\ntained with the ideal bulk interlayer distances and the\nrelaxed surface geometry (\"R\") are also compared in\nFig. 4. It can be seen that the on-site damping is in-\ncreasedinthesurfaceatomiclayer(L1), anddecreased\nin the subsurface (L2) and sub-subsurface (L3) atomic\nlayers upon atomic relaxation (\"R\") for both Fe and\nCo. The first-neighbor dampings ( α01) are of two\ntypes for the bcc Fe(001) and three types for the fcc\nCo(001), see caption of Fig. 4. All damping values\nare approaching their corresponding bulk value mov-\ning closer to the semi-infinite bulk (toward L9). In\nabsolute terms, for both Fe and Co the maximal sur-\nface effect is about 10−3for the on-site damping, and\n2×10−4for the first-neighbor dampings. Given the\ndamping values, the maximal relative change is about\n15% for the on-site damping, and the first-neighbor\ndampings can vary by more than 100% (and can even\nchangesign)inthevicinityofthesurfaceatomiclayer.\nNote that Thonig and Henk35studied layer-resolved\n(effective) damping at the surface of fcc Co within the\nbreathing Fermi surface model combined with a tight-\nbinding electronic structure approach. Although they\nstudied a different quantity compared to us, they also\nreported an increased damping value in the surface\natomic layer, followed by an oscillatory decay toward\nbulk Co.\nSo far the presented Gilbert damping results cor-\nrespond to spin moments pointing to the crystallo-\n1 3 5 7 90.81Fe\n1 92.43\nlayermsms\nmR\ns\nlayerα00[×10−2]\nα00\nαR\n00\n1 3 5 7 90.81\n1 91.71.8\nlayermsms\nmR\nsCo\nlayerα00[×10−2]α00\nαR\n00\n1 3 5 7 9−2−101Fe\nlayerα01[×10−4]\nα01+αR\n01+\nα01−αR\n01−\n1 3 5 7 91234\nCo\nlayerα01[×10−4]α01+αR\n01+\nα01−αR\n01−\nα01 αR\n01FIG. 4. Evolution of the layer-resolved Gilbert damping\nfrom the surface atomic layer (L1) of bcc Fe(001) and fcc\nCo(001) toward the bulk (L9), depending also on the out-\nof-plane atomic relaxation \"R\". On-site ( α00) and first\nneighbor ( α01) Gilbert damping values are shown in the\ntop two and bottom two panels, respectively. The broad-\nening is η= 0.68eV. The empty symbols belong to the\ncalculations with the ideal bulk interlayer distances, and\nthe full symbols to the relaxed surface geometry, denoted\nwith index \"R\". Note that α01is calculated for nearest\nneighbors of atomic sites in the neighboring upper, lower,\nand the same atomic layer (for fcc Co only), and they are\nrespectively denoted by \" +\" (L-(L+1)), \" −\" (L-(L −1)),\nand no extra index (L-L). The insets in the top two panels\nshow the evolution of the magnitudes of the layer-resolved\nspin magnetic moments ms. The horizontal dashed line in\nall cases denotes the corresponding bulk value.\ngraphic [001] ( z) direction, and the transverse compo-\nnents of the damping αxxandαyyare equivalent due\ntothe C4vsymmetryofthe(001)-orientedsurfaces. In\norder to study the effect of a different orientation of\nall spin moments on the transverse components of the\ndamping, we also performed calculations with an ef-\nfective field pointing along the in-plane ( x) direction:\n[100] for bcc Fe and [110] for fcc Co. In this case, due\nto symmetry breaking of the surface one expects an\nanisotropy in the damping, i.e., that the transverse\ncomponents of the damping tensor, αyyandαzz, are\nnot equivalent any more. According to our calcula-7\ntions, however, the two transverse components of the\non-site ( αyy\n00andαzz\n00) and nearest-neighbor ( αyy\n01and\nαzz\n01) damping tensor, at the Fe surface differed by less\nthan 0.1 % and at the Co surface by less than 0.2 %,\ni.e., despite the presence of the surface the damping\ntensor remained highly isotropic. The change of the\ndamping with respect to the orientation of the spin\nmoments in zorxdirection (damping anisotropy)\nturned out to be very small as well: the relative dif-\nference in αyy\n00was 0.1 % and 0.3 %, while 0.5 % and\n0.1 % in αyy\n01for the Fe and the Co surfaces, respec-\ntively. For the farther neighbors, this difference was\nless by at least two orders of magnitude.\nIV. CONCLUSIONS\nWe implemented an ab initio scheme of calculat-\ning diagonal elements of the atomic-site-dependent\nGilbert damping tensor based on linear response the-\nory of exchange torque correlation into the real-space\nKorringa-Kohn-Rostoker (KKR) framework. To val-\nidate the method, damping properties of bcc Fe and\nfcc Co bulk ferromagnets are reproduced in good com-\nparison with the available literature. The lattice com-\npression is also studied for Fe bulk, and important\nchanges for the Gilbert damping are found, most pro-\nnounced for the site-nonlocal dampings. By investi-\ngating (001)-oriented surfaces of ferromagnetic Fe andCo, we point out substantial variations of the layer-\nresolved Gilbert damping in the vicinity of the sur-\nfaces depending on various investigated parameters.\nThe effect of such inhomogeneous dampings should be\nincluded into future spin dynamics simulations aim-\ning at an improved accuracy, e.g., for 2D surfaces and\ninterfaces. We anticipate that site-nonlocal damping\neffects become increasingly important when moving\ntoward physical systems with even more reduced di-\nmensions (1D).\nACKNOWLEDGMENTS\nThe authors acknowledge discussions with Danny\nThonig. Financial support of the National Research,\nDevelopment, and Innovation (NRDI) Office of Hun-\ngary under Project Nos. 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In this paper, we study the Cauchy problem of the nonlinear wa ve equation with fractional Lapla-\ncian and time-dependent damping.\n∂2\ntu(t,x)+(−∆)σ/2u(t,x)+b(t)∂tu(t,x)=f(u(t,x)),(t,x)∈[0,T)×RN,\nu(0,x)=u0(x),∂tu(0,x)=u1(x), x∈RN.\nFirstly, we derive the weighted Sobolev estimate of the solu tion operators for the linear wave equation with\nthe damping of constant coe fficient, and prove the local existence and uniqueness in the we ighted Sobolev\nspace for the power-type nonlinearity and b(t)∈L∞, by the contraction mapping principle. Secondly, we\nconsider the case of the source nonlinearity f(u)≈|u|p. In the subcritical and critical cases 1 <p≤pc=\n1+σ\nN, based on the blow-up result on the ordinary di fferential inequality, we could prove the blow-up of the\nsolution and obtain the upper bound of the lifespan. And the u pper bound of the lifespan in the critical case\nis independent on the coe fficient of the time-dependent damping and is completely new ev en if the classical\ncase b(t)=1.\nMathematics Subject Classification (2010): 35B44; 35A01; 35L15;35L05\nKey words and phrases : Fractional Laplacian, time-dependent damping, local exi stence, blow-up, lifespan\nContents\n1 Introduction 1\n2 Proof of local existence and uniqueness 6\n3 Proof of blow-up in subcritical case 11\n4 Proof of blow-up and upper estimate of lifespan in the criti cal case 15\n1 Introduction\nIn the present paper we study the Cauchy problem of the σ-evolution equation with a dissipative term\nwith a time-dependent coe fficient and a power nonlinearity on the Euclidean space\n∂2\ntu(t,x)+(−∆)σ/2u(t,x)+b(t)∂tu(t,x)=f(u(t,x)),(t,x)∈[0,T)×RN,\nu(0,x)=u0(x),∂tu(0,x)=u1(x), x∈RN.(1.1)\n*Department of Mathematics and Science, School of Sciences, Zhejiang Sci-Tech University, 310018, Hangzhou, China,\nE-mail:jylin@zstu.edu.cn\n†Department of Mathematics, Faculty of Science and Technolo gy, Keio University, 3-14-1 Hiyoshi,\nKohoku-ku, Yokohama, 223-8522, Japan /Center for Advanced Intelligence Project, RIKEN, Japan, E- mail:\nmasahiro.ikeda@keio.jp/masahiro.ikeda@riken.jp\n1Here N∈Nis the spatial dimension, σ> 0 stands for the strength of di ffusion and (−∆)σ/2is the\nfractional Laplace operator on RNdefined by\n(−∆)σ/2f(x) :=F−1[|ξ|σF[f](ξ)](x),\nwhereFandF−1are the Fourier transform and its inverse respectively. Mor eover T∈(0,∞] is the\nexistence time of the function u,u: [0,T)×RN×Ris an unknown function. Here b: [0,∞)→R>0is a\ngiven positive function, satisfying\nb1(1+t)−β≤b(t)≤b2(1+t)−β, and|b′(t)|≤b3(1+t)−1b(t), (1.2)\nforβ∈Rand some constants b1,b2,b3>0, and f:R→Ris a nonlinear function satisfying f(0)=0\nand there exist p≥1 and C0such that for any z,w∈R,\n|f(z)−f(w)|≤C0(|z|p−1+|w|p−1)|z−w|. (1.3)\nAnd u0:RN→Ris the initial shape, while u1:RN→Ris the initial velocity.\nLetS/parenleftig\nRN/parenrightig\nbe the rapidly decaying function space. For f∈S(RN), we define the Fourier transform\noffas\nF/bracketleftbigf/bracketrightbig(ξ)=/hatwidef(ξ):=(2π)−N/2/integraldisplay\nRNf(x)e−ix·ξdx,\nand the inverse Fourier transform of fas\nF−1/bracketleftbigf/bracketrightbig(x):=(2π)−N/2/integraldisplay\nRNf(ξ)eix·ξdξ,\nand extend them to S′(RN) by duality. Next, we give the notations of function spaces. Fors∈Rand\nm≥0, we denote the weighted Sobolev space Hs,m(RN) with the norm\n/ba∇dblf/ba∇dblHs,m:=/parenleftigg/integraldisplay\nRN(/angb∇acketleftx/angb∇acket∇ightm/angb∇acketleft∇/angb∇acket∇ightsf)2dx/parenrightigg1/2\n,\nwhere/angb∇acketleft·/angb∇acket∇ight:=1+|·|.\nIn the last decades, the long-time behavior of the solution t o the Cauchy problem related to (1.1) has\nbeen investigated by many mathematicians. When σ=2 and b(t)=1, (1.1) is reduced to the semilinear\ndamped wave equation:\n∂2\ntu(x,t)−∆u(x,t)+∂tu(x,t)=|u(x,t)|p,t>0,x∈RN,\nu(x,0)=εu0(x),∂tu(x,0)=εu1(x),x∈RN,(1.4)\nG. Todorova and B. Yordanov [ 19, 20 ] studied the blow-up and global existence of the solution to the\nCauchy problem (1.4), and obtained the critical exponent pc=1+2\nN, which is same as the Fujita\nexponent for the heat equation. Precisely, they proved the g lobal existence for small initial data in the\nsupercritical caseρF(N)<ρ<N+2\n[N−2]+and blow up for the data positive on average in the subcritica l\ncase 1<ρ<ρ F(N). And in the critical case ρ=ρF(N), Qi S. Zhang [ 22] showed that the solution\nblows up within a finite time. Moreover, the estimates of the l ifespan T(ε) for the blow-up solution in\nthe subcritical and critical cases were derived by T. Li and Y . Zhou [ 14], K. Nishihara [ 15], M. Ikeda and\nT. Ogawa [ 9] and N. Lai and Y . Zhou [ 13] as follows:\nT(ε)≈Cε−(1\np−1−N\n2)−1, if 1<p<1+2\nN,\nexp(Cε−(p−1)), ifp=1+2\nN.\n2For the case thatσ=2 and b(t)=(1+t)−β, K. Nishihara [ 17] and J. Lin, K. Nishihara and J. Zhai\n[16] investigated the Cauchy problem for the semilinear wave eq uation with time-dependent damping\n/braceleftigg∂2\ntu(x,t)−∆u(x,t)+b(t)∂tu(x,t)=|u(x,t)|p,t>0,x∈RN,\nu(x,0)=εu0(x),∂tu(x,0)=εu1(x), x∈RN,(1.5)\nwith−1<β< 1, and they found that the critical exponent for (1.5) remain spc=1+2\nN. And the sharp\nestimate of the lifespan in the subcritical case 1 <p<pcwas obtained by M. Ikeda and Y . Wakasugi\n[11] K. Fujiwara, M. Ikeda and Y . Wakasugi [ 5]:\nT(ε)≈Cε−[(1\np−1−N\n2)(1+β)]−1.\nIn the critical case p=pc, the lower and upper estimates of the lifespan were obtained by M. Ikeda and\nY . Wakasugi [ 11], M. Ikeda, T. Ogawa [ 9] and M. Ikeda, M. Sobajima and Y . Wakasugi[ 10] as follows:\nT(ε)≈exp(Cε−(p−1)).\nMoreover, in the case β=−1, similar results were derived by K. Fujiwara, M. Ikeda and Y . Wakasugi\n[5] and M. Ikeda and T. Inui[ 8]:\nT(ε)≈Cexp (ε−(1\np−1−N\n2)−1), if 1<p<1+2\nN,\nexp(exp( Cε−(p−1))), ifp=1+2\nN.\nFor general effective damping, see M. D’Abbicco, S. Lucente and M. Reissig [ 2], M. D’Abbicco and S.\nLucente [ 1], M. Ikeda, M. Sobajima and Y . Wakasugi[ 10], etc.\nFor the equation with fractional Laplacian and damping\n/braceleftigg∂2\ntu(x,t)+(−∆)σ\n2u(x,t)+∂tu(x,t)=|u(x,t)|p,t>0,x∈RN,\nu(x,0)=εu0(x),∂tu(x,0)=εu1(x), x∈RN,(1.6)\nT. Dao and M. Reissig [ 4] proved that the solution would blow up in C([0,∞),Hσ) forσ≥2 in the\nsubcritical case 1<p<1+σ\nNand K. Fujiwara, M. Ikeda and Y . Wakasugi [ 6] obtained small data global\nwell-posedness in a weighted Sobolev space Hs∩H0,αin the case p>1+σ\nNwithα≥0 andσ> 0.\nCombining these two results, we could conclude that the crit ical exponent for the Cauchy problem (1.6)\nispc=1+σ\nN.For the weakly coupled system, see [ 3].\nIn this paper, we consider the Cauchy problem of semilinear w ave equation (1.1) with fractional\nLaplacian and time-dependent damping. Our first aim is to pro ve the local existence in the weighted\nSobolev space, in which we may obtain the global existence an d lower bound of the lifespan by the\nmethod of scaling variables, similar as [ 5] and [ 21]. We will investigate the global existence and the\nsharp lower bound of lifespan in the forthcoming paper. The s econd aim is to prove the solution would\nblow up in finite time even if the initial data is su fficiently small for the source nonlinearity, and get\nthe upper estimate of the lifespan, which is supposed to be sh arp, in the subcritical and critical cases,\nrespectively. We will discuss the sharpness of the upper bou nd in the forthcoming paper.\nBefore stating the local existence, let us introduce the defi nition of the mild solution.\nDefinition 1.1 (Mild solution) .u∈L∞(0,T;L2(RN)) is called the mild solution to (1.1) if usatisfies the\nfollowing integral equation\nu(t)=˜Sσ(t)u0+Sσ(t)u1−/integraldisplayt\n0Sσ(t−τ)(b(τ)−1)∂τudτ+/integraldisplayt\n0Sσ(t−τ)f(u)dτ. (1.7)\n3Here Sσ(t) is the solution operator of the linear damped wave equation ∂2\ntu(t,x)+(−∆)σ/2u(t,x)+\n∂tu(t,x)=0, that is,\nSσ(t) :=e−t\n2F−1L(t,ξ)F,\nwhere\nLσ(t,ξ)=sinh( t/radicalig\n1\n4−|ξ|σ)\n/radicalig\n1\n4−|ξ|σ, if|ξ|<2−σ\n2,\nsin(t/radicalig\n|ξ|σ−1\n4)\n/radicalig\n1\n4−|ξ|σ, if|ξ|>2−σ\n2.(1.8)\nAnd ˜Sσ(t) :=(∂t+1)Sσ.\nTheorem 1.1 (Local existence and uniqueness) .Let N∈N,σ∈(0,2]and1<p<N\nN−σ. Assume that\nb(t)∈L∞(0,1). Then there exists a unique mild solution u ∈C([0,T),Hσ\n2,m)∩C1([0,T),H0,m)for some\ntime T, if the initial data (u0,u1)∈Hσ\n2,m×H0,m.\nRemark 1.1.Note that the local existence and uniqueness theorem does no t require the assumption (1.2).\nTo state the blow-up results, we give the definition of the str ong solution to the Cauchy problem (1.1).\nDefinition 1.2 (Strong solution) .LetN∈N,σ∈(0,2), (u0,u1)∈Hσ×Hσ/2andT>0. We say that\nuis a strong solution on [0 ,T) ifu∈C2([0,T);L2)∩C1([0,T);Hσ\n2)∩C([0,T);Hσ) satisfies the initial\ncondition u(0)=u0and∂tu(0)=u1and satisfies the equation\n∂2\ntu+(−∆)σ/2u+b(t)∂tu=f(u)\nin the sense of C([0,T);L2).\nIn what follows, we would consider the case of the source nonl ineaity, that is,\nf(u)≥C1|u|p\nfor the exponent p>1 and the constant C1>0. For N<q<N+pσ, define\nA(N,p,σ,q) :=/parenleftigg2\nC1/parenrightiggp′−1\n(p′)−1\npp1−p′\np/ba∇dbl(−∆)σ\n2/angb∇acketleftx/angb∇acket∇ight−q/angb∇acketleftx/angb∇acket∇ightq\np/ba∇dblp′/p\nLp′/ba∇dbl/angb∇acketleftx/angb∇acket∇ight−q/ba∇dbl1\np′\nL1, (1.9)\nand\nR(ǫ)=A(N,p,σ,q)−p−1\nN(p−pc)/parenleftbiggǫ\n4I0/parenrightbigg p−1\nN(p−pc)(1.10)\nwith pc=1+σ\nN,p′=p\np−1, and I0defined in (1.11) below.\nLet us introduce the main result on the blow-up of the solutio n in the subcritical and critical cases.\nTheorem 1.2 (Blow up in the subcritical case) .Let−1≤β≤1and 1<p<pc. Let u be the\nstrong solution to the Cauchy problem (1.1) with the nonline arity f (u)satisfying f (u)≥C1|u|pfor some\nconstants C 1>0. Assume that the initial data (u0,u1)=ǫ(a0,a1)with (a0,a1)∈Hσ/2(RN)×L2(RN),\nsatisfy\nI0:=/integraldisplay\nRNa0(x)dx>0, I1:=/integraldisplay\nRNa1(x)dx>0. (1.11)\n4Letǫ0be a constant small enough such that\n/integraldisplay\nRN/angb∇acketleftx\nR(ǫ0)/angb∇acket∇ight−qa0(x)dx≥1\n2I0, (1.12)\n/integraldisplay\nRN/angb∇acketleftx\nR(ǫ0)/angb∇acket∇ight−qa1(x)dx≥1\n2I1, (1.13)\nǫ0I0≤/parenleftbiggC1\n2/parenrightbigg−1\np−1/ba∇dbl/angb∇acketleftx/angb∇acket∇ight−q/ba∇dblL1(RN)R(ǫ0)N. (1.14)\nThen, for any 0<ǫ≤ǫ0, the lifespan of the solution can be estimated by\nT0≤Cǫ−1/parenleftbigg1\np−1−N\n2/parenrightbigg\n(1+β), for−1<β≤1,\nexpCǫ−1\n1\np−1−Nσ, forβ=−1.(1.15)\nRemark 1.2.Noting that for 1<p<pc,R(ǫ)→∞ asǫ→0, sincep−1\nN(p−pc)<0. Thus, we have\n/integraldisplay\nRN/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qa0(x)dx→/integraldisplay\nRNa0(x)dx=I0,\n/integraldisplay\nRN/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qa1(x)dx→/integraldisplay\nRNa1(x)dx=I1,\nwhich implies the conditions (1.12) and (1.13) are satisfied . Moreover, it follows from\nǫR(ǫ)−N=A(N,p,σ,q)p−1\n(p−pc)/parenleftigg1\n4I0/parenrightiggp−1\npc−p\nǫ1+p−1\npc−p\nwith the power 1+p−1\npc−p>0 that there exists a constant ǫ0small enough such that the condition (1.14)\nholds.\nTheorem 1.3 (Blow up in the critical case) .Let−1≤β<1, and p=1+σ\nNwith 0<σ≤2. Assume\nthat u is a strong solution to the Cauchy problem (1.1) with the initial data (u0,u1)=ǫ(a0,a1)satisfying\n/integraldisplay\nRN(a1(x)+B0a0(x))dx>0,B0:=/integraldisplay∞\n0exp/parenleftigg\n−/integraldisplayt\n0b(s)ds/parenrightigg\ndt.\nThen, the solution u blows up in a finite time. Moreover, the li fespan of the solution can be estimated by\nT0≤exp ( Cǫ−(p−1), ifβ∈(−1,1),\nexp (exp ( Cǫ−(p−1)), ifβ=−1.(1.16)\nRemark 1.3.Note that the coefficient b(t) of the damping has no influence of the estimate of the lifespa n.\nWe briefly give the idea of the proof. To get the local existenc e, we write the equation as\n∂2\ntu+(−∆)σ/2u+∂tu=−(b(t)−1)∂tu+f(u),\n5to use the property of the solution operators Sσ(t) and ˜Sσ(t) for the linear wave equation with constant\ncoefficient damping. Then, we derive the weighted Sobolev estimat e of the operators, and treat the non-\nlinearity by the interpolation inequality for the weighted Sobolev norm in [ 7]. Applying the contraction\nmapping principle, we could obtain that the mild solution ex ists locally.\nTo prove the blow-up of the solution, we would apply the metho d based on the ordinary di fferential\ninequality. In the subcritical case, we follow the idea of K. Fujiwara, M. Ikeda and Y . Wakasugi [ 5].\nWe construct a weighted average of the solution/integraltext\nRNu(t,x)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qwith N<q<N+pσ. With the\nproperty of the fractional Laplacian on the weight /angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q, an ordinary differential inequality could be\nderived from the fractional nonlinear damped wave equation . From the definition of R(ǫ), the conditions\nof Proposition 2.3 in [ 5] are fulfilled for su fficiently smallǫ>0. Thus, by the blow-up and upper estimate\nof the lifespan of the solution to the ordinary di fferential inequality (Proposition 2.3 in [ 5]), we could\nprove that the solution uto the Cauchy problem (1.1) blows up in a finite time and obtain the upper bound\nof the lifespan. However, this method fails in the critical c ase, since the definition of R(ǫ) in (1.10) makes\nno sense. Instead, we use similar method as that in [ 14] and [ 8]. To overcome the di fficulties which come\nfrom the time-dependent damping, we multiply the equation b y a nonnegative function g(t)∈C1([0,∞))\nto the the divergence form:\n(g(t)u)tt+(−∆)σ/2(g(t)u)−(g′(t)u)t+ut=g(t)|u|ρ.\nThen, we treat the above equation as the semilinear fraction al heat equation with the nonlinearity F(x,t) :=\ng(t)|u|ρ−(g(t)u)tt+(g′(t)u)t, and write it as\nu(t)=Φ(G(t))∗u0+/integraldisplayt\n0Φ(G(t)−G(s))∗F(s)ds,\nwhereΦ(G(t)) is the Gaussian type function for the fractional Laplacia n. With the property of Φ(G(t)),\nwe could get ordinary di fferential inequalities. Applying Lemma A.1 and Lemma A.2 in [ 8], we could\nobtain the upper bound of the lifespan, which is independent onβin the damping.\nThis paper is organized as follows. In Section 2, we would pro ve Theorem 1.1 by the contraction\nmapping principle. Theorem 1.2 and 1.3 would be proved by the method based on the ordinary di fferen-\ntial inequality in Section 3 and Section 4, respectively.\n2 Proof of local existence and uniqueness\nIn this section, we apply the contraction mapping principle to get the local existence of the solution to\nthe Cauchy problem (1.1). The proof is based on the weighted S obolev estimate of the operators Sσ(t)\nand˜Sσ(t), which is stated as follows.\nLemma 2.1. Let m≥0and0≤s≤σ\n2,\n/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2≤C/ba∇dblf/ba∇dblH0,m. (2.1)\nProof. We divide the weighted Sobolev norm /ba∇dbl/angb∇acketleft·/angb∇acket∇ightm/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2into/ba∇dbl/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2and\n/ba∇dbl|·|m/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2.\nStep 1. We estimate /ba∇dbl/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2first. Note that\nF[/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f]=e−t\n2Lσ(t,ξ)/angb∇acketleftξ/angb∇acket∇ightsˆf.\n6When|ξ|<2−2\nσ,\nLσ(t,ξ)=sinh( t/radicalig\n1\n4−|ξ|σ)\n/radicalig\n1\n4−|ξ|σ≤Cet√1\n4−|ξ|σ.\nThus,\n/ba∇dblF[/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f]/ba∇dblL∞(|ξ|<2−2σ)\n≤/ba∇dble−t\n2Lσ(t,ξ)/angb∇acketleftξ/angb∇acket∇ights/ba∇dblL∞(|ξ|<2−2σ)/ba∇dblˆf/ba∇dblL2\n≤C/ba∇dblet(√1\n4−|ξ|σ−1\n2)/angb∇acketleftξ/angb∇acket∇ights/ba∇dblL∞(|ξ|<2−2σ)/ba∇dblˆf/ba∇dblL2\n≤C/ba∇dble−t|ξ|σ/angb∇acketleftξ/angb∇acket∇ights/ba∇dblL∞(|ξ|<2−2σ)/ba∇dblˆf/ba∇dblL2\n≤C/ba∇dblf/ba∇dblL2.(2.2)\nWhen|ξ|>2−2\nσ,\nLσ(t,ξ)=sin(t/radicalig\n|ξ|σ−1\n4)\n/radicalig\n|ξ|σ−1\n4.\nThus,\n/ba∇dble−t\n2Lσ(t,ξ)/angb∇acketleftξ/angb∇acket∇ights/ba∇dblL∞(|ξ|>2−2σ)<Ce−t\n2,\nwhich implies\n/ba∇dblF[/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f]/ba∇dblL2(|ξ|>2−2σ)\n≤/ba∇dble−t\n2Lσ(t,ξ)/angb∇acketleftξ/angb∇acket∇ights/ba∇dblL∞(|ξ|>2−2σ)/ba∇dblˆf/ba∇dblL2\n≤Ce−t\n2/ba∇dblf/ba∇dblL2.(2.3)\nCombining (2.2) and (2.3), we could conclude that\n/ba∇dbl/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2≤C/ba∇dblf/ba∇dblL2. (2.4)\nStep 2. To estimate /ba∇dbl|·|m/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2, we introduce the following smooth cut-o fffunction:\nχ(ξ)=1,|ξ|≤1,\n0,|ξ|>2.\nWith the cut-offfunctionχ(ξ), we would divide |·|m/angb∇acketleft∇/angb∇acket∇ightsSσ(t)finto low frequency term and high\nfrequency term, and estimate them as follow. For the low freq uency term,\n/ba∇dbl|·|mF−1[e−t\n2Lσ(t,ξ)χ(ξ)/angb∇acketleftξ/angb∇acket∇ightsˆf]/ba∇dblL2\n=/ba∇dbl|x|m/integraldisplay\nRNK(t,x−y)f(y)dy/ba∇dblL2\n≤C/ba∇dbl/integraldisplay\nRN/parenleftbig|x−y|m|K(t,x−y)|+|y|m|K(t,x−y)|/parenrightbigf(y)dy/ba∇dblL2\n≤C/ba∇dbl|·|mK(t)/ba∇dblL1/ba∇dblf/ba∇dblL2+C/ba∇dblK(t)/ba∇dblL1/ba∇dbl|·|mf/ba∇dblL2\n≤C/ba∇dblf/ba∇dblH0,m,(2.5)\n7where K(t,x) :=F−1[e−t\n2Lσ(t,ξ)χ(ξ)/angb∇acketleftξ/angb∇acket∇ights]. The last inequalities hold since\n/ba∇dbl|·|βK(t)/ba∇dblL1=/integraldisplay\nRN/angb∇acketleftx/angb∇acket∇ightβ|K(t,x)|dx\n≤/integraldisplay\nRN/angb∇acketleftx/angb∇acket∇ight2n−(N+1)|K(t,x)|dx\n≤/integraldisplay\nRN/angb∇acketleftx/angb∇acket∇ight−(N+1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n|ξ|<2/angb∇acketleft∇/angb∇acket∇ight2ne−t\n2Lσ(t,ξ)χ(ξ)/angb∇acketleftξ/angb∇acket∇ightseix·ξdξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingledx\n≤C,(2.6)\nfor anyβ>0. Here, the constant n∈Z+is chosen large enough such that |x|β≤/angb∇acketleftx/angb∇acket∇ight2n−(N+1).\nFor the high frequency term,\n/ba∇dbl|·|mF−1[e−t\n2Lσ(t,ξ)(1−χ(ξ))/angb∇acketleftξ/angb∇acket∇ightsˆf]/ba∇dblL2\n=/ba∇dbl|∇|me−t\n2Lσ(t,ξ)(1−χ(ξ))/angb∇acketleftξ/angb∇acket∇ightsˆf/ba∇dblL2\n≤Cn/summationdisplay\nj=1[m]+1/summationdisplay\nk=0/ba∇dbl∂k\nje−t\n2Lσ(t,ξ)(1−χ(ξ))/angb∇acketleftξ/angb∇acket∇ights/ba∇dblL∞/ba∇dbl/angb∇acketleft∇/angb∇acket∇ightmˆf/ba∇dblL2\n≤C/ba∇dbl/angb∇acketleft·/angb∇acket∇ightmf/ba∇dblL2=C/ba∇dblf/ba∇dblH0,m,(2.7)\nfor 0<σ≤2.\nTherefore, it follows from (2.5) and (2.7) that\n/ba∇dbl|·|m/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2≤C/ba∇dblf/ba∇dblH0,m. (2.8)\nFinally,\n/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2≤/ba∇dbl/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2+/ba∇dbl|·|m/angb∇acketleft∇/angb∇acket∇ightsSσ(t)f/ba∇dblL2≤C/ba∇dblf/ba∇dblH0,m. (2.9)\nSimilarly, we can get the following lemma.\nLemma 2.2. Let m≥0,\n/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm∂tSσ(t)f/ba∇dblL2≤C/ba∇dblf/ba∇dblH0,m, (2.10)\nand\n/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm∂2\ntSσ(t)f/ba∇dblL2≤C/ba∇dblf/ba∇dblHσ\n2,m, (2.11)\nProof. Lemma 2.2 could be proved in a similar manner as that of Lemme 2 .1. Here we only show\nthe difference if the operator Sσ(t) is differentiated once w.r.t. t.\nF[∂tSσ(t)f]=∂t/parenleftig\ne−t\n2Lσ(t,ξ)ˆf/parenrightig\n=−1\n2e−t\n2Lσ(t,ξ)ˆf+e−t\n2∂tLσ(t,ξ)ˆf(2.12)\nThe first term could be estimated by Lemma 2.1. And for the seco nd term, direct calculation implies\n∂tLσ(t,ξ)=cosh( t/radicalig\n1\n4−|ξ|σ)\n/radicalig\n1\n4−|ξ|σ·/radicalbigg\n1\n4−|ξ|σ, if|ξ|<2−σ\n2,\n−cos(t/radicalig\n|ξ|σ−1\n4)\n/radicalig\n1\n4−|ξ|σ·/radicalbigg\n|ξ|σ−1\n4, if|ξ|>2−σ\n2.(2.13)\n8Thus, the second term e−t\n2∂tLσ(t,ξ)ˆfcould be estimated similarly as e−t\n2Lσ(t,ξ)/angb∇acketleftξ/angb∇acket∇ightsˆfwith s=σ\n2.\nTo estimate the nonlinearity in the equation (1.1), we need t he following interpolation inequality.\nLemma 2.3. [Proposition 2.3,[ 7]] For any a,b>0,θ∈(0,1),\n/ba∇dbl/angb∇acketleftx/angb∇acket∇ightθa(1−∆)(1−θ)bf/ba∇dblL2≤/ba∇dbl(1−∆)bf/ba∇dbl1−θ\nL2/ba∇dbl/angb∇acketleftx/angb∇acket∇ightaf/ba∇dblθ\nL2. (2.14)\nProof of Theorem 1.1. Write the equation as\n∂2\ntu+(−∆)σ/2u+∂tu=−(b(t)−1)∂tu+f(u).\nThen, the solution could be expressed as\nu(t)=˜Sσ(t)u0+Sσ(t)u1−/integraldisplayt\n0Sσ(t−τ)(b(τ)−1)∂τudτ+/integraldisplayt\n0Sσ(t−τ)f(u)dτ. (2.15)\nFor the first two term in the right hand side, it follows from Le mma 2.1 and Lemma 2.2 that\n/ba∇dbl˜Sσ(t)u0/ba∇dblHσ\n2,m=/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm˜Sσ(t)/angb∇acketleft∇/angb∇acket∇ightσ\n2u0/ba∇dblL2\n≤C/ba∇dblu0/ba∇dblHσ\n2,m,\nand\n/ba∇dblSσ(t)u1/ba∇dblHσ\n2,m≤C/ba∇dblu1/ba∇dblH0,m.\nApplying Lemma 2.1,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0Sσ(t−τ)(b(τ)−1)∂τudτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleHσ\n2,m≤/integraldisplayt\n0/ba∇dblSσ(t−τ)(b(τ)−1)∂τu/ba∇dblHσ\n2,mdτ\n≤(/ba∇dblb/ba∇dblL∞+1)t/ba∇dbl∂tu/ba∇dblL∞([0,T),H0,m)\nFor the forth term, with Lemma 2.1, we get\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0Sσ(t−τ)f(u)dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleHσ\n2,m≤C0/integraldisplayt\n0/ba∇dblSσ(t−τ)|u|p/ba∇dblHσ\n2,mdτ\n≤C/integraldisplayt\n0/ba∇dbl|u|p/ba∇dblH0,mdτ\n≤Ct/ba∇dblu/ba∇dblp\nL∞([0,T),Hσ\n2,m),\nsince\n/ba∇dbl|u|p/ba∇dblH0,m=/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm|u|p/ba∇dblL2=/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm\npu/ba∇dblp\nL2p\n≤C/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm\npu/ba∇dblp\nHσ′\n2=C/ba∇dbl/angb∇acketleft∇/angb∇acket∇ightσ′\n2(/angb∇acketleft·/angb∇acket∇ightm\npu)/ba∇dblp\nL2\n≤C/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm\np(1−θ)u)/ba∇dbl(1−θ)p\nL2/ba∇dbl/angb∇acketleft∇/angb∇acket∇ightσ′\n2θu/ba∇dblθp\nL2\n≤C/ba∇dblu/ba∇dblp\nHσ\n2,m\nwhereσ′=σθwith the constantθsatisfying 0<θ≤p−1\np, by Soblov inequality and Lemma 2.3.\n9Note that Sσ(0)=0. Differentiating the equation (2.15) with respect to t, we have\n∂tu(t)=∂t˜Sσ(t)u0+∂tSσ(t)u1−/integraldisplayt\n0∂tSσ(t−τ)(b(τ)−1)∂τudτ+/integraldisplayt\n0��tSσ(t−τ)f(u)dτ. (2.16)\nIt follows from Lemma 2.2 that\n/ba∇dbl∂t˜Sσ(t)u0/ba∇dblH0,m=/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm∂t˜Sσ(t)u0/ba∇dblL2\n≤C/ba∇dblu0/ba∇dblHσ\n2,m,\nand\n/ba∇dbl∂tSσ(t)u1/ba∇dblH0,m≤C/ba∇dblu1/ba∇dblH0,m.\nApplying Lemma 2.2,\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0∂tSσ(t−τ)(b(τ)−1)∂τudτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH0,m≤/integraldisplayt\n0/ba∇dbl∂tSσ(t−τ)(b(τ)−1)∂τu/ba∇dblH0,mdτ\n≤(/ba∇dblb/ba∇dblL∞+1)t/ba∇dbl∂tu/ba∇dblL∞([0,T),H0,m)\nFor the forth term, with Lemma 2.2, we get\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt\n0∂tSσ(t−τ)f(u)dτ/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH0,m≤C0/integraldisplayt\n0/ba∇dbl∂tSσ(t−τ)|u|p/ba∇dblH0,mdτ\n≤C/integraldisplayt\n0/ba∇dbl|u|p/ba∇dblH0,mdτ\n≤Ct/ba∇dblu/ba∇dblp\nL∞([0,T),Hσ\n2,m).\nThus,\nsup\nt∈[0,T)/ba∇dbl(u,∂tu)/ba∇dblHσ\n2,m×H0,m\n≤C/ba∇dblu0/ba∇dblHσ\n2,m+C/ba∇dblu1/ba∇dblH0,m+C(T)/ba∇dbl∂tu/ba∇dblL∞([0,T),H0,m)+C(T)/ba∇dblu/ba∇dblp\nL∞([0,T),Hσ\n2,m).(2.17)\nTo get the local solution, we construct an approximate seque nce as follows:\nu1(t)=˜Sσ(t)u0+Sσ(t)u1, (2.18)\nand\nui(t)=˜Sσ(t)u0+Sσ(t)u1−/integraldisplayt\n0Sσ(t−τ)(b(τ)−1)∂τui−1dτ+/integraldisplayt\n0Sσ(t−τ)f(ui−1)dτ\nfori≥2, with the same initial data u0andu1. Based on the estimate (2.17), we could conclude that such\nsequence exists in the function space\nX(T) :=(u,∂tu)∈C([0,T),Hσ\n2,m)×C([0,T),H0,m)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesup\nt∈[0,T)/ba∇dbl(u,∂tu)/ba∇dblHσ\n2,m×H0,m<∞.\nNote that\n|f(u)−f(v)|≤C0(|u|p−1+|v|p−1)|u−v|, (2.19)\n10and\n/ba∇dbl|v|p−1|u−v|/ba∇dblH0,m=/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm|v|p−1|u−v|/ba∇dblL2\n=/ba∇dbl/angb∇acketleft·/angb∇acket∇ightp−1\npm|v|p−1/angb∇acketleft·/angb∇acket∇ight1\npm|u−v|/ba∇dblL2\n≤/ba∇dbl/angb∇acketleft·/angb∇acket∇ightp−1\npm|v|p−1/ba∇dbl\nL2p\np−1/ba∇dbl/angb∇acketleft·/angb∇acket∇ight1\npm|u−v|/ba∇dblL2p\n=/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm|v|p/ba∇dblp−1\np\nL2/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm|u−v|p/ba∇dbl1\np\nL2.(2.20)\nSimilarly as the estimate of (2.17), we get\n/ba∇dblui−ui−1/ba∇dblHσ\n2,m≤/integraldisplayt\n0/ba∇dblSσ(t−τ)(b(τ)−1)∂τ(ui−1−ui−2)/ba∇dblHσ\n2,mdτ\n+/integraldisplayt\n0/ba∇dblSσ(t−τ)(f(ui−1)−f(ui−2))/ba∇dblHσ\n2,mdτ\n≤(/ba∇dblb/ba∇dblL∞+1)t/ba∇dbl∂t(ui−1−ui−2)/ba∇dblL∞([0,T),H0,m)\n+C/integraldisplayt\n0/ba∇dbl(|ui−1|p−1+|ui−2|p−1)|ui−1−ui−2|/ba∇dblH0,mdτ\n≤(/ba∇dblb/ba∇dblL∞+1)t/ba∇dbl∂t(ui−1−ui−2)/ba∇dblL∞([0,T),H0,m)\n+C/integraldisplayt\n0(/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm|ui−1|p/ba∇dblp−1\np\nL2+/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm|ui−2|p/ba∇dblp−1\np\nL2)/ba∇dbl/angb∇acketleft·/angb∇acket∇ightm|ui−1−ui−2|p/ba∇dbl1\np\nL2dτ\n≤(/ba∇dblb/ba∇dblL∞+1)t/ba∇dbl∂t(ui−1−ui−2)/ba∇dblL∞([0,T),H0,m)\n+Ct/parenleftbigg\n/ba∇dblui−1/ba∇dblp−1\nL∞([0,T),Hσ\n2,m)+/ba∇dblui−2/ba∇dblp−1\nL∞([0,T),Hσ\n2,m)/parenrightbigg\n/ba∇dbl|ui−1−ui−2|/ba∇dblL∞([0,T),Hσ\n2,m)(2.21)\nAnd\n/ba∇dbl∂t(ui−ui−1)/ba∇dblH0,m≤/integraldisplayt\n0/ba∇dbl∂tSσ(t−τ)(b(τ)−1)∂τ(ui−1−ui−2)/ba∇dblH0,mdτ\n+/integraldisplayt\n0/ba∇dbl∂tSσ(t−τ)(f(ui−1)−f(ui−2))/ba∇dblH0,mdτ\n≤(/ba∇dblb/ba∇dblL∞+1)t/ba∇dbl∂t(ui−1−ui−2)/ba∇dblL∞([0,T),H0,m)\n+C/integraldisplayt\n0/ba∇dbl(|ui−1|p−1+|ui−2|p−1)|ui−1−ui−2|/ba∇dblH0,mdτ\n≤(/ba∇dblb/ba∇dblL∞+1)t/ba∇dbl∂t(ui−1−ui−2)/ba∇dblL∞([0,T),H0,m)\n+Ct/parenleftbigg\n/ba∇dblui−1/ba∇dblp−1\nL∞([0,T),Hσ\n2,m)+/ba∇dblui−2/ba∇dblp−1\nL∞([0,T),Hσ\n2,m)/parenrightbigg\n/ba∇dbl|ui−1−ui−2|/ba∇dblL∞([0,T),Hσ\n2,m)(2.22)\nBased on the above estimates, we can obtain the local existen ce and uniqueness by the contraction\nmapping principle.\n3 Proof of blow-up in subcritical case\nIn this section, we would apply the test function method to pr ove the blow-up of the solution in the\nsubcritical case. We focus on the deriving an ordinary di fferential inequality for the weighted average of\nthe solution. Then, by the comparison lemma in [ 5], we could obtain the blow-up result and the upper\nbound of the lifespan.\n11Introduce\nφ(x) :=/angb∇acketleftx/angb∇acket∇ight−q\nand\nφǫ(x) :=/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q,\nwhere R(ǫ) is defined in (1.10). And define\nIǫ(t) :=/integraldisplay\nRNu(t,x)/angb∇acketleftx\nR(ǫ)/angb∇acket��ight−qdx.\nHere, qis a constant satisfying N<q<N+pσ.\nMultiplying the the equation by φǫ(x)=/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q, it follows from H¨ older’s inequality and Young’s\ninequality that\nd2\ndt2Iǫ(t)+b(t)d\ndtIǫ(t)\n=/integraldisplay\nRN/parenleftig\n∂2\nt+b(t)∂t/parenrightig\nu(t,x)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx\n=−/integraldisplay\nRN(−∆)σ\n2u(t,x)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx+/integraldisplay\nRNf(u)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx\n≥−/integraldisplay\nRN(−∆)σ\n2u(t,x)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx+C1/integraldisplay\nRN|u|p/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx\n=−/integraldisplay\nRNu(t,x)(−∆)σ\n2/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx+C1/ba∇dblu(t)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q\np/ba∇dblp\nLp\n≥−/ba∇dbl (−∆)σ\n2/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/angb∇acketleftx\nR(ǫ)/angb∇acket∇ightq\np/ba∇dblLp′/ba∇dblu(t)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ightq\np/ba∇dblLp+C1/ba∇dblu(t)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q\np/ba∇dblp\nLp\n≥−/parenleftigg2\nC1/parenrightiggp′\np\n(p′)−1p−p′\np/ba∇dbl(−∆)σ\n2/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/angb∇acketleftx\nR(ǫ)/angb∇acket∇ightq\np/ba∇dblp′\nLp′−C1\n2/ba∇dblu(t)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q\np/ba∇dblp\nLp\n+C1/ba∇dblu(t)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q\np/ba∇dblp\nLp\n=−/parenleftigg2\nC1/parenrightiggp′−1\n(p′)−1p1−p′/ba∇dbl(−∆)σ\n2/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/angb∇acketleftx\nR(ǫ)/angb∇acket∇ightq\np/ba∇dblp′\nLp′+C1\n2/ba∇dblu(t)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q\np/ba∇dblp\nLp(3.1)\nApplying H¨ older’s inequality, we have\nIǫ(t)=/integraldisplay\nRNu(t,x)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx\n≤/parenleftigg/integraldisplay\nRN|u|p/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx/parenrightigg1\np/parenleftigg/integraldisplay\nRN/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx/parenrightigg1\np′\n,(3.2)\nwhich implies\n/ba∇dblu(t)/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q\np/ba∇dblp\nLp≤/parenleftigg/integraldisplay\nRN/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qdx/parenrightigg−p\np′\n(Iǫ(t))p. (3.3)\nThus,\n12d2\ndt2Iǫ(t)+b(t)d\ndtIǫ(t)\n≥−/parenleftigg2\nC1/parenrightiggp′−1\n(p′)−1p1−p′/ba∇dbl(−∆)σ\n2/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/angb∇acketleftx\nR(ǫ)/angb∇acket∇ightq\np/ba∇dblp′\nLp′+C1\n2/ba∇dbl/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/ba∇dbl1−p\nL1(Iǫ(t))p\n=C1\n2/ba∇dbl/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/ba∇dbl1−p\nL1/parenleftbigIǫ(t)p−Aǫ(N,p,σ,q)p/parenrightbig\n=C1\n2/ba∇dbl/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/ba∇dbl1−p\nL1(Iǫ(t)−Aǫ(N,p,σ,q))p.(3.4)\nHere, Aǫ(N,p,σ,q) :=/parenleftig2\nC1/parenrightigp′−1(p′)−1/pp1−p′\np/ba∇dbl(−∆)σ\n2/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/angb∇acketleftx\nR(ǫ)/angb∇acket∇ightq\np/ba∇dblp′\np\nLp′/ba∇dbl/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/ba∇dbl1\np′\nL1. Note that the last\ninequality holds under the condition that\nIǫ(t)−Aǫ(N,p,σ,q)>0. (3.5)\nIntroducing J(t)=Iǫ(t)−Aǫ(N,p,σ,q), we obtain\nd2\ndt2J(t)+b(t)d\ndtJ(t)≥C1\n2/ba∇dbl/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/ba∇dbl1−p\nL1J(t)p,for t∈[0,t0),\nJ(0)=Iǫ(0)−Aǫ(N,p,σ,q)\nJ′(0)=A1J(0)(3.6)\nwhere A1:=I′\nǫ(0)\nIǫ(0)−Aǫ(N,p,σ,q).\nDirect calculation implies\n/ba∇dbl/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/ba∇dblL1=R(ǫ)N/ba∇dbl/angb∇acketleftx/angb∇acket∇ight−q/ba∇dblL1. (3.7)\nIt follows from Lemma 3.2 in [ 18] that\n(−∆)σ\n2φǫ(x)=R(ǫ)−σ/parenleftig\n(−∆)σ\n2φ/parenrightig/parenleftiggx\nR(ǫ)/parenrightigg\n. (3.8)\nTherefore, we get\n/ba∇dbl(−∆)σ\n2/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/angb∇acketleftx\nR(ǫ)/angb∇acket∇ightq\np/ba∇dblLp′=R(ǫ)−σ+N\np′/ba∇dbl(−∆)σ\n2/angb∇acketleftx/angb∇acket∇ight−q/angb∇acketleftx/angb∇acket∇ightq\np/ba∇dblLp′ (3.9)\nComparing (3.7), (3.8), (1.9), (1.10) and the definition of Aǫ(N,p,σ,q), we obtain\nAǫ(N,p,σ,q)=A(N,p,σ,q)R(ǫ)−σp′\np+N=ǫ\n4I0. (3.10)\nFrom the assumption (1.12) and (3.10), we obtain\nIǫ(0)−Aǫ(N,p,σ,q)=ǫ/integraldisplay\nRN/angb∇acketleftx\nR(ǫ0)/angb∇acket∇ight−qa0(x)dx−Aǫ(N,p,σ,q)≥ǫ\n4I0. (3.11)\nOn the other hand, by the assumption (1.14), we have\nIǫ(0)−Aǫ(N,p,σ,q)=ǫ/integraldisplay\nRN/angb∇acketleftx\nR(ǫ0)/angb∇acket∇ight−qa0(x)dx−ǫ\n4I0\n≤ǫI0\n≤/parenleftbiggC1\n2/parenrightbigg−1\np−1/ba∇dbl/angb∇acketleftx/angb∇acket∇ight−q/ba∇dblL1R(ǫ)N\n=/parenleftbiggC1\n2/parenrightbigg−1\np−1/ba∇dbl/angb∇acketleftx\nR(ǫ0)/angb∇acket∇ight−q/ba∇dblL1(3.12)\n13since/integraltext\nRN/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−qa0(x)dx→/integraltext\nRNa0(x)dx=I0asǫ→0. Moreover, by the assumption (1.13) and (3.12),\nA1=I′\nǫ(0)\nIǫ(0)−Aǫ(N,p,σ,q)\n=ǫ/integraltext\nRN/angb∇acketleftx\nR(ǫ0)/angb∇acket∇ight−qa1(x)dx\nIǫ(0)−Aǫ(N,p,σ,q)\n≥ǫ\n2I1\nǫI0=I1\n2I0(3.13)\nNow, from (3.11)-(3.13), the conditions in Proposition 2.3 in [5] are fulfilled. Thus, applying this\nproposition, we get\nJ(t)≥J(0)(1−µ(p,b,β,A1)˜J(0)p−1B(t))−2\np−1, (3.14)\nand the upper bound of the lifespan\nT0≤B−1(µ(p,b,β,A1)−1˜J(0)1−p), (3.15)\nwhere ˜J(0)=/parenleftigC2\n2/parenrightig1\np−1/ba∇dbl/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/ba∇dbl−1\nL1J(0), and\nµ(p,b,β,A1)=min/braceleftigg\n1,p−1\n2b(0)A1,\n2(p+1)\n(p−1)2b−2\n1(21\n1+β(1+B4))max(0,2β)+2(b−1\n1b3+1)\np−1−1.\nHere, B(t) :=/integraltextt\n0b(s)ds, and B−1is the inverse function of B(t), satisfying\nB1(1+t)1/(1+β)≤B−1(t)≤B2(1+t)1/(1+β), for−1<β≤1,\nexp ( B1(1+t))≤B−1(t)≤exp ( B2(1+t)), forβ=−1,(3.16)\nwith some constants B1,B2>0.\nNow, let’s prove the condition (3.5) holds, that is, J(t)>0 for all t∈[0,T0). Let T∗>0 satisfy\nJ(t)>0 for t∈[0,T∗) and J(T∗)=0. Then, in the same manner as above, we have (3.14) for t∈[0,T0).\nHowever, the right hand side of (3.14) is still positive for t=T∗, which contradicts J(T∗)=0. Thus, we\ncould conclude that J(t)>0 for all t∈[0,T0), which implies that (3.14) and (3.15) hold for t∈[0,T0).\nNote that\n˜J(0)p−1=C1\n2/ba∇dbl/angb∇acketleftx\nR(ǫ)/angb∇acket∇ight−q/ba∇dbl1−p\nL1J(0)p−1\n≥C1\n2/ba∇dbl/angb∇acketleftx/angb∇acket∇ight−q/ba∇dbl1−p\nL1R(ǫ)N(1−p)/parenleftbiggǫ\n4I0/parenrightbiggp−1\n=C1\n2/ba∇dbl/angb∇acketleftx/angb∇acket∇ight−q/ba∇dbl1−p\nL1A(N,p,σ,q)(p−1)2\np−pc/parenleftbiggǫ\n4I0/parenrightbigg−(p−1)2\np−pc+p−1\n=C1\n2/ba∇dbl/angb∇acketleftx/angb∇acket∇ight−q/ba∇dbl1−p\nL1A(N,p,σ,q)(p−1)2\np−pc/parenleftbiggǫ\n4I0/parenrightbigg 1\n1\np−1−Nσ,(3.17)\n14since\n−(p−1)2\np−pc+p−1=−(p−1)2\np−1−σ\nN+p−1\n=(p−1)1−1\n1−σ\nN·1\np−1\n=(p−1)−σ\nN·1\np−1\n1−σ\nN·1\np−1\n=1\n1\np−1−N\nσ.\nThus, substituting the estimate (3.17) into (3.15), we obta in\nT0≤Cǫ−1/parenleftbigg1\np−1−N\n2/parenrightbigg\n(1+β), for−1<β≤1,\nexpCǫ−1\n1\np−1−Nσ, forβ=−1.(3.18)\n4 Proof of blow-up and upper estimate of lifespan in the criti cal case\nTo rewrite the equation into a divergence form, we introduce a function g: [0,∞)→R>0as the unique\nsolution to the initial value problem of the ordinary di fferential equation\n/braceleftigg−g′(t)+b(t)g(t)=1,t∈(0,∞),\ng(0)=B0,(4.1)\nwhere B0:=/integraltext∞\n0exp/parenleftig\n−/integraltextt\n0b(s)ds/parenrightig\ndt<∞.The solution gis explicitly given by\ng(t)=exp/parenleftigg/integraldisplayt\n0b(s)ds/parenrightigg/parenleftigg\nB0−/integraldisplayt\n0exp/parenleftigg\n−/integraldisplayτ\n0b(s)ds/parenrightigg\ndτ/parenrightigg\n. (4.2)\nHere we note that g′(0)=b(0)g(0)−1 and\nlim\nt→∞b(t)g(t)=1,C−1b(t)−1≤g(t)≤Cb(t)−1(0≤t<∞) (4.3)\nWith the property of the function g, we can get the following equation of divergence form.\nLemma 4.1. Let u be a strong solution. Then u satisfies\n(g(t)u)tt+(−∆)σ/2(g(t)u)−(g′(t)u)t+ut=g(t)f(u) (4.4)\nin the sense of C ([0,T);L2)\nSetF(x,t) :=g(t)f(u)−(g(t)u)tt+(g′(t)u)t. Thus, we could write (4.4) as\nut+(−∆)σ/2(g(t)u)=F(x,t).\nTaking the Fourier transform, we get\n∂tˆu+g(t)|ξ|σˆu=ˆF(t).\n15Solving this ordinary di fferential equation, we have\nˆu(t)=e−G(t)|ξ|σˆu(0)+/integraldisplayt\n0e−(G(t)−G(s))|ξ|σF(s)ds, (4.5)\nwhere G(t) :=/integraltextt\n0g(s)ds.\nDenote\nϕ(x) :=F−1(e−|ξ|σ)=1\n(2π)N/2/integraldisplay\nRNe−|ξ|σeiξ·xdξ,\nand\nΦ(t)=Φ(x,t) :=1\ntn/σϕ/parenleftbiggx\nt1/σ/parenrightbigg\n=F−1(e−t|ξ|σ).\nThen, taking the inverse Fourier transform of (4.5), we obta in\nLemma 4.2. Let u be a strong solution. Then\nu(t)=Φ(G(t))∗u0+/integraldisplayt\n0Φ(G(t)−G(s))∗F(s)ds. (4.6)\nTo prove blow-up in the critical case, we require the followi ng estimates ofϕ(x).\nLemma 4.3. Forσ∈(0,2), the following estimates\n/integraldisplay\nRNϕ(x)dx≤C, (4.7)\n/integraldisplay\nRN|∇ϕ(x)·x|dx≤C, (4.8)\nand /integraldisplay\nRN|x2∆ϕ(x)|dx≤C, (4.9)\nhold for some constant C.\nProof. It’s follows from [ 12] that\n|∇iϕ(x)|≤C\n|x|N+σ+i(4.10)\nfor|x|≥1 (i=0,1,2) with∇0ϕ(x)=ϕ(x). Thus,\n/integraldisplay\nRNϕ(x)dx=/integraldisplay\n|x|<1ϕ(x)dx+/integraldisplay\n|x|≥1ϕ(x)dx\n≤C+C/integraldisplay\n|x|≥1|x|−(N+σ+i)dx\n≤C(4.11)\nThe inequalities (4.8) and (4.9) could be obtained similarl y.\n16Proof of Theorem 1.3. Multiplying the equation (4.6) by ( G(t)+1)N/σΦ(x,G(t)+1) and integrating\nit onRN, we get\n/integraldisplay\nRNϕ/parenleftiggx\n(G(t)+1)1/σ/parenrightigg\nu(x,t)dx\n=(G(t)+1)N/σ/integraldisplay\nRNΦ(x,G(t)+1)Φ(G(t))∗u0dx\n+(G(t)+1)N/σ/integraldisplay\nRNΦ(x,G(t)+1)/integraldisplayt\n0Φ(G(t)−G(s))∗g(s)f(u)dsdx\n−(G(t)+1)N/σ/integraldisplay\nRNΦ(x,G(t)+1)/integraldisplayt\n0Φ(G(t)−G(s))∗(g(s)u(s))ssdsdx\n+(G(t)+1)N/σ/integraldisplay\nRNΦ(x,G(t)+1)/integraldisplayt\n0Φ(G(t)−G(s))∗(g′(s)u(s))sdsdx(4.12)\nIt follows from the property Φ(s)∗Φ(t)=Φ(s+t) for s,t>0, and Fubini-Tonelli theorem that\n/integraldisplay\nRNΦ(x,G(t)+1)Φ(G(t))∗u0dx\n=/integraldisplay\nRNΦ(G(t)+1)∗Φ(G(t))u0(y)dy\n=/integraldisplay\nRNΦ(2G(t)+1)u0(x)dx,(4.13)\nand /integraldisplay\nRNΦ(x,G(t)+1)/integraldisplayt\n0Φ(G(t)−G(s))∗(g(s)u(s))ssdsdx\n−/integraldisplay\nRNΦ(x,G(t)+1)/integraldisplayt\n0Φ(G(t)−G(s))∗(g′(s)u(s))sdsdx\n=/integraldisplay\nRNΦ(x,G(t)+1)/integraldisplayt\n0Φ(G(t)−G(s))∗(g(s)us(s))sdsdx\n=/integraldisplayt\n0/integraldisplay\nRNΦ(x,2G(t)−G(s)+1)(g(s)us(s))sdsdx\n=/integraldisplay\nRNΦ(G(t)+1,x)g(t)ut(x,t)dx−/integraldisplay\nRNΦ(x,2G(t)+1)g(0)ut(x,0)dx\n+/integraldisplayt\n0/integraldisplay\nRNg(s)2Φ′\n2(x,2G(t)−G(s)+1)us(s)dsdx(4.14)\nwhereΦ′\n2(x,t) :=∂\n∂tΦ(x,t).\n17Thus,/integraldisplay\nRNϕ/parenleftiggx\n(G(t)+1)1/σ/parenrightigg\nu(x,t)dx\n=(G(t)+1)N/σ/integraldisplay\nRNΦ(2G(t)+1)u0(x)dx\n+(G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRNΦ(2G(t)−G(s)+1)g(s)f(u(s))dxds\n−(G(t)+1)N/σ/integraldisplay\nRNΦ(x,G(t)+1)g(t)ut(x,t)dx\n+(G(t)+1)N/σ/integraldisplay\nRNΦ(x,2G(t)+1)g(0)ut(x,0)dx\n−(G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRNg(s)2Φ′\n2(x,2G(t)−G(s)+1)us(s)dsdx(4.15)\nLet\nA(t) :=/integraldisplay\nRNϕ/parenleftiggx\n(G(t)+1)1/σ/parenrightigg\nu(x,t)dx\nB(t) :=(G(t)+1)N/σ/integraldisplay\nRNΦ(x,G(t)+1)g(t)ut(x,t)dx\nC(t) :=(G(t)+1)N/σ/integraldisplay\nRNΦ(2G(t)+1)(u0+g(0)u1)(x)dx\nD(t) :=(G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRNΦ(2G(t)−G(s)+1)g(s)f(u(s))dxds\nE(t) :=−(G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRNg(s)2Φ′\n2(x,2G(t)−G(s)+1)us(s)dsdx\nThen, the equation (4.15) becomes A(t)+B(t)=C(t)+D(t)+E(t).\nWe estimate A(t) first. It follows from H¨ older’s inequality, we have\nA(t)=/integraldisplay\nRNϕ/parenleftiggx\n(G(t)+1)1/σ/parenrightigg\nu(x,t)dx\n≤/parenleftigg/integraldisplay\nRNϕ(x\n(G(t)+1)1/σ)dx/parenrightigg1/p′/parenleftigg/integraldisplay\nRNϕ(x\n(G(t)+1)1/σ)|u(t,x)|pdx/parenrightigg1/p\n≤C(G(t)+1)N/σp′/parenleftigg/integraldisplay\nRNϕ(x\n(G(t)+1)1/σ)|u(t,x)|pdx/parenrightigg1/p\n=:CR(t),(4.16)\nwhere p′is the H¨ older conjugate of pwith p′=p\np−1, since/integraltext\nRNϕ(x)dx≤Cby Lemma 4.3.\nAnd\nB(t)=(G(t)+1)N/σ/integraldisplay\nRNΦ(x,G(t)+1)g(t)ut(x,t)dx\n=g(t)A′(t)+g(t)2\nσ(G(t)+1)/integraldisplay\nRN∇ϕ(x\n(G(t)+1)1/σ)·x\n(G(t)+1)1/σu(x,t)dx.\n18Here, by H¨ older inequality,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg(t)2\nσ(G(t)+1)/integraldisplay\nRN∇ϕ(x\n(G(t)+1)1/σ)·x\n(G(t)+1)1/σu(x,t)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤g(t)2\nσ(G(t)+1)/integraldisplay\nRN/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇ϕ(x\n(G(t)+1)1/σ)·x\n(G(t)+1)1/σ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleϕ(x\n(G(t)+1)1/σ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−p′\npdx1/p′\n×/parenleftigg/integraldisplay\nRNϕ(x\n(G(t)+1)1/σ)|u(t,x)|pdx/parenrightigg1/p\n≤Cg(t)2\nG(t)+1(G(t)+1)N/σp′/parenleftigg/integraldisplay\nRNϕ(x\n(G(t)+1)1/σ)|u(t,x)|pdx/parenrightigg1/p\n=Cg(t)2\nG(t)+1R(t),(4.17)\nwhich implies\nB(t)≤g(t)A′(t)+Cg(t)2\nG(t)+1R(t). (4.18)\nWe estimate D(t) as follows.\nD(t)≥C1(G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRNΦ(2G(t)−G(s)+1)g(s)|u(x,s)|pdxds\n=C1/integraldisplayt\n0/parenleftiggG(t)+1\n2G(t)−G(s)+1/parenrightiggN/σ\ng(s)/integraldisplay\nRNϕ(x\n(2G(t)−G(s)+1))|u(x,s)|pdxds\n≥2−N/σC1/integraldisplayt\n0g(s)/integraldisplay\nRNϕ(x\n(2G(t)−G(s)+1))|u(x,s)|pdxds\n=2−N/σC1/integraldisplayt\n0g(s)(G(s)+1)−N p\nσp′R(s)pds\n=2−N/σC1/integraldisplayt\n0g(s)(G(s)+1)−N\nσ(p−1)R(s)pds\n19Then, for E(t), by the integration by parts, we have\nE(t)=−(G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRNg(s)2Φ′\n2(x,2G(t)−G(s)+1)us(s)dsdx\n=−(G(t)+1)N/σ/integraldisplay\nRNg(t)2Φ′\n2(x,G(t)+1)u(t,x)dx\n+(G(t)+1)N/σ/integraldisplay\nRNg(0)2Φ′\n2(x,2G(t)+1)u0(x)dx\n+(G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRN/bracketleftig\ng(s)2Φ′\n2(x,2G(t)−G(s)+1)/bracketrightig\nsu(x,s)dsdx\n=−(G(t)+1)N/σ/integraldisplay\nRNg(t)2Φ′\n2(x,G(t)+1)u(t,x)dx\n+(G(t)+1)N/σ/integraldisplay\nRNg(0)2Φ′\n2(x,2G(t)+1)u0(x)dx\n+2(G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRNg(s)g′(s)Φ′\n2(x,2G(t)−G(s)+1)u(x,s)dsdx\n−(G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRNg(s)3Φ′′\n22(x,2G(t)−G(s)+1)u(x,s)dsdx\n=:E1(t)+E2(t)+E3(t)+E4(t).\nwhereΦ′′\n22(x,t) :=∂2\n∂t2Φ(x,t).\nSince\nΦ′\n2(x,t)=∂\n∂tΦ(x,t)=−N\nσtΦ(x,t)−1\nσtN/σ+1∇ϕ(x\nt1/σ)·x\nt1/σ, (4.19)\nwe get\n|E1(t)|≤(G(t)+1)N/σ Ng(t)2\nσ(G(t)+1)/integraldisplay\nRN|Φ(x,G(t)+1)u(x,t)|dx\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleg(t)2\nσ(G(t)+1)/integraldisplay\nRN∇ϕ(x\n(G(t)+1)1/σ)·x\n(G(t)+1)1/σu(x,t)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤Cg(t)2\nG(t)+1R(t)(4.20)\nby (4.16) and (4.17), and\nE2(t)=−/parenleftiggG(t)+1\n2G(t)+1/parenrightiggN/σNg(0)2\nσ(2G(t)+1)/integraldisplay\nRN|ϕ(x\n(2G(t)+1)1/σ)u0(x)|dx\n+/parenleftiggG(t)+1\n2G(t)+1/parenrightiggN/σg(0)2\n2σ(G(t)+1)/integraldisplay\nRN∇ϕ(x\n(2G(t)+1)1/σ)·x\n(2G(t)+1)1/σu0(x)dx(4.21)\nhold.\n20It follows from (4.19) that\n|E3(t)|\n≤(G(t)+1)N/σ 2N\nσ(G(t)+1)/integraldisplay\nRN/integraldisplayt\n0g(s)|g′(s)||Φ(x,2G(t)−G(s)+1)u(x,s)|dsdx\n+2\nσ(G(t)+1)N\nσ−1/integraldisplayt\n0/integraldisplay\nRN(2G(t)−G(s)+1)−N+1\nσg(s)|g′(s)||∇ϕ(x\n(2G(t)−G(s)+1)1/σ)·x||u(x,s)|dxds\n=:E31(t)+E32(t).\n(4.22)\nBy H¨ older inequality and Young’s inequality, we get\nE31(t)≤2N\nσ(G(t)+1)N/(σp′)−1/parenleftigg/integraldisplay\nRN/integraldisplayt\n0g(s)|g′(s)|p′|Φ(x,2G(t)−G(s)+1)|dsdx/parenrightigg1/p′\n×(G(t)+1)N/(σp)/parenleftigg/integraldisplayt\n0/integraldisplay\nRNΦ(x,2G(t)−G(s)+1)g(s)|u(s)|pdxds/parenrightigg1/p\n≤C(G(t)+1)N/(σp′)−1/parenleftigg/integraldisplayt\n0g(s)|g′(s)|p′ds/parenrightigg1/p′\nD(t)1/p\n≤1\n8D(t)+C(G(t)+1)N/(σ)−p′/parenleftigg/integraldisplayt\n0g(s)|g′(s)|p′ds/parenrightigg(4.23)\nand\nE32(t)≤2\nσ(G(t)+1)N/σ−1/parenleftigg/integraldisplayt\n0/integraldisplay\nRNΦ(x,2G(t)−G(s)+1)g(s)|u(s)|pdxds/parenrightigg1/p\n×/parenleftigg/integraldisplayt\n0/integraldisplay\nRN(2G(t)−G(s)+1)−N+p′\nσg(s)|g′(s)|p′|∇ϕ(x\n(2G(t)−G(s)+1)1/σ)·x|p′|ϕ(x\n(2G(t)−G(s)+1)1/σ)|−p′\npdxds/parenrightigg1/p′\n≤C(G(t)+1)N/(σp′)−1D(t)1/p/parenleftigg/integraldisplayt\n0g(s)|g′(s)|p′ds/parenrightigg1/p′\n≤1\n8D(t)+C(G(t)+1)N/σ−p′/parenleftigg/integraldisplayt\n0g(s)|g′(s)|p′ds/parenrightigg\n(4.24)\nCombing the estimates (4.22)-(4.24), we obtain\n|E3(t)|≤1\n4D(t)+C(G(t)+1)N/σ−p′/parenleftigg/integraldisplayt\n0g(s)|g′(s)|p′ds/parenrightigg\n. (4.25)\nDirect computation shows\nΦ′′\n22(x,t)=N(N+1)\nσ2t2Φ(x,t)+2N+σ+1\nσ2t2+N/σ∇ϕ(x\nt1/σ)·x\nt1/σ\n+1\nσt2+N/σ∆ϕ(x\nt1/σ)|x|2\nt2/σ,\n21which implies\n|E4(t)|\n=G(t)+1)N/σ/integraldisplayt\n0/integraldisplay\nRNg(s)3|Φ′′\n22(x,2G(t)−G(s)+1)u(x,s)|dsdx\n≤C(G(t)+1)N/σ−2/integraldisplayt\n0/integraldisplay\nRNg(s)3|Φ(x,2G(t)−G(s)+1)u(x,s)|dsdx\n+C(G(t)+1)N/σ−2/integraldisplayt\n0/integraldisplay\nRNg(s)3(2G(t)−G(s)+1)−N+1\nσg(s)|g′(s)||∇ϕ(x\n(2G(t)−G(s)+1)1/σ)·x||u(x,s)|dxds\n+C(G(t)+1)N/σ−2/integraldisplayt\n0/integraldisplay\nRNg(s)3(2G(t)−G(s)+1)−N+1\nσg(s)|g′(s)||∆ϕ(x\n(2G(t)−G(s)+1)2/σ)|x|2||u(x,s)|dxds\n=:E41(t)+E42(t)+E43(t).\n(4.26)\nAgain, by H¨ older’s inequality and Young’s inequality, we g et\nE41(t)≤C(G(t)+1)N/σ−2/parenleftigg/integraldisplayt\n0/integraldisplay\nRNg(s)|Φ(x,2G(t)−G(s)+1)||u(x,s)|pdxds/parenrightigg1/p\n×/parenleftigg/integraldisplayt\n0/integraldisplay\nRNΦ(x,2G(t)−G(s)+1)g(s)2p′+1dxds/parenrightigg1/p\n≤C(G(t)+1)N/(σp′)−2/parenleftigg/integraldisplayt\n0g(s)2p′+1ds/parenrightigg1/p′\nD(t)1/p\n≤1\n12D(t)+C(G(t)+1)N/σ−2p′/parenleftigg/integraldisplayt\n0g(s)2p′+1ds/parenrightigg\n,(4.27)\nE42(t)≤C(G(t)+1)N/σ−2/parenleftigg/integraldisplayt\n0/integraldisplay\nRNg(s)|Φ(x,2G(t)−G(s)+1)||u(x,s)|pdxds/parenrightigg1/p\n×/parenleftigg/integraldisplayt\n0/integraldisplay\nRN(2G(t)−G(s)+1)−N+p′\nσg(s)2p′+1|∇ϕ(x\n(2G(t)−G(s)+1)1/σ)·x|p′|ϕ(x\n(2G(t)−G(s)+1)1/σ)|−p′\npdxds/parenrightigg1/p′\n≤C(G(t)+1)N/(σp′)−2/parenleftigg/integraldisplayt\n0g(s)2p′+1ds/parenrightigg1/p′\nD(t)1/p\n≤1\n12D(t)+C(G(t)+1)N/σ−2p′/parenleftigg/integraldisplayt\n0g(s)2p′+1ds/parenrightigg\n,\n(4.28)\nand\nE43(t)≤C(G(t)+1)N/σ−2/parenleftigg/integraldisplayt\n0/integraldisplay\nRNg(s)|Φ(x,2G(t)−G(s)+1)||u(x,s)|pdxds/parenrightigg1/p\n×/parenleftigg/integraldisplayt\n0/integraldisplay\nRN(2G(t)−G(s)+1)−N+2p′\nσg(s)2p′+1|∆ϕ(x\n(2G(t)−G(s)+1)1/σ)|x|2|p′|ϕ(x\n(2G(t)−G(s)+1)1/σ)|−p′\npdxds/parenrightigg1/p′\n≤C(G(t)+1)N/(σp′)−2/parenleftigg/integraldisplayt\n0g(s)2p′+1ds/parenrightigg1/p′\nD(t)1/p\n≤1\n12D(t)+C(G(t)+1)N/σ−2p′/parenleftigg/integraldisplayt\n0g(s)2p′+1ds/parenrightigg\n,\n(4.29)\n22since /integraldisplay\nRN|∆ϕ(x)|x|2|p′|ϕ(x)|−p′\npdx≤C.\nThus, combing the estimates (4.26)-(4.29), we get\n|E4(t)|≤1\n4D(t)+C(G(t)+1)N/σ−2p′/integraldisplayt\n0g(s)2p′+1ds. (4.30)\nTherefore, from the estimates of B(t),E(t) and D(t), we have\nA(t)+g(t)A′(t)+Cg(t)2\nG(t)+1R(t)≥C/integraldisplayt\n0g(s)R(s)p\n(G(s)+1)N\nσ(p−1)ds+ǫH(t)−I(t), (4.31)\nwhere H(t)=ǫ−1(C(t)−E2(t)) and\nI(t)=C(G(t)+1)N/σ−p′/integraldisplayt\n0g(s)|g′(s)|p′ds+C(G(t)+1)N/σ−2p′/integraldisplayt\n0g(s)2p′+1ds.\nDefineΓ(t) :=/integraltextt\n01\ng(s)ds. Multiplying both sides of the equation (4.31) by/parenleftig\neΓ(t)/parenrightig′, we get\n/parenleftig\neΓ(t)A(t)/parenrightig′+Cg(t)eΓ(t)\nG(t)+1R(t)≥C/parenleftig\neΓ(t)/parenrightig′/integraldisplayt\n0g(s)R(s)p\n(G(s)+1)N\nσ(p−1)ds+/parenleftig\neΓ(t)/parenrightig′(ǫH(t)−I(t)), (4.32)\nsince/parenleftig\neΓ(t)/parenrightig′=eΓ(t)\ng(t).\nIntegrating on [0,T],\neΓ(t)A(t)−A(0)+C/integraldisplayt\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτ\n≥C/integraldisplayt\n0/parenleftig\neΓ(τ)/parenrightig′/integraldisplayτ\n0g(s)R(s)p\n(G(s)+1)N\nσ(p−1)dsdτ+/integraldisplayt\n0/parenleftig\neΓ(τ)/parenrightig′(ǫH(τ)−I(τ))dτ\n=C/integraldisplayt\n0(eΓ(t)−eΓ(τ))g(s)R(s)p\n(G(s)+1)N\nσ(p−1)dτ+/integraldisplayt\n0/parenleftig\neΓ(τ)/parenrightig′(ǫH(τ)−I(τ))dτ(4.33)\nCombining (4.33) with (4.16), we have\neΓ(t)R(t)+C/integraldisplayt\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτ\n≥C/integraldisplayt\n0(eΓ(t)−eΓ(τ))g(s)R(s)p\n(G(s)+1)N\nσ(p−1)dτ+/integraldisplayt\n0/parenleftig\neΓ(τ)/parenrightig′(ǫH(τ)−I(τ))dτ+A(0).(4.34)\nMultiplying g(t)−1and integrating on [0 ,t], we obtain\n/integraldisplayt\n01\ng(s)eΓ(s)R(s)ds+C/integraldisplayt\n01\ng(s)/integraldisplays\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτds\n≥C/integraldisplayt\n01\ng(s)/integraldisplays\n0(eΓ(t)−eΓ(τ))g(s)R(s)p\n(G(s)+1)N\nσ(p−1)dτds+/integraldisplayt\n01\ng(s)/bracketleftigg/integraldisplayt\n0/parenleftig\neΓ(τ)/parenrightig′(ǫH(τ)−I(τ))dτ+A(0)/bracketrightigg\nds.\n(4.35)\nTo estimate the left hand side of the above equation, we intro duce the following Lemma, which is\nproved in [ 8].\n23Lemma 4.4. Let\nχβ:=1,ifβ=1,\n0,ifβ∈(−1,1).\nThen,\nd\nds/braceleftigg\n[log( s+1)+1]χβ(Γ(s)+1)/integraldisplays\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτ/bracerightigg\n≥C1\ng(s)eΓ(s)R(s)+C1\ng(s)/integraldisplays\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτds(4.36)\nThus, applying Lemma 4.4,\n/integraldisplayt\n01\ng(s)eΓ(s)R(s)ds+C/integraldisplayt\n01\ng(s)/integraldisplays\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτds\n≤C/integraldisplayt\n0/bracketleftigg1\ng(s)eΓ(s)R(s)+1\ng(s)/integraldisplays\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτ/bracketrightigg\nds\n≤/integraldisplayt\n0d\nds/braceleftigg\n[log( s+1)+1]χβ(Γ(s)+1)/integraldisplays\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτ/bracerightigg\nds\n=C[log( t+1)+1]χβ(Γ(t)+1)/integraldisplayt\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτ.(4.37)\nSet\nX(t) :=[log( t+1)+1]χβ(Γ(t)+1)/integraldisplayt\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτ,\nand\nY(t) :=/integraldisplayt\n01\ng(s)/integraldisplays\n0(eΓ(s)−eΓ(τ))g(τ)R(τ)p\n(G(τ)+1)N\nσ(p−1)dτds.\nCombing (4.35) and (4.37), we obtain\nC2X(t)≥C3Y(t)+/integraldisplayt\n01\ng(s)/bracketleftigg/integraldisplayt\n0/parenleftig\neΓ(τ)/parenrightig′(ǫH(τ)−I(τ))dτ+A(0)/bracketrightigg\nds. (4.38)\nNow, we estimate the second term of the right hand side in (4.3 8). Recall that\nH(t)=ǫ−1(C(t)−E2(t))\n=/parenleftiggG(t)+1\n2G(t)+1/parenrightiggN/σ/integraldisplay\nRNϕ(x\n(2G(t)+1)1/σ)(a0+g(0)a1)(x)dx\n+/parenleftiggG(t)+1\n2G(t)+1/parenrightiggN/σNg(0)2\nσ(2G(t)+1)/integraldisplay\nRNϕ(x\n(2G(t)+1)1/σ)a0(x)|dx\n−/parenleftiggG(t)+1\n2G(t)+1/parenrightiggN/σg(0)2\n2σ(G(t)+1)/integraldisplay\nRN∇ϕ(x\n(2G(t)+1)1/σ)·x\n(2G(t)+1)1/σa0(x)dx(4.39)\nNote that G(t)=/integraltextt\n0g(s)ds→∞ ast→+∞, since C−1b(t)−1≤g(t)≤Cb(t)−1. Therefore,\nlim\nt→+∞H(t)=2−N/σ/integraldisplay\nRN(a0+g(0)a1)(x)dx>0, (4.40)\n24by the assumption on the initial data. Then, by the similar ar gument as [ 8], we get\n/integraldisplayt\n01\ng(s)/bracketleftigg/integraldisplayt\n0/parenleftig\neΓ(τ)/parenrightig′(ǫH(τ)−I(τ))dτ+A(0)/bracketrightigg\nds≥C3ǫeΓ(t), (4.41)\nfort>tǫ, which implies\nC2X(t)≥C3Y(t)+C4ǫeΓ(t)for t>tǫ. (4.42)\nHere, tǫdepends polynomially on ǫ.\nOn the other hand, direct calculation implies\ng(t)Y′′(t)+g′(t)Y′(t)−Y′(t)=1\ng(t)/integraldisplayt\n0eΓ(τ))g(τ)R(τ)p\n(G(τ)+1)N\nσ(p−1)dτ. (4.43)\nAnd by H¨ older’s inequality,\nX(t)=[log( t+1)+1]χβ(Γ(t)+1)/integraldisplayt\n0g(τ)eΓ(τ)\nG(τ)+1R(τ)dτ\n≤[log( t+1)+1]χβ(Γ(t)+1)/integraldisplayt\n0g(τ)eΓ(τ)\n(G(τ)+1)N\nσ(p−1)R(τ)pdτ1\np/integraldisplayt\n0g(τ)eΓ(τ)\n(G(τ)+1)p′−N\nσdτ1\np′\n.(4.44)\nTherefore,\ng(t)Y′′(t)+g′(t)Y′(t)−Y′(t)\n=1\ng(t)/integraldisplayt\n0eΓ(τ))g(τ)R(τ)p\n(G(τ)+1)N\nσ(p−1)dτ\n≥1\ng(t)X(t)p/braceleftbig[log( t+1)+1]χβ(Γ(t)+1)/bracerightbig−p/integraldisplayt\n0g(τ)eΓ(τ)\n(G(τ)+1)p′−N\nσdτ−p\np′\n.(4.45)\nThus, combing the above inequality with (4.42), we get\ng(t)Y′′(t)+g′(t)Y′(t)−Y′(t)\n≥1\ng(t)X(t)p/braceleftbig[log( t+1)+1]χβ(Γ(t)+1)/bracerightbig−p/integraldisplayt\n0g(τ)eΓ(τ)\n(G(τ)+1)p′−N\nσdτ−p\np′\n≥C1\ng(t)(Y(t)+ǫeΓ(t))p/braceleftbig[log( t+1)+1]χβ(Γ(t)+1)/bracerightbig−p/integraldisplayt\n0g(τ)eΓ(τ)\n(G(τ)+1)p′−N\nσdτ−p\np′\n.(4.46)\nSetW(t) :=e−Γ(t)Y(t)+ǫ. Then, by direct calculations, we have\nW′′(t)+b(t)W′(t)\n≥CW(t)\ng(t)2e(p−1)Γ(t)/braceleftbig[log( t+1)+1]χβ(Γ(t)+1)/bracerightbig−p/integraldisplayt\n0g(τ)eΓ(τ)\n(G(τ)+1)p′−N\nσdτ−p\np′\n.(4.47)\nWe treat the right hand side of the above inequality by the sim ilar method in [ 8] to get that for t>tǫ,\n(t+1)βW′′(t)+CW′(t)≥CW(t)p\nt+1, ifβ∈(−1,1),\n(t+1)−1W′′(t)+CW′(t)≥CW(t)p\n(t+1)(log( t+1)+1, ifβ=−1.(4.48)\nThus, applying Lemma A.1 and Lemma A.2 in [ 8] respectively, we can conclude that\nT0≤exp ( Cǫ−(p−1), ifβ∈(−1,1),\nexp (exp ( Cǫ−(p−1)), ifβ=−1.(4.49)\n25Acknowedgements\nThe first author is supported by Science Foundation of Zhejia ng Sci-Tech University (ZSTU) under\nGrant No.13062120-Y . The second author is supported by JST C REST Grant Number JPMJCR1913,\nJapan and Grant-in-Aid for Young Scientists Research (No.1 9K14581), Japan Society for the Promotion\nof Science.\nReferences\n[1] M. D’Abbicco and S. Lucente, A modified test function meth od for damped wave equations. Adv.\nNonlinear Stud. 13 (2013), no. 4, 867-892.\n[2] M. D’Abbicco, S. Lucente and M. Reissig, Semi-linear wav e equations with e ffective damping.\nChin. Ann. Math. Ser. B 34 (2013), no. 3, 345-380.\n[3] T. Dao and T. Pham, Critical exponent for a weakly coupled system of semi-linear σ-evolution\nequations with frictional damping, arXiv:1911.01946.\n[4] T. Dao and M. Reissig, A blow-up result for semi-linear st ructurally dampedσ-evolution equations,\narXiv:1909.01181.\n[5] K. Fujiwara, M. Ikeda and Y . Wakasugi, Estimates of lifes pan and blow-up rates for the wave equa-\ntion with a time-dependent damping and a power-type nonline arity. Funkcial. Ekvac. 62 (2019), no.\n2, 157-189.\n[6] K. Fujiwara, M. Ikeda and Y . Wakasugi, The Cauchy problem of the semilinear second order evo-\nlution equation with fractional Laplacian and damping, Non linear Differ. Equ. Appl. 28 (2021), no.\n6, Paper No. 63, https: //doi.org/10.1007/s00030-021-00723-6.\n[7] I. Friedman, O.Ria˜ no, S. Roudenko, D. Son and K. Yang, We ll-posedness and dynamics of solutions\nto the generalized KdV with low power nonlinearity, Nonline arity, 36 (2023), 584-635.\n[8] M. Ikeda and T. Inui, The sharp estimate of the lifespan fo r the semi-linear wave equation with\ntime-dependent damping, Di fferential Integral Equations 32 (2019) 1-36.\n[9] M. Ikeda, T. Ogawa, Lifespan of solutions to the damped wa ve equation with a critical nonlinearity,\nJ. Differential Equations 261(3) (2016) 1880-1903.\n[10] M. Ikeda, M. Sobajima and Y . Wakasugi, Sharp lifespan es timates of blowup solutions to semi-\nlinear wave equations with time-dependent e ffective damping. J. Hyperbolic Di ffer. Equ. 16 (2019),\nno. 3, 495-517.\n[11] M. Ikeda and Y . Wakasugi, A note on the lifespan of soluti ons to the semilinear damped wave\nequation, Proc. Amer. Math. Soc. 143 (2015), no. 1, 163-171.\n[12] K. Kim and S. Lim, Asymptotic behaviors of fundamental s olution and its derivatives to fractional\ndiffusion-wave equations, J. Korean Math. Soc. 53 (2016), No. 4, 929-967.\n[13] N. A. Lai and Y . Zhou, The sharp lifespan estimate for sem i-linear damped wave equation with\nFujita critical power in higher dimensions, J. Math. Pures A ppl. 123 (2019) 229-243\n26[14] T. Li and Y . Zhou, Breakdown of solutions to /squareu+ut=u1+α, Discrete Cont. Dynam. Syst., 1\n(1995), 503-520.\n[15] K. Nishihara, Lp−Lqestimates of solutions to the damped wave equation in 3-dime nsional space\nand their application, Math. Z. 244 (2003) 631-649.\n[16] J. Lin, K. Nishihara and J. Zhai, Critical exponent for t he semi-linear wave equation with time-\ndependent damping, Discrete Contin. Dyn. Syst. 32 (2012) 43 07-4320.\n[17] K. Nishihara, Asymptotic behavior of solutions to the s emilinear wave equation with time-\ndependent damping. Tokyo J. Math. 34 (2011), no. 2, 327-343.\n[18] Y . Qian, T. Dao, On the Cauchy problem for a weakly couple d system of semi-linear σ-evolution\nequations with double dissipation, arXiv: 2311.06663v1.\n[19] G. Todorova and B. Yordanov, Critical exponent for a non linear wave equation with damping, C.\nR. Acad. Sci. Paris I, 330 (2000), 557-562.\n[20] G. Todorova and B. Yordanov, Critical exponent for a non linear wave equation with damping, J.\nDifferential Equations, 174 (2001), 464-489.\n[21] Y . Wakasugi, Scaling variables and asymptotic profiles for the semilinear damped wave equation\nwith variable coefficients, J. Math. Anal. Appl., 447 (2017), pp. 452-487.\n[22] Qi S. Zhang, A blow-up result for a nonlinear wave equati on with damping:the critical case, C.R.\nAcad. Sci. Paris I, 333 (2001),109-114.\n27"    },    {        "title": "2401.12022v1.Damping_Enhanced_Magnon_Transmission.pdf",        "content": "Damping-Enhanced Magnon Transmission\nXiyin Ye,1Ke Xia,2Gerrit E. W. Bauer,3, 4and Tao Yu1,∗\n1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China\n2School of Physics, Southeast University, Jiangsu 211189, China\n3WPI-AIMR and Institute for Materials Research and CSRN, Tohoku University, Sendai 980-8577, Japan\n4Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, Beijing 100190, China\n(Dated: January 23, 2024)\nThe inevitable Gilbert damping in magnetization dynamics is usually regarded as detrimental\nto spin transport. Here we demonstrate in a ferromagnetic-insulator–normal-metal heterostructure\nthat the strong momentum dependence and chirality of the eddy-current-induced damping causes\nalso beneficial scattering properties. Here we show that a potential barrier that reflects magnon wave\npackets becomes transparent in the presence of a metallic cap layer, but only in one direction. We\nformulate the unidirectional transmission in terms of a generalized group velocity with an imaginary\ncomponent and the magnon skin effect. This trick to turn presumably harmful dissipation into useful\nfunctionalities should be useful for future quantum magnonic devices.\nIntroduction .—Magnonic devices save power by ex-\nploiting the collective excitations of the magnetic or-\nder, i.e., spin waves or their quanta, magnons, for non-\nreciprocal communication, reprogrammable logics, and\nnon-volatile memory functionalities [1–10]. The possibil-\nity to modulate magnon states and their transport in fer-\nromagnets by normal metals or superconductors brings\nfunctionalities to spintronics [11–14], quantum informa-\ntion [15–21], and topological materials [22, 23]. The pre-\ndiction of inductive magnon frequency shifts by supercon-\nducting gates on magnetic insulators [24–30] have been\nexperimentally confirmed [31]. Normal metals are not\nequally efficient in gating magnons [32–35], but the stray\nfields of magnetically driven “eddy currents” [36–43] sig-\nnificantly brake the magnetization dynamics [36].\nThe intrinsic Gilbert damping seems to be detrimental\nto transport since it suppresses the magnon propagation\nlength. However, in high-quality magnets such as yt-\ntrium iron garnet (YIG) films, this is not such an issue\nsince the magnon mobility is often limited by other scat-\ntering processes such as two-magnon scattering by disor-\nder, and measurements can be carried out in far smaller\nlength scales.\nNatural and artificial potential barriers are impor-\ntant instruments in electronics and magnonics by confin-\ning and controlling the information carriers. They may\nguide magnon transport [31, 44], act as magnonic logic\ngate [45], induce magnon entanglement [18, 46], and help\ndetecting exotic magnon properties [47–50]. In the lin-\near transport regime, the transmission of electrons and\nmagnons through an obstacle has always been assumed\nto be symmetric, i.e., the same for a wave or particle\ncoming from either side.\nIn this Letter, we address the counter-intuitive ef-\nfect that the strong momentum-dependent eddy-current-\ninduced damping by a normal metal overlayer as shown\nin Fig. 1 may help surmount obstacles such as mag-\nnetic inhomogeneities [51], artificial potential barriers\nformed by surface scratches [52], or dc-current carryingwires [46]. Here we focus on the band edges of magnetic\nfilms that are much thinner than the extinction length of\nthe Damon-Eshbach surface states in thick slabs and are\ntherefore not chiral. Instead, the effect therefore origi-\nnates from the Oersted fields generated by the eddy cur-\nrents in the overlayer that act in only half of the recip-\nrocal space [7] and causes magnon accumulations at the\nsample edges or magnon skin effect [8, 9]. The trans-\nmission through a barrier that is small and symmetric\nfor magnons with opposite wave numbers in an uncov-\nered sample becomes unidirectional with the assistance\nof dissipative eddy currents.\nFIG. 1. Ferromagnetic insulator-normal metal heterostruc-\nture. An in-plane external magnetic field H0orients the\nmagnetization at an angle θwith the ˆz-direction. The yellow\nsheet between the normal metal and ferromagnetic insulator\nindicates suppression of the exchange interaction and conven-\ntional spin pumping.\nModel and non-perturbation theory .—We consider the\nferromagnetic insulator (FI)-normal metal (NM) het-\nerostructure with thickness 2 dFanddMand an in-plane\nmagnetic field H0in Fig. 1. The saturated equilib-\nrium magnetization Msmakes an angle θwith the ˆz-\ndirection such that the torques exerted by the external\nand anisotropy fields cancel. For convenience, we set\nθ= 0 in the following discussion and defer results forarXiv:2401.12022v1  [cond-mat.mes-hall]  22 Jan 20242\nfinite θto the Supplemental Material (SM) [53]. We gen-\neralize a previous adiabatic theory [7, 36] to the full elec-\ntrodynamics of the system by self-consistently solving the\nMaxwell equations coupled with the linearized Landau-\nLifshitz (LL) equations and Ohm’s Law. This treatment\nbecomes exact in the limit of an instantaneous response\nof the metal electrons and high-quality ultrathin mag-\nnetic films.\nThe driving force is an externally generated spatiotem-\nporal magnetization dynamics M(r, t) =M(r, ω)e−iωtat\nfrequency ω. According to Maxwell’s theory, the electric\nfieldEobeys the wave equation ∇2E(r, ω)+k2\n0E(r, ω) =\n−iωµ0JM, where the wave number k0=ω√µ0ε0,µ0(ε0)\nis the vacuum permeability (permittivity), and JM=\n∇×Mis the “magnetization current” [54]. Disregarding\nthe intrinsic Gilbert damping, the LL equation\niωM=−µ0γM×Heff[M] (1)\ngoverns the magnetization dynamics in the FI, where γ\nis the gyromagnetic ratio. The effective magnetic field\nHeff[M] =−δF[M]/δM(r), where the free energy Fis\na functional of the magnetization. It includes the static\nfieldH0, the dipolar field Hd, and (in the FI) the ex-\nchange field Hex=αex∇2Mthat depends on the spin-\nwave stiffness αex. In the presence of the NM layer,\nHeff[M] also contains the Oersted magnetic fields gen-\nerated by the “eddy” currents J=σE, where the elec-\ntrical conductivity σis real. This defines a closed self-\nconsistency problem that we solve numerically.\nWe consider a thin FI film with constant Ms=\n(0,0, Ms). The transverse fluctuations M(r, ω) =\n(Mx(k, ω), My(k, ω),0)eik·rwith in-plane wave vectors\nk= (0, ky, kz) are small precessions with iMx(k, ω) =\nakMy(k, ω), where the complex ellipticity akbecomes\nunity for circular motion.\nThe electric-field modes outside the magnet are plane\nwaves with wave numbers km=p\nω2µ0ε0+iωµ0σ,\nwhere σ= 0 in the absence of an NM layer. The continu-\nity of electric and magnetic fields provides the interface\nboundary conditions. The field in the FI\nEη={x,y,z}(−dF⩽x⩽dF)\n=E(0)\nη(−dF⩽x⩽dF) +RkE(0)\nη(x=dF)e−iAk(x−dF)\nis now modified by the reflection coefficient\nRk=\u0000\nA2\nk−B2\nk\u0001\neiBkdM−\u0000\nA2\nk−B2\nk\u0001\ne−iBkdM\n(Ak−Bk)2eiBkdM−(Ak+Bk)2e−iBkdM,(2)\nwhere E(0)is the solution of Eq. (1) inside the FI without\nthe NM cap [53], Ak=p\nk2\n0−k2, and Bk=p\nk2m−k2.\nThe reflection is isotropic and strongly depends on the\nwave vector. Naturally, Rk= 0 when dM= 0. On the\nother hand, when |k|= 0, the electric field cannot escape\nthe FI, since the reflection is total with Rk=−1.A corollary of Maxwell’s equation—Faraday’s Law—\nreads in frequency space iωµ0[Hd(r, t) +M(r, t)] =∇ ×\nE(r, t). When the magnetization of sufficiently thin mag-\nnetic films is uniform, the Zeeman interaction is propor-\ntional to the spatial average Hdover the film thickness.\nReferring to SM for details [53], we find\nHd,x=\u0014\n−Rk\n4A2\nkdFak(e2iAkdF−1)2(−iAkak+ky)\n+i\n2AkdF(e2iAkdF−1)\u0015\nMx≡ζx(k)Mx,\nHd,y=\"\n−Rk\n4iAkdF(e2iAkdF−1)2 \n−ky\niAkak+k2\ny\nA2\nk+ 2!\n+k2\ny\nA2\nk−k2\ny\nA2\nk1\n2iAkdF(e2iAkdF−1)#\nMy≡ζy(k)My.\nBy substitution into the LL equation (1), the spin wave\neigenfrequencies and ellipticities become\nω(k) =µ0γq\n(˜H0−ζx(k)Ms)(˜H0−ζy(k)Ms),(3a)\nak=q\n(˜H0−ζy(k)Ms)/(˜H0−ζx(k)Ms), (3b)\nwhere ˜H0=H0+αexk2Ms. Imω(k)̸= 0 because of the\nJoule heating due to the eddy currents in the cap layer.\nChiral damping and frequency shifts .—The stray elec-\ntric fields of spin waves propagating perpendicular to the\nmagnetization are chiral, i.e., they depend on their prop-\nagation direction by a hand rule. When kz= 0,E=Ezˆz\nis along the equilibrium magnetization and Ez∝My\nis complex only for positive ky. We illustrate the re-\nsults of the self-consistent calculations for dF= 100 nm,\ndM= 500 nm, conductivity σ= 6.0×107(Ω·m)−1\nfor copper at room temperature [55], applied magnetic\nfield µ0H0= 0.02 T, µ0Ms= 0.178 T, the exchange\nstiffness αex= 3×10−16m2for YIG [56], and γ=\n1.77×1011(s·T)−1. The presence of the NM cap lay-\ners shifts the relative phases between the stray electric\nfields and that of the generating spin waves. We focus\nhere on the wave numbers ky=±1µm−1in Fig. 2(a)\n[Fig. 2(b)] at which the electric field is in-phase (out-\nof-phase) with the transverse magnetization Myˆy. The\nresponse to an in-phase (out-of-phase) electric field is dis-\nsipative (reactive). Both components decay in the FI and\nthe vacuum as ∝1/|k|. In the NM, the in-phase compo-\nnent is screened only in the metal region on the scale of\na skin depth λ=p\n2/(ωµ0σ)∼1.5µm at ω= 11 GHz.\nThe out-of-phase electric field, on the other hand, cre-\nates only a reactive response and is therefore symmetric\nabove and below the metallic film. Also in this case the\ndamping is modulated for constant Gilbert damping by\nthe associated spin wave frequency shift in Fig. 2(b), an\neffect that cannot be captured by the adiabatic approxi-\nmation [7, 36].3\nFIG. 2. The system responds strongly to a phase difference\nbetween the spin waves and their wave vector-dependent ac\nelectric stray fields E. ReEcauses damping [(a)] and Im E\na frequency shift [(b)]. Im Ezgoverns the spin wave vector\ndependence of the chiral damping [(c)]. (d) illustrates the\nstrong ky-dependence of the damping of the lowest standing\nspin wave for Cu thicknesses dM={50,100,200,500}nm. (e)\nshows the real and imaginary parts of the reflection coefficient\nRkthat causes the frequency shifts plotted in (f).\nThe chirality of the radiated electric field controls the\nbackaction of the NM layer that modifies the magnon\ndispersion in a chiral fashion. Figure 2(c) illustrates\nthe strong wave vector-dependent damping coefficient\nαeff(k) =|Imωk|/Reωk. Spin waves propagating in the\npositive ˆy-direction decay much faster than those along\nthe negative direction, while the damping for positive\nand negative kzis the same. According to Fig. 2(d), the\ncalculated damping for kz= 0 in Fig. 2(c) increases (de-\ncreases) with the thickness of the Cu (YIG) film. The\nenhancement of the damping saturates for NM thick-\nnesses dN>1/p\nk2+ 1/λ2, depending on the skin depth\n(λ∼1.5µm) and the wave number 1 /kof the electric\nfield. Moreover, the Kittel mode at k= 0 in Fig. 2(e)\nis not affected by the metal at all because the reflection\ncoefficient Rk=−1, which implies that the dynamics\nof the FI and metal fully decouple. Indeed, recent ex-\nperiments do not find a frequency shift of the FMR by a\nsuperconducting overlayer [57, 58]. The additional damp-\ning by eddy currents reported by Ref. [39] is caused bythe width of the exciting coplanar waveguide, a finite-size\neffect that we do not address here.\nThe real part of Rkin Fig. 2(e) causes an in-phase\nOersted magnetic field that chirally shifts the spin wave\nfrequencies by as much as ∼1 GHz, see Fig. 2(f). Refer-\nence [59] indeed reports a frequency shift of perpendicular\nstanding spin wave modes in Bi-YIG films in the presence\nof thin metallic overlayers.\nThe predicted effects differ strongly from those caused\nby spin pumping due to the interface exchange coupling\nαsp= (ℏγ/M sdF)Reg↑↓, where g↑↓is the interfacial spin\nmixing conductance [60]. αspdoes not depend on the\nthickness of the metal and vanishes like 1 /dF. The fre-\nquency shift scales like Im g↑↓/dFand is very small even\nfor very thin magnetic layers. In contrast, the eddy\ncurrent-induced damping is non-monotonic, scaling like\n∝dFwhen 2 kdF≪1, vanishing for much thicker mag-\nnetic layers, and reaching a maximum at dF∼2λ.\nUnidirectional transmission of wave packets through a\npotential barrier .—The transmission of a wave packet im-\npinging from the left or right at a conventional potential\nbarrier is the same [61]. In the presence of a metal cap,\nthis does not hold for magnons in thin magnetic films.\nBefore turning to the potential scattering in this\nmodel, we have to address the effect of the edges. When\nmagnons propagate in the negative direction without\ndamping but decay quickly when propagating in the op-\nposite one, those reflected at the left boundary of the\nsample accumulate, which is a non-Hermitian skin ef-\nfect [62–65]. We substantiate this conclusion by nu-\nmerical calculations for a two-dimensional square lat-\ntice model with ˆ mi= (1 /√\nN)P\nkˆmkeik·ri, where ˆ mk\nis the annihilation operator of magnons with frequency\nωkfrom Eq. (3a) and ilabels the sites and Nis the\nnumber of sites. The Hamiltonian in the real space\nˆH0=P\nijtjiˆm†\njˆmi, where tji= (1/N)P\nkℏωkeik·(rj−ri)is\na hopping amplitude between possibly distant sites iand\njand the summation is over the first Brillouin zone. With\na coarse-grained lattice constant of ay=az= 0.1µm the\nreciprocal lattice vector 2 π/ay,zis much larger than the\nmagnon modes of interest (refer to the SM [53] for de-\ntails). When the frequencies ωkare complex, the Hamil-\ntonian is non-Hermitian, i.e.,tji̸=t∗\nij.\nFigure 3(a) shows the winding path of the real and\nimaginary eigenfrequencies with wave number. In the\ninterval ky= [−25,25]µm−1and an applied magnetic\nfield parallel to the boundary with θ= 0, the complex\ncomponent is hysteretic, indicating localization of modes\nat opposite boundaries. Figure 3(b)-(c) show the average\nspatial distributions W(r) = (1 /Nm)PNm\nl=1|ϕl(r)|2ofNm\nlowest-frequency eigenstates ϕl(r) for ky∈[−1,1]µm−1\nandkz∈[−1,1]µm−1. When the static magnetic field\naligns with the sample boundary z-axis, i.e.θ= 0 in\nFig. 3(b), the magnons tend to accumulate at the left4\nedge. In the antiparallel configurations θ=π[Fig. 3(c)],\nthe magnons aggregate at the right. In the noncollinear\nconfiguration with θ=π/4 [Fig. 3(d)], the maxima shifts\nto the upper-left corner. While Wis an average, we\nalso illustrate the localization of individual low-frequency\nmodes in SM [53].\nFIG. 3. The magnon skin effect caused by chiral damping.\n(a) Complex spectral winding under periodic boundary con-\nditions when kyevolves from −25 to 25 µm−1forθ= 0. (b)-\n(d) corresponds to the edge or corner aggregations of magnon\neigenstates for other magnetic configurations θ∈ {0, π, π/ 4}.\nWe now illustrate the effect of square potential barri-\ners of width dand height u0,ˆV(y) =u0[Θ(y+d/2)−\nΘ(y−d/2)], where Θ( x) is the Heaviside step function,\non the magnon transmission along ˆy(⊥Ms). With in-\ncoming ⟨y|k0⟩=eik0y, the scattered states |ψs⟩obey the\nLippmann-Schwinger formula [66]\n|ψs⟩=|k0⟩+1\niℏ∂t−ˆH0+i0+ˆV|ψs⟩. (4)\nwhere ˆH0=P\nkℏωkˆm†\nkˆmkis the magnon Hamiltonian for\nan extended film. The transmitted waves read\n⟨y|ψs⟩=\u001aT+(k0)eik0y,{y, k0}>0\nT−(k0)eik0y,{y, k0}<0. (5)\nIn the weak scattering limit |u0d| ≪ | ℏvk0|,\nT±(k0) = 1±\u0012iℏvk0\nu0d−vk0\n2|vk0|\u0013−1\n≈1∓iu0d\nℏvk0,(6)\nwhere vk0=∂ωk/∂k|k=k0ˆyis a generalized group ve-\nlocity that dissipation renders complex. The imaginary\npart of the group velocity and transmission amplitudes\ndepend on the direction of the incoming wave:\nD±(k0) =|T±(k0)|2≈1±2Im\u0012u0d\nℏvk0\u0013\n. (7)For example, with u0/ℏ= 30.5 GHz, d= 0.1µm,k0=\n±0.8µm−1,vk0>0= (2.32 + 0 .52i) km/s and vk0<0=\n−(2.64 + 0 .16i) km/s lead to T+(k0>0)≈0.6 while\nT−(k0<0)≈0.9, so even in the weak scattering limit the\nNM cap layer significantly and asymmetrically reduces\nthe transmission probability.\nWe can assess the strong scattering regime with |u0d|≳\n|ℏvk0|by numerical calculations but find dramatic ef-\nfects on the time evolution of a real-space spin-wave\npacket as launched, e.g., by a current pulse in a mi-\ncrowave stripline. We adopt a Gaussian shape Ψ( r,0) =\ne−(r−r0)2/(2η2)eiq0·rcentered at r0with a width η≫ay,z\nthat envelopes a plane wave with wave vector q0and\nˆV(r) = u0f(r) with either f(|y−˜y0|< d) = 1 or\nf(|z−˜z0|< d) = 1, where ˜ y0and ˜z0are the center of the\nbarriers. According to Schr¨ odinger’s equation Ψ( r, t) =\neiˆHt/ℏΨ(r, t= 0) with ˆH=ˆH0+ˆV(r). Numerical results\nin Fig. 4(a) and (b) u0d≪ |ℏvk0|agree with perturbation\ntheory (7) in the weak scattering regime. However, when\n|ℏvk0|≲u0dand|Im(v−k0)| ≪ | Im(vk0)|≲|Re(v±k0)|\nthe transmission and unidirectionality becomes almost\nperfect. Figure 4(c) and (d) show a nearly unidirectional\ntransmission of the wave packet through the potential\nbarrier for the Damon-Eshbach configuration q0⊥Ms;\nit is transparent for spin waves impinging from the left,\nbut opaque for those from the right. In the calculations,\nq0=q(0)\nyˆywith q(0)\ny=±5µm−1andη= 3µm≫d.\nThe potential barrier is peaked with d=ay,z= 0.1µm\nand its height u0/ℏ= 15 GHz is relatively weak (the\nregular on-site energy ∼13 GHz). Also, dM= 50 nm\nanddF= 20 nm. The results are insensitive to the de-\ntailed parameter values (see SM [53]). The red and blue\ncurves are the incident and reflected wave packets, re-\nspectively. When q(0)\ny<0, the barrier does not affect the\nwave packet that propagates freely through the poten-\ntial barrier and accumulates on the left edge [Fig. 4(c)].\nWhen q(0)\ny>0, as shown in Fig. 4(d), the barrier reflects\nthe wave packet nearly completely, which we associate\nagain with the skin effect since these magnons cannot ac-\ncumulate on the right side. The unidirectional transmis-\nsion is therefore a non-local phase-coherent phenomenon\nthat involves the wave function of the entire sample.\nSince we find the skin effect to be crucial, its absence\nin waves propagating in the ˆz-direction must affect the\ntransport over the barrier. Indeed, our calculations in\nFig. 4(e) and (f) find strong reflection for both propaga-\ntion directions, even when reducing the barrier height by\nan order of magnitude to u0= 1.5 GHz (see SM [53]).\nDiscussion and conclusion .—In conclusion, we cal-\nculate the chiral damping, chiral frequency shift, and\nanomalous transport of magnonic modes in ferromag-\nnetic films with NM cap layers beyond the adiabatic ap-\nproximations. We predict anomalous unidirectional spin\ntransport over potential barriers. This effect is rooted\nin the non-Hermitian magnon skin effect and reflects the5\nFIG. 4. Calculated transmissions [(a) and (b)] and time evolu-\ntion of spin-wave packets in the presence of a potential barrier\nat the origin when q0⊥Ms[(c) and (d)] and q0∥Ms[(e)\nand (f)], where Msand the applied magnetic field are parallel\nto the sample edge with θ= 0. The red and blue curves rep-\nresent, respectively, the incident and scattered wave packets\nwith propagation directions indicated by arrows.\nglobal response of the entire system to a local perturba-\ntion. 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To this end we compute the bulk viscosity and t he associated damping time in\nthree-flavor quark matter, considering both nonleptonic an d semileptonic electroweak processes.\nWe use two different equations of state of quark matter, more p recisely, the MIT bag model and\nperturbative QCD, including the leading order corrections in the strong coupling constant. We\nanalyze the dependence of our results on the density, temper ature and value of strange quark mass\nin each case. We then find that the maximum of the bulk viscosit y is in the range of temperature\nfrom 0.01 to 0.1 MeV for frequencies around 1 kHz, while the as sociated minimal damping times of\nthe density oscillations at those temperatures might be in t he range of few to hundreds milliseconds.\nOur results suggest that bulk viscous damping might be relev ant in the post-merger phase after the\ncollision of two neutron stars if deconfined matter is achiev ed in the process.\nKeywords: Three-flavor quark matter, Bulk viscosity, Dampi ng time\nI. INTRODUCTION\nThelong-debatedpossibilitythatquarkmattermaybe\npresent within the core of neutron stars, or adopting the\nformofquarkstars, hasbeen extensivelyexplored[1] (see\nalso [2] for a review and references). While most inves-\ntigations predominantly focused on assessing the mass-\nto-radius ratio of the stars, dictated by the equation of\nstate (EoS) linked to the stellar material, recent oppor-\ntunities for getting insights from neutron star interiors\nhavearisenthanksto the detection ofgravitationalwaves\n(GW) [3, 4]. In the events of neutron star mergers or in\nthe exploration of diverse stellar oscillation modes gener-\nating gravitational radiation [5–7], non-equilibrium pro-\ncesses unfold. The dynamics of these processes are in-\nfluenced by the material’s transport coefficients. A com-\nprehensive understanding of the transport coefficients of\nultra-dense matter becomes imperative, as these are de-\ntermined by the microscopic composition and the dom-\ninant interactions of its constituents. The knowledge of\nthe transport coefficients then brings a connection of the\nmicroscopic and macroscopic dynamics of the star.\nIn this work we will focus on the damping of density\noscillations of quark matter, which are relevant in the\nstudy of neutron star mergers [8]. We aim to dilucidate\nwhether in quark matter dissipative processes might af-\nfect the dynamics in neutron star mergers, in the event\nthat a deconfined phase is achieved in the process. The\ndamping of density oscillations is mainly governedby the\nbulk viscosity, which is the transport coefficient quantify-\ning the energy dissipation in a compressionor rarefaction\nof matter.\nThe bulk viscosityof quarkmatter has been previously\nstudied in [9–12] mainly to determine its effect on the\ndamping of the so called r-modes on isolated compact\nstars. The computation of the bulk viscosity of quarkmatter has also been recently reviewed in [13], improv-\ning the form of the EoS of quark matter, both using a\nperturbative QCD approach and taking into account a\nnonleptonic electroweak (EW) process, and also using\ntwo holographic models. Here we will also compute the\nbulk viscosity. We first consider the MIT bag model to\ndescribe quark matter, as this is extensively used in as-\ntrophysical settings, but we also use a QCD perturbative\napproach. We include both nonleptonic and semileptonic\nprocessesin our computation, as the last arerelevant in a\ncertain range of temperatures, as we will show. Unfortu-\nnately, at the densities one might expect to find in astro-\nphysicalsettings, whichwouldhardlyexceed10times the\nvalue of nuclear saturation density n0≈0.15fm−3, the\nQCD coupling constant αsis not so small, and higher or-\nder corrections than those we consider might be needed.\nIt has been claimed that the computations of the EoS\nfor quark matter only converge for values of the density\n≈40n0[14–16], values which however are not realistic\nin astrophysical settings. That is why one has to invoke\nsome modelling for the EoS of quark matter. It is how-\never instructive to compare those results from those pre-\ndicted in perturbation theory, as another possible model\nfor the description of quark matter in compact stars [17].\nWe will further compute the time scale associated to\nthe damping of the density oscillations. This brings rel-\nevant information on whether these dissipative processes\nmight be relevant or not, for example, in the inspiral\nphase of neutron star mergers, or its post-merger dynam-\nics. Attempts to include the effects of bulk viscosity in\nthe numerical modelling of viscous relativistic hydrody-\nnamics valid for neutron stars and neutron star mergers\nhave only being recently initiated [18, 19].\nOur study is complementary to the same study carried\nout for nuclear matter in Ref. [20, 21], or in [22]. Also\nfor nuclear matter the processes responsible for the bulk\nviscosity are mediated by the EW interactions, by either2\ndirect or by modified URCA processes. Also the value of\nthe bulk viscosity and the damping density oscillations\nstrongly depends on the EoS used to describe nuclear\nmatter.\nThis paper is structured as follows. In Sec. IIwe\npresent the general framework we use for the computa-\ntion of the bulk viscosity in the presence of a periodic\ndisturbance in three-flavor quark matter, and associated\nto different EW processes. We provide numerical values\nof the viscosity and corresponding damping times of den-\nsity oscillations in Sec. III, using the EoS associated to\nthe MIT bag model in Sec. IIIA, andalsoto perturbative\nQCD in Sec. IIIB. We present a discussion of our results\nin Sec.IV. And, finally, in Appendix Awe justify why we\nignore temperature effects in the EoS in the temperature\nrange we are considering for the MIT bag model.\nWe use natural units throughout the article, /planckover2pi1=c=\nKB= 1\nII. BULK VISCOSITY IN THREE-FLAVOR\nQUARK MATTER\nIn this section we study the bulk viscosity generated\nin three-flavor quark matter by nonleptonic and semilep-\ntonic weak processes. As a result, we get the bulk viscos-\nity associated to neutrino-free quark matter as a func-\ntion of temperature, T, the chemical potentials of the\nconstituent particles of the star, µi, and the frequency of\nthe oscillation mode, ω. In this article we will only focus\non the study of density oscillations, although our results\nfor the bulk viscosity might be also used for the study\nof the damping of different stellar oscillation modes. In\nthe normal phase and the neutrino-transparent regime,\nwe consider the following equilibration processes\nu+d↔u+s , (1)\nu+e−→d+νe, (2)\nd→u+e−+ ¯νe, (3)\nu+e−→s+νe, (4)\ns→u+e−+ ¯νe, (5)\nthat involve electrons ( e) and electronic neutrinos and\nantineutrinos ( νeand ¯νe, respectively) as well as up ( u),\ndown (d) and strange ( s) quarks.\nOn the one hand, fluctuations around the equilibrium\nvalue of the four-vector velocity ( uµ) and the particle\nnumber density ( nj) can be expressed as follows\nuµ=uµ\n0+δuµ, nj=nj,0+δnj, (6)\nsuch that, in beta equilibrium, we have\n∂µ(njuµ) = 0. (7)\nConsidering the LRF (Local Rest Frame) in the equilib-\nrium state, that is uµ\n0=uµ\nLRFand neglecting quadraticterms in the deviations, O(δ2), Eq. (7) implies that the\nparticle conservation law can be expressed as\nθnj,0+uµ\nLRF∂µδnj(t) = 0, (8)\nwhereθ=∂µδuµis the fluid expansion rate or equiva-\nlently\nθnj,0+∂\n∂tδnj(t) = 0. (9)\nOn the other hand, out-of-beta-equilibrium deviations of\nthe particle number density of the constituent particles\nare obtained from the following set of equations for the\nparticle density of strange quarks ( ns) and the particle\ndensity of electrons ( ne)\n∂\n∂tδns(t)+θns,0=−λ1µ1−λ2µ2,(10)\n∂\n∂tδne(t)+θne,0=λ2µ2+λ3µ3,(11)\nwhereλ1, λ2,andλ3aredefinedintermsoftheequilibra-\ntion rates of the nonleptonic and semileptonic processes\nas\nµ1λ1= Γs+u→d+u−Γd+u→s+u, (12)\nµ2λ2= Γs→u+e+¯νe−Γu+e→s+νe,(13)\nµ3λ3= Γd→u+e+¯νe−Γu+e→d+νe.(14)\nThe rates in the limit of massless up and down quarks\nhave been computed in several studies at tree level [23–\n25]. For pure massless up and down quarks, perturbative\ncorrectionsto the quark dispersion law in the strong cou-\npling constant αsare however needed in order to find a\nnon-vanishing value for λ3. One then finds\nλ1=64\n5π3G2\nFsin2ΘCcos2ΘCµ5\ndT2,(15)\nλ2=17\n40πG2\nFsin2ΘCµsm2\nsT4, (16)\nλ3=17\n15π2G2\nFcos2ΘCαsµdµuµeT4,(17)\nwhereGF= 1.166×10−5GeV−2is the Fermi coupling\nconstant, Θ C= 13.02◦is the Cabibbo angle, msis the\nstrange quark mass, and\nµ1=µs−µd, (18)\nµ2=µs−µe−µu, (19)\nµ3=µd−µe−µu. (20)\nNote that µ2−µ1=µd−µe−µu=µ3. Then, we obtain\nthat\n∂\n∂tδne(t)+θne,0= (λ2+λ3)µ2−λ3µ1.(21)\nThe oscillating parts of the particle density are taken to\nbe proportional to eiωt, so that\nδnj(t) =−θ\niωnj,0. (22)3\nThus the equation for the strange quark density can be\nexpressed as follows\niωδns(t)+θns,0=−λ1µ1−λ2µ2.(23)\nWe can then determine the out-of-beta-equilibrium de-\nviations of the particle number density of electrons and\nstrange quarks considering\nnd= 2nB−ns−ne, (24)\nnu=nB+ne. (25)\nHere we used the charge neutrality condition\nne+1\n3ns+1\n3nd=2\n3nu, (26)\nand the definition of the baryon density\nnB≡1\n3nu+1\n3nd+1\n3ns. (27)\nUsing the beta equilibrium and charge neutrality condi-\ntions in the out-of-beta-equilibrium particle number den-\nsities, we obtain\nδnd= 2δnB−δns−δne, (28)\nδnu=δnB+δne. (29)\nThe imbalance of the chemical potentials out-of-beta-\nequilibrium can be written in terms of deviations of the\nparticle number densities as follows\nµ1=Asδns−Adδnd, (30)\nµ2=Asδns−Aeδne−Auδnu,(31)\nwhereAjare the susceptibilities of the constituent par-\nticles, which can be determined in terms of the following\npartial derivatives\nAij=/parenleftbigg∂µi\n∂nj/parenrightbigg\n, (32)\noncewechooseanEoS.Employingtheserelations, weget\nthe conservationequationforthe particlenumber density\nof strange quarks\niωδns=−θns,0−λ1[Asδns−Ad(2δnB−δns−δne)]\n−λ2[Asδns−Aeδne−Au(δnB+δne)],(33)\nand a similar expression for the electrons\niωδne=−θne,0+(λ2+λ3)[Asδns−Au(δnB+δne)\n−Aeδne]−λ3[Asδns−Ad(2δnB−δns−δne)].\n(34)\nUsing Eqs.( 33) and (34) we are able to get an expres-\nsion forδns. Next, by employing the conservation of the\nbaryon number\nδnB=−θ\niωnB,0, (35)we obtain\nBδns=−θ\niω{iω[[iω+(λ2+λ3)A+λ3Ad]ns,0\n+ (λ2A−λ1Ad)ne,0]+[Ad(Au+2A)λQ\n+iω(2λ1Ad+λ2Au)]nB,0}, (36)\nwhere we define A≡Ae+Au,\nλQ≡λ1λ2+λ1λ3+λ2λ3, (37)\nand\nB≡iω[λ1(As+Ad)+λ2(A+As)+λ3(A+Ad)]\n+λQ[A(As+Ad)+AdAs]−ω2. (38)\nAnd by means of the Eq. ( 34) and (36), we have\nBδne=−θ\niω{iω[[iω+λ1(As+Ad)+λ2As]ne,0\n+ (λ2As−λ3Ad)ns,0]+[λ2As(λ2Au+2λ1Ad)\n−[λ2Au+λ3(Au−2Ad)][iω+(λ1+λ2)As]\n−AdAuλQ]nB,0}. (39)\nIn general, the deviations of the particle number density\ncan be expressed as\nδnj=δnj,0+δn′\nj, (40)\nwhereδnj,0is a fluctuation around beta equilibrium and\nδn′\njis an out-of-equilibrium deviation. For the former\none it follows\nδnj,0=−θ\niωnj,0. (41)\nThen, for the electron and the strange quark number\ndensity, we get\nδne=δne,0+δn′\ne, (42)\nδns=δns,0+δn′\ns. (43)\nThus\nBδn′\ns=θ\niω[iω(λ1C1+λ2C2)+(AC1+AdC2)λQ],(44)\nand\nBqδn′\ne=θ\niω{iω[(C1−C2)λ3−C2λ2]\n−[(Ad+As)C2−AsC1]λQ},(45)\nwhere\nC1≡ns,0As−nd,0Ad,\n=ns,0As−(2nB,0−ns,0−ne,0)Ad,(46)\nC2≡ns,0As−nu,0Au−ne,0Ae,\n=ns,0As−ne,0A−nB,0Au. (47)4\nOnce known the out-of-equilibrium fluctuations of the\nparticle number densities, we can calculate the bulk vis-\ncosity. First, we determine the out-of-equilibrium pres-\nsure as\np(nj(t)) =p(nj,0+δnj,0)+δp′(t) =p0(t)+δp′(t),(48)\nwhere the nonequilibrium part of the pressure is given by\nΠ =δp′=/summationdisplay\nj/parenleftbigg∂p\n∂nj/parenrightbigg\n0δn′\nj. (49)\nThis follows if we assume that the thermal equilibrium\nrate is much larger than the chemical equilibrium rate,\nso the temperature is constant (see the note [51] in [26]).\nUsing the Gibbs-Duhem relation\ndp=sdT+/summationdisplay\ninidµi, (50)\nwhich is valid out of equilibrium, and taking dT≈0, we\nhave\ncj≡/parenleftbigg∂p\n∂nj/parenrightbigg\n0=/summationdisplay\nini,0/parenleftbigg∂µi\n∂nj/parenrightbigg\n0=/summationdisplay\nini,0Aij.(51)\nThen the nonequilibrium pressure in quark matter can\nbe expressed as\nΠ =/summationdisplay\njcjδn′\nj=ceδn′\ne+cuδn′\nu+cdδn′\nd+csδn′\ns.(52)\nDue to the conservation of the baryon particle number\ndensity, its out-of-equilibrium deviation is zero\nδn′\nB= 0, (53)\nand, as a result, we get\nδn′\nu=δn′\ne, (54)\nδn′\nd=−δn′\ns−δn′\ne. (55)\nThus it follows that\nΠ = (cu−cd+ce)δn′\ne+(cs−cd)δn′\ns.(56)\nAccording to the Eq. ( 51), we find that\ncs=ns,0Ass, (57)\ncu=nu,0Auu, (58)\ncd=nd,0Add, (59)\nce=ne,0Aee. (60)\nAt this stage the model choice of three-flavor quark mat-\nter with electrons is required. For electrons we only con-\nsider diagonal terms for Aeebecause they form an ultra-\nrelativistic noninteracting gas. The same applies for the\nquarks (even if we consider the first correction in αs),\nso that the particle number density for each flavor doesnot depend on the chemical potential of the other flavors.\nThus,\nAu=Auu, Ad=Add, As=Ass, Ae=Aee,(61)\nwhere\ncs−cd=C1, (62)\ncs−cu−ce=C2, (63)\nso that\nΠ = (C1−C2)δn′\ne+C1δn′\ns, (64)\nwhich can be expressed as\nBΠ =θ\niω{iω[λ1C2\n1+λ2C2\n2+λ3(C1−C2)2]\n+ [AdC2\n2+AC2\n1+As(C1−C2)2]λQ}.(65)\nThe bulk viscosity in quark matter is given by\nζ≡ −Re[Π]\nθ, (66)\nwhere the real part of the nonequilibrium pressure can\nbe obtained from Eq. ( 65) as\nζ=κ1+κ2ω2\nκ3+κ4ω2+ω4, (67)\nwhere\nκ1≡λQ{C2\n1[(A+Ad)[A(λ2+λ3)+Adλ3]\n−Ad(Aλ2−Adλ1)]\n−2C1(C1−C2)[Ad[(Ad+As)λ1+(A+Ad)λ3]\n−AAsλ2]\n+ (C1−C2)2[λ1(Ad+As)2+λ2A2\ns+λ3A2\nd]},(68)\nκ2≡λ1C2\n1+λ2C2\n2+λ3(C1−C2)2, (69)\nκ3≡λ2\nQ[A(As+Ad)+AdAs]2, (70)\nκ4≡[(Ad+As)λ1+Asλ2]2+2(Aλ2−Adλ1)\n×[As(λ2+λ3)−(Ad+As)λ3]\n+ [Adλ3+A(λ2+λ3)]2. (71)\nIII. DAMPING TIME OF DENSITY\nOSCILLATIONS\nIn this section we determine the damping time asso-\nciated to the bulk viscosity coming from baryon number\ndensity oscillations in a medium. Let us assume a small\ndensity oscillation described by δnB=δnB,0Re(eiωt),\nwhereδnB,0andωare the magnitude and frequency of\nthe oscillation, respectively.\nThe energy density ǫstored in a baryonic oscillation\nwith amplitude δnBcan be obtained as\nǫ=1\n2∂2ε\n∂n2\nB(δnB)2, (72)5\nwhere the energy density is\nε= Ω+/summationdisplay\niniµi, (73)\nand the damping time is defined by\nτζ≡ǫ/(dǫ/dt). (74)\nThe energy dissipation time can be related with the bulk\nviscosity according to following expression\ndǫ\ndt=ω2ζ\n2/parenleftbiggδnB\nnB/parenrightbigg2\n. (75)\nAs a result, the damping time of baryon density oscilla-\ntions by bulk viscosity is given by\nτζ=n2\nB\nω2ζ∂2ε\n∂n2\nB. (76)\nConsidering an EoS in the limit of zero temperature re-\nsults in a simplification of the expression leaving the full\ntemperature-dependenceencodedonlyin thebulk viscos-\nity.\nA. MIT bag model\nIn this section we determine the values of the chemical\npotentials and particle number densities of three-flavor\nquark matter with electrons in the neutrino-transparent\nregime using as constraints the beta equilibrium and\ncharge neutrality. As a first approximation we can con-\nsider the simplest phenomenological bag model at zero\ntemperature. For finite-temperature corrections to the\nideal Fermi gas expressions, see the Appendix A. For this\nmodel the thermodynamic potential is given by\nΩ =/summationdisplay\ni=e,u,d,sΩ(0)\ni+Beff, (77)\nwhereBeffis the bag constant [27], and Ω(0)\niis the grand\ncanonical potential of massless electrons and light quarks\ndescribed as ideal Fermi gases\nΩ(0)\ne=−µ4\ne\n12π2, (78)\nand\nΩ(0)\nf=−Nc\n12π2/bracketleftbigg\nµfuf/parenleftbigg\nµ2\nf−5\n2m2\nf/parenrightbigg\n+3\n2m4\nfln/parenleftbiggµf+uf\nmf/parenrightbigg/bracketrightbigg\n, (79)\nhereafter f=u,d,s,Nc= 3 is the number of colors, mf\ndenotes the quark mass, and uf≡/radicalBig\nµ2\nf−m2\nf.From the thermodynamic potential we are able to get\nthe thermodynamic properties of quark matter. Particu-\nlarly, the number particle density for each particle specie\ncan be calculated by\nni=−/parenleftbigg∂Ω\n∂µi/parenrightbigg\nT,V. (80)\nAt zero temperature the number densities of quarks and\nelectrons can be written as follows\nnf=Nc\n3π2(µ2\nf−m2\nf)3/2, (81)\nand\nne=1\n3π2µ3\ne. (82)\nWith these expressions at hand, the beta equilibrium\nconditions can be expressed as\nµd=µs, (83)\nµs=µu+µe. (84)\nUsing Eq. ( 26) (the charge neutrality condition) and the\ndefinition of the baryon number density of Eq. ( 27), we\ncan determine the four chemical potentials and the four\nnumber densities ( µd,µs,µu,µe,nd,ns,nu, andne) for\na fixed value of the baryon number density.\nThesusceptibilitiescanbe obtainedfromEqs.( 81)and\n(82), and are given by\nAee=π2\nµ2e, (85)\nand\nAff=π2\n3µf/radicalBig\nµ2\nf−m2\nf, (86)\nfor electrons and quarks, respectively.\nFor this study we consider different values of the\nstrange quark mass and the baryon number density\nin terms of the nuclear saturation density, n0= 0.15\nfm−3[28]. The light quarks are considered massless, as\ntheirtinyvaluesdonothavearelevanteffectonthequan-\ntities of interest.\nTablesIandIIshow the values of the chemical po-\ntential and number density for quarks and electrons at\nms= 100 MeV and typical values of the baryon number\ndensity in neutron stars.\nnB,0/n0µu,0µd,0µs,0µe,0\n3324.31331.84331.847.53\n5384.52390.91390.916.39\n6408.61414.64414.646.03\nTable I: Chemical potentials in MeV for quark matter with\nelectrons imposing charge neutrality and beta equilibrium us-\ning the MIT bag model at ms= 100 MeV and varying the\nbaryonnumberdensity(normalized tonuclearsaturation de n-\nsity).6\nnB,0/n0nu,0/n0nd,0/n0ns,0/n0ne,0/n0\n33.003.212.791.25×10−5\n55.005.254.757.66×10−6\n66.006.275.736.42×10−6\nTable II: Particle number densities normalized to the nucle ar\nsaturation density imposing charge neutrality and beta equ i-\nlibrium using the MIT bag model at ms= 100 MeV and\nvarying the normalized baryon number density.\nIn Fig.1the bulk viscosity as a function of the tem-\nperature is depicted for different frequencies and baryon\nnumber densities for a fixed value of ms= 100 MeV. As\ncan be seen, increasing the baryon number density gen-\nerates a shift of the maximum of the bulk viscosity to\nlower temperatures, increasing slightly its value. This is\nclearlyseeninFig. 2, wherewezoomthe maximumofthe\nbulk viscosities for different densities at ω/2π= 1 kHz\nandms= 100 MeV. The values for the maxima for the\ndifferent densities at ω/2π= 1 kHz and ms= 100 MeV\nare given in Table III. In addition, in Fig. 1we consider\ndifferent values of the angular frequency around 1 kHz.\nWe observe that the larger the frequency is, the smaller\nthe value of the maximum of the bulk viscosity becomes\nwhereas it moves to larger temperatures.\n0.010.050.100.5015101024102510261027102810291030\nFigure 1: Bulk viscosity of three-flavor quark matter in the\nneutrino-free regime using the MIT bag model for different\nnormalized baryon number densities and normalized frequen -\ncies atms= 100 MeV.0.050.060.070.080.090.102.60×10282.65×10282.70×10282.75×10282.80×1028\nFigure 2: Zooming in of Fig. 1 to display the behavior of the\nmaximum bulk viscosities at ω/2π= 1 kHz, ms= 100 MeV\nand for different baryon number densities.\nIn order to study the damping times of baryon density\noscillations induced by the weak-interaction-driven bulk\nviscosity we resort to Eq. ( 76). The energy density in\nthe MIT bag model is given explicitly by Eqs. ( 73), (77),\n(81), and (82).\nFig.3displays the damping times associated to the\nbulk viscosities in Fig. 1. Note that the exact values for\nthe minimal damping times can be found in Table III.\nThe temperature dependence of the damping time is the\nsame as for the inverse of the bulk viscosity, as this fol-\nlows from the zero-temperature approximation for the\nthermodynamic potential. However, all other terms in-\nvolved in Eq.( 76) are relevant for determining the exact\nvalue of the damping times as a function of the baryon\nnumber density. In addition, the ω−2term modifies sig-\nnificantly the frequency-dependence from the inverse of\nthe bulk viscosity. We also note that at nB,0/n0= 3\nthe damping times seem to be independent of the fre-\nquencies considered in the low temperature regime for\napproximately T <20 keV.\n0.010.050.100.5015101100104106108\nFigure 3: Damping times from density oscillations using the\nMIT bag model for different normalized baryon number den-\nsities and frequencies at ms= 100 MeV.7\nnb,0/n0Tm ζmax τminω/2π\n32.5×10−22.73×10292132.420.1\n32.5×10−12.73×102721.3210\n37.9×10−22.73×1027213.241\n56.2×10−22.75×1028420.141\n65.7×10−22.76×1028535.371\nTable III: Maximum of the bulk viscosity (in gr cm−1s−1)\nand minimum of the damping times (in ms) for different nor-\nmalized baryon number densities and frequencies (in kHz) at\nms= 100 MeV according to Figs. 1 and 3. Here Tmdenotes\nthe temperature in MeV of the maximum (minima) of the\nbulk viscosity (damping time).\nIn addition, one can study the effect of the strange\nquark mass in the bulk viscosity. Fig. 4shows the bulk\nviscosity as a function of the temperature for different\nvalues of the strange quark mass at ω/2π= 1 kHz and\nnB,0/n0= 3. As the strange quark mass increases, not\nonly the bulk viscosity is larger but also the effect of the\nsemileptonic processes becomes more evident for temper-\naturesofafewMeVs. Thiseffect islinkedtothefactthat\nλ2∝m2\ns.\n0.010.050.100.5015101024102510261027102810291030\nFigure 4: Bulk viscosity of three-flavor quark matter in the\nneutrino-free regime using the MIT bag model for different\nvalues of the strange quark mass at ω/2π= 1 kHz and\nnB,0/n0= 3.\nFinally, in Fig. 5we depict the damping times asso-\nciated to varying the value of the strange quark mass,\ncorresponding to the viscosities in Fig. 4. Increasing the\nvalue of the mass of the strange quark has a drastic ef-\nfect in loweringthe damping times below10 milliseconds.\nThe minimal damping times of density oscillations for\nω/2π= 1 kHz thus can range from 3 to 200 milliseconds\nat a given temperature, but this depends strongly on the\nvalue of the strange quark mass. The values for the max-\nima of the bulk viscosity and the minimal damping time\nare given in Table IV.0.010.050.100.5015101101001000104105106\nFigure 5: Damping times from density oscillations using the\nMIT bag model for different values of the strange quark mass\natω/2π= 1 kHz and nB,0/n0= 3.\nmsTm ζmax τmin\n1007.9×10−22.73×1028213.24\n1507.5×10−21.34×102942.77\n2006.9×10−24.05×102913.93\n3005.7×10−21.73×10303.19\nTable IV:Maximumofthebulkviscosity (ingrcm−1s−1)and\nminimum of the damping times (in ms) for different masses\n(in MeV) according to Figs. 4 and 5. Here Tmdenotes the\ntemperature in MeV of the maximum (minima) of the bulk\nviscosity (damping time).\nB. Perturbative QCD\nA similar analysis can be performed for perturbative\nQCD at high density with a finite mass for the strange\nquark. As previously stated for the MIT model, a zero-\ntemperature limit for the number particle density and\nsusceptibility of the constituent particles is a good ap-\nproximation for the temperature region of interest. In\nperturbative QCD, the thermodynamic potential at fi-\nnite temperature and chemical potential up to O(αs) has\nbeen addressed in Ref. [29]. In this work we consider the\nzero-temperature limit of this expression which is given\nby\nΩ = Ω(0)\ne+/summationdisplay\nf=u,d,s/parenleftBig\nΩ(0)\nf+Ω(1)\nf/parenrightBig\n,(87)\nwhere the leading-order terms for massless electrons and\nnon-vanishing quark masses are shown in Eqs. ( 78) and\n(79), and the first-order correction in the MS scheme is\ngiven by8\nΩ(1)\nf=αs(N2\nc−1)\n16π3/braceleftbigg\n3/bracketleftbigg\nm2\nfln/parenleftbiggµf+uf\nmf/parenrightbigg\n−µfuf/bracketrightbigg2\n−2u4\nf+m2\nf/bracketleftbigg\nµfuf−m2\nfln/parenleftbiggµf+uf\nmf/parenrightbigg/bracketrightbigg\n×/bracketleftbigg\n6ln/parenleftbigg¯Λ\nmf/parenrightbigg\n+4/bracketrightbigg/bracerightbigg\n, (88)\nwhere¯Λistherenormalizationscaleand uf≡/radicalBig\nµ2\nf−m2\nf\nas in the previous section. The thermodynamic po-\ntential up to order αsdepends on the renormalization\nsubtraction point explicitly and implicitly through the\nscale dependence of the strong coupling constant and the\nmass [14, 29, 30]. Considering the massless approxima-\ntion for the light quarks, the scale dependence of the\ncoupling and the strange quark mass to first order in αs\ncan be expressed as\nαs(¯Λ) =4π\nβ0L/bracketleftbigg\n1−2β1\nβ2\n0ln(L)\nL/bracketrightbigg\n, (89)\nms(¯Λ) = ˆms/parenleftBigαs\nπ/parenrightBig4/9/parenleftBig\n1+0.895062αs\nπ/parenrightBig\n,(90)\nwithL= 2ln(¯Λ/ΛMS), the one-loop β-function coef-\nficientβ0= 11−2Nf/3, and the two-loop coefficient\nβ1= 51−19Nf/3 withNf= 3. ΛMSand the invari-\nant mass ˆ mscan be fixed by requiring αs≃0.3 and\nms≃100 MeV at ¯Λ = 2 GeV. As a result, one ob-\ntains ΛMS≃380 MeV and ˆ ms= 262 MeV. According\nto these constraints, the only undetermined parameter is\nthe value of the renormalization scale ¯Λ.\nThe particle number density for electrons is given in\nEq. (82) and for quarks up to O(αs) can be expressed as\nnf=n(0)\nf+n(1)\nf, (91)\nwhere\nn(0)\nf=Nc\n3π2(µ2\nf−m2\nf)3/2, (92)\nand\nn(1)\nf=−αs(N2\nc−1)\n4π3µfu2\nf/braceleftbigg\n1−3m2\nf\nµfufln/parenleftbiggµf+uf\nmf/parenrightbigg\n+m2\nf\n2µfuf/bracketleftbigg\n6ln/parenleftbigg¯Λ\nmf/parenrightbigg\n+4/bracketrightbigg/bracerightbigg\n. (93)\nAn alternative to handle this expression is not to con-\nsider the ¯Λ-dependent term in Eq. ( 93) as in Ref. [11],\nwhich is equivalent to set ¯Λ = exp( −2/3)mf. However,\nthe strong coupling constant and the strange quark mass\nin Eqs. ( 89) and (90) are fixed according to this choice.\nOther alternatives consider ¯Λ = 2µsand 3µs, which have\na relevant impact in the mass-to-radius ratio of com-\npact stars [17, 29], and ¯Λ = 2/summationtext\nfµf/Nfas in Ref. [14].In this work, we proceed setting ¯Λ = 2µsand imple-\nment the beta-equilibrium and the charge-neutralitycon-\nditions. Using this procedurethe strangequarkmassand\nthe strong coupling constant are decreasing functions of\nthe baryon particle density.\nWith theseexpressionsathand, wesolvethebeta equi-\nlibrium and charge neutrality conditions for different val-\nues of the baryon number density. The light quarks are\nconsidered massless. Tables VandVIlist the values\nof the chemical potential, the strange quark mass, the\nstrong coupling constant, and the number density of the\nconstituent particles in three-flavor quark matter with\nelectrons for different baryon number density.\nnB,0/n0µu,0µd,0µs,0µe,0msαs\n6470.47489.18489.1818.71138.460.54\n10544.80557.26557.2612.46127.120.47\n20671.16678.75678.757.58115.060.39\n40832.88837.75837.754.87106.010.33\nTable V: Input parameters for the bulk viscosity with pQCD\nat different normalized baryon number densities imposing\nbeta equilibrium and electric charge neutrality: chemical po-\ntentials in MeV, the strange quark mass in MeV and the\nstrong coupling constant.\nnB,0/n0nu,0/n0nd,0/n0ns,0/n0ne,0/n0\n66.006.745.251.92×10−4\n1010.0010.709.305.67×10−5\n2020.0020.6919.311.28×10−5\n4040.0040.7139.293.39×10−6\nTableVI:Normalized particle numberdensities withpQCDat\ndifferent normalized baryon number densities imposing beta\nequilibrium and electric charge neutrality.\nThe susceptibilities are given by Eq. ( 85) and\nA−1\nff=Nc\nπ2µf/radicalBig\nµ2\nf−m2\nf−αs(Nc−1)2\n4π3/braceleftbigg\n3µ2\nf−4m2\nf\n+m2\nfµf\nuf/bracketleftbigg\n2−3ln/parenleftbiggµf+uf\nmf/parenrightbigg\n+3ln/parenleftbigg¯Λ\nmf/parenrightbigg/bracketrightbigg/bracerightbigg\n,\n(94)\nfor electrons and quarks, respectively.\nIn Fig.6we plot the bulk viscosity as function of the\ntemperature at ω/2π= 1 kHz and ¯Λ = 2µsfor different\nbaryon number densities.9\n0.010.050.100.501510102410251026102710281029\nFigure 6: Bulk viscosity of three-flavor quark matter with\nelectrons using perturbative QCD for different normalized\nbaryon number densities at ω/2π= 1 kHz and and ¯Λ = 2µs.\nOur results seem to qualitatively agree with those re-\ncently presented in [13], valid for densities 40 n0, within\nperturbative QCD, even if in that reference higher order\ncorrections to the EoS were included, and only the non-\nleptonic process u+d↔u+swas considered. We have\nchecked that the value of the maximum value of the bulk\nviscosity as well as its location as a function of the tem-\nperature qualitatively agree, when computed at the same\norder of accuracy. In Ref. [13] higher order perturbative\ncorrections are included, changing slightly the position\nand the value of the maximum of the bulk viscosity. We\nhowever find some discrepancies with [13] that might be\ndue to the fact that in this reference the semileptonic\nprocesses were discarded.\nIn order to check the relevance of the semileptonic pro-\ncess we consider a similar approach to Ref. [11] comput-\ningthebulkviscositygeneratedonlybynonleptonicweak\nprocesses, ζnon. This can be obtained from the general\nexpression in Eq.( 67) setting λ2,λ3→0 and it is given\nby\nζnon=λ1C2\n1\n(Ad+As)2λ2\n1+ω2. (95)\nWe plot the ratio of the full bulk viscosity with that\narising only from the nonleptonic processes in Fig. 7. We\nconclude that there is certain range of temperatures, in\nthe region from 0.1 MeV to 2 MeV, where neglecting the\nsemileptonic processes is not a good approximation. We\nnote that in Ref. [11] it has been claimed that the regime\nwhere the semileptonic processes might be dropped de-\npends on the value of the frequency at a given value of\nthe density and temperature.1.01.21.41.61.82.0\n0.010.050.100.5015101.011.02\nFigure 7: Ratio ζ/ζnonas a function of the temperature using\nperturbative QCD for different frequencies at nB,0/n0= 6\nand 40. Note that the y-axis is different for each case.\nLastly, Fig. 8shows the damping times for different\nbaryon densities at ω/2π= 1 kHz and ¯Λ = 2µs. The\nvalues for the maxima of the bulk viscosity and the min-\nimal damping time for different densities are given in\nTableVII. As stated before, the strange quark mass de-\ncreases as the baryon number density gets larger. We\nsee that the maximum bulk viscosity decreases when the\nbaryon density increases. The damping time curves ex-\nhibit the same trend as in Fig. 5. AtnB,0/n0= 6 we\nhavems≈138 MeV and for higher values of the baryon\nnumber density up to nB,0/n0= 40 we get ms≈106\nMeV.\nNote that the maximum bulk viscosity and shortest\ndamping times exhibit the opposite behavior when in-\ncreasing the density in the MIT bag model, see Fig. 2.\nThis is linked to the fact that there the strange quark\nmass is a fixed parameter, which does not change with\nthe density. It should be possible to improve this feature\nin the modelling of the EOS, but we leave it for future\nwork.10\n0.010.050.100.5015101100104106108\nFigure 8: Damping times from density oscillations using\nperturbative QCD for different baryon number densities at\nω/2π= 1 kHz and ¯Λ = 2µs.\nnB,0/n0Tm ζmax τmin\n63.6×10−23.78×102943.65\n103.1×10−22.28×1029142.12\n202.4×10−21.29×1029626.32\n401.8×10−28.20×10282459.50\nTable VII: Maximum of the bulk viscosity (in gr cm−1s−1)\nand minimum of the damping times (in ms) for different\nbaryon number density according to Figs. 6 and 8. Here\nTmdenotes the temperature in MeV of the maximum (min-\nima) of the bulk viscosity (damping time).\nIV. OUTLOOK\nWe have studied the bulk viscosity and the damping\ntime of density oscillations of quark matter, using differ-\nent EoSs, and exploring their dependence on both the\nbaryon density, temperature and value of the strange\nquark mass. At the densities that could be attained in\nneutron stars we have considered the MIT bag model,\nand checked that the value of the bulk viscosity changes\nsignificatively with the value of the strange quark mass.\nWe have also used an EoS extracted from QCD at order\nαs. We have included all the relevant electroweak pro-\ncesses that equilibrate quark matter after a disturbance\nof the density, and checked in which temperature regime\nthe nonleptonic process is dominant.\nWhile we see that the numerical value of the bulk vis-\ncosity of quark matter depends on the form of EoS, on\nthe value of the strange quark mass, and the form of\nthe quark dispersion law, one might see some general\nfeatures from our results. In particular, we find that\nthe maximum value of the bulk viscosity, producing the\nshortest damping times of the density oscillations (in the\norder of the few to several hundreds of milliseconds, de-\npending on values of the density and the strange quark\nmass) occur at temperatures in the range from 0.01 to0.1 MeV, the precise value depends on the EoS describ-\ning quark matter. The bulk viscosity of nuclear matter,\nwhich also highly depends on the corresponding mod-\nelled EoS of nuclear matter, seems to have its maximum\nat much higher values of the temperature, in the order\nof few MeVs [20, 21]. Then one can clearly conclude the\nstrongest damping of density oscillations occur in dif-\nferent temperature regimes in quark or nuclear matter,\nwhile these different phases occur at different densities.\nOur results might be of interest so as to assess whether\nthe effect of the bulk viscosity should be included or\nnot in numerical simulations of mergers of neutron stars.\nSeveral of such numerical studies mention the possibility\nof reaching to a deconfined quark matter phase [31–35].\nAs the time scales associated to the initial stages of the\nmerger are of the order of few milliseconds, unless there\nare regions in the stars where the reached temperatures\nare in the range of 0.01 MeV, the effect of the bulk vis-\ncosity in the quark matter phase would be unnoticeable.\nThe effect might be more pronounced in the post-merger\nphase, as it seems also to be the case if one assumes only\nthe presence of nuclear matter, see [19]. However, the ef-\nfect in both cases, depends on the temperatures attained\nin the post-merger object.\nWe have not considered the possibility of Cooper pair-\ning of quarks in this article. In the so called color flavor\nlocked (CFL) [36] phase, and much below the supercon-\nducting transition the bulk viscosity is dominated by the\ninteraction of the superfluid phonons [37], and the kaons\n[38] and it was computed in [39, 40]. A further study of\nhow density oscillations are damped would be required,\nbut from the results found in [39, 40] one might predict\nthat damping times would be longer than in the normal\nphase.\nThe effect of the bulk viscosity of quark matter might\nbe also relevant in the study of the damping of the differ-\nent oscillation modes of isolated compact stars. We will\naddress them in a different publication.\nACKNOWLEDGMENTS\nWhile we were finishing this project, Ref. [13]\nappeared in the arxives, which has a clear over-\nlap with part of the content of this work. We\nacknowledge support from the program Unidad\nde Excelencia Mar´ ıa de Maeztu CEX2020-001058-\nM, from the projects PID2019-110165GB-I00 and\nPID2022-139427NB-I00 financed by the Spanish\nMCIN/AEI/10.13039/501100011033/FEDER, UE\n(FSE+), as well as from the Generalitat de Catalunya\nunder contract 2021SGR 171, by the EU STRONG-2020\nproject, under the program H2020-INFRAIA-2018-1\ngrant agreement no. 824093, and by the CRC-TR 211\n’Strong-interaction matter under extreme conditions’-\nproject Nr. 315477589 - TRR 211.11\nAppendix A: MIT model at finite temperature\nA typical approach to determine the chemical poten-\ntials and susceptibilities of quark matter with electrons\ninside neutron stars appeals to a zero-temperature limit\nof the thermodynamic potential. In this appendix we\nstudy the temperature-dependence of the ideal Fermi gas\nexpressions for the particle density and the susceptibili-\nties to infer its relevance in the temperature region of up\nto 10 MeV.\nThe leading order of the thermodynamic potential for\nthis model at finite temperature is given by [41, 42]\nΩ(0)\ni=−γiT\n2π2/integraldisplay∞\n0k2dk/braceleftBigg\nln/bracketleftbigg\n1+exp/parenleftbigg\n−Ei,k−µi\nT/parenrightbigg/bracketrightbigg\n+ ln/bracketleftbigg\n1+exp/parenleftbigg\n−Ei,k+µi\nT/parenrightbigg/bracketrightbigg/bracerightBigg\n, (A1)\nwithEi,k=/radicalbig\nk2+m2\niandγithe degeneracy factor, for\nelectrons, γe= 2, accounts their spin degrees of freedom\nand for quarks, γf= 2Nc, considers the spin and color\ndegrees of freedom. For electrons their mass, me= 0.511\nMeV, is small compared to the strange quark mass, re-\nsorting to the massless approximation. Then, the inte-\ngration in the thermodynamic potential of an ideal rela-\ntivistic Fermi gas can be carried out to give\nΩ(0)\ne=−1\n12/parenleftbiggµ4\ne\nπ2+2µ2\neT2+7\n15π2T4/parenrightbigg\n.(A2)\nUsing Eqs. ( A1) and (A2), the finite-temperature expres-\nsions for the number densities are given by\nnf=Nc\nπ2/integraldisplay\nk2dk/braceleftBigg\n1\n1+exp[( Ef,k−µf)/T]\n−1\n1+exp[( Ef,k+µf)/T]/bracerightBigg\n, (A3)\nand\nne=µe\n3/parenleftbigg\nT2+µ2\ne\nπ2/parenrightbigg\n, (A4)\nfor quarks and electrons, respectively.\nUsing Eqs. ( A3) and (A4) we compute the chemical\npotentials of quark matter with electrons imposing the\ncharge neutrality and beta equilibrium conditions at dif-\nferent temperatures.\nIn Fig.9we show the chemical potentials at finite tem-\nperature normalized by its value at zero temperature as\nwell as at nB,0/n0= 3 and ms= 100 MeV. Thermal\neffects reduce the chemical potential of the constituent\nparticles in these conditions. For temperatures below 10\nMeV the deviations compared to the value at zero tem-\nperature are up to 0 .3% for light quarks and up to 0 .1%\nfor electrons, while for temperatures up to 50 MeV thesedeviations can be up to 8% for light quarks and ∼1%\nfor electrons.\n0.050.100.501510500.920.940.960.981.00\nFigure 9: The chemical potentials of the different particle\nspeciesµiobtained by imposing beta equilibrium and electric\ncharge neutrality normalized by the same value obtained at\nzero temperature, µT=0. Additionally, we set nB,0/n0= 3\nandms= 100 MeV.\nTosumup, ourpredictionsforthe bulkviscosityvalues\nwould not be significantly affected by these thermal cor-\nrectionsfortemperaturesupto10MeV.Forhighervalues\nof temperatures these effects may be relevant. Also, at\nthese temperatures neutrinos might get trapped in the\nmedium and have to be taken into account in the beta-\nequilibrium conditions, thus changing the bulk viscosity\nand associated damping time [43].12\n[1]B. 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D 84, 034004 (2011) ,\narXiv:1101.3870 [hep-ph] ."    },    {        "title": "2403.08179v1.Effects_of_wave_damping_and_finite_perpendicular_scale_on_three_dimensional_Alfvén_wave_parametric_decay_in_low_beta_plasmas.pdf",        "content": "Effects of wave damping and finite perpendicular scale on\nthree-dimensional Alfvén wave parametric decay in low-beta plasmas\nFeiyu Li,1,a)Xiangrong Fu,2, 1and Seth Dorfman3, 4\n1)New Mexico Consortium, Los Alamos, NM 87544, USA\n2)Los Alamos National Laboratory, Los Alamos, NM 87545, USA\n3)Space Science Institute, Boulder, CO 80301, USA\n4)University of California Los Angeles, Los Angeles, CA 90095, USA\n(Dated: 14 March 2024)\nShear Alfvén wave parametric decay instability (PDI) provides a potential path toward significant wave dissipation\nand plasma heating. However, fundamental questions regarding how PDI is excited in a realistic three-dimensional\n(3D) open system and how critically the finite perpendicular wave scale—as found in both the laboratory and space\nplasmas—affects the excitation remain poorly understood. Here, we present the first 3D, open-boundary, hybrid kinetic-\nfluid simulations of kinetic Alfvén wave PDI in low-beta plasmas. Key findings are that the PDI excitation is strongly\nlimited by the wave damping present, including electron-ion collisional damping (represented by a constant resistivity)\nand geometrical attenuation associated with the finite-scale Alfvén wave, and ion Landau damping of the child acoustic\nwave. The perpendicular wave scale alone, however, plays no discernible role, with different wave scales exhibiting\nsimilar instability growth. These findings are corroborated by theoretical analysis and estimates. The new understanding\nof 3D kinetic Alfvén wave PDI physics is essential for laboratory study of the basic plasma process and may also help\nevaluate the relevance/role of PDI in low-beta space plasmas.\nI. INTRODUCTION\nAlfvén waves represent a fundamental magnetohydrody-\nnamic (MHD) mode with far-reaching implications for lab-\noratory, space, and astrophysical plasmas. The interaction of\nAlfvén waves with energetic particles is crucial to the per-\nformance of burning fusion plasmas1. Shear Alfvén waves\nare also an excellent carrier of significant magnetic and ki-\nnetic energy over large distances in space plasmas. Nonlinear\nprocesses associated with large-amplitude Alfvén waves are\nkey to understanding several major problems such as turbulent\ncascades and plasma energization. As a prominent example,\nparametric instabilities are thought to potentially contribute\nto solar coronal heating2, the observed spectrum and cross-\nhelicity of solar wind turbulence3–5, and damping of fast mag-\nnetosonic waves in fusion plasmas6,7. In particular, the para-\nmetric decay instability (PDI)8–10, well established in theory\nfor over half a century8–15, produces a forward propagating\nion acoustic wave (or sound wave which we use interchange-\nably hereafter) and a backward propagating Alfvén wave; this\nprocess may directly cause plasma heating and cascades of\nwave decays16,17. Theory also suggests a modulational insta-\nbility, which results in forward propagating upper and lower\nAlfvénic sidebands as well as a non-resonant acoustic mode\nat the sideband separation frequency15.\nObservational evidence of Alfvén wave PDI in space\nplasma has been reported. A satellite measurement in the\nion foreshock region found a number of possible PDI events,\nyet the results were inconclusive as the “decay line” signa-\ntures were missing in many intervals18. An analysis of WIND\nspacecraft data suggested that the fluctuations of magnetic\nfield and plasma density in the solar wind at 1 AU may be\na)Electronic mail: fyli.acad@gmail.comlimited by the PDI19. Hahn et al. recently reported an obser-\nvational evidence of PDI in the lower solar atmosphere using\nremote measurements of spectral lines20. However, space ob-\nservations can be limited (e.g. by a turbulent environment and\nthe lack of control over the process) and especially challeng-\ning in the near-Sun low-beta region, where the PDI is pre-\ndicted to have largest growth rates.\nTo fully elucidate the relevance and role of PDI in space\nplasma dynamics, more controlled studies using either labora-\ntory experiments or numerical modeling are needed. Such in-\nvestigations under dimensionless and scaled parameters simi-\nlar to that of some space plasma regions21–23will help validate\nPDI theories and gain new insights into the spatiotemporal\nbehavior and consequences of this basic plasma wave phe-\nnomenon. Experimental progress has been made over the past\ndecade in studying PDI-related physics with the Large Plasma\nDevice (LAPD), a flagship device hosted at UCLA uniquely\nsuited for studying space-relevant Alfvén waves in low-beta\nplasmas24–32. Using two counter-propagating Alfvén waves\nof comparable amplitudes, the three-wave coupling at the\nheart of PDI was verified by measuring a clear resonant\npeak in the acoustic beat wave response.31; more recently,\nPDI growth rates have been inferred from a reduction in\nthe damping of a small-amplitude, counter-propagating seed\nwave when a large-amplitude pump wave is turned on33. Nev-\nertheless, these experiments have thus far not been able to pro-\nduce PDI in its standard form driven by a single Alfvén wave.\nWhen using a single wave driver, Alfvén wave sidebands and\na low frequency nonresonant mode were produced32. How-\never, the spatial pattern of the child modes does not match pre-\ndictions for the standard perpendicular wavenumber k⊥=0\nmodulational instability, suggesting that perpendicular non-\nlinear forces play a key role in the observations. The standard\nk⊥=0 modulational instability is also predicted to have a sig-\nnificantly smaller growth rate than PDI under the chosen set\nof experimental parameters15.arXiv:2403.08179v1  [physics.plasm-ph]  13 Mar 20242\nOn the other hand, extensive numerical modeling of PDI\nhas been conducted, using either MHD simulations4,34, hybrid\nsimulations35–42, or even full particle simulations43,44. How-\never, a periodic infinite system has been routinely adopted by\nmany of these studies, lacking direct relevance to both the\nlaboratory and space plasma settings which feature an open\nsystem with wave injection. Preliminary open-boundary sim-\nulations showed distinct energy transfer and partition from\nusual periodic boundary interactions41. More critically, most\nof these simulations have focused on investigating the con-\nsequences of PDI, without addressing what conditions are\nneeded to excite PDI in the first place. This problem is non-\ntrivial as partly illustrated by the difficulty in demonstrating\nPDI in the laboratory, and directly determines the relevance\nof PDI in space plasmas.\nToward addressing the excitation problem, we have recently\ndeveloped quasi-1D open-boundary hybrid simulations focus-\ning on LAPD-relevant conditions42, and found the thresh-\nold amplitudes and frequencies of a planar zero- k⊥Alfvén\nwave required for exciting PDI under given plasma param-\neters. Physically, these thresholds were obtained by requir-\ning PDI to grow faster than Landau damping of the acoustic\nmode, as well as the convective motion of both child modes\nin a bounded plasma. While the result is of interest to both\nthe laboratory and space study at large perpendicular wave\nscales (i.e. the spatial extent across the background magnetic\nfield), Alfvén waves in both contexts can also develop sig-\nnificant wave k⊥. In the low-beta solar coronal region, large\nk⊥may be induced by transverse plasma gradients, resonance\nabsorption, and turbulent cascade17,45,46. In the laboratory,\nk⊥di≫1 (diis the ion inertial length) due to the finite perpen-\ndicular antenna size necessary to fit the wave in the laboratory\nplasma column47. These finite- k⊥kinetic Alfvén waves carry\nsignificant parallel electric current channels46,48, fundamen-\ntally different from the plane-wave scenario42. How the PDI\nexcitation may be modified by the new 3D features remains\npoorly understood. Furthermore, previous PDI theories were\nmostly derived for k⊥=0 plane waves9,10,15, although some\ntheoretical/numerical studies allowed for child/parent waves\nwith finite k⊥38,49,50. It is not clear, and no present theory\nexplores, how k⊥associated with a finite-perpendicular-scale\npump wave may influence PDI development.\nIn the present work, we present for the first time 3D open-\nboundary hybrid simulations of PDI driven by a single Alfvén\nwave of finite perpendicular scale. Our central new results are\nthat the PDI excitation is found to be strongly limited by 3D\nwave damping of the child modes, including both the Alfvén\nwave damping and acoustic wave damping. Currently consid-\nered by the simulations are i) electron-ion collisional damp-\ning (represented by a constant resistivity η51) and geomet-\nrical attenuation associated with the finite-frequency, finite-\nsource-size Alfvén wave47, and ii) ion Landau damping of the\nchild acoustic wave. On the other hand, for a given magnitude\nof wave damping, the PDI excitation in a low-beta plasma is\nfound to have no discernible dependence on k⊥alone, as long\nas the parallel ponderomotive force remains constant. This\nlack of a dependence on k⊥means that existing laboratory\nexperiments, which can only produce highly oblique Alfvénwaves, may still be capable of demonstrating PDI excitation.\nIn space plasmas, the effectiveness of PDI under large k⊥is\nimportant to establishing the relevance of PDI as wave energy\ncascades in the perpendicular direction towards a small dissi-\npation scale.\nII. 3D OPEN-BOUNDARY HYBRID SIMULATION OF PDI\nWITH A SINGLE FINITE-SCALE ALFVÉN WAVE\nWe start by introducing the 3D simulation setup (Fig. 1)\nbased on the H3D code52, which models kinetic ions plus\na massless electron fluid. The box/plasma occupies z=\n[0,100] dialong the background magnetic field ( B0) direction,\nand two field masks (used for absorbing Alfvén waves41) oc-\ncupy z= [0,30] diandz= [70,100] di. Only the central region\nz= [30, 70] di, containing actual Alfvén wave-plasma inter-\nactions, is displayed. The cell size along zis∆z=0.5di.\nIn this example, the perpendicular dimensions are of size\nLx=Ly=10diand sampled by 80 ×80 cells. The ions\nare sampled by 125 macro-particles per cell. The electron\nfluid follows the adiabatic equation of state Te/nγe−1\ne=const,\nwhere neis the electron density and γe=5/3. The time step\nis∆t=0.01Ω−1\nciwhere Ωciis the ion cyclotron frequency.\nThe injection fields of a finite-scale, left-hand circularly po-\nlarized Alfvén wave are obtained as follows. First, in Simula-\ntion #1, a linearly polarized Bxfield is prescribed at z=35di\nwith an amplitude δBx/B0=Acos[π(x−x0)/2rs]cos[π(y−\ny0)/2rs]cos(ω0t)forr≤rsandδBx/B0=0 otherwise, where\nA≪1 (to avoid nonlinear interactions), rs=Lx/8 is the wave\nsource radius, ( x0,y0) represents the center of the perpendicu-\nlar plane, and ˜ω0≡ω0/Ωci=0.31; then the downstream mag-\nnetic fields (both x,ycomponents) at the perpendicular plane\nz=40di,t=100Ω−1\nciare extracted as the first set of base\nfields (Bx1,By1). We repeat this process in Simulation #2,\nwhere we prescribe a linearly polarized Byfield and obtain\nthe second set of base fields (Bx2,By2). For actual physics\nruns, we inject the Alfvén wave by prescribing at z=35dithe\nfollowing combined base fields:\nBx=δBx\nB0\u0014Bx1\nBmax\n1cos(ω0t)−Bx2\nBmax\n2sin(ω0t)\u0015\n, (1a)\nBy=δBy\nB0\u0014By1\nBmax\n1cos(ω0t)−By2\nBmax\n2sin(ω0t)\u0015\n, (1b)\nwhere δBx/B0=δBy/B0≡δB/B0is the normalized wave\namplitude and Bmax\n1,Bmax\n2are the maximum value ofq\nB2\nx1+B2\ny1,q\nB2\nx2+B2\ny2, respectively. The injection con-\ntains a small ring-up time of 50 Ω−1\nciand lasts for 3000 Ω−1\nci\nThe dispersion relation of resulting Alfvén wave in the down-\nstream was checked and verified.\nThe transverse field patterns both at the injection and down-\nstream are displayed in Figs. 1(d1-d4). As the finite-scale\nwave propagates essentially in an Alfvén wave cone47, the\nwave pattern rotates and spans multiple cycles in the perpen-\ndicular plane, giving a dominant k⊥≃2.75/rs(following a\nBessel function fit47). This simulation case uses parameters3\nFIG. 1. 3D open-boundary hybrid simulation of PDI driven by a finite-scale, circularly polarized Alfvén wave. (a-c) 3D isosurfaces of the\nperpendicular wave magnetic field at t=200Ω−1\nci, parallel electric current density at t=200Ω−1\nci, and ion density fluctuations at t=2000Ω−1\nci,\nrespectively. (d1-d4) The xy-cut of perpendicular magnetic field at z=35,40,50,60di, respectively, taken at t=200Ω−1\nci. (d1) corresponds to\nthe plane at which the wave is injected. (e-g) The magnetic field envelope |δBy/B0|and the density fluctuations probed at (x,y,z) = ( 5,5,41)di\nand their corresponding Fourier spectra (obtained over the the full time window shown). In (e) the envelope of the field oscillation is shown,\nand its spectrum in (f) is performed on the fast field oscillations not shown in (e). The vertical dashed line in (f,g) refers, respectively, to the\nfrequency of the child Alfvén wave and ion acoustic wave, as predicted from PDI theories8–10.\n˜ω0=0.63,Te/Ti=9, and total beta β=βe+βi=1×10−3.\nFigure 1(a) shows the contour surface of the perpendicular\nwave field B⊥=q\nB2x+B2y(for an initial injection of am-\nplitude δB/B0=0.01) at t=200Ω−1\nci, well before the onset\nof PDI. This finite-scale wave contains two rotating parallel\nelectric currents [Fig. 1(b)]. The parallel currents exert an in-\nfluence on both the wave fields and ion dynamics through a\nconstant resistivity η=4πνei/ω2\npeused in the hybrid code to\nmimic electron-ion collisions, where νeiis the collisional rate\nandωpethe electron plasma frequency; see the Ohm’s law and\nion motion equation of the hybrid system51:\nE+ui×B\nc=ηJ+1\nqinicJ×B−1\nqini∇Pe, (2)\nmidvi\ndt=qi(E+vi×B/c)−eηJ, (3)\nwhere Eis the electric field, Peis the electron pressure tensor,\nJis the total plasma current, eis the elementary charge, cis\nthe light speed in a vacuum, and ( ui,vi,qi,mi,ni) are the en-\nsemble ion speed, individual ion speed, ion charge, ion mass,\nand ion density, respectively. In the present case, the normal-\nized resistivity used in the code is ˜η=ηω pi/4π=1×10−5\n(ωpiis the ion plasma frequency).\nThe evidence of PDI is partly illustrated by the density fluc-\ntuations shown at t=2000Ω−1\nci[Fig. 1(c)], after the instabil-\nity has sufficiently developed. The fluctuations are associatedwith the child acoustic wave, which co-propagates with the\npump Alfvén wave. The acoustic wave develops bowed iso-\nsurfaces as a result of nonlinear frequency shifts, i.e. cen-\ntral axis corresponds to larger wave amplitudes and more fre-\nquency shifts toward the smaller end. To further confirm the\nPDI signatures, we probe the temporal evolution of the field\nenvelope |δBy/B0|and ion charge-density fluctuations δn/n0\nat a fixed location and display the result in Fig. 1(e). These\nfluctuations start to emerge after a few hundreds of Ω−1\nciand\ncontinue to increase throughout the simulation. The Bfield\nenvelop oscillates due to the pump beating with the child\nAlfvén wave with a frequency difference ∆ω=ω0−ω1, and\nthe density fluctuation oscillates at the eigen acoustic wave\nfrequency ω. Their growth matches each other in time, and\ntheir similar oscillation frequency ∆ω≃ωverifies the fre-\nquency matching condition ω0=ω1+ωas required for PDI\ncoupling8–10. The frequency matching is also revealed in the\nspectra, Figs. 1(f, g), where the predicted frequencies of the\nchild waves are indicated by the vertical dashed lines. The\nexcitation is so strong in this case that PDI coupling with har-\nmonics of the acoustic mode Nω(Nis an integer) is also vis-\nible.\nIII. EFFECTS OF 3D WAVE DAMPING\nThe more realistic 3D finite-scale injection involves two\nnew features (compared to a plane wave injection42): i) The4\nFIG. 2. Effects of wave damping on PDI, while keeping the perpendicular wave scale k⊥difixed to 2.2. The simulations correspond to the\nsame beta β=5×10−4,Te/Ti=4, and ω0/Ωci=0.63, with the resistivity shown in the titles. Panel (a) shows the central xz-cuts of the wave\nfield component δBy/B0. Panel (b) shows the space-time evolution of the density fluctuation on the central axis (x,y) = ( 5,5)di. (c,d) The\nFourier spectra of δBy/B0and density fluctuations δn/n0probed at (x,y,z) = ( 5,5,41)difor all four cases. (e) Theoretical calculation of the\ndamping rate geometrical mean (Γ1Γ2)1/2/ω0versus varying ˜ηand its comparison with the growth rate γg/ω0.\nAlfvén wave itself suffers from damping, including resistive\ndamping [see Eq. (2)] and geometrical attenuation induced\nequivalent damping47. ii) The finite k⊥associated with the\nfinite perpendicular scale. While the wave damping is also\ndependent on k⊥(as we will see more clearly later), we will\nexplore the effects of these two new features separately. Iso-\nlating the effects of k⊥is of interest because k⊥is potentially\nan important parameter determining PDI growth38,50.\nWe first examine the effects of wave damping on PDI exci-\ntation using a set of simulations with the outcome summa-\nrized in Fig. 2. The magnitude of wave damping in these\nruns is controlled by the constant resistivity ˜η, while the wave\nk⊥is fixed as the cases have the same perpendicular dimen-\nsions and same source radius as used for Fig. 1 ( Lx,Ly=\n10di,rs=Lx/8=1.25di). The total beta is β=5×10−4\nwith Te/Ti=4. For each run, we display a snapshot of cen-\ntralxz-cut of the wave field component δBy/B0att=200Ω−1\nci\n(before PDI develops) in Fig. 2(a) and the space-time evolu-\ntion of on-axis density fluctuations till the end of the simula-\ntiontmax=3000Ω−1\nciin Fig. 2(b). It is seen that by increas-\ning˜ηfrom 1 ×10−5to 5×10−5, the damping of the Alfvén\nwave is indeed much enhanced. While the density fluctua-\ntion (evidence of PDI) is strong for ˜η=1×10−5, it becomes\nmuch weaker for ˜η=2×10−5and nearly disappeared when˜η≥3×10−5. Figures 2(c,d) show more quantitative evi-\ndence of PDI by looking at the probe data (as done for Fig. 1);\nsimilarly, prominent spectral peaks for the child Alfvén wave\n[Fig. 2(c)] and child acoustic mode [Fig. 2(c)] are found only\nfor˜η≤2×10−5.\nTo physically and more quantitatively understand the wave\ndamping effects, we estimate the damping rate for each damp-\ning mechanism. The resistive Alfvén wave damping essen-\ntially comes from the damping of the channel currents through\nelectron-ion collisions (represented by η). The total magnetic\nfield evolves according to∂B\n∂t=∇×(ui×B)−c2\n4πη∇×(∇×\nB) +c\n4πqi∇×[1\nniB×(∇×B)], obtained by substituting Fara-\nday’s law and Ampere’s law into Eq. (2). The first term on the\nright-hand side denotes motion of field lines frozen-in to the\nplasma, the second term denotes the resistivity-induced dif-\nfusion with the diffusion rate Dr=c2\n4πηk2, the third term is\nthe Hall term, and the electron pressure term is dropped as we\nconsider an isotropic pressure. A two-fluid analysis53finds\nthat the resistive damping rate is related to the diffusion rate\nasΓr=1\n2(1+k2zd2\ni+k2zd2e)Dr=1\n2(1+k2zd2\ni)Dr, where kzde→0 for\nour massless electron fluid. By normalizing to the pump wave5\nfrequency, the resistive damping rate can be cast as\nΓr\nω0=1\n21+k2\n⊥/k2\nz\n1−˜ω2\n0k2\n⊥ρ2\ni+k2\n⊥ρ2ωpi\nΩci˜ω0˜η, (4)\nwhere ρ2=ρ2\ns+ρ2\ni,ρsis the ion sound gyroradius, ρithe\nion gyroradius, and we have used the dispersion relation for a\nfinite-frequency, finite-scale kinetic Alfvén wave54\nω0\nkz=vAq\n1−˜ω2\n0(1+k2\n⊥ρ2\ni)+k2\n⊥ρ2≡vA√\nΞ. (5)\nThe equivalent Alfvén wave damping associated with geo-\nmetrical attenuation may be estimated as follows. The wave\nsource radius rsat the injection increases by drswithin dz\npropagation distance, according to the Alfvén wave cone an-\ngle tan θ=drs\ndz=vg,⊥\nvg,z=∂kz\n∂k⊥=q\nβ\n2k⊥ρs+√\nTi/Tek⊥ρi(1−˜ω2\n0)\nΞ3/2˜ω047\nwhere β=2(cs/vA)2,cs=q\nTe+Ti\nmiis the sound speed, and\nvg,⊥=∂ω0/∂k⊥,vg,z=∂ω0/∂kzare the perpendicular and\nparallel group velocity, respectively. Without wave dissi-\npation, the wave energy conservation at the two locations\nrequires (δB1)2r2\ns= (δB2)2(rs+drs)2, which givesδB2\nδB1≃\n1−drs/rs=1−tanθ\nrsdz. ComparingδB2\nδB1with an exponential\non-axis wave damping/reduction due to geometrical spread-\ningδB= (δB)0exp(−Sgdz)≃(δB)0(1−Sgdz), one has Sg=\ntanθ\nrs≃0.36k⊥tanθ. Converting to the temporal damping rate\ngives\nΓg\nω0=Sgvg,z\nω0=0.26k⊥\nkzk⊥ρs+p\nTi/Tek⊥ρi(1−˜ω2\n0)\nΞ3/2+(1+k2\n⊥ρ2\ni)˜ω2\n0Ξ1/2˜ω0p\nβ,\n(6)\nwhere vg,z=ω0/kz\n1+(1+k2\n⊥ρ2\ni)˜ω2\n0/Ξ.\nFinally, the ion acoustic wave (or sound wave) Landau\ndamping rate may be estimated as42\nΓs\nω0≃2p\nβr\nTi\nTe. (7)\nThe resistivity also incurs a friction force −eηJon the ion\nmotion, as seen in Eq. (3). However, the friction force works\nthrough the current channels, which reside off the central axis.\nTherefore, the potential impact of the friction force on the\ndamping of ion acoustic wave mostly located around the cen-\ntral axis can be neglected.\nIn a system with significant wave damping present, the PDI\nmay be excited only if the following condition is satisfied55,56:\nγg/ω0>p\nΓ1Γ2/ω0≡Γgm/ω0, (8)\nwhere γg≃1\n2(δB/B0)/β1/4is the PDI growth rate obtained\nfork⊥=0 and Γgmis the damping rate geometrical mean\nconstructed from the damping rates of the two child modes:\nΓ1=Γr+Γg,Γ2=Γs. The use of zero- k⊥growth rate will\nbe justified later where PDI excitation shows no discernible\ndependence on k⊥alone. Additionally, we essentially use\nthe pump wave damping to approximate child Alfvén wavedamping, because the two waves have the same nature except\nfor a minor frequency difference ∼2p\nβω0at low beta. To\ntest Eq. (8), we substitute the common simulation parameters\nω0/Ωci=0.63,ωpi/Ωci=300,β=5×10−4,Te/Ti=4 and\nk⊥di=2.2 into (Γr,Γg,Γs,γg). We take the wave amplitude\nδB/B0=0.8×10−2atz=41di(close to the injection, where\nPDI is probed) for the growth rate calculation; if PDI cannot\nbe excited close to the injection, it cannot be excited in the rest\nof the domain where the wave amplitudes are smaller. The re-\nsult shows that Γg/ω0≃2.9×10−4,Γs/ω0≃2.2×10−2, and\nγg/ω0≃2.7×10−2, i.e. γg>Γs≫Γg. Therefore, whether\nPDI can be excited is strongly dependent on the resistive\ndamping Γr/ω0. By varying ˜η(hence Γr/ω0), the two sides\nof Eq. (8) plotted in Fig. 2(e) shows that Eq. (8) is satisfied\nonly for ˜η<4×10−5. The good agreement with the 3D sim-\nulations confirms our physical understanding and underscores\nthe importance of wave damping for PDI excitation by a 3D\nfinite-scale kinetic Alfvén wave. Notice that Eq. (8) neces-\nsarily modifies the threshold Alfvén wave amplitude obtained\nin the plane-wave study42, due to the new 3D wave damping\npresented here.\nIV. EFFECTS OF FINITE PERPENDICULAR SCALE\nWe next explore the effects of finite perpendicular wave\nscale on PDI excitation, by varying k⊥and maintaining con-\nstant wave damping. Since the resistive damping [Eq. (4)] and\ngeometrical attenuation [Eq. (6)] also change with k⊥, we si-\nmultaneously adjust ˜ηin the simulations to keep Γgmclose\nto constant. Figure 3 displays the outcome of three cases sat-\nisfying the above requirements, where we increase the wave\nsource radius from 1 .25di, 5di, to 7 .5diwith corresponding\nnormalized resistivity adjusted to be ˜η=1,4,5×10−5, re-\nspectively. With these parameters and their common setup\nδB/B0≃0.8×10−2(at the probe point), ω0/Ωci=0.63,\nandωpi/Ωci=c/vA=300, the Alfvén wave damping for\nthe three cases, (Γr+Γg)/ω0, is kept at about 0.006. As\nshown in Figs. 3(a1-a3), the wave magnetic field δBx/B0in-\ndeed have similar spatial wave damping profile. The sound\nwave damping, Γs/ω0≃0.022, has no dependence on k⊥.\nTherefore, the wave damping geometrical mean for all three\ncases is Γgm/ω0≃0.012. The simulation results shown in\nFig. 3 reveal little difference in PDI excitation, both in terms\nof the space-time evolution of on-axis density fluctuations and\nprobed spectra, despite the factor of six difference in k⊥di\namong the three cases. This result strongly suggests that the\nPDI excitation has no discernible dependence on the perpen-\ndicular wave scale alone.\nThe result may seem to contradict some 1D/2D periodic-\nboundary simulations of a finite- k⊥plane Alfvén wave re-\nported previously38, where a cos θkBdependence of PDI\ngrowth rate was extracted ( θkB=arctan (k⊥/kz)is the normal\nangle of the oblique Alfvén wave). The three cases shown in\nFig. 3 have k⊥/kz≃2.71,0.68,0.45, corresponding to a wave\nnormal angle of 70, 34, 24 degrees and cos θkBof 0.35, 0.83,\n0.91, respectively, which span a variation large enough to dis-\ncern the potential consequences of the cos θkBdependence.6\nFIG. 3. Effects of perpendicular wave scale on PDI excitation under constant wave damping. The simulations correspond to the same beta\nβ=5×10−4andTe/Ti=4 but varying perpendicular wave scales (see the titles). To maintain constant damping, the normalized resistivity\nfor the cases shown from left to right is adjusted to be ˜η=1,4,5×10−5, respectively. Panels (a-d) have the same format with Fig. 2. (e)\nDependence of kzdion the perpendicular wave scale for a kinetic Alfvén wave, calculated for different βand same Te/Ti=4,ω0/Ωci=0.63.\nThe contradiction may be understood as follows. In their\nplane-wave scenario, the Alfvén wave was loaded with wave-\nlength λ0along z, and θkBwas introduced by tuning the B0di-\nrection away from z; as a result, the parallel wavelength along\nthe background field becomes λ∥=λ0/cosθkB, the perpen-\ndicular wave electric field is δE⊥=δEcosθkB, and the wave\nfrequency ω=k∥vAis cos θkBtimes smaller. This results in\ncosθkBtimes smaller parallel ponderomotive force which has\nthe form Fp,∥∝ω−2∇∥(δE⊥)2. However, in the present finite-\nscale scenario, the parallel wavelength or kzchanges with\nthe source scale according to the dispersion relation Eq. (5),\nwhich can be recast as\nkzdi=˜ω0/r\n1−˜ω2\n0(1+1\n2k2\n⊥d2\niβi)+1\n2k2\n⊥d2\ni(βi+β),(9)\nwhere we have used k2\n⊥ρ2\ns=1\n2k2\n⊥d2\niβandk2\n⊥ρ2\ni=1\n2k2\n⊥d2\niβi. A\ncalculation of kzdiversus a broad range of perpendicular wave\nscales under different βis shown in Fig. 3(e). It is seen that,\ndespite the large k⊥orθkB, the parallel wavelength depends\nonly weakly on the source size, especially for the low-beta\nregime with β<10−2. Meanwhile, the driving frequency is\nfixed and the perpendicular wave field δB⊥remains the same\nas the cases in Fig. 3 are tuned to have similar wave damping.\nTherefore, the parallel ponderomotive force in our scenario\nis similar, leading to similar PDI growth when varying the\nperpendicular wave scale alone.V. DISCUSSION\nThese 3D simulations and associated analyses suggest that\nthe wave damping is the major limiting factor for driving\nfinite-scale Alfvén wave PDI in a uniform background, while\nk⊥alone shows no discernible influence. As illustrated by\nEq. (4), Eq. (6) and Eq. (7, the wave damping depends on\nmultiple parameters, e.g. the electron-ion collisional rate\nνei, wave normal angle ∝k⊥/kz, driving frequency ˜ω0, to-\ntal plasma beta β, temperature ratio Te/Ti, and the absolute\nAlfvén speed vA/c. The explicit scaling versus these parame-\nters will help extrapolate to a broad range of parameters with\nrelevance to both the laboratory and space plasmas.\nThe low-beta laboratory plasma usually involves highly\noblique waves k⊥di∼O(10)andTe/Ti≫1,˜ω0≲1. With\na typical set of LAPD parameters, δB/B0=0.7×10−2,˜ω0=\n0.6,c/vA=660, ne=3.5×1012cm−3,β=1×10−3,Te/Ti=\n5.5,rs=0.18di,˜η=1×10−6(νei=8.6 MHz, electron skin\ndepth de≃3 mm), and the ion species He+, the calculated\nresistive damping is Γr/ω0≃0.084, geometrical attenuation\nrateΓg/ω0≃0.026, sound wave damping Γs/ω0≃0.027 and\nthe damping rate geometrical mean Γgm/ω0≃0.055, while\nthe growth rate under this set of parameters is only γg/ω0≃\n0.02. The damping rate is almost thrice larger, which may\nexplain why PDI has been difficult to excite on LAPD using7\na single wave driver. Actual experiments also involve elec-\ntron Landau damping of the Alfvén wave and ion-neutral col-\nlisional damping of the sound wave, which will further raise\nthe damping rates and thereby constrain PDI excitation.\nTo excite PDI essentially requires the PDI growth to over-\ncome the wave damping. While a smaller k⊥helps reduce\nAlfvén wave damping ΓrandΓgsignificantly, a new, next-\ngeneration laboratory facility would be required to launch an\nAlfvén wave with a much larger perpendicular size ( k⊥di<\n1)23. Yet our study shows that the PDI growth is not compro-\nmised by the large k⊥. Therefore, future optimization strate-\ngies should focus on reducing wave damping for currently\nachievable values of k⊥by varying other parameters. One\noption is to achieve higher electron temperatures, while main-\ntaining a cold ion population. Hotter electrons will reduce\nelectron Landau damping and electron-ion collisional damp-\ning (e.g. ηin our simulations) of the Alfvén wave; a larger\ntemperature ratio Te/Tiwill help suppress the sound wave\ndamping. Higher electron temperatures may also improve\nantenna-plasma coupling48, leading to larger driving wave\namplitudes δB/B0. To achieve PDI excitation in a bounded\nlaboratory plasma, care must also be taken to operate at suf-\nficiently high driving wave frequency42. Based on the 3D\ndamping effects elucidated in this paper, it may be benefi-\ncial to operate at as low of a pump frequency as possible to\nboth minimize Alfvén wave damping and increase the parallel\n(along z) ponderomotive force, i.e. Fp,z∝kz/˜ω2\n0∝1\n˜ω0√\n1−˜ω2\n0becomes bigger at smaller ˜ω0.\nThe low-beta space plasmas, on the other hand, have dis-\ntinct wave and plasma properties from the laboratory (al-\nbeit similar dimensionless/scaled parameters, for example, in\nthe solar coronal region57), such as a very low frequency\n˜ω0≪1, comparable electron/ion temperatures Te/Ti∼1, and\na large wave amplitude δB/B058,59. Taking δB/B0=0.2,\n˜ω0=0.01,c/vA=600, β=2×10−2,Te/Ti=1,k⊥di=\n0.55 and ˜η=1×10−8for example, a calculation based on\nthe present framework shows that the growth rate γg/ω0≃\n0.27 is much larger than the damping rate geometrical mean\nΓgm/ω0≃0.02, where Γr/ω0≃9×10−5,Γg/ω0≃1×10−3,\nandΓs/ω0≃0.28. Therefore, it is potentially much easier to\nexcite PDI in space plasma than in the laboratory. Interest-\ningly, while Γgmis small, a significant sound wave damping\nΓs∼γg≫Γgmis allowed owing to the way Γgmis constructed.\nIn other words, PDI in this space-relevant case is mainly fa-\ncilitated by the negligible Alfvén wave damping, while the\nappreciable sound wave damping facilitates final PDI energy\ndissipation. Certainly, the above estimates based on a uniform\nbackground may be complicated by the strong inhomogeneity\npresent in space plasmas. In particular, the Alfvén wave in the\nsolar coronal region may develop large k⊥17,45,46among other\ncomplications. The independence of PDI versus k⊥alone, as\ndiscovered in this work, will be of key importance to estab-\nlishing the relevance of PDI at small dissipation scales.VI. SUMMARY\nIn summary, we have presented the first 3D open-boundary\nhybrid simulations of PDI driven by a single Alfvén wave of\nfinite perpendicular scale. It is found that the PDI excitation\nis strongly limited by 3D wave damping, while the perpendic-\nular wave scale ( k⊥) alone plays no discernible influence on\nPDI. These results are crucial to understanding the excitation\ncriteria of Alfvén wave PDI in a practical 3D open system.\nIn the laboratory with very small-scale waves, PDI is mainly\nhindered by the strong wave damping. Strategies to optimize\nexperimental parameter to minimize the damping rates and\nenhance PDI growth rate are briefly discussed, which will be\nessential for demonstrating PDI in future laboratory experi-\nments. In low-beta space plasmas, PDI excitation will benefit\ngreatly from our finding that the PDI growth rate has no dis-\ncernible dependence on k⊥alone, and the Alfvén wave prop-\nerties in space are more likely to make PDI a relevant and im-\nportant scheme leading to wave dissipation at both large and\nsmall scales. Future studies would involve elaborating on the\ndynamics/consequences of finite-scale Alfvén wave PDI with\nthe 3D open system, as well as developing more comprehen-\nsive simulation models to include both electron Landau damp-\ning and ion-neutral collisional damping that are absent in the\ncurrent hybrid code. It would also be important to investigate\nthe effects of a nonuniform background (both in the perpen-\ndicular and parallel directions) which is commonly found in\nspace plasmas.\nVII. ACKNOWLEDGEMENT\nThis work was supported by the DOE grant DE-SC0021237\nthrough the NSF/DOE Partnership in Basic Plasma Science\nand Engineering program, the DOE grant DE-SC0023893,\nand the NASA grant 80NSSC23K0695. We acknowledge the\nTexas Advanced Computing Center (TACC) at The University\nof Texas at Austin and the National Energy Research Scien-\ntific Computing Center (NERSC) for providing the computing\nand visualization resources.\nVIII. 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Leon2*, Wenqing He 3, Zhongyu Liang1,  \nXiaohan L i3, Xiaoxiao Fang1, Wenyun Yang1, Licong Peng4,  \nJinbo Yang1,5*, Caihua Wan3, Gerrit E. W. Bauer6,7, Zhaochu Luo1,5* \n1State Key Laboratory of Artificial Microstructure and Mesoscopic Physics, Institute of Condensed \nMatter Physics and Materials, School of Physics, Peking University, 100871 Beijing, China.  \n2Departamento de Física, Facultad de Ciencias Naturales, Matemática y del Medio Ambiente, \nUniversidad Tecnológica Metropolitana, Las Palmeras 3360, Ñuñoa 780- 0003, Santiago, Chile.  \n3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese \nAcademy of Sciences, Chinese Academy of Sciences, 100190 Beijing, China.  \n4School of Materials Science and Engineering, Peking University, 100871 Beijing, China.  \n5Beijing Key Laboratory for Magnetoelectric Materials and Devices, 100871 Beijing, China  \n6Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, 100864 \nBeijing, China.  \n7Advanced Institute for Materials Research (AIMR), Tohoku University, 980- 8576 Sendai, Japan.  \n \n*Correspondence to: zhaochu.luo@pku.edu.cn (Z.Luo); aleonv@utem.cl  (A.L.);  \njbyang@pku.edu.cn  (J.Y .)  \n \nKeywords: Magnetic coupling; magnetic insulator; spin- orbit torques; spin -mixing \nconductance; spin -Hall magnetoresistance  \nAbstract: Owing to the unique features of low Gilbert damping, long spin -diffusion lengths \nand zero Ohmic losses, magnetic insulators are pro mising candidate materials for next -\ngeneration spintronic applications. However, due to the localized magnetic moments and the \ncomplex metal -oxide interface between magnetic insulators and heavy metals, spin -functional \nDzyaloshinskii –Moriya interactions or  spin Hall and Edelstein effects are weak, which \ndiminishes the performance of these typical building blocks for spintronic devices. Here, we \nexploit the exchange coupling between metallic and insulating magnets for efficient electrical \nmanipulation of heavy metal/magnetic insulator heterostructures. By inserting a thin Co layer, \nwe enhance the spin -orbit torque efficiency by more than 20 times, which significantly reduces \nthe switching current density. Moreover, we demonstrate field -free current -induced \nmagnetization switching caused by a symmetry- breaking non -collinear magnetic texture. Our \nwork launches magnetic insulators as an alternative platform of low -power spintronic devices.    \n 2 1. Introduction  \nEngineering the magnetic coupling plays a crucial role in determining the functionalities \nof spintronic devices. The interlayer Ruderman –Kittel–Kasuya –Yosida (RKKY) interaction[1] \nand exchange -bias effect[2] stabilize the magnetic reference layer[3,4] in the magnetic tunnel \njunctions of non- volatile magnetoresistive random -access memories (MRAM), while the \nintralayer dipolar and chiral couplings are essential for scalable spin logic[5-7] and neuromorphic \ncomputing devices[8-10]. Research on the physics and applications of these couplings focusses \non electric conductors such as magnetic/ non-magnetic and ferro magnetic/antiferromagnetic \nmetal bilayers, but  much of the cor responding phenomenology in magnetic insulators remains \nto be explored. Taking advantages of low Gilbert damping[11,12], long spin- diffusion lengths[13-\n15] and zero Ohmic losses, magnetic insulators, particularly the iron garnets, are promising \ncandidate materials for low -power and high -speed spintronic applications. Pioneering studies \non magnetic metal/magnetic insulator bilayers demonstrated injection and modulation of spin \nwaves[16-20], but due to the complexity of polycrystalline metal- oxide interfaces[21-23] a full \nunderstanding of the underlying mechanisms remains elusive.  \nHere, we report experiments on a high- quality magnetic heterostructure comprising of an \nultrathin Co film on an epitaxial terbium iron garnet (Tb 3Fe5O12, TbIG ) layer, which shows a \nstrong ferromagnetic exchange coupling of ~69.0 μ J/m2. By growing a heavy metal layer on \ntop of the Co/TbIG bilayer, we electrically detect and manipulate the TbIG magnetization via \nthe direct and inverse spin Hall and/or Edelstein effects. The interlayer exchange coupling \nsignificantly enhances the interfac ial spin -mixing conductance, leading to a large spin -Hall \nmagnetoresistance (SMR) and a high spin -orbit torque (SOT) efficiency. Moreover, the non-\ncollinear magnetization of the Co  layer breaks the mirror symmetry of the device, thereby \nfacilitating current -induced switching of the TbIG magnetization without external magnetic \nfields. Our work demonstrates an interplay between magnetic coupling and SOTs that can help to design reliable and efficient magnetic memory devices and pave a new pathway for magnetic \ninsulator spintronics.  \n2. Structural and magnetic properties of Pt/TbIG   \n 3 The insulating rare -earth iron garnet TbIG was epitaxially grown on (111) -oriented single -\ncrystalline gadolinium gallium garnet (Gd 3Ga5O12, GGG) substrates by pulsed laser deposition \n(PLD) (see Method s). The structure of TbIG is constructed with three magnetic sublattices: an \noctahedral Fe, a tetrahedral F e, and a dodecahedral Tb sublattice[24] (Figure 1a ). TbIG films \nwith various thicknesses ranging from 5 to 53 nm were deposited for different experiments. The \n5 nm -thick TbIG was used for the electrical tr ansport measurement, whereas the thick TbIG \nyielding a stronger structural signal was used for X -ray diffraction (XRD) analysis that gives \ninsight into the microstructure and strain state. As shown in Figure 1b, we present the symmetric \nXRD scans of a 53 n m-thick TbIG around the (444) peak of GGG substrate. The rocking curve \ngives a full -width at half -maximum (FWHM) of ~0.02 ° (see Supplementary Information S1), \nindicating a high- quality single crystallinity. We can further extract the lattice cons tant of d 444 \n= 0.1815 nm for the epitaxial TbIG  film which is slightly larger than that (~0.1795 nm) in a \nTbIG bulk[25], implying a strained state originating from the TbIG/GGG interface[26,27]. We then \nconducted the magnetization measurements using vibrating- sample magnetometer (VSM) and \nmagneto -optical Kerr microscopy (MOKE) to study the magnetic properties of TbIG films \n(Figure 1c). The magnetization hysteresis loops with out -of-plane magnetic fields reveal a \nstrong perpendicular magnetic anisotropy and a small saturation magnetization ( MS ~23 emu/cc ) \nas a result of the partial compensation of sublattice magnetizations.  \nWe sputtered a 5 nm -thick Pt layer on TbIG (5 nm)/GGG with a low deposition power (20 \nW) to mitigate the bombardment damage of Pt atoms on the TbIG surface. The cross- sectional \nhigh- resolution transmission electron microscopy (HR -TEM) image confirms the cl ean and \nsharp interface in the heterostructure that is p rerequisite to efficiently manipulate the \nmagnetization by spin currents (Figure 1d). The Pt/TbIG heterostructure was then patterned \ninto Hall bar structures with the channel width of 10 µ m using UV p hotolithography combined \nwith the ion milling process. We measured the electrical transport with out -of-plane magnetic \nfields and observed an anomalous Hall resistance of - 1.3 mΩ at room temperature (Figure 1e).  \nBy varying the measurement temperature, the coercivity exhibits a peak at a critical temperature \nat which the polarity of the anomalous Hall resistance is reversed, indicating magnetic moment compensation in TbIG around 270 K  (T\ncomp) (Figure 1f). Since the sign of the anomalous Hall \nresistance depends on the direction of the magnetic moment of  the tetrahedral Fe  sublattice[28],  \n 4 the sign is reversed  when the temperature varies across the compensation state due to the fact \nthat the alignment relation ship between the magnetic field and the magnetic moment of the \ntetrahedral Fe sublattice reverses at T comp (Figure 1e).  \n3. Spin Hall magnetoresistance and magnetic coupling in Pt/Co/TbIG  \nTo exploit the rich interfacial magnetic effect of heavy metal/metallic magnet such as high \nspin-mixing conductance and large Dzyaloshinskii –Moriya interaction (DMI), we inserted a \nthin Co layer ( tCo = 0.3, 0.6 and 0.9 nm as calculated by multiplying the deposition rate and \ntime) between the Pt and TbIG layers by DC magnetron sputtering ( Figure 2 a). With the \ninsertion of Co layers, a linear Hall resistance with positive slope emerges on top of the square -\nlike anomalous Hall loop, indicating an in- plane magnetic anisotropy of the Co layer (Figure \n2b). The Co magnetization gets fully perpendicular at  large out -of-plane magnetic fields (Figure \n2c). Interestingly, the magnitude of the square -like anomalous Hall loop is significantly \nenhanced for thicker Co films, accompanying a change of sign of the Hall resistance from \nnegative to positive (Figure 2d). Moreover, the anomalous Hall loop in Pt/Co ( 0.9 nm)/TbIG of \n52.0 mΩ is more than an order of magnitude larger than that in Pt/TbIG ( -1.3 mΩ) with the \nsame TbIG thickness. An ultrathin Co insertion therefore offers a sensitive method to \nelectrically detect the magnetization direction of insulators.  \nTwo mechanisms may cause the large anomalous Hall loop. On one hand, the exchange \ncoupling between Co and TbIG may force a non- collinear alignment of the Co and TbIG \nmagnetizations such that the out -of-plane component of the Co magnetization contributes to  \nthe anomalous Hall effect. In th is scenario, the longitudinal ( Rxx) and transverse Hall ( Rxy) \nresistance can be written as:  \n𝑅𝑅𝑥𝑥𝑥𝑥=𝑅𝑅0+∆𝑅𝑅AMR sin2𝜃𝜃Cosin2𝜑𝜑Co,                      (1)  \n𝑅𝑅𝑥𝑥𝑥𝑥=−∆𝑅𝑅AMR sin2𝜃𝜃Cosin2𝜑𝜑Co+𝑅𝑅AHECo𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃Co+𝑅𝑅OHE𝐻𝐻𝑧𝑧,             (2)  \nwhere R0 and ΔRAMR represent the magnetization -independent longitudinal resistance and the \nanisotropic magnetoresistance (AMR), respectively. 𝑅𝑅AHECo (ROHE) is the anomalous (ordinary) \nHall resistance. θ Co (φCo) is the angle between Co magnetization and z-axis ( y-axis) as defined \nin Figure 2a. We disregard the very small magnetic- proximity effect in Pt (see Supplementary \nInformation S2). The second mechanism would follow the spin Hall magnetoresistance (SMR)  \n 5 scenario as widely used to extract spin transport parameters in heavy metal/magnetic insulator \nbilayers[29,30] such as the interfacial spin -mixing conductance G. In such measurements, an \nelectric current is applied to the heavy metal layer and due to the spin Hall effect, it can generate a transverse spin current that will either be transmitted or reflected at the heavy metal/magnetic \ninsulator interface depending on the relative orientations between the spin current polarization \nσ and the magnetization m . The ratio of spin transmission/reflection at the interface will further \nmodulate the electric current in the heavy metal layer due to the inverse spin Hall effe ct. This \nleads to the presence of a resistance change that has a distinct angle dependence. In the high \nfield limit, the magnetizations of Co and TbIG are aligned along the field direction with angle θ (φ) with the z -axis ( y-axis). The longitudinal ( R\nxx) and transverse Hall ( Rxy) resistance then \nread[29,30]:  \n𝑅𝑅𝑥𝑥𝑥𝑥=𝑅𝑅0−∆𝑅𝑅SMR sin2𝜃𝜃cos2𝜑𝜑,                     (3) \n𝑅𝑅𝑥𝑥𝑥𝑥=−∆𝑅𝑅SMR sin2𝜃𝜃sin2𝜑𝜑+𝑅𝑅AHESMRcos𝜃𝜃+𝑅𝑅OHE𝐻𝐻𝑧𝑧,            (4)  \nwhere ΔRSMR and 𝑅𝑅AHESMR are the SMR and the SMR -induced anomalous Hall resistances of the \nTbIG/Co interfaces adjacent to Pt, respectively, which are a function of the Co thickness.  \nNote that the transverse SMR and the SMR -induced anomalous Hall effect (AHE) have \nthe same symmetry as the AMR -induced transverse (planar Hall) resistance and anomalous Hall \nresistance in Co, respectively. Hence, we cannot quantitively distinguish the cont ribution of \nanomalous Hall resistance from the SMR -induced AHE and conventional AHE by the \ntransverse resistance only (see Supplementary Information  S3). By contrast, the contributions \nof the AMR and SMR to the longitudinal resistance changes distinctly de pend on the \nmagnetization angles and can be resolved by angular -dependent measurements. As shown in \nFigures 2e -2g, we measured Rxx with a rotating magnetic field of 60 kOe in zx, zy and xy planes. \nAccording to the symmetry of SMR and AMR, a fit to the angular -dependent  data in zy plane \nyields ΔRSMR = 61.1, 87.7, 225.9 and 555.7 m Ω for different Co thickness of tCo=0, 0.3, 0.6 and \n0.9 nm, respectively, while a fit to the angular -dependent  data in zx  plane yields Δ RAMR = -0.9, \n1.4, 13.2 and 72.6 m Ω. The sizeable ΔRSMR implies a substantial absorption of the spin currents \nby the heavy metal/magnets interface.  \nInterfacial spin transmission/reflection can be quantified by the complex spin -mixing \nconductance G with a real ( Gr) and an imaginary ( Gi) part[29,30]. Gr is related to the damping - \n 6 like torques acting upon m, which is proportional to m ×(m×σ) and manifests itself as a \nlongitudinal SMR resistance Δ RSMR. On the other hand, the imaginary part G i is associated with \na field -like torque, which is proportional to m×σ and contributes a Hall resistance 𝑅𝑅AHESMR in Pt. \nThe experimental values of ΔRSMR and 𝑅𝑅AHESMR, and assuming spin Hall angle θ Pt = 0.08 and spin \ndiffusion length λPt =1.4 nm for Pt[30,32], we can estimate Gr = 4.3×1014 Ω-1m-2 and Gi = -1.2×1013 \nΩ-1m-2, which is consistent with reported values for Pt/magnetic garnet bilayers[30-32]. The \ninsertion of the Co layer greatly enhances the spin -mixing conductance up to one order of \nmagnitude to a G r for tCo of 0.6 and 0.9 nm of 3.9 ×1015 Ω-1m-2 and -5.8×1015 Ω-1m-2, respectively, \nmuch larger than that in simple Pt/TbIG or Pt/Co heterostructures[33], implying that the \nexchange coupling plays a critical role in the spin -dependent transport.  \nTo unravel the effect of the Co spacer, we recorded the hysteresis loop by sweeping the in -\nplane magnetic field. In Pt/TbIG, the Hall resistance exhibits the usual shape for a perpendicular \nmagnetization under an in -plane magnetic field ( Figure 3a ). By fitting the in -plane hysteresis \nloop, we can obtain the effective magnetic anisotropy field HK = 8 kOe and magnetic anisotropy \nenergy Ean = HKMS/2 = 9.2× 103 J m-3 (see Supplementary Information  S6). In contrast, Pt/Co \n(0.9 nm)/TbIG  shows a distinctly different shape with a sharp peak at small magnetic fields and \nsaturation at a higher magnetic field  (Figure 3b). This curve  contains information about the \nexchange coupling parameter J  between Co and TbIG. The magnetic energy per unit area of \nPt/Co/TbIG can be written as:  \n𝐸𝐸=−𝐽𝐽𝒎𝒎�Co∙𝒎𝒎�TbIG−𝐾𝐾TbIG𝑡𝑡TbIG𝒎𝒎�TbIG ,𝑧𝑧2−𝐾𝐾Co𝑡𝑡Co𝒎𝒎�Co,𝑧𝑧2−𝑯𝑯∙(𝒎𝒎Co+𝒎𝒎TbIG ),  (5)  \nwhere KCo(TbIG)  is the magnetic anisotropy coefficient and 𝒎𝒎�Co(TbIG )  is the unit vector of the \nmagnetization 𝒎𝒎Co(TbIG )  of Co (TbIG) . At zero magnetic field, the Co and TbIG \nmagnetization unit vectors 𝒎𝒎�Co and 𝒎𝒎�TbIG  are in -plane and out-of-plane , respectively . \nMinimizing Eq. (5) for H =0 and leading order in the interface exchange J  leads to:  \n𝑚𝑚�Co,𝑧𝑧=±𝐽𝐽(2|𝐾𝐾Co|𝑡𝑡Co) ⁄ ,                               \n𝑚𝑚�TbIG ,𝑥𝑥=𝐽𝐽(2|𝐾𝐾TbIG |𝑡𝑡TbIG ) ⁄ ,                         (6)   \nwhere the sign of ± holds for up - and downward TbIG  magnetization (see Supplement S4). A \nrelatively weak in -plane magnetic field overcomes  the exchange coupling and pulls the Co \nmagnetization into the plane, lead ing to a sharp decrease of the Hall resistance around zero \nmagnetic field (Figure 3c). Due to its large perpendicular magnetic anisotropy,  the TbIG  \n 7 magnetization persists to be perpendicular up to higher in- plane magnetic fields. The gradual \npull into the plane is accompanied by a reduced and ultimately vanishing Hall resistance. The \neffective magnetic anisotropy field H K = 53.0 kOe of TbIG in Co (0.9 nm)/TbIG is even larger \nthan that in pure TbIG (see Figure 3b). It has been reported that the interfaces of Pt/Co and \nCo/oxide can give large interfacial perpendicular magnetic anisotropy due to spin- orbit \ncoupling[34]. Owing to the interlayer exchange co upling and proximity effect, the perpendicular \nmagnetic anisotropy of TbIG may get enhancement from Pt/Co interfaces.  We can then deduce \na Hall resistance caused by the out -of-plane tilt of the Co magnetization ( 𝑅𝑅AHECo = 71.0 m Ω), \nwhich is larger than the high -field TbIG/Co SMR ( 𝑅𝑅AHESMR = -19.0 mΩ). We hence estimate G i \nin Pt/Co (0.9 nm)/TbIG  to be -7.2×1014 Ω-1 m-2, which is 60 times larger than that in Pt/TbIG. \nFrom the ratio of 𝑅𝑅AHECo at zero and that at the saturation out -of-plane magnetic fields (Figure \n2c) the competition between Co/TbIG exchange coupling and magnetic anisotropies leads to a \ntilt angle of the Co magnetization of 83.2 ° at zero magnetic fields, i.e. pulled out of the plane \nby 6.8° . Substituting the observed magnetic anisotropies into the macrospin model, we arrive \nat an  exchange coupling strength between Co and TbIG of J = 69.0 μ J m-2 (see Supplementary \nInformation S4 ).  \n4. Efficient current -induced magnetization switching  \nThe enhanced spin -mixing conductance implies efficient SOTs in current -biased \nPt/Co/TbIG structures. We measured the SOT efficiency by recording magnetic hysteresis loops as a function of a charge current bias and in -plane and out -of-plane magnetic fields,  a common \ntechnique for both metallic and insulating magnet/heavy metal bilayers\n[35-37]. As shown in \nFigure 4a , we can measure the damping -like SOTs by the current -induced shifts of the \nhysteresis loops since they give rise to out -of-plane effective fields H eff that act on the DMI -\nstabilized Néel -type domain walls[36]. We may define an SOT efficiency χ as: \n𝜒𝜒=𝐻𝐻eff/𝑗𝑗.                                 (7)  \nRepresentative hysteresis loops of Pt/Co (0.6 nm)/TbIG with H x = 300 Oe and j = ±4.3×1010 \nAm-2 are shown in Figure 4b. Changing the direction of the applied currents cause opposite \nshifts of the hysteresis loops as expected for a current -induced Heff caused by damping -like \ntorques. The hysteresis loops shifts increase with an in -plane magnetic field and saturate at a  \n 8 critical value that is governed by the DMI[32,33,35] (Figure 4c). In Pt/TbIG, the SOT efficiency \nsaturates to 0.9 ×10-14 TA-1m2 at Hx = 340 Oe. We can estimate the modulus of the effective DMI \nconstant |𝐷𝐷|=𝜇𝜇0𝑀𝑀sΔ|𝐻𝐻𝑥𝑥| from the saturation field H x, where  μ0 is the vacuum permeability \nand Δ = (A/Ku)1/2 the width of the domain wall. Here, A = 2.3 pJ m-1 is the exchange stiffness \nand Ku is the anisotropy energy obtained from the experiment , leading to |DPt/TbIG | =8.6 μJ m-2 \nconsistent with other  Pt/magnetic garnet bilayers[35,36,38 -41],. Interestingly, the insertion of a thin \nCo layer between TbIG and Pt significantly enhances the SOT efficiency as well as the effective \nDMI. For Pt/Co (0.6 nm)/TbIG 𝜒𝜒=24.0× 10-14 TA-1m2 at Hx = 500 Oe, more than an order of \nmagnitude larger than that of other heavy metal/magnetic insulator systems[35,42].  \nThe model of magnetization reversal by the motion and annihilation of Neel domain walls \nimplies that the spin -orbit torque is most efficient at the domain wall center (see Figure 4a) \nwhere the magnetizations of Co and TbIG are strictly parallel. We may the refore carry over the \nanalysis of the SMR in the collinear limit of high magnetic fields to understand the observed \nenhancement of the damping- like torque. Here the Co capping layer enhances the effective spin \nmixing conductance G r from a small value for p ure TbIG to the large one of Co. Since Gr \nmeasures the absorption of the transverse spin current by the ferromagnet, this result directly \nexplains the enhanced spin transfer torque. The interface exchange coupling subsequently \ncommunicates the torque to the TbIG which leads to the motion of the e ntire domain wall. The \nCo overlayer ensures non -chiral domain walls required for the magnetization reversal. An \ninteresting subject for future research is an analysis of the Co -thickness dependence of the spin \ntransfer, since partial trapping of the spins into quantum wells[43,44] and the associated multiple \nscattering at the TbIG/Co interface could additionally increase 𝐻𝐻eff.  \nSince large damping -like torques in Pt/Co/TbIG should reduce the critical currents that \nswitch the magnetization, we measure the changes in the transverse resistance under current \npulses with a duration of 0.2 ms and a DC current of up to 1 mA corresponding to a current \ndensity of 2.5 ×1010 Am-2 in the same direction (see Supplementary Information S5). In the \npresence of an in -plane magnetic field of 400 Oe  along the current direction, pulses with an \namplitude up to j  = 6.1×1010 Am-2 change the sign of the t ransverse resistance which indicates \na complete TbIG magnetization switching (Figure 4d). Reversing the direction of the in- plane \nmagnetic field leads to an anti -clockwise hysteresis loop (Figure 4e), which agrees with the  \n 9 sign of the spin Hall angle of Pt[45]. Increasing the in- plane magnetic fields reduces the energy \nbarrier of magnetization reversal and the critical switching current density of T bIG/Co (0.6 \nnm)/P t from 6.9×1010 to 5.0×1010 Am-2 (Figure 4f) half of that in Pt/TbIG (1.2 ×1011 Am-2 at H x \n= 204 Oe). The suppression of the critical switching current density is less than expected from \nthe enhanced SOT efficiency between Pt/Co/TbIG and Pt/TbIG deduced above, which suggests that Joule heating plays a role in the magnetization switching at high  current densities\n[46,47].  \n5. Field-free magnetization switching  \nCurrent -induced switching of perpendicular magnetization without the need to apply \nmagnetic fields is highly desirable in high -density magnetic memories. To this end, various \nschemes have been proposed in metallic systems, e.g., by breaking the mirror symm etry in \nasymmetric lateral designs[48,49] and non- collinear magnetic alignment[50,51]. These invoke \ncomplex  devices incorporating multiple exotic materials that might be difficult to realize with \nmagnetic insulators. Here we may take advantage of the exchange coupling between Co and TbIG that leads to a non- collinear magnetic texture that allows switching of a perpendicular \nmagnetization by electric currents without the assistance of magnetic fields ( Figure 5a ).  \nAs in Pt/Co (0.6 nm)/TbIG, we observed a full current -induced switching of magnetization \nin Pt/Co (0.9 nm)/TbIG  with the same switching polarity cycle as function of the in- plane \nmagnetic field sweeps (Figures 5b -5c). Taking the device out of the ele ctromagnet set -up, where \nthe residual magnetic field is less than 1.0 Oe, does not deteriorate the switching performance \n(Figures 5d -5e). The switching polarity is determined by the history of the applied in -plane \nmagnetic fields. When initialized with a p ositiv e (negative) in -plane magnetic field of 1 kOe, \nthe electric current can switch the magnetization with anti -clockwise (clockwise) polarity. To \nverify the reliability of current -induced field- free switching, we repeated measurements by \napplying alternating c urrent pulses of ± 1.43×10\n11 Am-2 in the absence of magnetic fields (Figure \n5f). The Hall resistance jumps between the resistance levels corresponding to up and down \nmagnetizations after every pulse. The switching polarity reverses when the direction of the  pre-\nset in -plane magnetic field changes from positive to negative, which is consistent with the \ncurrent -driven hysteresis loops.  \nThe SOT efficiency measurement in Pt/Co (0.9 nm)/TbIG  led to several interesting  \n 10 observations (Figure 5g). First, in contrast to Pt/Co (0.6 nm)/TbIG, the magnitude of SOT \nefficiency in Pt/Co (0.9 nm)/TbIG saturates at a low magnetic field, implying the existence of \nan effective in -plane magnetic field due to the coupling with the in- plane magnetized Co layer. \nSecond, the saturation SOT efficiency ( 2.6×10-14 TA-1m2) in Pt/Co (0.9 nm)/TbIG is lower than \nthat ( 24.0× 10-14 TA-1m2) in Pt/Co (0.6 nm)/TbIG, though the spin-mixing conductance in Pt/Co \n(0.9 nm)/TbIG is higher than that in Pt/Co (0.6 nm)/TbIG. However, a thicker Co layer also \ndissipates more spin currents, leading to a reduced SOT efficiency. Therefore, there is a trade -\noff of Co thickness to obtain the  optimum SOT efficiency. The SOT efficiency can be further \nenhanced by optimizing the thickness of TbIG[52] and tuning its magnetization compensation \nstate[53]. Moreover, there is a substantial  non-zero SOT efficiency at zero magnetic fields and \nits sign depe nds on the history of sweeping in- plane magnetic fields, supp orting the \nperformance of field -free magnetization switching.  \n6. Conclusion  \nIn summary, we  exploit the magnetic coupling between metallic and insulating magnets, \nto efficiently manipulate the magnetism in an insulator with perpendicular magnetization. The itinerant conduction electrons in metallic magnets provide rich and strong spin -related \ninterfacial effects in magnetic trilayers. By harnessing the magnetic coupling between metallic \nand insulating magnets, these interfacial effects can be imprinted into magnetic insulators, \nallowing for efficient electrical de tection and manipulation of magne tism. In addition, coating  \nthe interface with  a few metallic magnetic atoms significantly enhances the SOT efficiency as \nwell as the DMI. Furthermore, we demonstrate the performance of field -free current -induced \nmagnetization switching that results from symmetry -breaking non -collinear magnetic textures, \npaving the way for scalable magnetic memory devices. Therefore, our work offers a new avenue to engineer  efficient  spin memory and logic devices based on magnetic insulators.  \n \n  \n 11 Methods  \nGrowth of TbIG films : TbIG thin films were deposited on 5 mm × 5 mm Ga 3Gd5O12 (111) single -\nside-polished substrates via pulsed laser deposition (PLD) at a laser fluence of ~1.4 Jcm-2, and a \ntarget -to-substrate distance of ~6 cm. During deposition, the substrate temperature was heated to \n800°C and the oxygen pressure was 30 mtorr. After deposition, annealing process was performed to \npromote the epitaxial growth of TbIG films with high quality and the cooling rate of the chamber \nwas 20 °C min-1. Epitaxial growth of the films was confirmed via a high- resolution X -ray diffraction \n2θ scan of the (444) reflection. The thickness of thick TbIG films was determined by X -ray \nreflectometry, whereas the thickness of thin TbIG films (~5 nm) was calculated by the number of \nlaser pulses.  \nDevice fabrication and measurement : After the deposition of the TbIG thin film via PLD, metallic \nlayers (such as Pt and Co/P t) were deposited by DC magnetron sputtering at room temperature with \na base pressure <5 ×  10-8 torr. The deposition rate was 0.026 nms-1 for P t and 0.011 nms-1 for C o. \nThe thickness of Co and Pt layers was calculated according to the deposition rate. These multilayers \nwere patterned into Hall bars using a combination of UV photolithography and Ar ion milling \ntechnique with lateral dimensions of 10 µ m × 35 µm (width ×  length). The magnetic properties of \nTbIG films  were  measured by superconducting quantum interference device (SQUID) \nmagnetometer. 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Commun. 9, 3612 (2018).  \n53. H. Wu, Y. Xu, P. Deng, Q. Pan, S. A. Razavi, K. Wong, L. Huang, B. Dai, Q. Shao, G. Yu, X. Han, J.- C. Rojas -Sánchez, S. Mangin, K. L. Wang, Spin- Orbit Torque Switching \nof a Nearly Compensated Ferrimagnet by Topological Surface States. Adv. Mater.  31, \n1901681 (2019).  \n \n \n   \n 16 Figures and figure captions  \n \nFigure 1. Structural and magnetic properties of epitaxial TbIG films.  (a) Crystal structure \nof TbIG. The red, grey and blue balls represent the Tb, Fe and O atoms, respectively. Schematics \nof magnetic sublattices of TbIG illustrating the compensation state when the temperature is \nabove  or below the compensation temperature Tcomp, are shown . (b) X-ray diffraction patterns \n(2θ scan) of a 53 -nm-thick TbIG film grown on a (111) -oriented GGG substrate with peaks \nassigned to TbIG and GGG. ( c) Magnetization as a function of out -of-plane magnetic field Hz \nobtained by SQUID VSM (red curves) and MOKE (black curves) mea surement. ( d) Cross -\nsectional HR -TEM image of the T bIG/P t heterostructure on a GGG substrate. Scale bar: 10 nm. \n(e) AHE hysteresis loops measured at different temperatures in Pt/T bIG. (f) Measured \ncoercivity as a function of temperature measured in Pt/T bIG.  \n  \n \n 17  \nFigure 2. Transport properties of Pt/Co/TbIG heterostructures.  (a) Schematics of \nPt/Co/TbIG/GGG multilayer (not on scale) and optical image of a TbIG Hall bar device in a \nthree -dimensional rendering of the measurement setup with white s cale bar of 20 µm.  The \ncoordinate system and magnetic field angles are indicated. ( b) AHE hysteresis loops measured \nin Pt/Co/T bIG for different thicknesses of the Co layer. ( c) AHE hysteresis loops measured in \nPt/Co (0.9 nm)/T bIG up to large out -of-plane magnetic fields. The AHE hysteresis loops at \nsmall magnetic fields are shown in the inset. ( d) Magnitude of the anomalous Hall loops as a \nfunction of thickness of the Co layer. ( e-g) Angular -dependent longitudinal resistance in \nPt/Co/T bIG with an applied ma gnetic field of 60 kOe rotating in the xy, xz  and yz planes.  \n  \n \n 18  \nFigure 3. AHE hysteresis loops with in -plane magnetic fields. AHE hysteresis loops as a \nfunction of in- plane magnetic fields in ( a) Pt/TbIG and ( b) Pt/Co (0.9 nm)/TbIG. A fit to the \ndata according to the experimental parameters and the model of magnetic coupling is indicated. \n(c) Schematics illustrating the magnetization when the in -plane magnetic field is swept from \nnegative to positive in P t/Co (0.9 nm)/TbIG.  \n  \n \n 19  \nFigure 4. Efficient current -induced SOTs in Pt/Co (0.6 nm)/TbIG.  (a) Current -induced \ndomain- wall motion (domain expansion) with an in- plane magnetic field Hx used to realign \ndomain- wall moments. ( b) AHE measurement with currents of ±4.3×1010 Am-2 under an in-\nplane magnetic field Hx of 300 Oe. The horizontal shift of the hysteresis loops corresponds to \nthe current -induced effective field H eff. (c) SOT efficiency calculated from horizontal shifts for \ndifferent current densities at different in -plane magnetic fields in Pt/ TbIG and Pt/Co (0.6 \nnm)/TbIG. ( d) and ( e) Current -induced hysteresis loops with in- plane magnetic fields of H x = \n±400 Oe in Pt/Co ( 0.6 nm)/TbIG. The switching polarity is indicated by the arrows. ( f) Critical \nswitching current densit ies as a function of in -plane magnetic fields in Pt/ TbIG and Pt/Co ( 0.6 \nnm)/TbIG.  \n  \n \n 20  \nFigure 5. Field -free magnetization switching in Pt/Co (0.9 nm)/TbIG. (a) Mirror symmetry \nof the degenerate P t/Co/TbIG  magnetization configurations underlying field- free \nmagnetization switching. ( b) and ( c) dc current -induced hysteresis loops with in- plane magnetic \nfields of H x = ±140 Oe in Pt/Co ( 0.9 nm)/TbIG. ( d) and ( e): Current -induced hysteresis loops \nin the absence of magnetic fields in Pt/Co ( 0.9 nm)/TbIG after the pre -set with positive/negative \nin-plane magnetic fields of 1 kOe. ( f) Current -induced magnetization switching with 0.2 ms -\nlong current pulses of ±1.4×1011 Am-2 in the absence of an external magnetic field after the pre-\nset with positive/negative in -plane magnetic fields of 1 kOe. The magnetic field -induced \nhysteresis loops are shown in left, giving the resistance reference for magnetizations pointing \nup and down. ( g) SOT efficiency as a function of in -plane magnetic fields in Pt/Co (0.9 \nnm)/TbIG.   \n \n"    }]
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